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1,116,691,499,215 | arxiv |
\section{Introduction}
Sampling, i.e., the selection of a set of points that represent a probability distribution, is an essential problem in many areas such as statistics, machine learning, and nonlinear filtering.
For this purpose, randomized sampling algorithms such as importance sampling \cite{sarkkaBayesianFilteringSmoothing2013} and Markov Chain Monte Carlo methods (MCMC) methods including Metropolis-Hasting \cite{geyerPracticalMarkovChain1992}, and Gibbs Sampling \cite{gemanStochasticRelaxationGibbs1984,casellaExplainingGibbsSampler1992} are widely-used.
Recently, deterministic sampling methods \cite{hanebeckLocalizedCumulativeDistributions2008,frischEfficientDeterministicConditional2020} are becoming more and more relevant. Deterministically chosen samples can encode more information, thus fewer samples are required to represent a distribution \cite{chenSuperSamplesKernelHerding2010}.
Furthermore, deterministic algorithms require less testing, which is especially important in safety-critical areas such as autonomous driving \cite{koopmanChallengesAutonomousVehicle2016}.
The process of herding, introduced to generate pseudo-samples that match given moments \cite{wellingHerdingDynamicalWeights2009}, can be used to deterministically sample from discrete probability distributions. Moreover, herding has been applied to Gibbs sampling on graphical models, yielding a deterministic Gibbs sampler on discrete probability distributions \cite{chenHerdedGibbsSampling2016}. Herded Gibbs sampling also showed comparable results to randomized Gibbs Sampling considering data association in multi-object tracking \cite{wolfDeterministicGibbsSampling2020}. Bach et al. \cite{bachEquivalenceHerdingConditional2012} furthermore showed that the kernel herding procedure \cite{chenSuperSamplesKernelHerding2010} is equivalent to the Frank-Wolfe algorithm.
By applying kernels, the herding principle was extended to continuous densities, called kernel herding \cite{chenSuperSamplesKernelHerding2010}, yielding deterministic super-samples that contain more information than random i.i.d. samples. The herding methods will be discussed in more detail in Section~\ref{sec:background}.
Considering nonlinear filtering with deterministic samples, the unscented Kalman filter \cite{julierUnscentedFilteringNonlinear2004,wanUnscentedKalmanFilter2001} is widely-used, which employs $2d+1$ sigma-points to represent a $d$-dimensional Gaussian density. The number of samples, however, is limited and hence also the ability to represent arbitrary densities.
Deterministic Gaussian filters were created in \cite{huberGaussianFilterBased2008, steinbringS2KFSmartSampling2013} by optimizing the sample locations with respect to a suitable distance measure, where all samples are optimized simultaneously in a batch process.
Particle filters \cite{doucetSequentialMonteCarlo2000,sarkkaBayesianFilteringSmoothing2013}, which are based on sequential Monte Carlo sampling, are often applied to nonlinear filtering problems but they might suffer from particle degeneration such that resampling becomes necessary. In this context, particle flow filters \cite{daumExactParticleFlow2010, dingImplementationDaumHuangExactflow2012} avoid random sampling by systematically placing the particles based on solving partial differential equations. In \cite{lacoste-julienSequentialKernelHerding2015}, sequential Monte Carlo methods were combined with Frank-Wolfe optimization to use deterministic samples, which provided a better accuracy than random samples.
The main contribution of our paper is a new deterministic Gibbs sampling algorithm that systematically draws samples from continuous multivariate probability densities unlike the approach in \cite{chenHerdedGibbsSampling2016}, which works on discrete probability distributions and probabilistic graphical models.
For this purpose, we extend the kernel herding algorithm to Gibbs sampling, thus creating a herded Gibbs sampler for continuous densities.
The key idea is to apply kernel herding to sample from the conditional densities in a Gibbs sampling scheme, while employing the conditional kernel density estimate in order to incorporate all previous samples into the herding process.
Our algorithm is suited for high dimensions since only one-dimensional optimization problems have to be solved. Furthermore, it allows for generating samples from an unknown joint density, like randomized Gibbs sampling, as long as the conditional probability densities are known.
We provide experimental convergence results showing that the $L_2$ error decreases comparably to kernel herding while the computation time is significantly lower.
This paper is structured as follows:
In Section~\ref{sec:background} we revisit herding methods for deterministic sampling, including discrete herded Gibbs sampling and kernel herding. In Section~\ref{sec:chg} we then introduce our new algorithm called continuous herded Gibbs sampling and in Section~\ref{sec:evaluation} we present experimental results for Gaussian mixtures.
The paper ends with a brief conclusion and outlook in Section~\ref{sec:conclusion}.
\section{Background}
\label{sec:background}
In this section, we will review the basic herding \cite{wellingHerdingDynamicalWeights2009} procedure and herded Gibbs sampling \cite{chenHerdedGibbsSampling2016} for sampling from discrete distributions as well as kernel herding \cite{chenSuperSamplesKernelHerding2010} for the continuous case.
\begin{figure*}
\centering
\begin{tikzpicture}
\input{example_sampling}
\end{tikzpicture}
\caption{Illustration of 20 samples drawn from a Gaussian mixture with 5 components using kernel herding, continuous herded Gibbs sampling and random sampling.}
\label{fig:example_sampling}
\end{figure*}
\subsection{Herding discrete samples}
Herding \cite{wellingHerdingDynamicalWeights2009} is a deterministic process of generating samples that match given moments $\mu = \mathbb{E}_{\mat{x}\sim P} [\phi(\mat{x})]$ with respect to the features $\phi(\mat{x})$, where $P$ is a discrete probability distribution.
The sampling is carried out via maximizing a weight vector $\mat{w}$ w.r.t. $\phi$, which includes information on all previous samples. Then, the moments and current samples are incorporated into $\mat{w}$ in an update.
The procedure to compute the $t$\textsuperscript{th} sample $\mat{x}^{(t)}\in\mathcal{X}\subseteq\mathbb{R}^d$ is given by \cite{chenHerdedGibbsSampling2016}
\begin{align}
\mat{x}^{(t)} &= \argmax_{\mat{x}\in \mathcal{X}}\left\langle \mat{w}^{(t-1)},\rvec{\phi}(\mat{x})\right\rangle, \label{eq:herding_max}\\
\mat{w}^{(t)} &= \mat{w}^{(t-1)} + \mu - \rvec{\phi}\left(\mat{x}^{(t)}\right), \label{eq:herding_update}
\end{align}
with the weight vector $\mat{w}\in\mathcal{H}$ and feature map $\rvec{\phi}:\mathcal{X}\mapsto\mathcal{H}$. The feature space $\mathcal{H}$ is a Hilbert space with inner product $\langle\cdot,\cdot\rangle$. In order to sample from a probability distribution, i.e. $\mu=P$, the indicator vector is used as feature map $\rvec{\phi}$. The weight vector $\mat{w}$ can be initialized as $\mu$. With $\phi$ as the indicator vector, \eqref{eq:herding_max} boils down to maximizing $\mat{w}$, \eqref{eq:herding_update} results in a subtraction of $1$ at the corresponding element of $\mat{w}$. The probability of $\mat{x}$ determines how fast it will be sampled again, as $P$ is added to $\mat{w}$ after every sample.
In \cite{chenHerdedGibbsSampling2016}, the same process was applied to the full conditionals in Gibbs sampling, thus creating a deterministic Gibbs sampler for discrete probability distributions. However, herding has to be applied separately to each conditional probability distribution, maintaining separate weight vectors for all conditionals and resulting in a worst case exponential space complexity. Pseudocode generating $T$ samples using herded Gibbs sampling can be found in Alg.~\ref{alg:herdedgibbs}. We make use of the notation $\mat{\bar{x}}_i = (\mat{x}_1,\dots,\mat{x}_{i-1},\mat{x}_{i+1},\dots,\mat{x}_d)^\top$. The weights $\mat{w}^{i,\mat{\bar{x}}_i}$ are initialized with $P(\mat{x}_i|\mat{\bar{x}}_i)$. Note, that there is one weight vector for each combination of $i$ and $\mat{\bar{x}}_i$.
\begin{algorithm}
\caption{Discrete herded Gibbs sampling \cite{chenHerdedGibbsSampling2016}}
\label{alg:herdedgibbs}
\begin{algorithmic}[1]
\For{$t=1, \dots, T$}
\For{$i=1, \dots, d$}
\State $\mat{x}_i = \argmax \mat{w}^{i,\bar{\mat{x}}_i}$
\State $\mat{w}^{i,\bar{\mat{x}}_i} = \mat{w}^{i,\bar{\mat{x}}_i} + P\left(\mat{x}_i \mid \mat{\bar{x}}_i \right)$
\State $\mat{w}^{i,\bar{\mat{x}}_i}_{\mat{x}_i} = \mat{w}^{i,\bar{\mat{x}}_i}_{\mat{x}_i} -1$
\EndFor
\State $\mat{x}^{(t)}=\mat{x}$
\EndFor
\end{algorithmic}
\end{algorithm}
\subsection{Kernel herding}
The herding process was extended to continuous probability densities in \cite{chenSuperSamplesKernelHerding2010}. The objective is to deterministically determine $t$ samples $\mat{x}^{(1)},\dots,\mat{x}^{(t)}\in\mathbb{R}^d$ that represent a $d$-dimensional probability density $p$.
In the continuous case of herding, an infinite number of features needs to be considered, which is intractable.
By applying kernels $k(\mat{x},\mat{x'}) = \langle \phi(\mat{x}),\phi(\mat{x}')\rangle$, the feature vectors are incorporated implicitly. A weight vector is not needed anymore, it is replaced by a weight function, which incorporates the expectation of the kernel over $p$, the moments to be matched, as well as all previous samples and needs to be optimized. Sample $\mat{x}^{(t+1)}$ from a probability density $p(\mat{x})$ is then computed as follows, where both the sampling step and the weight update are combined into one optimization \cite{chenSuperSamplesKernelHerding2010}:
\begin{align}
\mat{x}^{(t+1)} = \argmax_{\mat{x}\in\mathcal{X}} \mathbb{E}_{\mat{x}'\sim p}\left[k(\mat{x},\mat{x}')\right]
- \frac{1}{t+1}\sum_{s=1}^t k\left(\mat{x},\mat{x}^{(s)}\right).
\label{eq:KernelHerding}
\end{align}
\section{Continuous herded Gibbs sampling}
\label{sec:chg}
In this work, we propose to combine herded Gibbs sampling with kernel herding to create a deterministic Gibbs sampler for continuous distributions. A naïve attempt would be to apply kernel herding directly to the conditional probabilities of Gibbs sampling. In \eqref{eq:KernelHerding} one would replace $\mat{x}^{(t+1)}$ by $\mat{x}^{(t+1)}_i$ and $p(\mat{x})$ by $p(\mat{x}_i|\mat{\bar{x}}_i)$ in \eqref{eq:KernelHerding}, where $\mat{\bar{x}}_i$ is defined as above.
In the summation in \eqref{eq:KernelHerding} only those samples $\mat{x}^{(s)}_i$ would be used, where $\mat{\bar{x}}_i^{(s)}=\mat{\bar{x}}_i$. This would be equivalent to having a separate weight vector for each combination of $i$ and $\mat{\bar{x}}_i$, as it is the case with discrete herded Gibbs sampling. However, since the number of possible $\mat{\bar{x}}_i$'s is uncountable in the continuous case, a second sample with $\mat{\bar{x}}_i^{(s')}=\mat{\bar{x}}_i^{(s)}$ will very unlikely appear again. This would result in starting a new herding process for almost every sample, which is undesirable, as the valuable information of previous samples is not used.
For this reason, our key idea is to include all previous samples into the herding process by replacing the kernel density estimate in kernel herding with the conditional kernel density estimate. For this purpose, we rewrite \eqref{eq:KernelHerding} as
\begin{equation}
\mat{x}^{(t+1)} = \argmax_{\mat{x}\in\mathcal{X}} \frac{t+1}{t}\mathbb{E}_{\mat{x}'\sim p}\left[k(\mat{x},\mat{x}')\right] - \tilde{p}_t(\mat{x}),
\end{equation}
with the kernel density estimate
\begin{equation}
\tilde{p}_t(\mat{x}) = \frac{1}{t} \sum_{s=1}^t k\left(\mat{x},\mat{x}^{(s)}\right).
\end{equation}
In order to sample from the conditional probability density, we consider the kernel density estimate conditioned on $\mat{\bar{x}}_i$
\begin{equation}
\tilde{p}_t(\mat{x}_i|\mat{\bar{x}}_i) = \frac{\tilde{p}_t(\mat{x})}{\tilde{p}_t(\mat{\bar{x}}_i)}
= \frac{\sum_{s=1}^t k\left(\mat{x},\mat{x}^{(s)}\right)}{\sum_{s=1}^t k\left(\mat{\bar{x}}_i,\mat{\bar{x}}_i^{(s)}\right)}.
\end{equation}
If the dimensions of the kernel are independent, as it is the case, e.g., for Gaussian kernels, we can rewrite the conditional kernel density estimate as
\begin{align}
\tilde{p}_t(\mat{x}_i|\mat{\bar{x}}_i) &= \frac{\sum_{s=1}^t k\left(\mat{x}_i,\mat{x}_i^{(s)}\right)k\left(\mat{\bar{x}}_i,\mat{\bar{x}}_i^{(s)}\right)}{\sum_{s=1}^t k\left(\mat{\bar{x}}_i,\mat{\bar{x}}_i^{(s)}\right)} \\
&= \frac{1}{c}\sum_{s=1}^t w_s k\left(\mat{x}_i,\mat{x}_i^{(s)}\right),
\end{align}
with $w_s = k\left(\mat{\bar{x}}_i,\mat{\bar{x}}_i^{(s)}\right)$ and $c= \sum_{s=1}^t w_s$.
We can then formulate the procedure for continuous herded Gibbs sampling as follows
\begin{align}
\mat{x}_i^{(t+1)} = &\argmax_{\mat{x}_i\in\mathbb{R}} \frac{t+1}{t} \mathbb{E}_{x'\sim p(\mat{x}_i|\mat{\bar{x}}_i)}[k(\mat{x}_i,x')]\notag\\
&\,- \tilde{p}_t(\mat{x}_i|\mat{\bar{x}}_i).
\end{align}
With independent kernels, this simplifies to
\begin{align}
\mat{x}_i^{(t+1)} = &\argmax_{\mat{x}_i\in\mathbb{R}} \frac{t+1}{t} \mathbb{E}_{x'\sim p(\mat{x}_i|\mat{\bar{x}}_i)}[k(\mat{x}_i,x')]\notag\\
&\,- \frac{1}{c}\sum_{s=1}^t w_s k\left(\mat{x}_i,\mat{x}_i^{(s)}\right).
\label{eq:ContHGsimple}
\end{align}
\eqref{eq:ContHGsimple} shows that all previous samples are incorporated into the herding process, however, they are weighted according to the distance between $\mat{\bar{x}}_i$ and $\mat{\bar{x}}_i^{(s)}$. Pseudocode for continuous herded Gibbs sampling can be found in Alg.~\ref{alg:contHG}.
Fig.~\ref{fig:example_sampling} shows samples drawn from a Gaussian mixture using kernel herding, continuous herded Gibbs sampling and random sampling from the joint density.
\begin{algorithm*}
\begin{algorithmic}
\For{$t=1, \dots, T$}
\For{$i=1, \dots, d$}
\State $\mat{x}_i = \argmax_{x_i\in\mathbb{R}} \frac{t+1}{t} \mathbb{E}_{x'\sim p(\mat{x}_i|\mat{\bar{x}}_i)}[k(\mat{x}_i,x')]
- \frac{1}{c}\sum_{s=1}^t w_s k\left(\mat{x}_i,\mat{x}_i^{(s)}\right)$
\EndFor
\State $\mat{x}^{(t)}=\mat{x}$
\EndFor
\end{algorithmic}
\caption{Continuous herded Gibbs sampling}
\label{alg:contHG}
\end{algorithm*}
\begin{figure}
\centering
\begin{tikzpicture}[scale=0.8]
\input{example}
\end{tikzpicture}
\caption{Visualization of continuous herded Gibbs sampling. Upper plot: two samples drawn from a Gaussian. The kernels and Gaussian are visualized with their $2\sigma$-bound. The red line depicts $\mat{\bar{x}}_1$. The lower plot shows the herding weight function \eqref{eq:ContHGsimple} (solid red line) and $\mathbb{E}_{x'\sim p(\mat{x}_1|\mat{\bar{x}}_1)}[k(\mat{x}_1,x')]$ (dashed blue line).}
\label{fig:example_chg}
\end{figure}
A visualization of our algorithm is provided in Fig.~\ref{fig:example_chg}. It shows two samples, $\mat{x}^{(1)}$ and $\mat{x}^{(2)}$ drawn from a Gaussian density using Gaussian kernels. In the next step $\mat{x}_1^{(3)}$ will be sampled along the red line in the upper plot, which depicts $\mat{\bar{x}}_1$.
The lower plot shows the expectation $\mathbb{E}_{x'\sim p(\mat{x}_1|\mat{\bar{x}}_1)}[k(\mat{x}_1,x')]$ (dashed blue line) and the herding weight function (solid red line), see \eqref{eq:ContHGsimple}. Note that the difference between both functions is given by $\frac{1}{c}\sum_{s=1}^t w_s k\left(\mat{x}_1,\mat{x}_1^{(s)}\right)$ in the Gaussian case and both functions are equal before any samples are drawn. The impact of a previous sample $\mat{x}^{(s)}$ on the herding weight function is determined by $w_s$, which resembles the distance of $\mat{x}^{(s)}$ to $\mat{\bar{x}}_1$ (solid red line in the upper plot). This can also be seen in Fig.~\ref{fig:example_chg}, where $\mat{x}^{(2)}$ has a higher impact on the weighting function with $w_2=0.997$ as opposed to $\mat{x}^{(1)}$ with $w_1=0.118$, which has a higher distance to $\mat{\bar{x}}_1$. In order to determine $\mat{x}_1^{(3)}$, the herding weight function is maximized. Afterwards, the process is repeated for the second dimension.
\section{Evaluation on Gaussian mixtures}
\label{sec:evaluation}
In order to demonstrate the performance of the proposed continuous herded Gibbs sampling algorithm, we provide numerical results for the problem of sampling from Gaussian mixture densities. A Gaussian mixture with $M$ components has the density function $p(\mat{x})=\sum_{i=1}^M \phi_i p_i(\mat{x})$, with $p_i(\mat{x})\sim \mathcal{N}(\rvec{\mu}_i,\Sigma_i)$, $\phi_i\geq 0$ and $\sum_{i=1}^M \phi_i=1$. We employed a Gaussian kernel
\begin{equation}
k_d(\mat{x},\mat{x}') = \frac{1}{\sqrt{ (2\pi)^d|\bar{\Sigma}|}} e^ {-\frac{1}{2}(\mat{x}-\mat{x}')^\top\bar{\Sigma}^{-1}(\mat{x}-\mat{x}')}
\end{equation}
with standard deviation $\sigma_k$ and $\bar{\Sigma} = \sigma_k^2\mat{I}_d$.
The expectation in \eqref{eq:KernelHerding} can then be calculated as
\begin{align}
&\mathbb{E}_{\mat{x}'\sim p}\left[k_d(\mat{x},\mat{x}')\right] = \int_{\mathbb{R}^d} k_d(\mat{x},\mat{x}')p(\mat{x}')\mathrm{d}\mat{x}' \notag\\
&\quad= \sum_{i=1}^M \phi_i \frac{1}{(2\pi)^d \left| \Sigma_i + \bar{\Sigma} \right|}
e^{-\frac{1}{2} (\mat{x}-\rvec{\mu}_i)^\top (\Sigma_i+\bar{\Sigma})^{-1}(\mat{x}-\rvec{\mu}_i)},
\label{eq:expectation_herding}
\end{align}
using \cite{jebaraBhattacharyyaExpectedLikelihood2003} and the identity $(\mat{A}^{-1} + \mat{B}^{-1})^{-1} = \mat{A}(\mat{A}+\mat{B})^{-1}\mat{B} = \mat{B}(\mat{A}+\mat{B})^{-1}\mat{A}$.
Since the conditional of a Gaussian mixture is also a Gaussian mixture, \eqref{eq:expectation_herding} can also be applied for the expectation in \eqref{eq:ContHGsimple}, which is a one-dimensional function.
\begin{figure*}[p]
\centering
\subfloat[Two-dimensional space.\label{fig:error_2d}]{
\begin{tikzpicture}
\input{error_d2}
\end{tikzpicture}
}
\\
\subfloat[Ten-dimensional space.\label{fig:error_10d}]{
\begin{tikzpicture}
\input{error_d10}
\end{tikzpicture}
}
\caption{Inverse herding error $1/\varepsilon_t$ and $L_2$ error for kernel herding (KH) and continuous herded Gibbs sampling (CHG) averaged over 10 random Gaussian mixtures each. The $L_2$ error is also computed for random samples (RND) drawn from the joint distribution. }
\label{fig:herding_error}
\end{figure*}
\begin{figure*}[p]
\centering
\subfloat[\label{fig:error_example_good}Continuous herded Gibbs converged in both the herding error and the $L_2$ error.]{
\begin{tikzpicture}[scale=0.95]
\input{error_example_good}
\end{tikzpicture}
}
\\
\subfloat[\label{fig:error_example_bad}Continuous herded Gibbs only converged in the $L_2$ error.]{
\begin{tikzpicture}[scale=0.95]
\input{error_example_bad}
\end{tikzpicture}
}
\caption{Two example Gaussian mixtures in two-dimensional space, with errors as in Fig.~\ref{fig:herding_error} and contour plot of the corresponding Gaussian mixture.}
\label{fig:error_example}
\end{figure*}
We used a heuristic approach to determine the starting points for the optimization, based on component means and covariances of the Gaussian mixtures.
This results in a higher number of starting points for kernel herding in higher dimensions, which, however, is needed to avoid local minima. For continuous herded Gibbs sampling the number of starting points does not depend on the dimension, as the weight function is one-dimensionl, but the number of optimizations per sample increases nonetheless.
We used the BFGS algorithm \cite{nocedalQuasiNewtonMethods2006} for optimization with default parameters as provided by scipy for kernel herding as well as continuous herded Gibbs sampling.
The first sample of the continuous herded Gibbs sampler was initialized as the first sample generated by kernel herding\footnote{Source code is available at \url{https://github.com/Fusion-Goettingen.}}.
First, we computed the herding error $\mathcal{E}_t$ \cite[eq. (9)]{chenSuperSamplesKernelHerding2010}, which measures the distance between $p$ and the set of samples in the feature space induced by the applied kernel, i.e.,
\begin{align}
\label{eq:herding_error}
\mathcal{E}^2_t = & \left|\left|\mathbb{E}_{x\sim p} [\phi(x)] - \frac{1}{t}\sum_{s=1}^t\rvec{\phi}\left(\mat{x}^{(s)}\right)\right|\right|^2 \notag \\
=& \:\mathbb{E}_{\mat{x},\mat{x}'\sim p}\left[k(\mat{x},\mat{x}')\right]
-\frac{2}{t}\sum_{s=1}^t \mathbb{E}_{\mat{x}\sim p}\left[k\left(\mat{x},\mat{x}^{(s)}\right)\right] \notag\\
&+\frac{1}{t^2}\sum_{s,s'=1}^t k\left(\mat{x}^{(s)},\mat{x}^{s'}\right) .
\end{align}
In the Gaussian case the first expectation in \eqref{eq:herding_error} is calculated analogous to \eqref{eq:expectation_herding}:
\begin{align}
&\mathbb{E}_{\mat{x},\mat{x}'\sim p}\left[k_d(\mat{x},\mat{x}')\right] = \int_{\mathbb{R}^d} k_d(\mat{x},\mat{x}')p(\mat{x})p(\mat{x}')\mathrm{d}\mat{x}\mathrm{d}\mat{x}' \notag\\
&\quad= \sum_{i,j=1}^M \phi_i \phi_j \frac{1}{(2\pi)^d \left| \Sigma_i + \Sigma_j + \bar{\Sigma} \right|} \;\times \notag \\
&\qquad \;\; e^{-\frac{1}{2} (\rvec{\mu}_i - \rvec{\mu}_j)^\top (\Sigma_i+\Sigma_j+\bar{\Sigma})^{-1}(\rvec{\mu}_i - \rvec{\mu}_j)}.
\label{eq:expectation_herding_error}
\end{align}
The herding error is minimized greedily by kernel herding~\cite{chenSuperSamplesKernelHerding2010}. We included it in order to determine if continuous herded Gibbs sampling can generate similar samples as kernel herding, even though it employs a different objective function.
We computed $\mathcal{E}_t$ on 10 randomly constructed two-dimensional and ten-dimensional Gaussian mixtures with 5 components. The samples were generated using a Gaussian kernel with $\sigma_k=0.1$.
Fig.~\ref{fig:error_2d} shows that $1/\mathcal{E}_t$ increases linearly with $t$ for kernel herding in the two-dimensional example, as was also found in \cite{chenSuperSamplesKernelHerding2010}. For continuous herded Gibbs sampling it increases sublinearly, indicating that continuous herded Gibbs sampling does not converge in the feature space of kernel herding. It has to be noted though, that these results highly depend on the chosen distribution. In some cases the convergence of continuous herded Gibbs sampling is as good as kernel herding (Fig.~\ref{fig:error_example_good}), whereas on others the herding error stagnates (Fig.~\ref{fig:error_example_bad}). The Gaussian mixture in Fig.~\ref{fig:error_example_bad} has a very narrow and long component as opposed to the Gaussian mixture in Fig.~\ref{fig:error_example_good}. This could lead to problems with our approach, if the kernel size is too big, as each dimension is sampled individually. A solution to this problem might be adapting the kernel size based on the covariances' eigenvalues. In the ten-dimensional example (Fig.~\ref{fig:error_10d}) both algorithms showed the same slow convergence in the herding error. This could be due to a too small kernel size for higher dimensions.
Second, we calculated the normalized $L_2$ distance as described in \cite{jensenEvaluationDistanceMeasures2007}, which calculates the distance in function space between the original Gaussian mixture density and the kernel density estimate that arises from the combination of the samples with the Gaussian kernel.
In contrast to the herding error, the $L_2$ error does not incorporate the kernels into the error computation, they are only applied to construct a density of the samples.
That way both herding variants can also be compared with random samples, where we averaged over 20~Monte Carlo runs. The random samples where directly drawn from the joint distribution. We used the same densities as above. Fig.~\ref{fig:error_2d} shows that continuous herded Gibbs sampling and kernel herding yield comparable results in two-dimensional space, with kernel herding yielding a better performance with few samples (less than 100) and continuous herded Gibbs sampling yielding a better performance with more than 100 samples. However, both seem to converge in the function space and show better results than random sampling. This shows that both herding variants are capable of representing a Gaussian mixture with fewer samples than random sampling. It has to be noted, that continuous herded Gibbs sampling even converges in the $L_2$ distance on the example in Fig.~\ref{fig:error_example_bad}, where the herding error did not converge. In ten-dimensional space (Fig.~\ref{fig:error_10d}), continuous herded Gibbs converges much faster than kernel herding, which almost does not converge just as random sampling. Furthermore, the $L_2$ error only slightly decreases. This could be caused by an inappropriate kernel size as well as a too small sample size for higher dimensions.
In order to compare the computation time, we randomly generated 10 Gaussian mixtures with 5 components and computed 100 samples with each method (kernel size $\sigma_k = 0.1$). We repeated this process for Gaussian mixtures from dimension 2 to 50. The results depicted in Fig.~\ref{fig:run_time} show that the average computation time of continuous herded Gibbs sampling increases linearly in the number of dimensions, while the average computation time for kernel herding is much higher below 10 and above 30 dimensions, in between it is a little bit faster than continuous herded Gibbs sampling.
\begin{figure}
\centering
\begin{tikzpicture}[scale=0.9]
\input{runtime}
\end{tikzpicture}
\caption{Computation time for kernel herding and continuous herded Gibbs sampling.}
\label{fig:run_time}
\end{figure}
\section{Conclusion and future work}\label{sec:conclusion}
In this paper, we proposed the continuous herded Gibbs sampler, a deterministic sampling algorithm for multivariate probability densities. As only one-dimensional optimization problems have to be solved, it is suited for high dimensional problems. The experiments showed that continuous herded Gibbs sampling can converge as fast as kernel herding, however, the performance in terms of the herding error heavily depends on the chosen Gaussian mixture although it always converges in the $L_2$ distance in our experiments.
Future research should include further investigating the convergence of the algorithm including strategies to find the optimal kernel size for a density. The results regarding the computation time indicate that continuous herded Gibbs sampling is also promising for high-dimensional applications. It should be investigated if the computation time can be optimized, e.g., by choosing a fixed number of random starting points for the optimization, which could be relevant for Gaussian mixtures with many components. Furthermore, the application to other families of distributions and kernels should be topic to future research.
\bibliographystyle{IEEEtran}
|
1,116,691,499,216 | arxiv |
\section{Method}
\label{sec:method}
Our generator network synthesizes images of a fixed resolution $H \times W$ and has the multi-layer perceptron-type architecture $G$ (see Fig.~\ref{fig:architecture}). In more detail, the synthesis of each pixel takes a random vector $\mathbf{z} \in \mathcal{Z}$ shared across all pixels, as well the pixel coordinates $\left(x, y\right) \in \left\{0\dots W-1\right\} \times \left\{0\dots H-1\right\}$ as input. It then returns the RGB value $\mathbf{c} \in \left[0, 1\right]^3$ of that pixel $G: \left(x, y, \mathbf{z}\right) \mapsto \mathbf{c}$.
Therefore, to compute the whole output image $I$, the generator $G$ is evaluated at at each pair $\left(x, y\right)$ of the coordinate grid, while keeping the random part $\mathbf{z}$ fixed:
\begin{align}
I = \left\{ G\left(x, y; \mathbf{z}\right) \mid \left(x, y \right) \in \texttt{mgrid}\left(H, W\right) \right\},
\label{eq:image_gen}
\end{align}
where
\begin{equation*}
\texttt{mgrid}\left(H, W\right) = \lbrace \left(x, y \right) \mid 0 \le x < W, \, 0 \le y < H \rbrace
\end{equation*}
is a set of integer pixel coordinates.
Following~\cite{stylegan1}, a mapping network $M$ (also a perceptron) turns $\mathbf{z}$ into a \emph{style} vector $\mathbf{w} \in \mathcal{W}$, $M: \mathbf{z} \mapsto \mathbf{w}$, and all the stochasticity in the generating process comes from this style component.
We then follow the StyleGANv2~\cite{stylegan2} approach of injecting the style $\mathbf{w}$ into the process of generation via weight modulation.
To make the paper self-contained, we describe the procedure in brief here.
Any modulated fully-connected (ModFC) layer of our generator (see Fig.~\ref{fig:architecture}) can be written in the form $\mathbf{\psi} = \hat{B} \mathbf{\phi} + \mathbf{b}$, where $\mathbf{\phi} \in \mathbb{R}^n$ is an input, $\hat{B}$ is a learnable weight matrix $B \in \mathbb{R}^{m \times n}$ modulated with the style $\mathbf{w}$, $\mathbf{b} \in \mathbb{R}^m$ is a learnable bias, and $\mathbf{\psi} \in \mathbb{R}^m$ is an output.
The modulation takes place as follows: at first, the style vector $\mathbf{w}$ is mapped with a small net (referred to as A in Fig.~\ref{fig:architecture}) to a scale vector $\mathbf{s} \in \mathbb{R}^n$
Then, the $\left(i, j\right)$-th entry of $\hat{B}$ is computed as
\begin{equation}
\hat{B}_{ij} = \frac{s_j B_{ij}}{\sqrt{\epsilon + \sum\limits_{k=1}^n \left(s_k B_{ik}\right)^2 }},
\end{equation}
where $\epsilon$ is a small constant.
After this linear mapping, a LeakyReLU function is applied to $\mathbf{\psi}$.
Finally, in our default configuration we add skip connections for every two layers from intermediate feature maps to RGB values and sum the contributions of RGB outputs corresponding to different layers. These skip connections naturally add values corresponding to the same pixel, and do not introduce interactions between pixels.
We note that the independence of the pixel generation process, makes our model parallelizable at inference time and, additionally, provides flexibility in the latent space $z$. E.g., as we show below, in some modified variants of synthesis, each pixel can be computed with a different noise vector $z$, though gradual variation in $z$ is needed to achieve consistently looking images.
\begin{figure}[t]
\centering
\includegraphics[width=\columnwidth]{figures/cips_model_scheme.pdf}
\caption{The Conditionally-Independent Pixel Synthesis (CIPS) generator architecture.
Top: the generation pipeline, in which the coordinates $\left(x, y\right)$ of each pixel are encoded (yellow) and processed by a fully-connected (FC) network with weights, modulated with a latent vector $\mathbf{w}$, shared for all pixels. The network returns the RGB value of that pixel. Bottom: The architecture of a modulated fully-connected layer (ModFC). Note: our default configuration also includes skip connections to the output (not shown here).
}
\label{fig:architecture}
\end{figure}
\subsection{Positional encoding}
The architecture described above needs an important modification in order to achieve the state-of-the-art synthesis quality.
Recently two slightly different versions of positional encoding for coordinate-based multi-layer perceptrons (MLP), producing images, were described in literature.
Firstly, SIREN~\cite{siren} proposed a perceptron with a principled weight initialization and sine as an activation function, used throughout all the layers.
Secondly, the Fourier features, introduced in \cite{fourier_features}, employed a periodic activation function in the very first layer only.
In our experiments, we apply a somewhat in-between scheme: the sine function is used to obtain Fourier embedding $e_{fo}$, while other layers use a standard LeakyReLU function:
\begin{equation}
e_{fo}\left(x, y\right) = \sin \left[B_{fo} \left(x', y'\right)^T \right],
\end{equation}
where $x' = \frac{2 x}{W-1}-1$ and $y' = \frac{2 y}{H-1} - 1$ are pixel coordinates, uniformly mapped to the range $\left[-1, 1\right]$ and the weight matrix $B_{fo} \in \mathbb{R}^{2 \times n}$ is learnable, like in SIREN paper.
However, only Fourier positional encoding usage turned out insufficient to produce plausible images. In particular, we have found out that the outputs of the synthesis tend to have multiple wave-like artifacts.
Therefore, we also train a separate vector $e_{co}^{\left(x, y\right)}$ for each spatial position and call them \textit{coordinate embeddings}. They represent $H \times W$ learnable vectors in total.
For comparison of these two embedding from the spectral point of view, see Sec.~\ref{sec:spectral_analysis}.
The full positional encoding $e\left(x, y\right)$ is a concatenation of Fourier features and a coordinate embedding
\begin{equation}
e\left(x, y\right) = \text{concat}\left[e_{fo}\left(x, y\right), e_{co}^{\left(x, y\right)}\right]
\end{equation}
and serve as an input for the next perceptron layer: $G\left(x, y; \mathbf{z}\right) = G' \left( e\left(x, y\right); M\left(\mathbf{z}\right) \right)$.
\section{Conclusion}
\label{sec:conclusion}
We presented a new generator model called \ganname, a high-quality architecture with conditionally independent pixel synthesis, such that the color value is computed using only random noise and coordinate position.
Our key insight is that the proposed architecture without spatial convolutions, attention or upsampling operations has the ability to compete in the model market and obtain decent quality in terms of FID and precision \& recall; such results have not been presented earlier for perceptron-based models.
Furthermore, in the spectral domain outputs of \ganname are harder to discriminate from real images.
Interestingly, \ganname-NE modification is weaker in terms of plausibility, yet has a more realistic spectrum.
Direct usage of a coordinate grid allows us to work with more complex structures, such as cylindrical panoramas, just by replacing the underlying coordinate system.
In summary, our generator demonstrates quality on par with state-of-the-art model StyleGANv2; moreover, it has applications in various diverse scenarios.
We have shown that the considered model could be successfully applied to foveated rendering and super-resolution problems in their generative interpretations.
Future development of our approach assumes researching these problems in their image-to-image formulations.
\section{Introduction}
\label{sec:introduction}
State-of-the-art in unconditional image generation is achieved using large-scale convolutional generators trained in an adversarial fashion~\cite{stylegan1,stylegan2,biggan}. While lots of nuances and ideas have contributed to the state-of-the-art recently, for many years since the introduction of DCGAN~\cite{dcgan} such generators are based around spatial convolutional layers, also occasionally using the spatial self-attention blocks~\cite{sagan}. Spatial convolutions are also invariably present in other popular generative architectures for images, including autoencoders~\cite{kingma2013auto}, autoregressive generators~\cite{van2016pixel}, or flow models~\cite{Dinh16,Kingma18}. Thus, it may seem that spatial convolutions (or at least spatial self-attention) is an unavoidable building block for state-of-the-art image generators.
Recently, a number of works have shown that individual images or collections of images of the same scene can be encoded/synthesized using rather different deep architectures (deep multi-layer perceptrons) of a special kind~\cite{nerf,siren}. Such architectures are not using spatial convolutions or spatial self-attention and yet are able to reproduce images rather well. They are, however, restricted to individual scenes. In this work, we investigate whether deep generators for unconditional image class synthesis can be built using similar architectural ideas, and, more importantly, whether the quality of such generators can be pushed to state-of-the-art.
Perhaps surprisingly, we come up with a positive answer (Fig.~\ref{fig:teaser}), at least for the medium image resolution (of $256\times{}256$). We have thus designed and trained deep generative architectures for diverse classes of images that achieve similar quality of generation to state-of-the-art convolutional generator StyleGANv2~\cite{stylegan2}, even surpassing this quality for some datasets.
Crucially, our generators are not using any form of spatial convolutions or spatial attention in their pathway. Instead, they use coordinate encodings of individual pixels, as well as sidewise multiplicative conditioning (weight modulation) on random vectors. Aside from such conditioning, the color of each pixel in our architecture is predicted independently (hence we call our image generator architecture \textit{Conditionally-Independent Pixel Synthesis} (CIPS) generators).
In addition to suggesting this class of image generators and comparing its quality with state-of-the-art convolutional generators, we also investigate the extra flexibility that is permitted by the independent processing of pixels. This includes easy extention of synthesis to non-trivial topologies (e.g.\ cylindrical panoramas), for which the extension of spatial convolutions is known to be non-trivial~\cite{cocogan,cohen2018spherical}.
Furthermore, the fact that pixels are synthesized independently within our generators, allows sequential synthesis for memory-constrained computing architectures. It enables our model to both improve the quality of photos and generate more pixel values in a specific areas of image (i.e.~to perform foveated synthesis).
\section{Related Work}
\label{sec:related}
Feeding pixel coordinates as an additional input to the neural network previously was successfully used in the widely known CoordConv technique~\cite{coordconv} to introduce the spatial-relational bias.
Recently, the same idea was employed by the COCO-GAN~\cite{cocogan} to generate images by parts or create "looped" images like spherical panoramas.
However, those models still used standard convolutions as the main synthesis operation. The synthesis process for neighboring pixels in such architectures is therefore not independent.
To the best of our knowledge, the problem of regressing a given image from pixel coordinates with a perceptron (that calculates each pixel's value independently) started from creating compositional patterns with an evolutionary approach~\cite{cppn}.
Those patterns, appealing for digital artists, were also treated as kind of differentiable image parametrization~\cite{mordvintsev_xy2rgb}.
However, this approach was not capable of producing photorealistic hi-res outputs (e.g., see the demo~\cite{karpathy_regression}).
Some machine learning blogs reported experiments with GANs, where the generator was a perceptron that took a random vector and pixel coordinates as an input, and returned that pixel's value as an output~\cite{otoro1,otoro2}. The described model was successfully trained on MNIST, but has not been scaled to more complex image data.
Scene-representation networks~\cite{srn} and later the neural radiance fields (NeRF) networks~\cite{nerf} have demonstrated how 3D content of individual scenes can be encoded with surprising accuracy using deep perceptron networks. Following this realization, systems \cite{siren} and \cite{fourier_features} considered the usage of periodic activation functions and so-called Fourier features to encode the pixel (or voxel) coordinates, fed to the multi-layer perceptron. In particular, the ability to encode high-resolution individual images in this way was demonstrated. All these works however have not considered the task of learning image generators, which we address here.
The very recent (and independent) Generative Radiance Fields (GRAF) system~\cite{graf} showed promising results at embedding the NeRF generator into an image generator for 3D aware image synthesis. Results for such 3D aware synthesis (still limited in diversity and resolution) for certain have been demonstrated. Here, we do not consider 3D-aware synthesis and instead investigate whether perceptron-based architectures can achieve high 2D image synthesis quality.
\section{Experiments}
\label{sec:experiments}
\subsection{Architecture details}
In our experiments, both Fourier features and coordinate embeddings had the dimension of 512.
The generator had 14 modulated fully-connected layers of width 512
We use leaky ReLU activation with the slope $0.2$.
We implement our experiments on top of the public code\footnote{\url{https://github.com/rosinality/stylegan2-pytorch}} for StyleGANv2.
Our model is trained with a standard non-saturating logistic GAN loss with $R_1$ penalty~\cite{r1_penalty} applied to the discriminator $D$.
The discriminator has a residual architecture, described in \cite{stylegan2} (we have deliberately kept the discriminator architecture intact).
Networks were trained by Adam optimizer~\cite{adam} with learning rate $2 \times 10^{-3}$ and hyperparameters: $\beta_0=0$, $\beta_1=0.99$, $\epsilon=10^{-8}$.
\subsection{Evaluation}
We now evaluate CIPS generators and their variations on a range of datasets. For the sake of efficiency, most evaluations are restricted to $256 \times 256$ resolution. The following datasets were considered:
\begin{itemize}
\item The \textit{Flickr Faces-HQ} (FFHQ)~\cite{stylegan1} dataset contains 70,000 high quality well-aligned, mostly near frontal human faces. This dataset is the most regular in terms of geometric alignement and the StyleGAN variants are known to perform very well in this setting.
\item The \textit{LSUN Churches}~\cite{lsun} contains 126,000 outdoor photographs of churches of rather diverse architectural style. The dataset is regular, yet images all share upright orientation.
\item The \textit{Landscapes} dataset contains 60,000 manually collected landscape photos from the Flickr website.
\item The \textit{Satellite-Buildings}\footnote{\url{https://www.crowdai.org/challenges/mapping-challenge}} dataset contains 280,741 images of $300 \times 300$ pixels (which we crop to $256 \times 256$ resolution and randomly rotate). This dataset has large size, and is approximately aligned in terms of scale, yet lacks consistent orientation.
\item Finally, the \textit{Satellite-Landscapes}\footnote{\url{https://earthview.withgoogle.com/}} contains a smaller curated collection of 2,608 images of $512 \times 512$ resolution of satellite images depicting various impressive landscapes found on Google Earth (which we crop to $256 \times 256$ resolution). This is the most ``textural'' dataset, that lacks consistent scale or orientation.
\end{itemize}
\begin{table}[t!]
\centering
\begin{tabular}{|l|c|c|c|}
\hline
& StyleGANv2 & \ganname (ours) \\
\hline
FFHQ & 3.83 & 4.38 \\
LSUN Churches & 3.86 & 2.92 \\
Landscapes & 2.36 & 3.61 \\
Satellite-Buildings & 73.67 & 69.67 \\
Satellite-Landscapes & 51.54 & 48.47 \\
\hline
\end{tabular}
\caption{FID on multiple datasets at resolution of $256^2$ for CIPS-skips model.
Note that \ganname is of comparable quality with state-of-the-art StyleGANv2, and better on Churches.
The value for CIPS model on FFHQ differs from the one reported in Tab.~\ref{tab:modifications} as we trained this model for more time and with larger batch size.
}
\label{tab:fid}
\end{table}
\begin{table}[t]
\centering
\begin{tabular}{|l|c|c|}
\hline
Model & Precision & Recall \\
\hline
StyleGANv2 & 0.609 & 0.513 \\
CIPS & 0.613 & 0.493 \\
\hline
\end{tabular}
\caption{Precision \& Recall measured on FFHQ at $256^2$.
The resulting quality of our model is better in terms of precision (corresponds to plausibility of images) and worse in recall (this points to the greater number of dropped modes).
}
\label{tab:ffhq_prec_rec}
\end{table}
For evaluation, we relied on commonly used metrics for image generation: Frechet Inception Distance (FID)~\cite{fid} as well as more recently introduced generative Precision and Recall measures~\cite{precision_recall,improved_prec_rec}.
Our main evaluation is thus against the state-of-the-art StyleGANv2 generator~\cite{stylegan2}. We took FID value for LSUN Churches directly from the original paper~\cite{stylegan2} and trained StyleGANv2 on other datasets in authors' setup but without style-mixing and path regularization of generator -- as noted in the original paper, these changes do not influence the FID metric.
The results of this key comparison are presented in Tab.~\ref{tab:fid} and \ref{tab:ffhq_prec_rec}. Neither of the two variants of the generator dominates the other, with StyleGANv2 achieving lower (better) FID score on FFHQ and Landscapes, while CIPS generator achieving lower score on LSUN Churches and both Satellite datasets.
\subsection{Ablations}
\label{sec:modifications}
\begin{table*}[h]
\centering
\begin{tabular}{|l|c|c|c|c|c|c|}
\hline
CIPS & ``base'' & ``No embed (NE)'' & ``No Fourier'' & \textbf{Main} & ``Residual'' & Sine \\
\hline
Fourier Features & + & + & -- & + & + & -- \\
Coordinate Embeddings & + & -- & + &+ & + & + \\
Residual blocks & -- & -- & -- &-- & + & -- \\
Skip connections & -- & -- & -- & + & -- & -- \\
Sine Activation & -- & -- & -- & -- & -- & + \\
\hline
FID & 6.71 & 12.71 & 10.18 & \textbf{6.31} & 6.52 & 10.0 \\
\hline
\end{tabular}
\caption{Effects of the modifications of CIPS generator on the FFHQ dataset in terms of Frechet Inception Distance (FID) score. Each column corresponds to a certain configuration, while rows correspond to present/missing features.
The simultaneous usage of Fourier features and coordinate embeddings is necessary for a good FID score.
Also, both residual connections and cumulative skip connections (default configuration) to the output outperform the plain multilayer perceptron.
}
\label{tab:modifications}
\end{table*}
\begin{figure}[t]
\centering
\includegraphics[width=\columnwidth]{figures/CIPS-vs-CIPS-wo-emb.jpg}
\caption{Image corresponding to the mean style vector in the space of $\mathcal{W}$ for \ganname (left) and \ganname-NE (no embeddings) generators (right). Left image has more plausible details like hair which confirms the results in Tab.~\ref{tab:modifications}.
}
\label{fig:cips-vs-cips-wo-emb}
\end{figure}
We then evaluate the importance of different parts of our model by its ablation on the FFHQ dataset (Tab.~\ref{tab:modifications}).
We thus consider removing Fourier features, coordinate embeddings (config referred to as \emph{\ganname-NE}) and replace LeakyReLU activation with sine function in all layers.
We also compare the variants with residual connections (we follow StyleGANv2~\cite{stylegan2} implementation adjusting variance of residual blocks with the division by $\sqrt{2}$) with our main choice of cumulative projections to RGB.
Additionally, the ``base'' configuration without skip connections and residual connections is considered.
In this comparison, all models were trained for 300K iterations with batch size of 16.
As the results show, coordinate embeddings, residual blocks and cumulative projection to RGB significantly improve the quality of the model.
The removal of coordinate embeddings most severely worsens the FID value, and affects the quality of generated images (Fig.~\ref{fig:cips-vs-cips-wo-emb}). We further investigate the importance of coordinate embeddings for the CIPS model below.
\subsection{Influence of positional encodings}
\begin{figure}[ht]
\centering
\setlength{\tabcolsep}{2pt}
\begin{tabular}{cc}
\includegraphics[width=0.49\columnwidth]{figures/encoding_comaparison/fourier_encoding_magnitudes.pdf} &
\includegraphics[width=0.49\columnwidth]{figures/encoding_comaparison/embed_encoding_magnitudes.pdf}
\\
\footnotesize (a) Fourier features &
\footnotesize (b) Coordinate embeddings
\end{tabular}
\caption{The spectrum magnitude for our two kinds of positional encoding (color scale is equal for both plots). The output of coordinate embeddings clearly has more higher frequencies.}
\label{fig:pos_encoding_magnitude}
\end{figure}
\begin{figure}[ht]
\centering
\setlength{\tabcolsep}{2pt}
\begin{tabular}{cc}
\includegraphics[width=0.49\columnwidth]{figures/encoding_comaparison/pca_fourier.pdf} &
\includegraphics[width=0.49\columnwidth]{figures/encoding_comaparison/pca_embed.pdf}
\\
\footnotesize (a) Fourier features &
\footnotesize (b) Coordinate embeddings
\end{tabular}
\caption{PCA plot (3 components) for two kinds of positional encoding of \ganname-base.
Coordinate embeddings contain not just more fine-grained details, but also key points of the averaged face.
}
\label{fig:pca_encoding}
\end{figure}
\begin{figure}[ht]
\centering
\setlength{\tabcolsep}{2pt}
\begin{tabular}{ccc}
\includegraphics[width=0.31\columnwidth]{figures/encoding_comaparison/spectrum_img_original.pdf} &
\includegraphics[width=0.31\columnwidth]{figures/encoding_comaparison/spectrum_img_wo_embed.pdf} &
\includegraphics[width=0.31\columnwidth]{figures/encoding_comaparison/spectrum_img_wo_fourier.pdf}
\end{tabular}
\caption{Influence of different types of positional encoding on the resulting image.
Left: original image.
Center: coordinate embeddings zeroed out (the image contains no fine-grained details).
Right: Fourier features zeroed out (only high-frequency details are present).
}
\label{fig:pos_encoding_zeroed}
\end{figure}
To analyze the difference between Fourier features $e_{fo}$ and coordinate embeddings $e_{co}$, we plotted the spectrum of these codes for the generator \ganname-base, trained on FFHQ.
As shown in Fig.~\ref{fig:pos_encoding_magnitude}, Fourier encoding generally carries low-frequency components, whereas coordinate embeddings resemble more high-frequency details.
The Principal Component Analysis (PCA) of the two encodings supports the same conclusion (Fig.~\ref{fig:pca_encoding})
The possible explanation is simple: coordinate embeddings are trained independently for each pixel, while $e_{fo} \left(x, y\right)$ is a learned function of the coordinates.
However, the next layers of the network could transform the positional codes and, for example, finally produce more fine-grained details, relying on Fourier features.
To demonstrate that this is not the case, we conducted the following experiment. We have zeroed out either the output of Fourier features or coordinate embeddings and showed the obtained images in Fig.~\ref{fig:pos_encoding_zeroed}. One can notice that the information about the facial hair's details as well as the forelock is located in the coordinate embeddings.
This proves that it is the coordinate embeddings that are the key to high-frequency details of the resulting image.
\subsection{Spectral analysis of generated images}
\label{sec:spectral_analysis}
\begin{figure}
\centering
\begin{subfigure}{\columnwidth}
\centering
\includegraphics[width=\linewidth]{figures/upd_spectrum_final.pdf}
\caption{Magnitude spectrum.
Our models produce less artifacts in high frequency components (note the grid-like pattern in StyleGANv2).
Two \ganname models are difficult to distinguish between (better zoom in).
}
\label{fig:2d_spectrum}
\end{subfigure}
\begin{subfigure}{\columnwidth}
\centering
\includegraphics[width=\linewidth]{figures/exp_azimuth1d_spectr_zoom_final.pdf}
\caption{Azimuthal integration over Fourier power spectrum.
The curve of StyleGANv2 has heavy distortions in most high frequency components.
Surprisingly, \ganname-NE demonstrates a more realistic and smooth tail than \ganname-base, while being worse in terms of FID.
}
\label{fig:azimuth_psd}
\end{subfigure}
\caption{Spectral analysis for models trained on FFHQ at resolution of $256^2$.
All results are averaged across 5000 samples.
We demonstrate that \ganname-NE is most similar to real images.
}
\label{fig:spectra}
\end{figure}
Recently, \cite{power_spectral_density} observed that the common convolutional upsampling operations can lead to the inability to learn the spectral distribution of real images, in spite of any generator architecture.
In contrast, \ganname operates explicitly with the coordinate grid and has no upscaling modules, which should to improved reproduction of the spectrum.
Indeed, we compare the spectrum of our models (CIPS-``base'' without residual and skip connections; CIPS-NE) to StyleGANv2 and demonstrate that CIPS generators design has the advantage in the spectral domain.
The analysis of magnitude spectra for produced images is given in Fig.~\ref{fig:2d_spectrum}.
The spectrum of StyleGANv2 has artifacts in high-frequency regions, not present in both CIPS generators under consideration.
Following prior works~\cite{power_spectral_density}, we also use the azimuthal integration (AI) over the Fourier power spectrum (Fig.~\ref{fig:azimuth_psd}).
It is worth noting that AI statistics of \ganname-NE are very close to the ones of real images.
However, adding the coordinate embeddings degrades a realistic spectrum while improving the quality in terms of FID~(Tab.~\ref{tab:modifications}).
We note that the introduction of skip connections in fact makes the spectra less similar to those of natural images.
\subsection{Interpolation}
We conclude the experimental part with the demonstration of the flexibility of \ganname. As well as many other generators, CIPS generators have the ability to interpolate between latent vectors with meaningful morphing (Fig.~\ref{fig:linear_interpolate}).
As expected, the change between the extreme images occurs smoothly and allows for the use of this property, in a similar vein as in the original works (e.g.~\cite{stylegan2}).
\begin{figure}[t]
\centering
\includegraphics[width=\columnwidth]{figures/fov_grid.png}
\caption{
Images generated using foveated synthesis. In each case, the CIPS generator was sampled on a 2D Gaussian distribution concentrated in the center of an image (standard deviation = $0.4*$image size).
Left to right: sampled pattern covers 5\% of all pixels, 25\%, 50\%, 100\% (full coordinate grid).
Missing color values have been filled via bicubic interpolation.
}
\label{fig:foveated_grid}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=\columnwidth]{figures/face_oversampling.png}
\caption{
Left: the generated image of resolution $256\times{}256$, upscaled with Lanczos upsampling scheme \cite{lanczos1950iteration} to $1024\times{}1024$.
Right: the image, synthesized by \ganname, trained at resolution of $256\times{}256$ on the coordinate grid of resolution $1024\times{}1024$.
Note the sharpness/plausibility of the eyelid and the more proper shape of the pupil.
}
\label{fig:oversampled_image}
\end{figure}
\begin{figure}[ht!]
\centering
\includegraphics[width=\linewidth]{figures/linear_interpolation.pdf}
\caption{Latent linear morphing between two sampled images -- the left-most and right-most ones.}
\label{fig:linear_interpolate}
\end{figure}
\subsection{Foveated rendering and interpolation}
One of the inspiring applications of our per-pixel generator is the foveated synthesis. The foveated synthesis ability can be beneficial for computer graphics and other applications, as well as mimics human visual system. In foveated synthesis, an irregular grid of coordinates is sampled first, more dense in the area, where the gaze is assumed to be directed to, and more sparse outside of that region. After that, \ganname is evaluated on this grid (its size is less than the full resolution), and color for missing pixels of the image is filled using interpolation. The demonstration of this method is provided in Fig.~\ref{fig:foveated_grid}.
Alongside the foveated rendering, we are also able to interpolate the image beyond the training resolution by simply sampling denser grids. Here we use a model, trained on images of $256\times{}256$ resolution to process a grid of $1024\times{}1024$ pixels and compare it with upsampling the results of upsampling the image synthesized at the $256\times{}256$ resolution with the Lanczos filter \cite{lanczos1950iteration}. As Fig.~\ref{fig:oversampled_image} suggests, more plausible details are obtained with denser synthesis than with Lanczos filter.
\subsection{Panorama synthesis}
\begin{figure*}[ht!]
\captionsetup[subfigure]{font=footnotesize}
\begin{minipage}{1.\columnwidth}
\centering
\subfloat[]{
\label{fig:panorama:a}
\includegraphics[scale=.24]{supplementary_figures/panorama_blends/panorama_25_1.png}
}
\end{minipage}%
\hspace{0.9cm}%
\begin{minipage}{1.\columnwidth}
\centering
\subfloat[]{\label{fig:panorama:b}\includegraphics[scale=.24]{supplementary_figures/panorama_blends/panorama_25_2.png}}
\end{minipage}
\par\medskip
\subfloat[]{\label{fig:panorama:result}\includegraphics[scale=.515]{supplementary_figures/panorama_blends/panorama_25_res.png}}
\caption{Panorama blending. We linearly blend two upper images from CISP generator trained on the Landscapes dataset with a cylindrical coordinate system. The resulting image contains elements from both original panoramas: land and water integrated naturally.}
\label{fig:panorama}
\end{figure*}
As \ganname is built upon a coordinate grid, it can relatively easily use non-Cartesian grids. To show this, we thus adopt a cylindrical system to produce landscape panoramas.
The training setup is as follows. We uniformly sample a crop $256 \times 256$ from the cylindrical coordinate grid and train the generator to produce images using these coordinate crops as inputs. A similar idea was also explored in \cite{cocogan}. We note, however, that during training we do not use any real panoramas in contrast to other coordinate-based COCO-GAN model~\cite{cocogan}.
Fig.~\ref{fig:panorama:a} and \ref{fig:panorama:b} provide examples of panorama samples obtained with the resulting model.
As each pixel is generated from its coordinates and style vector only, our architecture admits pixel-wise style interpolation (Fig.~\ref{fig:panorama:result}). In these examples, the style vector blends between the central part (the style of Fig.~\ref{fig:panorama:a}) and the outer part (the style of~\ref{fig:panorama:b}).
\subsection{Typical artifacts}
Finally, we show the typical artifacts that keep recurring in the results of \ganname generators (Fig. \ref{fig:failure_cassess}). We attribute the wavy texture (in hair) and repeated lines pattern (in buildings) to the periodic nature of sine activation function within the Fourier features. Also we note that sometimes \ganname produces a realistic image with a small part of the image being inconsistent with the rest and out of the domain. Our belief is that this behaviour is caused by the LeakyReLU activation function that divides the coordinate grid into parts. For each part, \ganname effectively applies its own inverse discrete Fourier transform. As \ganname generators do not use any upsampling or other pixel coordination, it is harder for the generator to safeguard against such behaviour.
\label{sec:failures}
\begin{figure}[ht!]
\centering
\includegraphics[width=\linewidth]{figures/failure_cases_all.pdf}
\caption{Examples of the most common kinds of artifacts on different datasets. They are best described as wavy textures on hair, background, and glowing blobs. see text for the discussion.
}
\label{fig:failure_cassess}
\end{figure}
\section{Coordinate embeddings}
\label{sec:coordinate_embeddings}
We also run the Principle Components Analysis (PCA) for coordinate embeddings of models trained on Landscapes and LSUN-Churches images (similar pattern for the FFHQ dataset is shown in the main paper).
Fig.~\ref{fig:pca_churches} provides the visualisation for the three main components. Note, that as these datasets are as aligned as FFHQ, there is considerably less spatial structural information in the learned embeddings.
\begin{figure}[t!]
\centering
\setlength{\tabcolsep}{2pt}
\begin{tabular}{ccc}
\includegraphics[width=0.31\columnwidth]{supplementary_figures/embeddings_pca/pca_embed_landscape.pdf} &
\includegraphics[width=0.31\columnwidth]{supplementary_figures/embeddings_pca/pca_embed_churches.pdf} &
\includegraphics[width=0.31\columnwidth]{supplementary_figures/embeddings_pca/captioned_church_original.pdf}
\end{tabular}
\caption{Visualisation of three main principal components of coordinate embeddings for CIPS models, trained on Landscapes (left) and LSUN-Churches (center). As these datasets are not as aligned as the face dataset, there is less recognizable structure in the learned coordinate embeddings. The bottom horizontal structure in the LSUN-Churches case is likely due to frequent watermark pattern in the dataset (a sample from the model with such watermark is shown on the right).
}
\label{fig:pca_churches}
\vspace{-1ex}
\end{figure}
\section{Architecture details}
\begin{figure}[h!]
\centering
\includegraphics[width=0.85\linewidth]{supplementary_figures/CIPS_full_scheme.pdf}
\caption{The diagram of the CIPS generator (default version).}
\label{fig:cips_scheme}
\end{figure}
\begin{table}[h!]
\centering
\begin{tabular}{|l|c|}
\hline
Modification & \# parameters (mln) \\
\hline
CIPS-``base'' & 43.8 \\
CIPS-NE & 10.2 \\
CIPS-Default & 45.9 \\
\hline
StyleGANv2 & 30.0 \\
\hline
\end{tabular}
\caption{The number of parameters for different version of the CIPS generator.
For reference, the number of parameters within the StyleGANv2 generator is also given.
}
\label{tab:num_params}
\end{table}
In this section we provide additional information about the default version of our CIPS generator (Fig.~\ref{fig:cips_scheme}). In total, its backbone contains 15 fully connected layers. The first layer projects concatenated coordinate embeddings and Fourier features into the joint space with the dimension of 512.
Next, the following layer pattern is repeated seven times. The representation is put through two modulated fully-connected layers and a projection to RGB color space is computed. The projections coming from the seven iterations are summed together to create the final image.
The number of parameters for the different modifications of the CIPS generator discussed in the paper are given in Tab.~\ref{tab:num_params}.
\section{Nearest neighbors}
\label{sec:neighbors}
To assess the generalization ability of CIPS architecture, we also show the samples from the model trained on the FFHQ face dataset alongside the most similar faces from the train dataset. To mine the most similar faces, we extract faces using the MTCNN model~\cite{mtcnn}, and then compute their embeddings using FaceNet~\cite{facenet} (the public implementation of these models\footnote{\url{https://github.com/timesler/facenet-pytorch}} was used).
Fig.~\ref{fig:neighbors} shows five nearest neighbors (w.r.t.\ FaceNet descriptors) for each samples. The samples generated by the model are clearly not duplicates of the training images.
\section{Additional samples}
\label{sec:samples}
In Fig.~\ref{fig:samples}, we provide additional samples from CIPS generators trained on different datasets. We also demonstrate more samples of cylindrical panoramas in Fig.~\ref{fig:panoramas}.
Although we do not apply mixing regularization~\cite{stylegan1} at train time, our model is still capable of layer-wise combination of latent variables at various depth (see Fig.~\ref{fig:style_mixing}). The examples suggest that similarly to StyleGAN, different layers of CIPS control different aspects of images.
\section{Patch-based generation}
\label{sec:patch_based}
\begin{table}[t]
\centering
\begin{tabular}{|l|c|c|}
\hline
Dataset & Patch size & FID \\
\hline
\multirow{3}{7em}{FFHQ} & 256 & 4.38 \\
& 128 & 9.08 \\
& 64 & 11.79 \\
\hline
\multirow{3}{7em}{LSUN-Churches} & 256 & 2.92 \\
& 128 & 7.08 \\
& 64 & 11.53 \\
\hline
\end{tabular}
\caption{Frechet Inception Distance (FID) values for CIPS models trained on patches of varying receptive field and fixed resolution ($64\times{}64$ and $128\times{}128$). The results for patch-based training are worse than the default training procedure, in which the discriminator observes the full $256\times 256$ image.
}
\label{tab:patches}
\end{table}
To show one benefit of coordinate-based approach, we demonstrate the results of \textit{memory-constrained} training, where the discriminator observes patches at lower resolution than the full image (inspired by the GRAF system~\cite{graf}). Since pixel generation is conditionally-independent, at each iteration only low-resolution patches need to be generated. Thus, only the following $K \times K$ patch is synthesized and submitted to the discriminator:
\small
\begin{align*}
P_{K, \sigma} \left( u, v \right) = \lbrace G\left(u+i\sigma, v+j\sigma; \mathbf{z}\right) \mid \left(i, j\right) \in \texttt{mgrid}\left(K, K \right)\rbrace,
\label{eq:patch_gen}
\end{align*}
\normalsize
where $0 \le u < W-\left(K-1\right)\sigma$ and $0 \le v < H-\left(K-1\right)\sigma$ are the coordinates of the corner pixel of the patch.
For $\sigma = 1$ this produces dense patch, while for $\sigma > 1$ dilated patch with increased receptive field is obtained. Applying this patch sampling to real images before putting them into the discriminator may be thought of as an example of a differentiable augmentation, the usefulness of which was recently proved by~\cite{aug_nvidia,aug_mit}.
Tab.~\ref{tab:patches} reports the quality (FID) for CIPS generators trained on patches of sizes $64\times{}64$ and $128\times{}128$, while the resolution of full images equals $256\times{}256$.
Fig.~\ref{fig:patch_samples} shows the outputs of models, trained with the patch-based pipeline. In our experiments, training with smaller size of patches degrades the overall quality of resulting samples.
\begin{figure}[t!]
\centering
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=\linewidth]{supplementary_figures/patches/churches_patch_samples.pdf} \\
LSUN-Churches
\label{fig:patch-church}
\vspace{10pt}
\end{subfigure}
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=1\linewidth]{supplementary_figures/patches/ffhq_patch_samples.pdf} \\
FFHQ
\label{fig:patch-ffhq}
\end{subfigure}
\caption{Samples from CIPS generators learned with memory-constrained \textbf{patch-based training}. Within every grid, the top row contains images from models trained with patches of size $128 \times 128$ and the bottom row represents outputs from training on $64 \times 64$ patches. While the samples obtained with such memory-constrained training are meaningful, their quality and diversity are worse compared to standard training.
}
\label{fig:patch_samples}
\end{figure}
|
1,116,691,499,217 | arxiv | \subsection*{Model 1}
Consider the pseudoscalar interaction
\begin{equation}
L_{\rm int} = if\overline\psi\gamma_5\psi\Phi + \lambda\Phi^4\,.
\end{equation}
Notice that the Lagrangian in this case obeys a parity symmetry
\begin{eqnarray}
\Phi \to -\Phi \,, \quad \psi \to \gamma_0 \psi \,,
\label{Psymm}
\end{eqnarray}
under which these are the only possible renormalizable
interaction terms. A cubic $\Phi^3$ interaction, for example, is
not invariant under this symmetry.
Using the free-field propagator in Eq.\ (\ref{S11}) for the
fermion field we obtain
\begin{eqnarray}
\label{pinonsingular}
\mbox{Re} \, {\mbox{\boldmath $\Pi$}}^{(\Phi)}_{11}(k_0, \vec k) & = &
4f^2 \int \frac{d^4p}{(2\pi)^3} \eta_F(p,\alpha_\psi)
\delta(p^2 - m_\psi^2)\nonumber\\
& & \quad\times\left\{\frac{p\cdot k}{k^2 + 2p\cdot k}
+ (k\rightarrow -k)\right\}\,,
\end{eqnarray}
which has the unique limit
\begin{eqnarray}
\mbox{Re} \, {\mbox{\boldmath $\Pi$}}^{(\Phi)}_{11}(0, \vec 0) & = &
4f^2 \int \frac{d^3p}{2E(2\pi)^3}(f_\psi + f_{\overline\psi})\,,
\end{eqnarray}
where we have denoted by $f_{\psi,\overline\psi} =
f_F(p,\pm\alpha_\psi)$ the fermion and
antifermion momentum distributions.
In Eq.\ (\ref{pinonsingular})
the vacuum contribution has been omitted,
as we will always do henceforth
whenever we
write explicit expressions for the self-energies.
In passing, we note that the property of a unique
zero-momentum limit does not follow
if the fermion-boson interaction is scalar rather than pseudoscalar.
On the other hand, if the fermion bilinear is either vector or
axial vector type, with a derivative coupling to the spin-0 boson,
the result of the 1-loop diagram gives
$\mbox{Re} \,
{\mbox{\boldmath $\Pi$}}^{(\Phi)}_{11}(0, \vec 0) = 0$
independently of how the limit is taken. Of course, such couplings
can arise only if the theory at hand is an effective one.
Nevertheless, it shows the possibility that
the zero momentum singularity
goes away
if the self-energy is calculated with an effective vertex,
that may itself be the result of an improved calculation,
instead of the fundamental one.
In the case of the vector coupling,
the parity transformation of Eq.\ (\ref{Psymm})
is not a symmetry, so
it is likely that the theory includes a trilinear coupling
$\Phi^3$, which gives a contribution to the self-energy that
contains the zero momentum singularity.
\subsection*{Model 2}
Consider a scalar $\Phi$ interacting with another (charged) scalar
$\phi$
with the following interaction Lagrangian
\begin{equation}
\label{model1scalar}
L_{\rm int} = (f\phi\phi\Phi^\ast + H.c.) + \lambda_\phi|\phi|^4
+ \lambda_\Phi|\Phi|^4
\end{equation}
It can be easily seen that these are the most general
renormalizable interaction terms if the Lagrangian obeys a
global U(1) symmetry under which the charges of $\phi$ and
$\Phi$ are 1 and 2 respectively. This symmetry automatically
rules out any cubic self-interaction of any of the scalar
fields.
This model is similar to the following one
\begin{equation}
\label{model1fermion}
L_{\rm int} = (f\psi_L^T C \psi_L\Phi^\ast + H.c.) + \lambda|\Phi|^4
\end{equation}
in which $\Phi$ interacts with a Weyl fermion.
Now consider, for example, the one-loop diagram for the $\Phi$
self-energy in a background
of $\phi$ particles, depicted in Fig.~\ref{f:piPhi}.
The result of calculating that diagram using the free-field
propagator given above for the $\phi$ field is
\begin{eqnarray}
\label{pisingular}
\mbox{Re} \, {\mbox{\boldmath $\Pi$}}^{(\Phi)}_{11}(k_0, \vec k) & = &
2f^2 \int \frac{d^4p}{(2\pi)^3} \eta_B(p,\alpha_\phi) \delta(p^2 - M_\phi^2)
\left(\frac{1}{k^2 - 2p\cdot k}\right)\nonumber\\
& = & 2f^2 \int \frac{d^3p}{(2\pi)^3 2E}\left[\frac{f_\phi}{k^2
- 2p\cdot k}
+ \frac{f_{\overline\phi}}{k^2 + 2p\cdot k}\right]\,,
\end{eqnarray}
where $p^\mu = (E,\vec p)$ with $E = \sqrt{\vec p^2 + M_\phi^2}$
and we have put $f_{\phi,\overline\phi} = f_B(p,\pm\alpha_\phi)$
to denote the momentum distributions.
The above formula reveals the problem to which we alluded
in Eq.\ (\ref{problem}). The same behaviour is obtained
for the photon self-energy in a background of electrons,
or charged scalars. The main observation of this paper is that,
as we already mentioned, this problem vanishes if the diagram
is evaluated employing the full propagator of the
$\phi$ field instead of the free-field propagator,
which is what we show now.
In what follows
we will treat in detail
only the two scalar model defined by Eq.\ (\ref{model1scalar}),
since the analysis and results are similar for the other model
of Eq.\ (\ref{model1fermion}).
As we mentioned earlier, this problem vanishes if the diagram
is evaluated employing the full propagator of the
$\phi$ field instead of the free-field propagator.
The full $\phi$ propagator, which we denote
by ${{\mbox{\boldmath $\Delta$}}}^{(\phi)\prime}(p)$,
can be written just like in Eq.\ (\ref{boseprop})
but with $\Delta_0$ replaced by
\begin{eqnarray}
\Delta^{(\phi)\prime}_0 = {1 \over p^2 - M_\phi^2 - \Pi^{(\phi)}_0},
\end{eqnarray}
where $\Pi^{(\phi)}_0$ is the self-energy function for the $\phi$ field.
Thus,
\begin{eqnarray}
{{\mbox{\boldmath $\Delta$}}}^{(\phi)\prime}(p) = {\mbox{\boldmath\cal $U$}}_B
\left( \begin{array}{cc} \Delta^{(\phi)\prime}_0 & 0 \\ 0 &
-\Delta_0^{(\phi)\prime\ast}
\end{array} \right) {\mbox{\boldmath\cal $U$}}_B \,,
\label{bosepropexact}
\end{eqnarray}
and in particular,
\begin{eqnarray}
{{\mbox{\boldmath $\Delta$}}}^{(\phi)\prime}_{11} (p) = \frac{1}{p^2 -
M_\phi^2 -
\Pi^{(\phi)}_0}
-2\pi i\rho^{(\phi)}(p)\eta_B(p,\alpha_\phi)\,,
\label{Delta'11}
\end{eqnarray}
with the spectral density $\rho^{(\phi)}$ given by
\begin{equation}
\label{rho}
\pi\rho^{(\phi)}(p) = -\frac{\mbox{Im}\,\Pi^{(\phi)}_0}
{(p^2 - M_\phi^2 - \mbox{Re}\,\Pi^{(\phi)}_0)^2 +
(\mbox{Im}\,\Pi^{(\phi)}_0)^2}\,.
\end{equation}
As the interactions are turned off, $\rho^{(\phi)}$ approaches
an on-shell delta function.
However, the existence of a non-zero $\mbox{Im}\,\Pi^{(\phi)}_0$
smears the delta function that is present
in Eq.\ (\ref{S11}). For this
reason it is the easy to see that,
if Diagram~\ref{f:piPhi} is calculated with the propagator
${{\mbox{\boldmath $\Delta$}}}^{(\phi)\prime}_{11}$ for the $\phi$ field
instead of
the free particle propagator,
then the problem of the non-analyticity of
${\mbox{\boldmath $\Pi$}}^{(\Phi)}_{11}(k)$ at zero momentum disappears.
Notice that the dispersive part of $\Pi^{(\phi)}_0$ plays no role
in this argument. It is not difficult
to see that retaining only the dispersive part of $\Pi^{(\phi)}_0$
and neglecting its absorptive part does not remove
the singularity at zero momentum of ${\mbox{\boldmath
$\Pi$}}^{(\Phi)}_{11}(k)$.
The next step is to calculate $\Pi^{(\phi)}_0$ and show that in general
it has an absorptive part.
To this end, we recall that the inverse of the full scalar
propagator is given by
\begin{eqnarray}\label{bospropinv}
{{\mbox{\boldmath $\Delta$}}}'^{-1} (p) = p^2 - M^2 -{\mbox{\boldmath
$\Pi$}}\,,
\end{eqnarray}
where ${\mbox{\boldmath $\Pi$}}$ is a $2\times 2$ matrix whose components
must be calculated using the Feynman rules of the theory.
Comparing Eqs.\ (\ref{bosepropexact}) and (\ref{bospropinv}), the
following relations are obtained~\cite{Nie90}:
\begin{eqnarray}
{\mbox{\boldmath $\Pi$}}_{11} &=& \Pi_0 + (\Pi_0 - \Pi_0^\ast )
\eta_B (p,\alpha) \\
{\mbox{\boldmath $\Pi$}}_{22} &=& -\Pi_0^\ast + (\Pi_0 - \Pi_0^\ast )
\eta_B (p,\alpha) \\
{\mbox{\boldmath $\Pi$}}_{12} &=& -(\Pi_0 - \Pi_0^\ast )
\epsilon (p\cdot u) f_B (p,\alpha)
\label{pi12}\\
{\mbox{\boldmath $\Pi$}}_{21} &=& -(\Pi_0 - \Pi_0^\ast )
\epsilon (-p\cdot u) f_B (-p,-\alpha)
\label{pi21}\,.
\end{eqnarray}
{}From these, it is easily seen that
\begin{eqnarray}\label{realparteq}
\mbox{Re}\, \Pi_0 (p) &=& \mbox{Re}\, {\mbox{\boldmath $\Pi$}}_{11}(p) \\
\mbox{Im}\, \Pi_0 (p) &=& {\epsilon(p\cdot u) {\mbox{\boldmath $\Pi$}}_{12}(p)
\over 2i
f_B(p,\alpha)} \,.\label{absparteq}
\end{eqnarray}
Therefore, to determine $\Pi^{(\phi)}_0$ we must calculate
${\mbox{\boldmath $\Pi$}}^{(\phi)}_{11}$ and
${\mbox{\boldmath $\Pi$}}^{(\phi)}_{12}$, which can be done by evaluating the
diagrams in
Fig.~\ref{f:piphi}. Since the dispersive part $\Pi^{(\phi)}_0$
is not relevant for
resolving the zero momentum problem of the $\Phi$
self-energy, we will not calculate it here. However,
for the consistency of our scheme it is important to stress that
since the internal lines in Fig.~\ref{f:piphi} correspond to particles
of different mass, the function $\mbox{Re}\,{\mbox{\boldmath
$\Pi$}}^{(\phi)}_{11}$
(and hence $\mbox{Re}\,\Pi^{(\phi)}_0$) does not suffer from
the zero momentum problem according
to the observation of AVBD~\cite{AVBD93}.
We now turn the attention to the calculation of the absorptive
part of $\Pi^{(\phi)}_0$. The simplest way to proceed is to
calculate
${\mbox{\boldmath $\Pi$}}^{(\phi)}_{12}$ and then use Eq.\ (\ref{absparteq}).
Writing
\begin{equation}
k = p^\prime + p \,,
\end{equation}
the application of the Feynman rules to the diagram of
Fig.~\ref{f:piphi} gives
\begin{eqnarray}
\label{eq.piphi1}
-i{\mbox{\boldmath $\Pi$}}^{(\phi)}_{12}(p) =
(if)(-if)\int\frac{d^4p'}{(2\pi)^4}
i{\mbox{\boldmath $\Delta$}}^{(\phi)}_{21}(p^\prime)
i{\mbox{\boldmath $\Delta$}}^{(\Phi)}_{12}(k)\,,
\end{eqnarray}
where the scalar propagators are given by
\begin{eqnarray}
{\mbox{\boldmath $\Delta$}}^{(\Phi)}_{12}(k) & = & -2\pi i\delta(k^2 -
M_\Phi^2)
f_B(k,\alpha_\Phi)\epsilon(k\cdot u)\,,\nonumber\\
{\mbox{\boldmath $\Delta$}}^{(\phi)}_{21}(p^\prime) & = & 2\pi
i\delta(p^{\prime 2} - M_\phi^2)
f_B(-p^\prime,-\alpha_\phi)\epsilon(p^\prime\cdot u)
\end{eqnarray}
Substituting these propagators
into Eq.\ (\ref{eq.piphi1}) and using Eq.\ (\ref{absparteq})
we obtain
\begin{eqnarray}\label{impi0}
\mbox{Im}\, \Pi^{(\phi)}_0(p) & = & (-\pi f^2)\epsilon(p\cdot u)
\int\frac{d^4 p'}{(2\pi)^3}\delta(k^2 - M_\Phi^2)\delta(p^{\prime 2} -
M_\phi^2)
\nonumber\\
& & \times \epsilon(p'\cdot u)\epsilon(k\cdot u)
(f_B(p^\prime,\alpha_\phi) - f_B(k,\alpha_\Phi))\,.
\end{eqnarray}
In writing this formula we have used the identity
\begin{equation}
f_B(-p^\prime,-\alpha_\phi)f_B(k,\alpha_\Phi)
= f_B(p,\alpha_\phi)[f_B(p^\prime,\alpha_\phi) - f_B (k,\alpha_\Phi)]\,,
\end{equation}
which follows from momentum conservation and the fact that
the chemical potentials satisfy
\begin{equation}
\alpha_\Phi = 2\alpha_\phi\,,
\end{equation}
as a consequence of charge conservation.
Eq.\ (\ref{impi0}) can be written in the form
\begin{equation}
\mbox{Im}\, \Pi^{(\phi)}_0 = - \left| p\cdot u \right| \Gamma(p)\,,
\end{equation}
where we have defined
\begin{eqnarray}\label{totalrate}
\Gamma(p) & \equiv & \frac{f^2}{2p\cdot u}
\int{\frac{d^3k}{(2\pi)^3 2\omega}\frac{d^3p'}{(2\pi)^3
2E'}}(2\pi)^4 \nonumber\\
& & \mbox{}\times\left\{\right.\delta^{(4)}(p + p^\prime - k)
[f_\phi(1 + f_\Phi) - f_\Phi(1 + f_\phi)]\nonumber\\
& & \mbox{} + \delta^{(4)}(p - p^\prime - k)
[(1 + f_{\overline\phi})(1 + f_\Phi) - f_{\overline\phi}f_\Phi)]\nonumber\\
& & \mbox{} + \delta^{(4)}(p + p^\prime + k)
[f_\phi f_{\overline\Phi} - (1 + f_\phi)(1 + f_{\overline\Phi})]\nonumber\\
& & \mbox{} + \delta^{(4)}(p - p^\prime + k)
[f_{\overline\Phi}(1 + f_{\overline \phi}) - f_{\overline\phi}(1 +
f_{\overline\Phi})]
\left.\right\}\,.
\end{eqnarray}
For the sake of brevity, we have put
\begin{eqnarray}
f_{\phi,\overline\phi} & = & f_B (p^\prime,\pm\alpha_\phi) \,, \nonumber\\
f_{\Phi,\overline\Phi} & = & f_B (k,\pm\alpha_\Phi) \,,
\end{eqnarray}
to denote the particle and antiparticle distributions,
and in addition,
\begin{eqnarray}
p^{\prime\mu} = (E',\vec p^{\,\prime})\,, &\qquad &
E' = \sqrt{\vec p^{\,\prime 2} + M_\phi^2}\,, \\
k^\mu = (\omega,\vec k)\,, &\qquad & \omega = \sqrt{\vec k^2 + M_\Phi^2}\,.
\end{eqnarray}
The formula for $\Gamma$ given in Eq.\ (\ref{totalrate})
is immediately recognized
as the total rate for a $\phi$ particle of energy $p^0$ and
momentum $\vec p$ (as seen from the rest frame of the medium)
with integrations over the phase space weighted by
the statistical factors appropriate for each process \cite{Wel83}.
Notice that $f^2$ is the
amplitude for the decay processes
$\phi\overline\Phi\rightarrow\overline\phi$ and
$\phi\rightarrow\overline\phi\Phi$,
for the annihilation processes
$\phi\phi\rightarrow\Phi$ and
$\phi\phi\overline\Phi\rightarrow 0$,
as well as for the inverse reactions of all of them.
For certain specific values
of $p^0$ and $\vec p$ some of these processes will
be kinematically forbidden, but in general $\Gamma$ is non-zero.
In conclusion, we have presented various models
that have similar particle
content but different interactions,
which exemplify several situations.
In some models the one-loop
self-energies are well defined at zero momentum while in others
the zero momentum singularity
dissapears if the coupling is taken to be an effective
vertex with a suitable momentum dependence.
In another type of model, the singularity
disappears if the self-energy is calculated
with improved propagators that include the absorptive
part.
Carrying over this idea to the case of QED,
which would require that we also use an improved
vertex in order to maintain the Ward identity and
guarantee a gauge invariant result, it implies that
the one-loop photon self-energy calculated with the full
charged particle propagator
instead of the free propagator
is defined at zero momentum. In particular, the absorptive
part of the charged particle self-energy, which physically
is related to the damping rate of the particle,
cannot be neglected if the photon self-energy
is evaluated at zero momentum. Then, the physical
picture that emerges is the following.
The traditional formulas that are given for
\begin{eqnarray}
\lim_{| \vec k | \to 0} \pi_{\mu\nu} (0, \vec k)
\end{eqnarray}
and
\begin{eqnarray}
\lim_{k^0 \to 0} \pi_{\mu\nu} (k^0, \vec 0)\,,
\end{eqnarray}
which are related to well known physical quantities
such as the plasma frequency and Debye radius,
are valid in the limiting cases
\begin{eqnarray}
k^0 = 0\,&;& \Gamma \ll| \vec k| \ll m \\
\vec k = 0\,&;& \Gamma \ll k^0 \ll m\,,
\end{eqnarray}
where $m$ stands for the charged particle mass.
Since the two limits correspond to two different
physical situations the results are different.
Traditionally $\Gamma$ is omitted in the above conditions,
but then it must be kept in mind that the formulas
cannot be taken literally all the way to zero momentum.
\newpage
|
1,116,691,499,218 | arxiv | \section{Introduction}
Automatic Speech Recognition (ASR) systems form a key component of various products across industry. Many of these ASR systems rely on a complex Acoustic Model (AM) whose output is rescored by a domain-specific Language Model (LM).
As we use ASR systems in new domains, the memory, maintenance and data-collection costs for these domain-specific LMs increase.
Particularly, with advent of parameter-heavy Transformer based LMs \citep{devlin-etal-2019-bert}, maintaining multiple domain-specific LMs is practically infeasible. While on the other hand, using a generic LM for all domains falls short in performance when compared to multiple domain-specific LMs. Therefore, a need for a middle ground between performance and costs is evident.
To overcome this problem, we bring forward a methodology based on recently proposed Prompt Tuning. \citet{lester2021power} introduced this idea of learning the token embeddings of the prompt used to prime a LM to a particular task. Prompts are special tokens describing a task which when appended to the input data sample, helps the model understand and use this problem description to better solve the task. For example, to solve the machine translation task, instead of fine-tuning the Transformer model with corresponding dataset, one can achieve comparable performance by just showing text describing machine translation to the powerful Transformer-based LM. In prompt tuning, instead of providing this prompt manually to the model, one learn it from the labelled examples from the task.
To the best of our knowledge, we are the first one to apply prompt-tuning for domain-adaptation of ASR systems. We find that one can learn the prompt-embeddings for not only different tasks but also different domains of the same task. This means there exists a domain description, which when passed to model as prefix, helps it better score sentences from that domain. We hypothesize that this is due to the fact that self-attention in Transformer model will create a positive interaction between domain embeddings (description) and the words from sentence, resulting in better prediction of the next token. This has an important usecase in multi-domain ASR systems, where learning a small number of parameters corresponding to domain prompt embedding can save costs associated with maintaining separate copies of LMs for each domain. Particularly, with Transformer models with millions of parameters, fine-tuning can be very slow and requires a large amount of domain data to prevent overfitting. However, with prompt-tuning one can achieve similar performance with <0.1\% of the parameters required for fine-tuning. We provide experimental evidence by comparing the performance of prompt-tuned and fine-tuned versions of LMs to score sentences from different domains of a dialog dataset. We also provide ablations on different sizes of prompt.
\section{Experiments}
\begin{figure*}
\centering
\includegraphics[width=0.8\textwidth]{result.png}
\caption{Comparing the performance, computation and memory costs of different methods of domain adaptation}
\label{fig:comp}
\end{figure*}
We design first set of experiments as a proof of concept to show that prompt-tuning works for domain adaptation of LM. For this, we used textual data from different domains of the publically available MultiDoGo dataset \citep{peskov-etal-2019-multi}. We split the data from each domain into train, dev and test. Then for each domain individually, we fine-tune (and prompt-tune) a pretrained GPT-2 \citep{radford2019language} using the train set, tune the hyper-parameters using dev set and compute the perplexity numbers on the test set. We also evaluate the test set for each domain using the same out-of-the-box pretrained GPT-2. We then compare the no-adaptation, fine-tuned and prompt-tuned versions for each domain for different performance, memory and computation time metrics. Rather than fixing the size of the prompt, we repeat the above experiment with different sizes to show how we can control the trade-offs between performance and costs using this hyper-parameter. Although, we summarize the results for one such domain (fastfood) in Fig. \ref{fig:comp}, the results are consistent across different domains. All the experiments were conducted using a machine with 16 Tesla K80 GPUs and the reported times are wall clock times in seconds. As we can see in Fig. \ref{fig:comp}, with as little as 5 prompt-tokens, the perplexity improves from 130 in out-of-the-box model (green line) to 37. With 400 prompt-tokens, we achieve a very similar perplexity to fully fine-tuned model (red line) but with number of learnable parameters reducing by large orders of magnitude. As one might expect, with increase in size of prompts, the performance improves but so do the memory and computation costs.
The above experiment proves us the effectiveness of prompt-tuning in domain adaptation. With comparable perplexity as fine-tuning, it is natural to expect that prompt-tuned model can perform at par with fully tuned model in domain specific rescoring of n-best hypothesis from an AM model in any ASR system. We verify this empirically by using the above models as first pass rescorers on an internal fastfood audio dataset for a hybrid AM model and measuring the Content Word Error Rates (CWER). While the fine-tuned version brings a relative improvement of 7.3\% over the no-adaptation baseline, training just 10 prompt tokens could already achieve 5.8\% relative improvement. Also, with 400 prompt tokens, we could almost exactly match the CWER of fine-tuned model. Since, ASR systems are used in tens of different domains, one can achieve a significant reduction in LMs that one need to maintain, by keeping a single copy of the base version of the model and saving the domain-specific prompt token embeddings.
\section{Future Work}
While the initial experiments gives us confidence about the potential use of prompt-tuning in LMs in ASR systems, we plan to explore a lot more as part of this ongoing work. We will be running multiple experiments to showcase the benefits in WER further in multiple domains, datasets, etc. While in our experiments, we used the complete training data to fine-tune (and prompt-tune) the models, we believe in limited domain data setting, prompt-tuned models might even outperform the corresponding fine-tuned versions as the number of learnable parameters are less. This will be especially critical for new domains in ASR where getting high-quality data is difficult. We initialized our prompt token embedding layer randomly but as suggested in \citep{lester2021power}, we plan to run an ablation on different choices of initializations. One such promising initialization can be embedding for the most frequent words in that domain. In addition to all these experiments, we also plan to compare our version of domain-adaptation with related works of AdapterHub \citep{pfeiffer-etal-2020-adapterhub}, meta-learning \citep{bansal-etal-2020-learning}, using semantic embeddings \citep{shenoy21_interspeech, shenoy2021asr} and freezing all but last layer of Transformer model. The analysis provided in this paper will act as a proof of concept on computation and memory efficiency of prompt-tuning guiding our future directions.
\bibliographystyle{acl_natbib}
|
1,116,691,499,219 | arxiv | \section{INTRODUCTION}
\label{sec:intro}
There are several unsolved knots in the realm of particle physics, e.g., the baryon asymmetry of the Universe, the dark matter content, the origin of neutrino masses and mixing, etc., and the understanding of these issues is one of the prime objectives of the present day research. In the last couple of decades, several diligent attempts have been made towards comprehending and resolving the issue of dynamical origin of the fermion masses and their mixing. Present scenario has taken us few steps ahead in terms of getting a convincing explanation of the origin of mass through Higgs mechanism while being within the domain of Standard Model (SM). However, it does not provide proper grounds to explain the origin of the observed neutrino masses and their mixing. Rather, very diverse approaches are made in order to gain an insightful resolution towards the above existing problems, and obviously the answer lies in going beyond standard model (BSM) physics. It should be emphasized that, certain well-defined patterns are observed in quark masses and mixing, the appreciation of which is still an enigma. Nonetheless, there are ample amount of research work present, which make an attempt to grasp their fundamental origin. In addition, perplexity to the problem has increased due to the observation of the neutrino masses and their sizeable mixing. The reason being, the order of magnitude of the observed neutrino masses are approximately twelve order smaller than that of electroweak (EW) scale. Also, there is immense difference in the pattern of leptonic and quark mixings, the former is having large mixing angles, while the later involves smaller mixing angles. Numerous experiments \cite{Bronner:2020eps,Pac:2018scx,Yu:2017woe,DoubleChooz:2019qbj} have corroborated the tininess of the neutrino masses and other parameters with high accuracy. The global fit values of the neutrino oscillation parameters are furnished in Refs. \cite{deSalas:2020pgw, Esteban:2020cvm}.
It is well-known that in the SM framework, the neutrino mass generation can not be explained through the standard Higgs mechanism due to the absence of the right-handed (RH) components. Still, if we could manage to add the RH neutrinos into SM by hand, and allow Dirac mass terms, the values of the required Yukawa couplings to be around ${\cal O}(10^{-12})$, i.e., which appear as aberrant. In contrast, there exist many BSM scenarios that help to generate tiny neutrino mass through the conventional seesaw mechanism. Some of the prominent seesaw mechanisms are categorized as type-I \cite{Mohapatra:1979ia,Brdar:2019iem,Branco:2020yvs,Bilenky:2010zza}, type-II \cite{Gu:2006wj,Luo:2007mq,Antusch:2004xy,Rodejohann:2004cg,Gu:2019ogb,McDonald:2007ka}, type-III \cite{Liao:2009nq,Ma:1998dn,Foot:1988aq,Dorsner:2006fx,Franceschini:2008pz,He:2009tf} and all of them require additional heavy fermions or scalars beyond the SM particle content. Literature survey shows there are many flavor symmetries either discrete $A_4$ \cite{King:2006np,Kalita:2015jaa,Altarelli:2007gb}, $S_3$ \cite{Kimura:2005sx,Mishra:2019keq,Meloni:2010aw,Pramanick:2019oxb}, $S_4$ \cite{Krishnan:2012me,Chakraborty:2020gqc,Vien:2016jkz} etc. or continuous $U(1)_{B-L}$ \cite{Ma:2014qra,Kanemura:2014rpa,Kanemura:2012zh,Mishra:2019gsr,Singirala:2017cch}, $U(1)_H$ \cite{Cai:2018upp,Nomura:2021adf,Dey:2019cts}, $U(1)_{L_e-L_\tau}$ \cite{Esmaili:2019pcy,Behera:2021nuk} etc., which can generate the tiny neutrino masses and also accommodate the observed neutrino oscillation data with the help of some additional scalars and perturbation (wherever required). As aforesaid, inclusion of flavons affects the neatness of the model and the predictability of the model is hampered because of the higher dimensional operators. These drawbacks can be eliminated through the recent approach of including modular symmetry, where the Yukawa couplings transform non-trivially under the discrete flavor symmetry group and have certain modular weight.\\
The modular group $\Gamma^{\prime}_5 \simeq A^{\prime}_5$ is a new and promising candidate, which corresponds to the special case of $N = 5$. People have done extensive studies on the essential properties of this $A^\prime_5$ finite group~\cite{Everett:2010rd,Hashimoto:2011tn,Chen:2011dn}, so here we bring up only the important points regarding $A^\prime_5$ modular symmetry. The $A^\prime_{5}$ group consists of 120 elements, which are likely to be originated by the generators: $S$, $T$ and $R$ gratifying the identities for $N = 5$ \cite{Wang:2020lxk}. So, categorization of these 120 elements are done into nine conjugacy classes which are represented by nine well defined irreducible representations, symbolized as ${\bf 1}$, $\widehat{\bf 2}$, $\widehat{\bf 2}^{\prime}_{}$, ${\bf 3}$, ${\bf 3}^{\prime}$, ${\bf 4}$, $\widehat{\bf 4}$, ${\bf 5}$ and $\widehat{\bf 6}$. Additionally, the conjugacy classes along with the character table of $A^\prime_5$, and the representation matrices of the generators are presented in Appendix of Ref.~\cite{Wang:2020lxk}. It ought to be noticed that the ${\bf 1}$, ${\bf 3}$, ${\bf 3}^{\prime}_{}$, ${\bf 4}$ and ${\bf 5}$ representations with $R = \mathbb{I}$ are identical to those for $A^{}_{5}$, while $\widehat{\bf 2}$, $\widehat{\bf 2}^{\prime}_{}$, $\widehat{\bf 4}$ and $\widehat{\bf 6}$ are different for $A^\prime_5$, with $R = -\mathbb{I}$. As we are working in the modular space of $\Gamma(5)$, hence, its dimension is $5k+1$, where, $k$ is the modular weight.
For $k=1$, the modular space $M_1[\Gamma(5)]$ will have six basis vectors i.e., ($\widehat{e}_i$, where $i=1,2,3,4,5,6$) whose $q$-expansion are given below and they are used in expressing the Yukawa coupling $Y^{(1)}_{\widehat{\bf6}}$ as shown in Appendix \ref{HOYC}:
\begin{eqnarray}
\widehat{e}_1&=&1+3q+4q^2+2q^3+q^4+3q^5+6q^6+4q^7-q^9+ \cdots \;,\nonumber\\
\widehat e_2 &=& q^{1/5} \left ( 1+2q+2q^2+q^3+2q^4+2q^5+2q^6+q^7+2q^8+2q^9+ \cdots\right )\;,\nonumber\\
\widehat e_3 &=& q^{2/5} \left ( 1+q+q^2+q^3+2q^4+q^6+q^7+2q^8+q^9+ \cdots\right )\;,\nonumber\\
\widehat e_4 &=& q^{3/5} \left ( 1+q^2+q^3+q^4-q^5+2q^6+2q^8+q^9+ \cdots\right )\;,\nonumber\\
\widehat e_5 &=& q^{4/5} \left ( 1-q+2q^2+2q^6-2q^7+2q^8+q^9+ \cdots\right )\;,\nonumber\\
\widehat e_6 &=& q\left ( 1-2q+4q^2-3q^3+q^4+2 q^5-2q^6+3q^8-2q^9+ \cdots\right )\;.
\label{eq:basisq}
\end{eqnarray}
Our aim here is to utilize the expediency of the modular $A^\prime_5$ symmetry by incorporating the linear seesaw mechanism in the context of supersymmetry, as we are quite familiar with the dynamics of TeV scale seesaw frameworks from numerous \cite{Grimus:2011mp,Ma:2009dk} literature. The deciding factor whether it will be linear seesaw or inverse seesaw is the position of the zero elements in the mass matrix under the basis of ($\nu$, $N_{R_i}$, $S_{L_i}$). It is quite evident when 11 and 33 elements of the mass matrix are zero, it gives the structure of linear seesaw. As mentioned above, introduction of three left-handed ($S_{L_i}$) alongside three right-handed ($N_{R_i}$)
neutral fermion superfields validates and generates the neutrino mass matrix structure of linear seesaw which has been widely studied in the context of discrete $A_4$ flavor symmetry \cite{Sruthilaya:2017mzt,Borah:2018nvu,Borah:2017dmk}. In this work, we are interested to implement it with $A^\prime_5$ modular symmetry. For this purpose, we consider the heavy fermions $S_{Li}$ and $N_{Ri}$ to transform as \textbf{$3^\prime$} under $A^\prime_5$ symmetry and the modular form of the Yukawa couplings leads to a constrained structure. After that we perform the numerical analysis to look for the region which is acceptable in order to fit the neutrino data. Hence, prediction for the neutrino sector is done after fixing the allowed parameter space.
\vspace{1mm}
The outline of the paper is as follows. In Sec. \ref{sec:linear}, we present the layout of the linear seesaw framework in the context of discrete $A^\prime_5$ modular symmetry.
Using the $A_5'$ product rules, we obtain the simple mass matrix structure for the charged leptons as well as the neutral fermions. After that we briefly discuss the light neutrino masses and mixing phenomena in this framework. The numerical analysis is performed in Sec. \ref{sec:results} followed by a brief comment on the non-unitarity effect. Sec. \ref{sec:lepto} contains the discussion on leptogenesis in the context of the present model and finally our results are summarized in Sec. \ref{sec:con}.
\section{The Model }
\label{sec:linear}
Here we work on a model under linear seesaw scenario in the context of supersymmetry (SUSY), where Table \ref{tab:fields-linear} provided below expresses the list of particles and their respective group charges. For exploring neutrino sector beyond standard model, we extend it with the discrete $A^\prime_5$ modular symmetry and a local $U(1)_{B-L}$ gauge symmetry. However, the local $U(1)_{B-L}$ becomes the auxillary symmetry which has been added to avoid certain undesirable terms in the superpotential. The advantage of working in BSM is that we can add right-handed neutrinos and extra fields, and hence, here we have included three SM singlet right-handed superfields ($N_{Ri}$), three left-handed superfields ($S_{Li}$) and one weighton ($\zeta$) in the particle gamut. The transformation of extra added superfields is taken as \textbf{$3^\prime$} under the modular group $A^\prime_5$. The $A^\prime_5$ and $U(1)_{B-L}$ symmetries are assumed to be broken at a very high scale, much greater than the scale of electroweak symmetry breaking \cite{Dawson:2017ksx}. Mass acquisition by the additional singlet superfield happens by assigning the vacuum expectation value (VEV) to the weighton $\zeta$. All the particles are assigned definite values of modular weights denoted as $k_I$. One of the key point of introducing modular symmetry is the curtailment of flavon (weighton) fields, which otherwise are traditionally required while working in BSM with discrete symmetries, since the Yukawa couplings transform non-trivially under $A^\prime_5$ modular symmetry group, and their transformation properties are present in \cite{Wang:2020lxk}.
\begin{center}
\begin{table}[h!
\centering
\begin{tabular}{|c||c|c|c|c|c|c||c|c|c|c|}\hline\hline
\textbf{Fields}& ~$e^c_R$~& ~$\mu^c_R$~ & ~$\tau^c_R$~& ~${L}_L$~& ~$N^c_R$~& ~$S_L$~& ~${H}_{u,d}$~&~$\zeta$ \\ \hline
$SU(2)_{L}$ & 1 & 1 & 1 & 2 & 1 & 1& 2 & 1 \\\hline
$U(1)_{Y}$ & 1 & 1 & 1 & $-\frac12$ & 0 & 0 & $\frac12, -\frac12$ & 0 \\\hline
$U(1)_{B-L}$ & 1 & 1 & 1 &$-1$ & 1 &0 &0 &1 \\\hline
$A^\prime_5$ & 1 & 1 & 1 & 3 & $3'$ & $3'$ & 1 & 1 \\ \hline
$k_I$ & 1 & 3 & 5 & 1 & 1 & 4 & 0 & 1\\
\hline
\end{tabular}
\caption{The particle spectrum and their charges under the symmetry groups $SU(2)_L\times U(1)_Y\times U(1)_{B-L} \times A^\prime_5$ while $k_I$ represents the modular weight.}
\label{tab:fields-linear}
\end{table}
\end{center}
\vspace{2cm}
The complete superpotential is given by
\begin{eqnarray}
\mathcal{W} &=& A_{M_l}\left[({L}_L l^c_R)_{\textbf{3}} Y^{k_Y}_{\textbf{3}}\right]H_d + \mu H_u H_d + G_D \left[(L_L N^c_R)_{\textbf{5}} Y_{5}^{(2)}\right] H_u + \\ \nonumber
&& G_{LS}\left[(L_L S_L)_4 H_u\sum\limits^{2}_{i=1} Y^{(6)}_{4,i}\right] \frac{\zeta}{\Lambda}+ B_{M_{RS}}\left[({S}_L N^c_R)_{\textbf{5}} \sum\limits^{2}_{i=1}Y_{\textbf{5},i}^{(6)}\right] \zeta^*, \label{super-p}
\end{eqnarray}
where $A_{\mathcal{M}_l}= (\alpha_{\mathcal{M}_l}, \beta_{\mathcal{M}_l}, \gamma_{\mathcal{M}_l}$), $l^c_R = (e^c_R, \mu^c_R, \tau^c_R$), $k_Y = (2, 4, 6)$, and $G_D = \rm{diag} \{g_{D_1}, g_{D_2}, g_{D_3}\}$, $G_{LS} = \rm{diag\{g_{LS_1},g_{LS_2}, g_{LS_3}\}}$, $B_{M_{RS}} = \rm{diag}\{\alpha_{RS_1}, \alpha_{RS_2}, \alpha_{RS_3}\}$ represent the coupling strengths of various interaction terms.
\subsection{ Mass terms for the charged leptons ($M_{\ell}$)}
For obtaining simplified and elegant structure for the mass matrix of charged leptons, we envisage that the three families of left-handed lepton doublets ($L_{Li}$) transform as triplets ($\bm{3}$) under the $A^\prime_5$ modular symmetry and singlet under $U(1)_{B-L}$. The right-handed charged leptons transform as singlets under both $A^\prime_5$ and $U(1)_{B-L}$ symmetries. However, ($e^c_R$, $\mu^c_R$, $\tau^c_R$) are given the modular weight as 1, 3, and 5 respectively. The Higgsinos ${H}_{u,d}$ are given charges 0 under $U_{B-L}$ and 1 for $A^\prime_5$ symmetries with zero modular weights. The VEVs of these Higgsinos ${H}_u$ and ${H}_d$ are represented as $v_u/\sqrt2$ and $v_d/\sqrt2$ respectively. Moreover, Higgsinos VEVs are associated to SM Higgs VEV as $v_H = \frac12 \sqrt{v^2_u + v^2_d}$ and the ratio of their VEVs is expressed as $\tan\beta= ({v_u}/{v_d})$ and we use its value to be 5 in our analysis. The relevant superpotential term for charged leptons obtained from (\ref{super-p}) is given by
\begin{align}
\mathcal{W}_{M_l}
&= \alpha_{M_l} \left[({L_L} e^c_R)_3 Y_3^{(2)} \right] H_d +\beta_{M_l} \left[({L_L} \mu^c_R)_3 Y_3^{(4)} \right] H_d + \gamma_{M_l} \left[({L_L} \tau^c_R)_3 \left\lbrace \sum\limits_{i=1}^2Y_{3,i}^{(6)} \right\rbrace \right] H_d.
\label{Eq:yuk-MD}
\end{align}
Working under $A^\prime_5$ modular group, its Kronecker product leaves us with a non diagonal charged lepton mass matrix after the spontaneous symmetry breaking. The mass matrix takes the form
\begin{align}
M_{l}&=\frac{v_d}{\sqrt2}
\left[\begin{matrix}
\left(Y^{(2)}_{\bf 3}\right)^{}_1 && \left(Y^{(4)}_{\bf 3}\right)^{}_1 && \left(\sum\limits_{i=1}^2Y^{(6)}_{{\bf 3},i}\right)_1 \\
\left(Y^{(2)}_{\bf 3}\right)^{}_3 &&\left(Y^{(4)}_{\bf 3}\right)^{}_3 &&\left(\sum\limits_{i=1}^2Y^{(6)}_{{\bf 3},i}\right)_3 \\
\left(Y^{(2)}_{\bf 3}\right)^{}_2 &&\left(Y^{(4)}_{\bf 3}\right)^{}_2 && \left(\sum\limits_{i=1}^2Y^{(6)}_{{\bf 3},i}\right)_2 \\
\end{matrix}\right]_{LR} \cdot
\left[\begin{array}{ccc}
\alpha_{M_l} & 0 & 0 \\
0 & \beta_{M_l} & 0 \\
0 & 0 & \gamma_{M_l} \\
\end{array}\right] .
\label{Eq:Mell}
\end{align}\\
The charged lepton mass matrix ${M}_l$ can be diagonalised by the unitary matrix $U_l$, giving rise to the physical masses $m_e,~m_\mu$ and $m_\tau$ as
\begin{equation}
U^\dagger_l {M}_l {M}^\dagger_l U_l = {\rm diag}(m^2_e, m^2_\mu, m^2_\tau)\;.
\end{equation}
Additionally, it also satisfies the following identities, which will be used for numerical analysis in section \ref{sec:results}:
\begin{eqnarray}
{\rm Tr} \left( {M}^{}_l {M}^{\dag}_l \right) &=& m^{2}_{e} + m^{2}_{\mu} + m^{2}_{\tau} \; , \nonumber\\
{\rm Det}\left( {M}^{}_l {M}^{\dag}_l \right) &=& m^{2}_{e} m^{2}_{\mu} m^{2}_{\tau} \; , \nonumber\\
\dfrac{1}{2}\left[{\rm Tr} \left({M}^{}_l {M}^{\dag}_l\right)\right]^2_{} - \dfrac{1}{2}{\rm Tr}\left[ ({M}^{}_l {M}^{\dag}_l)^2_{}\right] &= & m^{2}_{e}m^{2}_{\mu}+m^{2}_{\mu}m^{2}_{\tau}+m^{2}_{\tau}m^{2}_{e} \; . \label{eq:tr}
\end{eqnarray}
\subsection{Dirac as well as pseudo-Dirac mass terms for light neutrinos}
In addition to lepton doublet transformation, hitherto, the heavy fermion superfields, i.e., $N_{R_i}$ ($S_{L_i}$) transform as triplet $\bm{3^\prime}$ under $A^\prime_5$ modular group with $U(1)_{B-L}$ charge of $-1$ (0) along with modular weight $1$ ($4$) respectively. As discussed in Ref. \cite{Wang:2020lxk}, the choice of Yukawa couplings depends on the equation $k^{}_{Y} = k^{}_{I^{}_1} + k^{}_{I^{}_2} + \cdots + k^{}_{I^{}_n}$ where $k_Y$ is the modular weight of Yukawa couplings and $\Sigma^{I_n}_{i=1} k_{I_n}$ is sum of the modular weights of all other particles present in the definition of superpotential terms. These Yukawa couplings can be written in terms of Dedekind eta-function $\eta(\tau)$, and thus have $q$-expansion forms, in order to avoid the complexity in calculations.
The relevant superpotential term involving the active and right-handed heavy neutrino fields can be expressed as
\begin{align}
\mathcal{W}_{D}
&= G_D \left[({L_L} N^c_R)_5 Y_5^{(2)} \right] H_u\;,
\label{Eq:yuk-MD1}
\end{align}
where, $G_D$ is the diagonal matrix containing the free parameters and the modular weight of the Yukawa coupling is equal to the sum of the the modular weights of all other particles present in (\ref{Eq:yuk-MD1}). The choice of the Yukawa coupling is made based on the Kroncker product rules for $A^\prime_5$ modular symmetry such that superpotential remains singlet. Consequently, the Dirac neutrino mass matrix can be given as
\begin{align}
M_D&=\frac{v_u}{\sqrt30} G_D
\left[\begin{array}{ccc}
\sqrt3 (Y^{(2)}_\textbf{5})_1 & (Y_\textbf{5}^{(2)})_4 & (Y_\textbf{5}^{(2)})_3 \vspace{2mm}\\
(Y_\textbf{5}^{(2)})_5 &~~ -\sqrt2 (Y_\textbf{5}^{(2)})_3 &~~ -\sqrt2 (Y_\textbf{5}^{(2)})_2 \vspace{2mm}\\
(Y_\textbf{5}^{(2)})_2 &~~ -\sqrt2 (Y_\textbf{5}^{(2)})_5 &~~ -\sqrt2 (Y_\textbf{5}^{(2)})_4 \\
\end{array}\right]_{LR}.
\label{Eq:Mell}
\end{align}
As the transformation of the sterile fermion superfield $S_L$ is same as $N_R$ under $A^\prime_5$, it allows us to write the pseudo-Dirac mass term for the light neutrinos and the corresponding super potential can be written as
\begin{align}
\mathcal{W}_{LS}
&= G_{LS} \left[({L_L} S_L)_4 \sum\limits^{i=1}_2Y_{4,i}^{(6)} \right] H_u \left(\frac{\zeta}{\Lambda}\right)\;,
\label{Eq:yuk-MD}
\end{align}
where, $G_{LS}$ is a diagonal matrix containing three free parameters and the choice of Yukawa coupling depends upon the idea of keeping the superpotential singlet. Thus, we obtain the pseudo-Dirac mass matrix for the neutrinos as,
\begin{align}
M_{LS}&=\frac{v_u}{2\sqrt6}\left(\frac{v_\zeta}{\sqrt2 \Lambda}\right) G_{LS}
\left[\begin{array}{ccc}
0 & -\sqrt2 \left(\sum\limits_{i=1}^2 Y_\textbf{4,i}^{(6)}\right)_3 &-\sqrt2 \left(\sum\limits_{i=1}^2 Y_\textbf{4,i}^{(6)}\right)_2 \vspace{2mm}\\
\sqrt2 \left(\sum\limits_{i=1}^2 Y_\textbf{4,i}^{(6)}\right)_4 &- \left(\sum\limits_{i=1}^2 Y_\textbf{4,i}^{(6)}\right)_2 &\left(\sum\limits_{i=1}^2 Y_\textbf{4,i}^{(6)}\right)_1 \vspace{2mm}\\
\left(\sum\limits_{i=1}^2 Y_\textbf{4,i}^{(6)}\right)_1 &\left(\sum\limits_{i=1}^2 Y_\textbf{4,i}^{(6)}\right)_4 &-\left(\sum\limits_{i=1}^2 Y_\textbf{4,i}^{(6)}\right)_3 \\
\end{array}\right]_{LR}.
\label{Eq:yuk-LS}
\end{align}\\
\subsection{ Mixing between the heavy fermions $N_{Ri}$ and $S_{Li}$}
Introduction of extra symmetries, helps in allowing the mixing of heavy superfields but forbids the usual Majorana mass terms. Hence, below we show the mixing of these extra superfields i.e., $(N_{R}, S_{L})$ as follows
\begin{eqnarray}
\mathcal{W}_{M_{RS}}
&=&~ B_{M_{RS}}\left[({S_L} N^c_R)_{5} \sum\limits_{i=1}^2 Y_{5,i}^{(6)}\right] \zeta^*,
\label{Eq:yuk-M}
\end{eqnarray}
where, $B_{M_{RS}}$ is the free parameter and $\langle \zeta \rangle = v_\zeta/\sqrt{2}$ is the VEV of $\zeta$ and the superpotential is singlet under the $A^\prime_5$ modular symmetry product rule. Thus, one can obtain the mass matrix as follows:
\begin{align}
M_{RS}&=\frac{v_\zeta}{\sqrt60}B_{M_{RS}}
\left[\begin{array}{ccc}
2 \left(\sum\limits_{i=1}^2 Y_\textbf{5,i}^{(6)}\right)_1 & -\sqrt3\left(\sum\limits_{i=1}^2 Y_\textbf{5,i}^{(6)}\right)_4 & -\sqrt3\left(\sum\limits_{i=1}^2 Y_\textbf{5,i}^{(6)}\right)_3 \vspace{2mm}\\
-\sqrt3\left(\sum\limits_{i=1}^2 Y_\textbf{5,i}^{(6)}\right)_4 &\sqrt6 \left(\sum\limits_{i=1}^2 Y_\textbf{5,i}^{(6)}\right)_2 &-\left(\sum\limits_{i=1}^2 Y_\textbf{5,i}^{(6)}\right)_1 \vspace{2mm}\\
-\sqrt3 \left(\sum\limits_{i=1}^2 Y_\textbf{5,i}^{(6)}\right)_3 &-\left(\sum\limits_{i=1}^2 Y_\textbf{5,i}^{(6)}\right)_1 &\sqrt6 \left(\sum\limits_{i=1}^2 Y_\textbf{5,i}^{(6)}\right)_5 \\
\end{array}\right]_{LR}. \label{yuk:MRS}
\end{align}
As a consequence, one can write the masses for the heavy fermions in the $( N_R, S_L)^T$ basis as
\begin{eqnarray}
M_{hf}= \begin{pmatrix}
0 & M_{RS}\\
M^T_{RS} & 0
\end{pmatrix},\label{mrs matrix}
\end{eqnarray}
and after diagonalization, three doubly degenerate heavy superfield mass eigenstates can be obtained.
\vspace{1cm}
\subsection{Linear Seesaw framework for light neutrino mass}
In the present scenario of $A^\prime_5$ modular symmetry, the light neutrino masses can be generated in the framework of linear seesaw. Thus, the mass matrix in the flavor basis of $\left(\nu_L, N^c_R, S_L \right)^T$, can be manifested as
\begin{eqnarray}
\mathbb{M} = \left(\begin{array}{ccc}
0 ~&~ M_{D} &~ M_{LS} \vspace{1mm}\\
~~ M_D^T ~& 0 &~ M_{RS} \vspace{1mm}\\
~ M_{LS}^T ~&~ M_{RS}^T & 0
\end{array}
\right).
\label{eq:numatrix-complete}
\end{eqnarray}
Assuming that $ M_{RS} \gg M_D, M_{LS}$, one can write the linear seesaw mass formula for light neutrinos as
\begin{eqnarray}
m_\nu
&=& M_D M_{RS}^{-1} M_{LS}^{T}+{\rm transpose}.\label{mass}
\end{eqnarray}
Besides the neutrino masses, other relevant parameters which can play important role in the understanding of neutrino physics are the Jarlskog invariant, that signifies the measure of CP violation and the effective electron neutrino mass $m_{ee}$ in neutrinoless double beta decay. These parameters can be computed from the PMNS matrix elements as following:
\begin{eqnarray}
J_{CP}& =& \text{Im} [U_{e1} U_{\mu 2} U_{e 2}^* U_{\mu 1}^*] = s_{23} c_{23} s_{12} c_{12} s_{13} c^2_{13} \sin \delta_{CP},\\
m_{ee}&=&\left | m_{1} |U_{e1}|^2 + m_{2} |U_{e2}|^2 e^{i\alpha_{21}}+ m_{3} |U_{e3}|^2e^{i(\alpha_{31}-2\delta_{CP})} \right |\nonumber\\
&=&\left |m_{1} \cos^2\theta_{12} \cos^2\theta_{13}+ m_{2} \sin^2\theta_{12} \cos^2\theta_{13}e^{i\alpha_{21}}+ m_{3} \sin^2\theta_{13}e^{i(\alpha_{31}-2\delta_{CP})}\right |.
\end{eqnarray}
Many dedicated experients are are planned to measure the CP violation in neutrino sector as well as the neutrinoless double beta decay processes. Hopefully, CP violation will be precisely measured in the upcoming DUNE and T2HK experiments and the effective Majorana mass parameter $ m_{ee}$ can be measured by the KamLAND-Zen experiment in the near future~\cite{KamLAND-Zen:2016pfg}.
\section{Numerical Results}
\label{sec:results}
For numerical analysis, we use the neutrino oscillation parameters from the global-fit results \cite{deSalas:2017kay, Gariazzo:2018pei,Esteban:2018azc} obtained from various experiments, as given in Table \ref{table:expt_value}. The numerical diagonalization of the light neutrino mass matrix given in eqn.(\ref{mass}), is done through $U_\nu^\dagger {\cal M}U_\nu= {\rm diag}\left (m_1^2, m_2^2, m_3^2\right )$, with ${\cal M}=m_\nu m_\nu^\dagger$ and $U_\nu$ is the unitary matrix. Thus, one can write the lepton mixing matrix as $U=U_l^\dagger U_\nu$ and consequently obtain the mixing angles by using the standard relations:
\begin{eqnarray}
\sin^2 \theta_{12}= \frac{|U_{12}|^2}{1-|U_{13}|^2},~~~~~~~\sin^2 \theta_{23}= \frac{|U_{23}|^2}{1-|U_{13}|^2},~~~ ~{\rm and}~~~~\sin^2 \theta_{13}= |U_{13}|^2\;.
\end{eqnarray}
For explaining the observed neutrino oscillation data, we vary the model parameters in the following ranges:
\begin{align}
&{\rm Re}[\tau] \in [0,0.5],~~{\rm Im}[\tau]\in [0.5,3],~~G_{D} \in ~[10^{-7},10^{-6}],~~G_{LS} \in ~[10^{-4},10^{-3}] \quad v_\zeta \in \nonumber [10,100] \ {\rm TeV}, \nonumber \\ & \alpha_{RS} \in ~[10^{-3},10^{-2}], \quad \Lambda \in [10^4,10^5] \ {\rm TeV}. \label{input-values}
\end{align}
\begin{table}[htb]
\centering
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
\bf{Oscillation Parameters} & \bf{Best fit value} \bf{$\pm$ $1\sigma$} & \bf{ 2$\sigma$ range}& \bf{3$\sigma$ range} \\
\hline \hline
$\Delta m^2_{21}[10^{-5}~{\rm eV}^2]$& 7.56$\pm$0.19 & 7.20--7.95 & 7.05--8.14 \\
\hline
$|\Delta m^2_{31}|[10^{-3}~{\rm eV}^2]$ (NO) & 2.55$\pm$0.04 & 2.47--2.63 & 2.43--2.67\\
\hline
$\sin^2\theta_{12} / 10^{-1}$ & 3.21$^{+0.18}_{-0.16}$ & 2.89--3.59 & 2.73--3.79\\
\hline
$\sin^2\theta_{23} / 10^{-1}$ (NO)
& 4.30$^{+0.20}_{-0.18}$
& 3.98--4.78 \& 5.60--6.17 & 3.84--6.35 \\
& 5.98$^{+0.17}_{-0.15}$
& 4.09--4.42 \& 5.61--6.27 & 3.89--4.88 \& 5.22--6.41 \\
\hline
$\sin^2\theta_{13} / 10^{-2}$ (NO) & 2.155$^{+0.090}_{-0.075}$ & $1.98-2.31$ & $2.04-2.43$ \\
\hline
$\delta_{CP} / \pi$ (NO) & 1.08$^{+0.13}_{-0.12}$ & $0.84 - 1.42$ & $0.71 - 1.99$\\
\hline
\end{tabular}
\caption{The global-fit values of the oscillation parameters alongwith their 1$\sigma$/2$\sigma$/3$\sigma$ ranges \cite{deSalas:2017kay, Gariazzo:2018pei,Esteban:2018azc}.} \label{table:expt_value}
\end{table}
The input parameters are varied randomly in the ranges as provided in Eqn. (\ref{input-values}) and constrained by imposing the $3\sigma$ bounds on neutrino oscillation data, i.e., the solar and atmospheric mass squared differences and the mixing angles as presented in Table \ref{table:expt_value}, as well as the sum of the active neutrino masses: $\Sigma m_i < 0.12$ eV \cite{Planck:2019nip}. The allowed range of the modulus $\tau$ is found to be: 0\ $\lesssim\ $Re$[\tau]\lesssim$\ 0.5 and 1\ $\lesssim\ $Im$[\tau]\lesssim$\ 3, for normal ordering of neutrino masses. In Fig. \ref{mix_angles}, we show the variation of the sum of active neutrino masses ($\Sigma m_i$) with the reactor mixing angle $\sin^2 \theta_{13}$ (left panel), while the right panel demonstrates $\Sigma m_i$ versus $\sin^2\theta_{12}$ and $\sin^2\theta_{23}$. From these figures, it can be observed that the model predictions for the sum of neutrino masses as $\Sigma 0.058~{\rm eV} \leq m_i \leq 0.062$ eV for the allowed $3 \sigma$ ranges of the mixing angles.
\begin{figure}[htpb]
\begin{center}
\includegraphics[height=50mm,width=75mm]{ss13_sum_a5_lin.pdf}
\includegraphics[height=50mm,width=75mm]{ss12_23_sum.pdf}\\
\caption{the plot in the left (right) panel demonstrates the correlation between $\sin^2 \theta_{13}$ ($\sin^2\theta_{12}$ \& $\sin^2 \theta_{23}$) with the sum of active neutrino masses $\sum m_i$. The vertical lines represent the $3 \sigma $ allowed ranges of the mixing angles.}
\label{mix_angles}
\end{center}
\end{figure}
The variation of the effective mass parameter $m_{ee}$ of neutrinoless double beta decay with $\Sigma m_i$ is displayed in Fig. \ref{M23_mee}, from which the upper limit on $m_{ee}$ is found to be 0.025 eV satisfying KamLAND-Zen bound. Further, we display the variation of $\delta_{CP}$ and $J_{CP}$ in the left and right panel of Fig. \ref{dcpandjcpss13} respectively, where we obtain their limits as $100^\circ \leq \delta_{CP} \leq 250^\circ$ and $| J_{CP} | \leq 0.004$.
\begin{figure}[htpb]
\begin{center}
\includegraphics[height=50mm,width=75mm]{mee_mi_A5_lin.pdf}
\caption{Correlation plot between the effective neutrino mass $m_{ee}$ of neutrinoless double beta decay and the sum of active neutrino masses.}
\label{M23_mee}
\end{center}
\end{figure}
\begin{figure}[htpb]
\begin{center}
\includegraphics[height=50mm,width=75mm]{dcp_ss13_A5_lin.pdf}
\hspace*{0.2 true cm}
\includegraphics[height=48mm,width=72mm]{jcp_ss13.pdf}
\caption{Left (right) panel shows the plot of $\delta_{CP}$ ($J_{CP}$) with $\sin^2 \theta_{13}$ within its $3\sigma$ bound.}
\label{dcpandjcpss13}
\end{center}
\end{figure}
\section*{Comment on non-unitarity of leptonic mixing matrix}
\label{sec:non-unitarity}
Here, we present a brief discussion on the non-unitarity of neutrino mixing matrix $U'_{\rm PMNS}$ in the context of the present model. Due to the mixing between the light and heavy fermions, there will be small deviation from unitarity of the leptonic mixing matrix, which can be expressed as follows \cite{Forero:2011pc}
\begin{align}
U'_{\rm PMNS}\equiv \left(1-\frac12 FF^\dag\right) U_{\rm PMNS}.
\end{align}
Here, $U_{\rm PMNS}$ denotes the PMNS mixing matrix that diagonalises the light neutrino mass matrix and $F$ represents the mixing of active light neutrinos with the heavy one, which is approximated as $F\equiv (M^{T}_{NS})^{-1} M_D \approx \frac{g_D v}{\alpha_{RS} v_\zeta}$, and is a hermitian matrix.
The non-unitarity parameters are constrained through various electroweak precision measurements \cite{Antusch:2014woa,Blennow:2016jkn,Fernandez-Martinez:2016lgt}, such as the mass of $W$ boson ($M_W$), Weinberg mixing angle ($\theta_W$), ratios of various fermionic decays of $Z$ boson and its invisible decay, lepton flavour violations and the CKM unitarity bounds. In the context of the present model, we take into account the following approximated normalized order of masses for the Dirac, pseudo-Dirac and heavy fermions for generating the observed mass-squared differences as well as the sum of active neutrino masses of desired order as
\begin{eqnarray}
\left(\frac{m_\nu}{0.1 ~{\rm eV}} \right) \approx \left(\frac{M_D}{10^{-3}~~ {\rm GeV}}\right) \left(\frac{M_{RS}}{10^3~~ {\rm GeV}}\right)^{-1} \left( \frac{M_{LS}}{10^{-4}~~ {\rm GeV}}\right).
\end{eqnarray}
With these chosen order masses, we obtain the following approximated non-unitary mixing for the present model:
\begin{align}
|FF^\dag|\le
\left[\begin{array}{ccc}
4.5\times 10^{-13}~~~~ &~~~~ 2.3\times 10^{-13} ~~~~&~~ ~~6.2\times 10^{-13} \\
2.3\times 10^{-13} ~~~~&~~ ~~2.08\times 10^{-12} ~~~~ &~~~~ 4.5\times 10^{-12} \\
6.2\times 10^{-13} ~~~~ & ~~~~4.5\times 10^{-12} ~~~~&~~~~ 5.6\times 10^{-12} \\
\end{array}\right].
\end{align}
As the mixing between the active light and heavy fermions is quite small in our model, it generates a negligible contribution for the non-unitarity.
\iffalse
\section{Comment on $(g-2)_{e, \mu}$}
\label{g-2}
As we have used local $U(1)_{B-L}$ symmetry to forbid certain unwanted terms in the superpotential. Therefore, an associated gauge boson $Z^\prime_{B-L}$ can couple to lepton. To have muon $g-2$ we need to have an interaction term given by Eqn. (\ref{eq:amuZ})
\begin{figure}[htpb]
\begin{center}
\includegraphics[height=55mm,width=50mm]{g-2_fenyman.pdf}
\caption{Feynman diagram for muon anomalous g − 2 contribution arising from through $Z^\prime_{B-L}$ gauge boson with muons.}
\end{center}
\end{figure}
\begin{equation}
\mathcal{L}_{NC} = G_{g} \left[Y^\textbf{(2)}_5\overline{L}_{L} \gamma^\mu L_{L}\right]Z^\prime_{B-L}
\end{equation}
\begin{eqnarray}
\Delta a_\mu &=& \frac{g^{2}_{B-L}}{4\pi^2 } \int^1_0 \frac{x^2(1-x)}{x^2 + (M^2_{Z_{B-L}^\prime}/m^2_\mu)(1-x)} {\rm d}x \nonumber\\
&\approx& \frac{g_{B-L}^{2}}{12\pi^2} \frac{m^2_\mu}{ M^2_{Z^\prime_{B-L}}}.
\label{eq:amuZ}
\end{eqnarray}
\fi
\section{Leptogenesis}
\label{sec:lepto}
The present universe is clearly seen to be baryon dominated, with the ratio of the measured over-abundance of baryons over anti-baryons to the entropy density is found to be
\begin{equation}
Y_B = (8.56 \pm 0.22) \times 10^{-11}
\end{equation}
Considering the fact that the universe had started from an intially symmetric state of baryons and antibaryons, following three conditions have to be fulfilled for generating a non-zero baryon asymmetry. According to Sakharov \cite{Sakharov:1967dj}, these three criteria are: Baryon number violation, C and CP violation and departure from thermal equilibrium during the evolution of the universe. Though the SM assures all these criteria for an expanding Universe akin ours, the extent of CP violation found in the SM is quite small to accommodate the observed baryon asymmetry of the universe. Therefore, additional sources of CP violation are absolutely essential for explaining this asymmetry. The most common new sources of CP violation possibly could arise in the lepton sector, which is however, not yet firmly established experimentally. Leptogenesis is the phenomenon that furnishes a minimal set up to correlate the CP violation in the lepton sector to the observed baryon asymmetry, as well as imposes indirect constraints on the CP phases from the requirement that it would yield the correct baryon asymmetry.
It is seen that the scale of CP-asymmetry generated from the heavy neutral fermion decays can come down to as low as TeV \cite{Pilaftsis:1997jf,Bambhaniya:2016rbb,Pilaftsis:2003gt, Abada:2018oly} due to resonant enhancement. However, the present scenario is realized by involving six heavy states, which comprises three pairs of heavy neutrinos with doubly degenerate masses Eq.\eqref{mrs matrix}. Nevertheless, introduction of a higher dimensional mass terms for the Majorana fermions ($N_R$) can be made through the following superpotential
\begin{eqnarray}
\mathcal{W}_{M_R}=-G_R \left[\sum\limits_{i=1}^2Y^{(4)}_{5,i} {N^c_R} N^c_R \right]\frac{{\zeta^*}^2}{\Lambda}\;,
\label{rhnmatrix}
\end{eqnarray}
which gives rise to a petty mass splitting amid the heavy neutral fermions, and hence provides an enhancement in the CP asymmetry for generating the required lepton asymmetry \cite{Pilaftsis:2005rv,Asaka:2018hyk}.
Thus, from (\ref{rhnmatrix}) one can construct the Majorana mass matrix for the right-handed neutrinos $N_R$ as
\begin{equation}
M_R =\frac{G_R v^2_\zeta}{2\Lambda\sqrt{30}}
\left[\begin{array}{ccc}
2 \left(\sum\limits_{i=1}^2 Y_\textbf{5,i}^{(4)}\right)_1 & -\sqrt3\left(\sum\limits_{i=1}^2 Y_\textbf{5,i}^{(4)}\right)_4 & -\sqrt3\left(\sum\limits_{i=1}^2 Y_\textbf{5,i}^{(4)}\right)_3 \vspace{2mm}\\
-\sqrt3\left(\sum\limits_{i=1}^2 Y_\textbf{5,i}^{(4)}\right)_4 &\sqrt6 \left(\sum\limits_{i=1}^2 Y_\textbf{5,i}^{(4)}\right)_2 &-\left(\sum\limits_{i=1}^2 Y_\textbf{5,i}^{(4)}\right)_1 \vspace{2mm}\\
-\sqrt3 \left(\sum\limits_{i=1}^2 Y_\textbf{5,i}^{(4)}\right)_3 &-\left(\sum\limits_{i=1}^2 Y_\textbf{5,i}^{(4)}\right)_1 &\sqrt6 \left(\sum\limits_{i=1}^2 Y_\textbf{5,i}^{(4)}\right)_5 \\
\end{array}\right]_{LR}.
\end{equation}
The coupling $G_R$ is considered as exceptionally small to preserve the linear seesaw texture of the neutrino mass matrix (\ref{eq:numatrix-complete}), i.e., $ M_D, M_{LS}\gg M_R$ and hence, inclusion of such additional term does not alter the results obtained earlier. However, this added term generates a small mass difference. Hence, the $2 \times 2$ submatrix of \ref{eq:numatrix-complete} in the basis of $(N_R, S_L)$, becomes
\begin{eqnarray}
M=\begin{pmatrix}
M_R& M_{RS}\\
M_{RS}^T & 0
\end{pmatrix},
\end{eqnarray}
which can be block diagonalized by using the unitary matrix: $ \frac{1}{\sqrt 2}
\begin{pmatrix}
I & -I\\
I & I
\end{pmatrix} $ as
\begin{eqnarray}
M'=\begin{pmatrix}
M_{RS}+\frac{M_R}{2} & -\frac{M_R}{2}\\
-\frac{M_R}{2} & -M_{RS}+\frac{M_R}{2}
\end{pmatrix} \approx \begin{pmatrix}
M_{RS}+\frac{M_R}{2} & 0\\
0 & -M_{RS}+\frac{M_R}{2}
\end{pmatrix}.\label{block-diag}
\end{eqnarray}
Thus, one can express the mass eigenstates ($N^\pm$) in terms of the flavor states ($N_R, S_L$) as
\begin{equation}
\begin{pmatrix}
S_{Li}\\N_{Ri}
\end{pmatrix}= \begin{pmatrix}
\cos{\theta} ~~&~ -\sin{\theta}\\
\sin{\theta}~ &~ \cos{\theta}
\end{pmatrix} \begin{pmatrix}
N_i^+ \\ N_i^-
\end{pmatrix}.
\end{equation}
Assuming the mixing to be maximal, one can have
\begin{eqnarray}
N_{Ri} = \frac{(N_i^+ + N_i^-)}{\sqrt{2}},~~ S_{Li}= \frac{(N_i^+ - N_i^-)}{\sqrt{2}}.
\end{eqnarray}
Hence, the interaction Lagrangian (\ref{Eq:yuk-MD1}) can be manifested in terms of the new basis. One can obtain the mass eigenvalues of the new states $N^+$ and $N^-$ by diagonalizing the block-diagonal form of the heavy-fermion masses which are given as $\frac{M_R}{2} + M_{RS}$ and $\frac{M_R}{2} - M_{RS}$ (\ref{block-diag}).\\
The Dirac (\ref{Eq:yuk-MD}) and pseudo-Dirac (\ref{Eq:yuk-LS}) terms are now modified as
{\small{
\begin{eqnarray}
{\cal W}_D && = G_{D} {L}_{L} {H_u} \left[Y^{(2)}_5 \left(\frac{(N_i^+ + N_i^-)}{\sqrt{2}}\right)\right],
\end{eqnarray}}}
and
{\small{
\begin{eqnarray}
{\cal W}_{LS} &&=G_{LS} {L}_{L} {H_u} \left[\sum\limits^{i=1}_2Y_{4,i}^{(6)} \left(\frac{(N_i^+ - N_i^-)}{\sqrt{2}}\right)\right] \frac{\zeta}{\Lambda} .
\end{eqnarray}}}
Thus, one can symbolically express the block-diagonal matrix for the heavy fermions (\ref{block-diag}) as
\begin{align}
M_{RS} \pm \frac{M_R}{2}&=\frac{v_\zeta}{\sqrt60}B_{M_{RS}}
\begin{bmatrix}
2a~&~ d~&~e\\
d ~&~ b ~&~ f \\
e ~&~ f ~&~ c
\end{bmatrix} _{LR} \pm \frac{G_R v^2_\zeta}{2\Lambda\sqrt{30}}
\begin{bmatrix}
2a^\prime~&~ d^\prime~&~e^\prime\\
d^\prime ~&~ b^\prime ~&~ f^\prime \\
e^\prime ~&~ f^\prime ~&~ c^\prime
\end{bmatrix}_{LR},
\label{yuk:MRS}
\end{align}
where, the different matrix elements are defined as
\small{\begin{eqnarray}
a(a^\prime)=\left(\sum\limits_{i=1}^2 Y_\textbf{5,i}^{6(4)}\right)_1,~~ b(b^\prime)=\sqrt6 \left(\sum\limits_{i=1}^2 Y_\textbf{5,i}^{6(4)}\right)_2,~~ c(c^\prime)= \sqrt6 \left(\sum\limits_{i=1}^2 Y_\textbf{5,i}^{6(4)}\right)_5 \;,\nonumber\\
d(d^\prime)= -\sqrt3\left(\sum\limits_{i=1}^2 Y_\textbf{5,i}^{6(4)}\right)_4, ~~~e(e^\prime)= -\sqrt3\left(\sum\limits_{i=1}^2 Y_\textbf{5,i}^{6(4)}\right)_3, ~~~f(f^\prime)= -a(a^\prime)\;.
\end{eqnarray}}
One can obtain the diagonalized mass matrix from (\ref{yuk:MRS}) through rotation to the mass eigen-basis as: $(M^{\pm})_{\rm diag}=U_{\rm TBM} U_R \left(M_{RS}\pm \frac{M_{R}}{2}\right) U^T_R U^T_{\rm TBM}$. Consequently, three sets of approximately degenerate mass states can be obtained upon diagonalization. Further, we presume that the lightest pair among them with mass in the TeV range, contribute predominantly to the CP asymmetry. The small mass difference amid the lightest pair demonstrates that the CP asymmetry generated through the one-loop self energy contribution dominates over the vertex part in the decay of heavy particle.
Thus, the CP asymmetry can be expressed as \cite{Pilaftsis:1997jf,Gu:2010xc}
\begin{eqnarray}
\epsilon_{N^-_i}
&&\approx \frac{1}{32\pi^2 A_{N^-_i}}{\rm Im}\left[ \left(\frac{\tilde{M_D}}{v}-\frac{\tilde{M}_{LS}}{v}\right)^\dagger \left(\frac{\tilde{M}_D}{v}+\frac{\tilde{M}_{LS}}{v} \right)^2 \left(\frac{\tilde{M}_D}{v}-\frac{\tilde{M}_{LS}}{v}\right)^\dagger \right]_{ii} \frac{r_N}{r^2_N + 4 A^2_{N^-_i}}\;,\hspace*{0.5 true cm}
\end{eqnarray}
where $\tilde{M}_{D(LS)}= M_{D(LS)} U_{\rm TBM} U_R$, $\Delta M =M^+_i - M^-_i \approx M_R$, $v=v_u$ and $r_N$ and $A_{N^-}$ are given as
\begin{eqnarray}
&& r_N =\frac{{(M^+_i)}^2 - {(M^-_i)}^2}{M^+_i M^-_i} = \frac{\Delta M (M^+_i + M^-_i)}{M^+_i M^-_i} ,\nonumber \\
&& A_{N^-} \approx \frac{1}{16\pi}\left[\left(\frac{\tilde{M}_D}{v}-\frac{\tilde{M}_{LS}}{v}\right)\left(\frac{\tilde{M}_D}{v}+\frac{\tilde{M}_{LS}}{v}\right) \right]_{ii}.
\end{eqnarray}
In Fig. \ref{CP_var}, we depict the behavior of CP asymmetry with $r_N$, which satisfies both neutrino oscillation data and the CP asymmetry required for leptogenesis \cite{Davidson:2008bu,Buchmuller:2004nz}, which will be discussed in the next subsection.
\begin{figure}[h!]
\begin{center}
\includegraphics[height=48mm,width=73mm]{cp_asym_rn_1.pdf}
\caption{Correlation plot demonstrating the dependence of CP asymmetry with the parameter $r_N$.}
\label{CP_var}
\end{center}
\end{figure}
\begin{center}
\begin{table}[htpb]
\centering
\begin{tabular}{|c|c|c|c|c|c|} \hline
{$\epsilon^{e}_{N^-}$} & ~{ $\epsilon^{\mu}_{N^-}$}~& ~$\epsilon^{\tau}_{N^-}$~ & $\epsilon_{N^-}$ & $\Delta M$~(GeV) \\\hline
$-1.78\times 10^{-5}$ & $-2.6 \times 10^{-5}$ & $-4.85 \times 10^{-5}$ & $-6 \times 10^{-5}$ & $4\times 10^{-6}$ \\\hline
\end{tabular}
\caption{CP asymmetries and mass splitting obtained from the allowed range of model parameters which satisfy neutrino oscillation data.}
\label{tab:leptobench}
\end{table}
\end{center}
\subsection{Boltzmann Equations}
Boltzmann equations are invoked to solve for the lepton asymmetry. It should be reiterated that, the Sakharov criteria \cite{Sakharov:1967dj} require the decay of the parent heavy fermion which ought to be out of equilibrium for generating the lepton asymmetry. In order to implement this, one needs to confront the Hubble expansion rate with the decay rate as
\begin{equation}
K_{N^-_i} = \frac{\Gamma_{N^-_i}}{H(T=M^-_i)}.
\end{equation}
Here, $H = \frac{1.67 \sqrt{g_\star}~ T^2 }{M_{\rm Pl}}$ is the Hubble rate, with $g_{\star} = 106.75$ is the number of relativistic degrees of freedom in the thermal bath and $M_{\rm Pl} = 1.22 \times 10^{19}$ GeV is the Planck mass. Coupling strength becomes the deciding factor which assures that inverse decay would not come into thermal equilibrium. For instance, if the strength is below $10^{-7}$, it gives $K_{N^-} \sim 1$. The Boltzmann equations associated with evolution of the number densities of right-handed fermion field and lepton can be articulated in terms of the yield parameters, i.e., the ratio of number densities to entropy density, and are expressed as \cite{Plumacher:1996kc, Giudice:2003jh,Buchmuller:2004nz, Strumia:2006qk, Iso:2010mv}
\begin{eqnarray}
&& \frac{d Y_{N^-}}{dz}=-\frac{z}{s H(M_{N^-})} \left[\left( \frac{Y_{N^-}}{{Y^{\rm eq}_{N^-}}}-1\right)\gamma_D +\left( \left(\frac{{Y_{N^-}}}{{Y^{\rm eq}_{N^-}}}\right)^2-1\right)\gamma_S \right],\nonumber\\
&& \frac{d Y_{{B-L}}}{d z}= -\frac{z}{s H(M_{N^-})} \left[ 2\frac{Y_{{B-L}}}{{Y^{\rm eq}_{\ell}}}\gamma_{Ns}- \epsilon_{N^-} \left( \frac{Y_{N^-}}{{Y^{\rm eq}_{N^-}}}-1\right)\gamma_D \right],\label{Boltz1}
\end{eqnarray}
where $z = M^-_i/T$, $s$ is the entropy density, and the equilibrium number densities have the form \cite{Davidson:2008bu}
\begin{eqnarray}
Y^{\rm eq}_{N^-}= \frac{135 g_{N^-}}{16 {\pi}^4 g_\star} z^2 K_2(z), \hspace{3mm} {Y^{\rm eq}_\ell}= \frac{3}{4} \frac{45 \zeta(3) g_\ell}{2 {\pi}^4 g_{\star}}\,.\label{eq:39}
\end{eqnarray}
$K_{1,2}$ in Eq. (\ref{eq:39}) represent the modified Bessel functions, the lepton and RH fermion degrees of freedom take the values $g_\ell=2$ and $g_{N^-}=2$ and the decay rate $\gamma_D$ is given as
\begin{equation}
\gamma_D = s Y^{\rm eq}_{N^-}\Gamma_{N^-} \frac{K_1(z)}{K_2(z)}.
\end{equation}
While $\gamma_S$ represents the scattering rate of $N^-N^- \to \zeta \zeta$ \cite{Iso:2010mv} and $\gamma_{Ns}$ denotes the scattering rate of $\Delta L = 2$ process.
One can keep away the delicacy of the asymmetry being produced, even when the RH fermion field $N^-$ is in thermal equilibrium, by subtracting the contribution arising from the exchange of on-shell $N^-$, i.e., $\frac{\gamma_{D}}{4}$ from the total rate $\gamma_{Ns}$ and is given as $\gamma^{\rm sub}_{Ns} = \gamma_{Ns} - \frac{\gamma_D}{4}$~ \cite{Giudice:2003jh}.
\begin{figure}
\begin{center}
\includegraphics[height=65mm,width=95mm]{yield_NF.pdf}
\caption{Evolution of the yield parameters $Y_{N}$ and $Y_{B-L}$ as a function of $z \equiv M_{N^-}/{T}$.}\label{yield}
\end{center}
\end{figure}
The solution of Boltzmann eq. (\ref{Boltz1}) is displayed in Fig. \ref{yield}. For large coupling strength $Y_{N^-}$ (green-thick curve) almost traces $Y^{\rm eq}_{N^-}$ (black-solid curve) and the generated lepton asymmetry (red-dashed curve). The lepton asymmetry thus obtained can be converted into baryon asymmetry through the sphaleron transition process, and is given as \cite{Plumacher:1996kc}
\begin{equation}
Y_B = -\left(\frac{8N_f + 4 N_H}{22 N_f + 13 N_H}\right)Y_{\mathcal{L}},
\end{equation}
where $N_f$ represents the number of fermion generations, $N_H$ denotes the no. of Higgs doublets and $Y_{\mathcal L}=Y_{\ell} -Y_{\bar \ell}$.
The observed baryon asymmetry of the universe generally expressed in terms of baryon to photon ratio as \cite{Planck:2018vyg}
\begin{equation}
\eta = \frac{n_b - n_{\bar{b}}}{n_\gamma} = 6.08 \times 10^{-10}.
\end{equation}
The current bound on baryon asymmetry can be procured from the relation $Y_{B} = \eta/ 7.04$ as $Y_{B} = 8.6\times 10^{-11}$. Using the asymptotic value of the lepton asymmetry $Y_{\mathcal{L}}$ as ($8.77 \times 10^{-10}$) from Fig. \ref{yield}, we obtain the value of baryon asymmetry as $Y_{B} = -\frac{28}{79}~ Y_{\mathcal{L}} \sim 10^{-10}$.
\subsection{A note on flavor consideration}
\begin{figure}
\begin{center}
\includegraphics[height=58mm,width=75mm]{flavor_yield.pdf}
\vspace{7mm}
\includegraphics[height=58mm,width=75mm]{yield_enhance.pdf}
\caption{After including the flavor effects the yield is shown in left panel, whereas, right panel displays the Yeild enhancement due to flavor effects.}\label{yield_Flavor}
\end{center}
\end{figure}
In leptogenesis, one flavor approximation is sufficient when ($T>10^{12}$ GeV), meaning all the Yukawa interactions are out of equilibrium. But for temperatures $\ll 10^{8}$ GeV, several charged lepton Yukawa couplings come into equilibrium making flavor effects an important consideration for generating the final lepton asymmetry. For temperatures below $10^6$ GeV, all the Yukawa interactions are in equilibrium and the asymmetry is stored in the individual lepton flavor. The detailed investigation of flavor effects in type-I leptogenesis can be seen in myriad literature \cite{Pascoli:2006ci,Antusch:2006cw,Nardi:2006fx,Abada:2006ea,Granelli:2020ysj,Dev:2017trv}.
The Boltzmann equation describing the generation of lepton asymmetry in each flavor is \cite{Antusch:2006cw}
\begin{eqnarray}
\frac{d Y^{\alpha}_{ B-L_\alpha}}{d z}= -\frac{z}{s H(M_1^-)} \left[ \epsilon^\alpha_{N^-} \left( \frac{Y_{N^-}}{{Y^{eq}_{N^-}}}-1\right)\gamma_D-\left(\frac{\gamma^{\alpha}_D}{2}\right)\frac{A_{\alpha \alpha}Y^\alpha_{\rm B-L_\alpha}}{{Y^{eq}_{\ell}}}\right],
\end{eqnarray}
where, $\epsilon^\alpha_{N^-}$ i.e. $(\alpha=e,\mu,\tau)$ represents the CP asymmetry in each lepton flavor
\begin{equation}
\gamma_D^\alpha = s Y_{N^-}^{eq}\Gamma_{N^-}^\alpha \frac{K_1(z)}{K_2(z)}, \quad \gamma_D = \sum_\alpha \gamma^\alpha_D.\nonumber\\
\end{equation}
The matrix $A$ is given by \cite{Nardi:2006fx},
\begin{equation}
A=\begin{pmatrix}
-\frac{221}{711} && \frac{16}{711} && \frac{16}{711}\\
\frac{16}{711} && -\frac{221}{711} && \frac{16}{711}\\
\frac{16}{711} && \frac{16}{711} && -\frac{221}{711} \\
\end{pmatrix}.\nonumber\\
\end{equation}
From the benchmark considered in Table. \ref{tab:leptobench}, we estimate the $B-L$ yield with flavor consideration in the left panel of Fig. \ref{yield_Flavor}. It is quite obvious to notice that the enhancement in $B-L$ asymmetry is obtained in case of flavor consideration (blue line) over the one flavor approximation (red line), as displayed in the right panel. This is because, in one flavor approximation the decay of the heavy fermion to a particular lepton flavor final state can get washed away by the inverse decays of any flavor unlike the flavoured case \cite{Abada:2006ea}.
\section{Summary and Conclusion}
\label{sec:con}
In this work, we have investigated the implications of $A^\prime_5$ modular symmetry on neutrino phenomenology. The important feature of the modular flavor symmetry is that it minimizes the complications of using multiple flavon fields, which are usually associated with the use of discrete flavor symmetries. In the present model, we consider the SM to be extended by the $A^\prime_5$ modular symmetry along with a $U(1)_{B-L}$ local gauge symmetry. It encompasses three right-handed as well as three left-handed heavy fermion fields to realize the neutrino phenomenology in the linear seesaw framework. In addition, it contains a singlet scalar, which plays a key role in breaking the $U(1)_{B-L}$ symmetry spontaneously and additionally provides masses to the heavy fermions. Another important feature is that the Yukawa couplings are considered to transform non-trivially under the modular $A^\prime_5$ group, which basically replace the role of typical flavon fields. This in turn, induces a unique flavor structure for the neutrino mass matrix and provides a simplified way for exploring neutrino phenomenology. We further, diagonalized the neutrino mass matrix numerically and obtained the allowed regions for the model parameters by confronting with the current $3\sigma$ limit of the oscillation data. Additionally, our model predicts the CP violating phase $\delta_{CP}$ to be in the range of $(100^\circ-250^\circ)$ and the Jarlskog invariant to be ${\cal O}(10^{-3})$. The sum of active neutrino masses is found to be in the range $0.058~\rm{eV} \leq \Sigma m_i \leq0.062~\rm{eV}$ and the value of effective neutrinoless double beta decay mass parameter $m_{ee}$ as $(0.001-0.025)$ eV, which is below the current upper limits from KamLAND-Zen experiment i.e., $< (61-165)~ \rm{meV}$. In addition, the flavor structure of heavy fermion masses leads to three sets of doubly-degenerate mass states and hence, to incorporate leptogenesis, we have introduced a higher dimensional mass term for the RH neutrinos for generating a small mass difference between them. We then obtained the non-zero CP asymmetry from the lightest heavy fermion decay where the contribution arising from the self-energy diagram is partially enhanced because of the small mass splitting between the two lightest heavy fermions. Utilizing a particular benchmark of model parameters consistent with oscillation data, we solved the coupled Boltzmann equations to generate lepton asymmetry at TeV scale. The obtained lepton asymmetry is found to be of the of the order $\simeq 10^{-10}$, which is adequate to accommodate the present baryon asymmetry of the universe. Besides, we have additionally shed light on the increase in asymmetry due to flavor consideration.
\acknowledgments
MKB acknowledges DST for its financial help. RM would like acknowledge SERB, Government of India for the support received through grant No. EMR/2017/001448 and University of Hyderabad IoE project grant no. RC1-20-012. We also acknowledge the use of CMSD HPC facility of Univ. of Hyderabad for carrying out the computational work.
|
1,116,691,499,220 | arxiv | \section{Introduction}
This paper is a sequel to our paper \cite{AD} where we studied the
universal subspaces $V$ for the representation of a connected compact Lie
group $G$ on a finite-dimensional real vector space $U$. The meaning
of the word ``universal'' in this context is that every $G$-orbit
in $U$ meets the subspace $V$. The general results obtained in
that paper have been applied in particular to the conjugation actions
$A\to XAX^{-1}$, $X\in G$, of the classical compact Lie groups $G$,
i.e., ${\mbox{\rm U}}(n)$, ${\mbox{\rm SO}}(n)$ and ${\mbox{\rm Sp}}(n)$, on
the space of $n\times n$ matrices $M(n,{\bf C})$, $M(n,{\bf R})$ and
$M(n,{\bf H})$, respectively. (By ${\bf H}$ we denote the algebra of real
quaternions.)
In the present paper we restrict our scope to the
complex case, i.e., to $M(n)=M(n,{\bf C})$ and $G={\mbox{\rm U}}(n)$.
However, all results where we establish the nonsingularity (see Section
\ref{Nonsing-Obl} for the definition) of certain patterns are directly
applicable to the real and quaternionic cases.
Throughout the paper we denote by $L(n)\subseteq M(n)$ the subspace of
traceless matrices, and by $T_n\subseteq{\mbox{\rm U}}(n)$ the maximal
torus consisting of the diagonal matrices. We shall
consider only a very special class of complex subspaces of $M(n)$;
those that can be specified by requiring
that the matrix entries in specified positions $(i,j)$ vanish.
We denote the set of these positions $(i,j)$ by $I$ and
denote by $M_I(n)$ the corresponding subspace. We also set
$L_I(n)=L(n) \cap M_I(n)$. We refer to
$I$ as a (zero) pattern and denote the set of all such $I$'s by ${\mathcal P}_n$.
A pattern is strict if it contains no diagonal positions.
It is proper if it does not contain all the diagonal positions.
We say that a pattern $I\in{\mathcal P}_n$ is universal if the subspace $L_I(n)$
is universal in $L(n)$. We point out that, for a strict pattern
$I\in{\mathcal P}_n$, $L_I(n)$ is universal in $L(n)$
iff $M_I(n)$ is universal in $M(n)$.
The main question we consider, the universality problem, is to
determine all universal patterns in ${\mathcal P}_n$. In full generality,
this problem is solved only for $n\le3$.
There is a simple necessary condition for universality of a
proper pattern $I$: $|I|\le\mu_n=n(n-1)/2$ (see Proposition
\ref{RestrDim} below). We denote by ${\mathcal P}'_n$ the set of strict
patterns $I\in{\mathcal P}_n$ with $|I|=\mu_n$.
Theorem 5.1 of \cite{AD} provides a sufficient condition for
the universality of a pattern (see Theorem \ref{TeorAD} below).
We use this result to construct some infinite families of strict
universal patterns. The main results in this direction are
Theorems \ref{SchurAnalog} and \ref{Fam-J}.
In Section \ref{UnivObl} we define the universal patterns and state the
universality problem for patterns $I\in{\mathcal P}_n$. The case $n=2$ is easy:
All patterns $I\in{\mathcal P}_2$ of size 1 are universal. The nonsingularity
of all $I\in{\mathcal P}'_3$ has been established in \cite{AD}. In Proposition
\ref{3-Obl} we show that none of the proper nonstrict patterns $I\in{\mathcal P}_3$ of
size 3 is universal. Thus the universality problem is solved for $n\le3$.
A proper pattern $I\in{\mathcal P}_n$ is $n$-defective if the stabilizer of
$L_I(n)$ in ${\mbox{\rm U}}(n)$ has dimension larger than $n^2-2|I|$. Such patterns
are not universal.
In Section \ref{Nonsing-Obl} we introduce the nonsingular patterns.
We say that $I\in{\mathcal P}_n$ is $n$-nonsingular if $\chi_I\notin{\mathcal K}(n)$,
where $\chi_I\in{\bf R}[x_1,\ldots,x_n]$ is the product of all differences
$x_i-x_j$, $(i,j)\in I$, and ${\mathcal K}(n)$ is the ideal generated
by the nontrivial elementary symmetric functions of the $x_i$'s. The
basic fact, that nonsingular patterns are universal, was proved in
\cite{AD}. We say that a pattern $I$ is simple if $(i,j)\in I$
implies that $(j,i)\notin I$. All simple patterns are nonsingular,
and so universal. A pattern $I\in{\mathcal P}'_n$ is $n$-exceptional if it is
$n$-singular but not $n$-defective. There is no general method for
deciding whether an exceptional pattern $I\in{\mathcal P}'_n$ is universal.
Proposition \ref{FundPol} provides a simple method for testing
whether a pattern $I\in{\mathcal P}'_n$ is nonsingular. The inner product
${\langle}\cdot,\cdot{\rangle}$ used in the proposition is defined in the
beginning of the section.
In Section \ref{Ekv} we introduce two equivalence relations ``$\approx$''
and ``$\sim$'' in ${\mathcal P}'_n$. We refer to the former simply as
``equivalence'' and to the latter as ``weak equivalence''.
This is justified since $I\approx I'$ implies $I\sim I'$.
If $I\approx I'$ then $I$ is universal iff $I'$ is universal, but
we do not know if this also holds for weak equivalence.
However, if $I\sim I'$ then $I$ is nonsingular iff $I'$ is nonsingular.
For any pattern $I$ we define its complexity $\nu(I)$ as the number
of positions $(i,j)$ with $i\le j$ such that both $(i,j)$ and $(j,i)$
belong to $I$. The patterns of complexity 0 are exactly the simple patterns.
We show that for $n\ge4$ the set of patterns of complexity 1 in
${\mathcal P}'_n$ splits into two weak equivalence classes. One of these
classes is singular and the other nonsingular.
In Section \ref{Analog} we consider a particular sequence of
nonsingular patterns $\Lambda_n\in{\mathcal P}'_n$, $n\ge1$, of maximal complexity,
i.e., with $\nu(\Lambda_n)=[\mu_n/2]$. For convenience let us write
$n=4m+r$ where $m,r\ge0$ are integers and $r<4$. The pattern
$\Lambda_n$ consists of all positions $(i,j)$ with $i\ne j$ and
$i+j\le n+1$, except those of the form $(2i-1,n-2i+1)$ and
$(n-2i+1,2i-1)$ with $1\le i\le m$, and if $r=2$ or 3 we also
omit the position $(2m+2,2m+1)$. As $\Lambda_n$ is nonsingular,
it is also universal, i.e., every matrix $A\in M(n)$ is unitarily
similar to one in the subspace $M_{\Lambda_n}(n)$. Thus we can
view this result as an analogue of Schur's theorem. The whole
section is dedicated to the proof of this result.
In Section \ref{Analogue} we consider an infinite family of patterns
$J(\sigma,{\bf i})$ depending on an integer $n\ge1$, a permutation
$\sigma\in S_n$ and a sequence ${\bf i}=(i_1,i_2,\ldots,i_{n-1})$ of
distinct integers. This sequence has to be chosen so that,
for each $k$, $|i_k|\in\{\sigma(1),\sigma(2),\ldots,\sigma(k)\}$.
The pattern $J(\sigma,{\bf i})$ consists of all positions
$(i_k,\sigma(j))$ with $i_k>0$ and $(\sigma(j),-i_k)$ with $i_k<0$,
where in both cases $1\le k<j\le n$. The main result of this
section (Theorem \ref{Fam-J}) shows that all the patterns
$J(\sigma,{\bf i})$ are nonsingular. As a special case, we obtain
another analogue of Schur's theorem (see Proposition \ref{Primer-2}).
In Section \ref{Cetiri} we consider the exceptional patterns $I\in{\mathcal P}'_4$.
Up to equivalence, there are seven of them (see Table 2). We prove
that the first two of them are not universal while the third one is.
This is the unique example that we have of a strict pattern which is
singular and universal. For the remaining four patterns in Table 2
the universality question remains open. The same question for the
nonstrict patterns in ${\mathcal P}_4$ remains wide open.
There are other interesting questions that one can raise about
the subspaces $L_I(n)$ and the unitary orbits
${\mathcal O}_A=\{UAU^{-1}:\, U\in{\mbox{\rm U}}(n) \}$, $A\in L(n)$.
For instance, if $L_I(n)$ is not universal
we can ask for the characterization of the set ${\mbox{\rm U}}(n)\cdot L_I(n)$.
Another question of interest is to determine the number, $N_{A,I}$,
of $T_n$-orbits contained in the intersection ${\mathcal X}_A=L_I(n)\cap{\mathcal O}_A$.
For instance, if $L_I(n)$ is the space of upper (or lower) triangular
traceless matrices and $A\in L(n)$ has $n$ distinct eigenvalues
then $N_{A,I}=n!$.
In Section \ref{RedFlag} we consider a pattern $I\in{\mathcal P}'_n$ and a matrix
$A\in L(n)$ such that ${\mathcal X}_A\ne\emptyset$. The homogeneous space
${\mathcal F}_n={\mbox{\rm U}}(n)/T_n$ is known as the flag manifold and we refer to
its points as flags. If $g^{-1}Ag\in L_I(n)$,
$g\in{\mbox{\rm U}}(n)$, we say that the flag $gT_n$ reduces $A$ to $L_I(n)$.
We say that $A$ is generic if ${\mathcal X}_A$ and $L_I(n)$ intersect
transversally (see the next section for the definition).
For generic $A$, we show that $N_{A,I}$ is equal to the number of
flags which reduce $A$ to $L_I(n)$.
In Section \ref{CikObl} we consider the case $n=3$ and the cyclic pattern
$I=\{(1,3),(2,1),(3,2)\}$. We know that $I$ is
nonsingular and so ${\mbox{\rm U}}(3)\cdot L_I(3)=L(3)$. We show that the set
$\Theta$ of nongeneric matrices $A\in L(3)$ is contained in
a hypersurface $\Gamma$ defined by $P=0$, where $P$ is a homogeneous
${\mbox{\rm U}}(3)$-invariant polynomial of degree 24. This polynomial is
explicitly computed and we show that it is absolutely irreducible.
The restriction, $P_I$, of $P$ to $L_I$ factorizes as
$P_I=P_1^2P_2$, where $P_1$ and $P_2$ are absolutely irreducible
homogeneous polynomials of degree 6 and 12, respectively. Thus,
$\Gamma \cap L_I=\Gamma_1 \cup \Gamma_2$ where $\Gamma_i\subset L_I$
is the hypersurface defined by $P_i=0$, $i=1,2$. The hypersurface
$\Gamma_1$ consists of all matrices $A\in L(3)$ such that ${\mathcal O}_A$
and $L_I$ meet non-transversally at $A$.
We propose a conjecture (see Section \ref{Nonsing-Obl}) and
several open problems.
For any positive integer $n$ we set ${\bf Z}_n=\{1,2,\ldots,n\}$ and
$\mu_n=n(n-1)/2$.
If ${\mathcal A}$ is a ${\bf Z}$-graded algebra, we denote by ${\mathcal A}_d$ the
homogeneous component of ${\mathcal A}$ of degree $d$. We use the same
notation for the homogeneous ideals of ${\mathcal A}$.
We thank the referee for his suggestions regarding the
presentation and some minor corrections.
\section{Universal patterns} \label{UnivObl}
Let $M(n)$ denote the algebra of complex $n\times n$ matrices
and $L(n)$ its subspace of matrices of trace 0.
We are interested in subspaces of $M(n)$ or $L(n)$ which can be
specified by zero patterns. For that purpose we introduce
the notion of patterns.
A {\em position} is an ordered pair $(i,j)$ of positive integers.
We say that a position $(i,j)$ is {\em diagonal} if $i=j$.
A {\em pattern} is a finite set of positions.
A pattern is {\em strict} if it has no diagonal positions.
The {\em size} of a pattern $I$ is its cardinality, $|I|$.
If $I,I'$ are patterns and $I\subseteq I'$ then we say
that $I'$ is an {\em extension} of $I$.
We denote by ${\mathcal P}$ the set of all patterns and by ${\mathcal P}_n$ the set
of patterns contained in ${\bf Z}_n\times{\bf Z}_n$.
We denote by ${\mathcal P}'_n$ the subset of ${\mathcal P}_n$ consisting
of the strict patterns of size $\mu_n$.
The ``$n\times n$ zero patterns'' used in our paper \cite{AD}
are the same as the patterns in ${\mathcal P}'_n$.
We define an involutory map $T:{\mathcal P}\to{\mathcal P}$, called {\em transposition},
by setting $I^T=\{(j,i):\, (i,j)\in I\}$ for $I\in{\mathcal P}$.
We refer to $I^T$ as the {\em transpose} of $I$. We say that $I$ is
{\em symmetric} if $I^T=I$. The sets
${\mathcal P}_n$ and ${\mathcal P}'_n$ are $T$-invariant for all $n$.
Note that if $I\in{\mathcal P}_n$ is universal (or nonsingular)
then $I^T$ has the same property.
For $I\in{\mathcal P}_n$, we denote by $M_I(n)$, or just $M_I$ if $n$ is fixed,
the subspace of $M(n)$ which consists of all matrices $X=[x_{ij}]$
such that $x_{ij}=0$ for all $(i,j)\in I$. We also set
$L_I(n)=L(n) \cap M_I(n)$.
Some important patterns in ${\mathcal P}_n$ are the diagonal pattern
$\Delta_n=\{(i,i):i\in{\bf Z}_n\}$ and the four triangular patterns:
\begin{eqnarray*}
{\rm NE}_n &=& \{(i,j)\in{\bf Z}_n\times{\bf Z}_n: i<j \}, \\
{\rm SW}_n &=& \{(i,j)\in{\bf Z}_n\times{\bf Z}_n: i>j \}, \\
{\rm NW}_n &=& \{(i,j)\in{\bf Z}_n\times{\bf Z}_n: i+j<n+1 \}, \\
{\rm SE}_n &=& \{(i,j)\in{\bf Z}_n\times{\bf Z}_n: i+j>n+1 \}.
\end{eqnarray*}
The first is the {\em upper triangular} and the second
the {\em lower triangular} pattern. Note that, according to this terminology,
if $I$ is the upper triangular pattern, then $M_I$ is the space of lower
triangular matrices.
The unitary group ${\mbox{\rm U}}(n)$ acts on $L(n)$ by conjugation, i.e.,
unitary similarities.
\begin{definition} We say that a real subspace $V\subseteq L(n)$
is {\em universal} in $L(n)$ if every matrix in $L(n)$ is unitarily
similar to a matrix in $V$. We also say that a pattern $I\in{\mathcal P}_n$
is {\em $n$-universal} if the subspace $L_I(n)$ is universal
in $L(n)$.
\end{definition}
The prefix ``$n$-'' will be supressed if $n$ is clear from
the context. This convention shall apply to several other
definitions that we will introduce later.
It is obvious that a strict $n$-universal pattern
is also $m$-universal for all $m>n$.
The converse is not valid, e.g., the pattern $\{(1,2),(2,1)\}$ is
3-universal but not $2$-universal.
We are interested in the (pattern) universality problem, i.e.,
the problem of deciding which patterns $I\in{\mathcal P}_n$ are universal.
It is easy to see that, for a strict pattern $I\in{\mathcal P}_n$, the
subspace $L_I(n)$ is universal in $L(n)$ iff the subspace
$M_I(n)$ is universal in $M(n)$. The Schur's triangularization
theorem asserts that the triangular patterns ${\rm NE}_n$ and
${\rm SW}_n$ are $n$-universal.
It is well known that $\Delta_n$ is $n$-universal \cite[Theorem 1.3.4]{HJ}
for all $n$. However, if $i,j\in{\bf Z}_n$ and $i\ne j$ then the next example
implies that the pattern $\Delta_n \cup \{(i,j)\}$ is not universal.
\begin{example} \label{dij+1}
The pattern $I=\{(1,1),(1,2),(2,2)\}$ is not $n$-universal for $n\ge2$.
Let $D={\rm diag}(1,\ldots,1,1-n)\in L(n)$ and let $A\in L_I(n)$. As the rank
of $D-{\rm id}$ is 1 and that of $A-{\rm id}$ is at least 2,
$D$ and $A$ are not similar. Consequently, $L_I$ is not universal.
This implies that ${\rm NW}_n$ is not $n$-universal for $n\ge4$. For the case $n=3$
see Proposition \ref{3-Obl}.
\end{example}
We give another example of a nonuniversal nonstrict pattern.
\begin{example} \label{drugi-neuni}
The pattern $J=\{(i,1):1\le i<n\} \cup \{(1,n)\}$ is not $n$-universal for $n\ge2$.
Let $D$ be as above and let $A=[a_{ij}]\in L_I(n)$.
Assume that $D$ and $A$ are similar. Since $D-{\rm id}$ has rank 1,
$A-{\rm id}$ must also have rank 1. Thus
$$
\left| \begin{array}{cc} -1&0\\ a_{n1} & a_{nn}-1 \end{array} \right|=0 \quad
\text{ and } \quad
\left| \begin{array}{cc} -1 & a_{1j} \\ 0 & a_{jj}-1 \end{array} \right|=0
\text{ for } 1<j<n, $$
i.e., $a_{jj}=1$ for $1<j\le n$. As $A\in L_J$, we have $a_{11}=0$ contradicting
the fact that ${\,\rm tr\,}(A)=0$. Consequently, $L_I$ is not universal.
\end{example}
If $n-1$ of the diagonal entries of a matrix $A\in L(n)$ vanish
then all $n$ of them vanish. For that reason we introduce the
following definition: We say that a pattern $I\in{\mathcal P}_n$ is
$n$-{\em proper} if $(i,i)\notin I$ for at least one $i\in{\bf Z}_n$.
Observe that, for any pattern $I\in{\mathcal P}_n$, the subspace $L_I(n)$ is
stabilized by the maximal torus $T_n$. An easy dimension argument
shows that the following is valid, see \cite[Lemma 4.1]{AD}.
\begin{proposition} \label{RestrDim}
Let $I\in{\mathcal P}_n$ be proper and universal. Then the dimension of the
stabilizer of $L_I(n)$ in ${\mbox{\rm U}}(n)$ does not exceed $n^2-2|I|$.
In particular, $n\le n^2-2|I|$, i.e., $|I|\le\mu_n$.
\end{proposition}
Thus we have a simple condition that any proper universal
pattern $I\in{\mathcal P}_n$ must satisfy: $|I|\le\mu_n$.
We say that a proper pattern $I\in{\mathcal P}_n$ is $n$-{\em defective} if
the dimension of the stabilizer of $L_I(n)$ in ${\mbox{\rm U}}(n)$ is larger
than $n^2-2|I|$.
By the proposition, such patterns are not universal. Note that
any proper pattern $I\in{\mathcal P}_n$ with $|I|>\mu_n$ is defective.
Next, we show that some special extensions of strict universal patterns
are also universal. For $I\in{\mathcal P}$ and integers $m,n\ge0$ we denote by
$(m,n)+I$ the translate $\{(m+i,n+j):\,(i,j)\in I\}$ of $I$.
\begin{proposition} \label{UnivPros}
Let $I\in{\mathcal P}_n$ and $J\in{\mathcal P}_{m-n}$, $m>n$, be strict patterns and
assume that $I$ is $n$-universal and $J$ is $(m-n)$-universal.
Then the pattern
$$ I'=I \cup \left( (n,n)+J \right) \cup
\left( (0,n)+{\bf Z}_n\times{\bf Z}_{m-n} \right) $$
is $m$-universal.
\end{proposition}
\begin{proof}
Given a matrix $A\in L(m)$, choose $X\in{\mbox{\rm U}}(m)$ such that $B=XAX^{-1}$
is lower triangular. Let $B_1$ resp. $B_2$ denote the square submatrix
of $B$ of size $n$ resp. $m-n$ in the upper left resp. lower right
hand corner.
Since $I$ and $J$ are strict and universal, there exist matrices
$Y_1\in{\mbox{\rm U}}(n)$ and $Y_2\in{\mbox{\rm U}}(m-n)$ such that $Y_1B_1Y_1^{-1}\in M_I(n)$
and $Y_2B_2Y_2^{-1}\in M_J(m-n)$.
Thus if $Y=Y_1\oplus Y_2$, then $YBY^{-1}\in L_{I'}(m)$.
Hence $I'$ is $m$-universal.
\end{proof}
Let us fix a positive integer $n$ and a pattern $I\in{\mathcal P}'_n$.
For $A\in L(n)$ we denote by ${\mathcal O}_A$ the ${\mbox{\rm U}}(n)$-orbit through $A$,
i.e., ${\mathcal O}_A=\{UAU^{-1}:\, U\in{\mbox{\rm U}}(n) \}$. The set
${\mathcal X}_A=L_I(n)\cap{\mathcal O}_A$ is closed and $T_n$-invariant.
We say that ${\mathcal O}_A$ intersects $L_I$ {\em transversally at a point}
$B\in{\mathcal X}_A$ if the sum of $L_I(n)$ and the tangent space of ${\mathcal O}_A$
at $B$ is equal to the whole space $L(n)$.
If this is true for all points $B\in{\mathcal X}_A$, then we say that
${\mathcal O}_A$ and $L_I$ intersect {\em transversally},
and that the matrix $A$ and its orbit ${\mathcal O}_A$ are {\em $I$-generic}.
We shall denote by $N_{A,I}$ the cardinality of the set ${\mathcal X}_A/T_n$
(the set of $T_n$-orbits in ${\mathcal X}_A$). We note that $N_{A,I}$
is finite if $A$ is $I$-generic, see Section \ref{RedFlag} and
\cite[Section 4]{AD}.
Almost nothing is known about the universality of the subspaces
$L_I(n)$ of $L(n)$ for nonstrict patterns $I$, but see the above examples.
The case $n=2$ is easy and we leave it to the reader. Let us analyze
the case $n=3$. The case of strict patterns, ${\mathcal P}'_3$, has been handled
in \cite{AD}. It is easy to see that any pattern $I\in{\mathcal P}_3$ of size 2 is
universal. By taking into account the above examples and the fact
that $\Delta_3$ is universal, there are only four cases to consider:
\begin{eqnarray*}
&&
\left[ \begin{array}{ccc} 0&0&*\\ 0&*&*\\ *&*&* \end{array} \right],
\quad
\left[ \begin{array}{ccc} 0&*&*\\ *&*&0\\ *&0&* \end{array} \right],
\quad
\left[ \begin{array}{ccc} 0&0&*\\ *&*&*\\ *&0&* \end{array} \right],
\quad
\left[ \begin{array}{ccc} 0&0&*\\ *&*&0\\ *&*&* \end{array} \right].
\end{eqnarray*}
(The starred entries are arbitrary, subject only to the
condition that the matrices must have zero trace.)
We shall prove that none of them is universal and thereby complete
the solution of the universality problem for $n=3$.
\begin{proposition} \label{3-Obl}
For $n=3$, no proper nonstrict pattern of size 3 is universal.
\end{proposition}
\begin{proof}
We need only consider the four subspaces, $V$, mentioned above.
It turns out that in all four cases there exists a diagonal matrix
$D\in L(3)$ such that ${\mathcal O}_D\cap V=\emptyset$.
In the first two cases the proof consists in constructing an
${\mbox{\rm U}}(3)$-invariant polynomial function $P:L(3)\to{\bf R}$ which is
nonnegative on $V$ and negative on the diagonal matrices
$D={\rm diag}(u,v,-u-v)$ when $u$ and $v$ are
linearly independent over ${\bf R}$. The polynomial $P$ will be
expressed as a polynomial in the ${\mbox{\rm U}}(3)$-invariants $i_k$ given
in Appendix A. We shall write $u=u_1+iu_2$, $u_1,u_2\in{\bf R}$,
and similarly for other variables.
For the first subspace we define
\begin{eqnarray*}
P &=& (2i_4+3i_5-i_6)^2 +4i_1(i_2-i_3)(i_1i_3-6i_5) +4i_1^2(i_1i_5+i_8-i_7) \\
&& +4i_1i_2(i_6-i_4)+4i_8(5i_3-4i_2)+16i_3^2(2i_2-i_3) \\
&& +4i_7(2i_2-3i_3) +4i_2^2(i_2-5i_3)+8(i_1i_{11}-i_{13}).
\end{eqnarray*}
A computation using {\sc Maple} shows that for
$$ A=\left[ \begin{array}{ccc} 0&0&x\\ 0&z&y\\ u&v&-z
\end{array} \right] \in V $$
we have
$$ P(A)=(|u|^2-|x|^2)^2\left(
|x-\bar{u}|^2z_1^2-4(u_1x_2+u_2x_1)z_1z_2
+|x+\bar{u}|^2 z_2^2 \right)^2 . $$
On the other hand, we have $P(D)=-64(u_1v_2-u_2v_1)^2$.
For the second subspace we define $P=i_1^2+4(i_3-i_2)$.
One can easily verify that for
$$ A=\left[ \begin{array}{ccc} 0&x&y\\ u&z&0\\ v&0&-z
\end{array} \right] \in V $$
we have
$$ P(A)=(|u|^2+|v|^2-|x|^2-|y|^2)^2, $$
while $P(D)=-4(u_1v_2-u_2v_1)^2$.
For the remaining subspaces we have more elementary arguments.
In the third case let
$D={\rm diag}(1,\zeta,\zeta^2)\in L(3)$, $\zeta=(-1+i\sqrt{3})/2$.
Since $D$ is a normal matrix, its field of values, $F(D)$,
is the equilateral triangle with vertices $1,\zeta,\zeta^2$.
Assume that $D$ is unitarily similar to a matrix $A=[a_{ij}]\in L_I$.
As $A\in L_I$, $a_{22}$ is an eigenvalue of $A$ and so
$a_{22}\in\{1,\zeta,\zeta^2\}$. As ${\,\rm tr\,}(A)=0$ and $a_{11}=0$, we
deduce that $a_{33}=-a_{22}\notin F(D)$. Since $F(D)=F(A)$ and
$a_{33}$ is a diagonal entry of $A$, we have a contradiction.
In the last case let $D={\rm diag}(1,i,-1-i)\in L(3)$.
Assume that $D$ is unitarily similar to a matrix
$$ A=\left[ \begin{array}{ccc} 0&0&x\\ y&z&0\\ u&v&-z
\end{array} \right] \in V. $$
Since $D$ is a normal matrix, so is $A$. From $AA^*=A^*A$ we obtain
the system of equations
\begin{eqnarray*}
&& x\bar{z}=z\bar{u},\quad z\bar{y}=-v\bar{u},\quad y\bar{u}=-2z\bar{v}, \\
&& |x|^2=|y|^2+|u|^2,\quad |y|=|v|.
\end{eqnarray*}
Assume that $z=0$. Then $uy=0$. As $A$ must be nonsingular, we have
$y\ne0$ and $u=0$. Thus $A^3$ is a scalar matrix. Since $D^3$ is not,
we have a contradiction.
We conclude that $z\ne0$. The equation $x\bar{z}=z\bar{u}$ implies
that $|x|=|u|$, which entails that $y=v=0$. Hence $z$ is an eigenvalue
of $A$, and so $z\in\{1,i,-1-i\}$. By switching the first two rows
(and columns) of $A$, we obtain the direct sum $[z]\oplus B$
where $B=\left[ \begin{array}{cc} 0&x\\ u&-z \end{array} \right].$
Hence $0\in F(B)$. This is a contradiction since $B$ is normal, and so
$F(B)$ is the line segment joining two of the eigenvalues of $D$.
\end{proof}
The following theorem provides an infinite collection of universal
patterns. It is an easy consequence of a result of Ko\v{s}ir and
Sethuraman proven in \cite{HS}.
\begin{theorem} \label{Kos-Set}
The pattern
$$ \left( (0,1)+{\rm NE}_{n-1} \right) \cup
\left\{ (i,1): 2<i\le n \right\} \cup \left\{ (n,2) \right\},
\quad n\ge3, $$
is $n$-universal.
\end{theorem}
\begin{proof}
Denote this pattern by $J$.
Let $A\in M(n)$ be arbitrary. By \cite[Theorem A.4]{HS}, there exists
$S\in{\mbox{\rm GL}}(n,{\bf C})$ such that $SA^*S^{-1}\in M_{J'}(n)$ and
$SAS^{-1}\in M_{J''}(n)$, where
$J'=(1,0)+{\rm SW}_{n-1}$ and $J''=\{(i,1):2<i\le n\}\cup\{(n,2)\}$.
By \cite[Remark 1]{HS}, we can assume
that $S$ is unitary.
As $J=(J')^T \cup J''$, we deduce that $SAS^{-1}\in M_J(n).$
Hence $J$ is $n$-universal.
\end{proof}
\section{Nonsingular patterns} \label{Nonsing-Obl}
Before defining the singular and nonsingular patterns we introduce
some preliminary notions.
Let ${\bf R}[x_1,x_2,\ldots]$ resp. ${\bf R}[x_1^{\pm1},x_2^{\pm1},\ldots]$
be the polynomial ring resp. Laurent polynomial ring in countably
many commuting independent variables $x_1,x_2,\ldots$ over ${\bf R}$.
We introduce an inner product, ${\langle} .,. {\rangle}$, in
${\bf R}[x_1^{\pm1},x_2^{\pm1},\ldots]$
by declaring that the basis consisting of the Laurent monomials
is orthonormal. We also introduce the involution $f\to f^*$ and the
shift endomorphism $\tau$ of this Laurent polynomial ring where,
by definition,
$$ f^*(x_1,x_2,\ldots) = f(x_1^{-1},x_2^{-1},\ldots) $$
and $\tau(x_i)=x_{i+1}$ for all $i\ge 1$.
For any Laurent polynomial $f$, we denote by ${\mbox{\rm CT}}\{f\}$ the constant
term of $f$. It is easy to see that for
$f,g\in{\bf R}[x_1^{\pm1},x_2^{\pm1},\ldots]$ we have
$$ {\langle} f,g {\rangle} = {\mbox{\rm CT}}\{f^*g\} = {\mbox{\rm CT}}\{fg^*\}. $$
We shall denote by $\partial_i$ the partial derivative with respect
to the variable $x_i$. For any $f\in{\bf R}[x_1,x_2,\ldots]$ we set
$$ {\partial}_f=f({\partial}_1,{\partial}_2,\ldots). $$
To each pattern $I$ we associate the polynomial
$$ \chi_I=\prod_{(i,j)\in I}(x_i-x_j), $$
and the differential operator
$$ {\partial}_ I={\partial}_{\chi_I}=\prod_{(i,j)\in I}({\partial} _i-{\partial} _j). $$
Next we introduce special notation for certain symmetric polynomials in
the first $n$ variables $x_1,x_2,\ldots,x_n$:
\begin{equation} \label{ElemSim}
\sigma_{k,n}=\sum_{1\le i_1<i_2<\cdots<i_k\le n}
x_{i_1}x_{i_2}\cdots x_{i_k}, \quad k\in\{0,1,\ldots,n\},
\end{equation}
for the elementary symmetric functions, and
\begin{equation} \label{PotpSim}
h_{k,n}=\sum_{d_1+\cdots+d_n=k} x_1^{d_1}x_2^{d_2}\cdots x_n^{d_n},
\quad k\in\{0,1,\ldots,n\},
\end{equation}
for the complete symmetric functions.
We denote by ${\mathcal H}(n)$ the quotient of
${\bf R}[x_1,x_2,\ldots,x_n]$ modulo the ideal ${\mathcal K}(n)$ generated by the
$\sigma_{k,n}$, $k\in{\bf Z}_n$, and define
$\varphi_n:{\bf R}[x_1,x_2,\ldots,x_n]\to{\mathcal H}(n)$ to be the
natural homomorphism.
\begin{definition}
We say that $I\in{\mathcal P}_n$ is {\em $n$-singular} if
$\varphi_n(\chi_I)=0$, and otherwise that it is {\em $n$-nonsingular}.
We say that $I\in{\mathcal P}'_n$ is $n$-{\em exceptional} if it is
$n$-singular but not $n$-defective.
\end{definition}
Note that if a pattern $I$ contains a diagonal position
then $\chi_I=0$ (in particular, $I$ is singular).
\begin{lemma} \label{Pros-Lema}
Every $n$-nonsingular pattern is also $m$-nonsingular for all $m>n$.
\end{lemma}
\begin{proof}
Let $I\in{\mathcal P}_n$ be $m$-singular for some $m>n$.
Then there exist polynomials
$f_k\in{\bf R}[x_1,x_2,\ldots,x_m]$ such that
$$ \chi_I(x_1,x_2,\ldots,x_n)=\sum_{k=1}^m
f_k(x_1,x_2,\ldots,x_m)\sigma_{k,m}(x_1,x_2,\ldots,x_m) . $$
By setting $x_{n+1}=\cdots=x_m=0$, we see that $I$ is $n$-singular.
\end{proof}
We remark that the converse of this lemma is not valid. (All seven
patterns listed in Table 2 of Section \ref{Cetiri} are
counter-examples for $n=4$.)
To simplify the notation, we set
$$ \chi_n=\chi_{{\rm NE}_n}=\prod_{1\le i<j\le n} (x_i-x_j). $$
Its expansion is given by the well known formula
\begin{equation} \label{Hi_n}
\chi_n = \sum_{\sigma\in S_n} {\rm sgn}(\sigma) \prod_{i=1}^n x^{n-i}_{\sigma(i)}.
\end{equation}
Consequently, we have ${\langle} \chi_n,\chi_n {\rangle} =n!$.
The following result (see \cite[Proposition 2.3]{AD}) provides a
practical method for deciding whether a pattern is nonsingular.
In particular, it implies that ${\rm NE}_n$ is nonsingular.
\begin{proposition} \label{FundPol}
A pattern $I\in{\mathcal P}'_n$ is $n$-singular iff ${\langle} \chi_I,\chi_n {\rangle} =0$.
\end{proposition}
We now state two problems concerning the inner product in this
proposition.
\begin{problem} \label{Nejed}
Is it true that ${\langle} \chi_I,\chi_n {\rangle} \le n!$ for all $I\in{\mathcal P}_n$
and that equality holds iff $\chi_I=\chi_n$?
\end{problem}
\begin{problem} \label{Norma}
Is it true that ${\langle} \chi_I,\chi_I {\rangle} \ge n!$ for all $I\in{\mathcal P}'_n$
and that equality holds iff $\chi_I=\pm\chi_n$?
\end{problem}
The computations carried out for $n\le5$ show that, in these cases,
the answer is affirmative for both problems.
In connection with Theorem \ref{Kos-Set} we propose
\begin{conjecture} \label{Hess}
Let $J_{k,n}\in{\mathcal P}'_n$, $k\in{\bf Z}_{n-1}$, be the union of the translate
$(0,1)+{\rm NE}_{n-1}$, the product $\{n\}\times{\bf Z}_k$, and
$\{(i,1):k<i<n\}$. Then
$$ {\langle} \chi_{J_{k,n}}, \chi_n {\rangle} = (-1)^{n-1}\binom{n}{k}\binom{n-2}{k-1}. $$
\end{conjecture}
The assertion in the case $k=1$ will be proved in Example \ref{Primer-1}.
The universality of $J_{2,n}$ was proved in Theorem \ref{Kos-Set}.
The conjecture has been verified for $n\le10$ and all $k\in{\bf Z}_{n-1}$,
except for $(n,k)=(10,5)$ in which case our program ran out of memory.
We say that a pattern $I$ is {\em simply laced}, or just {\em simple},
if $I \cap I^T=\emptyset$,
and otherwise we say that $I$ is {\em doubly laced}.
In particular, a simple pattern is strict.
Any simple pattern $I\in{\mathcal P}_n$ can be extended to
a simple pattern $I'\in{\mathcal P}'_n$. As $\chi_I$ divides $\chi_{I'}$
and $\chi_{I'}=\pm\chi_n$, any simple pattern is nonsingular.
The following two results are extracted from Proposition 4.3 and
Theorem 4.2 of our paper \cite{AD}.
\begin{proposition} \label{Prosiri}
Every nonsingular pattern $I\in{\mathcal P}_n$ can be extended to
a nonsingular pattern $I'\in{\mathcal P}'_n$.
\end{proposition}
\begin{theorem} \label{TeorAD} {\rm (Generalization of Schur's
triangularization theorem)}
Every nonsingular pattern in ${\mathcal P}_n$ is $n$-universal.
In particular, every simple pattern in ${\mathcal P}_n$ is $n$-universal
\end{theorem}
In connection with these results we pose an open problem.
\begin{problem} \label{Pros-Str}
Let $I$ be a strict $n$-universal pattern.
Does $I$ extend to a strict $n$-universal pattern of size $\mu_n$?
\end{problem}
If we replace the word ``strict'' with ``proper'' then
Proposition \ref{3-Obl} shows that the answer is negative.
A special case of the general inner product identity that we prove in the next
theorem will be used in the proof of the subsequent proposition.
\begin{theorem} \label{KompGr}
Let $G$ be a connected compact Lie group, $T$ a maximal torus with Lie algebra
$\mathfrak{t}$ and $W$ the Weyl group.
Denote by $\mu$ be the number of positive roots and by $\chi$ their product.
Let ${\mathcal A}=\oplus_{n\ge0} {\mathcal A}_n$ be the algebra of polynomial functions $\mathfrak{t}\to{\bf R}$ with the degree gradation,
and let ${\langle}\cdot,\cdot{\rangle}$ be a $W$-invariant inner product on ${\mathcal A}$.
If $h\in\mathcal{A}$ is $W$-invariant, then
$$ \|\chi\|^2 {\langle} hf,h\chi {\rangle}=\|h\chi\|^2{\langle} f,\chi {\rangle},
\quad \forall f\in\mathcal{A}_\mu. $$
\end{theorem}
\begin{proof}
We may assume that $h\ne0$.
For the linear functional $L:{\mathcal A}_\mu\to{\bf R}$ defined by $L(f)={\langle} hf, h\chi {\rangle}$,
we have $\sigma\cdot L={\rm sgn}(\sigma)L$ for all $\sigma\in W$. There is a unique $g\in{\mathcal A}_\mu$
such that $L(f)={\langle} f,g {\rangle}$ for all $f\in{\mathcal A}_\mu$. Moreover, $\sigma\cdot g={\rm sgn}(\sigma)g$
for all $\sigma\in W$. As the sign representation of $W$ occurs only once in ${\mathcal A}_\mu$,
we must have $g=c\chi$ for some $c\in{\bf R}$. The equality $L(\chi)={\langle} \chi,c\chi {\rangle}$
completes the proof.
\end{proof}
The next proposition, a nonsingular analogue of Proposition \ref{UnivPros},
shows that some extensions of nonsingular
patterns are nonsingular. In a weaker form, it was originally
conjectured by Jiu-Kang Yu.
\begin{proposition} \label{Pros-Stav}
For $I\in{\mathcal P}'_n$, $J\in{\mathcal P}'_{m-n}$, $m>n$, and
$$ I'=I \cup \left( (n,n)+J \right) \cup
\left( (0,n)+{\bf Z}_n\times{\bf Z}_{m-n} \right) $$
we have
$$ {\langle} \chi_{I'},\chi_m {\rangle}={m \choose n} {\langle} \chi_I,\chi_n {\rangle}
{\langle} \chi_J,\chi_{m-n} {\rangle}. $$
\end{proposition}
\begin{proof}
We apply the above theorem to the case where $G={\mbox{\rm U}}(n)\times{\mbox{\rm U}}(m-n)$.
Then the algebra ${\mathcal A}$ can be identified with the polynomial algebra
${\bf R}[x_1,\ldots,x_m]$ so that ${\bf R}[x_1,\ldots,x_n]$ resp.
${\bf R}[x_{n+1},\ldots,x_m]$ is the corresponding algebra for
${\mbox{\rm U}}(n)$ resp. ${\mbox{\rm U}}(m-n)$. The polynomial $\chi$ factorizes as
$\chi=\chi_n\cdot\tau^n\chi_{m-n}$, where $\tau$ is the shift operator.
We take $f=\chi_I\cdot\tau^n\chi_J$ and for $h$ we take the polynomial
$$
\prod_{i=1}^n\prod_{j=n+1}^m(x_i-x_j),
$$
which is obviously invariant under $W=S_n\times S_{m-n}$.
Since $h\chi=\chi_m$ and $hf=\chi_{I'}$, the theorem gives the identity
$$ \|\chi\|^2 {\langle} \chi_{I'},\chi_m {\rangle}=\|\chi_m\|^2{\langle} f,\chi {\rangle}. $$
Since $\|\chi\|^2=n!(m-n)!$, $\|h\chi\|^2=\|\chi_m\|^2=m!$, and
$$ {\langle} \chi_I\cdot\tau^n\chi_J,\chi_n\cdot\tau^n\chi_{m-n} {\rangle}=
{\langle} \chi_I,\chi_n {\rangle} {\langle} \chi_J,\chi_{m-n} {\rangle}, $$
the assertion follows.
\end{proof}
\section{Equivalence and weak equivalence} \label{Ekv}
The symmetric group $S_n$ acts on ${\mathcal P}_n$ by
$\sigma(I)=\{(\sigma(i),\sigma(j)):\, (i,j)\in I\}$ for $\sigma\in S_n$.
For $\sigma\in S_n$ and $I\in{\mathcal P}_n$ we have $\sigma\cdot\chi_I=\chi_{\sigma(I)}$.
As ${\mathcal P}'_n$ is $S_n$-invariant, we obtain an action on ${\mathcal P}'_n$.
We denote by $\tilde{S}_n$ the group of transformations of
${\mathcal P}'_n$ generated by the action of $S_n$
and the restriction of transposition $T$ to ${\mathcal P}'_n$.
(Note that this restriction commutes with the action of $S_n$.)
As the inner product is $S_n$-invariant, we have
$$ {\langle} \chi_{\sigma(I)},\chi_n {\rangle}={\langle} \chi_I,\sigma\cdot\chi_n {\rangle}=
{\rm sgn}(\sigma) {\langle} \chi_I,\chi_n {\rangle},\quad I\in{\mathcal P}'_n,\quad \sigma\in S_n. $$
We say that the patterns $I,I'\in{\mathcal P}'_n$ are \emph{equivalent}
if they belong to the same orbit of $\tilde{S}_n$.
If so, we shall write $I\approx I'$. We denote by $[I]$
the equivalence class of $I\in{\mathcal P}'_n$.
Assume that $I\approx I'$. It is easy to see that
$I'$ is universal iff $I$ is universal. Since $\ker \varphi_n$ is
invariant under permutations of the variables $x_1,x_2,\ldots,x_n$,
we deduce that $I'$ is nonsingular iff $I$ is nonsingular.
We say that the class $[I]$ is {\em singular, nonsingular,
universal, defective} or {\em exceptional} if $I$ has
the same property. These terms are clearly well defined.
Let $I\in{\mathcal P}$ be any pattern and let us fix a position $(i,j)\in I$ such
that $(j,i)\notin I$. Denote by $I'$ the pattern obtained from $I$ by
replacing the position $(i,j)$ with $(j,i)$. We shall refer to
the transformation $I\to I'$ as a {\em flip}.
\begin{problem} \label{Flip}
Let $I\in{\mathcal P}'_n$ be $n$-universal and let $I\to I'$ be a flip.
Is it true that $I'$ is $n$-universal?
\end{problem}
We say that the patterns $I,I'\in{\mathcal P}_n$ are \emph{weakly equivalent}
if $I$ can be transformed to $I'$ by using flips and the
action of $S_n$. If so, we shall write $I\sim I'$. We denote by $[I]_w$
the weak equivalence class of $I\in{\mathcal P}_n$.
As the transposition map ${\mathcal P}_n\to{\mathcal P}_n$ can be realized
by a sequence of flips, we have $[I]\subseteq[I]_w$ for all $I\in{\mathcal P}_n$.
Note that the nonsingularity property is preserved by weak equivalence.
Let us define the {\em complexity}, $\nu(I)$, of a pattern $I$
as the number of positions $(i,j)\in I\cap I^T$ with $i\le j$.
The patterns of complexity 0 are precisely the simple patterns.
Observe that a pattern $I\in{\mathcal P}'_n$ has complexity 1 iff there exist
a unique $2$-element subset $\{i,j\}\subseteq{\bf Z}_n$ such that
$(i,j)$ and $(j,i)$ belong to $I$.
Similarly, $I\in{\mathcal P}'_n$ has complexity 1 iff there exist a unique
$2$-element subset $\{k,l\}\subseteq{\bf Z}_n$ such that neither
$(k,l)$ nor $(l,k)$ is in $I$.
It is natural to ask which patterns $I\in{\mathcal P}'_n$ of
complexity 1 are universal or nonsingular. We shall now give a
complete answer to the latter question. At the same time we classify,
up to weak equivalence, the patterns in ${\mathcal P}'_n$ having complexity 1.
\begin{theorem} \label{WE-Compl-1}
Let $I\in{\mathcal P}'_n$, $\nu(I)=1$, and let $\{i,j\}$ resp.
$\{k,l\}$ be the unique $2$-element subset of ${\bf Z}_n$ such that
$(i,j),(j,i)\in I$ resp. $(k,l),(l,k)\notin I$.
(a) If $i,j,k,l$ are not distinct then
$$ I \sim \left( {\rm NE}_n \setminus \{(1,2)\} \right)
\cup \{(3,1)\}, \quad n\ge3, $$
and we have ${\langle} \chi_I,\chi_n {\rangle}= \pm\; n!/2.$
(b) If $i,j,k,l$ are distinct then $I$ is singular and
$$ I \sim \left( {\rm NE}_n \setminus \{(1,2)\} \right)
\cup \{(4,3)\}, \quad n\ge4. $$
\end{theorem}
\begin{proof}
(a) We have, say, $i=k$. Let $J=\sigma(I)$, where $\sigma\in S_n$
is chosen so that $\sigma(i)=1,$ $\sigma(j)=3$ and $\sigma(l)=2$. For each
$(r,s)\in J$ with $r>s$ and $(r,s)\ne(3,1)$ we apply a flip to
replace $(r,s)$ with $(s,r)$. We obtain the desired pattern.
Proposition \ref{Pros-Stav} gives the formula for the inner product.
(b) The equivalence assertion is proved in the same manner as in (a)
and, since this new pattern is defective, $I$ must be singular.
\end{proof}
We conclude this section by providing some numerical data
about the equivalence classes in ${\mathcal P}'_n$ for $2\le n\le 5$.
For each $n$, Table 1 gives the cardinality of ${\mathcal P}'_n$, the
number of equivalence classes, and the number of nonsingular,
defective, and exceptional classes in that order. The
last column records the number of weak equivalence classes.
\begin{center}
\begin{tabular}{crrrrrr}
&\\
\multicolumn{6}{c}
{\bf Table 1: Equivalence classes} \\
&\\
$n$ & $|{\mathcal P}'_n|$ & Equ. & Nons. & Def. & Exc. & Weak \\
&\\ \hline &\\
2 & 2 & 1 & 1 & 0 & 0 & 1 \\
3 & 20 & 3 & 3 & 0 & 0 & 2 \\
4 & 928 & 30 & 19 & 4 & 7 & 12 \\
5 & 184956 & 880 & 619 & 66 & 195 & 110 \\
\end{tabular}
\end{center}
For any $n$ we have $|{\mathcal P}'_n|=\binom{2\mu_n}{\mu_n}$, but
a formula for the number of (weak) equivalence classes is not known.
\begin{problem} \label{Equ-Cl}
Find a formula for the number of (weak) equivalence classes in ${\mathcal P}'_n$.
\end{problem}
\section{An analogue of Schur's theorem} \label{Analog}
Recall the patterns $\Lambda_n\in{\mathcal P}'_n$, $n\ge1$, defined in the
Introduction. The zero entries required by $\Lambda_n$ for
$n=2,3,\ldots,7$ are exhibited below:
\begin{equation*}
\left[ \begin{array}{cc}
* & 0 \\ * & * \end{array} \right],\quad
\left[ \begin{array}{ccc}
* & 0 & 0 \\ * & * & * \\ 0 & * & * \end{array} \right],\quad
\left[ \begin{array}{cccc}
* & 0 & * & 0 \\ 0 & * & 0 & * \\ * & 0 & * & * \\ 0 & * & * & *
\end{array} \right],\quad
\left[ \begin{array}{ccccc}
* & 0 & 0 & * & 0 \\ 0 & * & 0 & 0 & * \\ 0 & 0 & * & * & * \\
* & 0 & * & * & * \\ 0 & * & * & * & *
\end{array} \right],
\end{equation*}
\begin{equation*}
\left[ \begin{array}{cccccc}
* & 0 & 0 & 0 & * & 0 \\ 0 & * & 0 & 0 & 0 & * \\
0 & 0 & * & 0 & * & * \\ 0 & 0 & * & * & * & * \\
* & 0 & * & * & * & * \\ 0 & * & * & * & * & *
\end{array} \right],\quad
\left[ \begin{array}{ccccccc}
* & 0 & 0 & 0 & 0 & * & 0 \\ 0 & * & 0 & 0 & 0 & 0 & * \\
0 & 0 & * & 0 & 0 & * & * \\ 0 & 0 & * & * & * & * & * \\
0 & 0 & 0 & * & * & * & * \\ * & 0 & * & * & * & * & * \\
0 & * & * & * & * & * & *
\end{array} \right].
\end{equation*}
The maximum of $\nu(I)$ over all $I\in{\mathcal P}'_n$ is
$[\mu_n/2]$. Let us write $n=4m+r$ where $m$ is a nonnegative integer
and $r\in\{0,1,2,3\}.$ Note that $\Lambda_n$ is symmetric for
$r\in\{0,1\}$, and otherwise there is a unique $(i,j)\in\Lambda_n$,
namely $(2m+1,2m+2)$, such that $(j,i)\notin\Lambda_n$. Hence
$\Lambda_n$ has the maximal complexity $[\mu_n/2]$.
Our objective in this section is to prove that $\Lambda_n$ is
nonsingular. As $\Lambda_n$ is a (necessary) minor modification of the
triangular pattern ${\rm NW}_n$, we consider this result as an analogue
of Schur's theorem.
\begin{theorem} \label{SchurAnalog}
We have $ {\langle} \chi_{\Lambda_n}, \chi_n {\rangle} = (-1)^s n!/2^s,$ where
$s=[(n+1)/4]$. In particular, $\Lambda_n$ is universal.
\end{theorem}
For the proof we need four lemmas and the following three facts which
follow immediately from \cite[Chapter III, Lemma 3.9]{He}:
(i) $x_i^k\in{\mathcal K}(n)$ for $k\ge n$, $1\le i\le n$;
(ii) If $f,g\in{\bf R}[x_1,x_2,\ldots,x_n]$ and $f\equiv g \pmod{{\mathcal K}(n)}$
then
\begin{equation} \label{fg-hi}
{\partial}_f\chi_n={\partial}_g\chi_n;
\end{equation}
(iii) If $f\in{\bf R}[x_1,x_2,\ldots,x_n]_{\mu_n}$ then
\begin{equation} \label{df-hi}
{\partial}_f\chi_n=\prod_{k=1}^{n-1}k!\cdot{\langle} f,\chi_n {\rangle}.
\end{equation}
By diferentiating the formula (\ref{PotpSim}), we obtain that
\begin{equation} \label{IzvPot}
{\partial}_i h_{k,n}=\sum_{d_1+\cdots+d_n=k-1} (1+d_i)x_1^{d_1}x_2^{d_2}\cdots
x_n^{d_n}, \quad i\le n.
\end{equation}
We prove first the following congruence.
\begin{lemma}\label{Kongr}
For $1\le r\le m\le n$ we have
$$
\prod_{m<i\le n} (x_r-x_i) \equiv {\partial}_r h_{n-m+1,m} \pmod{{\mathcal K}(n)}.
$$
\end{lemma}
\begin{proof}
Without any loss of generality we may assume that $r=1$.
Let $s,t$ be two additional commuting indeterminates. Observe that
for any polynomial $f(t)$, with coefficients in ${\bf R}[x_1,\ldots,x_n]$,
the constant term of
$$ F(s,t)=f(s^{-1})\cdot\frac{1}{1-st}, $$
when expanded into a formal Laurent series with respect to $s$,
is equal to $f(t)$. For
$$ f(t)=(t-x_1)\cdot\prod_{m<i\le n} (t-x_i) $$
we have
\begin{eqnarray*}
F(s,t) &=& \frac{\prod_{i=1}^n (1-sx_i)}
{s^{n-m+1} (1-st) \prod_{i=2}^m (1-sx_i)} \\
&=& \frac{1}{s^{n-m+1}}\cdot\sum_{k=0}^n (-s)^k \sigma_{k,n}\cdot
\sum_{l=0}^\infty h_{l,m}(t,x_2,\ldots,x_m)s^l.
\end{eqnarray*}
Hence
$$ f(t)=\sum_{k=0}^{n-m+1} (-1)^k \sigma_{k,n} h_{n-m+1-k,m}
(t,x_2,\ldots,x_m). $$
By evaluating the partial derivative with respect to $t$ at the
point $t=x_1$, we obtain that
$$ \prod_{m<i\le n} (x_1-x_i) = \sum_{k=0}^{n-m+1} (-1)^k \sigma_{k,n}
\partial_1 h_{n-m+1-k,m}(x_1,x_2,\ldots,x_m) $$
and the assertion of the lemma follows.
\end{proof}
Recall the endomorphism $\tau$ defined in the beginning of
Section \ref{Nonsing-Obl}.
\begin{lemma} \label{L:1}
For $P={\partial}_1 h_{2,n-1}(x_1,x_n)$, i.e.,
$$P=\sum_{k=0}^{n-2}(n-1-k)x_1^{n-2-k}x_n^k,\quad n\ge4,$$
we have
\begin{equation}\label{E:1}
\partial_P^2\chi_n=(-1)^nn!(n-2)!\left(\frac{n-1}{2}x_1+
\frac{n-3}{2}x_n-\sum_{k=2}^{n-1}x_k\right) \tau\chi_{n-2}.
\end{equation}
\end{lemma}
\begin{proof}
Since $x_1^n,x_n^n\in{\mathcal K}(n)$, we have
$$P^2 \equiv ax_1^{n-3}x_n^{n-1}+bx_1^{n-2}x_n^{n-2}+cx_1^{n-1}x_n^{n-3},
\pmod{{\mathcal K}(n)},$$
where
\begin{align*}
a &=\sum_{k=1}^{n-2}k(n-1-k)=n(n-1)(n-2)/6, \\
b &=\sum_{k=1}^{n-1}k(n-k)=n(n-1)(n+1)/6, \\
c &=\sum_{k=2}^{n-1}k(n+1-k)=n(n+5)(n-2)/6.
\end{align*}
By using the property (\ref{fg-hi}), we obtain that
$${\partial}_P^2\chi_n=(a{\partial}_1^{n-3}{\partial}_n^{n-1}+b{\partial}_1^{n-2}{\partial}_n^{n-2}
+c{\partial}_1^{n-1}{\partial}_n^{n-3})\chi_n.$$
If we omit from $\chi_n$ the terms $\pm x_1^{d_1}\cdots x_n^{d_n}$ with
$d_1+d_n<2n-4$, we obtain the polynomial $(-1)^n Q \tau\chi_{n-2}$ where
$$ Q=(x_1^{n-1}x_n^{n-2}-x_1^{n-2}x_n^{n-1})
-(x_1^{n-1}x_n^{n-3}-x_1^{n-3}x_n^{n-1})\sum_{k=2}^{n-1}x_k.
$$
Since the omitted terms are killed by ${\partial}_P^2$, we have
${\partial}_P^2\chi_n=(-1)^n C \tau\chi_{n-2}$ where
$$ C=(n-1)!\left[ (n-2)!((b-a)x_1+(c-b)x_n)
-(n-3)!(c-a)\sum_{k=2}^{n-1}x_k \right]. $$
It remains to plug in the values of $a,b$ and $c$.
\end{proof}
The next lemma gives another important identity.
\begin{lemma}\label{L:2}
Let $J_n=J'_n\cup (J'_n)^T$, $n\ge4$, where $J'_n$ is the union of
$\{(1,2),(1,n-1),(2,n)\}$ and $\{1,2\}\times\{3,4,\ldots,n-2\}$.
Then
$${\partial}_{J_n}\chi_n=\frac{1}{2}n!(n-1)!(n-2)!(n-3)!\tau^2\chi_{n-4}.$$
\end{lemma}
\begin{proof}
By using Lemma \ref{Kongr} we obtain the congruence
\begin{align*}
\chi_{J_n} =& -\left(\prod_{i=2}^{n-1}(x_1-x_i)\right)^2\left((x_2-x_n)
\prod_{i=3}^{n-2}(x_2-x_i)\right)^2 \\
\equiv & -P^2R^2 \pmod{{\mathcal K}(n)},
\end{align*}
where $P$ is defined as in the previous lemma and
$$R={\partial}_2 h_{n-2,3}(x_1,x_2,x_{n-1})=
\sum_{d_1+d_2+d_3=n-3}(1+d_2)x_1^{d_1}x_2^{d_2}x_{n-1}^{d_3}.$$
In view of the formula (\ref{E:1}), it suffices to prove that
\begin{equation}\label{E:2}
{\partial}_R^2F=\frac{(-1)^{n-1}}{2}(n-1)!(n-3)!\tau^2\chi_{n-4},
\end{equation}
where
$$F=\left(\frac{n-1}{2}x_1+\frac{n-3}{2}x_n-
\sum_{k=2}^{n-1}x_k\right)\tau\chi_{n-2}.$$
Since $\deg(R^2)=2n-6$, we need only consider the terms
in $F$ for which the sum of the exponents of $x_1, x_2$ and
$x_{n-1}$ is at least $2n-6$. Their sum is
$$F'=(-1)^n(\frac{n-1}{2}x_1-x_2-x_{n-1})
(x_2^{n-3}x_{n-1}^{n-4}-x_2^{n-4}x_{n-1}^{n-3})
\tau^2\chi_{n-4}.$$
Note also that the exponents of $x_1$ in $F'$ are $\le1$. So we need
only consider the terms in ${\partial}_R^2$ for which the exponent
of ${\partial}_1$ is 0 or 1. Their sum is $D_1^2+2{\partial}_1D_1D_2$ where
$$D_1=\sum_{k=0}^{n-3}(n-2-k){\partial}_2^{n-3-k}{\partial}_{n-1}^k,$$
$$D_2=\sum_{k=0}^{n-4}(n-3-k){\partial}_2^{n-4-k}{\partial}_{n-1}^k.$$
After expanding the products $D_1D_2$ and $D_1^2$:
\begin{eqnarray*}
D_1D_2 &=& a_1{\partial}_2^{n-3}{\partial}_{n-1}^{n-4}+b_1{\partial}_2^{n-4}{\partial}_{n-1}^{n-3}+\ldots, \\
D_1^2 &=& a_2{\partial}_2^{n-2}{\partial}_{n-1}^{n-4}+b_2{\partial}_2^{n-4}{\partial}_{n-1}^{n-2}+\ldots,
\end{eqnarray*}
the exhibited coefficients can be easily computed:
\begin{eqnarray*}
a_1 &=& \sum_{k=1}^{n-3}k(n-1-k)=(n-2)(n-3)(n+2)/6, \\
a_2 &=& \sum_{k=2}^{n-2}k(n-k)=(n-1)(n-3)(n+4)/6, \\
b_1=b_2 &=& \sum_{k=1}^{n-3}k(n-2-k)=(n-1)(n-2)(n-3)/6.
\end{eqnarray*}
Thus
\begin{eqnarray*}
D_1D_2(x_2^{n-3}x_{n-1}^{n-4}-x_2^{n-4}x_{n-1}^{n-3}) &=&
(n-3)!(n-4)!(a_1-b_1) \\
&=& (n-2)!(n-3)!/2, \\
D_1^2(x_2^{n-2}x_{n-1}^{n-4}-x_2^{n-4}x_{n-1}^{n-2}) &=&
(n-2)!(n-4)!(a_2-b_2) \\
&=& (n-1)!(n-3)!.
\end{eqnarray*}
Hence
\begin{eqnarray*}
{\partial}_R^2F &=& (D_1^2+2{\partial}_1D_1D_2)F'\\
&=& (-1)^n \left[ (n-1)D_1D_2(x_2^{n-3}x_{n-1}^{n-4}-x_2^{n-4}x_{n-1}^{n-3}) \right. \\
& & \quad \left. -D_1^2(x_2^{n-2}x_{n-1}^{n-4}-x_2^{n-4}x_{n-1}^{n-2}) \right]
\tau^2\chi_{n-4},
\end{eqnarray*}
which completes the proof.
\end{proof}
The fourth lemma follows easily from the previous one.
\begin{lemma}\label{L:3}
Let $I=J_n \cup \left( (2,2)+I' \right)$, $n\ge4$, where $J_n$ is defined
as in the previous lemma and $I'\in{\mathcal P}'_{n-4}$ is arbitrary. Then
$$ {\langle} \chi_I, \chi_n {\rangle} = \frac{1}{2}n(n-1)(n-2)(n-3){\langle} \chi_{I'},
\chi_{n-4} {\rangle}. $$
\end{lemma}
\begin{proof}
Since $\chi_{I}=\chi_{J_n} \tau^2 \chi_{I'}$, we have
${\partial}_I={\partial}_{J_n}{\partial}_{\tau^2 \chi_{I'}}$.
By using (\ref{df-hi}) and Lemma \ref{L:2}, we obtain that
\begin{eqnarray*}
{\langle}\chi_I,\chi_n{\rangle} &=& \frac{{\partial}_I\chi_n}{\prod_{k=1}^{n-1}k!} \\
&=& \frac {n(n-1)(n-2)(n-3) \tau^2 \left( {\partial}_{I'} \chi_{n-4} \right) }
{2\prod_{k=1}^{n-5}k!} \\
&=& \frac{1}{2}n(n-1)(n-2)(n-3){\langle} \chi_{I'}, \chi_{n-4} {\rangle}.
\end{eqnarray*}
\end{proof}
We can now prove the theorem itself.
\begin{proof}
We construct the patterns $\Lambda'_n$ inductively as
$\Lambda'_0=\Lambda'_1=\emptyset$, $\Lambda'_2=\{(1,2)\}$,
$\Lambda'_3=\{(2,1),(2,3),(3,2)\}$, and
$\Lambda'_n=J_n\cup((2,2)+\Lambda'_{n-4})$ for $n\geq4$. We claim
that ${\langle} \chi_{\Lambda'_n}, \chi_n {\rangle} =n!/2^s$.
We prove the claim by induction on $n$. It is straightforward to
verify the claim for $n=0,1,2,3$. For $n\ge 4$, by
Lemma \ref{L:3} and the induction hypothesis, we have
\begin{eqnarray*}
{\langle} \chi_{\Lambda'_n}, \chi_n {\rangle} &=& \frac{1}{2}n(n-1)(n-2)(n-3)
{\langle} \chi_{\Lambda'_{n-4}}, \chi_{n-4} {\rangle} \\
&=& \frac{1}{2}n(n-1)(n-2)(n-3)\cdot\frac{(n-4)!}{2^{s-1}}=\frac{n!}{2^s}.
\end{eqnarray*}
It remains to observe that $\sigma(\Lambda'_n)=\Lambda_n$, where
$$ \sigma=\prod_{k=1}^s(2k-1,2k) \in S_n. $$
\end{proof}
\section{A remarkable family of nonsingular patterns} \label{Analogue}
It is a challenging problem to construct an infinite family
of new nonsingular doubly laced patterns.
The main result of this section gives a construction of such a family.
It includes, as a special case, a new analogue of Schur's theorem
namely another modification of the triangular pattern ${\rm NW}_n$
(see Proposition \ref{Primer-2}).
We first introduce the notation that we need to state and prove our theorem.
Let $\sigma\in S_n$ and let ${\bf r}=(r_1,r_2,\ldots,r_{n-1})$ be a sequence
with $r_k\in\sigma({\bf Z}_k)$ for all $k\in{\bf Z}_{n-1}$.
For $k\in{\bf Z}_{n-1}$ we set
$$ I_k=\{(r_k,\sigma(j)): k<j\le n \}. $$
As usual, $\delta_{ij}$ will denote the Kronecker delta symbol.
The main technical tool is the following lemma.
\begin{lemma} \label{glavna}
Under the above hypotheses, for $0\le m<n$ we have
\begin{eqnarray*}
&& {\partial}_{I_1}{\partial}_{I_2}\cdots{\partial}_{I_m}\chi_n= \\
&& \quad {\rm sgn}(\sigma)\left(\prod_{k=1}^m
(n-k+\delta_{r_k,\sigma(k)})! \right) \chi_{n-m}(x_{\sigma(m+1)},\ldots,x_{\sigma(n)}).
\end{eqnarray*}
\end{lemma}
\begin{proof}
We use induction on $m$. The assertion obviously holds for $m=0$.
Assume that $m>0$. By the induction hypothesis we have
\begin{eqnarray*}
&& {\partial}_{I_1}{\partial}_{I_2}\cdots{\partial}_{I_m}\chi_n= \\
&& \quad {\rm sgn}(\sigma)\left(\prod_{k=1}^{m-1}(n-k+\delta_{r_k,\sigma(k)})!\right)
\cdot{\partial}_{I_m}\chi_{n-m+1}(x_{\sigma(m)},\ldots,x_{\sigma(n)}).
\end{eqnarray*}
By Lemma \ref{Kongr} we have
\begin{eqnarray*}
\chi_{I_m} &=& \prod_{m<i\le n} (x_{r_m}-x_{\sigma(i)}) \\
&\equiv& {\partial}_{r_m} h_{n-m+1,m}(x_{\sigma(1)},\ldots,x_{\sigma(m)})
\pmod{{\mathcal K}(n)}.
\end{eqnarray*}
By invoking the property (\ref{fg-hi}) and the identity (\ref{df-hi}),
we deduce that
\begin{eqnarray*}
&& {\partial}_{I_m}\chi_{n-m+1}(x_{\sigma(m)},\ldots,x_{\sigma(n)})= \\
&& \quad \sum_{d_1+\cdots+d_m=n-m} (1+d_{r_m}) {\partial}_{\sigma(1)}^{d_1} \cdots
{\partial}_{\sigma(m)}^{d_m} \chi_{n-m+1}(x_{\sigma(m)},\ldots,x_{\sigma(n)}).
\end{eqnarray*}
All terms in this sum vanish except the one for $d_1=\cdots=d_{m-1}=0$
and $d_m=n-m$. We infer that
\begin{eqnarray*}
&& {\partial}_{I_m}\chi_{n-m+1}(x_{\sigma(m)},\ldots,x_{\sigma(n)})= \\
&& \quad \left(1+(n-m)\delta_{r_m,\sigma(m)} \right){\partial}_{\sigma(m)}^{n-m}
\chi_{n-m+1}(x_{\sigma(m)},\ldots,x_{\sigma(n)}).
\end{eqnarray*}
Since
$$ {\partial}_{\sigma(m)}^{n-m} \chi_{n-m+1}(x_{\sigma(m)},\ldots,x_{\sigma(n)})=
(n-m)!\chi_{n-m}(x_{\sigma(m+1)},\ldots,x_{\sigma(n)}), $$
we are done.
\end{proof}
Let us fix a permutation $\sigma\in S_n$ and let
${\bf i}=(i_1,i_2,\ldots,i_{n-1})$ be a sequence
of distinct integers such that
$|i_k|\in\sigma({\bf Z}_k)$ for all $k\in{\bf Z}_{n-1}$. Next we set
$$ (i_k,\sigma(j))^+=\begin{cases}
(i_k,\sigma(j)) & \text{if } i_k>0; \\
(\sigma(j),-i_k) & \text{otherwise}.
\end{cases} $$
With these data at hand, we construct the strict pattern
\begin{equation} \label{Obl-J}
J(\sigma,{\bf i})=\{(i_k,\sigma(j))^+:1\le k<j\le n\}.
\end{equation}
We claim that the conditions imposed on {\bf i} imply that the map
sending $(k,j)\to(i_k,\sigma(j))^+$ for $k<j$ is injective, i.e., that
$|J(\sigma,{\bf i})|=\mu_n$. Indeed, assume that two different pairs
$(k,j)$ and $(r,s)$, with $k<j$ and $r<s$, have the same image, i.e.,
$(i_k,\sigma(j))^+=(i_r,\sigma(s))^+$.
Clearly, we must have $k\ne r$ and $i_k i_r<0$. Say, $k<r$.
Then $|i_k|=\sigma(s)\notin\sigma({\bf Z}_r)$, which contradicts the condition
$|i_k|\in\sigma({\bf Z}_k)$. This proves our claim, and so we have
$J(\sigma,{\bf i})\in{\mathcal P}'_n$.
Let ${\mathcal J}_n\subseteq{\mathcal P}'_n$ be the set of all patterns $J(\sigma,{\bf i})$.
For any $n$, let $\iota\in S_n$ be the identity permutation.
There are exactly $(n!)^2$ choices for the ordered pairs
$(\sigma,{\bf i})$. However the corresponding patterns $J(\sigma,{\bf i})$
are not all distinct. For instance, if $n=2$ we have
$J(\sigma,(2))=J(\iota,(-1))$ with $\sigma=(1,2)$.
The following is the main result of this section.
\begin{theorem} \label{Fam-J}
For $J=J(\sigma,{\bf i})$ we have
$${\langle} \chi_J,\chi_n {\rangle} =(-1)^d{\rm sgn}(\sigma)\prod_{k:|i_k|=\sigma(k)}(n-k+1), $$
where
$$ d=\sum_{k \;:\; i_k<0}(n-k). $$
\end{theorem}
\begin{proof}
We apply the lemma with ${\bf r}=(|i_1|,|i_2|,\ldots,|i_{n-1}|)$
and $m=n-1$. Thus we now have
$I_k=\{(|i_k|,\sigma(j)): k<j\le n \}$ for all $k\in{\bf Z}_{n-1}$.
Since $\chi_1=1$, the lemma gives
$$ {\partial}_{I_1}{\partial}_{I_2}\cdots{\partial}_{I_{n-1}}\chi_n=
{\rm sgn}(\sigma)\prod_{k=1}^{n-1}(n-k+\delta_{|i_k|,\sigma(k)})!. $$
By (\ref{df-hi}) we have
$$
{\partial}_{I_1}{\partial}_{I_2}\cdots{\partial}_{I_{n-1}}\chi_n=\prod_{k=1}^{n-1}(n-k)!\cdot
{\langle} \chi_{I_1}\chi_{I_2}\cdots\chi_{I_{n-1}}, \chi_n {\rangle}.
$$
Observe that $J$ is the disjoint union of the
$I_k$'s with $i_k>0$ and the $I_k^T$'s with $i_k<0$.
Therefore we have $\chi_J=(-1)^d \chi_{I_1}\chi_{I_2}\cdots\chi_{I_{n-1}}$
and the assertion follows.
\end{proof}
Let us give an example.
\begin{example} \label{Primer-1}
Let $n>1$ and let $\sigma\in S_n$ be the identity. We set $i_1=-1$ and
$i_k=k-1$ for $1<k<n$. The $i_k$'s are distinct and the condition $|i_k|\in{\bf Z}_k$
is satisfied for all $k$. In this case we obtain the pattern
$$ J=\{(i,j):1\le i<j-1\le n-1\} \cup \{(i,1):1<i\le n\}. $$
Only $i_1$ is negative and so $d=n-1$, and $|i_k|=\sigma(k)=k$ is valid
only for $k=1$.
By the theorem we have ${\langle} \chi_J,\chi_n {\rangle}=(-1)^{n-1}n.$
This proves the case $k=1$ of Conjecture \ref{Hess}.
\end{example}
Recall that ${\rm NW}_n$ is not $n$-universal for $n\ge3$
(see Example \ref{dij+1}). However, if we
modify this pattern to make it strict by replacing its diagonal
positions $(i,i)$ with $(i,n+1-i)$, we can show that the new pattern
$$ \Pi_n=\{(i,j):i+j\le n, i\ne j\} \cup \{(i,n-i+1):2i\le n\} $$
is nonsingular and, consequently, universal.
This is our second analogue of Schur's theorem.
The zero entries required by $\Pi_n$ for
$n=2,3,4,5$ are exhibited below:
\begin{equation*}
\left[ \begin{array}{cc}
* & 0 \\ * & * \end{array} \right],\quad
\left[ \begin{array}{ccc}
* & 0 & 0 \\ 0 & * & * \\ * & * & * \end{array} \right],\quad
\left[ \begin{array}{cccc}
* & 0 & 0 & 0 \\ 0 & * & 0 & * \\ 0 & * & * & * \\ * & * & * & *
\end{array} \right],\quad
\left[ \begin{array}{ccccc}
* & 0 & 0 & 0 & 0 \\ 0 & * & 0 & 0 & * \\ 0 & 0 & * & * & * \\
0 & * & * & * & * \\ * & * & * & * & *
\end{array} \right].
\end{equation*}
\begin{proposition} \label{Primer-2}
We have ${\langle} \chi_{\Pi_n}, \chi_n {\rangle}=n!!.$
\end{proposition}
\begin{proof}
This is in fact a special case of Theorem \ref{Fam-J}.
We take $\sigma\in S_n$ to be the permutation $1,n,2,n-1,\ldots.$
Thus $\sigma(2k-1)=k$ for $2k-1\le n$ and $\sigma(2k)=n+1-k$ for $2k\le n$.
We set ${\bf i}=(1,-1,2,-2,\ldots).$ The $i_k$'s are distinct. As
$i_{2k-1}=-i_{2k}=k=\sigma(2k-1)$, the condition
$|i_k|\in\sigma({\bf Z}_k)$ is satisfied for all $k$'s.
With this $\sigma$ and {\bf i} we have $\Pi_n=J(\sigma,{\bf i})$.
The equality $|i_k|=\sigma(k)$ holds iff $k$ is odd and the inequality
$i_k<0$ holds iff $k$ is even. Thus $d=\sum(n-2k)$, the sum being over
all positive integers $k$ such that $2k\le n$. Therefore $d$
is even for $n$ even and $d\equiv [n/2] \pmod{2}$ for $n$ odd.
One can easily verify that ${\rm sgn}(\sigma)=(-1)^d$ in all cases.
Hence, we obtain the formula given in the proposition.
\end{proof}
In several cases we used \cite{NS} to identify various sequences that
we have encountered, such as the double factorial sequence $A006882$
in the above proposition.
Let $J=J(\sigma,{\bf i})$ be the pattern (\ref{Obl-J}).
Note that $J^T=J(\sigma,-{\bf i})$,
where $-{\bf i}=(-i_1,-i_2,\ldots,-i_{n-1})$.
If $\rho\in S_n$ it is easy to verify that
$\rho(J)=J(\rho\sigma,{\bf j})$ where ${\bf j}=(j_1,j_2,\ldots,j_{n-1})$
with
$$ j_k=\begin{cases} \rho(i_k) & \text{if } i_k>0; \\
-\rho(-i_k) & \text{otherwise}.
\end{cases} $$
It follows that ${\mathcal J}_n$ is a union of equivalence classes.
\begin{problem} \label{oblici}
Determine the number of equivalence classes contained in ${\mathcal J}_n$.
\end{problem}
For $n=2,3,4,5$ the answers are $1,2,7,34$ respectively.
In the case when $\sigma=\iota$, the sequence
${\bf i}=(i_1,i_2,\ldots,i_{n-1})$
is subject only to the conditions:
(a) the integers $i_k$ are pairwise distinct and
(b) $i_k\in{\bf Z}_k$ for all $k\in{\bf Z}_{n-1}$.
In particular $i_1=\pm1$. To simplify the
notation, in this case we set $J({\bf i})=J(\iota,{\bf i})$.
From the previous discussion it is clear that each equivalence class
contained in ${\mathcal J}_n$ has a representative of the form $J({\bf i})$
with $i_1=1$. However, $J({\bf i})\approx J({\bf j})$ may hold
for two different sequences ${\bf i}$ and ${\bf j}$ with $i_1=j_1=1$.
For instance, for $n=3$ we have $J((1,2))\approx J((1,-2))$.
\section{The case $n=4$} \label{Cetiri}
In this section we fix $n=4$. Recall that the nonsingular
patterns are universal, and the defective ones are not.
In this section we shall exhibit a strict pattern which is singular
and universal (see Proposition \ref{Oblik 6} below).
There are $7$ exceptional equivalence classes in ${\mathcal P}'_4$,
their representatives are listed in Table 2. The last column of the
table shows what is known about the universality of the pattern.
\begin{center}
\begin{tabular}{ccl}
&\\
\multicolumn{3}{c}
{\bf Table 2: Exceptional classes for $n=4$} \\
&\\
No. & Representative pattern & Univ. \\
&\\ \hline &\\
1 & \{(1,2),(2,1),(1,3),(3,1),(2,3),(3,2)\} & No \\
2 & \{(1,2),(2,1),(1,3),(1,4),(2,4),(3,2)\} & No \\
3 & \{(1,2),(1,3),(1,4),(2,1),(3,4),(4,3)\} & Yes \\
4 & \{(1,2),(2,1),(1,3),(3,1),(2,4),(4,3)\} & ? \\
5 & \{(1,2),(2,1),(1,3),(2,3),(4,1),(4,2)\} & ? \\
6 & \{(1,2),(2,1),(1,4),(2,3),(3,1),(4,2)\} & ? \\
7 & \{(1,2),(2,1),(1,4),(3,1),(3,4),(4,3)\} & ? \\
\end{tabular}
\end{center}
\bigskip
We shall prove now that the first two patterns are not universal.
\begin{proposition} \label{Oblik 1}
The first pattern in Table 2 is not universal.
\end{proposition}
\begin{proof}
Denote this pattern by $I$. Let $A\in L(4)$ be the matrix whose entries
in positions $(1,2)$ and $(3,4)$ are 1 and all other entries are 0.
Note that $A$ has rank 2 and that $A^2=0$.
Assume that $A$ is unitarily similar to some $X=[x_{ij}]\in L_I$.
Thus $X$ has the form
$$ X=\left[ \begin{array}{cccc}
* & 0 & 0 & * \\ 0 & * & 0 & * \\ 0 & 0 & * & * \\ * & * & * & *
\end{array} \right]. $$
Since $X$ must be nilpotent of rank 2, it is clear that at least one
of $x_{14}$, $x_{24}$, $x_{34}$ is nonzero, and also at least one
of $x_{41}$, $x_{42}$, $x_{43}$ is nonzero.
We may assume that $x_{14}\ne0$.
Then $X^2=0$ implies that $x_{42}=x_{43}=0$, and so $x_{41}\ne0$.
As $x_{22}$ and $x_{33}$ are eigenvalues of $X$,
we must have $x_{22}=x_{33}=0$.
From $X^2=0$ we deduce that $x_{24}=x_{34}=0$.
Thus only the four corner entries of $X$ may be nonzero.
As $X$ is nilpotent of rank 2, we have a contradiction.
\end{proof}
\begin{proposition} \label{Oblik 3}
The second pattern in Table 2 is not universal.
\end{proposition}
\begin{proof}
Denote this pattern by $I$ and let
\begin{equation*}
A=\left[ \begin{array}{rrrr}
0&0&0&0 \\
0&-i&-i&0 \\
0&-1&1&1-i \\
1&2&0&i-1 \end{array} \right],
\end{equation*}
a nilpotent matrix of rank 3.
Assume that $AU=UX$ for some $X=[x_{ij}]\in L_I$ and some
$U=[u_{ij}]\in{\mbox{\rm U}}(4)$. Thus $X$ has the form
$$ X=\left[ \begin{array}{cccc}
* & 0 & 0 & 0 \\ 0 & * & * & 0 \\ * & 0 & * & * \\ * & * & * & *
\end{array} \right]. $$
Let $u_k$ denote the $k$-th column of $U$. By equating the entries in
$X=U^*AU$, we obtain that $x_{ij}=u_i^*Au_j$ for all $i,j$.
Since $X$ is nilpotent, we must have $x_{11}=0$. The first row of $X$
is zero, and the other three rows must be linearly independent because
$X$ has rank 3. Since the first row of $UX=AU$ is zero,
we conclude that $u_{12}=u_{13}=u_{14}=0$.
As $U$ is unitary, we also have $u_{21}=u_{31}=u_{41}=0$.
Without any loss of generality, we may assume that $u_{11}=1$.
Since $u_2^*Au_1=x_{21}=0$, we have $u_{42}=0$.
Assume that $u_{32}=0$. Then $u_{23}=u_{24}=0$, and we may assume
that $u_{22}=1$. Since $u_3^*Au_2=x_{32}=0$, we obtain that
$-\bar{u}_{33}+2\bar{u}_{43}=0$.
Since $u_3 \perp u_4$, we must have $u_{44}=-2u_{34}\ne0$.
Now we obtain a contradiction: $0=x_{24}=u_2^*Au_4=-iu_{34}$.
Thus we must have $u_{32}\ne0$. Then
$u_2=u_{32}(0,\xi,1,0)$ with $\xi=u_{22}/u_{32}$.
If $u_{24}=0$, then $u_2 \perp u_4$ implies that $u_{34}=0$ and
the condition $x_{24}=0$ again gives a contradiction.
Consequently, we must have $u_{24}\ne0$ and so
$u_4=u_{24}(0,1,-\bar{\xi},\eta)$ for some $\eta\in{\bf C}$. Then we have
\begin{eqnarray*}
Au_2 &=& u_{32}(0,-i(1+\xi),1-\xi,2\xi), \\
Au_4 &=& u_{24}(0,i(\bar{\xi}-1),-1-\bar{\xi}+(1-i)\eta,2+(i-1)\eta).
\end{eqnarray*}
As we must have $Au_4\perp u_2$, we obtain that
\begin{equation} \label{prva}
i\xi(1-\xi)-(1+\xi)+(1+i)\bar{\eta}=0.
\end{equation}
Since $U^*Au_2$ is the second column of $X$, $Au_2$ must be a linear
combination of $u_2$ and $u_4$. Therefore
\begin{equation*}
\left| \begin{array}{ccc}
\xi & -i(1+\xi) & 1 \\
1 & 1-\xi & -\bar{\xi} \\
0 & 2\xi & \eta \end{array} \right|=0,
\end{equation*}
i.e.,
\begin{equation} \label{druga}
\eta(\xi(1-\xi)+i(1+\xi))+2\xi(1+|\xi|^2)=0.
\end{equation}
From (\ref{prva}) and (\ref{druga}) we obtain that
\begin{eqnarray*}
&& \xi(1-\xi)+i(1+\xi)=(i-1)\bar{\eta}, \\
&& (1-i)|\eta|^2=2\xi(1+|\xi|^2).
\end{eqnarray*}
Hence, $\xi=\lambda(1-i)$ for some real $\lambda\ge0$.
It follows that
\begin{eqnarray*}
|\eta|^2 &=& 2\lambda(1+2\lambda^2), \\
(i-1)\bar{\eta} &=& i+2\lambda+2i\lambda^2, \\
2|\eta|^2 &=& 4\lambda^4+8\lambda^2+1, \\
0 &=& (1-2\lambda+2\lambda^2)^2,
\end{eqnarray*}
which is a contradiction.
\end{proof}
We now give the promised example of a pattern which
is universal and singular.
\begin{proposition} \label{Oblik 6}
The third pattern in Table 2 is universal.
\end{proposition}
\begin{proof} Let $A$ be any linear operator on ${\bf C}^4$
of trace 0.
We have to construct an orthogonal basis $\{a_1,a_2,a_3,a_4\}$
such that, with respect to
this new basis, the matrix of $A$ belongs to $L_I$.
For $a_1$ we choose an eigenvector of $A^*$.
The case when $a_1$ is also
an eigenvector of $A$ is easy and we leave it to the reader.
We extend $\{a_1\}$ to an orthogonal basis $\{a_1,b_1,b_2,b_3\}$
such that $Aa_1=\lambda a_1+b_3$ for some $\lambda\in{\bf C}$.
The matrix of $A$ with respect to this new basis has the form
\begin{equation*}
\left[ \begin{array}{c|ccc} \lambda&0&0&0 \\
\hline 0& & & \\ 0& &B& \\ 1& & & \end{array} \right].
\end{equation*}
In order to complete the proof, it suffices to show that there exist
nonzero vectors $x,y\in a_1^\perp$ such that
\begin{equation} \label{treca}
Ax\perp y,\quad x\perp Ay,\quad x\perp y,\quad b_3\in{\rm span}\{x,y\}.
\end{equation}
We shall now work with the $A$-invariant subspace $a_1^\perp$.
Let $b_{ij}$, $i,j\in\{1,2,3\}$ be the entries of the submatrix $B$.
We may assume that $b_{31}\bar{b}_{23}\ne b_{32}\bar{b}_{13}$ because
such matrices form a dense open subset of $M(3)$.
We shall write vectors in $a_1^\perp$ by using their coordinates with
respect to the basis $\{b_1,b_2,b_3\}$. We shall seek the vectors
$x$ and $y$ in the form
\[ x=(a+ib,c+id,1),\quad y=(a+ib,c+id,-a^2-b^2-c^2-d^2), \]
where $a,b,c,d\in{\bf R}$. Observe that the last two conditions in
(\ref{treca}) are automatically satisfied. The first two conditions give
\begin{eqnarray}
\label{cetvrta} && \left( b_{11}(a+ib)+b_{12}(c+id)+b_{13} \right)
(a-ib)+ \\
\notag && \left( b_{21}(a+ib)+b_{22}(c+id)+b_{23} \right) (c-id)- \\
\notag && \left( b_{31}(a+ib)+b_{32}(c+id)+b_{33} \right)
(a^2+b^2+c^2+d^2) =0
\end{eqnarray}
and
\begin{eqnarray} \label{peta}
&& \left( b_{11}(a+ib)+b_{12}(c+id)-b_{13}(a^2+b^2+c^2+d^2) \right)
(a-ib)+ \\
\notag && \left( b_{21}(a+ib)+b_{22}(c+id)-b_{23}(a^2+b^2+c^2+d^2)
\right)(c-id)+ \\
\notag && \left( b_{31}(a+ib)+b_{32}(c+id)-b_{33}(a^2+b^2+c^2+d^2) \right) \cdot1=0.
\end{eqnarray}
By taking the difference and cancelling the factor $1+a^2+b^2+c^2+d^2$,
we obtain the linear equation
\begin{equation} \label{sesta}
b_{13}(a-ib)+b_{23}(c-id)-b_{31}(a+ib)-b_{32}(c+id)=0.
\end{equation}
Thus our problem is reduced to showing that the equations (\ref{cetvrta})
and (\ref{sesta}) have a real solution for the unknowns $a,b,c,d$.
We now set $b_{ij}=b_{ij}'+ib_{ij}''$ where $b_{ij}',b_{ij}''\in{\bf R}$
and denote by $(S)$ the system of four equations obtained from
(\ref{cetvrta}) and (\ref{sesta}) by equating to zero their real and
imaginary parts. The first two of these four equations are not
homogeneous. By homogenizing these two equations we obtain the system
which we denote by $(S')$. Although we are interested in real solutions,
we shall now consider all complex solutions of $(S')$ in the complex
projective 4-space. By B\'{e}zout's theorem, there are 9 solutions in the
generic case (counting multiplicities). We are going to show that
exactly two of these solutions lie on the hyperplane at infinity.
Consequently, the system $(S)$ has exactly 7 solutions (counting
multiplicities). Since the non-real solutions come in complex conjugate
pairs, at least one of them has to be real.
Clearly, this will complete the proof.
To find the solutions in the hyperplane at infinity (a complex
projective 3-space), we have to solve yet another homogeneous
system, $(S'')$, which is obtained from $(S)$ by omitting the terms
of degree less than 3 in the first two equations and retaining the
last two (linear) equations. The two new cubic equations factorize
as follows:
\begin{eqnarray*}
&& (a^2+b^2+c^2+d^2)(b_{31}'a-b_{31}''b+b_{32}'c-b_{32}''d)=0, \\
&& (a^2+b^2+c^2+d^2)(b_{31}''a+b_{31}'b+b_{32}''c+b_{32}'d)=0. \\
\end{eqnarray*}
If we assume that $a^2+b^2+c^2+d^2\ne0$, then we obtain a system of
four linear equations. The condition
$b_{31}\bar{b}_{23}\ne b_{32}\bar{b}_{13}$ is just saying that the
determinant of this system of linear equations is not 0. Thus,
this system has only the trivial solution. Hence the
solutions of $(S'')$ are just the solutions of the
system of two linear equations of $(S)$ and the equation
$a^2+b^2+c^2+d^2=0$. But a line intersects a quadric in exactly
two points and we are done.
\end{proof}
For the remaining four exceptional classes the universality
remains undecided.
\begin{problem} \label{Obl-4}
Decide which of the last four patterns in Table 2 are universal.
\end{problem}
\section{Counting reducing flags} \label{RedFlag}
Recall that the homogeneous space
${\mathcal F}_n={\mbox{\rm U}}(n)/T_n$ is known as the {\em flag variety}. It is a real
smooth manifold of dimension $n(n-1)$. The points of this
manifold can be interpreted in several ways. By the above definition,
the points are cosets $gT_n$, $g\in{\mbox{\rm U}}(n)$. They can be viewed also as
complete flags
$$ 0=V_0\subset V_1\subset V_2\subset\cdots\subset V_n={\bf C}^n, $$
i.e., the increasing sequence of complex subspaces $V_k$ with
$\dim V_k=k$. We shall follow this practise and refer to the points
of ${\mathcal F}_n$ as {\em flags}.
Yet another often used interpretation is to consider
the points of ${\mathcal F}_n$ as ordered $n$-tuples $ (W_1,W_2,\ldots,W_n) $
of 1-dimensional complex subspaces of ${\bf C}^n$ which are orthogonal
to each other under the standard inner product.
Let $I\in{\mathcal P}'_n$ and $A\in L(n)$.
Assume that the orbit ${\mathcal O}_A$ meets $L_I=L_I(n)$ and let
${\mathcal X}_A={\mathcal O}_A\cap L_I$. The maximal torus $T_n$ acts on ${\mathcal X}_A$
(on the right) by conjugation, i.e., $(X,t)\to t^{-1}Xt$ where
$t\in T_n$ and $X\in{\mathcal X}_A$.
Let us also introduce the set
$$ {\mathcal U}_A=\{g\in{\mbox{\rm U}}(n):g^{-1}Ag\in L_I\}, $$
on which $T_n$ acts by right multiplication.
If $g\in{\mathcal U}_A$ then $g^{-1}Ag\in L_I$ and we say that the
flag $gT_n$ {\em reduces} $A$ to $L_I$. Thus the set ${\mathcal U}_A/T_n$
can be identified with the set of all reducing flags of the
matrix $A$.
The map
\begin{equation} \label{ekv-jed}
\theta:{\mathcal U}_A\to{\mathcal X}_A,\quad \theta(g)=g^{-1}Ag,
\end{equation}
is surjective and $T_n$-equivariant. Our main objective in this
section is to prove that the induced map
\begin{equation} \label{jed-jed}
\hat{\theta}:{\mathcal U}_A/T_n\to{\mathcal X}_A/T_n
\end{equation}
is bijective, i.e., that we have a natural bijective correspondence
between the reducing flags (for $A$) and the $T_n$-orbits
in ${\mathcal X}_A$.
We need three lemmas. The first one is valid for any connected
compact Lie group $G$. For any $g\in G$ let $Z_g$ denote the identity
component of the centralizer of $g$ in $G$.
\begin{lemma} \label{Centre}
Let $G$ be a connected compact Lie group and $H$ a connected closed
subgroup of maximal rank. For any $g\in G$, $g$ belongs to the center
of $Z_g$. If $g\in H$ then $H$ contains the center of $Z_g$.
\end{lemma}
\begin{proof}
Let $T$ be a maximal torus of $G$ such that $g\in T$.
As $T\subseteq Z_g$, we have $g\in Z_g$ and the first assertion
follows. If $g\in H$, we may assume that $T$ is chosen so that
$T\subseteq H$. Then $T$ is a maximal torus of $Z_g$, and so $T$
must contain the center of $Z_g$. As $T\subseteq H$, we are done.
\end{proof}
The next two lemmas deal with the case $G={\mbox{\rm U}}(n)$.
\begin{lemma} \label {Connect-1}
If $H_1$ and $H_2$ are connected closed subgroups of ${\mbox{\rm U}}(n)$
of rank $n$, then $H_1\cap H_2$ is connected.
\end{lemma}
\begin{proof}
In view of the above lemma, it suffices to show that the center of
$Z_g$ is connected for all $g\in{\mbox{\rm U}}(n)$. To prove this,
we may assume that $g$ is a diagonal matrix. It follows that
$$ Z_g \cong {\mbox{\rm U}}(n_1)\times{\mbox{\rm U}}(n_2)\times\cdots\times{\mbox{\rm U}}(n_k), \quad
n_1+\cdots+n_k=n. $$
Hence the center of $Z_g$ is a torus.
\end{proof}
\begin{lemma} \label {Connect-2}
The centralizer of any $A\in M(n)$ in ${\mbox{\rm U}}(n)$ is connected.
\end{lemma}
\begin{proof}
Let $A=A_1+iA_2$ where $A_1$ and $A_2$ are hermitian matrices.
Since $A_k$ is unitarily diagonalizable, its centralizer $H_k$ in
${\mbox{\rm U}}(n)$ is a closed connected subgroup of rank $n$. Hence,
the centralizer $H_1\cap H_2$ of $A$ in ${\mbox{\rm U}}(n)$ is connected
by Lemma \ref{Connect-1}.
\end{proof}
We can now prove the desired result. Recall that $A\in L(n)$ is
$I$-generic if ${\mathcal O}_A$ and $L_I$ intersect transversally.
\begin{theorem} \label{Corresp}
Let $I\in{\mathcal P}'_n$ and let $A\in L(n)$ be $I$-generic. Then the map
$\hat{\theta}$ defined by (\ref{ekv-jed}) and (\ref{jed-jed}) is
bijective. Consequently, $N_{A,I}$ is the number of flags which reduce
$A$ to $L_I$.
\end{theorem}
\begin{proof}
Since $\theta$ is surjective, so is $\hat{\theta}$. In order to prove
that $\hat{\theta}$ is injective, it suffices to show that if
$g_1,g_2\in{\mathcal U}_A$ are such that $g_1^{-1}Ag_1=g_2^{-1}Ag_2$, we denote
this matrix by $B$, then the element $h=g_1^{-1}g_2$ belongs to $T_n$.
If ${\mbox{\germ u}}(n)\subseteq M(n)$ is the space of skew-hermitian matrices,
the transversality hypothesis implies that
$$ \{X\in{\mbox{\germ u}}(n):[X,B]\in L_I\} $$
is the space ${\mbox{\germ t}}_n$ of the diagonal skew-hermitian matrices. Hence, the Lie
algebra of the centralizer of $B$ in ${\mbox{\rm U}}(n)$ is contained in ${\mbox{\germ t}}_n$.
By the above lemma this centralizer is connected, and so must be
contained in $T_n$. As $h$ commutes with $B$, we have $h\in T_n$.
\end{proof}
\begin{remark} \label{IntN}
It follows from the theorem that the number $N_{A,I}$ coincides
with the number denoted by $N(A,M_I(n,{\bf C}))$ in \cite{AD}.
\end{remark}
\section{The cyclic pattern in the case $n=3$} \label{CikObl}
In this section we consider only the case $n=3$ and the cyclic pattern
\[ I=\{(1,3),(2,1),(3,2)\}\in{\mathcal P}'_3. \]
By Theorem \ref{TeorAD}, $I$ is universal.
For $A\in L(3)$ let ${\mathcal O}_A=\{UAU^{-1}:U\in{\mbox{\rm U}}(3)\}$ be its orbit.
Recall that $A$ (or ${\mathcal O}_A$) is said to be $I$-generic
if ${\mathcal O}_A$ and $L_I=L_I(3)$ intersect transversally. Since $I$ is fixed,
we shall drop the prefix ``$I$-'' and say just that $A$ is generic.
We denote by $\Theta$ the set of nongeneric matrices in $L(3)$.
Clearly $\Theta$ is a closed ${\mbox{\rm U}}(n)$-invariant subset.
We shall study here the set $\Theta$, and the intersections
${\mathcal X}_A={\mathcal O}_A \cap L_I$ for generic $A\in L(3)$.
In particular, we are interested in the possible values
of the number, $N_A=N_{A,I}$, of $T_3$-orbits contained in ${\mathcal X}_A$.
First we consider the intersection $\Theta\cap L_I$.
This set contains all points $A\in L_I$ such that ${\mathcal O}_A$
and $L_I$ are not transversal at $A$.
An arbitrary matrix $A\in L_I$ can be written as
\begin{equation} \label{Mat-A}
A=\left[ \begin{array}{ccc} u & z & 0 \\ 0 & v & x \\ y & 0 & w
\end{array} \right], \quad w=-u-v.
\end{equation}
Define the homogeneous polynomial $P_1:L_I\to{\bf R}$ of degree 6 in the
real and imaginary parts of the complex variables $x,y,z,u,v$ by:
\begin{eqnarray} \label{Pol-P1}
P_1(A) &=&
|(v-w)x^2|^2+|(w-u)y^2|^2+|(u-v)z^2|^2 \\ \notag
&& +\left( |(v-w)x|^2+|yz|^2 \right)\cdot \left( |v|^2+|w|^2-5|u|^2 \right) \\
\notag
&& +\left( |(w-u)y|^2+|zx|^2 \right)\cdot \left( |w|^2+|u|^2-5|v|^2 \right) \\
\notag
&& +\left( |(u-v)z|^2+|xy|^2 \right)\cdot \left( |u|^2+|v|^2-5|w|^2 \right) \\
\notag
&& +|(u-v)(v-w)(w-u)|^2.
\end{eqnarray}
\begin{proposition} \label{Hip-pov}
For $A\in L_I$ as above, ${\mathcal O}_A$ and $L_I$ intersect
non-transversally at $A$ iff $A$ lies on the real hypersurface
$\Gamma_1\subset L_I$ defined by the equation $P_1=0$.
\end{proposition}
\begin{proof}
The tangent space to ${\mathcal O}_A$ at the point $A$ consists of all matrices
$[A,X]=AX-XA$ with $X^*=-X$ and ${\,\rm tr\,}(X)=0$. If $X$ is diagonal, then
$[A,X]\in L_I$. Let $S$ be the subspace of skew-Hermitian matrices
with all diagonal entries 0. We see immediately that ${\mathcal O}_A$ and
$L_I$ are transversal at $A$ iff $L_I+[A,S]=L(3)$.
A routine computation shows that the latter condition is
equivalent to the vanishing of a certain determinant, a homogeneous
polynomial of degree 6. It is not hard to verify that this polynomial
(unique up to a scalar factor) is $P_1$.
\end{proof}
\begin{corollary} \label{Gama-1}
We have
\begin{eqnarray}
\label{G-Teta} \Gamma_1 \subseteq \Theta \cap L_I, \\
\label{T-Gama} \Theta = \cup_{A\in\Gamma_1} {\mathcal O}_A.
\end{eqnarray}
\end{corollary}
\begin{proof}
The inclusion (\ref{G-Teta}) is obvious.
If $B\in\Theta$ then there exists $A\in{\mathcal X}_B$ such that
${\mathcal O}_B$ and $L_I$ intersect non-transversally at $A$. Hence $A\in\Gamma_1$
and $B\in{\mathcal O}_A$. Thus (\ref{T-Gama}) is valid.
\end{proof}
Our next objective is to construct the (unique) irreducible real hypersurface
$\Gamma$ in $L(3)$ containing the set $\Theta$. It is given by an equation
$P=0$, where $P:L(3)\to{\bf R}$ is a homogeneous polynomial of
degree 24 in 16 variables, the real and imaginary parts of the entries
of $X\in L(3)$ except the last entry.
The polynomial $P$ is invariant under the action of the direct product
${\mbox{\rm U}}(1)\times{\mbox{\rm SU}}(3)$, where ${\mbox{\rm U}}(1)$ acts by multiplication with scalars of
unit modulus and ${\mbox{\rm SU}}(3)$ acts by conjugation.
It can be expressed as a polynomial in the invariants $i_1,i_2,\ldots,i_{16}$
listed in Appendix A.
As this expression has 203 terms, we shall give it separately in Appendix B.
We warn the reader that $P$ is rather large.
Denote by $P_I$ its restriction to the subspace $L_I$.
We run out of memory if we try to evaluate $P_I$ at an
arbitrary matrix in $L_I$ and expand it as a polynomial
in the 10 real variables (the real and imaginary parts of the complex
variables $x,y,z,u,v$). However, when we set the imaginary parts of $y$
and $z$ to 0, then we can expand $P_I$ and obtain 130571 terms. If we additionally
set the imaginary part of $x$ to 0, then the number of terms goes down to 50583.
Recall that the hypersurface $\Gamma\subset L(3)$ is defined by
the equation $P=0$.
\begin{proposition} \label{Hip-Pov}
The restriction $P_I$ admits the factorization:
\begin{equation} \label{Faktor}
P_I=P_1^2 P_2,
\end{equation}
where $P_1$ is defined by (\ref{Pol-P1}) and $P_2$ is a homogeneous polynomial
of degree 12 with integer coefficients. Thus
$\Gamma \cap L_I=\Gamma_1 \cup \Gamma_2$, where $\Gamma_2$
is the hypersurface defined by $P_2=0$. We also have
$\Theta\subseteq\Gamma$.
\end{proposition}
\begin{proof}
The first assertion can be verified by using a computer and suitable software,
e.g., {\sc Maple} \cite{MGH}.
The assertion that $\Gamma \cap L_I=\Gamma_1 \cup \Gamma_2$
is now obvious. Finally, the assertion $\Theta\subseteq\Gamma$ follows from
Corollary \ref{Gama-1} and the fact that ${\mathcal O}_A\subseteq\Gamma$ for
$A\in\Gamma$.
\end{proof}
We prove next that $P_1$, $P_2$ and $P$ are irreducible.
In fact we have a stronger result.
\begin{proposition} \label{AbsIred}
The real polynomials $P_1$, $P_2$ and $P$ defined above are absolutely
irreducible (i.e., irreducible over ${\bf C}$).
\end{proposition}
\begin{proof}
The absolute irreducibility of $P_1$ can be proved by using the
``absfact.lib'' library in {\sc Singular} \cite{GPSO5}.
To make this computation easier,
it suffices to check that after setting $y=2$, $z=1$,
and setting the imaginary parts of $x$ and $v$ to 0 and 1, respectively,
the resulting polynomial (having 69 terms) still has degree 6
and is absolutely irreducible.
The same method works for $P_2$. In this case we set $u=1$, $y=2$, $z=1$,
and we also set the imaginary parts of $x$ and $v$ to 0 and 1, respectively.
We obtain an absolutely irreducible polynomial of degree 12
(having 47 terms).
It is much harder to prove that $P$ is also absolutely irreducible.
(We were not successful in using the same method as above.)
Assume that $P$ has a nontrivial factorization $P=QR$ over ${\bf C}$.
We can assume that $Q$ is irreducible.
Since ${\mbox{\rm U}}(3)$ is connected and $P$ is ${\mbox{\rm U}}(3)$-invariant, both
$Q$ and $R$ must be invariant. By restricting these polynomials to
$L_I$ and by using (\ref{Faktor}), we obtain
that $P_I=Q_I R_I=P_1^2 P_2.$ It follows that $Q_I$
is equal to $P_1$ or $P_2$ (up to a scalar factor).
We are going to show that this leads to a contradiction.
One can easily verify that, for
$$ A= \left[ \begin {array}{ccc} 3+3\,i&5&0\\\noalign{\medskip}0&3-3\,i&5
\\\noalign{\medskip}5&0&-6\end {array} \right]\in L_I, $$
we have $P_1(A)=0$ and $P_2(A)=-22675690800$, and so
$A\in\Gamma_1\setminus\Gamma_2.$
On the other hand, the matrix
$$ B= \left[ \begin {array}{ccc} -1&\frac12\,\sqrt {222+6\,\sqrt {69}}&0
\\\noalign{\medskip}0&\frac12\,(1-\sqrt {69})&0\\\noalign{\medskip}\frac12\,
\sqrt {222-6\,\sqrt {69}}&0&\frac12(1+\sqrt {69})\end {array} \right]\in L_I $$
satisfies $P_1(B)=89424$ and $P_2(B)=0$, and so
$B\in\Gamma_2\setminus\Gamma_1.$
We obtain a contradiction by showing that $B\in{\mathcal O}_A$.
An explicit unitary matrix $X$ satisfying $AX=XB$ is given in Appendix C.
\end{proof}
We note that, in the above proof, $A$ is a regular point of $\Gamma_1$,
while $B$ is singular on $\Gamma_2$.
The subspace $L_I$ is $Z$-invariant, for the cyclic matrix
$$ Z=\left[ \begin{array}{ccc} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0
\end{array} \right]. $$
Indeed, if $A\in L_I$ is given by (\ref{Mat-A}) then
$$ ZAZ^{-1}=\left[ \begin{array}{ccc} w & y & 0 \\ 0 & u & z \\ x & 0 & v
\end{array} \right]\in L_I. $$
Assume now that $A\in L_I$ and that $P(A)\ne0$. We claim that the two points
$A,ZAZ^{-1}\in{\mathcal X}_A$ are not $T_3$-conjugate. Otherwise
$ZAZ^{-1}=DAD^{-1}$ for some $D\in T_3$. This implies that $u=v=w$, and
$u+v+w=0$ forces that $u=v=w=0$. Consequently, $P_1(A)=0$ which
contradicts our assumption. We conclude that the three points of the
$Z$-orbit $\{A,ZAZ^{-1},Z^{-1}AZ\}$ belong to three different
$T_3$-orbits in ${\mathcal X}_A$. Since $P_1$ is $Z$-invariant, we have
$$ P_1(A)=P_1(ZAZ^{-1})=P_1(Z^{-1}AZ). $$
It follows that $N_A$ is divisible by 3. As $N_A=N(A,M_I(3))$
by the remark \ref{IntN} and as we know that in this case
$N(A,M_I(3))$ is even, we conclude that $N_A$ is divisible by 6.
Numerical computations indicate that $N_A$ is either 6 or 18
and that ${\mathcal X}_A/T_3$ can be split into 6-tuples such that
$P_1$ is constant on the union of the $T_3$-orbits
belonging to the same 6-tuple. We have no explanation for this
phenomenon.
Let us look at an example. The matrices
$$ A= \left[ \begin{array}{ccc} 1+i & 0 & 0 \\ 0 & -1 & 0 \\
1 & 0 & -i \end{array} \right] \quad \text{and} \quad
B= \left[ \begin{array}{ccc} -i & 0 & 0 \\ 0 & -1 & 0 \\
1 & 0 & 1+i \end{array} \right] $$
belong to $L_I$. As $P_1(A)=P_1(B)=45$ and $P_2(A)=P_2(B)=0$,
we have $A,B\in\Gamma_2\setminus\Gamma_1.$
One can easily verify that $A$ and $B$ are unitarily similar and
are regular points of $\Gamma_2$.
Hence ${\mathcal X}_A$ contains the six $T_3$-orbits with representatives
$Z^kAZ^{-k},Z^kBZ^{-k}$, $k=0,1,2$. We propose $A$ and $B$ as
examples of generic points which belong to $\Gamma$, which
would imply that $\Theta\ne\Gamma$.
\section{Appendix A: Generators of ${\mbox{\rm U}}(1)\times{\mbox{\rm SU}}(3)$-invariants} \label{UxSU}
Consider the representation of the direct product ${\mbox{\rm U}}(1)\times{\mbox{\rm SU}}(3)$
on $L(3)$, where ${\mbox{\rm U}}(1)$ acts by multiplication with scalars of
unit modulus and ${\mbox{\rm SU}}(3)$ acts by conjugation.
The algebra of real polynomial invariants for this action is
a subalgebra of the corresponding algebra for the conjugation action
of ${\mbox{\rm U}}(3)$ on $L(3)$.
A minimal set of homogeneous generators of the first algebra
consists of 16 invariants $i_1,i_2,\ldots,i_{16}$ listed below.
This fact is neither proved nor used in this paper.
However, these generators are used in Sections \ref{UnivObl} and
\ref{CikObl} to construct some other invariants that we need.
The 16 generators are
\begin{eqnarray*}
&& i_1={\,\rm tr\,}(XY),\, i_2={\,\rm tr\,}(X^2 Y^2),\, i_3=\frac14 | {\,\rm tr\,}(X^2) |^2, \\
&& i_4={\,\rm tr\,}(XYX^2 Y^2),\, i_5=\frac19| {\,\rm tr\,}(X^3) |^2,\, i_6=| {\,\rm tr\,}(X^2 Y) |^2, \\
&& i_7=\frac16\, \Re \left( {\,\rm tr\,}(Y^2)( 3{\,\rm tr\,}^2(X^2 Y)+{\,\rm tr\,}(X^3){\,\rm tr\,}(XY^2) ) \right), \\
&& i_8=\frac16\, \Re \left( {\,\rm tr\,}(Y^2)( 3{\,\rm tr\,}^2(X^2 Y)-{\,\rm tr\,}(X^3){\,\rm tr\,}(XY^2) ) \right), \\
&& i_9=\frac12\, \Im \left( {\,\rm tr\,}(Y^2){\,\rm tr\,}^2(X^2 Y) \right), \\
&& i_{10}=\frac16\, \Im \left( {\,\rm tr\,}(X^2){\,\rm tr\,}^2(X^2 Y){\,\rm tr\,}(Y^3) \right), \\
&& i_{11}=\frac1{12}\, \Re( {\,\rm tr\,}^2(X^2){\,\rm tr\,}(Y^3){\,\rm tr\,}(XY^2) ), \\
&& i_{12}=\frac1{12}\, \Im( {\,\rm tr\,}^2(X^2){\,\rm tr\,}(Y^3){\,\rm tr\,}(XY^2) ), \\
&& i_{13}=\frac1{72}\, \Re( {\,\rm tr\,}^2(X^3){\,\rm tr\,}^3(Y^2) ), \\
&& i_{14}=\frac1{72}\, \Im( {\,\rm tr\,}^2(X^3){\,\rm tr\,}^3(Y^2) ), \\
&& i_{15}=\frac13\, \Im \left( {\,\rm tr\,}^3(X^2 Y){\,\rm tr\,}(Y^3) \right), \\
&& i_{16}=\frac1{144}\, \Re( {\,\rm tr\,}^4(X^2){\,\rm tr\,}^2(Y^3){\,\rm tr\,}^2(XY^2) ),
\end{eqnarray*}
where $X\in L(3)$ is an arbitrary matrix, $Y=X^*$ is its adjoint, and ${\,\rm tr\,}$
is the matrix trace function.
The symbols $\Re$ and $\Im$ stand for ``real part'' and ``imaginary part'',
respectively.
\section{Appendix B: The polynomial $P$ } \label{Polinom-P}
Here we construct the polynomial $P$ used in Section \ref{CikObl}.
It will be given as a polynomial in the invariants $i_k$ listed in Appendix A.
For convenience, we collect $P$ with respect to the invariants $i_3$ and $i_6$
\begin{equation} \label{Pol-P}
P = \sum_{k,l} p_{kl}i_3^k i_6^l.
\end{equation}
The nonzero $p_{kl}$ are as follows:
\begin{eqnarray*}
p_{00} &=&
-6\, \left( 126\,{i_{{2}}}^{2}{i_{{1}}}^{4}-26\,i_{{2}}{i_{{1}}}^{6}+
336\,i_{{7}}i_{{2}}{i_{{1}}}^{2}-270\,{i_{{2}}}^{3}{i_{{1}}}^{2}+1536
\,{i_{{7}}}^{2} \right. \\
&& \left. +216\,{i_{{2}}}^{4}-11\,i_{{7}}{i_{{1}}}^{4}+2\,{i_{{1}
}}^{8}-1152\,i_{{7}}{i_{{2}}}^{2} \right) \left( i_{{8}}+i_{{7}}
\right) \\
p_{01} &=&
18432\,i_{{11}}i_{{7}}+186\,i_{{8}}{i_{{1}}}^{5}+2016\,i_{{11}}{i_{{1}
}}^{2}i_{{2}}+2160\,{i_{{2}}}^{2}i_{{1}}i_{{7}}-66\,i_{{11}}{i_{{1}}}^{4} \\
&& +2160\,i_{{8}}i_{{1}}{i_{{2}}}^{2}-1278\,i_{{8}}{i_{{1}}}^{3}i_{{2}
}+186\,{i_{{1}}}^{5}i_{{7}}-6912\,i_{{11}}{i_{{2}}}^{2}+3456\,i_{{1}}{
i_{{7}}}^{2} \\
&& +3456\,i_{{8}}i_{{1}}i_{{7}}-1278\,{i_{{1}}}^{3}i_{{2}}i_{{7}} \\
p_{02} &=&
3888\,i_{{8}}i_{{2}}+297\,{i_{{1}}}^{2}{i_{{2}}}^{2}-4608\,i_{{13}}-
324\,{i_{{2}}}^{3}-90\,{i_{{1}}}^{4}i_{{2}}+3888\,i_{{2}}i_{{7}} \\
&& +9\,{i_{{1}}}^{6}-1242\,{i_{{1}}}^{2}i_{{7}}-1242\,i_{{8}}{i_{{1}}}^{2}-3456
\,i_{{11}}i_{{1}} \\
p_{03} &=& 18\,i_{{1}} \left( -7\,{i_{{1}}}^{2}+27\,i_{{2}} \right) \\
p_{04} &=& 729 \\
p_{10} &=& 6912\,i_{{7}}i_{{4}}i_{{1}}i_{{2}}+2016\,i_{{2}}{i_{{1}}}^{2}i_{{13}}-
33\,i_{{4}}i_{{2}}{i_{{1}}}^{5}+1008\,i_{{4}}{i_{{2}}}^{2}{i_{{1}}}^{3} \\
&& -3456\,i_{{4}}i_{{1}}{i_{{2}}}^{3}-1008\,{i_{{4}}}^{2}i_{{2}}{i_{{1}}
}^{2}+31104\,{i_{{5}}}^{2}{i_{{2}}}^{2}+33\,i_{{5}}{i_{{1}}}^{7} \\
&& +297\,{i_{{5}}}^{2}{i_{{1}}}^{4}+9792\,{i_{{2}}}^{3}i_{{8}}-2304\,{i_{{4}}}^
{2}i_{{8}}-251\,{i_{{1}}}^{6}i_{{8}}-20736\,{i_{{5}}}^{2}i_{{8}} \\
&& -3024\,i_{{5}}i_{{4}}i_{{2}}{i_{{1}}}^{2}+25920\,i_{{7}}{i_{{2}}}^{3}
-55296\,{i_{{7}}}^{2}i_{{2}}-6912\,i_{{7}}{i_{{4}}}^{2} \\
&& +17760\,{i_{{7}}}^{2}
{i_{{1}}}^{2}+3456\,{i_{{4}}}^{2}{i_{{2}}}^{2}+9504\,i_{{5}}{i_{{2}}}^
{2}{i_{{1}}}^{3}-6912\,i_{{7}}i_{{5}}{i_{{1}}}^{3} \\
&& -66\,{i_{{1}}}^{4}i_{{13}}+4608\,i_{{16}}-24012\,i_{{7}}{i_{{2}}}^{2}{i_{{1}}}^{2}
-317\,i_{{7}}{i_{{1}}}^{6}-62208\,i_{{7}}{i_{{5}}}^{2} \\
&& +2304\,i_{{4}}i_{{2}}i_{
{1}}i_{{8}}-1206\,i_{{5}}i_{{2}}{i_{{1}}}^{5}+3012\,i_{{2}}{i_{{1}}}^{
4}i_{{8}}+5226\,i_{{7}}i_{{2}}{i_{{1}}}^{4} \\
&& -20736\,i_{{7}}i_{{5}}i_{{4
}}+8544\,i_{{7}}{i_{{1}}}^{2}i_{{8}}-6912\,{i_{{2}}}^{2}i_{{13}}+9216
\,i_{{7}}i_{{13}}+66\,{i_{{1}}}^{5}i_{{11}} \\
&& -9072\,{i_{{5}}}^{2}i_{{2}}
{i_{{1}}}^{2}-2304\,i_{{5}}{i_{{1}}}^{3}i_{{8}}-27648\,i_{{7}}i_{{2}}i
_{{8}}-6912\,i_{{5}}i_{{4}}i_{{8}} \\
&& -11052\,{i_{{2}}}^{2}{i_{{1}}}^{2}i_
{{8}}+13824\,i_{{5}}i_{{2}}i_{{1}}i_{{8}}+10368\,i_{{5}}i_{{4}}{i_{{2}
}}^{2}+1632\,{i_{{2}}}^{2}{i_{{1}}}^{6} \\
&& -5151\,{i_{{2}}}^{3}{i_{{1}}}^{
4}+7200\,{i_{{2}}}^{4}{i_{{1}}}^{2}-256\,i_{{2}}{i_{{1}}}^{8}+16\,{i_{
{1}}}^{10}-1728\,{i_{{2}}}^{5} \\
&& -2016\,i_{{2}}{i_{{1}}}^{3}i_{{11}}+
41472\,i_{{7}}i_{{5}}i_{{1}}i_{{2}}-18432\,i_{{7}}i_{{1}}i_{{11}}+6912
\,{i_{{2}}}^{2}i_{{1}}i_{{11}} \\
&& -20736\,i_{{5}}i_{{1}}{i_{{2}}}^{3}+99\,
i_{{5}}i_{{4}}{i_{{1}}}^{4}+33\,{i_{{4}}}^{2}{i_{{1}}}^{4} \\
p_{11} &=&
-4320\,i_{{5}}i_{{2}}{i_{{1}}}^{2}+6\,{i_{{1}}}^{3}i_{{8}}+6912\,i_{{7
}}i_{{4}}+1728\,{i_{{4}}}^{2}i_{{1}}-3450\,i_{{7}}{i_{{1}}}^{3} \\
&& -5088\,{i_{{1}}}^{2}i_{{11}}+14688\,i_{{1}}{i_{{2}}}^{3}+2721\,i_{{2}}{i_{{1}
}}^{5}+2304\,i_{{4}}i_{{8}}-4320\,i_{{2}}i_{{1}}i_{{8}} \\
&& +15552\,{i_{{5}
}}^{2}i_{{1}}+1152\,i_{{1}}i_{{13}}+11520\,i_{{5}}i_{{8}}-3456\,i_{{4}
}{i_{{2}}}^{2}+1440\,i_{{7}}i_{{1}}i_{{2}} \\
&& -20736\,i_{{5}}{i_{{2}}}^{2}
-272\,{i_{{1}}}^{7}+1530\,i_{{5}}{i_{{1}}}^{4}-33\,i_{{4}}{i_{{1}}}^{4
}+32256\,i_{{2}}i_{{11}} \\
&& +5184\,i_{{5}}i_{{4}}i_{{1}}-9792\,{i_{{2}}}^{
2}{i_{{1}}}^{3}-720\,i_{{4}}i_{{2}}{i_{{1}}}^{2}+39168\,i_{{7}}i_{{5}} \\
p_{12} &=&
-1728\,i_{{4}}i_{{1}}+2130\,{i_{{1}}}^{4}-10170\,i_{{2}}{i_{{1}}}^{2}+
15228\,{i_{{2}}}^{2}+1296\,i_{{8}} \\
&& +1296\,i_{{7}}-10368\,i_{{5}}i_{{1}} \\
p_{13} &=& -3726\,i_{{1}}
\end{eqnarray*}
\begin{eqnarray*}
p_{20} &=&
-4272\,i_{{4}}i_{{2}}{i_{{1}}}^{3}+16128\,i_{{4}}i_{{1}}{i_{{2}}}^{2}-
16128\,{i_{{4}}}^{2}i_{{2}}+12816\,i_{{5}}i_{{4}}{i_{{1}}}^{2} \\
&& -48384\,i_{{5}}i_{{4}}i_{{2}}+4272\,{i_{{4}}}^{2}{i_{{1}}}^{2}-27360\,{i_{{2}}
}^{2}i_{{8}}+23808\,i_{{7}}i_{{8}} \\
&& +80844\,i_{{7}}i_{{2}}{i_{{1}}}^{2}+
69888\,{i_{{7}}}^{2}-47808\,i_{{5}}i_{{2}}{i_{{1}}}^{3}+117504\,i_{{5}
}i_{{1}}{i_{{2}}}^{2} \\
&& -39168\,i_{{7}}i_{{5}}i_{{1}}+19500\,i_{{2}}{i_{{
1}}}^{2}i_{{8}}-8544\,{i_{{1}}}^{2}i_{{13}}+32256\,i_{{2}}i_{{13}} \\
&& +8544\,{i_{{1}}}^{3}i_{{11}}+384\,{i_{{1}}}^{8}+20160\,{i_{{2}}}^{4}-
128736\,i_{{7}}{i_{{2}}}^{2}-11607\,i_{{7}}{i_{{1}}}^{4} \\
&& +4470\,i_{{5}}{i_{{1}}}^{5}-145152\,{i_{{5}}}^{2}i_{{2}}
+38448\,{i_{{5}}}^{2}{i_{{1}}}^{2}-2799\,{i_{{1}}}^{4}i_{{8}} \\
&& -11520\,i_{{5}}i_{{1}}i_{{8}}-50016\,{i_{{2}}}^{3}{i_{{1}}}^{2}
-5279\,i_{{2}}{i_{{1}}}^{6}+26283\,{i_{{2}}}^{2}{i_{{1}}}^{4} \\
&& -32256\,i_{{2}}i_{{1}}i_{{11}} \\
p_{21} &=&
-22224\,i_{{7}}i_{{1}}-4272\,i_{{4}}{i_{{1}}}^{2}-37632\,i_{{11}}+
16128\,i_{{4}}i_{{2}}+96768\,i_{{5}}i_{{2}} \\
&& -2112\,{i_{{1}}}^{5}+18192\,i_{{2}}{i_{{1}}}^{3}-15264\,i_{{5}}{i_{{1}}}^{2}
-8400\,i_{{1}}i_{{8}}-40608\,i_{{1}}{i_{{2}}}^{2} \\
p_{22} &=& -10476\,i_{{2}}+10401\,{i_{{1}}}^{2} \\
p_{30} &=& 44448\,i_{{5}}{i_{{1}}}^{3}+31296\,i_{{8}}i_{{2}}
+4415\,{i_{{1}}}^{6}-76308\,{i_{{1}}}^{2}i_{{7}}-18816\,i_{{4}}i_{{1}}i_{{2}} \\
&& +169344\,{i_{{5}}}^{2}+56448\,i_{{5}}i_{{4}}
+128496\,{i_{{1}}}^{2}{i_{{2}}}^{2}+37632\,i_{{11}}i_{{1}} \\
&& -209664\,i_{{5}}i_{{1}}i_{{2}}+18816\,{i_{{4}}}^{2}-9108\,i_{{8}}{i_{{1}}}^{2}
+236352\,i_{{2}}i_{{7}} \\
&& -37632\,i_{{13}}-90624\,{i_{{2}}}^{3}-43032\,{i_{{1}}}^{4}i_{{2}} \\
p_{31} &=& -117504\,i_{{5}}+480\,i_{{2}}i_{{1}}+3312\,{i_{{1}}}^{3}
-18816\,i_{{4}} \\
p_{32} &=& -5196 \\
p_{40} &=& -146816\,i_{{2}}{i_{{1}}}^{2}-10384\,i_{{8}}
+112896\,i_{{5}}i_{{1}}+23948\,{i_{{1}}}^{4} \\
&& +195840\,{i_{{2}}}^{2}-142480\,i_{{7}} \\
p_{41} &=& 33632\,i_{{1}} \\
p_{50} &=& -202048\,i_{{2}}+65232\,{i_{{1}}}^{2} \\
p_{60} &=& 78400
\end{eqnarray*}
\section{Appendix C: The unitary matrix $X$}
The entries of the unitary matrix $X=[x_{ij}]$, used in the proof of
Proposition \ref{AbsIred}, are given by
\begin{eqnarray*}
x_{11} &=& \frac { 2 \left( \sqrt {69}-7 \right)
-i \left( 3+\sqrt {69} \right) }{ 6\, \sqrt {37-\sqrt {69}}}, \\
x_{12} &=& \frac {1}{12\,\sqrt {13}}(1+3i)(i\sqrt {6}-\sqrt {46}), \\
x_{13} &=& \frac {1}{12}\,(\sqrt {46}+\sqrt {6}), \\
x_{21} &=& \frac {(3-i)\sqrt {69} +34+27i}{15\,\sqrt {37-\sqrt {69}} }, \\
x_{22} &=& \frac {4}{15\,\sqrt {13}} ( \sqrt {46}-\sqrt {6} )
+\frac {i}{60\,\sqrt {13}}\,(23\,\sqrt {6}-3\,\sqrt {46}), \\
x_{23} &=& \frac {1}{60}\,(3-i)(3\,\sqrt {6}-i\,\sqrt {46}), \\
x_{31} &=& \frac {2(2-\sqrt{69})-i(3+\sqrt{69})} {6\,\sqrt{37-\sqrt{69}} }, \\
x_{32} &=& \frac {4}{15\,\sqrt{13}}\,(\sqrt {6}+\sqrt {46})
+\frac{i}{60\,\sqrt{13}}\,(23\,\sqrt {6}+3\,\sqrt {46}), \\
x_{33} &=& \frac{1}{12}(\sqrt {46}-\sqrt {6}).
\end{eqnarray*}
|
1,116,691,499,221 | arxiv | \section{Introduction}
Let $B, n \in \mathbb{N}_{>1}$ be such that
\begin{eqnarray}
\label{nosplit}
\varphi^{*}(B) := \varphi({\rm rad } \ (B)) \quad \hbox{ and } \quad (n,
\varphi^*(B)) = 1.
\end{eqnarray}
Here ${\rm rad } \ (B)$ is the radical of $B$ and the condition implies that
$B$ has no prime factors $t \equiv 1 \bmod n$. In particular, none of
its prime factors splits completely in the \nth{n} cyclotomic field.
More generally, for a fixed $B \in \mathbb{N}_{> 1}$ we let
\begin{eqnarray}
\label{expos}
\id{N}(B) = \{ n \in \mathbb{N}_{ > 1} \ | \ \exists \ k > 0 \hbox{ such that } n | {\varphi^*(B)}^k \}.
\end{eqnarray}
If $p$ is an odd prime, we shall denote by CF the combined condition
requiring that
\begin{enumerate}[I]
\item \label{i} The Vandiver Conjecture holds for $p$, so the class
number $h_p^+$ of the maximal real subfield of the cyclotomic field
$\mathbb{Q}[\zeta_p]$ is not divisible by $p$.
\item \label{ii} The index of irregularity of $p$ is small, namely $i_r(p)
< \sqrt{p} -1$, so there are $i_r(p)$ odd integers $k < p$ such that
the Bernoulli number $B_k \equiv 0 \bmod p$.
\end{enumerate}
The second condition was discovered by Eichler, as a sufficient
condition for FLT1 to be true. It is known from recent computations of
Buhler and Harvey \cite{BH} that the condition CF is satisfied by
primes up to $163 \cdot 10^6$.
We consider the binary Thue equation
\begin{eqnarray}
\label{bin}
X^n - 1 & = & B \cdot Z^n,
\end{eqnarray}
where solutions with $Z \in \{ -1, 0, 1\}$ are considered to be
trivial. The assertion that equation \rf{bin} has finitely many solutions
other than the trivial ones is a special case of the general Pillai conjecture
(Conjecture 13.17 of \cite{Bilu}). This equation is encountered as a
particular case of binary Thue equations of the type
\begin{eqnarray}
\label{mike}
a X^n - b Y^n = c,
\end{eqnarray}
see \cite{BGMP}. In a seminal paper \cite{Be}, Michael Bennett proves
that in the case of $c = \pm 1$ there is at most one solution for
fixed $(a, b; n)$ and deduces that the parametric family $(a+1, a; n)$
has the only solution $(1,1)$ for all $n$. The equation \rf{bin}
inserts naturally in the family of equations \rf{mike}, with $a = c =
\pm 1$.
A conjecture related directly to \rf{bin} states that
\begin{conjecture}
\label{bt}
Under \rf{nosplit}, Equation \rf{bin} has no other non-trivial solution than $(X, Y; B, n) =
(18, 7; 17, 3)$.
\end{conjecture}
Current results on \rf{bin} are restricted to values of $B$ which are
built up from small primes $p \leq 13$ \cite{G}. If expecting that the equation
has no solutions, -- possibly with the exception of some isolated
examples -- it is natural to consider the case when the exponent $n$
is a prime. Of course, the existence of solutions $(X, Z)$ for
composite $n$ imply the existence of some solutions with $n$ prime, by
raising $X, Z$ to a power.
The main contribution of this paper will be to relate \rf{bin}
in the case when $n$ is a prime and \rf{nosplit} holds, to the
diagonal Nagell -- Ljunggren equation,
\begin{eqnarray}
\label{dn}
\frac{X^n-1}{X-1} = n^e Y^n, \quad e = \begin{cases} 0 & \hbox{ if } X
\not \equiv 1 \bmod n, \\
1 & \hbox{ otherwise.} \end{cases}
\end{eqnarray}
This way, we can apply results from \cite{Mi1} and prove the following:
\begin{theorem}
\label{tbin}
Let $n$ be a prime and $B > 1$ an integer with $(\varphi^{*}(B), n ) =
1$. Suppose that the equation \rf{bin} has a non trivial integer
solution different from $n = 3$ and $(X, Z; B) = (18, 7; 17)$. Let $X
\equiv u \bmod n , 0 \leq u < n$ and $e = 1$ if $u = 1$ and $e = 0$
otherwise. Then:
\begin{enumerate}[A]
\item $n > 163 \cdot 10^6$. \label{A}
\item $X - 1 = \pm B/n^e$ and $B < n^n$. \label{B}
\item If $u \not \in \{ -1, 0, 1\}$, then condition CF \rf{ii} fails
for $n$ and \label{C}
\begin{eqnarray*}
\begin{array}{r c r r l}
2^{n-1} & \equiv & 3^{n-1} \equiv 1 & \bmod \ n^2, & \hbox{ and } \\
r^{n-1} & \equiv & 1 & \bmod \ n^2 & \hbox{ for all $ r | X(X^2-1)$}.
\end{array}
\end{eqnarray*}
If $u \in \{ -1, 0, 1\}$, then Condition CF \rf{i} fails for $n$.
\end{enumerate}
\end{theorem}
The particular solution $n = 3$ and $(X, Z; B) = (18, 7; 17)$ is
reminiscent of a solution of the diagonal Nagell solution; it is
commonly accepted that the existence of non trivial solutions tends to
render Diophantine equations more difficult to solve. Based on Theorem
\ref{tbin} we prove nevertheless the following
\begin{theorem}
\label{genexpo}
If equation \rf{bin} has a solution for a fixed $B$ verifying the
conditions \rf{nosplit}, then either $n \in \id{N}(B)$ or there is a
prime $p$ coprime to $\varphi^*(B)$ and a $m \in \id{N}(B)$ such that
$n = p \cdot m$. Moreover $X^m, Y^m$ are a solution of \rf{bin} for
the prime exponent $p$ and thus verify the conditions in Theorem
\ref{tbin}.
\end{theorem}
This is a strong improvement of the currently known results.
\begin{remark}
\label{remark}
Theorem \ref{tbin} uses criteria from the diagonal case of the
Nagell-Ljunggren equation, the relation being established by point
\rf{B} of the theorem. The criteria were proved in \cite{Mi1} and
are in part reminiscent from classical cyclotomic results on
Fermat's Last Theorem. Thus, the criteria for the First Case, which
are ennounced in point \rf{C} are the Eichler criterion CF \rf{ii}
and the criteria of Wieferich and Furtw\"angler (cf. Theorem 2 in \cite{Mi1}). For the Second Case
of Diagonal Nagell-Ljunggren, in point \rf{C}, it was possible to
restrict the two conditions proved by Kummer for the FLTII to the
single condition CF \rf{i}, namely Vandiver's conjecture (cf. Theorem 4 in \cite{Mi1}). This is a
consequence of the fact that unlike FLT, Nagell-Ljunggren is a
binary equation, a fact which allowed also to prove upper bounds for
the solutions, which are given in Theorem \ref{ubounds} below. The
fact that the Nagell-Ljunggren equation is not homogenous in $X$
makes it difficult to prove lower bounds, thus leaving a gap on the
way to a complete proof of Conjecture \ref{bt}.
The proofs in \cite{Mi1} used methods that generalize the ones that
helped proving the Catalan Conjecture \cite{Mi2}. A variant of these
methods will be applied for proving point \rf{B}. We
gathered the occasion of writing this paper to give in the
Appendix an extensive exposition of the computations on which the singular case
in the proof the respective estimate from \cite{Mi1} relies: some colleagues had
pointed out that they could not verify the computation on base of the arguments
in \cite{Mi1}, so this difficulty should be dealt with in the Appendix.
\end{remark}
The plan of the paper is as follows: in Section 2 we establish the
connection between equations \rf{bin} and \rf{tbin}, review some basic
properties of Stickelberger ideals and prove auxiliary technical
lemmata concerning coefficients of binomial series development.
With these prerequisites, we complete the proof of Theorem \ref{tbin}
in Section 3. Given the reduction to the Nagell-Ljunggren Diagonal
Case, the proof focuses on point \rf{B} of Theorem \ref{tbin}. In Section
4 we drop the condition that $n$ be a prime and use the proven facts
in order to deduce the results on \rf{bin} for arbitrary exponents $n$
which are stated in Theorem \ref{genexpo}. Finally, the Appendix
provides the details for an estimate used in \cite{Mi1}, as mentioned
in Remark \ref{remark}.
\section{Preliminary results}
The proof of Theorem \ref{tbin} emerges by relating the equation \rf{bin}
to the Diagonal Case of the Nagell -- Ljunggren conjecture. In this section
we shall recall this conjecture and several technical tools used for
reducing one conjecture to the other. The reduction is performed in the
next section.
\subsection{Link of \rf{bin} with the diagonal Nagell -- Ljunggren
equation}
We note that $\delta = \left(\frac{X^n-1}{X-1}, X-1\right)$ divides $ n$
and $\delta = n$ exactly when $X \equiv 1 \bmod n$. Indeed,
from the expansion
\[ \frac{X^n-1}{X-1} = \frac{((X-1)+1)^n - 1}{X - 1} = n + k (X-1), \]
with $k \in \mathbb{Z}$, one deduces the claim $\delta \big \vert n$. If $ u
\neq 1$, then $\delta=n$ and thus $n | (X - 1)$ must hold. Conversely,
inserting $X \equiv 1 \bmod n$ in the previous expression shows that
in this case $\delta = n$.
We first show that any solution of \rf{bin} leads to a solution of
\rf{dn}. For this, let $\prod_{i=1}^k
p_i$ be the radical of $\frac{X^n-1}{n^e(X-1)}$. Obviously, ${\rm rad } \ (\frac{X^n-1}{n^e(X-1)}) \ | \ {\rm rad } \ (X^n -
1)$. Let $\zeta \in \mathbb{C}$ be a primitive \nth{n} root of unity. Then the
numbers $\alpha_c = \frac{X - \zeta^c}{(1-\zeta^c)^e} \in \mathbb{Z}[\zeta]$
by definition of $e$, and $(\alpha_c, n) = 1$. Since for distinct $c, d
\not \equiv 0 \bmod n$ we have $( 1-\zeta^d)^e \cdot \alpha_d - (
1-\zeta^c)^e \cdot \alpha_c = \zeta^c - \zeta^d$, it follows that
$(\alpha_c, \alpha_d) \ | \ (1-\zeta)$ and in view of $(\alpha_c, n) =
1$, it follows that the $\alpha_c$ are coprime.
Let $F = \prod_{c=1}^{n-1} \alpha_c = \frac{X^n-1}{n^e(X-1)}$ and $q \ | \ F$ be a rational prime. In the
field $\mathbb{Q}[ \zeta ]$, it splits completely in the prime ideals $\eu{Q}_c
= (q, \alpha_c), \ c = 1, 2, \ldots, n-1$: these ideals are coprime,
as a consequence of the coprimality of the $\alpha_c$. Therefore $q
\equiv 1 \bmod n$ and it follows from \rf{nosplit} that $(q, B) = 1$,
so $q | Z$. Furthermore, \rf{bin} implies that there exists $j_q >0$
such that $q^{j_q n} || Z^n$ and thus $q^{j_q n} || F$. This holds
for all primes $q \ | \ {\rm rad } \ (F)$. It follows that \rf{dn} is verified
for $Y = \prod_{q | F} q^{j_q}$ and $Y \ | \ Z$. We have thus proved
that if $(X, Z)$ is a solution of \rf{bin} for the prime $n$, then
there exists $C \in \mathbb{Z}$ such that $Z = C \cdot Y$ with $Y$ as above,
and:
\begin{eqnarray}
\label{x}
\frac{X^n-1}{n^e (X-1)} & = & Y^n \quad \hbox{ and } \quad \\
\label{y} X-1 & = & B \cdot C^n / n^e.
\end{eqnarray}
We shall write from now on $D = X-1$.
From the above, we conclude that any integer solution of \rf{bin}
induces one of \rf{dn}. Conversely, if $(X, Y)$ is a solution of
\rf{dn}, then $\left(X, Y; n^e(X-1)\right)$ is a solution of
\rf{bin}. For instance, the particular solution $(X, Y; B) = (18, 7;
17)$ of \rf{bin} stems from
\begin{eqnarray*}
\label{part}
\frac{18^3-1}{18-1} = 7^3,
\end{eqnarray*}
which is supposed to be the only non trivial solution of \rf{dn}.
\begin{remark}
\label{minus}
Note that if $(X, Z)$ verify \rf{bin}, then $(-X, Z)$ is a solution of
$\frac{X^n + 1}{X + 1} = B Z^n$, so the results apply also to the
equation:
\begin{eqnarray*}
\label{tbins}
X^n + 1 = B Z^n.
\end{eqnarray*}
\end{remark}
\subsection{Bounds to the solutions of Equation \rf{dn}}
We shall use the following Theorem from \cite{Mi1}:
\begin{theorem}
\label{ubounds}
Suppose that $X, Y$ are integers verifying \rf{dn} with $n \geq 17$
being a prime. Let $u = (X \bmod n)$. Then there is an $E \in \mathbb{R}_+$
such that $| X | < E$. The values of $E$ in the various cases of the
equation are the following:
\begin{eqnarray}
\label{Bvals}
E = \begin{cases} \ 4 \cdot \left(\frac{n-3}{2}\right)^{\frac{n+2}{2}}
& \hbox{if $u \not \in \{-1, 0, 1\}$} \\ \ (4 n)^{\frac{n-1}{2}} &
\hbox{if $ u = 0$ }, \\ \ 4 \cdot
\left(n-2\right)^{n} & \hbox{otherwise}.
\end{cases}
\end{eqnarray}
\end{theorem}
By comparing the bounds \rf{Bvals} with \rf{y}, it follows that $| C |
< 2n-1$. In particular, the primes dividing $C$ do not split
completely in $\mathbb{Q}[\zeta_n]$ -- since a prime splitting in this field
has the form $r = 2 k n + 1 > 2n$.
\begin{remark}
\label{4}
Note that $| C | < 2 n-1$ implies a fortiori that for all primes $r |
C$, $r^2 \not \equiv 1 \bmod n$. If $d(r) \leq \mbox{Gal}(\mathbb{Q}[ \zeta ]/\mathbb{Q})$
is the decomposition group of the unramified prime $r$, it follows
that $| d(r) | \geq 3$; moreover, either $d(r)$ contains a subcycle
$d' \subset d(r)$ of odd order $| d' | \geq 3$ or it is a cyclic $2$-group
with at least $4$ elements.
\end{remark}
\subsection{A combinatorial lemma}
\begin{lemma}
\label{sm}
Let $p$ be an odd prime, $k \in \mathbb{N}$ with $1 < k < \log_2(p)$ and $P =
\{ 1, 2, \ldots, p-1\}$. If $S = \{ a_1, a_2, \ldots, a_k \} \subset
P$ be a set of numbers coprime to $p$ and such that $a_i \not \equiv
\pm a_j \bmod p$ for $i \neq j$. We set the bound $A = 2 \lceil
p^{1/k} \rceil$; then there are $k$ numbers $b_i \in \mathbb{Z}, \ i = 1, 2,
\ldots, k$, not all zero, with $0 \leq | b_i | \leq A$ and such that
\[ \sum_{i=1}^k a_i b_i \equiv 0 \bmod p. \]
For $k = 2$, we can choose the $b_i$ such that the additional condition
\[ \sum_{i=1}^2 b_i/a_i \not \equiv 0 \bmod p. \]
holds.
\end{lemma}
\begin{proof}
Let $T = \{ 1, 2, \ldots, A\} \subset P$. Consider the functional
$f : T^k \rightarrow \ZM{p}$ given by
\[ f(\vec{t}) \equiv \sum_{i=1}^k t_i a_i \ \ \bmod p, \quad \hbox{
with} \quad \vec{t} = (t_1, t_2, \ldots, t_k) \in T^k.\] Since $|
T^k | > p$, by the pigeon hole principle there are two vectors
$\vec{t} \neq \vec{t'}$ such that $f(\vec{t}) \equiv f(\vec{t'}) \bmod
p$. Let $b_i = t_i - t'_i$; by construction, $0 \leq | b_i | \leq A$
and not all $b_i$ are zero, since $\vec{t} \neq \vec{t'}$. The choice
of these vectors implies $\sum_{i=1}^k a_i b_i \equiv 0 \bmod p$, as
claimed.
We now turn to the second claim. If the claim were false, then
\[ a_1 b_1 + a_2 b_2 = 0 \hbox{ and } b_1/a_1 + b_2 / a_2 = 0, \]
a homogenous linear system $S$ with determinant
$\det(S) = \frac{a_1^2-a_2^2}{a_1 a_2}$, which is non vanishing
under the premise of the lemma. This would imply that
the solution $b_1, b_2$ is trivial, in contradiction with our construction.
This completes the proof.
\end{proof}
\subsection{Some notation}
We assume that $n$ is prime and let $\zeta$ be a primitive \nth{n}
root of unity, $\mathbb{K} = \mathbb{Q}(\zeta)$ the \nth{n} cyclotomic field and $G =
\mbox{Gal}(\mathbb{K}/\mathbb{Q})$ the Galois group. The automorphisms $\sigma_a \in G$ are
given by $\zeta \mapsto \zeta^a, \ a = 1, 2, \ldots, n-1$; complex
conjugation is denoted by $\jmath \in \mathbb{Z}[ G ]$. In the ring of
integers $\mathbb{Z} [\zeta]$, one has finite \textit{$\lambda$-adic} expansions:
for any $\alpha \in \mathbb{Z}[ \zeta ]$ and some $N > 0$ there are
$a_j \in \{ -(p-1)/2, \cdots, 0, 1, \cdots, (p-3)/2 \}, j = 0, 1, \ldots N$ such that:
\begin{equation}
\label{lambdaadic}
\alpha = \sum_{j=0}^{N} a_j (1-\zeta)^{j}.
\end{equation}
We shall use the algebraic $O( \cdot )$-notation, to suggest the
remainder of a power series. This occurs explicitly in the following
four contexts
\begin{itemize}
\item[ (i) ] In a $\lambda$-adic development of the type
\rf{lambdaadic}, we write $\alpha = x+ O(\lambda^m)$ to mean that
there is some $y \in \mathbb{Z}[ \zeta ]$ such that $\alpha - x = \lambda^m
y$. Since $(n) = (\lambda^{p-1})$, powers of $n$ can occur as well
as powers of $\lambda$ in this notation.
\item[ (ii) ] We also use formal power series, often written $f =
f(D) \in \mathbb{K}[[ D ]]$. For $f = \sum_{k=0}^{\infty} f_k D^k$ with
partial sum $S_m(f) = \sum_{k=0}^{m} f_k D^k$ we may also use the
$O( \cdot )$-notation and denote the remainder by $f(D) = S_m(D) +
O( D^{m+1} )$.
\item[ (iii) ] Suppose that $D$ is an integer and the formal power
series converges in the completion $\mathbb{K}_{\eu{P}}$ at some prime
$\eu{P} \subset \id{O}(\mathbb{K})$ dividing $D$. Suppose also that in this
case all coefficients of $f$ be integral: then the remainder $f(D) -
S_m(D)$ is by definition divisible by $\eu{P}^{m+1}$, so $O( D^{m+1}
)$ means in this context that the remainder is divisible by
$\eu{P}^{m+1}$.
\item[ (iv) ] If $f(D)$ converges at all the prime ideals dividing
some integer $a | D $, then $O( D^{m+1} )$ will denote a number
divisible by $a^{m+1}$. In this paper we shall use this fact in the
context in which $a = p$ is an integer prime dividing $D$ and such
that $f(D)$ converges at all prime ideals of $\mathbb{K}$ above $p$.
\end{itemize}
\subsection{Auxiliary facts on the Stickelberger module}
\label{SecStick}
The following results are deduced in \cite{Mi1}, Section 4, but see also
\cite{Mi2}, \S 2.1-2.3 and \S 4.1. The results shall only be
mentioned here without proof.
The Stickelberger module is $I = (\vartheta \cdot \mathbb{Z}[ G ])
\cap \mathbb{Z}[ G ]$, where $\vartheta = \frac{1}{n} \sum_{c=1}^{n-1} c \cdot
\sigma_c^{-1}$ is the Stickelberger element. For $\theta = \sum_c
n_c \sigma_c \in I$ we have the relation $\theta + \jmath \theta =
\varsigma(\theta) \cdot \mbox{\bf N}$, where $\varsigma(\theta) \in \mathbb{Z}$ is
called the \textit{relative weight} of $\theta$. The augmentation of
$\theta$ is then
\[ | \theta | = \sum_c n_c = \varsigma(\theta) \cdot \frac{n-1}{2}.\]
The Fueter elements are
\[ \psi_k = (1 + \sigma_{k} - \sigma_{k+1}) \cdot \vartheta =
\sum_{c=1}^{n-1} \left( \left[ \frac{(k+1)c}{n} \right] - \left[
\frac{kc}{n} \right] \right) \cdot \sigma_c^{-1}, \quad 1 \leq k
\leq (n-1)/2.\] Together with the norm, they generate $I$ as a $\mathbb{Z}$ -
module (of rank $(n+1)/2$) and $\varsigma(\psi_k) = 1$ for all $k$.
The Fuchsian elements are \[ \Theta_k = (k - \sigma_k) \cdot \vartheta
= \sum_{c=1}^{n-1} \left[ \frac{kc}{n} \right] \cdot \sigma_c^{-1},
\quad 2 \leq k \leq n.\] They also generate $I$ as a $\mathbb{Z}$ -
module. Note that $\Theta_n$ is the norm, and that we have the
following relationship between the Fueter and the Fuchsian elements:
\begin{equation*}
\psi_1 = \Theta_2 \text{ and }
\psi_k = \Theta_{k+1} - \Theta_k,\ k \geq 2
\end{equation*}
An element $\Theta = \sum_c n_c \sigma_c$ is \textit{positive}
if $n_c \geq 0$ for all $c \in \{ 1, 2, \ldots, p-1\}$. We write
$I^+ \subset I$ for the set of all positive elements. They
form a multiplicative and an additive semigroup.
The Fermat quotient map $I \rightarrow \ZM{n}$, given by $$\varphi :
\theta = \sum_{c=1}^{n-1} n_c \sigma_c \mapsto \sum_{c=1}^{n-1}
c n_c \bmod n,$$ is a linear functional, with kernel $I_f=\{ \theta
\in I : \zeta^{\theta} = 1\}$ (the \textit{Fermat module}), and enjoys
the properties:
\begin{eqnarray*}
\label{fc}
\zeta^{\theta} & = & \zeta^{\varphi(\theta)}, \nonumber \\
(1+\zeta)^{\theta} & = & \zeta^{\varphi(\theta)/2}, \\
(1-\zeta)^{\theta} & = & \zeta^{\varphi(\theta)/2} \cdot \left(
\lchooses{-1}{n} n\right)^{\varsigma(\theta)/2}, \nonumber
\end{eqnarray*}
where $\lchooses{-1}{n} $ is the Legendre symbol.
The last relation holds up to a sign which depends on the embedding of
$\zeta$. For a fixed embedding, we let $\nu = \sqrt{\lchooses{-1}{n}
n}$ be the generator of the quadratic subfield $\mathbb{Q}(\nu) \subset \mathbb{Q}(\zeta)$.
A short computation shows that $(1-\zeta)^{\theta} = \zeta^{\varphi(\theta)/2}
\nu$. Note that for $\theta \in I$ with $\varsigma(\theta) = 2$ we
have $(1-\zeta)^{2 \theta} = \zeta^{\varphi(\theta)} \cdot n^2$ for
any embedding.
We shall want to consider the action of elements of $\theta \in \mathbb{F}_n[G]$ on explicit algebraic
numbers $\beta \in \mathbb{K}$. Unless otherwise specified, an element $\theta = \sum_{c=1}^{n-1} m_c \sigma_c \in \mathbb{F}_n[G]$ is lifted to $\sum_{c=1}^{n-1} n_c \sigma_c$, where $n_c \in \mathbb{Z}$ are the unique integers with $0 \leq n_c < p$ and $n_c \equiv m_c \mod p$. In particular, lifts are always positive, of bounded weight $w(\theta) \leq (p - 1)^2$. Rather than introducing an additional notation for the lift defined herewith, we shall always assume, unless otherwise specified, that $\theta \in \mathbb{F}_p[G]$ acts upon $\beta \in \mathbb{K}$ via this lift.
Using this lift, we define the following additive maps:
\begin{equation*}
\rho_0 : \mathbb{F}_n [G] \rightarrow \mathbb{Q}(\zeta) \quad
\theta = \sum_{c=1}^{n-1} n_c \sigma_c \mapsto \sum_{c \in P} \frac{n_c}{1- \zeta^c},
\end{equation*}
and
\begin{equation*}
\rho : \mathbb{F}_n [G] \rightarrow \mathbb{Z}[\zeta] \quad
\theta \mapsto (1-\zeta) \cdot \rho_0[\theta].
\end{equation*}
The \nth{i} moment of an element $\theta = \sum_{c=1}^{n-1} n_c
\sigma_c$ of $\mathbb{Z}[G]$ is defined as:
\begin{equation*}
\phi^{(i)} (\theta) = \sum_{c=1}^{n-1} n_c c^i \mod n.
\end{equation*}
Note that $\phi^{(1)}$ is the {\em Fermat quotient map}: $\phi^{(1)} =
\varphi$. The moments are linear maps of $\mathbb{F}_p$-vector spaces and
homomorphism of algebras, verifying:
\begin{eqnarray}
\label{phiprop}
\begin{array}{l c l l c l}
\qquad \phi^{(i)}(a \theta_1 + b \theta_2) & = & a \phi^{(i)}(\theta_1) + b \phi^{(i)}(\theta_2), & \hbox{and} & \\
\qquad \phi^{(i)}(\theta_1 \theta_2) & = & \phi^{(i)}(\theta_1) \phi^{(i)}(\theta_2), & \hbox{with } & \theta_j \in \mathbb{F}_p[ G ]; a, b \in \mathbb{F}_p.
\end{array}
\end{eqnarray}
The linearity in the first identity is a straight-forward verification
from the definition. For the second, note that for $\theta = \sum_c
n_c \sigma_c$ we have
\[ \phi^{(i)}(\sigma_a \theta) = \phi^{(i)}\left(\sum_c n_c \sigma_{a c}\right) =
\sum n_c \cdot (a c)^i = a^i \cdot \phi^{(i)}(\theta). \]
Using the already established linearity, one deduces the multiplicativity of
$\phi^{(i)}$ as a ring homomorphism.
Let $\alpha = \frac{X-\zeta}{(1-\zeta)^e} \in \mathbb{Z}[ \zeta]$, as before,
and define $c_X \equiv 1/(X-1) \bmod n$ if $e = 0$ and $c_X = 0$ if $e
= 1$. For any $\theta \in I^+$, there is a \textit{Jacobi integer}
$\beta[\theta] \in \mathbb{Z}[ \zeta ]$ such that $\beta[\theta]^n =
(\zeta^{c_X} \alpha)^{\theta}$, normed by $\beta[\theta] \equiv 1
\bmod (1-\zeta)^2$ (Lemma 2 of \cite{Mi1}). The definition of $\varsigma(\theta)$
implies that
\begin{eqnarray}
\label{norm}
\beta[\theta] \cdot \overline{\beta[\theta]} =
\mbox{\bf N}_{\mathbb{K}/\mathbb{Q}}(\alpha)^{\varsigma(\theta)} = Y^{\varsigma(\theta)}.
\end{eqnarray}
We have for any $\theta \in I^+$,
\begin{eqnarray}
\label{beta1}
\beta[\theta]^n = (\zeta^{c_X} \alpha)^{\theta} = (\zeta^{c_X}
(1-\zeta)^{1-e})^{\theta} \cdot \left(1 + \frac{X-1}{1-\zeta}\right)^{\theta}
\end{eqnarray}
\begin{lemma}
\label{theta}
We remind that $D=X-1$. For any $\theta \in 2 \cdot I_f^+$, for any prime ideal $\eu{P} \ | \
D$, there is a $\kappa = \kappa_{\eu{P}}(\theta) \in \ZM{n}$ such that
\begin{equation*}
\label{base1}
\beta[ \theta ] \equiv \zeta^{\kappa} \cdot Y^{\frac{\varsigma(\theta)}{2}} \bmod \eu{P}.
\end{equation*}
\end{lemma}
\begin{proof}
Let $\theta_0$ be an element of $I^+_f$, and let $\theta = 2
\theta_0$. Note that from \rf{norm} we have $Y^{\varsigma(\theta_0)
n} = \beta[\theta_0]^n \cdot \overline \beta[ \theta_0 ]^n$. Thus
$ \beta[\theta]^n = \beta[\theta_0]^{2 n} = Y^{\varsigma(\theta_0)
n} \cdot \left ( \beta[\theta_0]/\overline \beta[\theta_0]
\right)^{n}$. Using \rf{beta1} and the previous observations, we
find:
\begin{eqnarray}
\beta[\theta]^n & = & Y^{\varsigma(\theta_0)n} \cdot \left(\zeta^{c_X} \cdot
(1-\zeta)^{1-e}\right)^{ (\theta_0 - \jmath \theta_0)} \cdot \left(1 +
\frac{X-1}{1-\zeta}\right)^{(\theta_0 - \jmath \theta_0)} \nonumber \\
& = & Y^{\varsigma(\theta_0) n} \cdot \zeta^{(2c_X + 1) \varphi(\theta_0)} \cdot \left(1 + D/(1-\zeta)\right)^{ (\theta_0 - \jmath
\theta_0)} \nonumber \\
\beta[\theta]^n & = & Y^{\varsigma(\theta_0) n} \cdot \left(1 + \frac{D}{1-\zeta}\right)^{(\theta_0 - \jmath
\theta_0)}. \label{pow}
\end{eqnarray}
Thus for any prime ideal $\eu{P} \ | \ D$ there is a $\kappa =
\kappa_{\eu{P}}(\theta) \in \ZM{n}$ such that
\begin{equation}
\label{kappa}
\beta[ \theta ] \equiv \zeta^{\kappa} \cdot Y^{\varsigma(\theta_0)} \bmod \eu{P}.
\end{equation}
\end{proof}
In the sequel, we indicate how to choose $\theta$ such that
$\kappa=0$. In this case, the relation \rf{beta1} leads to a
$\eu{P}$-adic binomial series expansion for $\beta[\theta]$.
We will use the Voronoi identities -- see Lemma 1.0 in \cite{Jha}
--, which we remind here for convenience:
\begin{lemma}
\label{Jha}
Let $m$ be an even integer such that $2 \leq m \leq n-1$. Let $a$ be
an integer, coprime to $n$. Then
\begin{equation}
\label{voronoi}
a^m \sum_{j=1}^{n-1} \left[ \frac{aj}{n} \right] j^{m-1} \equiv \frac{(a^{m+1}-a)B_m}{m} \mod n,
\end{equation}
where $B_m$ is the $m$-th Bernoulli number. In particular, for
$m=n-1$, we get
\begin{equation*}
\sum_{j=1}^{n-1} \left[ \frac{aj}{n} \right] j^{n-2} \equiv \frac{a^{n}-a}{n} \mod n,
\end{equation*}
which is the Fermat quotient map of the $a$-th Fuchsian element,
$\varphi(\Theta_a)$.
\end{lemma}
\begin{lemma}
\label{simple}
Let $\psi_k$ denote the $k$-th Fueter element. Then, there exists a
linear combination $\theta = \sigma \psi_k + \tau \psi_l \in I$ with
$\sigma, \tau \in G$ and $1 \leq k , l < n$ such that $\phi^{(1)}
(\theta) = 0$ and $\phi^{(-1)} (\theta) \neq 0$.
\end{lemma}
The proof of this Lemma is elementary, using the Voronoi relations
\rf{voronoi}; since the details are rather lengthy, they will
be given in the Appendix.
The following two lemmata contain computational information for binomial
series developments that we shall use below. First, we remind that $\rho_0$ is the following additive map:
\begin{equation*}
\rho_0 : \mathbb{F}_n [G] \rightarrow \mathbb{Q}(\zeta) \quad
\theta = \sum_{c=1}^{n-1} n_c \sigma_c \mapsto \sum_{c \in P} \frac{n_c}{1- \zeta^c}
\end{equation*}
\begin{lemma}
\label{dvpt}
Let $D$ be an indeterminate. Let $\theta = \sum_{c=1}^{n-1} n_c \sigma_c \in \mathbb{Z}[G]$ and $f[\theta]
= \left( 1 + \frac{D}{1-\zeta} \right)^{\theta/n} \in \mathbb{K}[[ D ]]$.
Let $0 < N < n$ be a fixed integer. Then,
\begin{equation*}
f[\theta] = 1 + \sum_{k=1}^N \frac{a_k[\theta]}{k! n^k} D^k + O(D^{N+1}),
\end{equation*}
where, for $1 \leq k \leq N$, we have
\begin{equation*}
a_k[\theta] = \rho_0^k [\theta] + O\left( \frac{n}{(1-\zeta)^k}\right).
\end{equation*}
In the above identity, $a_k[ \theta ], \rho_0^k[ \theta ] \in \mathbb{Z}[
\zeta, \frac{1}{n} ]$ are not integral, but their difference is an
algebraic integer $a_k[ \theta ] - \rho_0^k[ \theta ] \in
\frac{n}{(1-\zeta)^k} \cdot \mathbb{Z}[ \zeta ]$.
\end{lemma}
\begin{proof}
Let $\theta = \sum_c n_c \sigma_c$ and $m = m(\theta) = | \ \{ c \ :
\ n_c \neq 0 \} \ |$ be the number of non vanishing coefficients of
$\theta$. We prove this result by induction on $m$. First, note
that
\begin{equation*}
\binom{n_c/n}{k} = \frac{1}{k!} \cdot \frac{n_c^k}{n^k} \cdot (1 + O(n)).
\end{equation*}
Thus, if $\theta = n_c \sigma_c$ and $m = 1$, then:
\begin{equation*}
f[\theta] = 1 + \sum_{k=1}^{n-1} \frac{1}{k!} \cdot \frac{n_c^k}{n^k} \cdot (1 + O(n)) \cdot \frac{D^k}{(1-\zeta)^k} = 1 + \sum_{k=1}^N \frac{a_k[\theta]}{k! n^k} D^k + O(D^{N+1}),
\end{equation*}
where, for $1 \leq k \leq N$,
\begin{equation*}
a_k[\theta] = \rho_0^k [\theta] + O\left( \frac{n}{(1-\zeta)^k}\right),
\end{equation*}
which confirms the claim for $m = 1$. Suppose the claim holds for all
$j \leq m$ and let $\theta = \theta_1 + \theta_2$ with $m(\theta_i) <
m$ and $m(\theta) = m$. Then,
\begin{equation*}
\begin{array}{lcl}
f[\theta] & = & \left( 1 + \frac{D}{1-\zeta} \right)^{\theta_1/n} \cdot \left( 1 + \frac{D}{1-\zeta} \right)^{\theta_2/n}\\
& = & 1 + \sum_{k=1}^N \alpha_k[\theta] D^k + O(D^{N+1}),
\end{array}
\end{equation*}
where for $k < n-1$ we have
\begin{equation*}
\begin{array}{lcl}
\alpha_k[\theta] & = & \sum_{j=1}^k \frac{a_j [\theta_1]}{n^j j! (1-\zeta)^l} \cdot \frac{a_{k-j}[\theta_2]}{n^{k-j} (k-j)! (1-\zeta)^{k-l}} \cdot (1 + O(n))\\
& = & \frac{1}{k!n^k} \left( \rho_0 [\theta_1] + \rho_0 [\theta_2] \right)^{k} + O\left( \frac{n}{k! n^k(1-\zeta)^k}\right)\\
& = & \frac{1}{k!n^k} \cdot \rho_0^k [\theta] + O\left( \frac{n}{k! n^k(1-\zeta)^k}\right) =
\frac{1}{k!n^k} \cdot \left( \rho_0^k [\theta] + O( n/(1-\zeta)^k\right)
\end{array}
\end{equation*}
This proves the claim by complete induction.
\end{proof}
\begin{lemma}
\label{RemInt}
By proceeding like in Lema 8 of \cite{Mi2}, we notice that $\frac{a_k[\theta]}{k!} \in \mathbb{Z}[\zeta]$ (notation is different between both articles).
\end{lemma}
As a consequence, we may deduce that matrices built from the first
coefficients occurring in some binary series developments are regular.
\begin{lemma}
\label{regular}
Let $\theta = \sum_{c=1}^{n-1} n_c \sigma_c \in \mathbb{Z}[G]$ such that
$\phi^{(-1)}(\theta) \not\equiv 0 \mod n$, let $f[\theta] = \left( 1 +
\frac{D}{1-\zeta} \right)^{\theta/n}$ and $0<N<n-1$ be a fixed integer.
Then,
\[ f[\theta] = 1 + \sum_{k=1}^N \frac{b_k[\theta]}{k! n^k (1-\zeta)^k}
D^k + O(D^{N+1}), \quad \hbox{with} \quad \frac{b_k[\theta]}{k!} \in
\mathbb{Z}[\zeta]. \] Moreover, if $J \subset \mbox{Gal}(\mathbb{Q}[ \zeta ]/\mathbb{Q})$ is a
subset with $| J | = N$, then the matrix\footnote{We shall apply this
Lemma below, in a context in which $J$ satisfies the additional
condition that $i + j \neq n$ for any $i, j$ with $\sigma_i \in J$
and $\sigma_j \in J$.}
\[ A_N = \left(b_k[\sigma_c \theta] \right)_{ k=0;\sigma_c \in J}^{N-1}
\in \mbox{GL}(\mathbb{K}, N) \].
\end{lemma}
\begin{proof}
Let $\lambda = 1-\zeta$; we show that the determinant of $A_N$ is
not zero modulo $\lambda$. Using Lemma \ref{dvpt}, we know that we
have a development of symbolic power series
\begin{equation*}
f[\theta] = 1 + \sum_{k=1}^N \frac{a_k[\theta]}{k! n^k} D^k + O(D^{N+1}),
\end{equation*}
where
\begin{equation*}
a_k[\theta] = \rho_0^k [\theta] + O\left( \frac{n}{(1-\zeta)^k}\right).
\end{equation*}
By definition, $(1-\zeta)^k \cdot a_{k} [\sigma_c \theta] \in
\mathbb{Z}[\zeta]$ for all $\sigma_c \in G$. Let $b_k[\theta] = (1-\zeta)^k
\cdot a_{k} [\theta] \in \mathbb{Z}[\zeta]$. Then, according to Lemma
\ref{dvpt},
\begin{equation*}
\begin{array}{rcl}
b_k[\sigma_c \theta] & = & (1-\zeta)^k \cdot \left( \rho_0^k [\sigma_c \theta] + O\left( \frac{n}{(1-\zeta)^k}\right) \right)\\
& = & \rho^k [\sigma_c \theta] + O(n) = \left( \sum_{l=1}^{n-1} n_l \cdot \frac{1-\zeta}{1-\zeta^{lc}} \right)^k + O(n) \\
& \equiv & \left( \sum_{l=1}^{n-1} \frac{n_l}{lc} \right)^k \mod \lambda
\equiv \left( \frac{\phi^{(-1)} [\theta]}{c} \right)^k \mod \lambda.
\end{array}
\end{equation*}
Thus, $\det A_N \equiv \left| \left( \left( \frac{\phi^{(-1)}
[\theta]}{c} \right)^k\right)_{k=0,\sigma_c \in J}^{N-1} \right|
\mod \lambda$. We have obtained a Vandermonde determinant:
\begin{equation*}
\det A_N \equiv \left( \phi^{(-1)} [\theta] \right)^{N(N-1)/2} \cdot \prod_{i \neq j; \sigma_i, \sigma_j \in J} \left( \frac{1}{i} - \frac{1}{j}\right) \mod \lambda.
\end{equation*}
By hypothesis, $\phi^{(-1)} [\theta] \not\equiv 0 \mod n$,
and $1/i \not \equiv 1/j \bmod n$ for $\sigma_i, \sigma_j \in J$;
this implies finally that $\prod_{\sigma_i, \sigma_j \in J} \left(
\frac{1}{i} - \frac{1}{j}\right) \not\equiv 0 \mod n$, which
confirms our claim.
\end{proof}
\section{Proof of Theorem \ref{tbin}}
Theorem 4 in \cite{Mi1} proves that if CF holds, then \rf{dn} has no solution except for \rf{part}. The computations in \cite{BH} prove that CF holds for $n\leq 163.10^6$. This proves Theorem \ref{tbin}.\rf{A}.
Theorem \ref{tbin}.\rf{C} is also proved in Theorem 4 in \cite{Mi1}. In the sequel we shall show that the
only possible solutions are $X = \pm B/n^e + 1$. We may assume in particular that $n > 163 \cdot 10^6$.
We have already proved that $X-1 = B \cdot C^n/n^e$. If $C = \pm 1$,
then $X - 1 = \pm B/n^e$, as stated in point \rf{B} of Theorem \ref{tbin} and
$X$ is a solution of \rf{dn}. The bounds on $| X |$ in \rf{Bvals}
imply $| B | < n^n$, the second claim of \rf{B}.
Consequently, Theorem \ref{bin} will follow if we prove that $C = \pm
1$; we do this in this section. Assume that there is a prime $p | C$ with $p^{i} || C$.
Let $\eu{P} \subset \mathbb{Z}[\zeta]$ be a prime ideal lying
above $p$ and let $d(p) \subset G$ be its decomposition group. We shall use
Remark \ref{4} in order to derive some group ring elements which cancel
the exponents $\kappa$ occurring in \rf{kappa}.
Recall that $D = B \cdot C^n/n^e = X-1$, with $C$ defined by
\rf{y}. Note that \rf{y} implies that either $(n, D) = 1$, or $n^2 |
B$ and $(n, C) = 1$. Indeed, if $(n, D) \neq 1$, then $e = 1$ and $n^2
| n^e (X-1) = B C^n$ and since $(C, n) = 1$, it follows that $n^2 |
B$; the last relation follows from the bounds $C^n \leq E < 4
(n-2)^n$, hence $| C | < n$. In both cases $1/(1 - \zeta)$ is
congruent to an algebraic integer modulo $D/n^{v_n(D)} \cdot
\mathbb{Z}[\zeta]$.
According to Remark \ref{4}, we know that there are at least two elements, $\sigma^{'}_1, \sigma^{'}_2 \in
d(p)$ such that $\sigma^{'}_1 \neq \jmath \cdot \sigma^{'}_2$. Let
$\sigma^{'}_i(\zeta) = \zeta^{c_i}, \ c_i \in \ZMs{n}$. It
follows from Lemma \ref{sm} that, for $c_i \neq c_j$ when $i \neq j$, there are $h'_1, h'_2 \in \mathbb{Z}$ with $|
h'_i | \leq \sqrt{n}$ and $\sum_{i=1}^2 h'_i c_i \equiv 0 \bmod n$ while
$\sum_{i=1}^2 h'_i / c_i \not \equiv 0 \bmod n$.
We define
\begin{eqnarray}
\label{mudef}
\begin{array}{l c l}
\mu & = & \sum_{i=1}^2 h_i \sigma_i \in \mathbb{Z}[ d(p) ] \subset \mathbb{Z}[ G ], \quad \hbox{with} \\
h_i & = & \begin{cases}
h'_i & \hbox{if $h'_i > 0$ and }\\
- h'_i & \hbox{otherwise}, \quad \quad \quad \hbox{and}
\end{cases} \\
\sigma_i & = & \begin{cases}
\sigma^{'}_i & \hbox{if $h'_i > 0$ and }\\
\jmath \sigma^{'}_i & \hbox{otherwise}.
\end{cases}
\end{array}
\end{eqnarray}
By construction, $\mu$ is a positive element, i.e. the coefficients
$h_i \geq 0$. Let $\widehat{ \cdot } : G \rightarrow \ZMs{n}$ denote the
cyclotomic character and note that $h'_i \widehat{\sigma} = h_i
\widehat{\sigma'}$ for $h'_i < 0$ and thus $\phi^{(1)}(\mu) =
0$. In view of Lemma \ref{sm}, we also know that we can choose the
$h'_i$ and thus $\mu$, such that
\[ \phi^{(1)}(\mu) = 0, \quad \hbox{ but } \quad \phi^{(-1)}(\mu) \neq 0. \]
Since $\mathbb{K}/\mathbb{Q}$ is abelian, all the primes $\eu{P} | (p)$ have the same
decomposition group $d(p)$ and $\mu$ enjoys the following stronger
property: let $\eu{P} | (p)$ and $S \subset G$ be a set of
representatives of $G/d(p)$; let $\gamma \in \mathbb{Z}[\zeta]$ be such that
$\gamma \equiv \zeta^{c_{\sigma}} \bmod \sigma (\eu{P} )$ for all
$\sigma \in S$; then $\gamma^{\mu} \equiv 1 \bmod p \mathbb{Z}[\zeta]$, as
follows directly from $\zeta^{\mu} \equiv 1 \bmod \sigma(\eu{P})$, for
all $\sigma \in S$.
In view of Lemma \ref{simple} and the fact that Fueter elements are positive,
we also know that there is a $\theta_0 \in I_f^+$ such that
$\varsigma(\theta_0) = 2$ and $\phi^{(-1)} (\theta_0) \neq 0$.
Let
\begin{equation*}
\Theta = 2 \cdot \mu \cdot \theta_0.
\end{equation*}
In view of the properties \rf{phiprop} of moments and since for both
$\mu, \theta_0$, the Fermat quotient vanishes, while $\phi^{(-1)}$
is non-null, it follows that the same must hold for $\Theta$,
so $\Theta \in 2 \cdot I_f^+$ and $\phi^{(-1)} (\Theta)
\neq 0$. Let
\[ h = 2 \cdot \sum_{i=1}^l |h_i| = 2 . w( \mu ) , \]
where we defined the \textit{absolute weight} $w(\sum_c n_c \sigma_c)
= \sum_c | n_c |$. From subsection \ref{SecStick}, we know that there
exists a Jacobi integer $\beta[ 2 \theta_0 ] \in \mathbb{Z}[\zeta]$ such that
$\beta[ 2 \theta_0 ]^n = (\zeta^{c_X} (1-\zeta)^{1-e})^{\theta_0} \cdot
\left(1 + \frac{X-1}{1-\zeta}\right)^{\theta_0}$ (see \rf{beta1}). It
follows from \rf{base1} and Lemma \ref{theta}, that in both cases we
have $\beta[ 2 \theta_0 ] \equiv \zeta^{\kappa(\theta_0)} \cdot Y^4
\bmod \eu{P}$. We have chosen $\mu$ as a linear combination of two
elements from the decompostion group $D(\eu{P}) \subset G$, so $\mu$
acts on $\zeta \bmod \eu{P}$ by $\zeta \mod \eu{P} \mapsto \zeta^{\mu} \equiv 1 \bmod \eu{P}$.
Therefore, from $\beta[ \Theta ] = \beta[ 2 \theta_0 ]^{\mu}$ and thus,
by the choice of $\mu$, we have
\begin{eqnarray}
\label{unif}
\beta[ \Theta ] \equiv Y^h \bmod p \mathbb{Z}[ \zeta ].
\end{eqnarray}
Let $\Theta = 2 \sum_{c = 1}^{n-1} n_c \sigma_c$; for any prime
$\eu{P} | (p)$, the binomial series of the \nth{n} root of the right
hand side in \rf{pow} converges in the $\eu{P}$ - adic valuation and
its sum is equal to $\beta[ \Theta ]$ up to a possible \nth{n} root of
unity $\zeta^c$. Here we make use of the choice of $\Theta$: comparing
\rf{unif} with the product above, it follows that $\zeta^c = 1$ for
all primes $\eu{P} \ | \ (p)$. For any $N > 0$, we have $p^{i n N} | |
D^N$ and thus
\begin{eqnarray}
\label{beta}
\beta[ \Theta ] \equiv Y^h \prod_{c=1}^{n-1} \left(\sum_{k=0}^{N-1}
\binom{n_c/n}{k} \left(\frac{D}{1-\zeta^c}\right)^k \right) \quad \bmod p^{i n N}.
\end{eqnarray}
We develop the product in a series, obtaining an expansion which
converges uniformly at primes above $p$ and is Galois covariant; for
$N < n-1$ and $\sigma \in G$, we have:
\begin{eqnarray*}
\label{inhom}
\beta[ \sigma \Theta ] = Y^h \left(1 + \sum_{k = 1}^{N-1}
\frac{b_k[ \sigma \Theta ]}{(1-\zeta)^k n^k k!} \cdot D^k \right) + O
(p^{i n N}),
\end{eqnarray*}
with $b_k[ \Theta ] \in \mathbb{Z}[\zeta]$. Let $P \subset \{ 1, 2, \cdots ,
n-1 \}$ be a set of cardinal $1 < N < (n-1)/2$ such that if $c \in P$ then
$n-c \not \in P$, and $J \subset \mathbb{Z}[G]$ be the Galois automorphisms of
$\mathbb{K}$ indexed by $P$: $J=\{ \sigma_c \}_{c \in P}$. Consider the linear
combination $\Delta = \sum_{\sigma \in J} \lambda_{\sigma} \cdot
\beta[\sigma \cdot \Theta]$ where $\lambda_{\sigma} \in \mathbb{Q}[\zeta]$
verify the linear system:
\begin{eqnarray}
\label{hom}
\sum_{\sigma \in J} & & \lambda_{\sigma} \cdot b_k[\sigma \cdot \Theta] = 0 ,
\text{ for } k = 0, \ldots, N-1,\ k \neq \lceil N/2 \rceil \nonumber
\quad \text{ and } \\
\sum_{\sigma \in J} & & \lambda_{\sigma} \cdot b_{\lceil N/2 \rceil}[\sigma \cdot \Theta] = (1-\zeta)^{\lceil N/2 \rceil} n^{\lceil N/2 \rceil} \lceil N/2 \rceil !.
\end{eqnarray}
Applying Lemma \ref{dvpt} we observe that this system is regular for
any $N < n-1$. There exists therefore a unique solution which is not
null.
We recall that a power series $\sum_{k=0}^{\infty} a_k X^k \in
\mathbb{C}[[X]]$ is dominated by the series $\sum_{k=0}^{\infty} b_k X^k \in
\mathbb{R}[[X]]$ with non-negative coefficients, if for all $k \geq 0$, we
have $|a_k| \leq b_k$. The dominance relation is preserved by addition
and multiplication of power series.
Following the proof of Proposition 8.2.1 in \cite{Bilu2}, one shows
that if $r \in \mathbb{R}_{>0}$ and $\chi \in \mathbb{C}$, with $|\chi| \leq K$ with
$K \in \mathbb{R}_{>0}$, then the binomial series $(1+\chi T)^r$ is dominated
by $(1- K T)^{-r}$. From this, we obtain that $(1+ \chi T)^{\Theta/n}$ is
dominated by $(1 - K T)^{-w(\Theta)/n}$. In our case \rf{beta}, $T=D$, $\chi = \frac{1}{1 - \zeta^c}$ and
\[ K = \max_{1 \leq c < n} | 1/(1-\zeta^c) | = 1/\sin(\pi/n) \leq
n/\pi \cos(\pi/3) = 2n/\pi < n . \] Applying this to our selected
$\Theta$, whose absolute weight is bounded by $w \leq 4 n \sqrt{n}$,
we find after some computations that $|b_k[\sigma \cdot \Theta]| <
n^{k} \cdot \binom{-w/n}{k} \cdot k! < n^{3k}$ for $N < n/2$.
Let $A = \det \left( b_k[\sigma_c \cdot \Theta ] \right)_{k=0; c \in
I}^{N-1} \neq 0$ be the determinant of the matrix of the system
\rf{hom}, which is non vanishing, as noticed above: note that the
division by $k!$ along a complete row does not modify the regularity
of the matrix.
Let $\vec{d} = (1-\zeta)^{\lceil N/2 \rceil} n^{\lceil N/2 \rceil} \lceil N/2 \rceil ! \left( \delta_{k,\lceil N/2
\rceil}\right)_{k=0}^{N-1} $, where $\delta_{i,j}$ is Kronecker's
symbol. The solution to our system is $\lambda_\sigma = A_{\sigma} /
A$, where $A_\sigma \in \mathbb{Z}[\zeta]$ are the determinants of some minors
of $\left( b_k[\sigma_c \cdot \Theta ] \right)_{k=0;c \in I}^{N-1}$
obtained by replacing the respective column by $\vec{d}$.
Noticing that $|(1-\zeta)^{\lceil N/2 \rceil} n^{\lceil N/2 \rceil} \lceil N/2 \rceil !| < n^{3(N-1)}$, Hadamard's inequality implies that
\begin{eqnarray*}
|A_\sigma| & \leq & n^{3(N-1)(N-2)/2} \cdot (N-1)^{(N-1)/2} \leq n^{3N^2/2} \cdot N^{N/2} \quad \hbox{and} \\
|A| & \leq & n^{3N^2/2} \cdot N^{N/2}
\end{eqnarray*}
Let $\delta = A \cdot \Delta \in \mathbb{Z}[\zeta]$,
\[ \delta = \sum_{\sigma \in J} A_{\sigma} \cdot \beta[\sigma \cdot
\Theta] \in \mathbb{Z}[\zeta]. \]
We set $N = \lceil n^{3/4} \rceil$ and claim that for such $N$, $\delta \neq 0$. By choice of the $\lambda$'s, we have $\delta = A.p^{in \lceil N/2 \rceil}.u + p^{i n N} z$ for some $z \in \mathbb{Z}[ \zeta ]$, where $u = \frac{D^{\lceil N/2 \rceil}}{ p^{in \lceil N/2 \rceil}}.Y^h$ is a unit in $(\mathbb{Z}/p\mathbb{Z})^*$. Therefore, if we assume that $\delta = 0$, then necessarily $p^{in \lceil N/2 \rceil}$ divides $A$. However, $v_p(A) < n \lceil N/2 \rceil$. Indeed, the upper bound for $| A |$ implies a fortiori that $v_p(A) \leq \lceil N/2 \cdot \log N + \frac{3 N^2}{2}\log n \rceil$. Then, the assumption $\delta = 0$ would imply $n \leq 3\left[ n^{3/4} + \frac{1}{4}\right] \log n$, which is false for $n \geq 4,5.10^6$. This contradicts thus our initial assumption. Therefore $\delta \neq 0$.
Given the bounds on $A_{\sigma}$, we obtain $|\delta| \leq N Y^h
n^{3N^2/2} \cdot N^{N/2}$ and using the fact that $h < 4n^{1/2}$, $Y
< n^n$ (Theorem \ref{tbin}.\rf{B}) and $N = \lceil n^{3/4} \rceil$, we find
\begin{eqnarray*}
\label{up}
| \mbox{\bf N}_{\mathbb{K}/\mathbb{Q}}(\delta) | < \left( n^{\frac{11}{2} n^{3/2} + \frac{3}{8} n^{3/4} +
\frac{3}{4}}\right) ^{n-1}.
\end{eqnarray*}
The initial homogenous conditions in \rf{hom} imply $\delta \equiv 0
\bmod p^{i n \lceil N/2 \rceil }$, therefore $| \mbox{\bf N}_{\mathbb{K}/\mathbb{Q}}(\delta)
| \geq p^{i n (n-1) N/2}$. Combining this inequality with \rf{up} and
$n \geq 163 \cdot 10^6$, one finds that $\log p < 1.64$. This shows
that $p= 2, 3$ or $5$.
We consider the case $p \leq 5 $ separately as follows. We choose in this case $\mu = 1 + p \jmath
\sigma_p^{-1}$ and verify that $\varphi(\mu) = 0$, while
$\phi^{-1}(\mu) = 1 - p^2 \not \equiv 0 \bmod n$. Consequently
$\varsigma(\Theta) = 4(p+1)$ and the norm of $\delta$ is thus bounded by
\begin{eqnarray*}
p^{n (n-1) N/2} \leq | \mbox{\bf N}_{\mathbb{K}/\mathbb{Q}}(\delta) | < \left( n^{4(p+1) + 3N^2/2} \cdot N^{N/2+1} \right)^{n-1}.
\end{eqnarray*}
Letting $N = 48$, we obtain the inequality
\[ 2^n \leq n^{73} \cdot 48^{25/24} < 64 n^{73} \quad \Rightarrow
\quad \frac{n-6}{73} \leq \log(n)/\log(2), \] which is false for for
$n > 695$, and a fortiori for $n > 163.10^{6}$. We obtain a contradiction
in this case too, and thus $C = \pm 1$, which completes the proof of
Theorem \ref{tbin}.
\qed
\section{Consequences for the general case of the binary Thue equation \rf{bin}}
In this section we derive Theorem \ref{genexpo}. For this we
assume that \rf{bin} has a solution with $(\varphi^*(B),n) = 1$, since
our results only hold in this case, a fact which is reflected also in
the formulation of Theorem \ref{genexpo}.
Consider the case when $n = p \cdot q$ is the product of two distinct
primes. If $(n , B) = 1$, then Theorem \ref{tbin} holds for both $p$
and $q$ with the value $e = 0$. If $X, Z$ is a solution, then Theorem
\ref{tbin} . \rf{B} implies that $X^p = \pm B + 1$ and $X^q = \pm B
+1$. Consequently either $X^p + X^q = 2$ or $X^p - X^q = 2$. This is
impossible for $| X | > 2$ and a simple case distinction implies that
there are no solutions. As a consequence,
\begin{corollary}
\label{c1}
Consider Equation \rf{bin} for fixed $B$ and suppose that $n$ is
an integer which has two distinct prime divisors $p > q > 2$ with $(p, B) =
( q, B ) = 1$. Then \rf{bin} has no solutions for which \rf{nosplit} holds.
\end{corollary}
If all divisors of $n$ are among the primes dividing $B$, we are led
to the following equation: $p (X^q - 1) = q (X^p - 1)$, which has no
solutions in the integers other than $1$. Indeed, assume $X \neq 1$ to be a
solution of the previous equation, and $q=p+t,\ t \geq 0$. The
real function $f(t) = p(X^{p+t}-1) - (p+t)(X^p-1)$ is strictly
monotonous and $f(0)=0$. Therefore, the equation $p (X^q - 1) = q
(X^p - 1)$ has no solutions. There is only the case left in which $n$
is built from two primes, one dividing $B$ and one not. In this case,
one obtains that equation $p ( X^q - 1) = X^p - 1$ which can also be
shown not to have non trivial solutions, using the above remark, this
time with $f(t) = p(X^{p+t}-1) - (X^p-1)$. Hence:
\begin{corollary}
\label{c2}
The equation \rf{bin} has no solutions for exponents $n$ which are
divisible by more than one prime and for $B$ such that \rf{nosplit} holds.
\end{corollary}
We are left to consider the case of prime powers $n = p^c$ with $c >
1$. If $p \nmid B$, we obtain $X^{n/p} - 1 = B/p^e$, so in particular
$B/p^e+1 \geq 2^{p^{c-1}}$ is a \nth{p^{c-1}} power. Since in this
case, \rf{bin} has in particular a solution for the exponent $p$, the
Theorem \ref{tbin} implies that $B < p^p$; when $c > 2$, combining this with the
previous lower bound implies that there are no solutions.
For $c = 2$, we deduce that $| X | < p$ and, after applying the
Theorem \ref{tbin} again and letting $\xi = \zeta^{1/p}$ be a
primitive \nth{p^2} root of unity, we obtain the equation
\[ Y^{p^2} = \frac{X^{p^2} - 1}{p^e(X^p-1)} = \mbox{\bf N}_{\mathbb{Q}[ \xi ]/\mathbb{Q}}
(\alpha) \quad \quad \quad \alpha = \frac{X-\xi}{(1-\xi)^e} .\] As
usual, the conjugates of the ideal $(\alpha)$ are pairwise coprime. We
let $\eu{A} = (Y, \alpha)$ be an ideal with $N(\eu{A}) = (Y)$; moreover,
if $\eu{L} | \eu{A}$ is a prime ideal and $N(\eu{L}) = (\ell)$, then
the rational prime $\ell$ is totally split in $\mathbb{Q}[ \xi ]$, the factors
being the primes $(\ell, \sigma_c(\alpha))$. Being totally split, it
follows in particular that $\ell \equiv 1 \bmod p^2$ so $Y \geq \ell >
2 p^2$, in contradiction with $Y < X < p+1$. This shows that there are
no solutions for $n = p^2$. \qed
\begin{corollary}
If the Equation \rf{bin} in which $n = p^c$ is a prime power has non
trivial solutions for which \rf{nosplit} holds, then $c = 1$.
\end{corollary}
\qed
The primes dividing the exponent $n$ used in the above corollaries are
by definition coprime to $\varphi^*(B)$. As a consequence, if $n$ is
an exponent for which \rf{bin} has a solution and $m | n$ is the
largest factor of $n$ with $m \in \id{N}(B)$ -- as defined in
\rf{expos} -- then the corollaries imply that there is at most one
prime dividing $n/m$ and the exponent of this prime in the prime
decomposition of $n$ must be one. This is the first statement of
Theorem \ref{genexpo}, which thus follows from these corollaries and
Theorem \rf{bin}.
\section{Appendix}
The proof of Theorem \ref{tbin} is based on results from \cite{Mi1}. It has
been pointed out that the proof of Theorem 3 in \cite{Mi1} may require
some more detailed explanation in the case of a singular system of
equations in the proof of Lemma 14 of \cite{Mi1}. Since the statements
of \cite{Mi1} are correct and can even be slightly improved, while the
explanations may have seemed insufficient, we provide here for the readers
interested to understand the technicalities of the proofs in \cite{Mi1}
some additional details and explanation, confirming those claims and results.
\subsection{Clarification on the singular case of Theorem 3 in \cite{Mi1}}
Let $m \in \mathbb{Z}_{>0}$ be a positive integer, $\mathbb{K}$ a field, $V = \mathbb{K}^{m}$
as a $\mathbb{K}$-vector space and let $L \subsetneq V$ be a proper subspace of $V$ of dimension $r$. We
assume that there exists at least one vector $w_1 \in L$ which is free
of $0$-coefficients over the canonical base $\id{E}$. For $(x, y) \in V^2$, $x=(x_1, \cdots, x_m)$ and $y=(y_1,\cdots,y_m)$, the
Hadamard product is defined by $[ x, y ] = (x_1y_1, \cdots, x_m y_m)$. For
any subspace $W \subset V$ we define the $W$-\textit{bouquet} of $L$ by
\[ L_W = \langle \ \ \{ \ [ w, x ] \ : \ w \in W, \ x \in L \ \} \ \
\rangle_{\mathbb{K}} ,\] the $\mathbb{K}$-span of all the Hadamard products of
elements in $W$ by vectors from $L$.
\begin{lemma}
\label{bouquet}
Let $a_1 = (1, 1, \ldots, 1)$ over $\id{E}$, and $e_2 \in V$ such that its coordinates be pairwise distinct over $\id{E}$. Let $A_2 = \langle \{ a_1, a_2\} \rangle_{\mathbb{K}}$ be
the subspace generated by $e_1,e_2$. Let $L_{A_2}$ be the
resulting $A_2$-bouquet. Then $dim(L_{A_2}) > dim(L)$.
\end{lemma}
\begin{proof}
Obviously, $L \subset L_{A_2}$ (as $a_1 \in A$). We would like to
show that $L_{A_2} \neq L$. We know that the system $(w_1,
[w_1,a_2], [w_1,a_2^2], \cdots, [w_1,a_2^{m-1}])$ (the notion of
power of a vector here is to be understood as an "Hadamard power")
is free (as it induces a Vandermonde matrix over $\id{E}$,
$w_1$ does not have any zero among its coordinates and all
coordinates of $a_2$ are pairwise distinct). We know that $w_1 \in
L$; let us assume that $[w_1,a_2^i] \in L$ for $i \leq j$ (we know
that $j \leq m - r < m$). Then, $[w_1,a_2^j] \in L$ and
$[w_1,a_2^{j+1}] \notin L$. However, the Hadamard product of
$[w_1,a_2^j] \in L$ by $a_2$, that is $[w_1,a_2^{j+1}]$, belongs to
$L_{A_2}$. Thus, $\dim L_{A_2} > \dim L$.
\end{proof}
\subsubsection{Application of Lemma \ref{bouquet} to the proof
of the singular case in the argument on pages 266 -- 270 of \cite{Mi1}}
We apply here the lemma in the first case (that is $x \not\equiv s
\mod p$, where $s \in \{ -1, 0, 1\}$), the application to the second
case being similar.
Let all notation be like in Lemma 14 in \cite{Mi1}. As in \cite{Mi1},
we will assume that $\rg{A} = \left( \zeta^{-\kappa_{ac}/a}
\right)^{(p-1)/2}_{a,c=1}$ (where $\kappa_{ac}$ are the \textit{Galois
exponents}) is singular. Let $m = (p-1)/2$, $\mathbb{K} = \mathbb{Q}(\zeta_p)$ and
$r = {\rm rank } \ (\rg{A}) < (p-1)/2$. Without loss of generality, we assume
that a regular $r$-submatrix of $\rg{A}$ is built with the first $r$
rows and the first $r$ columns. Therefore, the first $r$ rows of
$\rg{A}$ are independent, and we denote by $W$ the sub-space of
$V=\mathbb{K}^{m}$ generated by the first $r$ row vectors $w_1, \cdots, w_r$
of $\rg{A}$. For $a_1 = (1, 1, \ldots, 1)$, we let $a_2$ be the vector
of $V$ whose components are \footnote{\text{In the context of \cite{Mi1}, $\eta$ corresponds to $b_1[\theta]$ in our context}} $\left( \eta (\sigma_c
\theta)\right)_{c=1}^{(p-1)/2}$ and $A_2=\{ a_1, a_2 \}$. Then,
according to Lemma \ref{bouquet}, there exists at least one vector
$\vec{v} \in L_{A_2}$ which is independent on the first $r$ vectors of
$\rg{A}$.
Let $\rg{S}$ be the $(r+1) \times (r+1)$ submatrix of $\rg{A}$
comprising the first $r$ rows and $r+1$ columns of $\rg{A}$, to which
we have added an additional row: the first $r+1$ components of
$\vec{v}$. Let $\vec{\lambda}'$ be the vector solution of $\rg{A}
\vec{\lambda}' = \vec{d}'$, where $\vec{d}' = \left( \delta_{c,r+1}
\right)_{c=1}^{r+1}$. We know that $\vec{\lambda}' \neq \vec{0}$, as
$\rg{S}$ is regular and $\vec{d}'$ is not the null vector. For $1 \leq
c \leq r + 1$, by Cramer's rule, $\lambda_c = \frac{S_c}{S}$, where
$S_c$ are the determinants of some minors of $\rg{S}$ obtained by
replacing the $c$-column by $\vec{d}'$, and $S = \det \rg{S}$.
Let $\vec{\lambda} \in V$ be a vector whose first $r+1$ coordinates
are those of $\vec{\lambda}'$ and the others are $0$. Let $\left(
\delta_{c,r+1} \right)_{c=1}^{m}$. Then, $\vec{\lambda}$ verifies:
$\rg{A} \vec{\lambda} = \vec{d}$.
Let $\delta = \sum_{c=1}^{r+1} \left( \lambda_c \cdot \beta_c +
\overline{\lambda_c \cdot \beta_c} \right)$. Using Hadamard's
inequality, we bound $|S_c| \leq \left(
\frac{p-3}{2}\right)^{\frac{p-3}{4}} = D_1$ and $|S| \leq \left(
\frac{p-1}{2}\right)^{\frac{p-1}{4}} = D_0$. Then, using the fact
that the choice of $\lambda_c$ eliminates the first term in the
expansion of $f_c$, we find that $|S| \cdot |\delta| \leq 2
x^{(p-1)/2p} \cdot \sum_{c=1}^{r+1} |S_c| |R_{c,0} (x)|$, where
$R_{c,0} (x) = f_c(x) - x^{(p-1)/2p}$. With the same arguments
as in \cite{Mi1}, we deduce: $$|S \delta| < 2(p-1) D_1 \cdot
\frac{1}{|x|^{(p+1)/2p}}.$$ This inequality holds for all conjugates
$\sigma_c(\delta)$, thus leading to:
\begin{equation*}
\left|\mbox{\bf N}\left(S \delta\right)\right| < \left( 2 (p-1) D_1\right)^{(p-1)/2}
\cdot \frac{1}{|x|^{\frac{(p-1)(p+1)}{4p}}}.
\end{equation*}
If $\delta \neq 0$, then $\left|\mbox{\bf N}\left(S \delta\right)\right| \geq
1$ and thus $|x| \leq 2^{5-p} \left( \frac{p}{2}
\right)^{\frac{p}{2}}$. If $\delta = 0$, then $0 = S \delta = S \cdot
|x|^{(p-1)/2} - \sum_{c=1}^{(p-1)/2} S_c R_{0,c}$, and thus:
\begin{equation*}
|x| \leq \sum_c |S_c|/|S| < (p-1) D_1 < 3
\left(\frac{p-3}{2}\right)^{(p-3)/2} .
\end{equation*}
These bounds are better than the ones in \cite{Mi1}, and this
concludes the clarification.
\subsection{Proof of Lemma \ref{simple}}
\begin{proof}
Let $\theta = \sigma_w \psi_u + \sigma_z \psi_v$. The conditions
required by the lemma lead to the following linear system of
equations over $\mathbb{F}_n$:
\begin{equation}
\label{sys1}
\left\{
\begin{array}{lcllcl}
w \cdot \varphi (\psi_u) & + & z \cdot \varphi (\psi_v) & = & 0 \\
1/w \cdot \phi^{(-1)} (\psi_u) & + & 1/z \cdot \phi^{(-1)} (\psi_v) & \neq & 0
\end{array}
\right.
\end{equation}
Considered as a linear system in the unknowns $w, z \in \mathbb{F}_n$, the
above system has the matrix $M = \left( \begin{array}{c c} \varphi
(\psi_u) & \varphi (\psi_v)\\ \phi^{(-1)} (\psi_v) & \phi^{(-1)}
(\psi_u) \end{array}\right)$. Assume that the product $P(t) =
\varphi(\psi_t) \cdot \phi^{(-1)}(\psi_t)$ is not constant for all $t
\in \ZMs{n}$. Then there are two elements $u, v \in \ZMs{n}$ such that
$P(u) \neq P(v)$; for such values $u, v$, the matrix $M$ is regular
over $\mathbb{F}_p$ and for any non vanishing right hand side in the second
equation, the system has a unique solution $(w, z)$. For this choice
of $u, v; w, z$, the element $\theta = \sigma_w \psi_u + \sigma_z
\psi_v$ satisfies the condition of the lemma.
We now show that $P(t) : \ZMs{n} \rightarrow \mathbb{F}_p$ is not a constant
function. The proof uses explicit computations which include divisions
by several constants which must be assumed to be non - null. Therefore
we suppose that $n \not \in E := \{ 3, 7 \}$ and shall verify
independently that the claim of the lemma holds for this exceptional
set.
Let $\varphi$ be the Fermat quotient map and $\Theta_k$ be the \nth{k}
Fuchsian. For any integer $1 < k < n-1$, we have:
\begin{equation*}
\begin{array}{lcll}
(n-k)^n - (n-k) & \equiv & -k^n - n + k & \mod n^2 \\
& \equiv & -n \left( \frac{k^n - k}{n} + 1 \right) & \mod n^2 .
\end{array}
\end{equation*}
Dividing both terms by $n$ and recalling from Lemma \ref{Jha} that
$\varphi(\Theta_k) = \varphi(k) \equiv \frac{k^n - k}{n} \bmod n$, we find:
\begin{equation}
\label{FuchsEq}
\varphi (\Theta_{n-k}) = n - \left( 1 + \varphi(\Theta_k) \right).
\end{equation}
Using now \rf{voronoi} from Lemma \ref{Jha}, with $m=2$, we find that:
\begin{equation*}
\phi^{(-1)} (\Theta_k) \equiv \frac{k^3 - k}{2k^2} B_{2} \equiv \frac{1}{12}\cdot
\left(k - \frac{1}{k} \right) \mod n,
\end{equation*}
where we used the fact that $B_2 = 1/6$. Finally, using that $\psi_k =
\Theta_{k+1} - \Theta_k$ for $k > 1$ while $\psi_1 = \Theta_2$, we
obtain the following expressions for the moments of interest:
\begin{eqnarray*}
\varphi(\psi_k) & = & \varphi(k+1)-\varphi(k), \\
\phi^{(-1)}(\psi_k) & \equiv & \frac{1}{12} \cdot \left( 1 + \frac{1}{k(k+1)}\right).
\end{eqnarray*}
Note that $\phi^{(-1)}(\psi_k) = 0$ iff $k^2 + k + 1 = 0$; if $n
\equiv 1 \bmod 6$, the equation has two solutions in $\mathbb{F}_n$, otherwise
it has none. In the latter case $\phi^{(-1)}(\psi_k) \neq 0$ for all
k.
We shall assume that $P$ is the constant function and shall show that
this assumption fully determines the Fermat quotient of integers in
dependence of $\varphi(2)$, and this determination is in contradiction
with \rf{FuchsEq}; the contradiction implies that $P$ cannot be
constant, thus completing the proof.
Let thus $C = \varphi(2) \cdot \phi^{(-1)}(\Theta_2) = \varphi(2)
\cdot \frac{1}{8}$. Assume first that $\varphi(2) = 0$ and recall from
\rf{FuchsEq} that $\varphi(k) + \varphi(n-k) + 1 = 0$. Therefore at
least $\frac{n-1}{2}$ of the values of $\varphi$ are
non-vanishing. Since $\phi^{(-1)}(k) \cdot (\varphi(k+1)-\varphi(k)) =
0$ for all $k$ we see that if $n \not \equiv 1 \bmod 6$, then
$\varphi$ is constantly vanishing, which is impossible.
If $n \equiv 1 \bmod 6$, let $l, m \in \mathbb{F}_n$ be the non trivial third
roots of unity, so $\phi^{(-1)}(\psi_l) = \phi^{(-1)}(\psi_m) = 0$,
while for all $k \not \in \{l, m\}$ we must have $\varphi(k+1) =
\varphi(k)$. In particular, if $l < m$, there are two integers $a, b$
such that
\[ \varphi(2) = 0 = \ldots = \varphi(l); \quad \varphi(l+1) = a =
\ldots = \varphi(m); \quad \varphi(m+1) = b = \ldots = \varphi(n-1).\]
But $\varphi(n-1) = -1$ while $\varphi(n-2) = -1 - \varphi(2) = -1$,
so $b = -1$. For symmetry reasons induced by \rf{FuchsEq}, we must
have $a = -1/2$ and $m = n-l$. This is absurd since $m^3 \equiv 1
\bmod n$ implies $l^3 = (n-m)^3 \equiv -m^3 \equiv -1 \bmod n$, so $n
= 2 \not \equiv 1 \bmod 6$. Thus $\varphi(2) \neq 0$ in this case
too. Since $\phi^{(-1)}(l) = 0$, it follows however that $C =
\varphi(l) \cdot \phi^{(-1)}(l) = 0$ and thus $C = 0 = \varphi(2) / 8$
and we should have $\varphi(2) = 0$, in contradiction with the facts
established above. Consequently, if $n \equiv 1 \bmod 6$, then $P$
cannot be constant.
We consider now the case $n \not \equiv 1 \bmod 6$, in which we know
that $C \neq 0$. By expressing $C = P(2) = P(k)$ we obtain the
following induction formula
\begin{eqnarray*}
C = \frac{1}{12} \cdot \frac{3 \varphi(2)}{2} & = &
\frac{1}{12} (\varphi(k+1) - \varphi(k)) \cdot \frac{k^2 + k + 1}{k(k+1)}, \quad \hbox{hence} \\
\varphi(k+1) - \varphi(k) & = & \frac{3 \varphi(2)}{2} \cdot \frac{k(k+1)}{k^2 + k + 1}, \\
\varphi(3) - \varphi(2) & = & \frac{9}{7} \varphi(2) \quad \Rightarrow \quad \varphi(3) = \frac{16}{7} \varphi(2).
\end{eqnarray*}
By eliminating $\varphi(2)$ from the above identity for two successive
values of $k$ one finds
\[ \varphi(k+1) = \frac{2k^3}{k^3-1} \cdot \varphi(k) +
\frac{k^3+1}{k^3-1} \cdot \varphi(k-1). \]
We shall use the reflexion formula \rf{FuchsEq} between the last and
the first values in the sequence $1, 2,\ldots, n-2, n-1$. Letting $k =
n-2$ in the above induction, we find
\begin{eqnarray*}
-1 & \equiv & \varphi(n-1) \equiv \frac{16}{9} \cdot \varphi(n-2) +
\frac{7}{9} \cdot \varphi(n-3) \\
& \equiv & \frac{16}{9} \cdot(-1- \varphi(2)) + \frac{7}{9} \cdot (-1-\varphi(3)) \bmod n, \\
9 & \equiv & 16 + 16 \varphi(2) + 7 + 7 \varphi(3) \equiv 23 +
(16+7 \cdot \frac{16}{7} ) \varphi(2) \bmod n, \quad \hbox{hence} \\
-7 & \equiv & 16 \cdot \varphi(2) \bmod n.
\end{eqnarray*}
Consequently $\varphi(2) \equiv -\frac{7}{16} \bmod n$ and thus
$\varphi(3) \equiv \frac{16}{7} \varphi(2) \equiv -1 \bmod n$. But
then \rf{FuchsEq} implies that $\varphi(n-3) = -1-\varphi(3) = 0$, and
thus $C = 0$, in contradiction with the previously obtained non
vanishing fact. This confirms that $P(t)$ is non constant in
this case too.
It remains to verify the claim for the exceptional primes in $E$. For
$n = 3$ the Stickelberger ideal is trivial, so there is nothing to
prove. For $n = 7$ one can repeat the proof of the case $n \equiv 1
\bmod 6$, which requires no division by $7$; this completes the proof
of the Lemma.
\end{proof}
\thanks{\textbf{Acknowledgements}: \textit{The first author is grateful to the Universities of Bordeaux and
G\"ottingen for providing a stimulating environment during the
development of this work. Both authors thank Mike Bennett and
Kalman Gy\H{o}ry for suggesting this interesting problem for an
algebraic investigation.}}
|
1,116,691,499,222 | arxiv | \section{Introduction}
Since its discovery as the first extragalactic source of very high energy (VHE,
$E>100\,$GeV) gamma rays, Mrk 421 has been intensively studied. The spectral
energy distribution (SED) shows the two characteristic non-thermal peaks of a
high energy peaked BL-Lac (HBL) system.
The VHE end of the spectrum is complicated by attenuation by the extragalactic
background light (EBL) which will add a characteristic absorption to the
spectrum \cite{Dwek05}. Spectral variability as a function of flux level has
been observed, the spectrum hardening with increasing flux level
\cite{Krennrich2002, Aharonian2002}, and curvature being readily
apparant in time averaged spectra at the highest flux levels
\cite{Krennrich2001, Daniel2005, Aharonian2005}. The most favoured spectral fit
form for the curvature has been that of a power-law with an exponential cut-off
\begin{equation}
\frac{\mathrm{d}N}{\mathrm{d}E} =
F_{c}
\exp{\left(-\frac{E}{E_{c}}\right)}
E^{-\alpha}
\end{equation}
where $F_{c}$ is the flux at 1\,TeV, $E_{c}$ is the characteristic cut-off
energy in TeV and $\alpha$ is the spectral index.
In spite of the spectral variability, the value for the cut-off has seemed a
relatively consistent feature between experiments and epochs (see
figures \ref{fig:CutoffByTime}, \ref{fig:CutoffByFlux}, \ref{fig:CutoffByIndex}) at an average value of $E_C = 3.57\pm0.16$\,TeV.
At low/intermediate flux levels there had not been sufficient improvement in a
$\chi^2$ fit to favour a curved spectrum over that of a pure power-law. That was
until measurements made by the new generation instrument MAGIC during 2005
showed compelling evidence in the time averaged spectrum favouring a curved
spectral shape \cite{Mazin2007}, although with a much lower value for a cut-off
than had been seen previously. Given that a change in the cut-off would
demonstrate spectral curvature to unequivocally be intrinsic to the source
rather than just attenuation due to the EBL it is important to confirm these
findings.
The Whipple 10\,m and the first of the VERITAS telescopes also observed
Mrk\,421 during 2005 and this paper compares the results of these three
independent sets of measurements of the time averaged spectrum.
\section{Observations and Data Analysis}
The Whipple 10\,m is described in \cite{Kildea07}.
The VERITAS T1 system is described in \cite{Holder06}, even though the array was
in the construction phase during this time frame this dataset was taken when T1
had begun taking regular observations. The MAGIC telescope is described in
\cite{Ferenc05}.
As a crosscheck between the different experiments and their analysis procedures
the reconstruction of the Crab spectrum for the equivalent epoch was compared,
as shown in figure~\ref{fig:Crab}, and demonstrates consistent results.
\begin{figure}[ht]
\includegraphics[width=0.48\textwidth]{Fig01.eps}
\caption{Values for the fit of a power law with
exponential cut-off term to the Mrk 421
spectrum available in the literature \cite{Krennrich2001,
Krennrich2002, Aharonian2002, Aharonian2003, Rebillot2006,
Daniel2005, Aharonian2005, Mazin2007}.
Plotted as a function of epoch.}
\label{fig:CutoffByTime}
\end{figure}
\begin{figure}
\includegraphics[width=.48\textwidth]{Fig02.eps}
\caption{as figure~\ref{fig:CutoffByTime},
but the cut-off energy is plotted as a function of flux level.}
\label{fig:CutoffByFlux}
\end{figure}
\begin{figure}
\includegraphics[width=.48\textwidth]{Fig03.eps}
\caption{as figure~\ref{fig:CutoffByTime},
but the cut-off energy is plotted as a function of the derived
spectral index.}
\label{fig:CutoffByIndex}
\end{figure}
\begin{figure}
\includegraphics[width=\textwidth]{Fig04.eps}
\caption{Comparison of the spectral reconstruction for the three different
telescopes for the standard candle of the Crab nebula.
}
\label{fig:Crab}
\end{figure}
The MAGIC observations of Mrk\,421 are detailed in \cite{Mazin2007} and consist
of 26 hours of on-source time taken between November 2004 and April 2005.
The Whipple 10\,m Mrk\,421 observations consist of 10 hours on-source data taken
during March-April of 2005 as part of the regular blazar monitoring program.
The data were reduced and spectrum derived according to the method described in
\cite{Grube2007}.
The VERITAS T1 data consist of 8 hours of on source observations also taken
during March and April 2005. It can be seen from the tables that whilst the
MAGIC observations overwhelmingly favour a curved spectrum the fits to the
other two show no improvement to those for a pure power-law.
The time averaged Mrk\,421 spectrum for the three experiments are shown together
in figure~\ref{fig:Spectra}.
The fits to a pure power-law function are summarised in
table~\ref{tab:power} and to a power-law with an exponential cut-off
are summarised in table~\ref{tab:cutoff} respectively.
A fit to all of the flux points
combined still favours a curved spectrum, but with a cut-off
$E_c = 1.65 \pm 0.31$\,TeV which is still much lower than previously measured
in the Mrk\,421 spectrum.
\section{Discussion \& Conclusions}
Even though the flux points between the different experiments agree well, from a
visual perspective at any rate, they have different conclusions on the best fit
of the spectral form: with the MAGIC data clearly favouring a curvature of the
low flux level spectrum that would appear different with previous results at
higher flux levels.
Whilst the data are all time averaged for the same epoch they do cover slightly
different time periods and are not truly simultaneous; if the cut-off in the
Mrk\,421 spectrum represent intrinsic changes in the emission region then the
different experiments could see different behaviour if sampling at different
times. The consistency of the previous generation instruments results does not
sit well with this conclusion even if the cut-off varies as a function of flux
level.
The obvious difference between these different single dish systems is the lower
threshold energy of MAGIC due to a much larger mirror area. When the MAGIC
points corresponding only to the energy range of the smaller instruments, and
the previous generation of instruments, are used then a curved spectrum is still
favoured, both individually and when combined with all flux points. The value
for this cut-off is $E_c = 3.24\pm2.13$\,TeV, now in agreement with the
historical and present data. The larger error bars down to a smaller number of
points over a much reduced energy range that are less constraining to the fit.
This clearly demonstrates that the spectrum of Mrk\,421 is not merely cut-off at
the highest energies, but exhibits curvature throughout the VHE spectrum. It may
be prudent to investigate other fit functions (such as a broken power law, or a
log-parabolic form found to improve fits in X-ray spectrum fits)
to describe the spectral curvature with the new generation of lower threshold
instruments, and particularly in light of the 5 orders of magnitude in energy
range that will be available in joint multiwavelength observations with GLAST.
\begin{theacknowledgments}
The author wishes to to thank Daniel Mazin for providing the MAGIC data and
helpful comments.
This research is supported by grants from the U.S. Department of
Energy, the U.S. National Science Foundation, and the Smithsonian
Institution, by NSERC in Canada, by PPARC in the UK and by Science
Foundation Ireland.
\end{theacknowledgments}
\bibliographystyle{aipproc}
|
1,116,691,499,223 | arxiv | \section{\label{sec:intro}Introduction}
Lattice and charge are definitely the two most essential degrees of freedom in condensed matter. The interplay between them, {\it i.e.}, the electron-phonon coupling (EPC), as a prototypical many-body problem, has received intensive theoretical and experimental investigations \cite{Grimvall.1981,Maksimov.1997,Pavarini.2017,Giustino.2017}. EPC is exceptionally important not only because it is closely bound up with macroscopic physical properties such as charge carrier dynamics, thermal conductivity, heat capacity and so on, but also due to its crucial roles in the microscopic mechanisms of many notable phenomena such as polarons, superconductivity, and charge density waves, {\it etc.}.
EPC changes the dispersion and lifetime of both the involving electrons and phonons. The effect of the EPC on the dispersion and lifetime of an electronic or phonon state is determined by the complex electron's or phonon's self-energy, respectively. The real part of the self-energy renormalizes the dispersion, while the imaginary part accounts for the finite lifetime arising from the interactions. One can analyze the renormalized dispersion of an electronic band, and then extract the electron's self-energy and evaluate the lifetime of relevant electrons \cite{McDougall.1995,Reinert.2004}, to determine the possible EPC. Strong EPC generally leads to kinks or quasiparticle peaks in the electronic band near the Fermi energy ($E_F$), as reported by previous angular-resolved photoemission spectroscopy (ARPES) surveys \cite{Hengsberger.1999b,Hengsberger.1999,Rotenberg.2000,Kondo.2013,Yang.2019,Zhou.2008}. Yet, the impacts on electronic properties from EPC usually average the contributions from all involving phonon modes. Consequently, it is rather indirect to assess the contribution from an individual phonon mode from the electron’s perspective only.
EPC also manifests itself from the perspective of phonons, through linewidth broadening \cite{Butler.1979, Wakabayashi.1986}, or the dispersion anomaly which is usually identified as the so-called Kohn anomaly. A Kohn anomaly is a drastic energy softening of a specific phonon branch when its momentum is located at $2k_F$ ($k_F$ the Fermi wave number), resulting from the coupling of the specific phonon mode with electrons near the Fermi surface \cite{Kohn.1959,Kohn.1962}. So Kohn anomaly can vail the quantification of EPC for an individual phonon mode. Kohn anomalies have been observed from phonon dispersion measurements in lots of materials, such as the typical one-dimensional systems with Peierls transitions \cite{Renker.1973,Shirane.1976,Pouget.1991,Guster.2019}, two-dimensional (2D) systems with strong EPC \cite{Politano.2015,Haddad.2018,Weber.2011}, and three-dimensional (3D) bulk systems \cite{Brockhouse.1962,Wong.2003,Agrestini.2004,Baron.2004}.
Similar to the 2D case, the EPC on surfaces, often viewed as quasi-2D, can lead to surface phonon anomalies. For example, the kinks in the phonon dispersions on the (111) surfaces of noble metals \cite{Kaden.1992,Doak.1983,Harten.1985,Harten.1985b,Jayanthi.1987}, are believed to arise from the coupling of surface electronic states to the ion displacement. And the sharp surface phonon anomalies in the hydrogen saturated (110) surfaces of molybdenum and tungsten \cite{Kroger.1997,Hulpke.1993,Kohler.1995, Hulpke.1992,Rotenberg.1998} have the origin due to the interactions with the surface electronic states from the chemical bonding \cite{Benedek.2018b}. More intriguingly, strong Kohn anomalies due to the interactions between surface phonons and the topologically protected surface electrons (or Dirac Fermions) have recently been observed in topological insulators \cite{Zhu.2012,Zhu.2011,Howard.2013} and topological crystalline insulators \cite{Kalish.2019}. In brief, Kohn anomalies should widely exist on a specific surface phonon branch, as long as there exist surface electronic states regardless of their origination.
Recently a new kind of surface states due to the reduction of bulk nonsymmorphic symmetry was discovered in ${\rm ZrSiS}$ \cite{Topp.2017}, which is a candidate of topological nodal line semimetal featured by its nontrivial bulk bands \cite{Schoop.2016,Neupane.2016,Wang.2016,Chen.2017,Hosen.2017,Fu.2019}. ${\rm ZrSiS}$ is considered to be one of the most promising topological materials with potential applications in electronics and spintronics, due to its very large energy range of linear band dispersion \cite{Schoop.2016}, extremely large non-saturating magnetoresistance and high mobility of charge carriers \cite{Ali.2016,Wang.2016,Lv.2016,Singha.2017,Hu.2017,Pezzini.2018}. However, despite the topological nature in its bulk band, it turns out the surface states of ${\rm ZrSiS}$ are not derived from the bulk topology. Instead, they are highly 2D, floating on top of the bulk, merely due to a reduced symmetry at the surface \cite{Topp.2017}. Consequently, it is natural to expect the unique surface states would also interact with phonons and show anomalies in surface phonon dispersions. The coupling between the surface floating 2D electrons and phonons, which could be signified in low dimensional devices, is of exceptional importance for any potential applications, but remains elusive.
Here, employing the technique of high-resolution electron energy loss spectroscopy (HREELS), we systematically mapped out the surface phonon dispersions of ${\rm ZrSiS}$. The results reveal a prominent signature of the surface EPC, manifesting as a strong Kohn anomaly of an optical surface phonon branch with a V-shaped minimum at approximately $2\bm{k}_{\bm{F}}$. Theoretical analysis attributes this strong renormalization of the surface phonon to the interactions with the surface floating 2D electrons of ${\rm ZrSiS}$. Then an average branch-specific EPC constant of 0.15 is obtained from the fitting to the experimental dispersion data.
\section{\label{sec:method}Method}
The ${\rm ZrSiS}$ crystals investigated were grown from the chemical vapor transport method with iodine as the agent. High-purity elemental zirconium $(99.99\%)$, silicon $(99.999\%)$ and sulfur $(99.99\%)$ were firstly mixed together and sintered at 1000 $^{\circ}$C for 5 days to attain polycrystalline powders. Then the polycrystalline powders and iodine were sealed in silica tubes under vacuum in a mass ratio of 1:0.07. The silica tubes were put in a gradient tube furnace with the source powders at 1050 $^{\circ}$C and the cold end at around 950 $^{\circ}$C for 7 days, to acquire rectangular plane-like ${\rm ZrSiS}$ crystals at the cold end.
Single crystalline ${\rm ZrSiS}$ was cleaved {\it in situ} in ultrahigh vacuum (better than $5\times10^{-10}$ Torr) with the surface quality and the crystallographic orientation examined by low energy electron diffraction (LEED). The HREELS measurements were performed in a state-of-the-art spectrometer with the capability of 2D energy-momentum mapping \cite{Zhu.2015}. Due to the special design of our facility, it is able to obtain a phonon spectrum for a certain direction through the Brillouin zone (BZ) in a single measurement without rotating sample, monochromator, or analyzer. All the measurements were done using an incident electron beam with energy of 110 eV and incident angle of 60$^{\circ}$ unless otherwise specified. The energy and momentum resolutions were $\Delta E\sim3$ meV and $\Delta k\sim 0.01\ {\rm \AA}^{-1}$ in this study. The low temperature was reached by continuous liquid helium flow through the manipulator and measured by a silicon diode on the stage behind the sample.
\begin{figure}[t]
\includegraphics[width=0.48\textwidth]{Fig1}
\caption{\label{fig:structures} \textbf{Sample characteristic and the surface floating 2D states}. (\textbf{a}) Crystal structure of ZrSiS. (\textbf{b}) Illustration of the bulk BZ and the SBZ. (\textbf{c}) The optical image of a sheeny surface of cleaved ZrSiS crystal pasted on a Mo sample holder. (\textbf{d}) Room temperature LEED pattern of a freshly cleaved (001) surface with an incident electron energy of 120 eV. The white triangle shows the least irreducible SBZ. (\textbf{e}) The Fermi surfaces from the surface electronic band calculations. The light blue lines are the projection from the bulk bands, while the dark blue ellipses denote the surface floating 2D states. (\textbf{f}) The cone-like surface floating 2D band around $\bar{\rm X}$ point. The orange arch is the elliptical Fermi surface with major axis $2k_a$ and minor axis $2k_b$.}
\end{figure}
First-principle lattice dynamical calculations were performed to obtain the phonon band structures and the corresponding surface local density of states (LDOS). Here, force constants were calculated by a $(3\times3\times3)$ supercell using Vienna $ab\ initio$ simulation package (VASP) \cite{Kresse.1996}, based on density functional perturbation theory (DFPT) \cite{Baroni.2001}. The generalized gradient approximation (GGA) of Perdew-Burke-Ernzerhof (PBE) \cite{Perdew.1992,Blochl.1994} type was used for the exchange-correlation functions. A plane wave cutoff energy of 420 eV with a $(6\times6\times3)$ Monkhorst-Pack $k$-mesh was employed for the integrations over the BZ. The phonon spectrum was obtained by an open-source package \cite{Togo.2015}, and the LDOS on the (001) surface was calculated by surface Green’s function \cite{Wu.2018}.
\section{\label{sec:res_disc}Results and Discussions}
\subsection{\label{subsec:exp_res}Experimental Results}
${\rm ZrSiS}$ has a tetragonal structure with space group $P4/nmm$ (No.129) and point group $D_{4h}$ \cite{KleinHaneveld.1964,Tremel.1987}, formed by stack of ${\rm S-Zr-Si-Zr-S}$ quintuple layers, with lattice constants $a=b=3.546(2)\ {\rm \AA}$, $c=8.055(4)\ {\rm \AA}$ \cite{Singha.2018}. Two adjacent ${\rm Zr-S}$ layers, with each S atom surrounded by four nearest Zr atoms, are sandwiched by two Si layers extending in the $ab$ plane, as shown in Fig. \ref{fig:structures}(a). Two kinds of nonsymmorphic symmetry can be recognized: a glide mirror in the plane formed by the square nets of the Si atoms and two screw axes $C_{2x}$$(C_{2y})$ along the $a(b)$ directions in the Si layer. The nonsymmorphic symmetry with glide mirror is crucial for the formation of the robust Dirac cone below $E_F$ \cite{Chen.2017} as well as the surface floating 2D bands \cite{Topp.2017}. The weak bonding energy between the two adjacent ${\rm Zr-S}$ layers \cite{Salmankurt.2017} allows easy cleavage along the $ab$ planes to obtain the (001) surface. The breaking of the translational symmetry after cleavage reduces the space group into $P4mm$, corresponding to a four-fold rotational symmetry along the $c$-axis. The bulk Brillouin zone (BZ) is reduced to a square surface Brillouin zone (SBZ), as shown in Fig. \ref{fig:structures}(b). Fig. \ref{fig:structures}(c) and (d) show the optical image and the LEED pattern of a freshly cleaved (001) surface, respectively. Most of the cleaved samples show bright sharp spots with dark background in the LEED patterns, indicating high-quality surfaces obtained for the HREELS measurements. All the LEED patterns show ($1\times1$) square lattices with no signal of surface reconstructions. In Fig. \ref{fig:structures}(e), the calculated Fermi surface on the SBZ illustrates the surface floating 2D bands. Such surface states show as ellipses around $\bar{\rm X}$ at the Fermi energy and shrink nearly linearly to a vertex at around 350 meV below the Fermi level \cite{Topp.2017,Neupane.2016,Hosen.2017}, as demonstrated in Fig. \ref{fig:structures}(f). So we approximate the surface floating 2D states as cone-like band with linear dispersions. The Fermi wave numbers of such cones are $k_a\approx0.15\ {\rm \AA}^{-1}$ along the major axis, and $k_b\approx0.11\ {\rm \AA}^{-1}$ along the minor axis, of the ellipses.
\begin{figure}[t]
\includegraphics[width=0.48\textwidth]{Fig2}
\caption{\label{fig:hreels_raw} \textbf{HREELS measurement results}. (\textbf{a}), (\textbf{b}) 2D momentum-energy mappings at RT along $\bar{\Gamma}\bar{\rm X}$ and $\bar{\Gamma}\bar{\rm M}$, respectively. The negative energy range corresponds to anti-Stokes peaks (phonon annihilation), while the positive range corresponds to Stokes peaks (phonon creation). The white dashed lines denote the SBZ boundaries. The brightness, representing the signal intensity, is plotted in logarithmical scale. (\textbf{c}), (\textbf{d}) Energy distributed curves along the red dash lines at $q=0.3\ {\rm \AA}^{-1}$ and $q=0.1\ {\rm \AA}^{-1}$ in (a), respectively. The orange background is the elastic peak. Phonon peaks are fitted by Gaussian functions and labelled by AP (acoustic phonon) or OP (optical phonon) according to their dispersions.}
\end{figure}
Figure \ref{fig:hreels_raw}(a) and (b) show the energy and momentum mappings of the surface phonons for ZrSiS obtained from the HREELS measurement at room temperature (RT) along $\bar{\Gamma}\bar{\rm X}$ and $\bar{\Gamma}\bar{\rm M}$, respectively. At least three phonon branches can be discerned in the energy range of 0 - 45 meV. The accurate energy and linewidth information of the phonon modes can be obtained by extracting energy distributed curves (EDCs) from the mapping. For example, two EDCs at $q=0.1\ {\rm \AA}^{-1}$ and $q=0.3\ {\rm \AA}^{-1}$ along $\bar{\Gamma}\bar{\rm X}$ [red dashed lines in Fig. \ref{fig:hreels_raw}(a)], are demonstrated in Fig. \ref{fig:hreels_raw}(c) and (d), respectively. After subtracting the background from elastic scattering, the phonon modes can be distinguished by fitting the energy loss peaks with Gaussian functions. At $q=0.3\ {\rm \AA}^{-1}$, one acoustic phonon (AP) mode and three optical phonon (OP) modes are clearly discerned. At $q=0.1\ {\rm \AA}^{-1}$, only the OP modes are clear, since the AP mode is so close to the elastic peak that its intensity is merged in the strong elastic scattering background.
\subsection{\label{subsec:rev_Kohn}Reveal of the Kohn Anomaly}
\begin{figure*}[t]
\includegraphics[width=0.9\textwidth]{Fig3}
\caption{\label{fig:phonon_disp} \textbf{Calculated and experimental phonon dispersions}. (\textbf{a}) Calculated bulk phonon dispersions along high symmetry directions of the bulk BZ. The green dots correspond to the Raman-active modes \cite{Singha.2018,Zhou.2017}. (\textbf{b}) Calculated LDOS for phonons of the $(001)$ surface along $\bar{\rm X}-\bar{\Gamma}-\bar{\rm M}-\bar{\rm X}$, with the intensity plotted in logarithmical scale. The yellow shades are the projection of bulk phonon dispersions onto the surface, and red sharp lines represent surface phonons with the LDOS larger than 2.5. The dash-dot lines denote the Rayleigh modes splitting from the bulk acoustic branch. (\textbf{c}) The second derivative image along $\bar{\rm M}-\bar{\Gamma}-\bar{\rm X}$ of the SBZ at RT obtained from the 2D HREELS mapping shown in Fig. \ref{fig:hreels_raw}. The superimposed red lines are the calculated surface phonon dispersions extracted from (b). The white dashed block designates the zone where the anomalous phonon softening occurs. The experimentally observed OP1 mode is compared with the theoretical $\alpha$ mode (white line). (\textbf{d}) The same comparison as (c) with the experimental data obtained at 35 ${\rm K}$.}
\end{figure*}
To analyze the measured phonon dispersions of ${\rm ZrSiS}$ in detail, we carried out first-principle bulk and surface lattice dynamical calculations. Figure \ref{fig:phonon_disp}(a) shows the calculated bulk phonon dispersions. The green dots superimposed on the energy axis denote the locations of the Raman active modes \cite{Singha.2018,Zhou.2017}, indicating good agreement with our calculations. Figure \ref{fig:phonon_disp}(b) shows the dynamical calculation results with the phonon local density of states (LDOS) projected onto the (001) surface using the method of surface Green’s function. The surface phonon modes contribute the highest LDOS manifest as sharp red lines, superimposed on the bulk-projected yellow bands. An elaborate comparison between the calculated and the measured surface phonon dispersions can be carried out by extracting the surface phonon modes from Fig. \ref{fig:phonon_disp}(b) and superimposing them on the second differential images of our experimental spectra. The results are demonstrated in Fig. \ref{fig:phonon_disp}(c) and (d) with experimental data collected at RT and 35 ${\rm K}$, respectively. Although the overall intensity at low temperature is weaker than that at RT, the measured phonon dispersions do not show significant temperature-dependence within the energy resolution, implying the phonon-phonon interactions are negligible. Due to the restriction of selection rules \cite{ibach.1982,Juan.2015}, not all the calculated phonon branches can be measured by HREELS. For those branches that are measured, most of them fit well with the calculations.
The most noticeable feature from the comparison is the optical mode encircled by the white rectangle (OP1 mode) in Fig. \ref{fig:phonon_disp}(c) and (d), which is dramatically discrepant from the calculated surface modes. This peculiar mode OP1 sets out from 25 meV at $\bar{\Gamma}$ point and softens sharply to near 16 meV at $q\approx0.3\ {\rm \AA}^{-1}$ along $\bar{\Gamma}\bar{\rm X}$, while at $q\approx0.25\ {\rm \AA}^{-1}$ along $\bar{\Gamma}\bar{\rm M}$. Then it inclines with increasing $q$ afterward. Although there are several calculated surface modes intersecting with the softened mode OP1 within the white rectangle, most of them are ruled out by the HREELS selection rules \cite{ibach.1982,Juan.2015}. Only the calculated optical surface mode labeled $\alpha$ in Fig. \ref{fig:phonon_disp}(c), coinciding with OP1 in energy at $\bar{\Gamma}$ point, is allowed to be detected by the selection rules. Thus, the observed OP1 mode is designated to the $\alpha$ mode but with energy softened with increasing $q$. At low temperature, the softening of this mode remains to the same extent despite the overall decreased intensity of the entire spectrum. This indicates that the softening originates from EPC rather than phonon-phonon interactions since the latter should be strongly temperature-dependent.
\subsection{\label{subsec:EPC_Model}Electron-phonon Coupling Model}
Similar softening of surface optical phonon modes, interpreted as strong Kohn anomalies, have been observed in quite a few topological materials \cite{Howard.2013,Kalish.2019,Zhu.2012,Zhu.2011}. Such strong Kohn anomalies descend from an abrupt change in the electron screening of atomic vibrations induced by surface electrons, indicating strong surface electron-phonon interactions. For ZrSiS, the surface floating 2D bands around $\bar{\rm X}$ [Fig. \ref{fig:structures}(e)] are the only surface electronic states \cite{Topp.2017,Neupane.2016,Schoop.2016,Wang.2016,Chen.2017,Hosen.2017}. Here, we set up a model in which the surface optical phonon interacts with the cone-like surface floating 2D electrons, to explain the observed Kohn anomaly and describe the detailed EPC picture.
The impact of EPC on phonons can be taken into account using the Dyson equation that describes the relation between perturbed and bare phonons,
\begin{equation}\label{eq:Dyson_equation_def}
\mathcal{D}_s\left(\bm{q},i\omega_n\right)=\frac{\mathcal{D}_s^{\left(0\right)}\left(\bm{q},i\omega_n\right)}{1-\mathcal{D}_s^{\left(0\right)}\left(\bm{q},i\omega_n\right)\Pi\left(\bm{q},\omega_n\right)}
\end{equation}
Here, $\mathcal{D}_s^{(0)}(\bm{q},i\omega_n)$ and $\mathcal{D}_s(\bm{q},i\omega_n)$ are the bare phonon and perturbed phonon Matsubara Green functions, respectively, and $\omega_n$ is the Matsubara frequency for phonons. Under random phase approximation (RPA) the phonon self-energy $\Pi(\bm{q},\omega_n)$ can be substituted by
\begin{equation}\label{eq:SelfEnergy_def}
\Pi\left(\bm{q},\omega_n\right)=\left|g_{\bm{q},s}\right|^2\frac{\mathcal{P}\left(\bm{q},i\omega_n\right)}{\varepsilon\left(\bm{q},i\omega_n\right)}
\end{equation}
where $|g_{\bm{q},s}|$ is the electron-phonon interaction matrix, $\mathcal{P}(\bm{q},i\omega_n)$ is the electron polarization function and $\varepsilon(\bm{q},i\omega_n)=1-v_c(\bm{q})\mathcal{P}(\bm{q},i\omega_n)$ is the RPA dielectric function. For surface electrons,$\ v_c(\bm{q})$ is chosen to be the 2D Fourier transformation of the Coulomb potential, $v_c(\bm{q})=\frac{2\pi e^2}{\kappa|\bm{q}|}$. The electron polarization function can be calculated from the bare electron Matsubara function $\mathcal{G}^{(0)}(\bm{k},ik_n)$ by
\begin{equation}\label{eq:Polarization_Func_def}
\mathcal{P}\left(\bm{q},\omega_n\right)=\frac{1}{\mathcal{A}}\frac{2}{\beta}\sum_{\bm{k}}\sum_{ik_n}{\mathcal{G}^{\left(0\right)}\left(\bm{k},ik_n\right)\mathcal{G}^{\left(0\right)}\left(\bm{k}+\bm{q},ik_n+i\omega_n\right)}
\end{equation}
which describes an electron of momentum $\bm{k}$ and energy $k_n$ scattered to a state of momentum $\bm{k}+\bm{q}$ and energy $k_n+\omega_n$ by a phonon of wave vector $\bm{q}$ and energy $\omega_n$.
The following analysis focuses on the phonon dispersion along $\bar{\Gamma}\bar{\rm X}$, which involves two orthogonal axes of the Fermi ellipse at two inequivalent $\bar{\rm X}$ [ Fig. \ref{fig:structures}(e)]. Taking phonons with wave vectors along the major axis of the Fermi ellipse for example, due to the constraints of energy and momentum conservation, there are three cases of the electron-phonon scattering processes with different energy and momentum ranges [Fig. \ref{fig:epc_model}(b), (c) and (d)]. The allowed initial states of electrons reside on the royal blue hyperbola within the light blue area. Careful analysis with details for these cases can be found in the Appendix. This analysis yields three constraint conditions for phonons involved in the EPC,
\begin{flushright}
$
\left\{
\begin{aligned}
\Omega & <Q \qquad\qquad\qquad & {\rm (Cst.1)}\\
\Omega & >Q-2 \qquad\qquad\qquad & {\rm (Cst.2)}\\
\Omega & <2-Q \qquad\qquad\qquad & {\rm (Cst.3)}
\end{aligned}
\right.
$
\end{flushright}
where $\Omega=\frac{\omega}{E_F}$ and $Q=\frac{q}{k_F}$ are the normalized frequency and normalized wave vector of phonons. These constraints require the phonons to fall into the regions (3), (2) and (1) in Fig. \ref{fig:epc_model}(a), corresponding to Fig. \ref{fig:epc_model}(b), (c) and (d), respectively. After an analytical continuation $i\omega_n\rightarrow\ \omega_{\bm{q},s}+i\delta$ in Eqn.(\ref{eq:Polarization_Func_def}), its imaginary part can be obtained by performing the integration within the allowed areas of Fig. \ref{fig:epc_model}(a), and its real part is achieved by conducting a Kramers-Kronig transformation. A similar process can be applied when the phonon wave vector is along the minor axis of Fermi ellipse. The total polarization is just the sum of the two individual polarizations along the two axes.
\begin{figure}[t]
\includegraphics[width=0.5\textwidth]{Fig4}
\caption{\label{fig:epc_model} \textbf{Illustrations of the electron-phonon scattering process}. (\textbf{a}) The allowed $(\Omega, Q)$ space for phonons defined by the constraint conditions (Cst. 1), (Cst. 2) and (Cst. 3). Regions (3), (2) and (1) correspond to (b), (c) and (d), respectively. (\textbf{b}), (\textbf{c}) and (\textbf{d}) Schematic drawings of the initial and final Fermi ellipses before and after interacting with phonons in three cases of different momentum and energy transfer. The light blue zones denote the allowed area of initial states for electron-phonon interactions, while the dark blue zones are the banned area. The royal blue hyperbolas are the lines that satisfy the conservation of energy.}
\end{figure}
The renormalized phonon dispersion is given by the real part of the singularity of the Dyson equation in Eqn. (\ref{eq:Dyson_equation_def}),
\begin{equation}\label{eq:phonon_renorm}
(\hbar\omega_{\bm{q},s})^2=
(\hbar\omega_{(\bm{q},s)}^{(0)})^2 + 2(\hbar\omega_{\bm{q},s}^{(0)})\left|g_{\bm{q},s}\right|^2 \textup{Re}\left[\frac{\mathcal{P}(\bm{q},\omega_{\bm{q},s})}{\varepsilon(\bm{q},\omega_{\bm{q},s})}\right],
\end{equation}
where $\omega_{\bm{q},s}$ and $\omega_{\bm{q},s}^{(0)}$ correspond to the real and bare frequency for a specific phonon branch $s$ at wave vector $\bm{q}$ with or without EPC, respectively. For the surface phonons, we take the assumption $\left|g_{\bm{q},s}\right|=\sqrt{\frac{N\hbar}{2M\omega_{\bm{q},s}^{\left(0\right)}}}(\gamma_\bot+\frac{\left|\bm{q}\right|}{{2k}_F}\gamma_{||})$ \cite{Zhu.2011} in the momentum range $\bm q<2k_F$, where $M$ is the unit cell mass and $N$ is the number of unit cells. $\gamma_\bot$ and $\gamma_{||}$ are treated as out-of-plane and in-plane interaction parameters to fit Eqn. (\ref{eq:phonon_renorm}) self-consistently. Details of the parameter fitting are described in the Appendix. Here $\omega_{\bm{q},s}^{(0)}$ is obtained from the calculated results of $\alpha$ in Fig. \ref{fig:phonon_disp}(c), and $\omega_{\bm{q},s}$ obtained from the model can be directly compared with the experimentally measured dispersion of the OP1 branch. The renormalized dispersion is depicted in Fig. \ref{fig:epc_parameters}(a).
\subsection{\label{subsec:EPC_Constant}Mode-Specific EPC constant}
From the perspective of phonons, one may assess the branch-specific coupling constant $\lambda_s(\bm{q})$ as a function of momentum $\bm{q}$ for specific phonon branch $s$ by \cite{Butler.1979,Zhu.2012}
\begin{equation}\label{eq:branch_lambda_def}
\lambda_s\left(\bm{q}\right)=-\frac{\textup{Im}\left[\Pi\left(\bm{q},\omega_{\bm{q},s}\right)\right]}{\pi\mathcal{N}\left(E_F\right)\left(\hbar\omega_{\bm{q},s}\right)^2},
\end{equation}
where $\mathcal{N}(E_F)$ is the electron density of states at the Fermi energy
Shown in Fig. \ref{fig:epc_parameters}(b) are the real part and the imaginary part of the phonon self-energy, as well as the mode-specific EPC constant $\lambda_s(\bm{q})$ for the branch OP1 as a function of phonon momentum. The two conspicuous peaks in the imaginary part of the phonon self-energy and $\lambda_{\textup{OP1}}(\bm{q})$ correspond to the two orthogonal axes of the Fermi ellipse. This is different from the conventional 2D Kohn anomaly model \cite{Kohn.1959,Kohn.1962} where an isotropic Fermi circle generates identical $2k_F$ along all the momentum directions. Here, even along one specific direction in the SBZ, $2k_F$ has two maxima at $2k_a$ and $2k_b$. Yet, the physical essence of the EPC is the same, i.e., the EPC constant $\lambda_{\textup{OP1}}(\bm{q})$ declines quickly when $q>2k_a$ or $q>2k_b$ due to the Fermi ellipse, analogous to the case of the Fermi circle with $\lambda(\bm{q})$ declining when $q>2k_F$. The peak at $q\sim2k_a$ is higher than the peak at $q\sim2k_b$, implying a more significant contribution from electrons with momentum along the major axis of the Fermi ellipse. $\lambda_{\textup{OP1}}(\bm{q})$ declines to zero immediately when $q>2k_a$ and the phonon dispersion inclines afterwards.
\begin{figure}[t]
\includegraphics[width=0.48\textwidth]{Fig5}
\caption{\label{fig:epc_parameters} \textbf{Renormalized dispersion and fitting results from the EPC model}. (\textbf{a}) The dispersions of the OP1 mode. The black line is the calculated bare surface phonon dispersion from first principle calculation, while the blue dots with error bars (using the instrument resolution here) are the extracted experimental data. The yellow line is the renormalized dispersion from the EPC model. (\textbf{b}) Calculated real (light blue) and imaginary (green) parts of the phonon self-energy and the EPC constant (magenta) for the OP1 mode.}
\end{figure}
The averaged EPC constant over ${\bm q}$ for the branch OP1 is evaluated to be $\bar{\lambda}\approx0.15$. This implies the renormalized mass of involved electrons is $m^\ast=(1+\bar{\lambda})m_e\approx1.15m_e$, where $m_e$ is the bare electron mass. The quantum oscillation experiment has confirmed an unconventional mass enhancement in ZrSiS in the range of $1\sim1.5\ m_e$ of electrons within the bulk “dog-bone-like” Fermi pockets under magnetic field \cite{Pezzini.2018}. The enhancement arises presumably from small-momentum density-wave correlations, which requires strongly enhanced Coulomb interactions between electrons \cite{Huh.2016,Roy.2017}. However, our findings here suggest a mass enhancement of electrons of the surface floating band, due to the interaction with surface optical phonons, which cannot be probed by the conductivity quantum oscillations. Different from the bulk electron mass enhancement that was observed only under the magnetic field, the surface EPC discovered here should always exist and will potentially affect any low-dimensional electronic transporting properties. Recently, quasiparticle interference between different faces of surface states around $\bar{\rm X}$ point has been confirmed using a scanning tunneling microscope \cite{Butler.2017,Su.2018,Zhu.2018}, which also supports our proposition that electrons of surface states around $\bar{\rm X}$ interact with optical phonons.
\section{\label{sec:summary}Summary}
In conclusion, we measured the surface phonon dispersions of ${\rm ZrSiS}$, a candidate of topological nodal line semimetal, along the high symmetry directions in the SBZ. Comparing with DFT calculations, we verified an obvious softening of a surface optical mode due to EPC. Detailed theoretical analyses, via an EPC model within RPA, revealed that the strong renormalization of the surface phonon originates from the interactions with the surface floating 2D electrons of ${\rm ZrSiS}$. Under this model, we calculated the polarization function, evaluated the phonon self-energy, and obtained the mode-specific and $q$-dependent EPC constant. The average EPC constant for the softened surface optical phonon branch is $\bar{\lambda}\approx0.15$, indicating the effective mass of renormalized electrons due to surface EPC is $m^\ast\approx1.15\ m_e$. Similar renormalizations of phonons and EPC behaviors are also expected to be observed in isologues like ZrSiSe, ZrSiTe and HfSiSe, for the similarity of structure and consequent surface floating states. The findings and the detailed studies of EPC in this work will be important for potential applications of these semimetal materials.
\section*{\label{sec:acknow}Acknowledgement}
This work was supported by the National Key Research \& Development Program of China (Nos. 2016YFA0302400, 2017YFA0303600, and 2017YFA0302901). X. Z. was supported by the National Natural Science Foundation of China (No. 11874404) and the Youth Innovation Promotion Association of Chinese Academy of Sciences (No. 2016008).
J. G. was partially supported by BAQIS Research Program (No. Y18G09). Y. S. was partially supported by the Beijing Natural Science Foundation (No. Z180008)
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1,116,691,499,224 | arxiv | \section*{\LARGE\bf #1}%
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\begin{document}
\head{Neutrino - Photon Interactions at Low Energy }
{Ara N.~Ioannisian}
{Yerevan Physics Institute, Yerevan 375036, Armenia and \\
Max-Planck-Institut f\"ur Physik (Werner-Heisenberg-Institut),
F\"ohringer Ring 6, 80805 M\"unchen, Germany}
\subsection*{Abstract}
We discuss $\nu - \gamma$ interactions in the presence
of a homogeneous magnetic field with energies less than pair production
threshold. The neutrinos are taken to be
massless with only standard-model couplings. The magnetic field
fulfills the dual purpose of inducing an effective neutrino-photon
vertexes and of modifying the photon dispersion relation.
Our conclusion is $\nu - \gamma$ interactions are too weak to be of
importance for pulsar physics.
\vspace{0.5cm}
\subsection*{Introduction}
By now it is well known that in many astrophysical environments the
absorption, emission, or scattering of neutrinos occurs in the presence of
strong magnetic fields ~\cite{r}. Of particular conceptual interest are
$\nu-\gamma$
interactions.
These interactions do not occur in vacuum because neutrinos do not couple
to photons and they are kinematically forbidden. In the presence of
an external field, neutrinos acquire an effective coupling to photons by
virtue of intermediate charged particles.
In addition, external fields modify the dispersion relations of all
particles so that phase space is opened for neutrino-photon reactions of
the type $1\to 2+3$.
The processes, which we'll discuss here, are related to the process of
photon splitting that may
occur in magnetic fields ~\cite{adler}.
In photon splitting the magnetic field plays
the dual role of providing an effective three-photon vertex which does not
exist in vacuum, and of modifying the dispersion relation of the
differently polarized modes such that $\gamma\to\gamma\gamma$ becomes
kinematically allowed for certain polarizations of the initial and final
states.
We would like to stress that $\nu-\gamma$ interactions are important when
typical energies of the particles involving in the reactions are less than
$2\ m_e \simeq 1MeV$.
If $E > 1MeV$ it becomes important reactions where involve real
electron-positrons and $\nu-\gamma$ interactions become less important. In
the last case one may consider $\nu-\gamma$ processes as radiational
corrections to those where photons do not exist.
Therefore, we are interested to $\nu-\gamma$ processes with typical
energies $E << 1MeV$ ( numerically our results work for $E < 0.5MeV$).
Energy limitation
allows us to use the limit of infinitely heavy
gauge bosons and thus an effective four-fermion interaction,
\begin{equation}
{\cal L}_{\rm eff}=
-\frac{G_F}{\sqrt{2}}\,\bar{\nu} \gamma_{\mu}(1-\gamma_5)\nu\,
\bar{E}\gamma^{\mu}(g_V-g_A \gamma_5)E.
\end{equation}
Here, $E$ stands for the electron field,
$\gamma_5=i\gamma^0\gamma^1\gamma^2\gamma^3$,
$g_V=2\sin^2\theta_W+\frac{1}{2}$ and
$g_A=\frac{1}{2}$ for $\nu_e$, and
$g_V=2\sin^2\theta_W-\frac{1}{2}$ and
$g_A=-\frac{1}{2}$ for $\nu_{\mu,\tau}$.
In our subsequent calculations we will always use
$\sin^2\theta_W=\frac{1}{4}$ for the weak mixing angle so that the
vector coupling will identically vanish for $\nu_\mu$ and
$\nu_\tau$.
We'll consider following amplitudes
$\bar\nu \nu$, $\bar\nu \nu \gamma$, $\bar\nu \nu \gamma \gamma$
and related processes.
In the Standard Model neutrino current couple to the electron via vector
or
axial-vector couplings.
For vector coupling, one may get $\nu- \gamma$ interaction amplitudes
using Euler-Heisenberg effective lagrangian of constant electromagnetic
field.
\begin{eqnarray}
{\cal L}=-{\cal F} -\frac{1}{8\pi^2}\int_0^{\infty}
\frac{ds}{s^3}e^{-m^2_es}[(es)^2L -c.t. ]\\
L=i{\cal G}\frac{\cosh [es\sqrt{2({\cal F}+i{\cal G})}]+
\cosh[es\sqrt{2({\cal F}-i{\cal G})}]}
{\cosh [es\sqrt{2({\cal F}+i{\cal G})}]-
\cosh[es\sqrt{2({\cal F}-i{\cal G})}]}\\
{\cal F}=\frac{1}{4}F_{\mu\nu}^2,
{\cal G}=\frac{1}{4}F_{\mu\nu}\widetilde{F}_{\mu\nu},
\widetilde{F}^{\mu\nu}=\frac{1}{2}\epsilon^{\mu \nu \rho \sigma}F_{\rho\sigma}
\end{eqnarray}
In the Euler-Heisenberg lagrangian we'll assume
\begin{equation}
F_{\mu\nu}=\bar{F_{\mu\nu}}+N_{\mu\nu}+f^1_{\mu\nu}+f^2_{\mu\nu}+...
\end{equation}
here $\bar{F_{\mu\nu}}$ is the external constant electromagnetic field,
$N_{\mu\nu}=
g_v\frac{G_F}{\sqrt{2}}
\{
\partial_{\mu}\bar\nu\gamma_{\nu}(1-\gamma_5)\nu -
\partial_{\nu}\bar\nu\gamma_{\mu}(1-\gamma_5)\nu\}$and $f^i_{\mu\nu}$ are
the fields of real photons involving in the process. We'll get
$\nu-\gamma$ interaction
amplitudes for vector coupling expanding the Euler-Heisenberg lagrangian
to the Tylor series with respect to the weak fields ($N_{\mu\nu}$,
$f^i_{\mu\nu}$).
Unfortunately we could not use the same procedure for axial-vector
coupling. Therefore we calculated each amplitude separately for
axial-vector coupling.
\subsection*{The $\bf \bar{\nu}\nu$ effective term}
This amplitude will allow us to get dispersion relation of neutrinos in an
external field.
One may expand to the Tylor series the Heisenberg-Euler lagrangian and get
vector part of this
amplitude. We have calculated$^1$ axial-vector part of this amplitude
~\cite{acr}
\begin{eqnarray}
{\it
L}_A&=&\frac{g_AG_F}{\sqrt{2}32\pi^2}\int_0^{\infty}\frac{ds}{s^2}e^{-m^2_es}
[A\partial_{\nu}+B_{\mu\nu}\partial_{\mu}]\bar{\nu}\gamma_{\nu}(1-\gamma_5)\nu\\
A&=&-4e^2s^2{\cal G}\nonumber\\
B_{\mu\nu}&=&\frac{4i}{e^2s^2}\frac{dC_1}{dF_{\mu\lambda}}
\frac{dC_2}{dF_{\lambda\nu}}\\
C_1&=&\frac{2ie^2s^2{\cal G}}{\cosh [es\sqrt{2({\cal F}+i{\cal G})}]-
\cosh[es\sqrt{2({\cal F}-i{\cal G})}]}\\
C_2&=&-2[\cosh [es\sqrt{2({\cal F}+i{\cal G})}]-
\cosh[es\sqrt{2({\cal F}-i{\cal G})}]]
\end{eqnarray}
It is easy to see that the effect is very small for external fields
$E,B \ll \frac{m^2_{W}}{e}$
\subsection*{The $\bar{\nu}\nu \gamma$ effective vertex}
It is shown in ~\cite{ar}
that the vector part of this matrix element is
proportional to the effective charge radius and may be neglected. Axial
vector part of the amplitude has the form ~\cite{ar}
\begin{equation}
\label{m}
M_5=\frac{g_Ae^3G_F}{\sqrt{2}4\pi^2m_e^2\,e}Z\varepsilon_{\mu}
\bar{\nu}\gamma_{\nu}(1-\gamma_5)\nu\,
\Bigl\{-C_\|\,k_{\|}^{\nu}(\widetilde{F} k)^{\mu}\
+ \ C_\bot\,\Bigl[k_{\bot}^{\nu}(k\widetilde{F})^{\mu}
+k_{\bot}^{\mu}(k\widetilde{F})^{\nu}-
k_{\bot}^2\widetilde{F}^{\mu \nu}\Bigr]\Bigr\},
\end{equation}
where $\varepsilon$ is the photon
polarization vector. At low energy $C_\bot$ and $C_\|$ are
real functions on $B$.
For $E_{\gamma}<2m_e$ the photon refractive
index always obeys the Cherenkov condition $n>1$ ~\cite{adler}.
Therefore only the Cherenkov process $\nu\to \nu \gamma$ is kinematically
allowed.
The four-momenta conservation constrains the photon emission angle to have
the value
\begin{equation}\label{emissionangle}
\cos \theta = \frac{1}{n} \
\left[1+(n^2-1)\frac{\omega}{2E}\right],
\end{equation}
where $\theta$ is the angle between the emitted photon and incoming
neutrino.
It turns out that for all situations of practical interest we have
$|n-1|\ll 1$ ~\cite{adler}.
This reveals that the outgoing photon
propagates parallel to the original neutrino direction.
It is interesting to compare this finding with the standard
plasma decay process $\gamma\to\bar\nu\nu$ which is dominated by the
$\Pi^{\mu \nu}$. Therefore, in the approximation
$\sin^2\theta_W=\frac{1}{4}$ only the electron
flavor contributes to plasmon decay. Here the Cherenkov rate is equal for
(anti)neutrinos of all flavors.
For neutrinos which propagate perpendicular to the magnetic
field, a Cherenkov emission rate can be written in the form
\begin{eqnarray}\label{finalresult}
\Gamma\ \approx = \
2.0\times10^{-9}~{\rm s}^{-1}~\left(\frac{E_{\nu}}{2m_e}\right)^5
\left(\frac{eB}{m^2_e}\right)^2 h(B),
\end{eqnarray}
where
\begin{equation}
h(B)=
\cases{(4/25)\,(eB/m^2_e)^4&for $eB\ll m^2_e$,\cr
1&for $eB\gg m^2_e$.\cr}
\end{equation}
Turning next to the case $E>2m_e$ we note that in the presence of a
magnetic field the electron and positron wavefunctions are Landau
states so that the process $\nu\to\nu e^+e^-$ becomes kinematically
allowed. Therefore, neutrinos with such large energies will
lose energy primarily by pair production rather than by Cherenkov
radiation ~\cite{ara10}.
\subsection*{The $\bar{\nu}\nu \gamma \gamma$effective vertex}
In the vacuum this amplitude is highly suppressed. The axial-vector
coupling is zero due to Landau-Yang theorem ( two photons cannot have
total spin equal to one) in
four-Fermi limit.
Beyond four-Fermi limit this amplitude is double Fermi suppressed.
In ~\cite{dr1} it was found that $\bar{\nu}\nu \to \gamma \gamma$ (or
crossed versions) cross-section is
$\sigma \sim 10^{-68}(\frac{\omega}{MeV})^6 cm^2$.
However, this amplitude is not double Fermi suppressed in a magnetic field.
In ~\cite{s} it was shown that in a weak magnetic field
($B \ll \frac{m_e^2}{e}$)
vector coupling is dominant and cross section of the process is
$\sigma \sim 10^{-50}(\frac{\omega}{MeV})^6 (\frac{eB}{m_e^2})^2cm^2$.
In a strong magnetic field \cite{ak} contribute both- vector and axial
vector couplings. Since $B \gg \frac{m_e^2}{e}$ in the amplitude survive
only lowest Landau levels and we lost dependence on magnetic field
$\sigma \sim 10^{-50}(\frac{\omega}{MeV})^6cm^2$.
\subsection*{The $\bar{\nu}\nu \gamma\gamma\gamma$ effective vertex}
In this amplitude contribute only vector coupling ~\cite{dr}. The cross
section,~\cite{dr}
of
the process $ \bar{\nu}\nu \to \gamma\gamma\gamma$ (or crossed processes
$\nu \gamma \to \nu \gamma \gamma$ and
$\gamma\gamma \to \bar{\nu}\nu\gamma$), is
$\sigma \sim 10^{-50}(\frac{\omega}{MeV})^{10} cm^2$.
\subsection*{Conclusions}
The
strongest magnetic fields known in nature are near pulsars. However,
they have a spatial extent of only tens of kilometers. Therefore, even
if the field strength is as large as the critical one, most neutrinos
escaping from the pulsar or passing through its magnetosphere will not
interact with photons. Thus, the magnetosphere of a pulsar is quite
transparent to neutrinos.
Therefore our main conclusion is $\nu-\gamma$ interactions are too weak to
be of
practical importance for pulsar physics.
\subsection*{Acknowledgments}
It is pleasure to thanks the organizers of the Neutrino Workshop at the
Ringberg Castle for organizing a very interesting workshop. A.I.
acknowledges support from the A.von Humboldt Foundation.
|
1,116,691,499,225 | arxiv | \section{Introduction}
\label{intro}
The $\pi$ and $\rho$ are similar states consisting of an $u$-quark and a $\bar{d}$-antiquark except for the spins as $\pi$ is in the spin-singlet state while $\rho$ is in the triplet.
Therefore we can easily attribute any differences to the spin-dependent nature of the interaction in the two states.
However their mass difference is quite large in comparison with the mass differences between other meson pairs such as $\eta_C$ and $J/\Psi$ ($c\bar{c}$) or $D_0$ and $D^*_0$ ($c\bar{u}$).
In this ground the large mass difference might also be due to the small mass of $\pi$, which is explained as a result of the chiral symmetry breaking in QCD.
Therefore it will be helpful to identify the origin of the small mass of $\pi$ and to trace the big mass difference between $\pi$ and $\rho$.
Both $\pi$ and $\rho$ are quark-antiquark systems which can be described within non-relativistic framework.
However quarks are light particles and interact strongly; they should be treated relativistically.
Furthermore, spin-dependence is purely relativistic in nature and cannot be derived non-relativistically.
Therefore we need a relativistic description rather than a non-relativistic description, for the quark-antiquark systems.
Todorov has suggested the two-body relativistic equations for spinless two-body systems \cite{todorov}.
Based on the Dirac's constraint formalism and a minimal interaction structure the TBDE has been tested and confirmed in classical \cite{crater92} and quantum field theory \cite{jallouli97}.
Within this formalism, two coupled 16-component Dirac equation with 2 fermion basis is reduced to the single 4-component 2nd-order differential equation and the radial part of this equation are simply Schr\"{o}dinger-type with various terms calculated from a basic radial potential, which can be phenomenologically adopted.
We have constructed the two-body Dirac equations (TBDE) \cite{alstine82,yoon09} for two interacting fermions in a form that is as easy to handle as the non-relativistic Schr\"{o}dinger equation.
The brief description of this formalism is summarized in Sec. 2.
For the interactions between a quark and an antiquark we adopt some phenomenological potential, which will be described in Sec. 3.
Final numerical results and discussions will be presented in Sec. 4.
\section{The Relativistic Schr\"{o}dinger-like equation}
\label{sec:2}
Van Alstine and Crater utilized the two body Dirac constraint dynamics for two interacting fermions with mass-shell constraint to reduce the relative energy and time degrees of freedom \cite{crater92,alstine82}.
They applied the Pauli reduction and scale transformation and succeeded in reducing the 16 component Dirac equation into a 4 component relativistic Schr\"{o}dinger-type equation,
\begin{equation}
\label{schreq}
{\cal{H}} \psi = {{\epsilon_1 {\cal H}_1 + \epsilon_2 {\cal H}_2} \over w} \psi
= \left\{ p^2 + \Phi_w (\sigma_1, \sigma_2, p_\perp, A(r), S(r)) \right\} \psi
=b^2(w) \psi,
\end{equation}
where $w$ is the invariant mass and $b^2(w)=\epsilon^2_w-m^2_w$ with the relativistic energy $\epsilon_w$ and the relativistic reduced mass $m_w$.
$\Phi_w$ plays a role of interaction in the Schr\"{o}dinger-type equation (\ref{schreq}) and includes many relativistic terms: Darwin($D$), spin-spin($SS$), tensor($T$), spin-orbit($SO$), etc.
For a scalar potential $S$ and a time-like vector potential $A$,
$\Phi_w$ can be expressed as follows \cite{yoon09} :
\begin{eqnarray}
\label{interaction}
\Phi_w &=& 2m_wS+S^2+2\epsilon_wA-A^2+\Phi_D+\L\cdot(\sigma_1+\sigma_2)\Phi_{SO}
+\sigma_1\cdot\sigma_2\Phi_{SS} \\
\nonumber
&&+\sigma_1\cdot\hat{r}\sigma_2\cdot\hat{r} \L\cdot(\sigma_1+\sigma_2)\Phi_{SOT}
+(3\sigma_1\cdot\hat{r}\sigma_2\cdot\hat{r}-\sigma_1\cdot\sigma_2)\Phi_{T} \\
\nonumber
&&+\L\cdot(\sigma_1-\sigma_2)\Phi_{SOD}
+i\L\cdot(\sigma_1\times\sigma_2)\Phi_{SOX}.
\end{eqnarray}
If $S$ and $A$ are central radial potentials, this equation is reduced further into two-coupled radial equations.
For $\pi$ which is in a spin-singlet state, the differential equation becomes
\begin{equation}
\label{pieq}
\left\{ -{d^2\over dr^2}+2m_wS+S^2+2\epsilon_wA-A^2+\Phi_D-3\Phi_{SS}
\right\} v_0=b^2v_0.
\end{equation}
$\rho$ is in a spin-triplet state, which is a superposition of the S- and D-states.
These two states are coupled by the following differential equations
\begin{equation}
\label{rhoeq1}
\left\{ -{d^2\over dr^2}+2m_wS+S^2+2\epsilon_wA-A^2+\Phi_D+\Phi_{SS}
\right\} u_S
+{2\sqrt{2} \over 3}(3\Phi_T-6\Phi_{SOT})u_D =b^2 u_{S}, \\
\end{equation}
\begin{eqnarray}
\label{rhoeq2}
\left\{ -{d^2\over dr^2}+{6\over r^2}+2m_wS+S^2+2\epsilon_wA-A^2+\Phi_D-6\Phi_{SO}+\Phi_{SS}-2\Phi_{T}+2\Phi_{SOT}
\right\} u_D \\ \nonumber
+{2\sqrt{2} \over 3}(3\Phi_T)u_S =b^2 u_{D}.
\end{eqnarray}
Here $v_0=\psi_\pi(r)/r$, and $u_S$ and $u_D$ are corresponding quantities of $\rho$ for the S-state and D-state, respectively.
Solving these equations are not easy but at least manageable numerically.
\section{Phenomenological Potential}
\label{sec:3}
To complete the differential equation, we need a basic potentials between a quark and an antiquark.
Commonly used potential for two quarks are the static Cornell potential,
which is composed of a static Coulomb-type potential and a linear potential \cite{eichten80}.
However this potential does not reflect the asymptotic freedom which is one of the important characteristics of the $q-\bar{q}$ interaction. Richardson\cite{richardson79} has suggested the following form for heavy quarkonia,
\begin{equation}
\tilde{V}(q) = -{ 16\pi\over 27}{1 \over {q^2 \ln (1+q^2 /\Lambda^2)}}
\end{equation}
in momentum space.
After Fourier transform to coordinate system, it becomes
\begin{equation}
V(r) = { 8\pi\Lambda^2r \over 27}-{ 8\pi f(\Lambda r) \over 27r}.
\end{equation}
The function $f(\Lambda r)$ needs to approach $-1/\ln (\Lambda r)$ as $r \rightarrow 0$ and to approach 1 as $r \rightarrow \infty$ for the potential satisfying both crucial characteristics in QCD, the asymptotic freedom and the confinement.
We modified this potential using 3 parameters, $\Lambda$, $K$, and $B$,
\begin{equation}
\label{modRpot}
V(r) = { 8\pi\Lambda^2r \over 27}-{ 16\pi \over 27r \ln (Ke^2+B/(\Lambda r)^2 )}.
\end{equation}
These 3 parameters will be determined to fit the experimental values of meson masses.
The one feature of this method is that there is no cutoff parameter introduced.
Everything is treated non-perturbatively and it gives good results for heavy quark meson systems as well as light two quark systems.
Partitioning the confinement term in $V(r)$ of Eq.(\ref{modRpot}) as an invariant $S$ responsible for the scalar potential and the Coulomb-like term as an invariant $A$ responsible for the full four-vector potential, we can obtain the specific forms for the various terms in Eq.(\ref{interaction}). The readers can refer the Eqs.(A17)-(A20) in the Ref.\cite{yoon09} for their explicit forms.
\section{Results and Conclusion}
\label{sec:3}
\begin{figure}[b]
\centering
\includegraphics[width=0.55\textwidth]{Fig1.pdf}
\caption{The wave functions of $\pi$ and $\rho$.
The solid (black in color) line is $\pi$ wave function.
The dashed (red in color) curve is the S-state wave function of $\rho$ and
the dotted (blue in color) curve is the D-state wave function of $\rho$.}
\label{fig:1}
\end{figure}
We fit the invariant masses of 32 mesons and determine 3 parameters in the potential. Their values are $\Lambda=0.4218$ GeV, $B=0.05081$, $K=4.198$.
Here the quark masses used to give the best fit are $m_u=0.0557$ GeV, $m_d=0.0553$ GeV.
Then using these values we solve the differential equations, Eq.(\ref{pieq}) for $\pi$ and Eqs.(\ref{rhoeq1}) and (\ref{rhoeq2}) for $\rho$,
and obtain their wave functions.
They are shown in Fig.1.
The wave functions for both mesons are not singular at short distances and the asymptotic behavior of D-state wave function of $\rho$ is proportional to that of its S-state wave function at small radii.
With the given wave functions we can obtain the eigenvalues, $b^2$, which in turn give the invariant masses of 0.159 GeV for $\pi$ and 0.792 GeV for $\rho$, while their experimental values are 0.140 GeV and 0.775 GeV, respectively.
We tabulated the contribution of each term in Eqs.(\ref{pieq})-(\ref{rhoeq2}) in Table \ref{tab:0}.
The spin-spin contribution ($<-3\Phi_{SS}>$) for pion is larger than the other spin-dependent terms by an order of magnitude.
However for $\rho$ the same contribution ($<\Phi_{SS}>$) is smaller than the other spin-dependent contributions by an order of magnitude.
Therefore the spin-dependent term plays crucial role for the mass difference between $\pi$ and $\rho$.
The major contribution for pion mass comes from the Darwin ($\Phi_{D}$) and spin-spin interaction terms. The magnitudes of these two terms are about same but opposite in sign, and therefore they almost cancel each other, resulting in a small pion mass.
However for the $\rho$, the Darwin term is not dominant at all.
It is about the same as or less than the off-diagonal contributions from tensor ($\Phi_{T}$) or spin-orbit-tensor ($\Phi_{SOT}$) terms.
Besides that the spin-spin contribution is much smaller than the Darwin or the
off-diagonal terms, it contributes in same direction as the Darwin term, as distinct from the pion case.
Overall the $\rho$ mass is not that small.
We can also notice that the kinetic energy
contributions ($<-d^2/dr^2 + J(J+1)/r^2>$) are
almost 1 GeV for $\pi$ and 0.5 GeV for $\rho$.
These numbers are much larger than the quark masses of $\sim 0.0557$ GeV and
it justifies to treat the quarks relativistically.
Finally we can conclude that the cancellation between the Darwin and spin-spin interaction terms occurs in $\pi$ mass but not in $\rho$ mass and this is the main source of a big difference in two masses.
Also we can say that our theory is consistent with the chiral symmetry breaking which causes the non-zero small mass of pion.
\begin{table}[t]
\caption{The contributions of all terms in Eqs.(\ref{pieq})-(\ref{rhoeq2}).
The column $<00>$ is for $\pi$ with S-state only
and the remaining four columns are the diagonal and off-diagonal terms
for $\rho$ with both S- and D-state wave functions.
The numbers in () are experimental values.
All terms except $w$ are in a unit of $\mbox{GeV}^2$ and $w$ is in
a unit of GeV. }
\centering
\label{tab:0}
\begin{tabular}{c||c||r|r|r|r}
\hline\noalign{\smallskip}
& $\pi$ & \multicolumn{4}{c}{$\rho$} \\
\cline{2-6}
Terms & $<00>$ & $<SS>$ & $<DD>$ & $<SD>$ & $<DS>$ \\
\tableheadseprule \hline
$<-d^2/dr^2 + J(J+1)/r^2>$ & {\bf 0.8508} &
{\bf 0.3085} & {\bf 0.2812} & & \\
\hline
$\matrix{<2m_w S> \cr <S^2> \cr <2 \epsilon_w A> \cr <-A^2 > \cr
{\bf \mbox{Sub Total}}}$ &
$\matrix{~~0.0103 \cr ~~0.0942 \cr -0.0598 \cr -0.4279 \cr {\bf -0.3832} }$ &
$\matrix{~~0.0026 \cr ~~0.1631 \cr -0.2091 \cr -0.1247 \cr {\bf -0.1680}}$ &
$\matrix{~~0.0006 \cr ~~0.0446 \cr -0.0211 \cr -0.0094
\cr ~~{\bf 0.0146}}$ & &
\\ \hline
$\matrix{<\Phi_D> \cr <-3\Phi_{SS}> \cr <\Phi_{SS}> \cr <-2\Phi_T>
\cr <-6\Phi_{SO}> \cr <2\Phi_{SOT}> \cr {\bf \mbox{Sub Total}}}$ &
$\matrix{-3.804 \cr ~~3.340 \cr\cr\cr\cr\cr {\bf -0.4643}}$ &
$\matrix{ -0.2790 \cr\cr -0.0364 \cr \cr \cr \cr {\bf -0.3154}}$ &
$\matrix{ -0.0436 \cr\cr -0.0095 \cr -0.0417 \cr ~~0.1133 \cr ~~0.0215
\cr ~{\bf 0.0346}}$ & & \\
\hline
$\matrix{(2\sqrt{2}/3)<3\Phi_T> \cr (2\sqrt{2}/3)<-6\Phi_{SOT}>
\cr {\bf \mbox{Sub Total}}}$ & & & &
$\matrix{ -0.3088 \cr ~~0.6177 \cr ~~{\bf 0.3090}}$ &
$\matrix{ -0.3088 \cr \cr {\bf -0.3088}}$ \\
\tableheadseprule \hline
Total Sum & {\bf 0.0033} & {\bf -0.1750} & {\bf 0.3158} & {\bf 0.3090}
& {\bf -0.3088} \\
\hline
$b^2$ & {\bf 0.0033} & \multicolumn{4}{c}{ {\bf 0.1411}} \\
\hline
$w(\mbox{GeV})$ & {\bf 0.159 (0.140)} & \multicolumn{4}{c}{~ {\bf 0.792 (0.775)}} \\
\hline
\end{tabular}
\end{table}
\vspace{-0.3cm}
\begin{acknowledgements}
This work was supported by National Research Foundation of Korea under the program number 2011-0003707 and by the Office of Nuclear Physics, U.S. Department of Energy.
\end{acknowledgements}
\vspace{-0.6cm}
|
1,116,691,499,226 | arxiv | \section{Introduction}
If $Y$ is a normed space and $X \subset Y$, then we set ${\rm dist}_Y (f, X) = \inf_{g \in X} \| f - g \|_Y$. If the
space $Y$ is clear from the context, we write simply ${\rm dist}(f, X)$. In the problems we are going to consider,
$X$ itself is going to be a (quasi)-Banach space.
We denote by $H(\Omega)$ the space of all holomorphic functions on an open set $\Omega \subset \mathbb C^n$.
In this paper we consider distance problems in weighted Bergman spaces over the upper half-plane
$\mathbb C_+ = \{x + iy : y > 0 \}$ and over a bounded strictly pseudoconvex domain $\Omega \subset \mathbb C^n$
with smooth boundary.
The weighted Bergman space $A^p_\alpha (\mathbb C_+)$ consists of all functions $f\in H(\mathbb C_+)$ such that
$$\| f \|_{p,\alpha} = \left(\int_0^\infty \int_{-\infty}^\infty |f(x+iy)|^p y^\alpha dx dy \right)^{\frac{1}{p}}
< \infty,$$
where $\alpha > -1$ and $0 < p < \infty$ (see \cite{MDAD} and \cite{DK}). The above spaces are Banach spaces for $p \geq 1$ and complete metric spaces for $0<p<1$. It is natural to consider the space $A^\infty_\nu = A^\infty_\nu(\mathbb C_+)$ of all holomorphic functions $F$ on $\mathbb C_+$ satisfying
$$\| F \|_{A^\infty_\nu} = \sup_{y>0} \sup_{x \in \mathbb R} |F(x+iy)|y^{\nu} < \infty,$$
where $\nu > 0$, this space is also a Banach space.
Let $D$ be a bounded strictly pseudoconvex domain in $\mathbb C^n$ with smooth boundary and let $\delta(z)$ denote the distance from $z \in D$ to the boundary of $D$ with respect to some Riemannian metric, see \cite{Be}. We define
$$L^p_s(D) = L^p(D, \delta(z)^{s-\frac{n+1}{p}}dV(z)),$$
where $sp>n$, $0<p<\infty$ and $dV(z)$ is the volume element on $D$, (see \cite{Be}). Set
$$A_s^p(D) = L^p_s(D) \cap H(D),\;\;\;\; sp>n, \;\; 0<p<\infty,$$
$$A^\infty_s(D) = \{ f \in H(D) : \sup_{z\in D} |f(z)|\delta^s (z) < \infty \},\;\; s \geq 0.$$
These spaces are Banach spaces for $1 \leq p \leq \infty$ and complete metric spaces for $0<p<1$.
In this paper we investigate the following two problems: estimate the distance, in
$A^\infty_{\frac{\nu+2}{q}}(\mathbb C_+)$ norm,
from $f \in A^\infty_{\frac{\nu+2}{q}}(\mathbb C_+)$ to $A^q_\nu(\mathbb C_+)$ and estimate the distance,
in $A^\infty_s(D)$ norm, from $f \in A^\infty_s(D)$ to $A^q_s$. In both cases we give sharp results. The next section
deals with the upper half-plane case, the last one deals with strongly pseudoconvex domains. Techniques used to obtain
our results in these two different settings are similar, and the same ideas were used to study analogous problems for analytic Besov spaces in the unit ball and polydisc (see \cite{ShM} and \cite{SM}). The literature on the extremal
problems in spaces of analytic functions is extensive, even in the case of the unit disk, a classical exposition of
these problems treated by duality methods developed by S. Havinson, W. Rogosinski and H. Shapiro can be found in \cite{Du}.
\section{Distance problems in $A^p_{\frac{\nu + 2}{p}}(\mathbb C_+)$ spaces}
The main tool in our investigation is Bergman representation formula. We first collect the results needed in the proofs
of our theorems. The following result is contained in \cite{MDAD}:
\begin{lem}\label{linre}
If $f \in A^p_\alpha (\mathbb C_+)$, $0<p<\infty$ and $\alpha > -1$ then
$$f(z) = \frac{\beta + 1}{\pi} \int_0^\infty \int_{-\infty}^\infty
\frac{f(w) (\Im w)^\beta}{(\overline w - z)^{2+\beta}} dm_2(w),$$
where $0 < p \leq 1$, $\beta \geq \frac{2+\alpha}{p} -2$ or $1 \leq p < \infty$, $\beta \geq
\frac{1+\alpha}{p}-1$.
\end{lem}
Here and in the following $m_2$ denotes two dimensional Lebesgue measure. The next two results are contained in \cite{DK}.
\begin{lem}\label{lcub}
Let $f \in H(\mathbb C_+)$. Assume $\mathbb C_+$ is covered by dyadic squares: $\mathbb C_+ = \cup_{k=1}^\infty
\Delta_k$ and let $(\Delta_k^\star)$ be the corresponding family of enlarged squares (see \cite{DK}). Then
$\Delta_k^\star$ is a finitely overlapping covering of $\mathbb C_+$ and
$$\sup_{z \in \Delta_k} |f(z)|^p (\Im z)^\alpha \leq \frac{C}{|\Delta_k^\star|} \int_{\Delta_k^\star}
|f(z)|^p (\Im w)^\alpha dm_2(w)$$
for $0<p<\infty$ and $\alpha > -1$.
\end{lem}
\begin{lem}\label{lines}
Let $\Delta_k$ and $\Delta_k^\star$ are as in the previous lemma, let $w_k$ be the center of the diadic square $\Delta_k$. Then we have:
$$(\Im w_k)^2 \asymp | \Delta_k | = m_2(\Delta_k) \asymp m_2(\Delta_k^\star),$$
$$|\overline w - z | \asymp |\overline w_k - z|, \;\; w\in \Delta_k \;\;\;, z \in \mathbb C_+,$$
$$\Im w \asymp \Im w_k, \;\;\; w \in \Delta_k$$
and the following integral estimate:
$$\int_{\mathbb C_+} \frac{(\Im z)^\alpha dm_2(z)}{|\overline w - z|^{(2+\beta)p}} \leq C
(\Im w)^{\alpha + 2 - (\beta + 2)p},\;\;\; w \in \mathbb C_+$$
valid for all $\beta$ satisfying $(\beta + 2)p - 2 > \alpha$, $\alpha > -1$.
\end{lem}
\begin{thm} {\rm (see \cite{BB} and \cite{BBG})} Let $0< p < \infty$, $\nu > 0$. Then there is a constant
$C = C(p, \nu) > 0$ such that for all $x+iy \in \mathbb C_+$ and all $F \in A^p_\nu(\mathbb C_+)$ we have
$$|F(x+iy)| \leq C y^{-\frac{\nu + 2}{p}} \| F \|_{A^p_\nu(\mathbb C_+)}.$$
Also, there is a constant $C = C(p, \nu) > 0$ such that for all $y > 0$ and all $F \in A^p_\nu(\mathbb C_+)$ we have
$$\left(\int_{-\infty}^\infty |F(x+iy)|^p dx \right)^{1/p} \leq C y^{-\frac{\nu + 1}{p}} \| F \|_{A^p_\nu(\mathbb C_+)}.$$
\end{thm}
We clearly have
$$\| F \|_{A^\infty_{\frac{\nu + 2}{p}}} \leq C \| F \|_{A^p_\nu}$$
for $0<p<\infty$, $\nu > 0$ and $F \in A^p_\nu(\mathbb C_+)$.
The next two theorems show that ${\rm dist}_{A^q_{\frac{\nu + 2}{q}}}(f, A^q_\nu)$ can be explicitly given in the case
of the upper halfplane $\mathbb C_+$. We treat separately cases $0<q\leq 1$ and $q>1$. The distance we are looking for
is described using the following sets:
$$V_{\epsilon, t}(f) = \{ z = x+iy \in \mathbb C_+ : |f(x+iy)|y^t \geq \epsilon \}.$$
\begin{thm}\label{qg1}
Let $q > 1$, $\nu > -1$, $t=\frac{\nu + 2}{q}$, $\beta > \max (\frac{\nu}{q}, \frac{\nu + 2}{q} - 1)$ and
$f \in A^\infty_{\frac{\nu + 2}{q}}(\mathbb C_+)$. Then $l_1 = l_2$ where
$$l_1 = {\rm dist}_{A^\infty_{\frac{\nu + 2}{q}}}(f, A^q_\nu),$$
\begin{equation}\label{ggeps}
l_2 = \inf \left\{ \epsilon > 0 : \int_{\mathbb C_+} \left( \int_{V_{\epsilon, t}(f)}
\frac{ (\Im w)^{\beta -t} dm_2(w)}{|\overline w - z|^{\beta + 2}} \right)^q (\Im z)^\nu dm_2(z) < \infty \right\}.
\end{equation}
\end{thm}
\begin{thm}\label{ql1}
Let $0<q\leq 1$, $\nu > -1$, $t=\frac{\nu +2}{q}$, $\beta > \frac{\nu + 2}{q} - 2$ and
$f \in A^\infty_{\frac{\nu + 2}{q}}(\mathbb C_+)$. Then $t_1 = t_2$ where
$$t_1 = {\rm dist}_{A^\infty_{\frac{\nu + 2}{q}}}(f, A^q_\nu),$$
\begin{equation}\label{gleps}
t_2 = \inf \left\{ \epsilon > 0 : \int_{\mathbb C_+} \left( \int_{V_{\epsilon, t}(f)}
\frac{ (\Im w)^{\beta -t} dm_2(w)}{|\overline w - z|^{\beta + 2}} \right)^q (\Im z)^\nu dm_2(z) < \infty \right\}.
\end{equation}
\end{thm}
We present now the proofs of the above theorems, since they significantly overlap a unified presentation is possible. Assume $l_1 < l_2$ or $t_1 < t_2$. Then there are $\epsilon > \epsilon_1 > 0$ and $f_{\epsilon_1} \in A^q_\nu (\mathbb C_+)$ such that $\| f - f_{\epsilon_1} \|_{A^\infty_{\frac{\nu +2}{q}}} \leq \epsilon_1$ and
$$ \int_{\mathbb C_+} \left( \int_{V_{\epsilon, t}(f)}
\frac{(\Im w)^{\beta - t} dm_2(w)}{|\overline w - z|^{\beta + 2}} \right)^q (\Im z)^\nu dm_2(z) = +\infty.$$
Since $\| f - f_{\epsilon_1} \|_{A^\infty_{\frac{\nu +2}{q}}} \leq \epsilon_1$, from the definition of the
set $V_{\epsilon, t}(f)$ we conclude that
$$(\epsilon - \epsilon_1) \chi_{V_{\epsilon, t}(f)}(z) (\Im z)^{-\frac{\nu + 2}{q}} \leq |f_{\epsilon_1}(z)|$$
and therefore we have
\begin{eqnarray*}
+\infty & = & \int_{\mathbb C_+} \left( \int_{\mathbb C_+}
\frac{\chi_{V_{\epsilon, t}(f)}(w) (\Im w)^{\beta -t} dm_2(w)}{|\overline w - z|^{\beta + 2}} \right)^q
(\Im z)^\nu dm_2(z) \cr
& \leq & \int_{\mathbb C_+} \left( \int_{\mathbb C_+}
\frac{|f_{\epsilon_1}(w)| (\Im w)^\beta}{|\overline w - z|^{2+\beta}} dm_2(w)\right)^q (\Im z)^\nu dm_2(z) = M. \cr
\end{eqnarray*}
Note that this estimate is valid for $0<q<\infty$ and therefore works for both of the above theorems. In both cases we
are going to arrive at the contradiction by proving $M < +\infty$. Let us first consider the case $q>1$. Using
H\"older's inequality and Lemma \ref{lines} (with $\alpha = 0$) we obtain
\begin{eqnarray*}
I(z) & = & \left( \int_{\mathbb C_+} \frac{|f_{\epsilon_1}(w)|(\Im w)^\beta}{|\overline w - z|^{2+\beta}}
dm_2(w) \right)^q \cr
& \leq & \int_{\mathbb C_+} \frac{|f_{\epsilon_1}(w)|^q (\Im w)^{\beta q}}{|\overline w - z|^{\beta q - \epsilon q
+2}} dm_2(w) \cdot \left( \int_{\mathbb C_+} \frac{dm_2(w)}{|\overline w -z|^{\epsilon p + 2}} \right)^{q/p} \cr
& \leq & C \int_{\mathbb C_+} \frac{|f_{\epsilon_1}(w)|^q (\Im w)^{\beta q}}{|\overline w - z|^{\beta q - \epsilon q
+2}} dm_2(w) (\Im z)^{-\epsilon q}\cr
\end{eqnarray*}
for $w \in \mathbb C_+$, $\epsilon > 0$. Using this estimate, Fubini's theorem and Lemma \ref{lines} we obtain
\begin{eqnarray*}
M & \leq C & \int_{\mathbb C_+} |f_{\epsilon_1}(w)|^q (\Im w)^{\beta q} \left( \int_{\mathbb C_+}
\frac{(\Im z)^{\nu - \epsilon q}}{|\overline w - z|^{\beta q - \epsilon q + 2}} dm_2(z) \right) dm_2(w)\cr
& \leq & C \int_{\mathbb C_+} |f_{\epsilon_1}(w)|^q (\Im w)^\nu dm_2(w) < \infty \cr
\end{eqnarray*}
where $\epsilon > 0$ is small enough so that $\nu - \epsilon q > -1$.
Now let us turn to the case $q \leq 1$. We have, using Lemma \ref{lcub} and Lemma \ref{lines},
\begin{eqnarray*}
I(z) & = & \left( \int_{\mathbb C_+} \frac{|f_{\epsilon_1}(w)|(\Im w)^\beta}{|\overline w - z|^{2+\beta}}
dm_2(w) \right)^q \cr
& = & \left( \sum_k \int_{\Delta_k} \frac{|f_{\epsilon_1}(w)|(\Im w)^\beta}{|\overline w - z|^{2+\beta}}
dm_2(w) \right)^q \cr
& \leq & C \sum_{k=1}^\infty \max_{w \in \Delta_k} |f_{\epsilon_1}(w)|^q (m_2(\Delta_k))^q
\frac{(\Im w_k)^{\beta q}}{|\overline w - z|^{(2+\beta)q}} \cr
& \leq & C \int_{\mathbb C_+} \frac{|f_{\epsilon_1}(w)|^q (\Im w)^{\beta q + 2q - 2} dm_2(w)}{|\overline w - z|^{(2+
\beta)q}},\cr
\end{eqnarray*}
in the last estimate we used finite overlapping property of the family of enlarged cubes. Now we get $M < \infty$ as
in the case $q>1$, namely by applying Funini's theorem and integral estimate from Lemma \ref{lines}.
The reverse inequalities $l_1 \leq l_2$ and $t_1 \leq t_2$ can be proved simultaneously. We
fix $\epsilon > 0$ such that the integrals in (\ref{ggeps}), respectively (\ref{gleps}), are finite (if there
are no such $\epsilon > 0$, the inequality is trivial) and use integral representation from Lemma \ref{linre}. This integral representation is valid for $p = \infty$, $\alpha > 0$ and $\beta$ sufficiently large.
\begin{eqnarray*}
f(z) & = & \frac{\beta +1}{\pi}\left( \int_{\mathbb C_+ \setminus V_{\epsilon, t}(f)}
\frac{f(w) (\Im w)^\beta dm_2(w)}{(\overline w - z)^{2+\beta}} + \int_{V_{\epsilon, t}(f)}
\frac{f(w) (\Im w)^\beta dm_2(w)}{(\overline w - z)^{2+\beta}} \right) \cr
& = & \frac{\beta +1}{\pi}(f_1(z) + f_2(z)) \cr
\end{eqnarray*}
where $\beta$ is sufficiently large. The estimate for $f_1(z)$ is immediate, using Lemma \ref{lines}:
$$|f_1(z)| \leq \epsilon \int_{\mathbb C_+} \frac{(\Im w)^{\beta - t} dm_2(w)}{|\overline w - z|^{2+\beta}}
\leq C\epsilon (\Im z)^{-t},$$
and this means $f_1 \in A^\infty_{\frac{\nu + 2}{q}}$ with $\| f_1 \|_{A^\infty_t} \leq C \epsilon$. Next, by the
choice of $\epsilon > 0$ we have:
\begin{eqnarray*}
\| f_2 \|_{A^q_\nu} & \leq & C \| f \|_{A^\infty_t} \int_{\mathbb C_+} \left( \int_{V_{\epsilon, t}(f)}
\frac{(\Im w)^{\beta - t} dm_2(w)}{|\overline w - z|^{2+\beta}} \right)^q (\Im z)^\nu dm_2(z) \cr
& \leq & C \| f \|_{A^\infty_t}.
\end{eqnarray*}
This completes the proofs of both theorems.
We end this section with an observation that the upper halfplane is the simplest tube domain. Analysis on tube domains
over symmetric cones is an active area of research (see \cite{BeBe}, \cite{BB} and \cite{BBG}) and availability of integral representations points to possibility of extending our results to this more general setting.
\section{Distance problems in $A^q_s(\Omega)$ spaces}
We now consider Bergman type spaces in $D \subset \mathbb C^n$, where $D$ is smoothly bounded relatively compact
strictly pseudoconvex domain, providing also sharp results in this case. Our proofs are heavily based on the estimates from \cite{Be}, where more general situation was considered.
Since $|f(z)|^p$ is subharmonic (even plurisubharmonic) for a holomorphic $f$, we have $A^p_s(D) \subset A^\infty_t(D)$ for $0<p<\infty$, $sp > n$ and $t = s$. Also, $A^p_s(D) \subset A^1_s(D)$ for $0 < p \leq 1$ and $A^p_s(D) \subset A^1_t(D)$ for $p > 1$ and $t$ sufficiently large. Therefore we have an integral representation
\begin{equation}\label{inre}
f(z) = \int_{D} f(\xi)K(z,\xi)\delta^t(\xi)dV(\xi),
\end{equation}
where $K(z,\xi)$ is a kernel of type $n+t+1$, that is a measurable function on $D \times D$ such that
$|K(z,\xi)| \leq C |\tilde \Phi(z,\xi)|^{-(n+1+t)}$, where $\tilde \Phi(z, \xi)$ is so called Henkin-Ramirez function
for $D$. From now on we work with a fixed Henkin-Ramirez function $\tilde \Phi$ and a fixed kerenel $K$ of type
$n+t+1$. We are going to use the following results from \cite{Be}.
\begin{lem}\label{be1}{\rm (\cite{Be}, Corollary 5.3.)}
If $r>0$, $0<p\leq 1$, $s>-1$, $p(s+n+1)>n$ and $f \in H(D)$, then we have
$$\left( \int_{D} |f(\xi)| |\tilde \Phi(z,\xi)|^r \delta^s(\xi) dV(\xi) \right)^p \leq C \int_{D}
|f(\xi)|^p |\tilde \Phi(z,\xi)|^{rp} \delta^{p(s+n+1)-(n+1)}(\xi) dV(\xi).$$
\end{lem}
\begin{lem}\label{be2}{\rm (\cite{Be}, Corollary 3.9.)}
Assume $T(z,\xi)$ is a kernel of type $\beta$, and $\sigma > 0$ satisfies
$\sigma + n -\beta < 0$. Then we have
$$\int_{D} T(z, \xi)\delta^{\sigma - 1}(z) dV(z) \leq C\delta^{\sigma + n - \beta}(\xi).$$
\end{lem}
A natural problem is to estimate ${\rm dist}_{A^\infty_s(D)}(f, A^q_s(D))$ where
$0<q<\infty$, $sq>n$ and $f \in A^\infty_s(D)$. We give sharp estimates below, treating cases $0<q\leq 1$ and
$q>1$ separately.
\begin{thm}
Let $0<q\leq 1$, $sq>n$, $f \in A^\infty_s(D)$ and $t > s$ is sufficiently large. Then $\omega_1 = \omega_2$
where
$$\omega_1 = {\rm dist}_{A^\infty_s(D)}(f, A^q_s(D)),$$
$$\omega_2 = \inf \left\{ \epsilon > 0 : \int_{D} \left( \int_{\Omega_{\epsilon, s}} |K(z,\xi)| \delta^{t-s}
(\xi) dV(\xi) \right)^q \delta^{sq-n-1}(z)dV(z) < \infty \right\},$$
where $K(z,\xi)$ is the above kerenel of type $n+t+1$ and
$$\Omega_{\epsilon, s} = \{ z \in D : |f(z)|\delta^s(z) \geq \epsilon \}.$$
\end{thm}
{\it Proof.} Let us prove that $\omega_1 \leq \omega_2$. We fix $\epsilon > 0$ such that the above integral
is finite and use (\ref{inre}):
$$f(z) = \int_{D \setminus \Omega_{\epsilon, s}} f(\xi)K(z,\xi) dV(\xi) + \int_{\Omega_{\epsilon, s}}
f(\xi)K(z,\xi) dV(\xi) = f_1(z) + f_2(z).$$
We estimate $f_1$:
\begin{eqnarray*}
|f_1(z)| & \leq & C \epsilon \int_{D} |K(z,\xi)| \delta^{t-s} dV(\xi) \cr
& \leq & C \epsilon \int_{D} \frac{\delta^{t-s}(\xi) dV(\xi)}{|\tilde \Phi(z,\xi)|^{n+t+1}} \cr
& \leq & C \epsilon \delta^{-s}(z), \cr
\end{eqnarray*}
where the last estimate is contained in \cite{Be} (see p. 375). Next,
\begin{eqnarray*}
\| f_2 \|_{A^q_s}^q & = & \int_{D} |f_2(z)|^q \delta^{sq-n-1}(z)dV(z) \cr
& \leq & C \int_{D} \left(\int_{\Omega_{\epsilon, s}} |f(\xi)| K(z,\xi) \delta^t(\xi) dV(\xi) \right)^q
\delta^{sq-n-1}(z) dV(z)\cr
& \leq & C^\prime \| f \|^q_{A^\infty_s}.\cr
\end{eqnarray*}
Now we have
$${\rm dist}_{A^\infty_s(D)}(f, A^q_s(D)) \leq \| f - f_2 \|_{A^\infty_s(D)} =
\| f_1 \|_{A^\infty_s(D)} \leq C \epsilon.$$
Now assume that $\omega_1 < \omega_2$. Then there are $\epsilon > \epsilon_1 > 0$ and $f_{\epsilon_1} \in A^q_s (D)$ such that $\| f - f_{\epsilon_1} \|_{A^\infty_s} \leq \epsilon_1$ and
$$I = \int_{D} \left( \int_{\Omega_{\epsilon, s}} |K(z,\xi)|\delta^{t-s}(\xi) dV(\xi) \right)^q
\delta^{sq-n-1}(z) dV(z) = \infty.$$
As in the case of the upper half-plane one uses $\| f - f_{\epsilon_1} \|_{A^\infty_s} \leq \epsilon_1$ to obtain
$$(\epsilon - \epsilon_1) \chi_{\Omega_{\epsilon, s}}(z) \delta^{-s}(z) \leq C|f_{\epsilon_1}(z)|.$$
Now the following chain of estimates leads to a contradiction:
\begin{eqnarray}\label{cont}
I & = & \int_{D} \left( \int_{D} \chi_{\Omega_{\epsilon, s}}(\xi) \delta^{t-s}(\xi)K(z,\xi) dV(\xi) \right)^q \delta^{sq-n-1}(z) dV(z) \cr
& \leq & C \int_{D} \left( \int_{D} |f_{\epsilon_1}(\xi)| \delta^t(\xi) K(z,\xi) dV(\xi) \right)^q
\delta^{sq-n-1}(z) dV(z) \cr
& \leq & C \int_{D} \left( \int_{D} |f_{\epsilon_1}(\xi)| \delta^t(\xi)
\frac{dV(\xi)}{|\tilde \Phi(z, \xi)|^{n+t+1}} \right)^q \delta^{sq-n-1}(z) dV(z)\cr
& \leq & C \int_{D} \int_{D} |f_{\epsilon_1}(\xi)|^q
\frac{\delta^{sq-n-1}(z) \delta^{q(t+n+1)-(n+1)}(\xi)}{|\tilde \Phi(z, \xi)|^{q(n+t+1)}} dV(z) dV(\xi) \cr
& \leq & C \int_{D} |f_{\epsilon_1}(\xi)|^q \delta^{sq-n-1}(\xi) dV(\xi) < \infty, \cr
\end{eqnarray}
where we used Lemma \ref{be1} and Lemma \ref{be2} with $\beta = q(n+t+1)$, $\sigma = sq-n$. $\Box$
Next theorem deals with the case $1 < q < \infty$.
\begin{thm}
Let $q>1$, $sq > n$, $t>s$, $t > \frac{s+n+1}{q}$ and $f \in A^\infty_s(D)$. Then $\omega_1 = \omega_2$ where
$$\omega_1 = {\rm dist}_{A^\infty_s(D)}(f, A^q_s(D)),$$
$$\omega_2 = \inf \left\{ \epsilon > 0 : \int_{D} \left( \int_{\Omega_{\epsilon, s}} |K(z,\xi)| \delta^{t-s}
(\xi) dV(\xi) \right)^q \delta^{sq-n-1}(z)dV(z) < \infty \right\}.$$
\end{thm}
{\it Proof.} An inspection of the proof of the previous theorem shows that it extends to this case also, provided
one can prove the estimate:
\begin{eqnarray*}
J & = & \int_{D} \left( \int_{D} |f_{\epsilon_1}(\xi)| \delta^t(\xi) K(z,\xi) dV(\xi) \right)^q
\delta^{sq-n-1}(z) dV(z) \cr
&\leq & C \int_{D} |f_{\epsilon_1}(\xi)|^q \delta^{sq-n-1}(\xi) dV(\xi) < \infty\cr
\end{eqnarray*}
where $q>1$. Using H\"older's inequality and Lemma \ref{be2}, with $\sigma = 1$ and
$\beta = n+1+p\epsilon$, we obtain
\begin{eqnarray*}
I(z) & = & \left( \int_{D} |f_{\epsilon_1}(\xi)| \delta^t(\xi) K(z,\xi) dV(\xi) \right)^q \cr
& \leq & \int_{D} \frac{|f_{\epsilon_1}(\xi)|^q \delta^{tq}(\xi) dV(\xi)}{|\tilde \Phi(z,\xi)|^{n+1+tq-
\epsilon q}} \cdot \left( \int_{D} \frac{dV(\xi)}{|\tilde \Phi(z, \xi)|^{n+1+p\epsilon}} \right)^{q/p}\cr
& \leq & C \int_{D} \frac{|f_{\epsilon_1}(\xi)|^q \delta^{tq}(\xi) dV(\xi)}{|\tilde \Phi(z,\xi)|^{n+1+tq-
\epsilon q}}\; \delta^{-q\epsilon}(z), \cr
\end{eqnarray*}
and this gives
\begin{eqnarray*}
J & \leq & C \int_{D} \int_{D} \frac{|f_{\epsilon_1}(\xi)|^q \delta^{tq}(\xi)\delta^{-q\epsilon +sq-n-1}(z)}{|\tilde \Phi(z,\xi)|^{n+1+tq-\epsilon q}} dV(z)\,dV(\xi) \cr
& \leq & C \int_{D} |f_{\epsilon_1}(\xi)|^q \delta^{sq-n-1}(\xi) dV(\xi) < \infty,\cr
\end{eqnarray*}
where we again used Lemma \ref{be2}, with $\beta = n+1+tq-\epsilon q$ and $\sigma = q(s-\epsilon)-n>0$. $\Box$
{\it Remark.} We note that most results of this paper and the previous one (\cite{SM}) on distances can be extended
to bounded symmetric domains $\Omega \subset \mathbb C^n$. Indeed, the methods of proofs are based on Bergman
representation formula, asymptotic properties of the Bergman kernel and Forelli-Rudin type estimates for integrals
$$\int_\Omega K^\alpha (w,w) |K(z,w)|^\beta dV(w), \;\; \alpha > 0, \; \beta > 0 \;\; z \in \Omega,$$
where $K(z,w)$ is a Bergman reproducing kernel for the weighted Bergman space $A^2_\alpha (\Omega)$. The relevant
estimates can be found in \cite{BeBe} and \cite{FK}.
|
1,116,691,499,227 | arxiv | \section{Introduction}
\label{sec:introduction}
Distributed optimization captures a variety of estimation and learning problems
over networks, including distributed data classification and localization in
smart sensor networks, to name a few. The term ``online'' refers to scenarios
in which the problem data is not available a-priori, but rather it arrives
dynamically while the optimization process is executed. In this paper, we
consider an online distributed optimization set-up in which agents in a network
must cooperatively minimize the sum of local cost functions that depend both on
a local optimization variable and on a global variable obtained by performing
some kind of aggregation of all the local variables (as, e.g., the mean). This
\emph{aggregative optimization} set-up was introduced in the pioneering
work~\cite{li2020distributed}.
The framework is fairly general and embraces several control applications
of interest such as cooperative robotics and multi-vehicle surveillance.
Originally, it stems from distributed aggregative games~\cite{koshal2016distributed,
liang2017distributed,gadjov2018passivity,yi2019operator,belgioioso2020distributed},
where however the objective is to compute a (generalized) Nash equilibrium
rather than to find the optimal solution cooperatively.
\subsection{Related works}
There exists a vast literature on distributed online optimization, which
addresses two main optimization set-ups known as cost-coupled (or consensus
optimization) and constraint-coupled, see, e.g.,~\cite{notarstefano2019distributed}. In the cost-coupled framework
the goal is to optimize a cost function given by the sum of several local
functions with a common decision variable.
In~\cite{cavalcante2013distributed} an online optimization algorithm based on a
distributed subgradient scheme is proposed.
In \cite{towfic2014adaptive} an adaptive diffusion algorithm is proposed to
address changes regarding both the cost function and the constraints
characterizing the problem. A class of coordination algorithms that generalize
distributed online subgradient descent and saddle-point dynamics is proposed in~\cite{mateos2014distributed} for network scenarios modeled by jointly-connected
graphs. An algorithm consisting of a subgradient flow combined with a push-sum
consensus is studied in~\cite{akbari2015distributed} for time-varying directed
graphs. Cost uncertainties and switching communication topologies are addressed
in~\cite{hosseini2016online} by using a distributed algorithm based on dual
subgradient averaging. A distributed version of the mirror descent algorithm is
proposed in~\cite{shahrampour2017distributed} to address online optimization
problems. In~\cite{yuan2017adaptive} a consensus adaptive primal dual
subgradient method is used to address distributed constrained optimization in
online setting. Distributed online optimization framework is used in~\cite{zhou2017incentive} to handle the distribution grids
problem. In~\cite{akbari2019individual} an online algorithm based on the
alternating direction method of multipliers is proposed.
In the more recent constraint-coupled optimization framework, each local objective
function depends on a local decision variable, but all the variables are coupled
through separable coupling constraints. In~\cite{lee2017sublinear} this set-up
is addressed in an online setting and a sublinear regret bound is ensured by
using a distributed primal dual algorithm. The same result is achieved in~\cite{li2020distributedConstraints} by introducing a push-sum mechanism to allow
for directed graph topologies.
Time-varying inequality constraints have
been taken into account in~\cite{yi2020distributed}, in which a distributed primal dual
mirror descent algorithm is proposed.
We recall that the above works are suited for cost-coupled (or consensus
optimization) and constraint-coupled setups.
The aggregative optimization framework addressed in this paper has been only
recently introduced in works~\cite{li2020distributed} and~\cite{li2020distributedOnline} which consider respectively a static
unconstrained framework and an online constrained one. The distributed
algorithms proposed in these two papers leverage a tracking action to reconstruct
both the aggregative variable and the gradient of the whole cost function.
The ``tracking action'' used in the proposed scheme is based on dynamic
average consensus (see~\cite{zhu2010discrete,kia2019tutorial}) and has been
introduced in the cost-coupled optimization set-up.
The so-called gradient tracking scheme uses this mechanism to obtain a local
estimate of the gradient of the whole cost function. The scheme has been
proposed in several variants and studied under different problem
assumptions~\cite{shi2015extra,varagnolo2015newton,dilorenzo2016next,nedic2017achieving,qu_harnessing_2018,xu2017convergence,xi2017add,xin2018linear,scutari2019distributed}.
In~\cite{zhang2019distributed}, the gradient tracking scheme, which uses this
mechanism to obtain a local estimate of the gradient of the whole cost function,
has been applied to online optimization problems.
In~\cite{notarnicola2020personalized} a dynamic gradient tracking update is
combined with a recursive least squares scheme to solve in a distributed way
online optimization problems in which the local functions are partially
unknown.
Finally, in~\cite{carnevale2020distributed} a gradient tracking algorithm
with adaptive momentum is proposed to address distributed online
optimization problems.
\subsection{Contributions}
The contributions of this paper are as follows.
We focus on an online constrained aggregative optimization set-up over
peer-to-peer networks of agents. The optimization set-up we consider
generalizes the one proposed in~\cite{li2020distributedOnline} in that we allow
for time-varying aggregation rules and constraint sets, and we drop the
assumption on bounded gradients.
These generalizations pose new challenges that must be appropriately handled in
the regret analysis, since they introduce additional unpredictable terms related
to time variations.
We propose a distributed optimization algorithm handling such modifications.
Specifically, each
agent projects the updated local solution estimate on its time-varying
constraint set and then performs a convex combination with the current
estimate. Moreover, the trackers of the aggregative variable are generalized to
handle time-variation of the aggregation rules.
Under mild assumptions on the cost and constraint variations, we
prove a linear regret for the proposed scheme.
In order to obtain
this result we study a dynamical system describing the algorithmic evolution
of: (i) the error of the solution estimate with respect to the minimum, (ii)
the consensus error of the aggregative variable trackers, and (iii) the
consensus error of the global gradient trackers. Such dynamics is characterized
by a Schur system matrix and a constant input, by which we are then able to
draw conclusions on the dynamic regret.
For the static case
(with constant cost and constraints), we show that the algorithm iterates converge to
the optimal solution with linear rate. Notably, our algorithm allows for
a constant step-size.
To corroborate the theoretical analysis, we show numerical simulations from two
cooperative robotics scenario: the first one dealing with a set of agents
representing defensive players of a robotic basketball team, and the
second one in which robots have to accomplish a surveillance task.
The proposed scenarios are dynamic and intrinsically characterized by time-varying
cost functions, aggregation functions and constraints, thus they cannot
be addressed by using state-of-art techniques.
It is worth pointing out that the theoretical results we provide
improve existing results both
in the online set-up and in the static case.
The regret bound provided by~\cite{li2020distributedOnline} is of the
type $O(\sqrt{T}) + O(T)$, while in our analysis the term $O(\sqrt{T})$
is improved to $O(1)$ thanks to the constant step size and the different
line of analysis (the term $O(T)$ is related to time variations of the
problem in both cases).
As a consequence, in the static case (i.e. when the term $O(T)$ is not present),
we are able to guarantee a linear convergence rate (instead of a sublinear one)
to the optimal solution.
\subsection{Organization and Notation}
The rest of the paper is organized as follows. In Section~\ref{sec:problem_formaulation} we formally
describe the distributed online aggregative optimization framework and we present
the distributed algorithm with its convergence properties. The algorithm analysis is performed
in Section~\ref{sec:analysis}. Finally, Section~\ref{sec:numerical_experiments} shows numerical examples
on control applications described in terms of distributed online aggregative optimization.
\paragraph*{Notation}%
We use $\textsc{col} (v_1, \ldots, v_n)$ to denote the vertical concatenation of the column
vectors $v_1, \ldots, v_n$. %
The Kronecker product is denoted by
$\otimes$. The identity
matrix in $\mathbb{R}^{m\times m}$ is $I_m$, while $0_m$ is the zero matrix in
$\mathbb{R}^{m\times m}$. The column vector of $N$ ones is denoted by $1_N$ and
we define $\mathbf{1} \triangleq 1_N \otimes I_d$.
Dimensions are omitted whenever they are clear from the context. We use $P_X[y]$ to denote the
projection of a vector $y$ on a closed convex set $X$, namely $P_X[y] = \arg \min_{x \in X}\norm{x-y}$. Given $x \in \mathbb{R}^n$, we use $[x]^+$ to denote $\max\{0,x\}$ in a component-wise sense. Given a function of two variables $f: \mathbb{R}^{n_1} \times \mathbb{R}^{n_2} \to \mathbb{R}$,
we denote as $\nabla_1 f \in \mathbb{R}^{n_1}$ the gradient of $f$ with respect to its
first argument and as $\nabla_2 f \in \mathbb{R}^{n_2}$ the gradient of $f$ with respect
to its second argument. We denote as $\nabla f \in \mathbb{R}^{n_1 + n_2}$ the gradient
of $f$ with respect to both its arguments.
\section{Problem Formulation and Algorithm Description}
\label{sec:problem_formaulation}
In this paper, we consider distributed online aggregative optimization problems
that can be written as
\begin{align}
\label{eq:online_aggregative_problem}
\begin{split}
\min_{(x_1,\dots,x_N) \in X} \: & \: \sum_{i=1}^{N}f_{i,t}(x_i,\sigma_t(x))
\\
\text{subj. to} \: & \: h_{i,t}(x_{i,t}) \leq 0, \hspace{0.7cm} \forall i \in {I},
\end{split}
\end{align}
in which $x := \textsc{col}(x_1, \dots, x_N) \in \mathbb{R}^n$ is the global decision vector,
with each $x_i \in \mathbb{R}^{n_i}$ and $n = \sum_{i=1}^{N} n_i$. The global decision
vector at time $t$ is constrained to belong to a set $X \subseteq \mathbb{R}^n$ that can
be written as $X = (X_{1} \times \ldots \times X_{N})$, where each
$X_i \subseteq \mathbb{R}^{n_i}$. The functions $f_{i,t}: \mathbb{R}^{n_i} \times \mathbb{R}^d \to \mathbb{R}$
represent the local objective functions at time $t$, while each function
$h_{i,t}: \mathbb{R}^{n_i} \to \mathbb{R}^S$ represents the local constraint function at time $t$,
and the \emph{aggregation function} $\sigma_t(x)$ has the form
\begin{align}
\sigma_t(x) := \frac{\sum_{i=1}^{N}\phi_{i,t}(x_i)}{N},
\end{align}
where each $\phi_{i,t} : \mathbb{R}^{n_i} \to \mathbb{R}^d$ is the $i$-th contribution to the aggregative variable at time $t$.
In the following, we denote the cost function of problem~\eqref{eq:online_aggregative_problem}
more compactly as $f_{t}(x, \mathbf{1}\sigma_t(x))$,
where $f_{t} : \mathbb{R}^{n} \times \mathbb{R}^{Nd} \to \mathbb{R}$ is defined as
$f_{t}(x, \textsc{col}(\xi_1, \ldots, \xi_N)) := \sum_{i=1}^N f_{i,t}(x_i,\xi_i)$
with each $\xi_i \in \mathbb{R}^d$ for all $i \in \{1, \ldots, N\}$.
In problem \eqref{eq:online_aggregative_problem}, the global objective function
is not known to any agent, and each of them can only privately access the
information on $f_{i,t}$, $X_i$, $h_{i,t}$, and $\phi_{i,t}$. It is important to remark that each
agent $i$ accesses its private information $f_{i,t}$, $h_{i,t}$ and $\phi_{i,t}$ only once
it has been computed its estimate $x_{i,t}$. The idea is to solve problem~\eqref{eq:online_aggregative_problem} in a distributed way over a network of $N$
agents communicating according to a graph
$\mathcal{G} := ({I},\mathcal{E}, \mathcal{A})$, where ${I} := \{1,\dots,N\}$ represents the set
of agents of the network, $\mathcal{E} \in {I} \times {I}$ represents the
edges set, and $\mathcal{A} \in \mathbb{R}^{N \times N}$ is the weighted adjacency
matrix. In this framework, each agent $i$ can exchange data only with its
neighboring agents defined by $\mathcal{E}$.
The goal is to design distributed algorithms to seek a minimum for problem~\eqref{eq:online_aggregative_problem}. In the next, we will denote as
$\nabla_1f_{i,t}(\cdot,\cdot)$ and as $\nabla_2f_{i,t}(\cdot,\cdot)$ the
gradient of $f_{i,t}$ with respect to respectively the first argument and the second
argument. Let $x_{i,t}$ be the solution estimate of the problem at time $t$
maintained by agent $i$, and let $x_t^\star$ be a minimizer of
$f_{t}(x, \mathbf{1}\sigma_t(x))$ over the set $\Xk$ defined as
\begin{equation*}
\Xk := X \cap \{x \in \mathbb{R}^n| h_{i,t}(x_i) \leq 0, \quad \forall i \in {I}\}.
\end{equation*}
Then, given a finite value $T > 1$, the agents want to minimize the dynamic regret:
\begin{align}\label{eq:regret}
R_T := \sum_{t=1}^T f_{t}(x_{t},\mathbf{1}\sigma_t(x_{t})) - \sum_{t=1}^Tf_{t}(x_t^\star, \mathbf{1}\sigma_t^\star),
\end{align}
where we have defined $\sigma_t^\star := \sigma_t(x_t^\star)$ for brevity. To this end,
we propose our \algo/ algorithm.
As customary in this kind of approaches,
each agent $i$ maintains for each time instant $t$ an estimate $x_{i,t}$ of the
component $i$ of a minimum $x_t^\star$ of problem~\eqref{eq:online_aggregative_problem}.
Moreover, each agent $i$ maintains
auxiliary variables $s_{i,t}$ and $y_{i,t}$ whose role is to reconstruct, through the
implementation of a tracking mechanism, respectively the aggregative variable
$\sum_{i=1}^{N}\frac{\phi_{i,t}(x_{i,t})}{N}$ and the global information
$\sum_{i=1}^{N}\nabla_2
f_{i,t}\left(x_{i,t},\sum_{j=1}^{N}\frac{\phi_{j,t}(x_{j,t})}{N}\right)$. A
pseudo-code of the \algo/ algorithm is reported in Algorithm~\ref{table:constrained_aggregative_GT}
from the perspective of agent $i$, in which $\alpha$ is a positive constant
step-size, $\delta \in (0,1)$ is a constant algorithm parameter, and each
element $a_{ij}$ represents the $(i,j)$ entry of the weighted adjacency matrix
$\mathcal{A}$ of the network.
\makeatletter
\xpatchcmd{\algorithmic}
{\ALG@tlm\z@}{\leftmargin\z@\ALG@tlm\z@}
{}{}
\makeatother
\begin{algorithm}%
\begin{algorithmic}
\State choose: $\alpha>0$, $\delta \in (0,1)$
\State initialization:
\begin{align*}
x_{i,0} \in X_{i,0}, \:\:\:\: s_{i,0} = \phi_{i,0}(x_{i,0}), \:\:\:\: y_{i,0} = \nabla_2f_{i,0}(x_{i,0},s_{i,0})
\end{align*}
\For{$t=0, 1, \dots$}
\begin{align}
\tilde{x}_{i,t} &\!=\! P_{X_{i,t}} \!\big[x_{i,t} \!-\! \alpha(\!\nabla_1 f_{i,t}(x_{i,t},s_{i,t}) \!+\!\! \nabla\phi_{i,t}(x_{i,t})y_{i,t})\!\big]\label{eq:feasible_direction}\\
x_{i,t+1} &\!=\! x_{i,t} + \delta(\tilde{x}_{i,t} - x_{i,t})\label{eq:convex_combination}\\
s_{i,t+1} &\!=\! \sum_{j=1}^{N}a_{ij}s_{j,t} + \phi_{i,t+1}(x_{i,t+1}) - \phi_{i,t}(x_{i,t})\\
y_{i,t+1} &\!=\! \sum_{j=1}^{N}a_{ij}y_{j,t} + \nabla_2 f_{i,t+1}(x_{i,t+1}, s_{i,t+1})\notag\\
&\hspace{1.84cm}- \nabla_2 f_{i,t}(x_{i,t},s_{i,t})\label{eq:y_update}
\end{align}
\EndFor
\end{algorithmic}
\caption{\algo/ (Agent $i$)}
\label{table:constrained_aggregative_GT}
\end{algorithm}
We highlight the scalability property of our algorithm, allowing for an
arbitrary dimension of the agents number $N$. Moreover, we remark that each
agent $i$ is only aware of its decision variable $x_{i,t}$, allowing the privacy
preservation of the other agents estimate. In fact, each agent $i$ exchanges
with its neighbors only the local aggregative variable estimate $s_{i,t}$ and the
local gradient estimate $y_{i,t}$. In the next, we will theoretically establish a
parametric bound for the choice of the step-size $\alpha$.
We highlight that \algo/ only requires knowledge of the constraint set $\Xk$.
\section{Convergence Analysis}
\label{sec:analysis}
In this section we study the convergence properties of the proposed distributed algorithm.
\subsection{\algo/ Reformulation and Assumptions}
First of all let us rewrite all the agents' updates, according to
Algorithm~\ref{table:constrained_aggregative_GT}, in a stacked vector form as
\begin{align}
\tilde{x}_{t} &= P_{\Xk}[x_{t} - \alpha(\nabla_1f_{t}(x_{t},s_{t}) + \nabla\phi_{t}(x_{t})y_{t})]\label{eq:tilde_gloabl_update}\\
x_{t+1} &= x_{t} + \delta(\tilde{x}_{t} - x_{t})\label{eq:x_global_update}\\
s_{t+1} &= As_{t} + \phi_{t+1}(x_{t+1}) - \phi_{t}(x_{t})\label{eq:sigma_global_update}\\
y_{t+1} &= Ay_{t} + \nabla_2 f_{t+1}(x_{t+1}, s_{t+1}) - \nabla_2f_{t}(x_{t},s_{t})\label{eq:y_global_update},
\end{align}
with $x_{t} := \textsc{col}(x_{1,t}, \dots, x_{N,t})$, $s_{t} := \textsc{col}(s_{1,t}, \dots, s_{N,t})$, $y_{t} := \textsc{col}(y_{1,t}, \dots, y_{N,t})$, $A := \mathcal{A} \otimes I$ and
\begin{align*}
\nabla_1 f_{t}(x_{t},s_{t}) &:= \begin{bmatrix}
\nabla_1 f_{1,t}(x_{1,t},s_{1,t})\\
\vdots\\
\nabla_2 f_{N,t}(x_{N,t},s_{N,t})
\end{bmatrix},\\
\nabla_2 f_{t}(x_{t},s_{t}) &:= \begin{bmatrix}
\nabla_2 f_{1,t}(x_{1,t},s_{1,t})\\
\vdots\\
\nabla_2 f_{N,t}(x_{N,t},s_{N,t})
\end{bmatrix},\\
\nabla \phi_{t}(x_{t}) &:= \begin{bmatrix}
\nabla\phi_{1,t}(x_{1,t}) &\\
& \ddots \\
& & \nabla\phi_{N,t}(x_{N,t})
\end{bmatrix}.
\end{align*}
In order to perform the convergence analysis, we derive bounds for the quantities
$\norm{x_{t+1} - x_{t}}$, $\norm{x_{t+1} - x_{t+1}^\star}$, $\norm{y_{t+1} - \mathbf{1}\bar{y}_{t+1}}$, and
$\norm{s_{t+1} - \mathbf{1}\bar{s}_{t+1}}$, in which $\bar{y}_t := \frac{1}{N}\sum_{i=1}^Ny_{i,t}$ and
$\bar{s}_t := \frac{1}{N}\sum_{i=1}^Ns_{i,t}$ denote the mean vectors of $y_{t}$ and $s_{t}$
respectively, and $\mathbf{1} := 1_N \otimes I$. We start by noticing that
\begin{align}
\bar{s}_{t+1} &= \bar{s}_t + \frac{1}{N}\sum_{i=1}^{N}(\phi_{i,t+1}(x_{i,t+1}) - \phi_{i,t}(x_{i,t}))\notag\\
& = \frac{1}{N}\mathbf{1}^\top s_{t} + \frac{1}{N}\mathbf{1}^\top (\phi_{t+1}(x_{t+1}) - \phi_{t}(x_{t}))\label{eq:sigma_mean_update}
\end{align}
and
\begin{align}
\small
\bar{y}_{t+1}\notag
& = \bar{y}_t + \!\frac{1}{N}\!\sum_{i=1}^{N}(\nabla_2f_{i,t+1}(x_{i,t+1},s_{i,t+1}) \!- \!\nabla_2f_{i,t}(x_{i,t},s_{i,t})\!)\notag\\
& = \frac{1}{N}\mathbf{1}^\topy_{t} + \frac{1}{N}\mathbf{1}^\top (\nabla_2f_{t+1}(x_{t+1},s_{t+1}) - \nabla_2f_{t}(x_{t},s_{t}))\label{eq:y_mean_update}.
\end{align}
Moreover, if we initialize $\sigma$ and $y$ as $\sigma_0 := \phi_0(x_0)$ and
$y_0 := \nabla_2f_0(x_0,s_0)$, then, from~\eqref{eq:sigma_mean_update}
and~\eqref{eq:y_mean_update}, it holds for all $t \in \mathbb{N}$
\begin{align}
\bar{s}_t = \frac{1}{N}\sum_{i=1}^{N}\phi_{t}(x_{i,t}) := \sigma_t(x_{t})\label{eq:s_mean}\\
\bar{y}_t = \frac{1}{N}\sum_{i=1}^{N}\nabla_2f_{i,t}(x_{i,t},s_{i,t})\label{eq:y_mean}.
\end{align}
Next, we state the main assumptions needed to prove the convergence properties
of the algorithm.
\begin{assumption}[Communication graph]\label{ass:network}
The graph $\mathcal{G}$ is undirected and connected. Moreover,
the associated weighted adjacency matrix $\mathcal{A}$ is doubly stochastic,
namely it holds $\mathcal{A}1_N = 1_N$ and $1_N^\top\mathcal{A} = 1_N^\top$. \oprocend
%
%
%
%
%
%
%
\end{assumption}
\begin{assumption}[Convexity]\label{ass:convexity}
For all $i \in {I}$ the set $X_i \subseteq \mathbb{R}^{n_i}$ is nonempty, closed and convex
and the components of the function $h_{i,t}$ are convex for all $t \ge 0$. Moreover
the global objective function $f_{t}(x, \mathbf{1}\sigma_t(x))$ is $\mu$-strongly convex. \oprocend
\end{assumption}
\begin{assumption}[Function Regularity]\label{ass:lipschitz}
The global objective function $f_{t}(x, \mathbf{1}\sigma_t(x))$ is differentiable and has $L_1$-Lipschitz continuous gradient, that is
\begin{equation*}
\norm{\nabla f_{t}(x,\mathbf{1}\sigma_t(x)) - \nabla f_{t}(x',\mathbf{1}\sigma_t(x'))} \leq L_1\norm{x - x'},
\end{equation*}
for all $x, x' \in \mathbb{R}^n$ and $t \ge 0$. Moreover, the function $\nabla_1 f_{t}(x,y) + \nabla \phi_{t}(x)\mathbf{1}\frac{1}{N}\sum_{i=1}^{N}\nabla_2 f_{i,t}(x_i,y_i)$ is $L_1$-Lipschitz, that is
\begin{align*}
\bigg\|& \nabla_1 f_{t}(x,y) + \nabla \phi_{t}(x)\mathbf{1}\frac{1}{N}\sum_{i=1}^{N}\nabla_2 f_{i,t}(x_i,y_i)
\\
& \hspace{0.5cm}-\nabla_1 f_{t}(x,y) - \nabla \phi_{t}(x)\mathbf{1}\frac{1}{N}\sum_{i=1}^{N}\nabla_2 f_{i,t}(x_i,y_i)\bigg \|
\\
& \leq L_1\norm{\textsc{col}(x-x',y-y')},
\end{align*}
%
for all $x, x' \in \mathbb{R}^n$ and $y, y' \in \mathbb{R}^{Nd}$ and $t \ge 0$. Furthermore, the function $\nabla_2 f_{t}(x,y)$ is $L_2$-Lipschitz continuous, that is,
%
\begin{equation*}
\norm{\nabla_2f_{t}(x,y) - \nabla_2f_{t}(x',y')} \leq L_2\norm{\textsc{col}(x -
x',y - y')},
\end{equation*}
%
for all $x, x' \in \mathbb{R}^n$ and $y, y' \in \mathbb{R}^{Nd}$ and $t \ge 0$.
The aggregation functions $\phi_{i,t}$ are differentiable
and $L_3$-Lipschitz continuous, that is,
%
\begin{equation*}
\norm{\phi_{i,t}(x) - \phi_{i,t}(y)} \leq L_3\norm{x - y},
\end{equation*}
%
for all $x_i,y_i \in \mathbb{R}^{n_i}$ and $t \ge 0$. \oprocend
\end{assumption}
The next assumption characterizes variations of the cost,
aggregation, and constraint functions.
\begin{assumption}[Bounded variations]\label{ass:bdd_var}
There exist constants $\eta, \omega, \gamma, \zeta \ge 0$ such that, for all $t \ge 0$,
it holds
%
\begin{align*}
%
\eta_{t}&:=\max_{i \in {I}} \max_{x\in X_i,z \in \mathbb{R}^d} \|\nabla_2
f_{i,t+1}(x,z) - \nabla_2 f_{i,t}(x,z)\| \leq \eta
\\
\omega_{t}&:=\max_{i \in {I}} \max_{x\in X_i} \|\phi_{i,t+1}(x) - \phi_{i,t}(x)\|\leq \omega
\\
\gamma_t &:= \max_{i \in {I}} \max_{x \in X_i} \|h_{i,t+1}(x) -
h_{i,t}(x)\| \leq \gamma,
\end{align*}
%
and moreover
%
\begin{equation*}
\zeta_t:=\|x_{t+1}^\star-x_t^\star\|\leq\zeta. \hfill \tag*{$\square$}
\end{equation*}
%
\end{assumption}
Note that a particular case in which Assumption~\ref{ass:bdd_var} is
automatically satisfied is when the sets $X_i$ are bounded, however it is not
required for our approach to work.
Depending on the structure of the constraints, in some cases
(see, e.g., the robotic basketball scenario in Section~\ref{sec:numerical_experiments})
one can use a restricted version of the sets to guarantee recursive feasibility.
\subsection{Preparatory Lemmas}
Here we present four preparatory Lemmas that we need to prove Theorem~\ref{th:convergence}.
\begin{lemma}\label{lemma:xk_xstar}
Let Assumptions~\ref{ass:network},~\ref{ass:convexity},~\ref{ass:lipschitz},~\ref{ass:bdd_var} hold. If $\alpha \leq \frac{1}{L_1}$, then %
%
\begin{align*}
\norm{x_{t+1} - x_{t+1}^\star} &\leq(1-\delta\mu\alpha)\norm{x_{t}-x_t^\star} + \delta\alpha L_1\norm{s_{t}-\mathbf{1}\bar{s}_t}\\
& \hspace{0.5cm}+ \delta\alpha L_3\norm{y_{t} - \mathbf{1}\bar{y}_t} + \zeta.
\end{align*}
\end{lemma}
\begin{proof}
The proof of Lemma~\ref{lemma:xk_xstar} can be found in Appendix~\ref{sec:proof_xk_xkstar}.
\end{proof}
\begin{lemma}\label{lemma:xkp_xk}
Let Assumptions~\ref{ass:network},~\ref{ass:convexity},~\ref{ass:lipschitz},~\ref{ass:bdd_var} hold. Then
\begin{align*}
\norm{x_{t+1} - x_{t}} &\leq \delta(2 + \alpha L_1 + \alpha L_1L_3)\norm{x_{t} -x_t^\star}\notag\\
&\hspace{0.5cm}+ \delta\alpha L_1\norm{s_{t}-\mathbf{1}\bar{s}_t}+\delta\alpha L_3\norm{y_{t} - \mathbf{1}\bar{y}_t}.
\end{align*}
\end{lemma}
\begin{proof}
The proof of Lemma~\ref{lemma:xkp_xk} can be found in Appendix~\ref{sec:proof_xkp_xk}.
\end{proof}
\begin{lemma}\label{lemma:sk_skavg}
Let Assumptions~\ref{ass:network},~\ref{ass:convexity},~\ref{ass:lipschitz},~\ref{ass:bdd_var} hold. Then
%
\begin{align*}
\norm{s_{t+1}-\mathbf{1}\bar{s}_{t+1}} &\leq \rho\norm{s_{t} - \mathbf{1}\bar{s}_t} + \delta\alpha L_1L_3\norm{s_{t}-\mathbf{1}\bar{s}_t}\notag
\\
&\hspace{0.5cm}+\delta(2 L_3 + \alpha L_1L_3 + \alpha L_1L_3^2)\norm{x_{t} -x_t^\star}\notag
\\
&\hspace{0.5cm}+\delta\alpha L_3^2\norm{y_{t} - \mathbf{1}\bar{y}_t} + \sqrt{N}\omega.
\end{align*}
\end{lemma}
\begin{proof}
The proof of Lemma~\ref{lemma:sk_skavg} can be found in Appendix~\ref{sec:proof_sk_skavg}.
\end{proof}
\begin{lemma}\label{lemma:yk_ykavg}
Let Assumptions~\ref{ass:network},~\ref{ass:convexity},~\ref{ass:lipschitz},~\ref{ass:bdd_var} hold. Then
%
\begin{align}
&\norm{y_{t+1} - \mathbf{1}\bar{y}_{t+1}} \leq \rho\norm{y_{t}-\mathbf{1}\bar{y}_t}\notag
\\
& \hspace{0.5cm}+ \delta\alpha L_3(L_2 + L_2L_3)\norm{y_{t} - \mathbf{1}\bar{y}_t}\notag
\\
& \hspace{0.5cm}+ \delta(2 + \alpha L_1 + \alpha L_1)(L_2 + L_2L_3)\norm{x_{t} -x_t^\star}\notag
\\
&\hspace{0.5cm}+ \delta\alpha L_1(L_2 + L_2L_3)\norm{s_{t}-\mathbf{1}\bar{s}_t}\notag
\\
&\hspace{0.5cm}+ 2L_2\norm{s_{t}-\mathbf{1}\bar{s}_t} + L_2\sqrt{N}\omega + \sqrt{N}\eta.\notag
\end{align}
\end{lemma}
\begin{proof}
The proof of Lemma~\ref{lemma:yk_ykavg} can be found in Appendix~\ref{sec:proof_yk_ykavg}.
\end{proof}
\subsection{Regret analysis}
Now we state the main theoretical results of the paper.
In the next theorem we provide a bound on the cost error and
constraint violation of the iterates generated by the \algo/
distributed algorithm in the general, online set-up~\eqref{eq:online_aggregative_problem}.
\begin{theorem}\label{th:convergence}
Let Assumptions~\ref{ass:network},~\ref{ass:convexity},~\ref{ass:lipschitz} and~\ref{ass:bdd_var} hold.
Let $\{x_{t}\}$ be the sequence of estimates generated by Algorithm~\ref{table:constrained_aggregative_GT}
initialized as specified in the algorithm table.
If $\alpha \leq \frac{1}{L_1}$, then, for a sufficiently small $\delta \in (0,1)$,
there exists a positive constant $\lambda < 1$ for which
%
\begin{align*}
&f_{t}(x_{t},\mathbf{1}\sigma_t(x_{t})) - f_{t}(x_t^\star,\mathbf{1}\sigma_t^\star)
\\
&\hspace{1cm}\leq \frac{L_1}{2}\left(\lambda^{2t} U_0^2 +2\lambda^{t}\frac{U_0Q}{1-\lambda}+ \frac{Q^2}{(1-\lambda)^2}\right),
\end{align*}
%
where
%
\begin{align*}
U_0 &:= \sqrt{\norm{x_0 - x_0^\star}^2 + \norm{s_0 - \mathbf{1}\bar{s}_0}^2 + \norm{y_0 - \mathbf{1}\bar{y_0}}^2},
\\
Q &:= \sqrt{\zeta^2 + N(\eta + \omega)^2+ N(\eta + L_2\omega)^2}.
\end{align*}
%
Moreover, the constraint violation is bounded by
%
\begin{equation}\label{eq:constraint_violation}
\sum_{i=1}^N[h_{i,t}(x_{i,t})]^+ \leq \gamma N\frac{1 - (1-\delta)^t}{\delta}1_S.
\end{equation}
\end{theorem}
\begin{proof}
Let us define
\begin{equation*}
z_t := \begin{bmatrix}
\norm{x_{t}-x_t^\star}\\
\norm{s_{t} - \mathbf{1}\bar{s}_t}\\
\norm{y_{t} - \mathbf{1}\bar{y}_t}
\end{bmatrix}, \quad u := \begin{bmatrix}
\zeta\\
\sqrt{N}\eta\\
\sqrt{N}\omega
\end{bmatrix}.
\end{equation*}
Then, by combining Lemma~\ref{lemma:xk_xstar},~\ref{lemma:sk_skavg}, and~\ref{lemma:yk_ykavg}, it is possible to write an upper bound regarding the evolution of $z_t$ trough the following dynamical system
\begin{equation}\label{eq:z}
z_{t+1} \leq M(\delta)z_t + Bu,
\end{equation}
in which
\begin{equation*}
M(\delta) := M_0 + \delta E, \quad B := \begin{bmatrix}
1& 0& 0\\
0& 1& 1\\
0& 1& L_2
\end{bmatrix},
\end{equation*}
where
\begin{align*}
M_0 &:= \begin{bmatrix}
1& 0& 0\\
0& \rho& 0\\
0& 2L_2& \rho
\end{bmatrix}, \\
E &:= \begin{bmatrix}
-\mu\alpha& \alpha L_1& \alpha L_3\\
E_{21}& \alpha L_1L_3& \alpha L_3^2\\
E_{31}& E_{32}& E_{33}\\
\end{bmatrix},
\end{align*}
with
\begin{align*}
E_{21} &:= 2 L_3 + \alpha L_1L_3 + \alpha L_1L_3^2,\\
E_{31} &:= (2 + \alpha L_1 + \alpha L_1)(L_2 + L_2L_3),\\
E_{32} &:= \alpha L_1(L_2 + L_2L_3),\\
E_{33} &:= \alpha L_3(L_2 + L_2L_3).
\end{align*}
Being $M_0$ triangular, it is easy to see that its spectral radius is $1$
since $\rho \in (0, 1)$. Denote by $\lambda(\delta)$ the eigenvalues of $M(\delta)$ as a
function of $\delta$. Call $v$ and $w$ respectively the right and left eigenvectors of $A_0$
associated to the eigenvalue $1$. It is easy to see that
\begin{equation*}
v = \begin{bmatrix}
1\\
0\\
0
\end{bmatrix}, \quad w = \begin{bmatrix}
1\\
0\\
0
\end{bmatrix}.
\end{equation*}
Since the eigenvalue $1$ is a simple eigenvalue of $M(0)$, from Lemma~\ref{lemma:horn} it holds
\begin{equation}\nonumber
\frac{d\lambda(\delta)}{d\delta}\bigg|_{\lambda=1,\delta=0} = \frac{w^\top Ev}{w^\top v} = -\mu\alpha < 0
\end{equation}
and then, by continuity of eigenvalues with respect to the matrix entries, $\lambda(\delta)$ is strictly less
than $1$ for sufficiently small $\delta>0$. Then, it is always possible to find $\delta>0$
that keeps the remaining eigenvalues inside the unit circle. For the sake of readability, from now on we will omit the dependency of $M$ and its eigenvalue $\lambda$ from $\delta$. Thus, since $z_t\geq 0$ for all $t$, and $M$ and $Bu$ have only non-negative entries, one can use~\eqref{eq:z} to write
\begin{equation}\nonumber\label{eq:y_evolution}
z_{t} \leq M^{t} z_0 + \sum_{k=0}^{t-1} M^{k}Bu.
\end{equation}
Here the first term decreases linearly with rate $\lambda$ (equal to the
spectral radius of $M$), while the second one is bounded. In particular,
\begin{align}\nonumber
\norm{z_{t}} &\leq \norm{M^{t} z_0 + \sum_{k=0}^{t-1} M^{k}Bu}
\\
&\stackrel{(a)} \leq \norm{M^{t} z_0 } + \norm{\sum_{k=0}^{t-1} M^{k}Bu}\nonumber
\\
&\stackrel{(b)} \leq \lambda^{t}\norm{z_0} + \norm{Bu}\norm{\sum_{k=0}^{t-1}\lambda^{k}}\nonumber
\\
&\stackrel{(c)} \leq \lambda^{t}\norm{z_0} + \norm{Bu} \norm{\frac{1-\lambda^{t}}{1-\lambda}}\nonumber
\\
&\leq \lambda^{t}\norm{z_0} + \norm{Bu} \norm{\frac{1}{1-\lambda}},\label{eq:bound_z}
\end{align}
where \emph{(a)} uses the triangle inequality, \emph{(b)} uses the spectral radius $\lambda$ of $M$, \emph{(c)} uses the geometric series property. By noting that
\begin{align}\label{eq:xz}
\norm{x_{t} - x_t^\star} \leq \norm{z_{t}},
\end{align}
and by exploiting the Lipschitz continuity of the gradients of $f_{t}(\cdot,\mathbf{1}\sigma_t(\cdot))$, we have that
\begin{align}\label{eq:f_minus_fstar}
f_{t}(x_{t},\mathbf{1}\sigma_t(x_{t})) - f_{t}(x_t^\star,\mathbf{1}\sigma_t^\star) \leq \frac{L_1}{2}\norm{x_{t} - x_t^\star}^2.
\end{align}
By combining~\eqref{eq:bound_z},~\eqref{eq:xz} and~\eqref{eq:f_minus_fstar}, we get
\begin{align*}\nonumber
&f_{t}(x_{t},\mathbf{1}\sigma_t(x_{t})) - f_{t}(x_t^\star,\mathbf{1}\sigma_t^\star)
\\
& \leq \!\frac{L_1}{2}\!\!\left(\!\lambda^{2t} \norm{z_0}^2\! +\!2\lambda^{t}\frac{\norm{z_0}\norm{Bu}}{1-\lambda}\!+ \!\frac{\norm{Bu}^2}{(1-\lambda)^2}\!\right)\!.
\end{align*}
The proof follows by noticing that $U_0 = \norm{z_0}$ and $\norm{Bu} = Q$.
As regards the result~\eqref{eq:constraint_violation}, for all $i \in {I}$ and $t \ge 0$, it holds
\begin{align}
&\proj{h_{i,t+1}(x_{i,t+1})} = [h_{i,t+1}(x_{i,t+1})\notag
\\
&\hspace{3cm}- h_{i,t}(x_{i,t+1}) + h_{i,t}(x_{i,t+1})]^+\notag
\\
&\stackrel{(a)} \leq [h_{i,t}(x_{i,t+1})]^+ + \proj{h_{i,t+1}(x_{i,t+1}) - h_{i,t}(x_{i,t+1})}\notag
\\
&\stackrel{(b)} \leq [h_{i,t}(x_{i,t+1})]^+ + \norm{h_{i,t+1}(x_{i,t+1}) - h_{i,t}(x_{i,t+1})}1_S\notag
\\
&\stackrel{(c)} \leq \proj{h_{i,t}(x_{i,t+1})} + \gamma 1_S\notag
\\
&\stackrel{(d)} = \proj{h_{i,t}(x_{i,t} + \delta(\tilde{x}_{i,t} - x_{i,t}))} + \gamma 1_S, \label{eq:constraint_violation_before_convexity}
\end{align}
where in \emph{(a)} we use the fact that for any $a, b \in \mathbb{R}^S$ it holds $\proj{a+b} \leq \proj{a} + \proj{b}$, \emph{(b)} uses the fact that for any $a\in\mathbb{R}^S$ it holds $\proj{a} \leq \norm{a}1_S$, \emph{(c)} uses Assumption~\ref{ass:bdd_var}, and \emph{(d)} uses the update~\eqref{eq:x_global_update}. Now we notice that, being each $h_{i,t}$ convex for all $t \ge 0$ by Assumption~\ref{ass:convexity} and being $\delta \in (0,1)$, it holds $h_{i,t}(x_{i,t} + \delta(\tilde{x}_{i,t} - x_{i,t})) \le (1-\delta)h_{i,t}(x_{i,t}) + \delta(h_{i,t}(\tilde{x}_{i,t}))$. Thus,~\eqref{eq:constraint_violation_before_convexity} can be rewritten as
\begin{align}
\proj{h_{i,t+1}(x_{i,t+1})} \!&\le (1-\delta)\!\proj{h_{i,t}(x_{i,t})} \!+ \delta\!\proj{h_{i,t}(\tilde{x}_{i,t})} %
\!+ \gamma 1_S\notag
\\
& \le (1-\delta)\!\proj{h_{i,t}(x_{i,t})} + \gamma 1_S,\label{eq:contraint_violation_dynamic}
\end{align}
where we have used $h_{i,t}(\tilde{x}_{i,t}) \leq 0$ because $\tilde{x}_{i,t} \in X_{i,t}$ by construction.
By applying recursively the inequality in~\eqref{eq:contraint_violation_dynamic} from $0$ up to $t-1$, we obtain
\begin{align}
\proj{h_{i,t}(x_{i,t})} &\leq (1 - \delta)^{t}\proj{h_{i,0}(x_{i,0})} + \sum_{k=0}^{t-1} (1 - \delta)^{k}\gamma 1_S\notag
\\
&\stackrel{(a)} = \gamma 1_S \sum_{k=0}^{t-1} (1 - \delta)^{k}\notag
\\
&\stackrel{(b)} = \gamma 1_S \frac{1 - (1-\delta)^t}{1-(1-\delta)}\notag
\\
&= \gamma 1_S \frac{1 - (1-\delta)^t}{\delta}\label{eq:constraint_violation_last}
\end{align}
where \emph{(a)} relies on the initialization $x_{i,0} \in X_{i,0}$
(i.e. $h_{i,0}(x_{i,0}) \le 0$) and
\emph{(b)} uses the geometric series property since $(1-\delta) \leq 1$.
The proof follows by summing~\eqref{eq:constraint_violation_last} over all the agents $i \in {I}$.
\end{proof}
Thanks to Theorem~\ref{th:convergence}, it is possible to derive more specific results
for the (average) dynamic regret and for the static case with time-invariant cost
and constraints (which is addressed in Section~\ref{sec:analysis_static_case}).
In the following corollary, we characterize the dynamic regret of \algo/.
\begin{corollary}[Dynamic regret]\label{cor:dynamic_regret}
Under the same assumptions of Theorem~\ref{th:convergence},
the dynamic regret (cf.~\eqref{eq:regret}) is equal to $R_T = O(1) + O(T)$,
in particular,
%
\begin{equation}
R_T \le \frac{L_1}{2}\left(\frac{U_0^2}{1-\lambda^2} + \frac{2U_0Q}{(1 - \lambda)^2} + T\frac{Q^2}{(1-\lambda)^2}\right),
\label{eq:dyn_regret_bound}
\end{equation}
%
where $R_T$ is defined as in~\eqref{eq:regret} and $U_0$, $\lambda$ and $Q$ are the
same as in~Theorem~\ref{th:convergence}. Moreover, the regret on the constraint violation is bounded by
%
\begin{align}
\sum_{t=1}^T\!\sum_{i=1}^N[h_{i,t}(x_{i,t})]^+ \!\le\! \frac{\gamma N\big( T\delta \!+\! \delta \!-\! 1 \!-\! (1-\delta)^{T+1}\big)}{\delta^2} 1_S.
\label{eq:cumulative_constraint_violation}
\end{align}
\end{corollary}
\begin{proof}
Recall the definition~\eqref{eq:regret} of $R_T$, i.e.,
%
\begin{align}
R_T
&= \sum_{t=1}^T [f_{t}(x_{t},\mathbf{1}\sigma_t(x_{t})) - f_{t}(x_t^\star, \mathbf{1}\sigma_t^\star)]
\notag
\\
&\le \frac{L_1}{2}\bigg(\sum_{t=1}^T \lambda^{2t} U_0^2 +\sum_{t=1}^T2\lambda^{t}\frac{U_0Q}{1-\lambda}+ \underbrace{\sum_{t=1}^T\frac{Q^2}{(1-\lambda)^2}}_{= \frac{TQ^2}{(1-\lambda)^2}}\bigg)
\label{eq:dyn_regret_sum}
\end{align}
%
where we used the bound provided by Theorem~\ref{th:convergence}.
%
As regards the term $\sum_{t=1}^T \lambda^{2t} U_0^2$, since $\lambda < 1$ it holds
\begin{align}
\sum_{t=1}^{T}\lambda^{2t}U_0^2 &= U_0^2\sum_{t=1}^{T}\lambda^{2t}\notag
\\
&\stackrel{(a)} = U_0^2\frac{1 - \lambda^{2T}}{1 - \lambda^2}\notag
\\
&\stackrel{(b)}\leq \frac{U_0^2}{1 - \lambda^2},\label{eq:dyn_regret_first_term}
\end{align}
in which \emph{(a)} is justified by the properties of the geometric series and
\emph{(b)} is due by the fact that $0 < \lambda < 1$ implies also $1-\lambda^{2T} < 1$.
By following similar steps, we can derive a bound for the term $\sum_{t=1}^T2\lambda^{t}\frac{U_0Q}{1-\lambda}$, i.e.,
\begin{align}
\sum_{t=1}^{T}2\lambda^{t}\frac{U_0Q}{1-\lambda}
&\leq \frac{2U_0Q}{(1 - \lambda)^2}.\label{eq:dyn_regret_cross_term}
\end{align}
By plugging~\eqref{eq:dyn_regret_first_term} and~\eqref{eq:dyn_regret_cross_term}
in~\eqref{eq:dyn_regret_sum}, we obtain the bound~\eqref{eq:dyn_regret_bound}.
As regards the regret on the constraint violation, let us sum~\eqref{eq:constraint_violation}
over $t = 1, \ldots, T$, i.e.,
\begin{align*}
&\sum_{t=1}^T \sum_{i=1}^N[h_{i,t}(x_{i,t})]^+
\leq \sum_{t=1}^T \gamma N\frac{1 - (1-\delta)^t}{\delta}1_S
\notag
\\
&\hspace{1cm} =
\frac{\gamma N}{\delta} 1_S \bigg( T - \sum_{t=1}^T (1-\delta)^t \bigg)
\notag
\\
&\hspace{1cm} \stackrel{(a)} =
\frac{\gamma N}{\delta} 1_S \bigg( T + 1 - \sum_{t=0}^T (1-\delta)^t \bigg)
\notag
\\
&\hspace{1cm} \stackrel{(b)} =
\frac{\gamma N}{\delta} 1_S \bigg( T + 1 - \frac{1 - (1-\delta)^{T+1}}{\delta}\bigg)
\notag
\\
&\hspace{1cm} =
\frac{\gamma N}{\delta^2} 1_S \big( T\delta + \delta - 1 - (1-\delta)^{T+1}\big),
\end{align*}
where in \emph{(a)} we reformulated the term in parenthesis so as to have
the sum starting from $0$ and in \emph{(b)} we used the property of the geometric series.
The proof follows.
\end{proof}
By using the result of Corollary~\ref{cor:dynamic_regret},
we obtain the following corollary on the average dynamic regret.
\begin{corollary}[Bounded average dynamic regret]
\label{cor:avg_regret}
Under the same assumptions of Theorem~\ref{th:convergence}, the asymptotic
average dynamic regret is bounded by
\begin{equation}\label{corollary:avg_regret}
\limsup_{T\to\infty} \: \frac{R_T}{T} \leq \frac{L_1}{2}\frac{Q^2}{(1-\lambda)^2},
\end{equation}
where $R_T$ is defined as in~\eqref{eq:regret} and $\lambda$ and $Q$ are the
same as in~Theorem~\ref{th:convergence}.
\oprocend
\end{corollary}
\subsection{Linear convergence in the static set-up}
\label{sec:analysis_static_case}
In this subsection we study the convergences properties of \algo/ in the
particular case in which problem~\eqref{eq:online_aggregative_problem} is static.
Namely, we assume constraints, cost and aggregation functions do not vary over time,
i.e. $f_{t} = f$, $h_{i,t} = h_i$, $\phi_{t} = \phi$ for all $t \ge 0$.
Thus, problem~\eqref{eq:online_aggregative_problem} becomes
\begin{align}
\label{eq:static_aggregative_problem}
\begin{split}
\min_{x \in X} \: & \: \sum_{i=1}^{N}f_i(x_i,\sigma(x))
\\
\text{subj. to} \: & \: h_i(x_i) \leq 0, \hspace{0.7cm} \forall i \in {I},
\end{split}
\end{align}
where
\begin{align*}
\sigma(x) := \frac{\sum_{i=1}^{N}\phi_i(x_i)}{N}.
\end{align*}
In the following corollary, we assess that in the static case
the \algo/ distributed algorithm converges to the optimal
solution with a linear rate.\footnote{%
A sequence $\{x_t\}$ converges linearly to $\bar{x}$
if there exists a number $\eta \in (0,1)$ such that
$\frac{\| x^{t+1} - \bar{x} \|}{\| x^{t} - \bar{x} \|} \to \eta$ as $t \to \infty$.}
\begin{corollary}[Static set-up]\label{cor:static}
Let Assumptions~\ref{ass:network},~\ref{ass:convexity} and~\ref{ass:lipschitz} hold
in the static case $f_{t} = f$, $h_{i,t} = h_i$, $\phi_{t} = \phi$ for all $t \ge 0$.
%
Let $\{x_t\}$ be the sequence of estimates generated by
Algorithm~\ref{table:constrained_aggregative_GT}.
If $\alpha \leq \frac{1}{L_1}$, then, for a sufficiently small $\delta$,
there exists a positive constant $\lambda < 1$ for which
%
\begin{equation}
f(x_{t},\mathbf{1}\sigma(x_{t})) - f(x^\star,\mathbf{1}\sigma_t^\star) \le \frac{L_1 U_0^2}{2} \lambda^{2t},
\label{eq:static_linear_convergence}
\end{equation}
%
where $x^\star$ is the optimal solution of problem~\eqref{eq:static_aggregative_problem}.
\end{corollary}
\begin{proof}
The proof directly derives from Theorem~\ref{th:convergence}.
In particular, in the static set-up Assumption~\ref{ass:bdd_var}
is trivially satisfied with $\eta = \omega = \gamma = \zeta = 0$.
%
Thus, one can apply Theorem~\ref{th:convergence} to obtain
%
\begin{align*}
&f_{t}(x_{t},\mathbf{1}\sigma_t(x_{t})) - f_{t}(x_t^\star,\mathbf{1}\sigma_t^\star)
\\
&\hspace{1.5cm}\leq \frac{L_1 U_0^2}{2} \lambda^{2t} + \lambda^{t}\frac{U_0 Q L_1}{1-\lambda} + \frac{L_1 Q^2}{2 (1-\lambda)^2}.
\end{align*}
%
By definition, in the considered static case $Q$ is equal to zero and thus the previous
inequality reduces to~\eqref{eq:static_linear_convergence}. The proof follows.
\end{proof}
\section{Numerical experiments}%
\label{sec:numerical_experiments}
In this section we consider a robotic basketball scenario and a robotic
surveillance scenario to test the effectiveness of our \algo/. It is worth
nothing that both these dynamic scenarios are characterized by time-varying
aggregation functions and constraints. These degrees of freedom are used to
adapt the behavior of each agent according to external factors, as specified
next.
\subsection{Robotic Basketball scenario}
In this subsection we consider a dynamic robotic scenario related to a basketball game.
The scenario concerns a team of $N = 5$ robotic agents that play the
role of the defending team, while the offending team, having also $5$ players,
pursues pre-defined trajectories.
The optimization variables $x_{i,t} \in \mathbb{R}^2$ represent the location of
each defending player $i$ at time $t$.
The defending team runs Algorithm~\ref{table:constrained_aggregative_GT}
to determine the target positions $x_{i,t}$ for all $t \ge 0$.
We assume that, between two consecutive time units, a low-level control
steers each defending player $i$ from $x_{i,t}$ to the location $x_{i,t+1}$.
This assumption is fairly realistic since one can always decrease the step size
to ensure that the displacements $\|x_{i,t+1} - x_{i,t}\|$ are sufficiently small.
Each defending player wants to follow the associated offending player,
located at time $t$ at some $p_{i,t} \in \mathbb{R}^2$. Moreover, the defending
team aims to %
maintain the weighted center of mass, which is modeled
with the aggregative variable $\sigma_t(x)$, as close as possible to the
ball, whose position at time $t$ is denoted by $b_{t} \in \mathbb{R}^2$.
Such defending strategy is captured by defining for each player $i$ a
local cost function $f_i: \mathbb{R}^2 \to \mathbb{R}$ of the form
\begin{equation*}
f_{i,t}(x_i,\sigma_t(x)) = \norm{x_i - p_{i,t}} + \frac{\gamma}{N}\norm{\sigma_t(x) - b_{t}},
\end{equation*}
where $\gamma > 0$ is a constant and the weighted center of mass
$\sigma_t(x)$ is defined as
\begin{equation*}
\sigma_t(x) = \frac{\sum_{i=1}^N\beta_{i,t} x_i}{N}.
\end{equation*}
In the definition of the weighted center of mass, the weights
$\beta_{i,t} > 0$ are chosen inversely proportional to the distance
between the defended basket and the $i$-th offending player.
Namely, by denoting with $\textsc{col}(h_B, v_B)$ the coordinate of the
defended basket, the weights are defined as
$\beta_{i,t} = \min\{1/\|p_{i,t} - \textsc{col}(h_B, v_B)\|^2, \beta^\text{max}\}$,
where we further introduced a saturation to $\beta^\text{max}$
to avoid too large numbers and ensure that $\beta_{i,t} < \beta^\text{max}$.
In this way, we force the defending players to stay close to the ball when
the associated opponents become close to the basket.
Let us introduce the constraints. We define $X_{\text{field}}$ as the set
of positions contained within the basketball field, with origin in the lower-left
corner. In order to protect the basket, we impose that
each defending player is ahead of the corresponding opponent, i.e.
its horizontal position must satisfy $[x_{i,t}]_h \ge [p_{i,t}]_h$,
where $[\cdot]_h$ denotes the horizontal component of the vectors.
However, instead of imposing this constraint directly, we enforce
a more stringent one (cf. the discussion after Assumption~\ref{ass:bdd_var}),
namely
\begin{subequations}
\begin{align}
[x_{i,t}]_h \ge [p_{i,t}]_h + \epsilon_h,
\label{eq:basket_con_1}
\end{align}
where $\epsilon_h > 0$ is an additional tolerance. Indeed,
by assuming the position of offending players changes
less than $\epsilon_h \delta$ from one iteration to the other
(i.e. $|[p_{i,t+1}]_h - [p_{i,t}]_h| \le \epsilon_h\delta$ for all $t \ge 0$),
one can see that, thanks to the projection operation~\eqref{eq:feasible_direction}
and the convex combination~\eqref{eq:convex_combination} step,
the iterates $x_{i,t}$ always satisfy the desired constraint
$[x_{i,t}]_h \ge [p_{i,t}]_h$ for all $t \ge 0$.
Moreover, we impose that the vertical position of the defending
player is between the corresponding opponent and the defended basket,
i.e., its vertical position must satisfy
\begin{align}
\begin{cases}
v_B \le [x_{i,t}]_v \le [p_{i,t}]_v & \text{if } [p_{i,t}]_v \le v_B,
\\
[p_{i,t}]_v \le [x_{i,t}]_v \le v_B & \text{if } [p_{i,t}]_v > v_B,
\end{cases}
\label{eq:basket_con_2}
\end{align}\end{subequations}
where $[\cdot]_v$ denotes the vertical component of the vectors.
The overall feasible set $X_{i,t}$ of each
agent $i$ at time $t$ is then
\begin{align}
X_{i,t} = X_{\text{field}} \cap \{x_{i,t} \in \mathbb{R}^2 \mid \eqref{eq:basket_con_1}, \eqref{eq:basket_con_2} \text{ are satisfied} \}.
\end{align}
It is worth remarking that this particular defending strategy can be
implemented only thanks to the fact that both aggregation functions
$\phi_{i,t}$ and the feasible set $X_{i,t}$ can change over time,
which would not be implementable with previously developed approaches
in the literature.
To simulate a realistic behavior, each offending player
moves along a piece-wise linear trajectory randomly initialized in
the left midfield, while the ball is randomly passed from an offending player
to another one during the simulation time. As regards the algorithm
parameters, we set $\alpha = 0.1$ and $\delta = 0.1$.
Fig.~\ref{fig:screen} shows four different time instants
of the match.
Note that, from the third to the fourth image, the center of mass of the
defending team moves towards the top, following the movement of the ball,
however at each time instant the defending players stay ahead of the
opponent team (horizontally) and between the opponent and the basket
(vertically), which is modeled thanks to the proposed constraints.
\begin{figure}
\centering
\subfloat[][Iteration $t = 130$.]
{\includegraphics[width=.23\textwidth]{figs/screen1.png}}\quad
\subfloat[][Iteration $t = 300$.]
{\includegraphics[width=.23\textwidth]{figs/screen2.png}} \\
\subfloat[][Iteration $t = 800$.]
{\includegraphics[width=.23\textwidth]{figs/screen3.png}}\quad
\subfloat[][Iteration $t = 1000$.]
{\includegraphics[width=.23\textwidth]{figs/screen4.png}}
\caption{Snapshots from the robotic basketball match.
Ball trajectory is depicted in green, while defending players are in red
and offending players are in blue.}
\label{fig:screen}
\end{figure}
\subsection{Multi-robot surveillance scenario}
\label{subsec:surveillance}
In this subsection we provide Monte Carlo numerical computations
on a multi-robot surveillance scenario.
We first consider the online set-up and then the static one.
\subsubsection{Online set-up}
\label{subsec:dynamic_case}
Let us consider a network of cooperating robots that aim to protect a
target with location $\mathcal{T}_t \in \mathbb{R}^2$ at time $t$ from some intruders.
As done for the previous example, the optimization variables $x_{i,t} \in \mathbb{R}^2$
represent the position of robots at each time $t$ and each robot $i$ is able to move from
$x_{i,t}$ to $x_{i,t+1}$ using a local controller.
We associate to each robot $i$ an intruder located at $p_{i,t} \in \mathbb{R}^2$
at time $t$. The dynamic protection strategy applied by each robot consists
of staying simultaneously close to the protected target and to the associated
intruder. Meanwhile, the whole team of robots tries to keep its
weighted center of mass close to the target. By denoting with
$b_t$ the position of the target, this strategy is obtained by solving
problem~\eqref{eq:online_aggregative_problem} with the following
cost functions,
\begin{align}\label{eq:simulation_cost}
f_{i,t}(x_i,\sigma_t(x)) &= \frac{1}{2}\norm{x_i - p_{i,t}}^2+ \frac{\gamma_1}{2}\norm{x_i - b_{t}}^2\notag
\\
& \hspace{0.4cm} + \frac{\gamma_2}{2N}\norm{\sigma_t(x) - b_t}^2,\notag
\end{align}
with each $x_i \in \mathbb{R}^2$ %
and $\gamma_1, \gamma_2 > 0$ (in particular, $\gamma_1 = 1$ and $\gamma_2 = 10$).
Here, the aggregative variable is defined as
\begin{equation}\label{eq:simulation_aggregative}
\sigma_t(x) = \frac{\sum_{i=1}^{N}\beta_{i,t}x_i}{N},\notag
\end{equation}
where the weights $\beta_{i,t} > 0$ are chosen similarly to the previous example,
in particular they are inversely proportional to the distance $\norm{p_{i,t} - b_t}^2$
with a saturation to $\beta^\text{max}$, i.e.
$\beta_{i,t} = \min\{\norm{p_{i,t} - b_t}^2, \beta^\text{max}\}$.
In this way, at each iteration the center of mass $\sigma_t(x_{t})$ is
positioned in such a way that it gives priority to the intruders
close to the target.
Let us graphically explain this concept by considering a simple
example with $N = 5$ agents and intruders. Fig.~\ref{fig:aggregation_function}
shows two final configurations obtained with two different sets of weights $\beta_{i,t}$.
On the left, we show the positions negotiated by the agents
when each $\beta_{i,t} = 1$ for all $i \in {I}$ and $t \ge 0$,
i.e., with time-invariant aggregation functions $\phi_{i,t} \equiv \phi_i$.
Even though the upper-right intruder is close to the target,
this choice of the weights does not force the formation to condensate
around the target.
On the right, we show the positions negotiated by the agents
with the previously discussed weights $\beta_{i,t}$, i.e.,
with time-varying aggregation functions $\phi_{i,t}$.
The figure clearly shows that the network composes a smarter
formation which better protects the target.
This simple example shows the benefit of our extension
that allows for time-varying aggregation functions.
\begin{figure}
\centering
\subfloat[][Time-invariant weights $\beta_{i,t} = 1$.] %
{\includegraphics[scale=0.95]{figs/aggregation_fixed.pdf}}\hfill
\subfloat[][Time-varying weights, inversely proportional to $\norm{p_{i,t} - b_t}^2$.]
{\includegraphics[scale=0.95]{figs/aggregation_varying.pdf}}
\caption{Comparison between two sets of weights $\beta_i$. The red star represents the target, circles represent agents and squares represent intruders.}
\label{fig:aggregation_function}
\end{figure}
Now, let us consider a larger scenario with $N = 50$ agents and intruders.
As regards the constraints, each $x_i$ must belong to
a common time-varying box constraint $X_{i,t} = \bar{X}_{t}$ given by
\begin{equation*}
\bar{X}_{t} := \{x \in \mathbb{R}^2 \mid d_{t} \leq x \leq u_{t} \},
\end{equation*}
where $d_{t}, u_{t} \in \mathbb{R}^2$ are randomly updated according to a piece-wise linear
trajectory and are respectively initialized as $[0, 0]^\top$ and $[20, 20]^\top$.
Each intruder $i$ moves along a
circle of radius $r = 1$ according to the following law
\begin{equation*}
p_{i,t} = p_{i,c} + r\begin{bmatrix}
\cos(t/100)\\
\sin(t/100)
\end{bmatrix},
\end{equation*}
where $p_{i,c} \in \mathbb{R}^2$ is a randomly generated center.
The target $b_{i,t}$ moves according to a similar law
with a randomly generated center $b_{i,c} \in \mathbb{R}^2$.
As regards the algorithm parameters, we set $\alpha = 1$ and $\delta = 0.5$.
We performed $100$ Monte Carlo trials. Fig.~\ref{fig:regret_avg} shows the
performance of \algo/ in terms of average dynamic regret.
As predicted by Corollary~\ref{cor:avg_regret}, the average dynamic regret
asymptotically converges to a constant term.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/regret_avg.pdf}
\caption{Multi-robot surveillance application -- Online case.
Mean of the average Dynamic regret and $1$-standard deviation band
over $100$ Monte Carlo trials.}
\label{fig:regret_avg}
\end{figure}
\subsubsection{Static set-up}
Here we address a static instance of the problem. Namely, we
freeze the positions of the intruders and of the target
and consider a single realization of the constraints,
so that it holds $f_{t} = f$, $\phi_{t} = \phi$, and $\Xk = X$
for all $t \ge 0$.
We perform a Monte Carlo simulation consisting of $100$ trials
on the same network of $N =50$ agents.
The algorithm parameters are $\alpha = 1$ and $\delta = 0.5$
and the initialization $x_{i,0}$ is random.
As predicted by Corollary~\ref{cor:static}, Fig.~\ref{fig:static}
shows an exponential decay of the relative error
$\frac{\norm{x_{t} - x^\star}}{\norm{x^\star}}$, where $x^\star$ is
the minimizer of $f$ over $X$ computed by a centralized solver.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/static.pdf}
\caption{Multi-robot surveillance application -- Static case.
Mean of the relative error and $1$-standard deviation band
obtained with $100$ Monte Carlo trials.}
\label{fig:static}
\end{figure}
\section{Conclusions}
\label{sec:conclusions}
In this paper, we focused on online instances of the emerging distributed constrained aggregative optimization framework.
We proposed \algo/, a distributed algorithm to solve the online optimization problem that allows for time-varying feasible sets and aggregation rules. We perform a regret analysis of the scheme by which we conclude that, in the online case, the dynamic regret is bounded by a constant term and a linear term related to time variations, while in the static case (i.e. with constant cost and constraints), the solution estimates computed by agents converge with a linear rate to the optimal solution.
%
Numerical computations confirmed the theoretical findings.
\begin{appendix}
In the next, for brevity we will denote as $d_t$, $g_t$, and $g_t^\star$ respectively
\begin{subequations}
\begin{align}
d_t &:= \nabla_1 f_{t}(x_{t},s_{t}) + \nabla\phi_{t}(x_{t})y_{t}\label{eq:dk}
\\
g_t &:= \nabla_1 f_{t}(x_{t},\mathbf{1}\sigma_t(x_{t}))\notag
\\
& \hspace{0.5cm}+ \nabla\phi_{t}(x_{t})\mathbf{1}\frac{1}{N}\sum_{i=1}^{N}\nabla_2f_{i,t}(x_{i,t},\sigma_t(x_{t}))\label{eq:gk}
\\
g_t^\star &:= \nabla_1f_t(x_t^\star,\mathbf{1}\sigma_t^\star)
\notag
\\
& \hspace{0.5cm} + \nabla\phi_{t}(x_t^\star)\mathbf{1}\frac{1}{N}\sum_{i=1}^{N}\nabla_2f_{i,t}(x_{i,t}^\star,\sigma_t^\star),\label{eq:dkstar}
\end{align}
\end{subequations}
where $x_{i,t}^\star$ is the component $i$ of the minimizer $x_t^\star$ of
problem~\eqref{eq:online_aggregative_problem}.
Note that $d_t$ is the descent direction used within the update~\eqref{eq:tilde_gloabl_update}.
Moreover, $g_t$ represents the gradient of $f_{t}(\cdot,\mathbf{1}\sigma_t(\cdot))$
evaluated at the current state $x_{t}$, while
$g_t^\star$ represents the gradient of same function evaluated
at the optimal solution $x_t^\star$.
\subsection{Preliminaries}
\begin{lemma}[\cite{li2020distributed}]\label{lemma:convexity}
Let $f : \mathbb{R}^n \to \mathbb{R}$ be $\mu$-strongly convex and assume $\nabla f$
is $L$-Lipschitz continuous.
Let $\alpha \in (0, 1/L)$.
Then, for all $x,y \in \mathbb{R}^n$ it holds,
%
\begin{align*}
\norm{x - \alpha\nabla f(x) - (y - \alpha\nabla f(y))}
\leq (1 - \mu\alpha)\norm{x - y}.
\tag*{$\square$}
\end{align*}
%
\end{lemma}
\begin{lemma}[{\cite[Theorem~6.3.12]{horn2012matrix}}]\label{lemma:horn}
Let $M_0$, $E \in \mathbb{R}^{n \times n}$ and let $\lambda$ be a simple eigenvalue of $M_0$. Let $v$ and $w$ be, respectively, the right and left eigenvectors of $M_0$ corresponding to the eigenvalue $\lambda$. Then, for each $\epsilon > 0$, there exists a $\Delta > 0$ such that, for all $\delta \in \mathbb{R}$ with $|\delta| < \Delta$, there is a unique eigenvalue $\lambda(\delta)$ of $M_0 +\delta E$ such that
\begin{equation*}
\left| \lambda(\delta) - \lambda - \delta\frac{ w^HEv}{w^Hv}\right| \leq \left|\delta\right|\epsilon,
\end{equation*}
in which $w^H$ denotes the Hermitian of $w$. Moreover $\lambda(\delta)$ is continuous at $\delta = 0$ and
\begin{equation*}
\lim_{\delta \to 0} \lambda(\delta) = \lambda.
\end{equation*}
Moreover $\lambda(\delta)$ is differentiable at $\delta = 0$ and it holds
\begin{equation*}
\frac{d\lambda(\delta)}{d\delta} \bigg|_{\delta=0} = \frac{ w^HEv}{w^Hv}.
\tag*{$\square$}
\end{equation*}
\end{lemma}
\subsection{Proof of Lemma \ref{lemma:xk_xstar}}
\label{sec:proof_xk_xkstar}
We begin by using~\eqref{eq:x_global_update}, which leads to
%
\begin{align}
&\norm{x_{t+1} - x_{t+1}^\star} = \norm{x_{t} + \delta(\tilde{x}_{t} - x_{t})- x_{t+1}^\star} \notag
\\
&\stackrel{(a)} \leq \norm{x_{t} + \delta(\tilde{x}_{t} - x_{t})- x_{t+1}^\star + x_t^\star -x_t^\star}\notag
\\
&\stackrel{(b)} \leq \norm{x_{t} + \delta(\tilde{x}_{t} - x_{t})- x_t^\star} +\norm{x_{t+1}^\star -x_t^\star}\notag
\\
&\stackrel{(c)} \leq \norm{x_{t} + \delta(\tilde{x}_{t} - x_{t})- x_t^\star} + \zeta,\label{eq:before_fixed_point}
\end{align}
%
where in \emph{(a)} we add and subtract the term $x_t^\star$,
in \emph{(b)} we use the triangle inequality,
and in \emph{(c)} we use Assumption~\ref{ass:bdd_var}.
We recall that $x_t^\star$ is the optimal solution of $f_{t}$ over the set $\Xk$,
then it holds %
$P_{\Xk}\left[x_t^\star - \alphag_t^\star\right] = x_t^\star$, with $g_t^\star$
defined according to~\eqref{eq:dkstar}. For this reason, we can add the
null quantity $\delta\left(P_{\Xk}\left[x_t^\star - \alphag_t^\star\right] - x_t^\star\right)$
in the first term of~\eqref{eq:before_fixed_point} obtaining
%
\begin{align}
&\norm{x_{t+1} - x_{t+1}^\star}\notag
\\
&\leq \norm{x_{t} + \delta(\tilde{x}_{t}-x_{t}) - \left[x_t^\star + \delta\left(P_{\Xk}\left[x_t^\star - \alphag_t^\star\right] - x_t^\star\right)\right]} + \zeta\notag
\\
&\stackrel{(a)}\leq\norm{(1-\delta)x_{t} + \delta\tilde{x}_{t} - \left[(1-\delta)x_t^\star + \delta P_{\Xk}\left[x_t^\star - \alphag_t^\star\right]\right]} + \zeta\notag
\\
&\stackrel{(b)}\leq(1-\delta)\norm{x_{t}-x_t^\star} + \delta\norm{\tilde{x}_{t} - P_{\Xk}\left[x_t^\star - \alphag_t^\star\right]} + \zeta\notag
\\
&\stackrel{(c)}\leq(1-\delta)\norm{x_{t}-x_t^\star}\notag
\\
&\hspace{0.5cm}+ \delta\norm{P_{\Xk}[x_{t} - \alphad_t] - P_{\Xk}\left[x_t^\star - \alphag_t^\star\right]}\notag+ \zeta\notag
\\
&\stackrel{(d)}\leq(1-\delta)\norm{x_{t}-x_t^\star} + \delta\norm{x_{t} - \alphad_t - \left(x_t^\star - \alphag_t^\star\right)} + \zeta\label{eq:after_projection},
\end{align}
%
where in \emph{(a)} we algebraically rearrange the terms, \emph{(b)} uses the triangle inequality, \emph{(c)} uses~\eqref{eq:tilde_gloabl_update} to substitute $\tilde{x}_{t}$, and \emph{(d)} uses the non-expansiveness of the projection, see~\cite{bertsekas2015convex}. Now, let us add and subtract within the second norm the term $\alphag_t$, with $g_t$ defined as in~\eqref{eq:gk}. This operation allows us to apply a triangle inequality to rewrite~\eqref{eq:after_projection} as
%
\begin{align}
&\norm{x_{t+1} - x_{t+1}^\star}\notag\\
&\leq(1-\delta)\norm{x_{t}-x_t^\star}+ \delta\norm{x_{t} - \alphag_t - \left(x_t^\star - \alphag_t^\star\right)}\notag
\\
&\hspace{0.5cm}+\delta\alpha\norm{d_t - g_t} + \zeta\notag\\
&\stackrel{(a)}\leq (1-\delta)\norm{x_{t}-x_t^\star}+ \delta(1 - \mu\alpha)\norm{x_{t} - x_t^\star}\notag
\\
&\hspace{0.5cm}+ \delta\alpha\norm{d_t - g_t} + \zeta\notag\\
&\stackrel{(b)}\leq(1-\delta\mu\alpha)\norm{x_{t}-x_t^\star}+ \delta\alpha\norm{d_t - g_t} + \zeta,\notag
\end{align}
%
where \emph{(a)} uses Lemma~\ref{lemma:convexity}, and in \emph{(b)} we group the common terms. Now, we substitute the expression of $d_t$ and we add and subtract within the second norm the term $\nabla\phi(x_{t})\mathbf{1}\frac{1}{N}\sum_{i=1}^{N}\nabla_2f_i(x_{i,t},s_{i,t})$. By leveraging the triangle inequality, we obtain
%
\begin{align}
&\norm{x_{t+1} - x_{t+1}^\star}\notag\\
&\leq(1-\delta\mu\alpha)\norm{x_{t}-x_t^\star}\notag
\\
&\hspace{0.1cm}+ \delta\alpha\norm{\nabla_1 f_{t}(x_{t},s_{t}) + \nabla\phi_{t}(x_{t})\mathbf{1}\frac{1}{N}\sum_{i=1}^{N}\nabla_2f_{i,t}(x_{i,t},s_{i,t}) - g_t}\notag\\
&\hspace{0.1cm}+\delta\alpha\norm{\nabla\phi_{t}(x_{t})\left(y_{t} -\mathbf{1}\frac{1}{N}\sum_{i=1}^{N}\nabla_2f_{i,t}(x_{i,t},s_{i,t})\right)} + \zeta\notag
\\
&\stackrel{(a)} = (1-\delta\mu\alpha)\norm{x_{t}-x_t^\star}\notag
\\
&\hspace{0.1cm}+ \delta\alpha\norm{\nabla_1 f_{t}(x_{t},s_{t}) + \nabla\phi_{t}(x_{t})\mathbf{1}\frac{1}{N}\sum_{i=1}^{N}\nabla_2f_{i,t}(x_{i,t},s_{i,t}) - g_t}\notag
\\
&\hspace{0.1cm}+\delta\alpha\norm{\nabla\phi_{t}(x_{t})\left(y_{t} -\mathbf{1}\bar{y}_t\right)} + \zeta,\label{eq:xkp_xkpstar_before_lipschitz}
\end{align}
%
where in \emph{(a)} we use the initialization of each $y_{i,0}$ to write $\frac{1}{N}\sum_{i=1}^{N}\nabla_2f_{i,t}(x_{i,t},s_{i,t}) = \bar{y}_t$. Consider the term $\|\nabla_1 f_{t}(x_{t},s_{t}) + \nabla\phi_{t}(x_{t})\mathbf{1}\frac{1}{N}\sum_{i=1}^{N}\nabla_2f_{i,t}(x_{i,t},s_{i,t}) - g_t\|$. By using~\eqref{ass:lipschitz} it is possible to write
%
\begin{align}
&\bigg\|\nabla_1 f_{t}(x_{t},s_{t})\notag
\\
&+\! \nabla\phi_{t}(x_{t})\mathbf{1}\frac{1}{N}\!\sum_{i=1}^{N}\!\nabla_2f_{i,t}(x_{i,t},\!s_{i,t}\!) \!-\! g_t\!\bigg\| \leq \!L_1 \!\norm{s_{t}\!-\! \mathbf{1}\bar{s}_t\!},\label{eq:dkavg_gk}
\end{align}
%
which can be combined with~\eqref{eq:xkp_xkpstar_before_lipschitz} to get
%
\begin{align}
\norm{x_{t+1} - x_{t+1}^\star}& \leq(1-\delta\mu\alpha)\norm{x_{t}-x_t^\star}+ \delta\alpha L_1\norm{s_{t} - \mathbf{1}\bar{s}_t}\notag
\\
&\hspace{0.5cm}+\delta\alpha\norm{\nabla\phi_{t}(x_{t})(y_{t} -\mathbf{1}\bar{y}_t} + \zeta\notag
\\
&\stackrel{(a)}\leq(1-\delta\mu\alpha)\norm{x_{t}-x_t^\star}+ \delta\alpha L_1\norm{s_{t} - \mathbf{1}\bar{s}_t}\notag
\\
&\hspace{0.5cm}+\delta\alpha\norm{\nabla\phi_{t}(x_{t})}\norm{y_{t} -\mathbf{1}\bar{y}_t} + \zeta\notag
\\
&\stackrel{(b)}\leq(1-\delta\mu\alpha)\norm{x_{t}-x_t^\star} + \delta\alpha L_1\norm{s_{t}-\mathbf{1}\bar{s}_t}\notag
\\
&\hspace{0.5cm}+ \delta\alpha L_3\norm{y_{t} - \mathbf{1}\bar{y}_t} + \zeta,\notag
\end{align}
%
where in \emph{(a)} we use the Cauchy-Schwarz inequality, and in \emph{(b)} we use the bound $\norm{\nabla\phi_{t}(x)} \le L_3$ for all $x \in \mathbb{R}^d$ (which directly derives from Assumption~\ref{ass:lipschitz}). The proof follows.
\oprocend
\subsection{Proof of Lemma \ref{lemma:xkp_xk}}
\label{sec:proof_xkp_xk}
We can use~\eqref{eq:x_global_update} to write
%
\begin{align}
&\norm{x_{t+1} - x_{t}}\notag
\\
&= \norm{x_{t} + \delta(\tilde{x}_{t} - x_{t}) - x_{t}}\notag
\\
& = \delta\norm{\tilde{x}_{t} - x_{t}}\notag
\\
&\stackrel{(a)} =\delta\norm{P_{\Xk}[x_{t} - \alphad_t] - x_{t}}\notag,
\\
&\stackrel{(b)} =\delta\norm{P_{\Xk}[x_{t} - \alphad_t] -\left(P_{\Xk}\left[x_t^\star - \alphag_t^\star\right] - x_t^\star\right) - x_{t}}\notag,
\end{align}
%
where in \emph{(a)} we have used the update~\eqref{eq:tilde_gloabl_update} and in \emph{(b)} we have added the null quantity $\left(P_{\Xk}\left[x_t^\star - \alphag_t^\star\right] - x_t^\star\right)$ within the norm. Now we apply the triangle inequality to write
%
\begin{align}
&\norm{x_{t+1} - x_{t}}\notag
\\
&\leq\delta\norm{P_{\Xk}[x_{t} - \alphad_t] - P_{\Xk}\left[x_t^\star - \alphag_t^\star\right]} + \delta\norm{x_{t} - x_t^\star} \notag
\\
&\stackrel{(a)}\leq\delta\norm{x_{t} - \alphad_t - \left(x_t^\star - \alphag_t^\star\right)} + \delta\norm{x_{t} - x_t^\star} \notag
\\
&\stackrel{(b)}\leq2\delta\norm{x_{t} -x_t^\star} + \delta\alpha\norm{d_t - g_t^\star},\notag
\end{align}
%
where in \emph{(a)} we use a projection property, and in \emph{(b)} we use the triangle inequality and we group common terms. Now, we substitute the expression of $d_t$ and we add and subtract within the norm the term $\nabla\phi_{t}(x_{t})\mathbf{1}\sum_{i=1}^{N}\nabla_2f_{i,t}(x_{i,t},s_{i,t})$ to write
%
\begin{align}
&\norm{x_{t+1} - x_{t}}\notag
\\
&\leq 2\delta\norm{x_{t} -x_t^\star}\notag
\\
&\hspace{0.5cm}+ \delta\alpha\norm{\nabla_1 f(x_{t},s_{t}) + \nabla\phi_{t}(x_{t})\mathbf{1}\sum_{i=1}^{N}\nabla_2f_{i,t}(x_{i,t},s_{i,t}) - g_t^\star}\notag\\
&\hspace{0.5cm}+ \delta\alpha\norm{\nabla\phi_{t}(x_{t})\left(y_{t} - \mathbf{1}\sum_{i=1}^{N}\nabla_2f_{i,t}(x_{i,t},s_{i,t})\right)}\notag
\\
&\stackrel{(a)} = 2\delta\norm{x_{t} -x_t^\star}\notag
\\
&\hspace{0.5cm}+ \delta\alpha\norm{\nabla_1 f(x_{t},s_{t}) + \nabla\phi_{t}(x_{t})\mathbf{1}\sum_{i=1}^{N}\nabla_2f_{i,t}(x_{i,t},s_{i,t}) - g_t^\star}\notag\\
&\hspace{0.5cm}+ \delta\alpha\norm{\nabla\phi_{t}(x_{t})\left(y_{t} - \mathbf{1}\bar{y}_t\right)}\notag
\\
&\stackrel{(b)} = 2\delta\norm{x_{t} -x_t^\star}\notag
\\
&\hspace{0.5cm}+ \delta\alpha\norm{\nabla_1 f(x_{t},s_{t}) + \nabla\phi_{t}(x_{t})\mathbf{1}\sum_{i=1}^{N}\nabla_2f_{i,t}(x_{i,t},s_{i,t}) - g_t^\star}\notag\\
&\hspace{0.5cm}+ \delta\alpha \norm{\nabla\phi_{t}(x_{t}}\norm{y_{t} - \mathbf{1}\bar{y}_t}\notag
\\
&\stackrel{(c)} = 2\delta\norm{x_{t} -x_t^\star}\notag
\\
&\hspace{0.5cm}+ \delta\alpha\norm{\nabla_1 f(x_{t},s_{t}) + \nabla\phi_{t}(x_{t})\mathbf{1}\sum_{i=1}^{N}\nabla_2f_{i,t}(x_{i,t},s_{i,t}) - g_t^\star}\notag\\
&\hspace{0.5cm}+ \delta\alpha L_3\norm{y_{t} - \mathbf{1}\bar{y}_t},\label{eq:before_dkstar}
\end{align}
%
where in \emph{(a)} we use the initialization of each $y_{i,0}$ to write $\frac{1}{N}\sum_{i=1}^{N}\nabla_2f_{i,t}(x_{i,t},s_{i,t}) = \bar{y}_t$, in \emph{(b)} we use the Cauchy-Schwarz inequality, and in \emph{(c)} we use the bound $\norm{\nabla\phi_{t}(x)} \le L_3$. Consider the term $\norm {\nabla_1 f(x_{t},s_{t}) + \nabla\phi_{t}(x_{t})\mathbf{1}\sum_{i=1}^{N}\nabla_2f_{i,t}(x_{i,t},s_{i,t}) - g_t^\star}$. In view of Assumption~\ref{ass:lipschitz}, it holds
%
\begin{align}
&\norm{\nabla_1 f_{t}(x_{t},s_{t}) + \nabla\phi_{t}(x_{t})\mathbf{1}\frac{1}{N}\sum_{i=1}^{N}\nabla_2f_{i,t}(x_{i,t},s_{i,t}) - g_t^\star}\notag
\\
& \leq L_1\norm{x_{t} - x_t^\star} + L_1 \norm{s_{t}- \mathbf{1}\sigma_t^\star}.\label{eq:dkavg_gkstar}
\end{align}
%
We can combine~\eqref{eq:before_dkstar} and~\eqref{eq:dkavg_gkstar} to write
%
\begin{align}
&\norm{x_{t+1} - x_{t}}\notag
\\
&\leq \delta(2 +\alpha L_1) \norm{x_{t} -x_t^\star} %
+ \delta\alpha L_1\norm{s_{t}-\mathbf{1}\sigma_t^\star}\notag\\
&\hspace{0.5cm}+\delta\alpha L_3\norm{y_{t} - \mathbf{1}\bar{y}_t} \notag
\\
&\stackrel{(a)} \leq \delta(2 +\alpha L_1) \norm{x_{t} -x_t^\star} %
+ \delta\alpha L_1\norm{s_{t}-\mathbf{1}\bar{s}_t}\notag\\
&\hspace{0.5cm}+\delta\alpha L_1\norm{\mathbf{1}\bar{s}_t - \mathbf{1}\sigma_t^\star}+\delta\alpha L_3\norm{y_{t} - \mathbf{1}\bar{y}_t},\label{eq:before_sigmak_sigmastar}
\end{align}
%
where in \emph{(a)} we add and subtract $\mathbf{1}\bar{s}_t$ and we apply the triangle inequality.
Now consider the term $\norm{\mathbf{1}\bar{s}_t - \mathbf{1}\sigma_t^\star}$. By using
the definition of $\sigma_t$ and the Lipschitz continuity of $\phi_{i,t}$ (cf. Assumption~\ref{ass:lipschitz}),
it can be seen that (see also~\cite{li2020distributed}) ,
%
\begin{align}
\norm{\mathbf{1}\bar{s}_t - \mathbf{1}\sigma_t^\star} &\leq L_3\norm{x_{t} - x_t^\star},\label{eq:sigmak_sigmastar}
\end{align}
%
The result~\eqref{eq:sigmak_sigmastar} can combined with~\eqref{eq:before_sigmak_sigmastar} to write
%
\begin{align}
&\norm{x_{t+1} - x_{t}}\notag
\\
&\stackrel{(b)} \leq \delta(2 +\alpha L_1 + \alpha L_1L_3) \norm{x_{t} -x_t^\star} %
+ \delta\alpha L_1\norm{s_{t}-\mathbf{1}\bar{s}_t}\notag\\
&\hspace{0.5cm}+\delta\alpha L_3\norm{y_{t} - \mathbf{1}\bar{y}_t},\notag
\end{align}
%
and the proof is given.
\oprocend
\subsection{Proof of Lemma \ref{lemma:sk_skavg}}
\label{sec:proof_sk_skavg}
By applying~\eqref{eq:sigma_global_update} and~\eqref{eq:sigma_mean_update}, we can write
\begin{align}
&\norm{s_{t+1} - \mathbf{1}\bar{s}_{t+1}}\notag
\\
&=\norm{As_{t} - \mathbf{1}\bar{s}_t + \left(I -\frac{\mathbf{1}\1^\top}{N}\right)(\phi_{t+1}(x_{t+1}) - \phi_{t}(x_{t}))}\notag
\\
&=\norm{\left(A - \frac{\1\1^\top}{N}\right)s_{t} + \left(I -\frac{\mathbf{1}\1^\top}{N}\right)(\phi_{t+1}(x_{t+1}) - \phi_{t}(x_{t}))}\notag
\\
&\stackrel{(a)}\leq\norm{\left(A - \frac{\1\1^\top}{N}\right)s_{t}}\notag
\\
&\hspace{0.5cm}+ \norm{\left(I - \frac{\1\1^\top}{N}\right)(\phi_{t+1}(x_{t+1}) - \phi_{t}(x_{t}))}\notag
\\
&\stackrel{(b)}\leq\norm{\left(A - \frac{\1\1^\top}{N}\right)\left(s_{t} - \mathbf{1}\bar{s}_t\right)}\notag
\\
&\hspace{0.5cm}+ \norm{\left(I - \frac{\1\1^\top}{N}\right)(\phi_{t+1}(x_{t+1}) - \phi_{t}(x_{t}))}\notag
\end{align}
%
where \emph{(a)} uses the triangle inequality and \emph{(b)} uses the fact that $\mathbf{1} \in \ker\left(A-\frac{\1\1^\top}{N}\right)$. Now, we add and subtract within the second norm the term $\phi_{t+1}(x_{t})$ and we apply the triangle inequality obtaining
%
\begin{align}
&\norm{s_{t+1} - \mathbf{1}\bar{s}_{t+1}}\notag
\\
&\leq\norm{\left(A - \frac{\1\1^\top}{N}\right)\left(s_{t} - \mathbf{1}\bar{s}_t\right)}\notag
\\
&\hspace{0.5cm}+ \norm{\left(I - \frac{\1\1^\top}{N}\right)(\phi_{t+1}(x_{t+1}) - \phi_{t+1}(x_{t}))}\notag
\\
&\hspace{0.5cm}+ \norm{\left(I - \frac{\1\1^\top}{N}\right)(\phi_{t+1}(x_{t}) - \phi_{t}(x_{t}))}\notag
\\
&\stackrel{(a)}\leq \rho\norm{s_{t} - \mathbf{1}\bar{s}_t} + L_3\norm{x_{t+1} - x_{t}} + \sqrt{N}\omega\notag
\\
&\stackrel{(b)}\leq\rho\norm{s_{t} - \mathbf{1}\bar{s}_t} + \delta(2 L_3 + \alpha L_1L_3 + \alpha L_1L_3^2)\norm{x_{t} -x_t^\star}\notag
\\
&\hspace{0.5cm}+ \delta\alpha L_1L_3\norm{s_{t}-\mathbf{1}\bar{s}_t} \notag %
+\delta\alpha L_3^2\norm{y_{t} - \mathbf{1}\bar{y}_t} + \sqrt{N}\omega,\notag
\end{align}
where in \emph{(a)} we use the maximum eigenvalue $\rho$ of the matrix $A-\frac{\1\1^\top}{N}$ (we have $\rho < 1$ as a consequence of Assumption~\ref{ass:network}), and we use also Assumptions~\ref{ass:lipschitz} and~\ref{ass:bdd_var}, and $\norm{I - \frac{\1\1^\top}{N}} = 1$, while in \emph{(b)} we use Lemma~\ref{lemma:xkp_xk} to bound $\norm{x_{t+1} - x_t^\star}$, and the proof follows.\oprocend
\subsection{Proof of Lemma \ref{lemma:yk_ykavg}}
\label{sec:proof_yk_ykavg}
We use~\eqref{eq:y_global_update} and~\eqref{eq:y_mean_update} to write
\begin{align}
&\norm{y_{t+1} - \mathbf{1}\bar{y}_{t+1}}\notag
\\
&\leq\norm{Ay_{t} - \bar{y}_t}\notag
\\
&\hspace{0.5cm}+ \norm{\left(I-\frac{\1\1^\top}{N}\right)(\nabla_2f_{t+1}(x_{t+1},s_{t+1}) - \nabla_2f_{t+1}(x_{t},s_{t})) }\notag
\\
&\stackrel{(a)}\leq\norm{\left(A-\frac{\1\1^\top}{N}\right)(y_{t}-\mathbf{1}\bar{y}_t)}\notag\\
&\hspace{0.5cm}+ \norm{\left(I- \frac{\1\1^\top}{N} \right)(\nabla_2f_{t+1}(x_{t+1},s_{t+1}) - \nabla_2f_{t+1}(x_{t},s_{t}))}\notag\\
&\hspace{0.5cm}+ \norm{\left(I- \frac{\1\1^\top}{N} \right)(\nabla_2f_{t+1}(x_{t},s_{t}) - \nabla_2f_{t}(x_{t},s_{t}))}\notag
\end{align}
%
where in \emph{(a)} we use the fact that $\mathbf{1} \in \ker\left(A-\frac{\1\1^\top}{N}\right)$ and we apply the triangle inequality after adding and subtracting the term $\left(I- \frac{\1\1^\top}{N} \right)\nabla_2f_{t+1}(x_{t},s_{t})$ within the norm. By using Assumptions~\ref{ass:lipschitz} and~\ref{ass:bdd_var}, we get
%
\begin{align}
&\norm{y_{t+1} - \mathbf{1}\bar{y}_{t+1}}\notag
\\
&\leq\norm{\left(A-\frac{\1\1^\top}{N}\right)(y_{t}-\mathbf{1}\bar{y}_t)}\notag
\\
&\hspace{0.5cm}+ L_2\norm{x_{t+1} - x_{t}} + L_2\norm{s_{t+1} - s_{t}} + \sqrt{N}\eta\notag
\\
&\stackrel{(a)}\leq\rho\norm{y_{t}-\mathbf{1}\bar{y}_t} + L_2\norm{x_{t+1} - x_{t}} \!+\! L_2\norm{s_{t+1} - s_{t}} \!+\! \sqrt{N}\eta\notag
\end{align}
%
where \emph{(a)} uses the maximum eigenvalue $\rho$ of the matrix $A-\frac{\1\1^\top}{N}$ (we have $\rho < 1$ as a consequence of Assumption~\ref{ass:network}). Now, we can use~\eqref{eq:sigma_global_update} to get
%
\begin{align*}
&\norm{y_{t+1} - \mathbf{1}\bar{y}_{t+1}}
\\
&\leq\rho\norm{y_{t}-\mathbf{1}\bar{y}_t} + L_2\norm{x_{t+1} - x_{t}}\notag
\\
& \hspace{0.5cm}+ L_2\norm{(A-I)s_{t} + \phi_{t+1}(x_{t+1}) - \phi_{t}(x_{t})}+ \sqrt{N}\eta\notag
\end{align*}
\begin{align}
&\stackrel{(a)}\leq\rho\norm{y_{t}-\mathbf{1}\bar{y}_t} + L_2\norm{x_{t+1} - x_{t}} + L_2\norm{(A-I)(s_{t}-\mathbf{1}\bar{s}_t)}\notag\\
&\hspace{0.5cm}+ L_2\norm{\phi_{t+1}(x_{t+1}) - \phi_{t}(x_{t})} + \sqrt{N}\eta\notag
\end{align}
%
where in \emph{(a)} we apply the triangle inequality and the fact that $\mathbf{1} \in \ker\left(A-\frac{\1\1^\top}{N}\right)$. Now, by adding and subtracting within the norm the term $\phi_{t+1}(x_{t})$ and by applying the triangle inequality, it is possible to write
%
\begin{align}
&\norm{y_{t+1} - \mathbf{1}\bar{y}_{t+1}}\notag
\\
&\leq\rho\norm{y_{t}-\mathbf{1}\bar{y}_t} + L_2\norm{x_{t+1} - x_{t}} + L_2\norm{(A-I)(s_{t}-\mathbf{1}\bar{s}_t)}\notag
\\
&\hspace{0.5cm}+ L_2\norm{\phi_{t+1}(x_{t+1}) - \phi_{t+1}(x_{t})}\notag
\\
&\hspace{0.5cm} + L_2\norm{\phi_{t+1}(x_{t}) - \phi_{t}(x_{t})} + \sqrt{N}\eta\notag
\\
&\stackrel{(a)}\leq\rho\norm{y_{t}-\mathbf{1}\bar{y}_t} + L_2\norm{x_{t+1} - x_{t}} + L_2\norm{(A-I)(s_{t}-\mathbf{1}\bar{s}_t)}\notag
\\
&\hspace{0.5cm}+ L_2L_3\norm{x_{t+1} - x_{t}} + L_2\sqrt{N}\omega + \sqrt{N}\eta\notag
\\
&\stackrel{(b)}\leq\rho\norm{y_{t}-\mathbf{1}\bar{y}_t} + 2L_2\norm{s_{t}-\mathbf{1}\bar{s}_t}\notag
\\
& \hspace{0.5cm}+ \delta(2 + \alpha L_1)(L_2 + L_2L_3)\norm{x_{t} -x_t^\star}\notag
\\
&\hspace{0.5cm}+ \delta\alpha L_1(L_2 + L_2L_3)\norm{s_{t}-\mathbf{1}\bar{s}_t}\notag
\\
&\hspace{0.5cm}+\delta\alpha L_1L_3(L_2 + L_2L_3)\norm{x_{t} - x_t^\star}\notag
\\
&\hspace{0.5cm}+\delta\alpha L_3(L_2 + L_2L_3)\norm{y_{t} - \mathbf{1}\bar{y}_t} + L_2\sqrt{N}\omega + \sqrt{N}\eta,\notag
\end{align}
where in \emph{(a)} we use Assumption~\ref{ass:lipschitz} and Assumption~\ref{ass:bdd_var}, while in \emph{(b)} we use the fact that $\norm{(A-I)(s_{t}-\mathbf{1}\bar{s}_t)} \le \lambda_\text{max}(A - I)\norm{s_{t}-\mathbf{1}\bar{s}_t}$ where $\lambda_\text{max}(A - I) \le 2$ is the maximum eigenvalue of $A - I$ (in absolute value), and we also apply Lemma~\ref{lemma:xkp_xk} to bound $\norm{x_{t+1}-x_{t}}$. The proof is given.
\oprocend
\end{appendix}
\bibliographystyle{IEEEtran}
|
1,116,691,499,228 | arxiv | \section{Introduction}
\label{intro}
To generate the baryon number asymmetry, Sakharov's criteria \cite{sakharov} have to be fulfilled.
One of them, the CP violation, is actually insufficient in the standard model (SM), and contributions from new physics beyond the SM is required.
As a good probe of CP violation beyond the SM, the electric dipole moment (EDM) \cite{engeledmreview,yamanakabook,atomicedmreview,chuppreview} is widely studied.
Here we discuss the EDM of light nuclei which is expected to be accurately measured using storage rings \cite{storage1,storage2,anastassopoulos,devriesedmreview,jedi,yamanakanuclearedmreview}.
Theoretically, the nuclear EDM presents many advantages.
The bare nucleus has no atomic electrons, so there is no suppression of hadronic CP violation due to Schiff's screening \cite{schiff}.
The nuclear EDM is also almost free from the CKM contribution \cite{yamanakasmedm1,yamanakasmedm2}, so that it has very small SM backgrounds to be considered.
An interesting question is whether the sensitivity of the nuclear EDM on CP violation beyond the SM is enhanced by many-body effects.
Here we discuss the EDM of light nuclei, which can be treated in the cluster model \cite{clusterreview3} with good accuracy.
By analyzing the EDM of several light nuclei ($^2$H, $^3$He, $^6$Li, $^7$Li, $^9$Be, $^{11}$B, and $^{13}$C), we will try to deduce an approximate counting rule.
The preliminary result of the EDM of $^{11}$B is shown for the first time.
This proceeding is organized as follows.
In the next section, we show the setup of the cluster model.
In Section \ref{sec:nuclearedm}, we summarize the current results on the EDM of light nuclei, and deduce an approximate counting rule.
Section \ref{sec:summary} is devoted to the summary.
\section{The cluster model and the CP-odd nuclear force}
Let us first introduce the CP-even interaction used in the cluster model.
In our cluster model, the nucleons, the $\alpha$ ($^4$He) and triton ($^3$H, denoted as $t$) clusters are the relevant degrees of freedom.
For the $N-N$ interaction required for the $^6$Li nucleus, the A$v$8' potential \cite{av18} is used.
For the CP-even $\alpha -N$ and $\alpha -\alpha$ interactions, we use the Kanada-Kaneko potential \cite{kanada} and the modified Hasegawa-Nagata potential \cite{hasegawa}, respectively, which were obtained by fitting the data of low energy scattering experiments.
For the calculation of $^{13}$C, we use the Kanada-Kaneko and Schmid-Wildermuth \cite{schmid} potentials, augmented by phenomenological three- and four-cluster interactions to reproduce the binding energies of the ground state as well as those of subsystems \cite{yamada13c}.
We use the interaction of Nishioka {\it et al}. \cite{nishioka} for the CP-even $\alpha -t$ potential, required in the calculation of $^7$Li and $^{11}$B.
For the $^{11}$B nucleus, we also introduce a phenomenological $\alpha -\alpha -t$ interaction to reproduce the energy levels of the ground state ($3/2_1^-$) and the $1/2_1^+$ states \cite{yamada11b}.
In our cluster model calculations, the orthogonality condition model \cite{saito1,saito2,saito3} is applied to exclude forbidden states.
Let us now model the CP-odd interaction.
For the bare $N-N$ system, the leading CP-odd Hamiltonian is given by the following one-pion exchange potential with three possible isospin structures \cite{pvcpvhamiltonian3}:
\begin{eqnarray}
H_{P\hspace{-.45em}/\, T\hspace{-.5em}/\, }^\pi
& = &
\bigg\{
\bar{G}_{\pi}^{(0)}\,{\vc{\tau}}_{1}\cdot {\vc{\tau}}_{2}\, {\vc{\sigma}}_{-}
+\frac{1}{2} \bar{G}_{\pi}^{(1)}\,
( \tau_{+}^{z}\, {\vc{\sigma}}_{-} +\tau_{-}^{z}\,{\vc{\sigma}}_{+} )
\nonumber\\
&&\hspace{6em}
+\bar{G}_{\pi}^{(2)}\, (3\tau_{1}^{z}\tau_{2}^{z}- {\vc{\tau}}_{1}\cdot {\vc{\tau}}_{2})\,{\vc{\sigma}}_{-}
\bigg\}
\cdot
\frac{ \vc{r}}{r} \,
V(r)
,
\label{eq:CPVhamiltonian}
\end{eqnarray}
where $\vc{r} \equiv \vc{r}_1 - \vc{r}_2$ is the relative coordinate.
The spin and isospin matrices are defined by ${\vc{\sigma}}_{-} \equiv {\vc{\sigma}}_1 -{\vc{\sigma}}_2$, ${\vc{\sigma}}_{+} \equiv {\vc{\sigma}}_1 + {\vc{\sigma}}_2$, ${\vc{\tau}}_{-} \equiv {\vc{\tau}}_1 -{\vc{\tau}}_2$, and ${\vc{\tau}}_{+} \equiv {\vc{\tau}}_1 + {\vc{\tau}}_2$.
The radial behavior of the CP-odd potential is given by
\begin{equation}
V (r)
=
-\frac{m_\pi}{8\pi m_N} \frac{e^{-m_\pi r }}{r} \left( 1+ \frac{1}{m_\pi r} \right)
\ .
\end{equation}
Its shape is shown in Fig. \ref{fig:folding}.
In the point-of-view of the chiral perturbation theory, the CP-odd potential of Eq. (\ref{eq:CPVhamiltonian}) is not complete.
In the leading order, there are also P and CP violating contact CP-odd $N-N$ interactions \cite{cpveft} as well as a CP-odd three-nucleon interaction \cite{cpveft2}.
The contribution of the former to the nuclear EDM, however, has a large uncertainty due to the poorly known short-range physics of nuclei \cite{bsaisou}, so we do not consider it.
The three-nucleon interaction is also neglected due to its small contribution to the EDM of light nuclei \cite{bsaisou}.
This is due to the fact that light nuclei have a dominant configuration with paired nucleons, whereas the CP-odd three-nucleon force requires three nucleons with aligned spins.
We also note that the isotensor CP-odd nuclear force [term with $\bar{G}_{\pi}^{(2)}$ in Eq. (\ref{eq:CPVhamiltonian})] is subleading in chiral perturbation, but we display it by habit.
\begin{figure}
\includegraphics[width=0.75\textwidth]{vcp_folded_LEAP2018.eps}
\caption{
Radial dependence of the CP-odd nuclear interactions.
}
\label{fig:folding}
\end{figure}
The CP-odd $\alpha -N $ and $\alpha -t$ potentials are modeled by folding \cite{horiuchi} the CP-odd $N-N$ interaction (\ref{eq:CPVhamiltonian}).
The isoscalar and isotensor CP-odd interactions cancel due to the closure of the spin and isospin shells of the $\alpha$-clusters.
For the oscillator parameters, we take $b = 1.358$ fm ($\alpha - N$) and $b=1.482$ fm ($\alpha - t$).
The shape of the folding CP-odd potentials are displayed in Fig. \ref{fig:folding}.
\section{The nuclear electric dipole moment: definition and results}
\label{sec:nuclearedm}
The nuclear EDM is generated by two leading contributions.
The first one is given by the intrinsic EDM of the nucleon:
\begin{eqnarray}
d_{A}^{\rm (Nedm)}
&=&
\sum_{i=1}^{A} \frac{1}{2}
\langle \, \Phi_J (A) \, |\, (1+\tau_i^z ) \, \sigma_{iz} \, | \, \Phi_J (A) \, \rangle
,
\label{eq:polarizationedm}
\end{eqnarray}
where $|\, \Phi_{J} (A)\, \rangle$ is the state vector of the polarized nucleus $A$.
The second process is the polarization of the nucleus by the CP-odd nuclear force:
\begin{eqnarray}
d_{A}^{\rm (pol)}
&=&
\sum_{i=1}^{A} \frac{e}{2}
\langle \, \Phi_J (A) \, |\, (1+\tau_i^z ) \, r_{iz} \, | \, \Phi_J (A) \, \rangle
.
\label{eq:polarizationedm}
\end{eqnarray}
Since the nucleon EDM and the CP-odd nuclear force are very small, the nuclear EDM can be expressed in the leading order of perturbation as
\begin{eqnarray}
d_{A}
&=&
\langle \sigma_p \rangle d_p
+\langle \sigma_n \rangle d_n
+\bar G_\pi^{(0)}
a_\pi^{(0)}
+\bar G_\pi^{(1)}
a_\pi^{(1)}
+\bar G_\pi^{(2)}
a_\pi^{(2)}
.
\label{eq:polarizationedm}
\end{eqnarray}
Evidently, the most interesting nuclei are those which have large coefficients.
\begin{table}
\caption{
The linear coefficients of the nuclear EDM for several nuclei.
The symbol ``$-$'' means that either the coefficient cancels or it cannot be determined with sufficient accuracy.
}
\label{tab:nuclearedm}
\begin{center}
\begin{tabular}{llllll}
\hline\noalign{\smallskip}
& $\langle \sigma_p \rangle$ & $\langle \sigma_n \rangle$ &$a_\pi^{(0)}$ ($ e$ fm) & $a_\pi^{(1)}$ ($e$ fm) & $a_\pi^{(2)}$ ($ e$ fm) \\
\noalign{\smallskip}\hline\noalign{\smallskip}
$^{2}$H \cite{liu,yamanakanuclearedm} & 0.91 & 0.91 & $-$ & $0.0145 $ & $-$ \\
$^{3}$He \cite{bsaisou,yamanakanuclearedm}& $-0.04$ & 0.89 & $0.0059$ & 0.0108 & 0.0168 \\
$^{3}$H \cite{bsaisou,yamanakanuclearedm} & 0.88 & $-0.05$ & $-0.0059$ & 0.0108 & $-0.0170$ \\
$^{6}$Li \cite{yamanakanuclearedm}& 0.86 & 0.86 & $-$ & 0.022 & $-$ \\
$^{7}$Li \cite{PPNS} & 0.9 & $-$ & $-0.006$ & 0.016 & $-0.017$ \\
$^{9}$Be \cite{yamanakanuclearedm} & $-$ & 0.75 & $-$ & $0.014$ & $-$ \\
$^{11}$B & 0.7 & $-$ & $-0.004$ & $0.02$ & $-0.01$ \\
$^{13}$C \cite{c13edm} & $-$ & $-0.33$ & $-$ & $-0.0020 $ & $-$ \\
$^{129}$Xe \cite{yoshinaga1,yoshinaga2} & $-$& 0.2 & $7 \times 10^{-5}$ & $7 \times 10^{-5} $ & $4 \times 10^{-4}$ \\
\noalign{\smallskip}\hline
\end{tabular}
\end{center}
\end{table}
In Table \ref{tab:nuclearedm}, we list the coefficients of the EDMs of $^6$Li \cite{yamanakanuclearedm}, $^7$Li \cite{PPNS}, $^9$Be \cite{yamanakanuclearedm}, $^{11}$B, and $^{13}$C \cite{c13edm}, calculated in the cluster model.
The result of $^{11}$B is new and preliminary.
Those of the deuteron \cite{liu,yamanakanuclearedm} and three-nucleon systems \cite{bsaisou,yamanakanuclearedm}, calculated with the phenomenological nuclear force Argonne $v18$ \cite{av18}, have also been displayed for comparison.
We see that several nuclei, such as the $^6$Li, $^7$Li, or $^{11}$B, have larger isovector coefficients than that of the deuteron.
This enhancement can be understood as the constructive interference between the EDM of the deuteron or triton cluster and the polarization from the CP-odd $\alpha -N$ or $\alpha -t$ interactions (see Fig. \ref{fig:counting_rule}).
We determine the latter by equating the following system of equations:
\begin{eqnarray}
d_{^{6}{\rm Li}}
& = &
2\times (\alpha -N \, \mbox{polarization})
+ d_{^{2}{\rm H}}
,
\nonumber\\
d_{^{7}{\rm Li}}
& = &
1 \times (\alpha -N \, \mbox{polarization})
+d_{^{3}{\rm H}}
,
\nonumber\\
d_{^{9}{\rm Be}}
& = &
2 \times (\alpha -N \, \mbox{polarization})
,
\nonumber\\
d_{^{11}{\rm B}}
& = &
2 \times (\alpha -N \, \mbox{polarization})
+d_{^{3}{\rm H}}
.
\end{eqnarray}
After using the values of Table \ref{tab:nuclearedm}, one obtains $(\alpha -N \, \mbox{polarization}) \sim (0.005 - 0.007)\, \bar G_\pi^{(1)}e$ fm.
This result forms an approximate counting rule for the EDM of light nuclei.
From this, we can predict the EDM of heavier nuclei.
For instance, we have
\begin{eqnarray}
d_{^{10}{\rm B}}
& \sim &
4\times (\alpha -N \, \mbox{polarization})
+ d_{^{2}{\rm H}}
\sim
0.03 \, \bar G_\pi^{(1)}e \, {\rm fm}
,
\nonumber\\
d_{^{14}{\rm N}}
& \sim &
6 \times (\alpha -N \, \mbox{polarization})
+ d_{^{2}{\rm H}}
\sim
0.04 \, \bar G_\pi^{(1)}e \, {\rm fm}
.
\end{eqnarray}
The contribution from the EDM of the deuteron cluster in $^6$Li and that of the triton in $^{11}$B are slightly decreased due to the mixing of angular momentum configurations.
The dependence on the isoscalar and isotensor nuclear forces of the EDM of $^7$Li and $^{11}$B is due to the triton cluster.
It may additionally be given by the intrinsic EDM of the nucleon, although we do not discuss it in this work.
It is also important to note that the two spins of the neutrons in the triton are likely to form a singlet, so they do not contribute to the CP-odd $\alpha - t$ polarization.
\begin{figure}
\includegraphics[width=1.0\textwidth]{counting_rule.eps}
\caption{
Counting rule for the EDM of light nuclei.
The EDM is induced by the constructive interference between the EDM of the deuteron or triton clusters and the CP-odd $\alpha -N$ or $\alpha -t$ polarizations.
}
\label{fig:counting_rule}
\end{figure}
For the EDM of $^{13}$C, however, we observe that the isovector coefficient is small \cite{c13edm}, and the counting rule cannot be applied.
The $^{13}$C nucleus has a ground state ($1/2^-_1$) composed by a dominant configuration of a $^{12}$C core with angular momentum two.
It happens that this state cannot easily make transition with the closest opposite parity state $1/2^+_1$ which is 3.1 MeV above it through CP-odd operators \cite{c13edm}.
The ground state is rather coupled with a state which is separated by about 10 MeV, which suppresses the EDM.
This feature is also expected to be relevant for $^{15}$N which also has a similar level structure.
What will happen when we increase the nucleon number?
Na\"{i}vely, we can imagine that the contribution from the CP-odd $\alpha -N$ polarization will grow.
This will however not be realized in real nuclei, since the configuration mixing, which makes destructive interference between angular momentum configurations of valence nucleons, is relevant in heavy nuclei.
In Table \ref{tab:nuclearedm}, we see that the result for the nuclear EDM of $^{129}$Xe is much smaller than those of light nuclei \cite{yoshinaga1,yoshinaga2}.
Moreover, heavy nuclei are more difficult to handle in storage ring experiments, so they have no remarkable advantages.
\section{Summary}
\label{sec:summary}
In this proceeding, we presented the calculations of the EDM of light nuclei.
The preliminary result of the EDM of $^{11}$B was shown for the first time.
By equating the sensitivity of the EDM of $^{6}$Li, $^{7}$Li, $^{9}$Be, and $^{11}$B on the isovector CP-odd nuclear force, we could infer an approximate counting rule.
We could then estimate the EDM of other unknown nuclei such as $^{10}$B or $^{14}$N which are estimated to be more sensitive than known ones.
On the other hand, there are other nuclei such as $^{13}$C for which the sensitivity on the isovector CP violation is suppressed by their nuclear structure.
Finally, increasing the number of nucleons is not likely to provide us sensitive nuclei on CP violation, since the configuration mixing will suppress the polarization.
We conclude for the moment that light nuclei are the most suitable to be measured in storage ring experiments.
\begin{acknowledgements}
The author is supported by the JSPS Postdoctoral Fellowships for Research Abroad.
\end{acknowledgements}
|
1,116,691,499,229 | arxiv | \section{Introduction}
The purpose of this note is to combine three standard results in homotopy theory:
\begin{enumerate}[label=(\arabic*)]
\item the construction of elements of order $p^{r+1}$ in the homotopy groups of the mod $p^r$ Moore space $P^n(p^r)$, as described in \cite{CMN1};
\item the stable splitting of $\Omega^2P^n(p^r)$ first proved by Snaith \cite{S}; and
\item the introduction of $v_1$-periodic self-maps by Adams in his work on the image of the $J$-homomorphism.
\end{enumerate}
In their fundamental work on the homotopy theory of Moore spaces, Cohen, Moore and Neisendorfer \cite{CMN1,CMN2, N, N1} proved that $P^n(p^r)=S^{n-1}\cup_{p^r}e^n$ has homotopy exponent exactly $p^{r+1}$ when $p$ is a prime number greater than $3$. We will refer to elements of this maximal possible order in $\pi_\ast(P^n(p^r))$ as \emph{higher torsion} elements. The main results of this paper give additional infinite families of higher torsion elements in the homotopy groups of odd primary Moore spaces which are different from
those constructed via Samelson products in \cite{CMN1}. The main technical ingredient is a slightly finer stable decomposition of $\Omega^2P^{2n+1}(p^r)$ which essentially follows from a combination of results (1) and (2) above.
The reason that this question arose is because of computations of Roman Mikhailov and Jie Wu, who asked about ``functorial elements" of order $p^{r+1}$ in the homotopy groups of mod $p^r$ Moore spaces not given by those in \cite{CMN1}.
By considering the integral homology of the double loop space of a Moore space, it is clear that certain spherical homology classes force the classical Snaith splitting of $\Omega^2P^{2n+1}(p^r)$ to stably decompose further then previously described. This new stable splitting allows for the construction of new higher torsion elements which are detected by $K$-theory but not detected in the ordinary homology of any iterated loop space of a Moore space, unlike the elements of order $p^{r+1}$ in \cite{CMN1} which have non-trivial Hurewicz images in the homology of $\Omega^2P^{2n+1}(p^r)$ (see Lemma~\ref{odd higher Bockstein}).
The main results are described next. Recall that the Snaith splitting gives a functorial stable homotopy equivalence
\[ \Sigma^\infty\Omega^2 \Sigma^2X \simeq \Sigma^\infty\bigvee_{j=1}^\infty D_j(\Omega^2\Sigma^2X) \]
for any path-connected CW-complex $X$, where the stable summands are given by suspension spectra of the extended powers
$D_j(\Omega^2\Sigma^2X) = \mathcal{C}_2(j)_+ \wedge_{\Sigma_j} X^{\wedge j}$
and $\mathcal{C}_2(j)$ denotes the space of~$j$ little $2$-cubes disjointly embedded in $\mathbb{R}^2$.
In the case that $X$ is an odd-dimensional sphere $S^{2n-1}$, the stable summands $D_j(\Omega^2S^{2n+1})$ of $\Omega^2S^{2n+1}$ have been well studied; they are $p$-locally contractible unless $j \equiv 0$ or $1$ mod $p$, in which case they can be identified with suitably suspended Brown--Gitler spectra. In particular, after localizing at a prime $p$, they are stably indecomposable. Below we consider the case of an odd-dimensional Moore space and the stable summands $D_{p^k}(\Omega^2P^{2n+1}(p^r))$ which map naturally onto these Brown--Gitler spectra by the map $\Omega^2\Sigma^2q$ where $q\colon P^{2n-1}(p^r) \to S^{2n-1}$ is the pinch map.
\begin{thm} \label{splitting thm}
Suppose $p$ is prime and $n>1$.
\begin{enumerate}
\item If $p\ge 3$ and $r \ge 1$, then $D_{p^k}(\Omega^2P^{2n+1}(p^r))$ is stably homotopy equivalent to
\[ P^{2np^k-2}(p^{r+1}) \vee X_{p^k} \]
for some finite CW-complex $X_{p^k}$ for all $k\ge 1$.
\item If $p=2$ and $r>1$, then $D_2(\Omega^2P^{2n+1}(2^r))$ is homotopy equivalent to
\[ P^{4n-2}(2^{r+1}) \vee X_2 \]
for some $4$-cell complex $X_2=P^{4n-3}(2^r) \cup CP^{4n-2}(2)$.
\item If $p=2$ and $r=1$, then $D_2(\Omega^2P^{2n+1}(2))$ is a stably indecomposable $6$-cell complex.
\end{enumerate}
\end{thm}
The reason the stable splitting of $\Omega^2P^{2n+1}(p^r)$ described by Theorem \ref{splitting thm} has implications for the unstable homotopy groups of $P^{2n+1}(p^r)$ is that maps $P^{2np^k-2}(p^{r+1}) \to \Omega^2P^{2n+1}(p^r)$ admitting stable retractions exist unstably when $p$ is odd (and in only a few cases when $p=2$; see Section \ref{new families section}).
\begin{thm} \label{factorization thm}
Let $p$ be an odd prime, $r\ge 1$ and $n>1$. Then for every $k\ge 1$ there exist homotopy commutative diagrams
\[
\xymatrix{
P^{2np^k-2}(p^{r+1}) \ar[r] \ar[rd]_{E^\infty} & \Omega^2P^{2n+1}(p^r) \ar[d] \\
& QP^{2np^k-2}(p^{r+1})
} \qquad\qquad
\xymatrix @C=1pc{
P^{(4n-2)p^k-2}(p^{r+1}) \ar[r] \ar[rd]_{E^\infty} & \Omega^2P^{2n}(p^r) \ar[d] \\
& QP^{(4n-2)p^k-2}(p^{r+1})
}
\]
where $E^\infty$ is the stabilization map (i.e., unit of the adjunction $\Sigma^\infty \dashv \Omega^\infty$).
\end{thm}
The loop space decompositions of odd primary Moore spaces given in \cite{CMN1,N1} imply that the stable homotopy groups $\pi_\ast^s(P^n(p^r))$ are in a certain sense retracts of the unstable homotopy groups $\pi_\ast(P^n(p^r))$. Different loop space decompositions were used in \cite{CW} to obtain the same result for $2$-primary Moore spaces, and other examples of spaces whose stable and unstable homotopy groups share this property are given in \cite{BW}. As a consequence of Theorem~\ref{factorization thm}, the stable homotopy groups of the mod $p^{r+1}$ Moore spectrum retract off the unstable homotopy groups of each mod $p^r$ Moore space in a similar sense when $p$ is an odd prime (see Corollary~\ref{stable unstable}).
This observation clearly suggests that $\pi_\ast(P^n(p^r))$ contains many $\mathbb{Z}/p^{r+1}$ summands when~$p$ is odd. To generate explicit examples, in Section \ref{new families section} we use desuspensions of Adams maps in conjunction with Theorem~\ref{factorization thm} to construct infinite $v_1$-periodic families of higher torsion elements and obtain the following. Let $q=2(p-1)$.
\begin{thm} \label{new odd families}
Let $p$ be an odd prime, $r\ge 1$ and $n>1$. Then for all sufficiently large $k$,
\[ \pi_{2np^k-1+tqp^r}(P^{2n+1}(p^r)) \, \text{ and }\ \pi_{(4n-2)p^k-1+tqp^r}(P^{2n}(p^r)) \]
contain $\mathbb{Z}/p^{r+1}$ summands for every $t\ge 0$.
\end{thm}
\begin{rem}
A lower bound on $k$ in Theorem \ref{new odd families} is required to ensure the existence of unstable Adams maps $v_1\colon P^{\ell+qp^r}(p^{r+1}) \to P^\ell(p^{r+1})$ which induce isomorphisms in $K$-theory. See Section \ref{new families section} for a more precise statement. In particular, in the most interesting case when $r=1$, we only require $k\ge 1$.
\end{rem}
For $p=2$, unstable maps analogous to those in Theorem~\ref{factorization thm} rarely exist for reasons related to the divisibility of the Whitehead square. When $r>1$, the inclusion of the stable summand $P^{4n-2}(2^{r+1})$ of $\Omega^2P^{2n+1}(2^r)$ given by Theorem~\ref{splitting thm}(b) exists unstably if and only if $n=2$ or $4$. As examples in these cases, we describe unstable $v_1$-periodic families of higher torsion elements in $\pi_\ast(P^5(2^r))$ and $\pi_\ast(P^9(2^r))$ for some small values of $r$ (Theorem~\ref{new even families}).
\begin{rem}
When $p$ is odd, the $r>1$ case of Theorem~\ref{splitting thm}(a) follows quickly from an unstable product decomposition of $\Omega^2P^{2n+1}(p^r)$ proved by Neisendorfer. More precisely, \cite[Theorem~1]{N} shows that $\Omega S^{2np^k-1}\{p^{r+1}\}$ is a retract of $\Omega^2P^{2n+1}(p^r)$ for all $k\ge 1$, and it is readily checked that $P^{2np^k-2}(p^{r+1})$ is a stable retract of $\Omega S^{2np^k-1}\{p^{r+1}\}$ (cf.\ \cite[Proposition~4.1]{Am}). The $r=1$ case of Theorem~\ref{splitting thm}(a) above gives some evidence for conjectures surrounding the unstable homotopy type of $\Omega^2P^{2n+1}(p)$ considered in \cite{CMN2,G2,N double,T}.
\end{rem}
\begin{ack}
This material is based upon work partly supported by the National Science Foundation under Grant No.\ DMS-1929284 while the first author was in residence at the Institute for Computational and Experimental Research in Mathematics in Providence, RI.
\end{ack}
\section{Homology of the stable summands $D_j(\Omega^2P^{2n+1}(p^r))$}
The homology of $\Omega^n\Sigma^nX$ taken with field coefficients
as a filtered algebra was worked out in \cite{CLM}. A short summary of that information elucidates the homology of the stable Snaith summands, usually denoted $D_{n,j}X$. In the applications below where $n=2$, the $j^\mathrm{th}$ stable summand $D_{2,j}X$ of $\Omega^2\Sigma^2X$ will be denoted $D_j(\Omega^2\Sigma^2X)$. All homology groups have $\mathbb{Z}/p$ coefficients unless indicated otherwise.
Start with the connected graded vector space $$V=\bar{H}_\ast(X)$$ given by the reduced mod~$p$ homology of a path-connected space~$X$. Next, consider the reduced homology of the suspension $\Sigma X$,
denoted $\sigma V$. Form the free graded Lie algebra generated by $\sigma V$, denoted \[L[\sigma V].\]
In addition, consider the free graded restricted Lie algebra \[L^p[\sigma V].\]
This restricted Lie algebra is isomorphic to the module of primitive elements in the tensor algebra generated by $\sigma V$, so the tensor algebra is a primitively generated Hopf algebra.
A basis for $L^p[\sigma V]$ is given by the union of:
\begin{enumerate}[label=(\arabic*)]
\item a basis $\mathcal B =\{b_{\alpha} \mid \alpha \in I\}$ for $ L[\sigma(V)]_{odd}$, the elements of odd degree in $L[\sigma(V)]$;
\item a basis $\mathcal C = \{c_{\gamma} \mid \gamma \in J\}$ for $L[\sigma(V)]_{even}$, the elements of even degree in $L[\sigma(V)]$; and
\item a basis for the $(p^k)^\mathrm{th}$ powers of $L[\sigma(V)]_{even}$, say
$\mathcal P\mathcal C = \{{c_{\gamma}}^{p^k} \mid \gamma \in J,\; k\ge 1\}$.
\end{enumerate}
It follows from the Bott--Samelson theorem that a basis for the module of primitives in the mod~$p$ homology of $\Omega \Sigma^2X$ is given by
\[ {\mathcal B} \cup {\mathcal C} \cup { \mathcal P\mathcal C}. \]
The mod $p$ homology of $\Omega^2\Sigma^2X$ can now be described using the preparations of the previous paragraph.
First, for each $(p^k)^\mathrm{th}$ power $x = {c_{\gamma}}^{p^k} \in \mathcal P \mathcal C$, let $\sigma^{-1}x$ denote the formal desuspension, lowering the degree of $x$ by one. Let $\beta\sigma^{-1}x$ denote the formal first Bockstein of $\sigma^{-1}{c_{\gamma}}^{p^k}$, with degree $|x|-2$. Let $\Psi$ denote the set of elements
given by \[ \Psi = \{ \sigma^{-1}x, \beta\sigma^{-1}x \mid x\in \mathcal P \mathcal C \}. \]
\begin{thm}[\cite{CLM}] \label{CLM thm}
If $X$ is a path-connected CW-complex and $p$ is an odd prime, then the mod~$p$ homology $H_\ast(\Omega^2\Sigma^2X)$ is isomorphic as an algebra to the free graded commutative algebra generated by
\[ \sigma^{-1} {\mathcal B} \cup \sigma^{-1} {\mathcal C} \cup \Psi. \]
\end{thm}
\begin{rem}
The result in case $p=2$ is different and mildly simpler (see \cite{CLM} and the remarks preceeding Lemma~\ref{Steenrod squares} below).
\end{rem}
The homology of iterated loop-suspensions are free $E_n$-algebras naturally equipped with more algebraic structure than we will need here, but we briefly mention two homology operations which will often be used below to label elements of $H_\ast(\Omega^2\Sigma^2X)$ when $X=P^{2n-1}(p^r)$; namely, the Dyer--Lashof operation
\[ Q_1\colon H_n(\Omega^2\Sigma^2X) \longrightarrow H_{np+p-1}(\Omega^2\Sigma^2X) \]
and the Browder bracket
\[ \lambda\colon H_n(\Omega^2\Sigma^2X) \otimes H_m(\Omega^2\Sigma^2X) \longrightarrow H_{n+m+1}(\Omega^2\Sigma^2X). \]
The images of $Q_1$ and $\lambda$ contain the transgressions of~$(p^k)^\mathrm{th}$ powers and iterated commutators, respectively, of primitive elements in the tensor Hopf algebra $H_\ast(\Omega\Sigma^2X)$. See \cite{CLM} for a description of the generators in $\sigma^{-1} {\mathcal B} \cup \sigma^{-1} {\mathcal C} \cup \Psi$ in terms of these operations.
We recall that the set of indecomposables $\sigma^{-1} {\mathcal B} \cup \sigma^{-1} {\mathcal C} \cup \Psi$ is also graded by weights as in \cite{CLM} in such a way that the reduced mod $p$ homology of $D_j(\Omega^2\Sigma^2X)$ is spanned by the monomials of weight $j$ in the free graded commutative algebra generated by $\sigma^{-1} {\mathcal B} \cup \sigma^{-1} {\mathcal C} \cup \Psi$.
Explicitly, for $X=P^{2n-1}(p^r)$, let $\{u,v\}$ be a basis for the graded vector space $V=\bar{H}_\ast(P^{2n-1}(p^r))$ with $|u|=2n-2$, $|v|=2n-1$ and $\beta^{(r)}v=u$. Then, if $p$ is odd, Theorem~\ref{CLM thm} implies that $H_\ast(\Omega^2P^{2n+1}(p^r))$ is a free graded commutative algebra on generators
\begin{alignat*}{2}
&\left. u,\ v,\ \lambda(u,u),\ \lambda(u,v),\ \lambda(u,\lambda(u,v)),\ \lambda(v, \lambda(u,v)), \ldots \right. &&\in\sigma^{-1}{\mathcal B}\cup\sigma^{-1}{\mathcal C} \\
&\left.\begin{aligned}
&Q_1v,\ \beta^{(1)}Q_1v,\ Q_1^2v,\ \beta^{(1)}Q_1^2v, \ldots & \\
&Q_1\lambda(u,u),\ \beta^{(1)}Q_1\lambda(u,u),\ Q_1^2\lambda(u,u),\ \beta^{(1)}Q_1^2\lambda(u,u), \ldots & \\
&Q_1\lambda(u,\lambda(u,v)),\ \beta^{(1)}Q_1\lambda(u,\lambda(u,v)), \ldots & \\
&\vdotswithin{Q_1}
\end{aligned} \right\} &&\in \Psi
\end{alignat*}
with weights defined by
\begin{align*}
\mathrm{wt}(u) &= \mathrm{wt}(v)=1, \\
\mathrm{wt}(Q_1^kx) &= \mathrm{wt}(\beta^{(1)}Q_1^kx) = p^k\mathrm{wt}(x), \\
\mathrm{wt}(\lambda(x,y)) &= \mathrm{wt}(x)+\mathrm{wt}(y),
\end{align*}
and extended to all monomials by $\mathrm{wt}(xy)=\mathrm{wt}(x)+\mathrm{wt}(y)$.
We will often use the same notation $u$, $v$ to denote generators of the bottom two mod $p$ homology groups of $P^{2n+1}(p^r)$, $\Omega P^{2n+1}(p^r)$ and $\Omega^2P^{2n+1}(p^r)$, indicating degrees with subscripts when necessary.
To prove the splittings in Theorem~\ref{splitting thm}, we will need to know the top two homology groups of the Snaith summands $D_{p^k}(\Omega^2P^{2n+1}(p^r))$ explicitly. Although the list of generators above is specific to the case of $p$ odd, the following lemma holds for all primes $p$.
\begin{lem} \label{top homology}
For each $k\ge 0$, $D_{p^k}(\Omega^2P^{2n+1}(p^r))$ is a $(2np^k-2p^k-1)$-connected $(2np^k-1)$-dimensional space with:
\begin{enumerate}
\item $H_{2np^k-1}(D_{p^k}(\Omega^2P^{2n+1}(p^r))) = \mathrm{span}\{ Q_1^kv \}$,
\item $H_{2np^k-2}(D_{p^k}(\Omega^2P^{2n+1}(p^r))) = \mathrm{span}\{ \beta^{(1)}Q_1^kv, \; \mathrm{ad}_\lambda^{p^k-1}(v)(u) \}$, \label{part b}
\end{enumerate}
where $\mathrm{ad}_\lambda^{p^k-1}(v)(u)$ denotes the $(p^k-1)$-fold iterated Browder bracket $\lambda(v, \lambda(v, \ldots, \lambda(v,u)\ldots ))$.
\end{lem}
\begin{proof}
Since the reduced mod $p$ homology of $D_{p^k}(\Omega^2P^{2n+1}(p^r))$ consists of the elements of homogeneous weight $p^k$ in $H_\ast(\Omega^2P^{2n+1}(p^r))$, the connectivity and dimension of $D_{p^k}(\Omega^2P^{2n+1}(p^r))$ follow from the fact that the weight $p^k$ monomials of lowest and highest homological degree are $u^{p^k}$ and $Q_1^kv$, respectively, with $|u^{p^k}|=2np^k-2p^k$ and $|Q_1^kv|=2np^k-1$.
Observe that any nonzero iterated Browder bracket with arguments in $\{u,v\}$ must involve $u\in H_{2n-2}(\Omega^2P^{2n+1}(p^r))$ since $\lambda(v,v)=0$ (being the transgression of the graded commutator of an even degree element with itself in the tensor algebra $H_\ast(\Omega P^{2n+1}(p^r))$). Parts (a) and (b) now follow easily by inspection of monomials of weight $p^k$ in homological degrees $2np^k-1$ and $2np^k-2$.
\end{proof}
\begin{rem}
Note that for $k=0$, the span in Lemma~\ref{top homology}\ref{part b} is $1$-dimensional since $\beta^{(1)}v$ and $\mathrm{ad}_\lambda^0(v)(u)$ coincide if $r=1$, and $\beta^{(1)}v=0$ if $r>1$. Of course, in this case $D_{p^k}(\Omega^2P^{2n+1}(p^r))=D_1(\Omega^2P^{2n+1}(p^r))$ is simply $P^{2n-1}(p^r)$. For all $k\ge 1$, $\dim H_{2np^k-2}(D_{p^k}(\Omega^2P^{2n+1}(p^r))) = 2$.
\end{rem}
In the $p=2$ case we will need to know the homology of $D_2(\Omega^2P^{2n+1}(2^r))$ as a module over the Steenrod algebra. The mod $2$ homology generators of weight~$2$ differ somewhat from those appearing in the list above for odd primes. First, since $H_\ast(\Omega^2P^{2n+1}(2^r))$ is a polynomial algebra, we have the quadratic generator $v^2$ in addition to $u^2$ and $uv$. Second, the Browder bracket $\lambda(u,u)$ is trivial since this class represents the transgression of the commutator $[u,u]=u^2+u^2=0$ in the tensor algebra $H_\ast(\Omega P^{2n+1}(2^r))=T(u,v)$ over~$\mathbb{Z}/2$. On the other hand, since $u^2\in H_{4n-2}(\Omega P^{2n+1}(2^r))$ is primitive, this class transgresses to a generator $Q_1u$ in $H_{4n-3}(\Omega^2P^{2n+1}(2^r))$, unlike in the odd primary case.
It follows that $D_2(\Omega^2P^{2n+1}(2^r))$ is a $6$-cell complex with
\begin{align*}
H_{4n-1}(D_2(\Omega^2P^{2n+1}(2^r))) &= \mathrm{span}\{ Q_1v \}, \\
H_{4n-2}(D_2(\Omega^2P^{2n+1}(2^r))) &= \mathrm{span}\{ v^2, \; \lambda(u,v) \}, \\
H_{4n-3}(D_2(\Omega^2P^{2n+1}(2^r))) &= \mathrm{span}\{ uv, \; Q_1u \}, \\
H_{4n-4}(D_2(\Omega^2P^{2n+1}(2^r))) &= \mathrm{span}\{ u^2 \}.
\end{align*}
\begin{lem} \label{Steenrod squares}
The action of the Steenrod algebra on $H_\ast(D_2(\Omega^2P^{2n+1}(2^r)))$ is determined by:
\begin{enumerate}
\item $Sq^1_\ast Q_1v = \begin{cases} v^2+\lambda(u,v) & \text{if } r=1 \\ v^2 & \text{if } r>1, \end{cases}$ \\
\item $Sq^2_\ast Q_1v = \begin{cases} Q_1u & \text{if } r=1 \\ 0 & \text{if } r>1, \end{cases}$ \\
\item $Sq^2_\ast v^2 = Sq^1_\ast uv = \begin{cases} u^2 & \text{if } r=1 \\ 0 & \text{if } r>1. \end{cases}$ \\
\end{enumerate}
\end{lem}
\begin{proof}
By \cite[III.3.10]{CLM}, $\beta^{(1)}Q_1x=x^2+\lambda(x,\beta^{(1)}x)$ for $x\in H_\ast(\Omega^2\Sigma^2X;\mathbb{Z}/2)$ with $|x|$ odd. Part (a) follows since $\beta^{(1)}v=u$ if $r=1$, and $\beta^{(1)}v=0$ if $r>1$.
Part (b) follows from the Nishida relation $Sq^2_\ast Q_1=Q_1Sq^1_\ast$.
For part (c), the Cartan formula and the fact that $Sq^1_\ast=\beta^{(1)}$ is a derivation on the Pontryagin ring $H_\ast(\Omega^2P^{2n+1}(2^r))$ imply that $Sq^2_\ast v^2$ and $Sq^1_\ast uv$ are as claimed.
Since the Browder bracket satisfies the Cartan formula \cite[III.1.2(7)]{CLM}
\[ Sq^n_\ast \lambda(x,y) = \sum_{i+j=n}\lambda(Sq^i_\ast x,Sq^j_\ast y), \]
we have $Sq^2_\ast \lambda(u,v) = Sq^1_\ast \lambda(u,v) = 0$. The relations above therefore determine all nontrivial Steenrod operations in $H_\ast(D_2(\Omega^2P^{2n+1}(2^r)))$.
\end{proof}
\section{Review of the work of Cohen, Moore and Neisendorfer}
To prepare for the proofs of Theorems \ref{splitting thm} and \ref{factorization thm} in the next section, we briefly review some of the work of Cohen, Moore and Neisendorfer \cite{CMN1, N} on torsion in the homotopy groups of Moore spaces.
Recall that the mod $p^r$ homotopy groups of a space $X$ are defined by $\pi_n(X;\mathbb{Z}/p^r) = [P^n(p^r),X]$. Provided $p^r>2$, there are splittings \cite[Proposition 6.2.2]{N2}
\[ P^n(p^r) \wedge P^m(p^r) \simeq P^{n+m}(p^r) \vee P^{n+m-1}(p^r) \]
for $n,m\ge 2$ which allow for the definition of mod $p^r$ Samelson products
\[ \pi_n(\Omega X; \mathbb{Z}/p^r) \otimes \pi_m(\Omega X; \mathbb{Z}/p^r) \longrightarrow \pi_{n+m}(\Omega X; \mathbb{Z}/p^r). \]
Together with the Bockstein differential, this gives $\pi_\ast(\Omega X; \mathbb{Z}/p^r)$ the structure of a differential graded Lie algebra when $p\ge 5$, $r\ge 1$ and a differential graded quasi-Lie algebra when $p=3$, $r\ge 2$ (see \cite{N, N2}).
The mod $p$ Hurewicz map \[ h\colon \pi_\ast(\Omega X;\mathbb{Z}/p) \longrightarrow H_\ast(\Omega X;\mathbb{Z}/p) \] intertwines mod $p$ Samelson products with commutators in the Pontryagin ring and commutes with Bockstein differentials, thereby inducing a morphism of spectral sequences from the mod~$p$ homotopy Bockstein spectral sequence $(E_\pi^s(\Omega X), \beta^{(s)})$ to the mod~$p$ homology Bockstein spectral sequence $(E_H^s(\Omega X), \beta^{(s)})$.
Consider $\pi_\ast(\Omega P^{2n+1}(p^r);\mathbb{Z}/p)$. In degrees $2n$ and $2n-1$, denote the mod $p$ reduction of the adjoint of the identity map and its $r^\text{th}$ Bockstein by
\begin{align*}
\nu &\colon P^{2n}(p) \longrightarrow \Omega P^{2n+1}(p^r) \;\text{ and} \\
\beta^{(r)}\nu=\mu &\colon P^{2n-1}(p) \longrightarrow \Omega P^{2n+1}(p^r),
\end{align*}
respectively. Then the Hurewicz images $h(\nu)=v$ and $h(\mu)=u$ generate $H_{2n}(\Omega P^{2n+1}(p^r);\mathbb{Z}/p)$ and $H_{2n-1}(\Omega P^{2n+1}(p^r);\mathbb{Z}/p)$, respectively, and by the Bott--Samelson theorem,
\begin{equation} \label{tensor alg}
H_\ast(\Omega P^{2n+1}(p^r); \mathbb{Z}/p) \cong T(u,v) \cong UL(u,v),
\end{equation}
where $L(u,v)$ is the free differential graded Lie algebra on two generators $u$, $v$ with differential~$\beta^{(r)}v=u$.
In any graded (quasi-)Lie algebra $L$ (more generally, any graded module with an antisymmetric bracket operation), let $\mathrm{ad}(x)(y)=[x,y]$ for $x,y\in L$. Define $\mathrm{ad}^0(x)(y)=y$ and inductively define $\mathrm{ad}^k(x)(y)=\mathrm{ad}(x)(\mathrm{ad}^{k-1}(x)(y))$ for $k\ge 1$.
To detect higher torsion in $\pi_\ast(P^{2n+1}(p^r))$, Cohen, Moore and Neisendorfer \cite{CMN1} consider the mod $p$ Samelson products
\begin{align*}
\tau_k(\nu) &= \mathrm{ad}^{p^k-1}(\nu)(\mu), \\
\sigma_k(\nu) &= \frac{1}{2p}\sum_{j=1}^{p^k-1}\binom{p^k}{j}[\mathrm{ad}^{j-1}(\nu)(\mu), \mathrm{ad}^{p^k-j-1}(\nu)(\mu)]
\end{align*}
in $\pi_\ast(\Omega P^{2n+1}(p^r); \mathbb{Z}/p)$ and their mod $p$ Hurewicz images $\tau_k(v)$, $\sigma_k(v)$ defined similarly in terms of graded commutators. Since the tensor algebra \eqref{tensor alg} is acyclic with respect to the differential $\beta^{(r)}$, the homology Bockstein spectral sequence collapses at the~$(r+1)^\text{st}$ page and no higher differentials in the homotopy Bockstein spectral sequence can be detected by the Hurewicz map:
\[ E_H^1(\Omega P^{2n+1}(p^r))=\cdots=E_H^r(\Omega P^{2n+1}(p^r))=T(u,v), \quad \bar{E}_H^{r+1}(\Omega P^{2n+1}(p^r))=0. \]
In particular, $\tau_k(v)\in H_{2np^k-1}(\Omega P^{2n+1}(p^r))$ is killed by the differential $\beta^{(r)}v^{p^k}=\tau_k(v)$ for all~$k\ge 0$. To tease out higher torsion, Cohen, Moore and Neisendorfer instead compute the homology Bockstein spectral sequence of the loops on the fibre $F^{2n+1}(p^r)$ of the pinch map~$q\colon P^{2n+1}(p^r) \to S^{2n+1}$, where lifts $\tau'_k(\nu)$, $\sigma'_k(\nu) \in \pi_\ast(\Omega F^{2n+1}(p^r); \mathbb{Z}/p)$ of $\tau_k(\nu)$, $\sigma_k(\nu)$ and their Hurewicz images $\tau'_k(v)$, $\sigma'_k(v)$ are shown to survive to the $(r+1)^\text{st}$ page, at least when~$p$ is odd.
\begin{thm}[{\cite[Theorem~10.3]{CMN1}}] \label{Bockstein in fibre}
Let $p$ be an odd prime and $r\ge 1$. Then there is an isomorphism of differential graded Hopf algebras
\[ E_H^{r+1}(\Omega F^{2n+1}(p^r)) \cong \Lambda(\tau'_0(v), \tau'_1(v), \tau'_2(v),\ldots) \otimes \mathbb{Z}/p\left[\sigma'_1(v),\sigma'_2(v),\ldots\right] \]
where $|\tau'_k(v)|=2np^k-1$, $|\sigma'_k(v)|=2np^k-2$ and \[ \beta^{(r+1)}\tau'_k(v) = \ell\sigma'_k(v), \ \ell\neq 0,\] for $k\ge 1$.
\end{thm}
Since the homology classes $\tau'_k(v)=h(\tau'_k(\nu))$ support nontrivial Bocksteins $\beta^{(r+1)}$ for $k\ge 1$, the same is true of the mod $p$ homotopy classes $\tau'_k(\nu) \in \pi_{2np^k-1}(\Omega F^{2n+1}(p^r); \mathbb{Z}/p)$. It follows that there exist maps
\begin{equation} \label{delta' map}
\delta'_k \colon P^{2np^k-1}(p^{r+1}) \longrightarrow \Omega F^{2n+1}(p^r)
\end{equation}
for each $k\ge 1$ which satisfy $(\delta'_k)_\ast(v_{2np^k-1}) = \tau'_k(v)$ in mod $p$ homology and induce split monomorphisms in integral homology.
To exhibit nontrivial classes in the image of $\beta^{(r+1)}$ in $E_\pi^{r+1}(\Omega P^{2n+1}(p^r))$, Cohen, Moore and Neisendorfer show that the composition of $\beta^{(r+1)}\tau'_k(\nu)$ with $\Omega F^{2n+1}(p^r) \to \Omega P^{2n+1}(p^r)$ does not represent zero in $E_\pi^{r+1}(\Omega P^{2n+1}(p^r))$, thereby proving the following.
\begin{thm}[\cite{CMN1,N}] \label{CMN torsion}
Let $p$ be an odd prime and $r\ge 1$. Then $\pi_{2np^k-1}(P^{2n+1}(p^r))$ contains a $\mathbb{Z}/p^{r+1}$ summand for every $k\ge 1$.
\end{thm}
\begin{rem}
It follows from Theorem \ref{CMN torsion} and the loop space decomposition for even-dimensional odd primary Moore spaces \cite[Theorem~1.1]{CMN1}
\begin{equation} \label{even decomposition}
\Omega P^{2n+2}(p^r) \simeq S^{2n+1}\{p^r\} \times \Omega \left( \bigvee_{j=0}^\infty P^{4n+2nj+3}(p^r) \right)
\end{equation}
that $\pi_\ast(P^n(p^r))$ contains $\mathbb{Z}/p^{r+1}$ summands for all $n\ge 3$ when $p$ is odd.
\end{rem}
\section{Splittings of $D_{p^k}(\Omega^2P^{2n+1}(p^r))$}
In this section we prove Theorem \ref{splitting thm} in a series of lemmas and discuss the stable homotopy type of $\Omega^2P^{2n+1}(p^r)$. We assume throughout that $n>1$.
The higher torsion discussed in the previous section is not reflected in the homology of the single loop space of a Moore space since $H_\ast(\Omega P^n(p^r))$ is acyclic with respect to $\beta^{(r)}$. The next lemma shows that it becomes visible in homology after looping twice.
\begin{lem} \label{odd higher Bockstein}
Let $p$ be an odd prime and $r\ge 1$. Then in the mod $p$ homology Bockstein spectral sequence of $\Omega^2P^{2n+1}(p^r)$, we have
\[ \beta^{(r+1)} \mathrm{ad}_\lambda^{p^k-1}(v)(u) \neq 0 \]
in $E_H^{r+1}(\Omega^2P^{2n+1}(p^r))$ for each $k\ge 1$. Moreover, there exist maps
\[ \delta_k \colon P^{2np^k-2}(p^{r+1}) \longrightarrow \Omega^2P^{2n+1}(p^r) \]
for each $k\ge 1$ which satisfy $(\delta_k)_\ast(v_{2np^k-2}) = \mathrm{ad}_\lambda^{p^k-1}(v)(u)$ in mod $p$ homology and induce split monomorphisms in integral homology.
\end{lem}
\begin{proof}
Consider the composite
\[ P^{2np^k-1}(p^{r+1}) \stackrel{\delta'_k}{\longrightarrow} \Omega F^{2n+1}(p^r) \longrightarrow \Omega P^{2n+1}(p^r), \]
where $\delta'_k$ is the map from \eqref{delta' map} and the second map is the fibre inclusion of the looped pinch map $\Omega q\colon \Omega P^{2n+1}(p^r) \to \Omega S^{2n+1}$. In mod $p$ homology, $(\delta'_k)_\ast(v_{2np^k-1})=\tau'_k(v)$ and $(\delta'_k)_\ast(u_{2np^k-2})=\ell\sigma'_k(v)$, $\ell\neq 0$, by Theorem~\ref{Bockstein in fibre} and naturality of the Bockstein. Since these classes map to $\tau_k(v)$, $\ell\sigma_k(v) \in H_\ast(\Omega P^{2n+1}(p^r))$, the composite above induces a monomorphism in mod~$p$ homology (with Bocksteins acting trivially on the image).
Define $\delta_k \colon P^{2np^k-2}(p^{r+1}) \to \Omega^2P^{2n+1}(p^r)$ to be the adjoint of the composite above. Let $\tau^\lambda_k(v)$ denote the iterated Browder bracket
\[ \tau^\lambda_k(v) = \mathrm{ad}_\lambda^{p^k-1}(v)(u) \in H_{2np^k-2}(\Omega^2P^{2n+1}(p^r)) \]
which is the transgression of the iterated commutator
\[ \tau_k(v)=\mathrm{ad}^{p^k-1}(v)(u) \in H_{2np^k-1}(\Omega P^{2n+1}(p^r)). \]
It follows that $(\delta_k)_\ast(v_{2np^k-2}) = \tau^\lambda_k(v)$. Similarly, $(\delta_k)_\ast(u_{2np^k-3}) = \ell\sigma^\lambda_k(v) \neq 0$ where $\sigma^\lambda_k(v)$ denotes the transgression of $\sigma_k(v) \in H_{2np^k-2}(\Omega P^{2n+1}(p^r))$.
We now have a map $\delta_k\colon P^{2np^k-2}(p^{r+1}) \to \Omega^2P^{2n+1}(p^r)$ inducing
\[
\xymatrix @R=1.2pc @C=5pc{
\quad v_{2np^k-2} \ar@/_1pc/[d]_-{\beta^{(r+1)}} \ar@{|->}[r] & \tau^\lambda_k(v) \qquad \\
\quad u_{2np^k-3} \ar@{|->}[r] & \ell\sigma^\lambda_k(v) \qquad
}
\]
in homology and it remains to show that $\beta^{(r+1)}\tau^\lambda_k(v) \neq 0$. It suffices by naturality of $\beta^{(r+1)}$ to show that $\sigma^\lambda_k(v)$ does not represent zero in $E_H^{r+1}(\Omega^2P^{2n+1}(p^r))=H_\ast(E_H^r(\Omega^2P^{2n+1}(p^r)), \beta^{(r)})$. First note that $\tau^\lambda_k(v)$ and $\sigma^\lambda_k(v)$ are $\beta^{(s)}$-cycles for $s\le r$ since $v_{2np^k-2}$ and $u_{2np^k-3}$ are. To see that they are not $\beta^{(s)}$-boundaries for any $s\le r$, consider the Snaith splitting of $\Omega^2P^{2n+1}(p^r)$. Since all Bocksteins must respect the induced splitting in homology and $\tau^\lambda_k(v)$, $\sigma^\lambda_k(v)$ lie in the homology of the stable summand $D_{p^k}(\Omega^2P^{2n+1}(p^r))$, it follows from Lemma~\ref{top homology} that for degree reasons the only class $x$ which could potentially satisfy $\beta^{(s)}x=\sigma^\lambda_k(v)$ is a linear combination of $\beta^{(1)}Q_1^kv$ and $\tau^\lambda_k(v)$. But $\beta^{(s)}\beta^{(1)}Q_1^kv=\beta^{(s)}\tau^\lambda_k(v)=0$ for all $s\le r$. Therefore $\tau^\lambda_k(v)$, $\sigma^\lambda_k(v)$ represent nontrivial classes in $E_H^{r+1}(\Omega^2P^{2n+1}(p^r))$, where the differential $\beta^{(r+1)}\tau^\lambda_k(v) = \ell\sigma^\lambda_k(v)$ is forced.
\end{proof}
As a partial $2$-primary analogue of Lemma~\ref{odd higher Bockstein}, we show that the class $\mathrm{ad}_\lambda^{2^k-1}(v)(u)$ supports a higher Bockstein when $k=1$.
\begin{lem} \label{even higher Bockstein}
Let $p=2$ and $r\ge 1$. Then in the mod $2$ homology Bockstein spectral sequence of $\Omega^2P^{2n+1}(2^r)$, we have
\[ \beta^{(r+1)}\lambda(u,v) = Q_1u \]
in $E_H^{r+1}(\Omega^2P^{2n+1}(2^r))$.
\end{lem}
\begin{proof}
We give a direct chain level calculation similar to the proof of \cite[III.3.10]{CLM}. Consider the $\Sigma_2$-invariant map $\theta\colon\mathcal{C}_2(2)\times X\times X \to X$ given by the action of the little $2$-cubes operad on $X=\Omega^2P^{2n+1}(2^r)$. Let $e_k$ and $\alpha$ be as defined in \cite[Section 6]{M} and let $a$, $b$ be chains representing $v$, $u$, respectively, with $d(a)=2^rb$. Then
\begin{align*}
d((\alpha +1)e_1\otimes a\otimes b) &= (\alpha^2-1)e_0\otimes a\otimes b - 2^r(\alpha+1)e_1\otimes b\otimes b \\
&= - 2^{r+1}e_1\otimes b\otimes b.
\end{align*}
Since $\theta_\ast$ commutes with $d$, it follows from the definitions of $\lambda$ and $Q_1$ (cf.\ \cite{M, CLM}) that $\beta^{(s)}\lambda(u,v)=0$ for $s\le r$ and $\beta^{(r+1)}\lambda(u,v)=Q_1u$.
\end{proof}
Let $i\colon S^{n-1} \to P^n(p^r)$ denote the inclusion of the bottom cell and $\eta\colon S^n \to S^{n-1}$ the Hopf map.
\begin{lem} \label{homotopy groups}
Let $n\ge 4$ and $r\ge 1$. Then
\begin{enumerate}
\item $\pi_{n-1}(P^n(p^r)) = \mathbb{Z}/p^r \left\langle i \right\rangle$, \smallskip
\item $\pi_n(P^n(p^r)) = \begin{cases} \mathbb{Z}/2 \left\langle i\eta \right\rangle & \text{if } p=2 \\ 0 & \text{if } p \text{ is odd.} \end{cases}$
\end{enumerate}
\end{lem}
\begin{proof}
Both parts follow immediately from the sequence
\[ \pi_j(S^{n-1}) \longrightarrow \pi_j(S^{n-1}) \longrightarrow \pi_j(P^n(p^r)) \longrightarrow \pi_j(S^n) \longrightarrow \pi_j(S^n) \]
induced by the cofibration defining $P^n(p^r)$, which is exact for $j=n-1$, $n$ by the Blakers--Massey theorem. Note that the degree $p^r$ map on $S^{n-1}$ induces multiplication by $p^r$ on $\pi_n(S^{n-1})=\mathbb{Z}/2 \left\langle \eta \right\rangle$ since $\eta$ is a suspension for $n\ge 4$ (whereas $S^2\xrightarrow{2}S^2$ induces multiplication by $4$ on $\pi_3(S^2)$, implying $\pi_3(P^3(2))=\mathbb{Z}/4$, e.g.).
\end{proof}
We are now ready to prove the splittings of Theorem~\ref{splitting thm}, parts (a) and (b) of which are restated below as Lemma~\ref{odd lemma} and Lemma~\ref{even lemma}, respectively.
\begin{lem} \label{odd lemma}
If $p$ is an odd prime and $r \ge 1$, then $D_{p^k}(\Omega^2P^{2n+1}(p^r))$ is stably homotopy equivalent to
\[ P^{2np^k-2}(p^{r+1}) \vee X_{p^k} \]
for some finite CW-complex $X_{p^k}$ for all $k\ge 1$.
\end{lem}
\begin{proof}
Suppose $p$ is an odd prime, $r\ge 1$ and let $k\ge 1$. By Lemma~\ref{odd higher Bockstein}, the map \[ \delta_k \colon P^{2np^k-2}(p^{r+1}) \longrightarrow \Omega^2P^{2n+1}(p^r) \] induces a monomorphism in mod $p$ homology with
\[ (\delta_k)_\ast(v_{2np^k-2}) = \mathrm{ad}_\lambda^{p^k-1}(v)(u), \quad (\delta_k)_\ast(u_{2np^k-3}) = \beta^{(r+1)}\mathrm{ad}_\lambda^{p^k-1}(v)(u). \]
Since these elements have weight $p^k$ in $H_\ast(\Omega^2P^{2n+1}(p^r))$, by stabilizing $\delta_k$ and composing with the Snaith splitting we obtain a stable map $P^{2np^k-2}(p^{r+1}) \to D_{p^k}(\Omega^2P^{2n+1}(p^r))$ with the same image in homology.
It therefore suffices to produce a map $f_k \colon D_{p^k}(\Omega^2P^{2n+1}(p^r)) \to P^{2np^k-2}(p^{r+1})$ with
\begin{equation} \label{f_k}
(f_k)_\ast(\mathrm{ad}_\lambda^{p^k-1}(v)(u)) = v_{2np^k-2}.
\end{equation}
By collapsing the $(2np^k-4)$-skeleton of $D_{p^k}(\Omega^2P^{2n+1}(p^r))$ to a point, we are left with a complex with cells only in dimensions $2np^k-3$, $2np^k-2$ and $2np^k-1$ (by Lemma~\ref{top homology}) of the form
\begin{equation} \label{cell decomp}
\left(\bigvee_{i=1}^d S^{2np^k-3}\right) \cup e^{2np^k-2} \cup e^{2np^k-2} \cup e^{2np^k-1},
\end{equation}
where $d=\dim H_{2np^k-3}(D_{p^k}(\Omega^2P^{2n+1}(p^r)))$ and the top three cells carry the homology classes $Q_1^kv$, $\beta^{(1)}Q_1^kv$ and $\mathrm{ad}_\lambda^{p^k-1}(v)(u)$. Since $\mathrm{ad}_\lambda^{p^k-1}(v)(u) \in H_{2np^k-2}(D_{p^k}(\Omega^2P^{2n+1}(p^r)))$ supports a nontrivial $(r+1)^\text{st}$ Bockstein by Lemma~\ref{odd higher Bockstein}, we may assume (altering by a self-homotopy equivalence if necessary) that the inclusion of one of the bottom cells in \eqref{cell decomp} has Hurewicz image $\beta^{(r+1)}\mathrm{ad}_\lambda^{p^k-1}(v)(u)$. Then by further collapsing a wedge $\vee_{i=1}^{d-1}S^{2np^k-3}$ of the other bottom cells to a point, we obtain a $4$-cell complex $C$ with mod $p$ homology Bockstein spectral sequence given by
\[
\xymatrix @R=1.2pc @C=1pc{
2np^k-1\qquad & Q_1^kv \ar@/_1pc/[d]_-{\beta^{(1)}} & \\
2np^k-2\qquad & \beta^{(1)}Q_1^kv & \mathrm{ad}_\lambda^{p^k-1}(v)(u) \qquad\quad \ar@/^1pc/[d]^-{\beta^{(r+1)}} \\
2np^k-3\qquad & & \beta^{(r+1)}\mathrm{ad}_\lambda^{p^k-1}(v)(u) \qquad\quad
}
\]
and a map $D_{p^k}(\Omega^2P^{2n+1}(p^r)) \to C$ inducing an epimorphism in homology.
It follows from the description of $H_\ast(C)$ above that $C \simeq P^{2np^k-2}(p^{r+1}) \cup_\alpha e^{2np^k-2} \cup_\gamma e^{2np^k-1}$ for some attaching maps $\alpha$, $\gamma$. Since $\beta^{(1)}Q_1^kv \in H_{2np^k-2}(C)$ is a permanent cycle in the Bockstein spectral sequence and every nonzero element $\alpha \in \pi_{2np^k-3}(P^{2np^k-2}(p^{r+1})) = \mathbb{Z}/p^{r+1}$ is detected by a Bockstein, we conclude that $\alpha$ is trivial. Next we consider
\[ \gamma \in \pi_{2np^k-2}(P^{2np^k-2}(p^{r+1}) \vee S^{2np^k-2}) = \pi_{2np^k-2}(P^{2np^k-2}(p^{r+1})) \oplus \pi_{2np^k-2}(S^{2np^k-2}). \]
By Lemma \ref{homotopy groups}, $\pi_{2np^k-2}(P^{2np^k-2}(p^{r+1}))=0$ since $p$ is odd, and since the top homology class $Q_1^kv \in H_{2np^k-1}(C)$ supports a nontrivial first Bockstein differential, it follows that the second component of $\gamma$ is of degree $\pm p$. Therefore $C \simeq P^{2np^k-2}(p^{r+1}) \vee P^{2np^k-1}(p)$. Finally, using this splitting we define the map $f_k$ by the composite
\[ f_k\colon D_{p^k}(\Omega^2P^{2n+1}(p^r)) \longrightarrow C \simeq P^{2np^k-2}(p^{r+1}) \vee P^{2np^k-1}(p) \stackrel{\pi_1}{\longrightarrow} P^{2np^k-2}(p^{r+1}), \]
where the first map is the quotient map described in the previous paragraph and $\pi_1$ is the projection onto the first wedge summand. By construction, $f_k$ satisfies \eqref{f_k} so the assertion follows.
\end{proof}
\begin{lem} \label{even lemma}
If $r>1$, then there is a homotopy equivalence
\[ D_2(\Omega^2P^{2n+1}(2^r)) \simeq P^{4n-2}(2^{r+1}) \vee X_2 \]
for some $4$-cell complex $X_2=P^{4n-3}(2^r) \cup CP^{4n-2}(2)$.
\end{lem}
\begin{proof}
Let $r>1$ and note that the mod $2$ homology generators $u$, $v \in H_\ast(\Omega^2P^{2n+1}(2^r))$ in respective degrees $2n-2$, $2n-1$ give a basis for the homology of the first stable summand $D_1(\Omega^2P^{2n+1}(2^r)) = P^{2n-1}(2^r)$ of $\Omega^2P^{2n+1}(2^r)$. Next, a basis for the quadratic part of $H_\ast(\Omega^2P^{2n+1}(2^r))$ is given by the classes
\[
\xymatrix @R=1.2pc @C=2.2pc{
4n-1\qquad & Q_1v \ar@/_1pc/[d]_-{\beta^{(1)}} & \\
4n-2\qquad & v^2 & \lambda(v,u) \quad \ar@/^1pc/[d]^-{\beta^{(r+1)}} \\
4n-3\qquad & uv \ar@/_1pc/[d]_-{\beta^{(r)}} & Q_1u \\
4n-4\qquad & u^2 &
}
\]
with Bockstein differentials acting as indicated by Lemma~\ref{Steenrod squares}, Lemma~\ref{even higher Bockstein} and the fact that $\beta^{(r)}v=u$.
It follows that $D_2(\Omega^2P^{2n+1}(2^r))$ has the homotopy type of a $6$-cell complex
\[ D_2(\Omega^2P^{2n+1}(2^r)) \simeq P^{4n-3}(2^r) \cup_\alpha e^{4n-3} \cup_\gamma e^{4n-2} \cup_\delta e^{4n-2} \cup_\epsilon e^{4n-1} \]
with homology as above, where the bottom Moore space carries the homology classes $uv$ and~$u^2$. As in the proof of Lemma \ref{odd lemma}, the attaching map $\alpha$ is null homotopic since every nonzero element of $\pi_{4n-4}(P^{4n-3}(2^r))=\mathbb{Z}/2^r$ is detected by a Bockstein and $Q_1u$ is a permanent cycle in the homology Bockstein spectral sequence. The next attaching maps $\gamma$ and $\delta$ may therefore be regarded as elements of $\pi_{4n-3}(P^{4n-3}(2^r)\vee S^{4n-3}) = \mathbb{Z}/2 \oplus \mathbb{Z}$, where the first summand is generated by $i\eta$ by Lemma \ref{homotopy groups}. Naturality and the morphism of cofibrations
\[
\xymatrix{
S^{4n-3} \ar[r]^\eta \ar@{=}[d] & S^{4n-4} \ar[r] \ar[d]^-i & \Sigma^{4n-6}\mathbb{C}P^2 \ar[d] \\
S^{4n-3} \ar[r]^-{i\eta} & P^{4n-3}(2^r) \ar[r] & C_{i\eta}
}
\]
imply that $i\eta$ is detected by $Sq^2_\ast$ since $\eta$ is. Since $Sq^2_\ast$ acts trivially on $H_\ast(D_2(\Omega^2P^{2n+1}(2^r)))$ when $r>1$ by Lemma~\ref{Steenrod squares}, the first components of $\gamma$ and $\delta$ must therefore be trivial. Without loss of generality, we may assume the second components of $\gamma$ and $\delta$ are trivial and degree~$\pm 2^{r+1}$, respectively, since we have a basis $\{v^2, \lambda(v,u)\}$ of $H_{4n-2}(D_2(\Omega^2P^{2n+1}(2^r)))$ where $v^2$ is a permanent cycle and $\lambda(v,u)$ supports a nontrivial $\beta^{(r+1)}$.
We now have a homotopy equivalence
\[ D_2(\Omega^2P^{2n+1}(2^r)) \simeq (P^{4n-3}(2^r) \vee P^{4n-2}(2^{r+1}) \vee S^{4n-2}) \cup _\epsilon e^{4n-1} \]
where $v^2 \in H_{4n-2}(D_2(\Omega^2P^{2n+1}(2^r)))$ corresponds to the fundamental class of the $(4n-2)$-sphere on the right. Denote the components of the attaching map $\epsilon$ by
\[ \epsilon=(\epsilon_1,\epsilon_2,\epsilon_3) \in \pi_{4n-2}(P^{4n-3}(2^r)) \oplus \pi_{4n-2}(P^{4n-2}(2^{r+1})) \oplus \pi_{4n-2}(S^{4n-2}). \]
It suffices to show that $\epsilon_2$ and $\epsilon_3$ are trivial and degree $\pm 2$, respectively. Clearly the first Bockstein $\beta^{(1)}Q_1v =v^2$ on the top class of $D_2(\Omega^2P^{2n+1}(2^r))$ implies $\epsilon_3$ is of degree $\pm 2$. To see that $\epsilon_2$ is trivial, collapse the bottom Moore space $P^{4n-3}(2^r)$ to a point and repeat the argument above analyzing the attaching map $\gamma$.
\end{proof}
To conclude the proof of Theorem~\ref{splitting thm}, it remains to show that $D_2(\Omega^2P^{2n+1}(2))$ is a stably indecomposable $6$-cell complex. This follows immediately from homological considerations: by Lemma~\ref{Steenrod squares} and Lemma~\ref{even higher Bockstein}, $H_\ast(D_2(\Omega^2P^{2n+1}(2)))$ clearly does not admit any nontrivial decomposition respecting Steenrod and higher Bockstein operations.
\section{Proof of Theorem~\ref{factorization thm}}
In this section we derive Theorem~\ref{factorization thm} from Theorem~\ref{splitting thm} and discuss some implications for the unstable homotopy groups of odd primary Moore spaces.
\begin{proof}[Proof of Theorem \ref{factorization thm}]
Suppose $p$ is an odd prime and $r\ge 1$. Then for each $k\ge 1$, the map $\delta_k\colon P^{2np^k-2}(p^{r+1}) \to \Omega^2P^{2n+1}(p^r)$ from Lemma~\ref{odd higher Bockstein} admits a stable retraction by Lemma~\ref{odd lemma}. Taking adjoints in the resulting homotopy commutative diagram
\[
\xymatrix{
\Sigma^\infty P^{2np^k-2}(p^{r+1}) \ar[r]^{\Sigma^\infty\delta_k} \ar@{=}[rd] & \Sigma^\infty\Omega^2P^{2n+1}(p^r) \ar[d] \\
& \Sigma^\infty P^{2np^k-2}(p^{r+1})
}
\]
yields the desired factorization of the unstable map $E^\infty\colon P^{2np^k-2}(p^{r+1}) \to QP^{2np^k-2}(p^{r+1})$ through $\Omega^2P^{2n+1}(p^r)$.
To similarly factor the stabilization map of a mod $p^{r+1}$ Moore space through $\Omega^2P^{2n}(p^r)$, we reduce to the odd-dimensional case using the fact that $\Omega^2P^{4n-1}(p^r)$ is an unstable retract of $\Omega^2P^{2n}(p^r)$ by the loop space decomposition~\eqref{even decomposition}. Explicitly, there is a map
\[ P^{(4n-2)p^k-2}(p^{r+1}) \stackrel{\delta_k}{\longrightarrow} \Omega^2P^{4n-1}(p^r) \longrightarrow \Omega^2P^{2n}(p^r) \]
admitting a stable retraction, so the argument above implies that the stabilization map of $P^{(4n-2)p^k-2}(p^{r+1})$ factors through $\Omega^2P^{2n}(p^r)$.
\end{proof}
Note that if $p$ is prime and $\pi_j(P^{2np^k-2}(p^{r+1}))$ is in the stable range so that the map
$E^\infty\colon P^{2np^k-2}(p^{r+1}) \to QP^{2np^k-2}(p^{r+1})$
is an isomorphism on $\pi_j(\;)$, then Theorem~\ref{factorization thm} implies that $\pi_j(P^{2np^k-2}(p^{r+1}))$ is a summand of $\pi_{j+2}(P^{2n+1}(p^r))$. Since for any given $j\in \mathbb{Z}$ we have $\pi_j^s(P^{2np^k-2}(p^{r+1})) = \pi_j(P^{2np^k-2}(p^{r+1}))$ for $k$ sufficiently large, it follows that every stable homotopy group of a mod $p^{r+1}$ Moore space is a summand of $\pi_\ast(P^{2n+1}(p^r))$.
Rephrasing a little, we have the following consequence of Theorem~\ref{factorization thm}. Let $\mathbb{S}/p^{r}$ denote the mod $p^r$ Moore spectrum, that is, the cofibre of $\mathbb{S} \xrightarrow{p^r} \mathbb{S}$ where $\mathbb{S}$ is the sphere spectrum.
\begin{cor} \label{stable unstable}
Let $p$ be an odd prime and $r\ge 1$. Then for each $j\in \mathbb{Z}$, $\pi_j(\mathbb{S}/p^{r+1})$ is a summand of $\pi_{2np^k+j-1}(P^{2n+1}(p^r))$ for every sufficiently large $k$.
\end{cor}
\begin{proof}
For each $j\in \mathbb{Z}$, we have
\[ \pi_j(\mathbb{S}/p^{r+1}) = \pi_{j+2np^k-3}^s(P^{2np^k-2}(p^{r+1})) = \pi_{j+2np^k-3}(P^{2np^k-2}(p^{r+1})) \]
for all sufficiently large $k$. Therefore the first commutative diagram in Theorem~\ref{factorization thm} implies that $\pi_j(\mathbb{S}/p^{r+1})$ retracts off $\pi_{j+2np^k-3}(\Omega^2P^{2n+1}(p^r)) = \pi_{j+2np^k-1}(P^{2n+1}(p^r))$.
\end{proof}
\begin{rem}
The second commutative diagram in Theorem~\ref{factorization thm} implies that similar results hold for the unstable homotopy groups of even-dimensional odd primary Moore spaces.
\end{rem}
\section{$v_1$-periodic families} \label{new families section}
In this section we construct new infinite families of higher torsion elements in the unstable homotopy groups of Moore spaces using Theorem~\ref{factorization thm} and periodic self-maps
\[ v_1\colon P^{n+q_r}(p^r) \longrightarrow P^n(p^r), \]
as introduced by Adams \cite{A} in his study of the image of the $J$-homomorphism. Here,
\[ q_r = \begin{cases} qp^{r-1} & \text{if } p \text{ is odd} \\ \max(8, 2^{r-1}) & \text{if } p=2, \end{cases} \]
where $q=2(p-1)$ and $v_1$ induces an isomorphism in $K$-theory. Such maps exist unstably provided $n\ge 2r+3$ by \cite{DM} and desuspend further to $P^3(p)$ in case $p$ is odd and $r=1$ by \cite{CN}.
Restricting each iterate $v_1^t = v_1 \circ \Sigma^{q_r}v_1 \circ\cdots\circ \Sigma^{tq_r}v_1$ of $v_1$ to the bottom cell gives an infinite family of maps
\[ S^{n+tq_r-1} \longrightarrow P^{n+tq_r}(p^r) \stackrel{v_1^t}{\longrightarrow} P^n(p^r) \]
which generate $\mathbb{Z}/p^r$ summands in $\pi_{n+tq_r-1}(P^n(p^r))$ for $t\ge 0$, and composing with the pinch map $q\colon P^n(p^r) \to S^n$ gives rise to the first studied infinite families in the stable homotopy groups of spheres. For example, if $p$ is odd, these composites form the $\alpha$-family and generate the $p$-component of the image of $J$ in $\pi_{tq_r-1}(\mathbb{S})$ (see \cite[Proposition~1.1]{CK}). To generate $\mathbb{Z}/p^{r+1}$ summands in $\pi_\ast(P^n(p^r))$ when $p$ is odd, we apply the same procedure to mod $p^{r+1}$ Moore spaces and compose into $P^n(p^r)$ along the maps in the diagrams of Theorem~\ref{factorization thm}.
When $p=2$, analogous unstable maps $\delta_1\colon P^{4n-2}(2^{r+1})\to \Omega^2P^{2n+1}(2^r)$ realizing the stable splittings of Theorem~\ref{splitting thm} only exist when $n=2$ or $4$, as we show below. In the $n=2$ case, no Adams self-map of the mod $2^{r+1}$ Moore spectrum desuspends far enough to precompose $\delta_1$ with. Instead, we show that an infinite family of elements of order $8$ in the homotopy groups of spheres constructed in \cite{KR} factors through $P^6(8)$ and injects along $\delta_1\colon P^6(8) \to \Omega^2P^5(4)$ (see Theorem~\ref{new even families} below).
\subsection{The odd primary case}
Let $n>1$. As usual, for an odd prime $p$ we let $q=2(p-1)$. The following is a more precise statement of Theorem~\ref{new odd families}.
\begin{thm}
Let $p$ be an odd prime and $r\ge 1$.
\begin{enumerate}
\item If $k \ge \log_p(\frac{r+4}{n})$, then $\pi_{2np^k-1+tqp^r}(P^{2n+1}(p^r))$ contains a $\mathbb{Z}/p^{r+1}$ summand for every $t\ge 0$.
\item If $k \ge \log_p(\frac{r+3}{2n-1})$, then $\pi_{(4n-2)p^k-1+tqp^r}(P^{2n}(p^r))$ contains a $\mathbb{Z}/p^{r+1}$ summand for every $t\ge 0$.
\end{enumerate}
\end{thm}
\begin{proof}
By Theorem~\ref{factorization thm}, $E^\infty\colon P^{2np^k-2}(p^{r+1}) \to QP^{2np^k-2}(p^{r+1})$ factors as a composite
\[ P^{2np^k-2}(p^{r+1}) \stackrel{\delta_k}{\longrightarrow} \Omega^2P^{2n+1}(p^r) \longrightarrow QP^{2np^k-2}(p^{r+1}). \]
Note that the restriction of $\delta_k$ to the bottom cell defines an element of $\pi_{2np^k-1}(P^{2n+1}(p^r))$ of order $p^{r+1}$. The bound on $k$ ensures that $2np^k-2 \ge 2(r+1)+4$, which implies that an unstable representative $v_1\colon P^{2np^k-2+qp^r}(p^{r+1}) \to P^{2np^k-2}(p^{r+1})$ of the Adams map exists by \cite[Proposition~2.11]{DM}. That the restriction of any iterate $v_1^t$ to the bottom cell has order~$p^{r+1}$ follows from the fact that $v_1^t$ induces an isomorphism in $K$-theory. The composite
\[ S^{2np^k-3+tqp^r} \longrightarrow P^{2np^k-2+tqp^r}(p^{r+1}) \stackrel{v_1^t}{\longrightarrow} P^{2np^k-2}(p^{r+1}) \stackrel{\delta_k}{\longrightarrow} \Omega^2P^{2n+1}(p^r) \]
therefore also has order $p^{r+1}$ for all $t\ge 0$ since composing further into $QP^{2np^k-2}(p^{r+1})$ gives the adjoint of the restriction of $\Sigma^\infty v_1^t$ to the bottom cell. Part (b) is proved similarly.
\end{proof}
\begin{rem}
The proof above shows that each $\delta_k$ generates an infinite $v_1$-periodic family in $\pi_\ast(P^{2n+1}(p^r); \mathbb{Z}/p^{r+1})$ giving rise to an infinite family of higher torsion elements in $\pi_\ast(P^{2n+1}(p^r))$. We point out that in the loop space decomposition \cite{CMN3}
\[ \Omega P^{2n+1}(p^r) \simeq T^{2n+1}\{p^r\} \times \Omega \left( \bigvee_\alpha P^{n_\alpha}(p^r) \right), \]
each of these elements lands in the homotopy of the bottom indecomposable factor $T^{2n+1}\{p^r\}$, and many more infinite families than are indicated here can be obtained by applying the Hilton--Milnor theorem to the second factor and iterating our construction above.
\end{rem}
\subsection{The $2$-primary case}
We consider next the problem of desuspending the inclusion of the stable summand $P^{4n-2}(2^{r+1})$ of $\Omega^2P^{2n+1}(2^r)$ given by Theorem~\ref{splitting thm}(b) and mimicking the construction above of unstable $v_1$-periodic families of higher odd primary torsion elements.
Note that a homotopy commutative diagram
\begin{equation} \label{diagram}
\begin{gathered}
\xymatrix{
P^{4n-2}(2^{r+1}) \ar[r] \ar[rd]_{E^\infty} & \Omega^2P^{2n+1}(2^r) \ar[d] \\
& QP^{4n-2}(2^{r+1})
}
\end{gathered}
\end{equation}
cannot exist unless $r>1$ since $D_2(\Omega^2P^{2n+1}(2))$ is stably indecomposable by Theorem~\ref{splitting thm}(c). Furthermore, such a factorization implies $Q_1u \in H_{4n-3}(\Omega^2P^{2n+1}(2^r))$ is spherical since only this class lies in the image of the $(r+1)^\text{st}$ Bockstein in degree $4n-3$. For $r=1$, it follows from the proposition below that this class is spherical only in Kervaire invariant dimensions.
\begin{prop}[{\cite[Proposition~2.21]{W}}] \label{Wu's prop}
The class $u^2\in H_{4n-2}(\Omega P^{2n+1}(2))$ is spherical if and only if the Whitehead square $w_{2n-1}\in \pi_{4n-3}(S^{2n-1})$ is divisible by $2$.
\end{prop}
For $r>1$, the same argument leads to the following.
\begin{prop} \label{spherical classes}
Let $r>1$. The following conditions are equivalent:
\begin{enumerate}
\item $Q_1u \in H_{4n-3}(\Omega^2P^{2n+1}(2^r))$ is spherical;
\item $u^2 \in H_{4n-2}(\Omega P^{2n+1}(2^r))$ is spherical;
\item $n=1$, $2$ or $4$.
\end{enumerate}
\end{prop}
\begin{proof}
If a map $f\colon S^{4n-3} \to \Omega^2P^{2n+1}(2^r)$ has mod $2$ reduced Hurewicz image $Q_1u$, then the adjoint of $f$ factors as
\[ f'\colon S^{4n-2} \stackrel{\Sigma f}{\longrightarrow} \Sigma\Omega^2P^{2n+1}(2^r) \stackrel{\sigma}{\longrightarrow} \Omega P^{2n+1}(2^r), \]
where $\sigma$ induces the homology suspension $\sigma_\ast \colon H_\ast(\Omega^2P^{2n+1}(2^r)) \to H_{\ast+1}(\Omega P^{2n+1}(2^r))$. Thus $\sigma_\ast(Q_1u)=u^2$ is the Hurewicz image of $f'$.
Conversely, given $g'\colon S^{4n-2} \to \Omega P^{2n+1}(2^r)$ with $g'_\ast(\iota_{4n-2})=u^2$, the adjoint of $g'$ factors as
\[ g\colon S^{4n-3} \stackrel{E}{\longrightarrow} \Omega S^{4n-2} \stackrel{\Omega g'}{\longrightarrow} \Omega^2P^{2n+1}(2^r). \]
Consider the morphism of path-loop fibrations induced by $g'$. Since $u^2$ transgresses to $Q_1u$ in the Serre spectral sequence associated to the path-loop fibration over $\Omega P^{2n+1}(2^r)$, it follows by naturality that $g_\ast(\iota_{4n-3})=Q_1u$. Therefore conditions (a) and (b) are equivalent.
If $n=1$, $2$ or $4$, then the adjoint of the Hopf invariant one map $S^{4n-1} \to S^{2n}$ has Hurewicz image $\iota_{2n-1}^2 \in H_{4n-2}(\Omega S^{2n})$, so the composite $S^{4n-2} \to \Omega S^{2n} \xrightarrow{\Omega i\,} \Omega P^{2n+1}(2^r)$ has Hurewicz image $u^2$.
Conversely, if $u^2 \in H_{4n-2}(\Omega P^{2n+1}(2^r))$ is spherical, then the proof given in \cite{W} of Proposition~\ref{Wu's prop} above shows that $i\circ w_{2n-1}$ is null homotopic in the diagram
\[
\xymatrix{
\Omega P^{2n}(2^r) \ar[r] & E \ar[r]^-f & S^{2n-1} \ar[r]^-i & P^{2n}(2^r) \\
& & S^{4n-3} \ar[u]_{w_{2n-1}} \ar@{-->}[ul]^-\ell
}
\]
where the top row is a fibration sequence. By \cite[Lemma~21.1]{C course}, $[u,v] \in H_{4n-3}(\Omega P^{2n}(2^r))$ is spherical and so is its image in $H_{4n-3}(E)$. As in \cite{W}, it follows that the $(4n-3)$-skeleton of~$E$ is homotopy equivalent to $S^{2n-1}\vee S^{4n-3}$ and $f|_{S^{4n-3}}$ is null homotopic. Therefore a lift~$\ell$ may be chosen to factor through $f|_{S^{2n-1}}$, which is of degree $2^r$. Since the degree $2$ map induces multiplication by $2$ on $\pi_{4n-3}(S^{2n-1})$ by Barratt's distributivity formula \cite[Proposition~4.3]{C course}, the Whitehead square $w_{2n-1}$ is divisible by $2^r$, which implies $n\in \{1,2,4\}$ since $r>1$.
\end{proof}
By Proposition~\ref{spherical classes}, diagrams of the form \eqref{diagram} inducing the stable splittings of Theorem~\ref{splitting thm}(b) (where $n>1$ is assumed) cannot exist if $n\neq 2$ or $4$. We verify that such diagrams do exist in these two exceptional dimensions.
\begin{thm} \label{even factorization thm}
Let $r>1$. Then there exist homotopy commutative diagrams
\[
\xymatrix{
P^6(2^{r+1}) \ar[r] \ar[rd]_{E^\infty} & \Omega^2P^5(2^r) \ar[d] \\
& QP^6(2^{r+1})
} \qquad\qquad
\xymatrix{
P^{14}(2^{r+1}) \ar[r] \ar[rd]_{E^\infty} & \Omega^2P^9(2^r) \ar[d] \\
& QP^{14}(2^{r+1}).
}
\]
\end{thm}
\begin{proof}
Let $n=2$ or $4$ and let $f\colon S^{4n-3} \to \Omega^2P^{2n+1}(2^r)$ be a map with mod $2$ reduced Hurewicz image $Q_1u$. Then by Lemma~\ref{even higher Bockstein}, the integral Hurewicz image of $f$ is a generator of $H_{4n-3}(\Omega^2P^{2n+1}(2^r); \mathbb{Z})\cong \mathbb{Z}/2^{r+1}$, so $f$ has order at least $2^{r+1}$. That $f$ has order at most~$2^{r+1}$ follows from \cite[Proposition~13.3]{C course}, so $f$ extends to a map $\bar{f}\colon P^{4n-2}(2^{r+1}) \to \Omega^2P^{2n+1}(2^r)$ with $\beta^{(r+1)}\bar{f}_\ast(v_{4n-2})=Q_1u$. Since $r>1$, the Snaith splitting and Theorem~\ref{splitting thm}(b) give a composite
\[ \Omega^2P^{2n+1}(2^r) \longrightarrow QD_2(\Omega^2P^{2n+1}(2^r)) \longrightarrow QP^{4n-2}(2^{r+1}) \]
which is an epimorphism on $H_{4n-2}(\:)$ and $H_{4n-3}(\:)$. It follows that the composition of $\bar{f}$ with the composite above is $(4n-2)$-connected and hence homotopic to the stabilization map $E^\infty\colon P^{4n-2}(2^{r+1}) \to QP^{4n-2}(2^{r+1})$.
\end{proof}
\begin{thm} \label{new even families}
Let $r=2$ or $3$. Then
\begin{enumerate}
\item $\pi_{3+8t}(P^5(4))$ contains a $\mathbb{Z}/8$ summand for every $t\ge 1$;
\item $\pi_{7+8t}(P^9(2^r))$ contains a $\mathbb{Z}/2^{r+1}$ summand for every $t\ge 1$.
\end{enumerate}
\end{thm}
\begin{proof}
We begin with part (b). Consider the second diagram in Theorem~\ref{even factorization thm} and let $r=2$ or~$3$. Then for stability reasons, an unstable Adams map $v_1\colon P^{n+8}(2^{r+1}) \to P^n(2^{r+1})$ exists for $n=14$. As in the odd primary case, restricting any iterate $v_1^{t-1}\colon P^{6+8t}(2^{r+1}) \to P^{14}(2^{r+1})$ to the bottom cell yields a homotopy class of order $2^{r+1}$ and stable order $2^{r+1}$. The resulting composition with the map $P^{14}(2^{r+1}) \to \Omega^2P^9(2^r)$ therefore generates a $\mathbb{Z}/2^{r+1}$ summand in $\pi_{5+8t}(\Omega^2P^9(2^r))=\pi_{7+8t}(P^9(2^r))$ by Theorem~\ref{even factorization thm}.
For part (a), we use the fact that an unstable Adams map $v_1\colon P^{n+8}(8) \to P^n(8)$ exists for $n\ge 9$ with the property that the composite
\[ N_t\colon S^{8t} \stackrel{i}{\longrightarrow} P^{1+8t}(8) \stackrel{v_1^{t-1}}{\longrightarrow} P^9(8) \stackrel{\nu^\sharp}{\longrightarrow} S^5 \]
has order $8$ in $\pi_{8t}(S^5)$ for all $t\ge 1$ by \cite[Theorem~E]{KR}. Here $\nu^\sharp$ denotes an extension of $\nu\colon S^8\to S^5$. Suspending once, an extension $\nu^\sharp\colon P^{10}(8) \to S^6$ of $\nu\colon S^9 \to S^6$ can be chosen to factor through the pinch map $q\colon P^6(8)\to S^6$ (we postpone a proof of this claim to Lemma~\ref{nu factorization} below). Combining this with the first diagram in Theorem~\ref{even factorization thm}, we obtain a homotopy commutative diagram
\[
\xymatrix{
P^{2+8t}(8) \ar[r]^{v_1^{t-1}} & P^{10}(8) \ar[r] & P^6(8) \ar[r] \ar[rd]_{E^\infty} \ar[d]^q & \Omega^2P^5(4) \ar[d] \\
S^{1+8t} \ar[u]_i \ar[rr]^{\Sigma N_t} & & S^6 \ar[rd]_{E^\infty} & QP^6(8) \ar[d]^{Qq} \\
& & & QS^6.
}
\]
The composite $E^\infty \circ \Sigma N_t$ is adjoint to $\Sigma^\infty N_t \in \pi_{8t-5}(\mathbb{S})$ and therefore has order $8$ since the proof of \cite[Theorem~E]{KR} shows $N_t$ has real $e$-invariant $\frac{b}{8}$ where $b$ is odd. Hence the composite $S^{1+8t} \to \Omega^2P^5(4)$ has order at least $8$. Since $8i=0$, the theorem follows.
\end{proof}
It remains to prove the following factorization of $\nu\in \pi_9(S^6)$ used in the proof above.
\begin{lem} \label{nu factorization}
There is a homotopy commutative diagram
\[
\xymatrix{
P^{10}(8) \ar[r] & P^6(8) \ar[d]^q \\
S^9 \ar[u]_i \ar[r]^\nu & S^6.
}
\]
\end{lem}
\begin{proof}
Since $8\circ \nu=\nu\circ 8=0$ in $\pi_9(S^6)$, $\nu$ lifts to the fibre $S^6\{8\}$ of the degree $8$ map, and since $\mathrm{sk}_9(S^6\{8\})\simeq P^6(8)$, it follows that $\nu$ factors as $S^9 \stackrel{\ell\,}{\to} P^6(8) \stackrel{q\,}{\to} S^6$. It suffices to show that $\ell$ has order $8$. The fibre $F$ of the pinch map $q$ has the homotopy type of a CW-complex $S^5 \cup e^{10} \cup e^{15}\cup\cdots$ where the first attaching map is $8w_5=0$ by \cite[Corollary~5.8]{G}. It follows that $\pi_9(F)=\mathbb{Z}/2$. The short exact sequence
\[ 0=\pi_{10}(S^6) \longrightarrow \pi_9(F) \longrightarrow \pi_9(P^6(8)) \stackrel{q_\ast}{\longrightarrow} \pi_9(S^6) \stackrel{0}{\longrightarrow} \pi_8(F) \]
therefore implies $\pi_9(P^6(8))=\mathbb{Z}/2 \oplus \mathbb{Z}/8$. In particular, $\ell$ has order $8$.
\end{proof}
|
1,116,691,499,230 | arxiv | \section{Definitions and notations}
Consider a planar three-body problem with equal masses $m_\ell=1$,
$\ell=0,1,2$,
defined by the Lagrangian
\begin{equation}
\label{Lagrangian}
L=\frac{1}{2}\sum_\ell \left(\frac{dq_\ell}{dt}\right)^2 + \sum_{i \ne j} V(|q_i-q_j|),\
\end{equation}
$q_\ell=(q_\ell^x, q_\ell^y) \in \mathbb{R}^2$.
The potential $V$ may be an extended Newton potential
\begin{equation}
\label{homogeneousPotential}
V_\alpha(|q_i-q_j|)=
\begin{cases}
\alpha^{-1}/|q_i-q_j|^\alpha & (\alpha \ne 0)\\
\log |q_i-q_j| & (\alpha=0)
\end{cases}
\end{equation}
or a Lennard-Jones type potential
\begin{equation}
V=\frac{a}{|q_i-q_j|^\alpha} -\frac{b}{|q_i-q_j|^\beta}.
\end{equation}
Choreographic solutions are periodic solutions to the equation of motion
\begin{equation}
\frac{d^2 q_\ell}{dt^2} = \frac{\partial}{\partial q_\ell}\sum_j V(|q_j-q_\ell|)
\end{equation}
that satisfy
\begin{equation}
(q_0,q_1,q_2)=(q(t), q(t+T/3), q(t-T/3)),
\end{equation}
where $T$ is the period.
Some choreographic solutions are known.
The famous figure-eight solution was discovered numerically by
C.~Moore \cite{Moore} in 1993.
Then, A.~Chenciner and R.~Montgomery \cite{CM} in 2000
rediscovered this solution and gave a proof for existence.
M.~\v{S}uvakov and V.~Dmitra\v{s}inovi\'{c} in 2013
found many ``slalom solutions''.
L.~Sbano \cite{Sbano} in 2004 found the figure-eight solution
under the Lennard-Jones type potential.
H.~Fukuda, T.~Fujiwara and H.~Ozaki \cite{LJ1} in 2017 found
many choreographic solutions
under the Lennard-Jones type potential.
Almost all these choreographic solutions have the same symmetries
as the figure-eight solutions.
Exceptional cases are $k=$ even slaloms,
where $k$ is the ``power of slalom'' \cite{SuvakovShibayama}.
The symmetries that we use in this note
will be shown in the section \ref{sectionSymmetries}.
Method in this note is applicable to
the original figure-eight solution, $k=$ odd slaloms,
figure-eight solutions by Sbano et al., and Fukuda et al.
\subsection{Definition of the Hessian}
Consider the variation of action integral at a solution $q_\ell(t)$ to second order of
variation $q_\ell(t) \to q_\ell(t)+\delta q_\ell(t)$,
\begin{equation}
S[q+\delta q]=S[q] + \frac{1}{2}\int_0^T dt \Bigg(
\sum_{\ell}\left(\frac{ d\delta q_\ell}{dt}\right)^2
+ \sum_{i,j}\delta q_i \left(\frac{\partial^2}{\partial q_i \partial q_j}\sum V\right) \delta q_j
\Bigg).
\end{equation}
Here,
the first order term $\delta S$ is zero as $q_\ell$ is a solution.
We restrict ourselves to consider the variational function $\delta q_\ell$ be
periodic function with period $T$. Then, partial integration makes
\begin{equation}
\int_0^T dt \sum_\ell \left(\frac{ d\delta q_\ell}{dt}\right)^2
=\int_0^T dt \sum_\ell \delta q_\ell \left(-\frac{d^2}{dt^2}\right)\delta q_\ell.
\end{equation}
Thus, the variation is given by
\begin{equation}
S[q+\delta q]=S[q] + \frac{1}{2}\int_0^T dt\ {}^t\! \Psi \mathcal{H} \Psi,
\end{equation}
where
\begin{equation}
\Psi
=\left(\begin{array}{c}\delta q_0^x \\\delta q_0^y \\
\delta q_1^x \\\delta q_1^y \\
\delta q_2^x \\\delta q_2^y\end{array}\right)
\end{equation}
is $\mathbb{R}^6$ column vector
and $\mathcal{H}$ is the Hessian that has the form
\begin{equation}
\begin{split}
\label{defOfHessian}
\mathcal{H}= -\frac{d^2}{dt^2} + {}^t\!\Delta \mathcal{U} \Delta,\
\mathcal{U}=\left(\begin{array}{ccc}u_{12} & 0 & 0 \\0 & u_{20} & 0 \\0 & 0 & u_{01}\end{array}\right).
\end{split}
\end{equation}
Here, $u_{ij}$ is $2\times 2$ matrix and $\Delta$ is $6\times 6$ anti-symmetric matrix
\begin{equation}
u_{ij}=\left(\begin{array}{ll}
u_{ij}^{xx} & u_{ij}^{xy} \\
u_{ij}^{yx} & u_{ij}^{yy}
\end{array}\right),
\Delta = \left(\begin{array}{rrr}
0 & E_2 & -E_2 \\-E_2 & 0 & E_2 \\E_2 & -E_2 & 0\end{array}\right),
E_2=\left(\begin{array}{cc}1 & 0 \\0 & 1\end{array}\right),
\end{equation}
and ${}^t\!\Delta$ stands for the transpose of $\Delta$.
The explicit expression of $u_{ij}$ for extended Newton potential, for example, is
\begin{equation}
\begin{split}
u_{ij}&=\frac{\alpha+2}{r_{ij}^{\alpha+4}}
\left(\begin{array}{cc}(q_i^x-q_j^x)^2 & (q_i^x-q_j^x)(q_i^y-q_j^y)\\
(q_i^x-q_j^x)(q_i^y-q_j^y) & (q_i^y-q_j^y)^2\end{array}
\right)
-\frac{1}{r_{ij}^{\alpha+2}}E_2,\\
r_{ij}&=|q_i-q_j|.
\end{split}
\end{equation}
When there is no confusion, we will use the following abbreviated notations
\begin{equation}
\Psi
=\left(\begin{array}{c}\delta q_0 \\\delta q_1 \\\delta q_2\end{array}\right)
\mbox{ and }
\Delta = \left(\begin{array}{rrr}0 & 1 & -1 \\-1 & 0 & 1 \\1 & -1 & 0\end{array}\right),
\end{equation}
and similar.
In this note, we consider the eigenvalue problem
\begin{equation}
\label{eigenValueProblem}
\mathcal{H} \Psi = \lambda \Psi.
\end{equation}
For linear operator $\mathcal{O}$, we express its eigenvalue $\mathcal{O}'$.
If $\mathcal{O}$ satisfies $\mathcal{O}^2=1$, the eigenvalue is one of
$\mathcal{O}'=\pm 1$.
While,
if $\mathcal{O}^3=1$ is satisfied, then $\mathcal{O}'=1, \omega, \omega^2$
with $\omega = (-1\pm i\sqrt{3})/2$.
For any two functions $\Phi$, $\Psi$ and any operator $\mathcal{O}$,
we define and write the inner product
\begin{equation}
\label{defOfInnerProduct}
\braket{\Phi|\Psi}=\frac{2}{3T}\int_0^T dt\ {}^t\!\Phi(t)\Psi(t),\
\braket{\Phi|\mathcal{O}|\Psi}=\frac{2}{3T}\int_0^T dt\ {}^t\!\Phi(t)\mathcal{O}\Psi(t).
\end{equation}
The factor $2/(3T)$ is for later convenience.
The inner product satisfy
\begin{equation}
\braket{\Phi|\Psi}=\braket{\Psi|\Phi},\
\braket{\Phi|\mathcal{O}|\Psi}
=\braket{\Phi|\mathcal{O}\Psi}
=\braket{{}^t\!\mathcal{O}\Phi|\Psi}.
\end{equation}
For writing basis vectors,
it is convenient to write $x$ and $y$ basis together
in a matrix form,
\begin{equation}
\left(\begin{array}{cc}
x_1 & 0 \\
0 & y_1 \\
x_2 & 0\\
0 & y_2 \\
x_3 & 0 \\
0 & y_3\\
\end{array}\right),
\end{equation}
with
\begin{equation}
\frac{2}{3T}\int_0^T dt (x_1^2+x_2^2+x_3^2)
=\frac{2}{3T}\int_0^T dt (y_1^2+y_2^2+y_3^2)
=1.
\end{equation}
Then
\begin{equation}
a^x \left(\begin{array}{c}
x_1\\
0\\
x_2\\
0\\
x_3\\
0\\
\end{array}\right)
+a^y \left(\begin{array}{c}
0\\
y_1\\
0\\
y_2\\
0\\
y_3\\
\end{array}\right)
=\left(\begin{array}{cc}
x_1 & 0 \\
0 & y_1 \\
x_2 & 0\\
0 & y_2 \\
x_3 & 0 \\
0 & y_3\\
\end{array}\right)
\left(\begin{array}{c}
a^x\\
a^y
\end{array}\right).
\end{equation}
If $x_k=y_k=z_k$, we simply write
\begin{equation}
\left(\begin{array}{cc}
z_1 & 0 \\
0 & z_1 \\
z_2 & 0 \\
0 & z_2 \\
z_3 & 0 \\
0 & z_3
\end{array}\right)
=
\left.\left(\begin{array}{c}
z_1 \\z_2 \\z_3\end{array}\!\!\!\right)\!\!\!\!\right).
\end{equation}
It should be stressed that
this is not a constraint but a matrix representation of two independent basis vectors.
For example,
\begin{equation}
\Phi_n
=
\left.\left(\begin{array}{l}
\cos(n\nu t) \\\cos(n\nu t + 2\pi/3) \\\cos(n\nu t - 2\pi/3)\end{array}\!\!\!\right)\!\!\!\!\right)
\end{equation}
is a matrix representation of two independent basis
\begin{equation}
\left(\begin{array}{l}
\cos(n\nu t)\\
0\\
\cos(n\nu t + 2\pi/3)\\
0\\
\cos(n\nu t - 2\pi/3)\\
0
\end{array}\right)
\mbox{ and }
\left(\begin{array}{ll}
0\\
\cos(n\nu t)\\
0\\
\cos(n\nu t + 2\pi/3)\\
0\\
\cos(n\nu t - 2\pi/3)
\end{array}\right)
\end{equation}
and $\braket{\Phi_m|\Phi_n}=\delta_{mn}$
for $\nu=2\pi/T$.
\subsection{Decomposition of function space by orthogonal projection operator}
A linear operator $\mathcal{P}$ is called ``orthogonal projection operator''
if it satisfies
\begin{equation}
\mathcal{P}(\mathcal{P}-1)=0 \mbox{ and } {}^t\!\mathcal{P}=\mathcal{P}.
\end{equation}
This operator decompose function space $W$ into two orthogonal subspaces
$W_1$ and $W_0$,
\begin{equation}
W_1 = \{ f |f \in W, \mathcal{P} f = f\}, W_0=\{ f | f \in W, \mathcal{P} f =0\},
W=W_1 \oplus W_0.
\end{equation}
\begin{figure}
\centering
\includegraphics[width=6cm]{decompositionByProjectionOP.eps}
\caption{An schematic image of decomposition of function space
$W=W_1 \oplus W_0$
by orthogonal projection operator $\mathcal{P}$ that satisfy
$\mathcal{P}(\mathcal{P}-1)=0$, ${}^t\! \mathcal{P}=\mathcal{P}$}
\label{fig:decompositionOfFunctionSpace}
\end{figure}
The subspaces $W_1$ and $W_0$ are mutually orthogonal compliment,
because arbitrary elements $f \in W_1$ and $g \in W_0$ are orthogonal
by $\braket{g|f}=\braket{g|\mathcal{P} f}=\braket{{}^t\!\mathcal{P} g|f}=0$,
and arbitrary element $\Psi \in W$ will be decomposed by the identities
$\Psi = \mathcal{P} \Psi + (1-\mathcal{P}) \Psi$.
Here,
$\mathcal{P} \Psi \in W_1$ and $(1-\mathcal{P}) \Psi \in W_0$ since
$\mathcal{P}(\mathcal{P} \Psi )=\mathcal{P} \Psi$ and $\mathcal{P}((1-\mathcal{P}) \Psi )=0$.
See figure \ref{fig:decompositionOfFunctionSpace}.
If the Hessian $\mathcal{H}$ is commutable with $\mathcal{P}$,
matrix representation of the Hessian is also decomposed into two blocks
for $W_1$ subspace and $W_0$ subspace,
because
the matrix elements of $\mathcal{H}$ between $W_1$ subspace and $W_0$ subspace are zero,
\begin{equation}
\braket{g|\mathcal{H}|f}=\braket{g|\mathcal{H}|\mathcal{P} f}=\braket{{}^t\! \mathcal{P} g|\mathcal{H}|f}=0
\mbox{ for } f \in W_1 \mbox{ and } g \in W_0.
\end{equation}
In the following sections,
we will find some orthogonal projection operators
that are commutable each other and $\mathcal{H}$.
These projection operators decompose the function space into small subspaces
and
make the matrix representation of $\mathcal{H}$ small blocks.
\section{Four zero eigenvalues and quadruply degenerated trivial eigenvalues}
\label{section:zeroAndTrivialMode}
In this section,
we describe $4$ zero eigenvalues and quadruply degenerated trivial eigenvalues,
which always exist for the Hessian (\ref{defOfHessian}).
Corresponding to the conservation law,
the linear momentum for $x$ and $y$ direction, angular momentum
and the energy,
the Hessian always has $4$ zero eigenvalues
for variations
$\delta q_\ell = {}^t\!(1,0)$, ${}^t\!(0,1)$, ${}^t\!(-q_\ell^y, q_\ell^x)$ and $ dq_\ell/dt$.
Since the function space is composed by all periodic functions with period $T$,
it contains a trivial eigenfunction $\Psi(t) = {}^t\!(\delta q(t),\delta q(t),\delta q(t))$
that describes three bodies move coherently.
Since, $\Delta \Psi=0$ for this function,
the eigenvalue problem is reduced to the simple form
\begin{equation}
H \Psi
= -\frac{d^2}{dt^2} \Psi
= \lambda \Psi,
\end{equation}
that yields quadruply degenerated eigenvalue $\lambda = 4\pi^2/T^2 \times k^2$.
Namely
$\delta q={}^t\!(\cos 2\pi k t/T, 0)$, ${}^t\!(0, \cos 2\pi k t/T)$,
${}^t\!(\sin 2\pi k t/T, 0)$, ${}^t\!(0, \sin 2\pi k t/T)$
for positive integers $k$.
The two trivial functions for $k=0$ are already counted
for translational function in the previous paragraph.
Let $\sigma$ be the cyclic permutation operator
$\sigma\ {}^t\!(\delta q_0, \delta q_1, \delta q_2)={}^t\!(\delta q_1, \delta q_2, \delta q_0)$,
or in the matrix form,
\begin{equation}
\sigma=\left(\begin{array}{ccc}
0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0\end{array}\right)
\mbox{ and } {}^t\!\sigma = \sigma^{-1}.
\end{equation}
Since $\sigma^3=1$,
it defines the orthogonal projection operator
\begin{equation}
\mathcal{P}_\sigma=\frac{1}{3}(1+\sigma+\sigma^2),\
\mathcal{P}_\sigma(\mathcal{P}_\sigma-1)=0.
\end{equation}
This operator splits the function space into
subspace with the eigenvalue
$\mathcal{P}_\sigma'=1$ or $\mathcal{P}_\sigma'=0$.
The subspace whose element has $\mathcal{P}_\sigma'=1$ is the trivial subspace.
The complemental subspace is composed of functions
with $\mathcal{P}_\sigma'=0$.
We call this subspace ``zero-centre-of-mass'' subspace,
because each function in this subspace keeps the center of mass at the origin,
\begin{equation}
0
=\frac{1}{3}(1+\sigma+\sigma^2)
\left(\begin{array}{c}\delta q_0 \\\delta q_1 \\\delta q_2 \end{array}\right)
=\frac{1}{3}
\left(\begin{array}{c}
\delta q_0+\delta q_1+\delta q_2\\
\delta q_0+\delta q_1+\delta q_2\\
\delta q_0+\delta q_1+\delta q_2
\end{array}\right).
\end{equation}
The figure-eight solution belongs to zero-center-of-mass subspace.
Now, let us proceed to the symmetry of the Hessian
at choreographic solutions
with figure-eight symmetry.
\section{Symmetries of the figure-eight solution}
\label{sectionSymmetries}
The figure-eight solution with proper $x, y$ coordinates and origin of time
has the following symmetries,
\begin{equation}
\label{choreographicSymmetry}
\mbox{choreographic symmetry: }
\left(\begin{array}{c}
q_0(t+T/3) \\q_1(t+T/3) \\q_2(t+T/3)
\end{array}\right)
=
\left(\begin{array}{c}
q_1(t) \\q_2(t) \\q_0(t)
\end{array}\right),
\end{equation}
\begin{equation}
\label{timeReversalSymmetry}
\mbox{time reversal symmetry: }
\left(\begin{array}{c}
q_0(-t) \\q_1(-t) \\q_2(-t)
\end{array}\right)
=
-\left(\begin{array}{c}
q_0(t) \\q_2(t) \\q_1(t)
\end{array}\right),
\end{equation}
and
\begin{equation}
\label{mirrorSymmetry}
\mbox{time shift in $T/2$:}
\left(\begin{array}{c}
q_0(t+T/2) \\q_1(t+T/2) \\q_2(t+T/2)
\end{array}\right)
=
-\left(\begin{array}{c}
\mu\ q_0(t) \\\mu\ q_1(t) \\ \mu\ q_2(t)
\end{array}\right),
\mu \left(\begin{array}{c}x \\y\end{array}\right)
=\left(\begin{array}{c}-x \\y\end{array}\right).
\end{equation}
\subsection{Choreographic symmetry}
Let $\mathcal{R}^{1/3}$ be a time displacement operator
$\mathcal{R}^{1/3} \delta q(t) =\delta q(t+T/3)$
and $\mathcal{C}=\sigma^{-1}\mathcal{R}^{1/3}$.
Then the choreographic symmetry in (\ref{choreographicSymmetry}) is equivalent to
\begin{equation}
\mathcal{C}
\left(\begin{array}{c}
q_0(t) \\q_1(t) \\q_2(t)
\end{array}\right)
=\left(\begin{array}{c}
q_0(t) \\q_1(t) \\q_2(t)
\end{array}\right).
\end{equation}
\begin{figure}
\centering
\includegraphics[width=8cm]{decompositionByProjectionOPC2.eps}
\caption{Decomposition of function space $W=W_\mathcal{C} \oplus W_z$
by orthogonal projection operator $\mathcal{P}_\mathcal{C}$.
We call a function $\Psi$ ``non-choreographic'' \emph{function} if $(1-\mathcal{P}_\mathcal{C})\Psi \ne 0$.
While, we call the subspace $W_z$ ``zero-choreographic'' \emph{subspace}.
}
\label{fig:decompositionOfFunctionSpaceByOPC}
\end{figure}
Since the function space has period $T$,
\begin{equation}
\int_0^T dt f(t)g(t+T/3)=\int_0^T dt f(t-T/3)g(t).
\end{equation}
Namely, ${}^t\! R^{1/3}=R^{-1/3}$
and ${}^t\!\mathcal{C}=\mathcal{C}^{-1}$.
Since the operator $\mathcal{C}$ satisfies $\mathcal{C}^3=1$,
it defines the orthogonal projection operator
\begin{equation}
\mathcal{P}_\mathcal{C}=\frac{1}{3}(1+\mathcal{C}+\mathcal{C}^2),\
\mathcal{P}_\mathcal{C}(\mathcal{P}_\mathcal{C}-1)=0.
\end{equation}
Thus the operator $\mathcal{P}_\mathcal{C}$ decomposes
the function space $W$
into two orthogonal subspaces;
the choreographic subspace $W_c$
that made of functions that belong to the eigenvalue $\mathcal{P}_\mathcal{C}'=1$
(i.e. $\mathcal{C}'=1$) and its orthogonal complement.
We call this complement ``zero-choreographic'' subspace $W_z$.
\begin{equation}
\begin{split}
&W_c=\set{\Psi | \mathcal{P}_\mathcal{C} \Psi = \Psi},\
W_z=\set{\Psi | \mathcal{P}_\mathcal{C} \Psi = 0},\\
&W=W_c \oplus W_z
\end{split}
\end{equation}
The figure-eight solution belongs to choreographic subspace.
\emph{In this note, we call the subspace $W_z$ ``zero-choreographic'' subspace
to avoid confusion with ``non-choreographic'' function.}
A function is called ``non-choreographic'',
if it is not choreographic.
See figure \ref{fig:decompositionOfFunctionSpaceByOPC}.
The eigenvalue of $\mathcal{H}$ in zero-choreographic subspace
is doubly degenerated.
Because $\mathcal{H}$ and $\mathcal{C}$ commute,
if $\Psi_\lambda\ne0$ belongs to the eigenvalue $\mathcal{H}'=\lambda$
then $\Psi_\lambda$ and
$\Phi_\lambda = (\mathcal{C}-\mathcal{C}^2)\Psi_\lambda/\sqrt{3}$
belong to the same eigenvalue.
Since $\Psi_\lambda$ satisfies $(1+\mathcal{C}+\mathcal{C}^2)\Psi_\lambda=0$,
$\braket{\Psi_\lambda|\Phi_\lambda}=0$
and $\braket{\Phi_\lambda|\Phi_\lambda}=\braket{\Psi_\lambda|\Psi_\lambda}\ne0$.
Actually,
\begin{equation}
\begin{split}
\braket{\Psi_\lambda|\Phi_\lambda}
&=\braket{\Psi_\lambda|\mathcal{C}-\mathcal{C}^2|\Psi_\lambda}/\sqrt{3}
=\big(\braket{\Psi_\lambda|\mathcal{C}\Psi_\lambda}-\braket{\mathcal{C}\Psi_\lambda|\Psi_\lambda}\big)/\sqrt{3}
=0,\\
\braket{\Phi_\lambda|\Phi_\lambda}
&=\braket{\Psi_\lambda|(\mathcal{C}^2-\mathcal{C})(\mathcal{C}-\mathcal{C}^2)|\Psi_\lambda}/3
=\braket{\Psi_\lambda|1|\Psi_\lambda}.
\end{split}
\end{equation}
Therefore $\Psi_\lambda$ and $\Phi_\lambda$ are
orthogonal bases of two dimensional subspace for each $\lambda$.
This proves the statement of this paragraph.
\subsection{Time reversal symmetry}
Let $\Theta$ be the time reversal operator $\Theta(\delta q_k(t))=\delta q_k(-t)$,
$\tau$ be the operator for exchange of the second and third row,
\begin{equation}
\tau=\left(\begin{array}{ccc}1 & 0 & 0 \\0 & 0 & 1 \\0 & 1 & 0\end{array}\right),
\end{equation}
and let $\mathcal{T}=\tau\Theta$.
Then the time reversal symmetry in (\ref{timeReversalSymmetry})
for the figure-eight solution is
\begin{equation}
\mathcal{T} \left(\begin{array}{c}
q_0(t) \\q_1(t) \\q_2(t)
\end{array}\right)
=-\left(\begin{array}{c}
q_0(t) \\q_1(t) \\q_2(t)
\end{array}\right).
\end{equation}
Since, $\mathcal{T}^2=1$, this operator defines the orthogonal projection operator
\begin{equation}
\mathcal{P}_{\mathcal{T}}=\frac{1}{2}(1+\mathcal{T}),\ \mathcal{P}_\mathcal{T}(\mathcal{P}_\mathcal{T}-1)=0.
\end{equation}
This operator decomposes the function space into two subspaces
with $\mathcal{P}_\mathcal{T}'=1$ (i.e. $\mathcal{T}'=1$)
and with $\mathcal{P}_\mathcal{T}'=0$ ($\mathcal{T}'=-1$).
The figure-eight solution belongs to the subspace with $\mathcal{T}'=-1$.
\subsection{Time shift symmetry in $T/2$}
Let $R^{1/2}$ be a time displacement operator
$R^{1/2}\delta q(t)=\delta q(t+T/2)$,
$\mu$ be the mirror operator with respect to $y$ axis
$\mu {}^t\!(\delta q_k^x, \delta q_k^y)={}^t\!(-\delta q_k^x, \delta q_k^y)$
and $\mathcal{M}=\mu R^{1/2}$.
Then the time shift symmetry in $T/2$ is
\begin{equation}
\mathcal{M} \left(\begin{array}{c}
q_0(t) \\q_1(t) \\q_2(t)
\end{array}\right)
=\left(\begin{array}{c}
q_0(t) \\q_1(t) \\q_2(t)
\end{array}\right).
\end{equation}
Since $\mathcal{M}$ satisfies $\mathcal{M}^2=1$, the eigenvalues are $\pm 1$,
and the orthogonal projection operator $P_\mathcal{M}$ is
\begin{equation}
P_\mathcal{M}=\frac{1}{2}(1+\mathcal{M}),\ P_\mathcal{M}(P_\mathcal{M}-1)=0.
\end{equation}
This operator decompose the function space
into two subspaces with $P_\mathcal{M}'=1$ (i.e. $\mathcal{M}'=1$)
and $P_\mathcal{M}'=0$ ($\mathcal{M}'=-1$).
The figure-eight solution belongs to the subspace with $P_\mathcal{M}'=1$.
\begin{table}
\caption{Zero-center-of-mass subspace is decomposed into $2^3$ subspaces
by three projection operators $\mathcal{P}_\mathcal{C}$, $\mathcal{P}_\mathcal{T}$ and $\mathcal{P}_\mathcal{M}$.
Correspondence between the name of subspaces and the eigenvalues are shown.
The last $\pm$ sign correspond to two eigenvalue of $\mathcal{P}_\mathcal{M}'=1,\ 0$
respectively.
}
\label{nameOfSubspaces}
\centering
\begin{tabular}{c|cc}
&$\mathcal{P}_\mathcal{C}'=1$&$\mathcal{P}_\mathcal{C}'=0$\\
\hline
$\mathcal{P}_\mathcal{T}'=1$&choreographic cos $\pm$&zero-choreographic cos $\pm$\\
$\mathcal{P}_\mathcal{T}'=0$&choreographic sin $\pm$&zero-choreographic sin $\pm$
\end{tabular}
\end{table}
\subsection{Summary of symmetries for the figure-eight solution}
In the section \ref{section:zeroAndTrivialMode}
and previous subsections,
we introduced operators $\sigma$, $\mathcal{C}$, $\mathcal{T}$, $\mathcal{M}$
and corresponding projection operators $\mathcal{P}_\sigma$,
$\mathcal{P}_\mathcal{C}$, $\mathcal{P}_\mathcal{T}$, $\mathcal{P}_\mathcal{M}$.
All these projection operators commute each other.
Therefore, these operators decompose the function space
into $2^4$ subspaces.
Actually, we don't divide the trivial subspace $\mathcal{P}_\sigma'=1$.
The zero-centre-of-mass subspace $\mathcal{P}_\sigma'=0$ will be divided into $2^3$ subspaces.
See table \ref{nameOfSubspaces}.
The figure-eight solution belongs to the subspace
$\mathcal{P}_\sigma'=0$ and
$\mathcal{P}_\mathcal{C}'=1$ ($\mathcal{C}'=1$) and
$\mathcal{P}_\mathcal{T}'=0$ ($\mathcal{T}'=-1$) and
and $\mathcal{P}_\mathcal{M}'=1$ ($\mathcal{M}'=1$).
The eigenspace of $\mathcal{H}$ is also divided into these subspaces,
because all these projection operators commute with $\mathcal{H}$.
In the next section,
we decompose the function space by these four symmetries.
\section{Subspaces and base functions}
\label{sectionDecomposition}
Before decompose the function space,
we count the degree of this space.
To make the degree finite, we introduce a cutoff
in Fourier series,
$1$, $\cos(\nu t)$, $\cos(2\nu t)$, $\cos(3\nu t),~\dots$,
$\cos(3N\nu t)$
and $\sin(\nu t), \sin(2\nu t), \sin(3\nu t)\dots, \sin(3N\nu t)$,
$\nu = 2\pi/T$
and $N=2^M$ with some integer $M$.
Then the degrees are $3$ for three bodies, $2$ for $x$ and $y$ components,
$6N+1$ for Fourier components.
So, total degree is $6(6N+1)\sim 2 \times 18N$.
We will decompose this space.
See figure \ref{blocksOfH}.
In this figure, trivial subspace is not decomposed.
The compliment, zero-centre-of-mass subspace, will be decomposed into $2^3$ subspaces.
\begin{figure}
{\small
\begin{equation*}
M_\mathcal{H}=
\left(
\begin{array}{cccccccccccccccccc}
\cdot&&&&&&&&&&&&&&&&&\\
&\cdot&&&&&&&&&&&&&&&&\\
&&\cdot&&&&&&&&&&&&&&&\\
&&&\cdot&&&&&&&&&&&&&&\\
&&&&\cdot&&&&&&&&&&&&&\\
&&&&&\cdot&&&&&&&&&&&&\\
&&&&&&*&&&&&&&&&&&\\
&&&&&&&*&&&&&&&&&&\\
&&&&&&&&*&&&&&&&&&\\
&&&&&&&&&*&&&&&&&&\\
&&&&&&&&&&*&*&&&&&&\\
&&&&&&&&&&*&*&&&&&&\\
&&&&&&&&&&&&*&*&&&&\\
&&&&&&&&&&&&*&*&&&&\\
&&&&&&&&&&&&&&*&*&&\\
&&&&&&&&&&&&&&*&*&&\\
&&&&&&&&&&&&&&&&*&*\\
&&&&&&&&&&&&&&&&*&*
\end{array}\right)
\begin{array}{l}
\mbox{trivial}\\
\\
\\
\\
\\
\\
\mbox{choreographic cos }+\\
\phantom{\mbox{choreographic cos }}-\\
\mbox{choreographic sin }+\\
\phantom{\mbox{choreographic sin }}-\\
\mbox{zero-choreographic cos }+\\
\\
\mbox{zero-choreographic cos }-\\
\\
\mbox{zero-choreographic sin }+\\
\\
\mbox{zero-choreographic sin }-\\
\\
\end{array}
\end{equation*}
}
\caption{Decomposition of matrix representation of the Hessian,
whose total size is $(2 \times 18N)^2$.
Here, each $\cdot$ stands $(2 \times N)^2$ diagonal matrix,
and $*$ stands $(2 \times N)^2$ matrix.
So, the size of each 4 choreographic blocks are $(2 \times N)^2$,
and zero-choreographic blocks are $(2 \times 2N)^2$.
All no marked elements are $0$.
}
\label{blocksOfH}
\end{figure}
Although we are considering the function space of $\mathbb{R}^6$,
it is convenient to consider the function space of $\mathbb{C}^6$
in the following two subsections.
This is because operator $\mathcal{O}$ with $\mathcal{O}^3=1$
has the eigenvalue only $\mathcal{O}'=1$ in $\mathbb{R}$.
This makes our decomposition not straightforward.
It has three eigenvalues $\mathcal{O}'=1, \omega, \omega^2$,
$\omega=(-1+i\sqrt{3})/2$ in $\mathbb{C}$ instead.
We will use $e^{-i 3N\nu t}, \dots, e^{-i \nu t}, 1, e^{i\nu t}, \dots, e^{i 3N\nu t}$
instead of $\sin$ and $\cos$ for the Fourier series.
We again have $6N+1$ Fourier components.
\subsection{Decomposition by $\mathcal{P}_\sigma$}
Since $\sigma^3=1$,
the eigenvalue of $\sigma$ are $\sigma'=1, \omega, \omega^2$
with $\omega = (-1+i\sqrt{3})/2$.
Then, the basis for $\mathcal{P}_\sigma=1$ are trivial,
\begin{equation}
\sigma'=1\leftrightarrow
\left.\left(\begin{array}{c}1 \\1\\1 \\\end{array}\!\!\!\right)\!\!\!\!\right) f(t)
\end{equation}
with arbitrary periodic function $f(t)$.
While function for $\mathcal{P}'=0$ are zero-centre-of-mass,
\begin{equation}
\sigma'=\omega \leftrightarrow
\left.\left(\begin{array}{c}1 \\\omega\\\omega^2 \\\end{array}\!\!\!\right)\!\!\!\!\right) f(t),
\mbox{ or }
\sigma'=\omega^2 \leftrightarrow
\left.\left(\begin{array}{c}1 \\\omega^2\\\omega \\\end{array}\!\!\!\right)\!\!\!\!\right) f(t).
\end{equation}
Therefore $1/3$ of whole basis are trivial
and $2/3$ are zero-centre-of-mass.
Namely, $2(6N+1)$ basis are trivial. We don't decompose them more.
In the following sections, we will decompose
$4(6N+1)$ basis by other projection operators.
\subsection{Decomposition by $\mathcal{P}_\mathcal{C}$}
Since $(R^{1/3})^3=1$, the eigenvalues of $R^{1/3}$
are $1, \omega$ and $\omega^2$.
The eigenfunctions are
$e^{i3k\nu t}, e^{i(3k+1)\nu t}$ and $e^{i(3k+2)\nu t}$
with $k \in \mathbb{Z}$
respectively.
More precisely,
\begin{equation}
\begin{split}
&(R^{1/3})'=1 \leftrightarrow
e^{i(-3N)\nu t}, e^{i(-3N+3)\nu t},\dots, e^{i(-3)\nu t}, 1, e^{i3\nu t},e^{i6\nu t},
\dots, e^{i(3N)\nu t},\\
&(R^{1/3})'=\omega \leftrightarrow
e^{i(-3N+1)\nu t}, e^{i(-3N+4)\nu t},\dots, e^{i(-2)\nu t}, e^{i\nu t},e^{i4\nu t},
\dots, e^{i(3N-2)\nu t},\\
&(R^{1/3})'=\omega^2 \leftrightarrow
e^{i(-3N+2)\nu t}, e^{i(-3N+5)\nu t},\dots, e^{-i\nu t}, e^{i2\nu t},e^{i5\nu t},
\dots, e^{i(3N-1)\nu t}.
\end{split}
\end{equation}
Each subspace has $2N+1$, $2N$ and $2N$ functions.
Then, choreographic basis belong to
$\mathcal{P}_\mathcal{C}'=1 \leftrightarrow \mathcal{C}'=(\sigma^{-1})'(R^{1/3})'=1 \leftrightarrow
(\sigma', (R^{1/3})')=(\omega, \omega)$ or $(\omega^2, \omega^2)$.
Namely, the following $8N$ basis are choreographic,
\begin{equation}
\label{choreographicModes}
\left.\left(\begin{array}{l}1 \\\omega\\\omega^2 \\\end{array}\!\!\!\right)\!\!\!\!\right)
e^{i(3k+1)\nu t}
\mbox{ and }
\left.\left(\begin{array}{l}1 \\\omega^2\\\omega \\\end{array}\!\!\!\right)\!\!\!\!\right)
e^{i(3k+2)\nu t}.
\end{equation}
It should be stressed again that
\begin{equation}
\left.\left(\begin{array}{l}1 \\\omega\\\omega^2 \\\end{array}\!\!\!\right)\!\!\!\!\right)
\end{equation}
represents two independent basis
\begin{equation}
\left(\begin{array}{l}1 \\0 \\\omega \\0 \\\omega^2 \\0\end{array}\right),\
\left(\begin{array}{l}0\\1\\0 \\\omega \\0 \\\omega^2\end{array}\right).
\end{equation}
So,
\begin{equation}
\left.\left(\begin{array}{l}1 \\\omega\\\omega^2 \\\end{array}\!\!\!\right)\!\!\!\!\right)
e^{i(3k+1)\nu t},\
k=-N, -N+1,\dots, -1,0,1,\dots N-1
\end{equation}
stands for $2\times 2N$ base functions.
Zero-choreographic basis $\mathcal{P}_\mathcal{C}'=0$ are composed of
$\mathcal{C}'=(\sigma^{-1})'(R^{1/3})'=\omega \mbox{ or }\omega^2$.
Namely,
$\mathcal{C}'=(\sigma^{-1})'(R^{1/3})'=\omega
\leftrightarrow
(\sigma', (R^{1/3})')=(\omega, \omega^2) \mbox{ or } (\omega^2, 1)
\leftrightarrow$
\begin{equation}
\label{zeroChoreographicModesI}
\left.\left(\begin{array}{l}1 \\\omega\\\omega^2 \\\end{array}\!\!\!\right)\!\!\!\!\right)
e^{i(3k+2)\nu t}
\mbox{ and }
\left.\left(\begin{array}{l}1 \\\omega^2\\\omega \\\end{array}\!\!\!\right)\!\!\!\!\right)
e^{i3k\nu t},
\end{equation}
and
$\mathcal{C}'=(\sigma^{-1})'(R^{1/3})'=\omega^2
\leftrightarrow
(\sigma', (R^{1/3})')=(\omega, 1), (\omega^2, \omega)
\leftrightarrow$
\begin{equation}
\label{zeroChoreographicModesII}
\left.\left(\begin{array}{l}1 \\\omega\\\omega^2 \\\end{array}\!\!\!\right)\!\!\!\!\right)
e^{i3k\nu t}
\mbox{ and }
\left.\left(\begin{array}{l}1 \\\omega^2\\\omega \\\end{array}\!\!\!\right)\!\!\!\!\right)
e^{i(3k+1)\nu t}.
\end{equation}
So, there are $16N+4$ zero-choreographic basis.
Thus,
total $4(6N+1)$ basis are split into $8N$ choreographic basis
and $16N+4$ zero-choreographic basis.
\subsection{Decomposition by $\mathcal{P}_\mathcal{T}$}
Now consider the action $\mathcal{T}=\tau \Theta$ to the above basis
(\ref{choreographicModes}),
(\ref{zeroChoreographicModesI}) and
(\ref{zeroChoreographicModesII}).
The operator $\tau$ exchange $\omega$ and $\omega^2$
and $\Theta$ exchange $e^{ik\nu t}$ and $e^{-ik\nu t}$.
As the result, $\mathcal{T}=\tau \Theta$ exchange a base
and its complex conjugate.
Therefore, the projection operator $\mathcal{P}_\mathcal{T}=(1+\mathcal{T})/2$
picks its real part,
while $1-\mathcal{P}_\mathcal{T}=(1-\mathcal{T})/2$ picks its imaginary part.
Thus, the basis for
$(\mathcal{P}_\sigma', \mathcal{P}_\mathcal{C}', \mathcal{D}_\mathcal{T}')=(0,1,1)$ are
\begin{equation}
\left.\left(\begin{array}{l}
\cos((3n+1)\nu t) \\\cos((3n+1)\nu t+2\pi/3) \\\cos((3n+1)\nu t-2\pi/3)\end{array}
\!\!\!\right)\!\!\!\!\right)
=
\left.\left(\begin{array}{l}
\cos((3n+1)\nu t) \\\cos((3n+1)\nu (t+T/3)) \\\cos((3n+1)\nu (t-T/3)\end{array}
\!\!\!\right)\!\!\!\!\right)
\end{equation}
or
\begin{equation}
\left.\left(\begin{array}{l}
\cos((3n+2)\nu t) \\\cos((3n+2)\nu t-2\pi/3) \\\cos((3n+2)\nu t+2\pi/3)\end{array}
\!\!\!\right)\!\!\!\!\right)
=
\left.\left(\begin{array}{l}
\cos((3n+2)\nu t) \\\cos((3n+2)\nu (t+T/3)) \\\cos((3n+2)\nu (t-T/3)\end{array}
\!\!\!\right)\!\!\!\!\right).
\end{equation}
Explicitly,
\begin{equation}
\begin{split}
\left.\left(\begin{array}{l}
\cos(\nu t) \\\cos(\nu (t+T/3)) \\\cos(\nu (t-T/3)\end{array}
\!\!\!\right)\!\!\!\!\right),&
\left.\left(\begin{array}{l}
\cos(2\nu t) \\\cos(2\nu (t+T/3)) \\\cos(2\nu (t-T/3)\end{array}
\!\!\!\right)\!\!\!\!\right),\\
\left.\left(\begin{array}{l}
\cos(4\nu t) \\\cos(4\nu (t+T/3)) \\\cos(4\nu (t-T/3)\end{array}
\!\!\!\right)\!\!\!\!\right),&
\left.\left(\begin{array}{l}
\cos(5\nu t) \\\cos(5\nu (t+T/3)) \\\cos(5\nu (t-T/3)\end{array}
\!\!\!\right)\!\!\!\!\right),\\
&\vdots\\
\left.\left(\begin{array}{l}
\cos((3N-2)\nu t) \\\cos((3N-2)\nu (t+T/3)) \\\cos((3N-2)\nu (t-T/3)\end{array}
\!\!\!\right)\!\!\!\!\right),&
\left.\left(\begin{array}{l}
\cos((3N-1)\nu t) \\\cos((3N-1)\nu (t+T/3)) \\\cos((3N-1)\nu (t-T/3)\end{array}
\!\!\!\right)\!\!\!\!\right).
\end{split}
\end{equation}
Let $k_n=\{1,2,4,5,7,8,10,\dots,3N-2, 3N-1\}$, namely,
the series of positive integers without multiple of 3.
This series is given by
\begin{equation}
k_n=\frac{1}{4}\Big( 6n-(-1)^n-3\Big),\
n=1,2,3,\dots,2N-1,2N.
\end{equation}
There are $4N$ basis for $(\mathcal{P}_\sigma', \mathcal{P}_\mathcal{C}', \mathcal{T}')=(0,1,1)$,
and $4N$ basis for $(\mathcal{P}_\sigma', \mathcal{P}_\mathcal{C}',\mathcal{T}')=(0,1,-1)$.
We write them
\begin{equation}
(\mathcal{P}_\sigma', \mathcal{P}_\mathcal{C}', \mathcal{T}')=(0,1,1)
\leftrightarrow
cc_n=\left.\left(\begin{array}{l}
\cos(k_n\nu t) \\\cos(k_n\nu (t+T/3)) \\\cos(k_n\nu (t-T/3)\end{array}
\!\!\!\right)\!\!\!\!\right)
\end{equation}
and
\begin{equation}
(\mathcal{P}_\sigma', \mathcal{P}_\mathcal{C}', \mathcal{T}')=(0,1,-1)
\leftrightarrow
cs_n=\left.\left(\begin{array}{l}
\sin(k_n\nu t) \\\sin(k_n\nu (t+T/3)) \\\sin(k_n\nu (t-T/3)\end{array}
\!\!\!\right)\!\!\!\!\right),
\end{equation}
for $n=1,2,3,\dots,2N-1,2N$.
For zero-choreographic subspace, $(\mathcal{C}',\mathcal{T}')=(0,1)$ basis are
\begin{equation}
\begin{split}
&\left.\left(\begin{array}{r}1 \\-1/2 \\-1/2\end{array}\!\!\!\right)\!\!\!\!\right)\!\!,
\left.\left(\begin{array}{l}\cos(\nu t) \\\cos(\nu t -2\pi/3) \\\cos(\nu t+2\pi/3)\end{array}\!\!\!\right)\!\!\!\!\right)\!\!,
\left.\left(\begin{array}{l}\cos(2\nu t) \\\cos(2\nu t +2\pi/3) \\\cos(2\nu t-2\pi/3)\end{array}\!\!\!\right)\!\!\!\!\right)\!\!,
\left.\left(\begin{array}{l}\cos(3\nu t) \\\cos(3\nu t -2\pi/3) \\\cos(2\nu t+2\pi/3)\end{array}\!\!\!\right)\!\!\!\!\right),\\
&\left.\left(\begin{array}{l}\cos(3\nu t) \\\cos(3\nu t +2\pi/3) \\\cos(3\nu t-2\pi/3)\end{array}\!\!\!\right)\!\!\!\!\right)\!\!,
\left.\left(\begin{array}{l}\cos(4\nu t) \\\cos(4\nu t -2\pi/3) \\\cos(4\nu t+2\pi/3)\end{array}\!\!\!\right)\!\!\!\!\right)\!\!,
\left.\left(\begin{array}{l}\cos(5\nu t) \\\cos(5\nu t +2\pi/3) \\\cos(5\nu t-2\pi/3)\end{array}\!\!\!\right)\!\!\!\!\right)\!\!,
\left.\left(\begin{array}{l}\cos(6\nu t) \\\cos(6\nu t -2\pi/3) \\\cos(6\nu t+2\pi/3)\end{array}\!\!\!\right)\!\!\!\!\right),\\
&\vdots\\
&\hdots,
\left.\left(\begin{array}{l}\cos((3N-2)\nu t) \\\cos((3N-2)\nu t -2\pi/3) \\\cos((3N-2)\nu t+2\pi/3)\end{array}\!\!\!\right)\!\!\!\!\right)\!\!,
\left.\left(\begin{array}{l}\cos((3N-1)\nu t) \\\cos((3N-1)\nu t +2\pi/3) \\\cos((3N-1)\nu t-2\pi/3)\end{array}\!\!\!\right)\!\!\!\!\right)\!\!,
\left.\left(\begin{array}{l}\cos(3N\nu t) \\\cos(3N\nu t -2\pi/3) \\\cos(3N\nu t+2\pi/3)\end{array}\!\!\!\right)\!\!\!\!\right),\\
&\left.\left(\begin{array}{l}\cos(3N\nu t) \\\cos(3N\nu t +2\pi/3) \\\cos(3N\nu t-2\pi/3)\end{array}\!\!\!\right)\!\!\!\!\right).
\end{split}
\end{equation}
Using series $\ell_n=\{0,1,2,3,3,4,5,6,6,7,8,9,9,10,11,\dots,3N-2,2N-1,3N,3N\}$
which is generated by
\begin{equation}
\begin{split}
\ell_n=\frac{1}{8}\Big( 6n+(-1)^n+(1-i)(-i)^n+(1+i)i^n-3\Big)\\
\mbox{for }n=1,2,3,\dots,4N+1,
\end{split}
\end{equation}
the basis are
\begin{equation}
\label{ZC}
(\mathcal{P}_\sigma', \mathcal{P}_\mathcal{C}', \mathcal{T}')=(0,0,1)
\leftrightarrow
zc_n=\left.\left(\begin{array}{l}
\cos(\ell_n \nu t) \\
\cos(\ell_n \nu t -(-1)^n 2\pi/3) \\
\cos(\ell_n \nu t +(-1)^n 2\pi/3)
\end{array}\!\!\!\right)\!\!\!\!\right),
\end{equation}
for $n=1,2,3,\dots,4N+1$.
There are $4N+1$ basis.
Similarly,
\begin{equation}
\label{ZS}
(\mathcal{P}_\sigma', \mathcal{P}_\mathcal{C}', \mathcal{T}')=(0,0,-1)
\leftrightarrow
zs_n=(-1)^{n+1}\left.\left(\begin{array}{l}
\sin(\ell_n \nu t) \\
\sin(\ell_n \nu t -(-1)^n 2\pi/3) \\
\sin(\ell_n \nu t +(-1)^n 2\pi/3)
\end{array}\!\!\!\right)\!\!\!\!\right),
\end{equation}
for $n=1,2,3,\dots,4N+1$.
There are again $4N+1$ basis.
The reason of the factor $(-1)^{n+1}$ will be shown soon.
We can easily check the relation
\begin{equation}
\label{rotationForZCandZS}
\mathcal{C} \left(\begin{array}{c}zc_n \\zs_n\end{array}\right)
=\left(\begin{array}{rr}-1/2 & \sqrt{3}/2 \\-\sqrt{3}/2 & -1/2\end{array}\right)
\left(\begin{array}{c}zc_n \\zs_n\end{array}\right).
\end{equation}
Namely, the operator $\mathcal{C}$ makes $2\pi/3$ rotation
in the subspace spanned by $zc_n$ and $zs_n$ for each $n$.
To prove this relation,
it is useful to note that
the behavior of $n, \ell_n$ and $\ell_n \nu T/3=2\pi\ell_n/3$ are
summarized in the following table.
\begin{equation}
\begin{array}{ccccc}
n \bmod 4 & 1 & 2 & 3 & 0 \\
\ell_n \bmod 3 & 1 & 2 & 0 & 0 \\
2\pi\ell_n/3 \bmod 2\pi &2\pi/3 & -2\pi/3 & 0 & 0
\end{array}
\end{equation}
So, to prove the relation in (\ref{rotationForZCandZS}),
it is sufficient to prove the four cases $n=1,2,3,4$.
The factor in (\ref{ZS}) is chosen to satisfy (\ref{rotationForZCandZS}).
Now, consider a function $\Psi_+$ in $(\mathcal{P}_oC', \mathcal{P}_\mathcal{T}')=(0,1)$
and $\Psi_-$ in $(\mathcal{P}_oC', \mathcal{P}_\mathcal{T}')=(0,-1)$,
\begin{equation}
\Psi_{+}=\sum_n zc_n\ a_n,\ \Psi_{-}=\sum_n zs_n\ a_n
\end{equation}
with the same coefficients $a_n$.
Then, we have
\begin{equation}
\mathcal{C}\left(\begin{array}{c}\Psi_{+} \\\Psi_{-}\end{array}\right)
=\left(\begin{array}{rr}-1/2 & \sqrt{3}/2 \\-\sqrt{3}/2 & -1/2\end{array}\right)\left(\begin{array}{c}\Psi_{+} \\\Psi_{-}\end{array}\right).
\end{equation}
So, if $\Psi_{+}$ belongs to $\mathcal{H}'=\lambda$ then $\Psi_{-}$ belongs to the same $\lambda$,
because $\mathcal{C}(\mathcal{H}-\lambda)\Psi_{+}=0$ implies
\begin{equation}
0
= (\mathcal{H}-\lambda)\mathcal{C}\Psi_{+}=(\mathcal{H}-\lambda)\left(-\frac{1}{2}\Psi_{+} + \frac{\sqrt{3}}{2}\Psi_{-}\right)
=\frac{\sqrt{3}}{2}(\mathcal{H}-\lambda)\Psi_{-}.
\end{equation}
Inverse is also true. This proves explicitly the double degeneracy of zero-choreographic subspace.
\subsection{Decomposition by $\mathcal{P}_\mathcal{M}$}
For integer $m$,
$\cos(m\nu (t+T/2))=\cos(m\nu t + m \pi)=(-1)^m \cos(m\nu t)$ and
$\sin(m\nu (t+T/2))=(-1)^m \sin(m\nu t)$.
So, for $x$ and $y$ components,
\begin{equation}
\mathcal{M}'=1
\leftrightarrow
\left(\begin{array}{cc}
\cos((2m+1)\nu t) & 0 \\
0 & \cos(2m' \nu t)
\end{array}\right)\!\!,
\left(\begin{array}{cc}
\sin((2m+1)\nu t) & 0 \\
0 & \sin(2m' \nu t)
\end{array}\right)
\end{equation}
and
\begin{equation}
\mathcal{M}'=-1
\leftrightarrow
\left(\begin{array}{cc}
\cos(2m\nu t) & 0 \\
0 & \cos((2m'+1)\nu t)
\end{array}\right)\!\!,
\left(\begin{array}{cc}
\sin(2m\nu t) & 0 \\
0 & \sin((2m'+1) \nu t)
\end{array}\right).
\end{equation}
Decompose $k_n$
\begin{equation}
k_n
=\{1,2,4,5,7,8,10,11,13,14,16,\dots,3N-2,3N-1\}
\end{equation}
into odd set and even set,
\begin{equation}
\begin{split}
k^o_n
&=\{1,5,7,11,13,17,\dots,3N-1\}\\
&=\frac{1}{2}\Big(6n+(-1)^n -3\Big),\ n=1,2,3,\dots, N
\end{split}
\end{equation}
and
\begin{equation}
\begin{split}
k^e_n
&=\{2,4,8,10,14,16,\dots,3N-2\}\\
&=\frac{1}{2}\Big(6n-(-1)^n -3\Big),\ n=1,2,3,\dots, N.
\end{split}
\end{equation}
Since $k^o_n \bmod 3=\{1,2,1,2,\dots\}$
and $k^e_n \bmod 3=\{2,1,2,1\dots\}$,
\begin{equation}
\label{sink2pidiv3}
\sin(k^o_n 2\pi/3)=(-1)^{n+1}\frac{\sqrt{3}}{2}
\mbox{ and }
\sin(k^e_n 2\pi/3)=(-1)^{n}\frac{\sqrt{3}}{2}.
\end{equation}
Similarly,
\begin{equation}
\ell_n
=\{0,1,2,3,3,4,5,6,6,7,\dots,3N-2,3N-1,3N,3N\}
\end{equation}
into odd set and even set,
\begin{equation}
\begin{split}
\ell^o_n
&=\{1,3,3,5,7,9,9,11,13,15,15,\dots,3N-1\}\\
&=\frac{1}{4}\Big(6n+(-1)^n -(1-i)(-i)^n-(1+i)i^n-3\Big),\ n=1,2,3,\dots, 2N
\end{split}
\end{equation}
and
\begin{equation*}
\begin{split}
\ell^e_n
&=\{0,2,4,6,6,8,10,12,12,\dots,3N-2,3N,3N\}\\
&=\frac{1}{4}\Big(6n+(-1)^n+(1-i)(-i)^n+(1+i)i^n-3\Big),\ n=1,2,3,\dots,2N+1.
\end{split}
\end{equation*}
\emph{To make the number of elements of $\ell^o_n$ and $\ell^e_n$ equal,
we just omit the last element of $\ell^e_n$.
} So, we use
\begin{equation}
\begin{split}
\ell^e_n
&=\{0,2,4,6,6,8,10,12,12,\dots,3N-2,3N\}\\
&=\frac{1}{4}\Big(6n+(-1)^n+(1-i)(-i)^n+(1+i)i^n-3\Big),\ n=1,2,3,\dots,2N.
\end{split}
\end{equation}
The series with $\ell^o_n \nu t$ in (\ref{ZC}) is
\begin{equation}
\label{seriesNCOddn}
\begin{split}
\left(\begin{array}{l}\cos(\nu t) \\\cos(\nu t-2\pi/3) \\\cos(\nu t+2\pi/3)\end{array}\right),&\
\left(\begin{array}{l}\cos(3\nu t) \\\cos(3\nu t-2\pi/3) \\\cos(3\nu t+2\pi/3)\end{array}\right),\
\left(\begin{array}{l}\cos(3\nu t) \\\cos(3\nu t+2\pi/3) \\\cos(2\nu t-2\pi/3)\end{array}\right),\
\left(\begin{array}{l}\cos(5\nu t) \\\cos(5\nu t+2\pi/3) \\\cos(5\nu t-2\pi/3)\end{array}\right),\\
\left(\begin{array}{l}\cos(7\nu t) \\\cos(7\nu t-2\pi/3) \\\cos(7\nu t+2\pi/3)\end{array}\right),&\
\left(\begin{array}{l}\cos(9\nu t) \\\cos(9\nu t-2\pi/3) \\\cos(9\nu t+2\pi/3)\end{array}\right),\
\left(\begin{array}{l}\cos(9\nu t) \\\cos(9\nu t+2\pi/3) \\\cos(9\nu t-2\pi/3)\end{array}\right),\
\left(\begin{array}{l}\cos(11\nu t) \\\cos(11\nu t+2\pi/3) \\\cos(11\nu t-2\pi/3)\end{array}\right),\\
\dots\\
&=
\left(\begin{array}{l}
\cos(\ell^o_n\nu t) \\
\cos(\ell^o_n\nu t+(-1)^{\lfloor (n+1)/2 \rfloor} 2\pi/3) \\
\cos(\ell^o_n\nu t-(-1)^{\lfloor (n+1)/2 \rfloor} 2\pi/3)
\end{array}\right)
\mbox{ for } n=1,2,3,\dots, 2N.
\end{split}
\end{equation}
Similarly, the series with $\ell^e_n \nu t$ in (\ref{ZC}) is
\begin{equation}
\label{seriesNCEvenn}
\begin{split}
\left(\begin{array}{r}1 \\-1/2 \\-1/2\end{array}\right),&\
\left(\begin{array}{l}\cos(2\nu t) \\\cos(2\nu t+2\pi/3) \\\cos(2\nu t-2\pi/3)\end{array}\right),\
\left(\begin{array}{l}\cos(4\nu t) \\\cos(4\nu t-2\pi/3) \\\cos(4\nu t+2\pi/3)\end{array}\right),\
\left(\begin{array}{l}\cos(6\nu t) \\\cos(6\nu t-2\pi/3) \\\cos(6\nu t+2\pi/3)\end{array}\right),\\
\left(\begin{array}{l}\cos(6\nu t) \\\cos(6\nu t+2\pi/3) \\\cos(6\nu t-2\pi/3)\end{array}\right),&\
\left(\begin{array}{l}\cos(8\nu t) \\\cos(8\nu t+2\pi/3) \\\cos(8\nu t-2\pi/3)\end{array}\right),\
\left(\begin{array}{l}\cos(10\nu t) \\\cos(10\nu t-2\pi/3) \\\cos(10\nu t+2\pi/3)\end{array}\right),\
\left(\begin{array}{l}\cos(12\nu t) \\\cos(12\nu t-2\pi/3) \\\cos(12\nu t+2\pi/3)\end{array}\right),\\
\dots\\
&=
\left(\begin{array}{l}
\cos(\ell^e_n\nu t) \\
\cos(\ell^e_n\nu t-(-1)^{\lfloor (n+1)/2 \rfloor} 2\pi/3) \\
\cos(\ell^e_n\nu t+(-1)^{\lfloor (n+1)/2 \rfloor} 2\pi/3)
\end{array}\right)
\mbox{ for } n=1,2,3,\dots, 2N.
\end{split}
\end{equation}
\subsection{Summary for the base functions}
\subsubsection{Choreographic subspace}
Using $k^o_n$ and $k^e_n$,
\begin{equation}
(\mathcal{P}_\sigma', \mathcal{P}_\mathcal{C}',\mathcal{T}',\mathcal{M}')=(0,1,1,1)\leftrightarrow
cc_n^+
=\left(\begin{array}{ll}
\cos(k^o_n \nu t) & 0 \\
0 & \cos(k^e_n \nu t) \\
\cos(k^o_n \nu (t+T/3)) & 0 \\
0 & \cos(k^e_n \nu (t+T/3)) \\
\cos(k^o_n \nu (t-T/3)) & 0 \\
0 & \cos(k^e_n \nu (t-T/3)) \\
\end{array}\right),
\end{equation}
and
\begin{equation}
(\mathcal{P}_\sigma', \mathcal{P}_\mathcal{C}',\mathcal{T}',\mathcal{M}')=(0,1,1,-1)\leftrightarrow
cc_n^-
=\left(\begin{array}{ll}
\cos(k^e_n \nu t) & 0 \\
0 & \cos(k^o_n \nu t) \\
\cos(k^e_n \nu (t+T/3)) & 0 \\
0 & \cos(k^o_n \nu (t+T/3)) \\
\cos(k^e_n \nu (t-T/3)) & 0 \\
0 & \cos(k^o_n \nu (t-T/3)) \\
\end{array}\right),
\end{equation}
for $n=1,2,3,\dots,N$.
Similarly,
\begin{equation}
(\mathcal{P}_\sigma', \mathcal{P}_\mathcal{C}',\mathcal{T}',\mathcal{M}')=(0,1,-1,1)\leftrightarrow
cs_n^+
=\left(\begin{array}{ll}
\sin(k^o_n \nu t) & 0 \\
0 & \sin(k^e_n \nu t) \\
\sin(k^o_n \nu (t+T/3)) & 0 \\
0 & \sin(k^e_n \nu (t+T/3)) \\
\sin(k^o_n \nu (t-T/3)) & 0 \\
0 & \sin(k^e_n \nu (t-T/3)) \\
\end{array}\right)
\end{equation}
and
\begin{equation}
(\mathcal{P}_\sigma', \mathcal{P}_\mathcal{C}',\mathcal{T}',\mathcal{M}')=(0,1,-1,-1)\leftrightarrow
cs_n^-
=\left(\begin{array}{ll}
\sin(k^e_n \nu t) & 0 \\
0 & \sin(k^o_n \nu t) \\
\sin(k^e_n \nu (t+T/3)) & 0 \\
0 & \sin(k^o_n \nu (t+T/3)) \\
\sin(k^e_n \nu (t-T/3)) & 0 \\
0 & \sin(k^o_n \nu (t-T/3)) \\
\end{array}\right)
\end{equation}
for $n=1,2,3,\dots,N$.
It may useful to define a function
\begin{equation}
\Phi_c(f, k^x, k^y, n)
=\left(\begin{array}{ll}
f(k^x_n\nu t) & 0 \\
0 & f(k^x_n\nu t) \\
f(k^x_n\nu (t+T/3)) & 0 \\
0 & f(k^x_n\nu (t+T/3))\\
f(k^x_n\nu (t-T/3)) & 0 \\
0 & f(k^x_n\nu (t-T/3)t)
\end{array}\right).
\end{equation}
Then,
\begin{equation}
\begin{array}{ll}
cc^+_n=\Phi_c(\cos, k^o, k^e,n),&cc^-_n=\Phi_c(\cos, k^e, k^o,n),\\
cs^+_n=\Phi_c(\sin, k^o, k^e,n),&cc^-_n=\Phi_c(\sin, k^e, k^o,n)
\end{array}
\end{equation}
\subsubsection{Zero-choreographic subspace}
\begin{equation}
\begin{split}
&(\mathcal{P}_\sigma', \mathcal{P}_\mathcal{C}', \mathcal{T}',\mathcal{M}')=(0,0,1,1)\leftrightarrow\\
&zc^+_n=\left(\begin{array}{ll}
\cos(\ell^o_n\nu t) & 0 \\
0 & \cos(\ell^e_n\nu t) \\
\cos(\ell^o_n\nu t+(-1)^{\lfloor (n+1)/2\rfloor} 2\pi/3) & 0 \\
0 & \cos(\ell^e_n\nu t-(-1)^{\lfloor (n+1)/2\rfloor} 2\pi/3) \\
\cos(\ell^o_n\nu t-(-1)^{\lfloor (n+1)/2\rfloor} 2\pi/3) & 0 \\
0 & \cos(\ell^e_n\nu t+(-1)^{\lfloor (n+1)/2\rfloor} 2\pi/3)
\end{array}\right),
\end{split}
\end{equation}
and
\begin{equation}
\begin{split}
&(\mathcal{P}_\sigma', \mathcal{P}_\mathcal{C}', \mathcal{T}',\mathcal{M}')=(0,0,1,-1)\leftrightarrow\\
&zc^-_n=\left(\begin{array}{ll}
\cos(\ell^e_n\nu t) & 0 \\
0 & \cos(\ell^o_n\nu t) \\
\cos(\ell^e_n\nu t-(-1)^{\lfloor (n+1)/2\rfloor} 2\pi/3) & 0 \\
0 & \cos(\ell^o_n\nu t+(-1)^{\lfloor (n+1)/2\rfloor} 2\pi/3) \\
\cos(\ell^e_n\nu t+(-1)^{\lfloor (n+1)/2\rfloor} 2\pi/3) & 0 \\
0 & \cos(\ell^o_n\nu t-(-1)^{\lfloor (n+1)/2\rfloor} 2\pi/3)
\end{array}\right)
\end{split}
\end{equation}
for $n=1,2,3,\dots, 2N$.
Similarly,
\begin{equation}
\begin{split}
&(\mathcal{P}_\sigma', \mathcal{P}_\mathcal{C}', \mathcal{T}',\mathcal{M}')=(0,0,-1,1)\leftrightarrow\\
&zs^+_n=(-1)^{n+1}\left(\begin{array}{ll}
\sin(\ell^o_n\nu t) & 0 \\
0 & \sin(\ell^e_n\nu t) \\
\sin(\ell^o_n\nu t+(-1)^{\lfloor (n+1)/2\rfloor} 2\pi/3) & 0 \\
0 & \sin(\ell^e_n\nu t-(-1)^{\lfloor (n+1)/2\rfloor} 2\pi/3) \\
\sin(\ell^o_n\nu t-(-1)^{\lfloor (n+1)/2\rfloor} 2\pi/3) & 0 \\
0 & \sin(\ell^e_n\nu t+(-1)^{\lfloor (n+1)/2\rfloor} 2\pi/3)
\end{array}\right),
\end{split}
\end{equation}
and
\begin{equation}
\begin{split}
&(\mathcal{P}_\sigma', \mathcal{P}_\mathcal{C}', \mathcal{T}',\mathcal{M}')=(0,0,-1,-1)\leftrightarrow\\
&zs^-_n=(-1)^{n+1}\left(\begin{array}{ll}
\sin(\ell^e_n\nu t) & 0 \\
0 & \sin(\ell^o_n\nu t) \\
\sin(\ell^e_n\nu t-(-1)^{\lfloor (n+1)/2\rfloor} 2\pi/3) & 0 \\
0 & \sin(\ell^o_n\nu t+(-1)^{\lfloor (n+1)/2\rfloor} 2\pi/3) \\
\cos(\ell^e_n\nu t+(-1)^{\lfloor (n+1)/2\rfloor} 2\pi/3) & 0 \\
0 & \sin(\ell^o_n\nu t-(-1)^{\lfloor (n+1)/2\rfloor} 2\pi/3).
\end{array}\right)
\end{split}
\end{equation}
for $n=1,2,3,\dots, 2N$.
It may useful to define a function
\begin{equation}
\Psi_z(f, \ell^x, \ell^y, \epsilon, n)
=\left(\begin{array}{ll}
f(\ell^x_n \nu t) & 0 \\
0 & f(\ell^y_n \nu t) \\
f(\ell^x_n \nu t + \epsilon (-1)^{\lfloor (n+1)/2\rfloor} 2\pi/3) & 0\\
0 & f(\ell^y_n \nu t -\epsilon (-1)^{\lfloor (n+1)/2\rfloor} 2\pi/3)\\
f(\ell^x_n \nu t - \epsilon (-1)^{\lfloor (n+1)/2\rfloor} 2\pi/3) & 0\\
0 & f(\ell^y_n \nu t +\epsilon (-1)^{\lfloor (n+1)/2\rfloor} 2\pi/3)
\end{array}\right).
\end{equation}
Then, the base functions are
\begin{equation}
\begin{array}{ll}
zc^+_n=\Psi(\cos, \ell^o, \ell^e, 1,n),\
&zc^-_n=\Psi(\cos, \ell^e, \ell^o, -1,n),\\
zs^+_n=(-1)^{n+1}\Psi(\sin, \ell^o, \ell^e, 1,n),\
&zs^-_n=(-1)^{n+1}\Psi(\sin, \ell^e, \ell^o, -1,n).
\end{array}
\end{equation}
\section{Matrix elements of Hessian}
\subsection{Definition of matrix elements}
Let $\phi_n$ be $6 \times 2$ matrix that describes bases
and $a_n$ be $2$ column vector that describes degrees of freedom.
For,
\begin{equation}
\phi_n = \left(\begin{array}{cc}
\delta q_{0n}^x & 0 \\
0 & \delta q_{0n}^y \\
\delta q_{1n}^x & 0 \\
0 & \delta q_{1n}^y \\
\delta q_{2n}^x & 0 \\
0 & \delta q_{2n}^y
\end{array}
\right)
\
\mbox{ and }
a_n=\left(\begin{array}{c}a_{n}^x \\a_{n}^y\end{array}\right),
\end{equation}
the product $\phi_n a_n$ represents the vector
\begin{equation}
\phi_n a_n
=\left(\begin{array}{c}
\delta q_{0n}^x a_{n}^x\\
\delta q_{0n}^y a_{n}^y\\
\delta q_{1n}^x a_{n}^x\\
\delta q_{1n}^y a_{n}^y\\
\delta q_{2n}^x a_{n}^x\\
\delta q_{2n}^y a_{n}^y
\end{array}\right).
\end{equation}
Note that we have only two degrees of freedom $a_{n}^x$ and $a_{n}^y$ for
given $n$.
Now, consider the eigenvalue problem
\begin{equation}
\mathcal{H} \sum_{n=1,2,3,\dots} \phi_n a_n = \lambda \sum_{n=1,2,3,\dots} \phi_n a_n.
\end{equation}
Substituting the expression
\begin{equation}
\sum_{n=1,2,3,\dots} \phi_n a_n
=(\phi_1, \phi_2, \phi_3, \dots)\left(\begin{array}{c}a_1 \\ a_2 \\ a_3 \\\vdots\end{array}\right)
\end{equation}
into the eigenvalue problem,
and multiply ${}^t\!(\phi_1, \phi_2, \phi_3, \dots)$ from the left
and taking the inner product,
we get the eigenvalue problem in a matrix representation
\begin{equation}
M_\mathcal{H} \left(\begin{array}{c}a_1 \\ a_2 \\ a_3 \\\vdots\end{array}\right)
=\lambda \left(\begin{array}{c}a_1 \\ a_2 \\ a_3 \\\vdots\end{array}\right).
\end{equation}
Where $M_\mathcal{H}$ is the matrix representation of the Hessian,
\begin{equation}
M_\mathcal{H}=
\left(\begin{array}{cccc}
\braket{\phi_1|\mathcal{H}|\phi_1} & \braket{\phi_1|\mathcal{H}|\phi_2} & \braket{\phi_1|\mathcal{H}|\phi_3} & \dots\\
\braket{\phi_2|\mathcal{H}|\phi_1} & \braket{\phi_2|\mathcal{H}|\phi_2} & \braket{\phi_2|\mathcal{H}|\phi_3} & \dots\\
\braket{\phi_3|\mathcal{H}|\phi_1} & \braket{\phi_3|\mathcal{H}|\phi_2} & \braket{\phi_3|\mathcal{H}|\phi_3} & \dots\\
\vdots & \vdots & \vdots & \ddots
\end{array}\right).
\end{equation}
The inner product is defined by (\ref{defOfInnerProduct}).
\subsection{Preparations}
Since we are considering the Hessian
at a choreography $q_0(t)=q(t)$, $q_1(t)=q(t+T/3)$ and $q_2(t)=q(t-T/3)$,
the function $u_{ij}$ satisfy
$u_{20}(t)=u_{12}(t+T/3)$ and $u_{01}(t)=u_{12}(t+T/3)$.
Let $u_{12}(t)=u(t)$, then $u_{20}(t)=R^{1/3}u(t)R^{-1/3}$ and
$u_{01}(t)=R^{-1/3}u(t)R^{1/3}$.
Then, ${}^t\!\Delta \mathcal{U} \Delta$ in (\ref{defOfHessian}) is
\begin{equation}
{}^t\!\Delta \mathcal{U} \Delta
={}^t\!\Delta
\left(\begin{array}{ccc}
1 & 0 & 0 \\
0 & R^{1/3} & 0 \\
0 & 0 & R^{-1/3}
\end{array}\right)
\left(\begin{array}{ccc}
u & 0 & 0 \\
0 & u & 0 \\
0 & 0 & u
\end{array}\right)
\left(\begin{array}{ccc}
1 & 0 & 0 \\
0 & R^{-1/3} & 0 \\
0 & 0 & R^{1/3}
\end{array}\right)
\Delta.
\end{equation}
In the following subsections, we use the notation
\begin{equation}
u=\left(\begin{array}{cc}u^{xx} & u^{xy} \\u^{yx} & u^{yy}\end{array}\right).
\end{equation}
We will use the lower case letters $a, b, \dots$ to express $x$ or $y$.
\subsection{Elements for choreographic subspace}
For the matrix elements for choreographic subspace,
consider
\begin{equation}
\begin{split}
&\left(\begin{array}{ccc}
1 & 0 & 0 \\
0 & R^{-1/3} & 0 \\
0 & 0 & R^{1/3}
\end{array}\right)
\Delta
\left(\begin{array}{l}
\cos(k\nu t) \\\cos(k\nu(t+T/3)) \\\cos(k\nu(t-T/3))\end{array}
\right)\\
&=\left(\begin{array}{ccc}
1 & 0 & 0 \\
0 & R^{-1/3} & 0 \\
0 & 0 & R^{1/3}
\end{array}\right)
\left(\begin{array}{l}
\cos(k\nu(t+T/3))-\cos(k\nu(t-T/3)) \\
\cos(k\nu(t-T/3))-\cos(k\nu t) \\
\cos(k\nu t)-\cos(k\nu(t+T/3))
\end{array}\right)\\
&=\big(\cos(k\nu(t+T/3))-\cos(k\nu(t-T/3))\big)\left(\begin{array}{c}1 \\1 \\1\end{array}\right)\\
&=-2 \sin(2\pi k/3)\sin(k\nu t){}^t\!(1,1,1).
\end{split}
\end{equation}
Similarly,
\begin{equation}
\begin{split}
&\left(\begin{array}{ccc}
1 & 0 & 0 \\
0 & R^{-1/3} & 0 \\
0 & 0 & R^{1/3}
\end{array}\right)
\Delta
\left(\begin{array}{l}
\sin(k\nu t) \\\sin(k\nu(t+T/3)) \\\sin(k\nu(t-T/3))\end{array}
\right)\\
&=\big(\sin(k\nu(t+T/3))-\sin(k\nu(t-T/3))\big) {}^t\!(1,1,1)\\
&=2 \sin(2\pi k/3)\cos(k\nu t) {}^t\!(1,1,1).
\end{split}
\end{equation}
Therefore,
$ab$ component of
$\braket{\Phi_c(\cos,k^x,k^y,m|{}^t\!\Delta\mathcal{U}\Delta|\Phi(k^x,k^y,n)}$
is
\begin{equation}
\begin{split}
&\braket{\Phi_c(\cos,k^x,k^y,m|{}^t\!\Delta\mathcal{U}\Delta|\Phi(k^x,k^y,n)}^{ab}\\
&=\sin(2\pi k^a_m/3)\sin(2\pi k^b_n/3)
\times\frac{8}{T}\int_0^T dt\ u^{ab}(t)
\sin(k^a_m\nu t)\sin(k^b_n\nu t)
\end{split}
\end{equation}
The relation (\ref{sink2pidiv3}) yields
\begin{equation}
\begin{split}
&\sin(2\pi k^o_m/3)\sin(2\pi k^o_n/3)
=\sin(2\pi k^e_m/3)\sin(2\pi k^e_n/3)
=(-1)^{m+n}\frac{3}{4},\\
&\sin(2\pi k^o_m/3)\sin(2\pi k^e_n/3)
=-(-1)^{m+n}\frac{3}{4}.
\end{split}
\end{equation}
So, we finally get the matrix elements
\begin{equation}
\label{uforCCPlus}
\begin{split}
&\braket{cc_m^+|{}^t\!\Delta \mathcal{U} \Delta |cc_n^+}\\
&=(-1)^{m+n}\frac{6}{T}\int_0^T dt
\left(\begin{array}{rr}
u^{xx}\sin(k^o_m\nu t)\sin(k^o_n\nu t) & -u^{xy}\sin(k^o_m\nu t)\sin(k^e_n\nu t) \\
-u^{xy}\sin(k^e_m\nu t)\sin(k^o_n\nu t) & u^{yy}\sin(k^e_m\nu t)\sin(k^e_n\nu t)
\end{array}\right),
\end{split}
\end{equation}
\begin{equation}
\label{uforCCMinus}
\begin{split}
&\braket{cc_m^-|{}^t\!\Delta \mathcal{U} \Delta |cc_n^-}\\
&=(-1)^{m+n}\frac{6}{T}\int_0^T dt
\left(\begin{array}{rr}
u^{xx}\sin(k^e_m\nu t)\sin(k^e_n\nu t) & -u^{xy}\sin(k^e_m\nu t)\sin(k^o_n\nu t) \\
-u^{xy}\sin(k^o_m\nu t)\sin(k^e_n\nu t) & u^{yy}\sin(k^o_m\nu t)\sin(k^o_n\nu t)
\end{array}\right),
\end{split}
\end{equation}
\begin{equation}
\label{uforCSPlus}
\begin{split}
&\braket{cs_m^+|{}^t\!\Delta \mathcal{U} \Delta |cs_n^+}\\
&=(-1)^{m+n}\frac{6}{T}\int_0^T dt
\left(\begin{array}{rr}
u^{xx}\cos(k^o_m\nu t)\cos(k^o_n\nu t) & -u^{xy}\cos(k^o_m\nu t)\cos(k^e_n\nu t) \\
-u^{xy}\cos(k^e_m\nu t)\cos(k^o_n\nu t) & u^{yy}\cos(k^e_m\nu t)\cos(k^e_n\nu t)
\end{array}\right),
\end{split}
\end{equation}
and
\begin{equation}
\label{uforCSMinus}
\begin{split}
&\braket{cs_m^-|{}^t\!\Delta \mathcal{U} \Delta |cs_n^-}\\
&=(-1)^{m+n}\frac{6}{T}\int_0^T dt
\left(\begin{array}{rr}
u^{xx}\cos(k^e_m\nu t)\cos(k^e_n\nu t) & -u^{xy}\cos(k^e_m\nu t)\cos(k^o_n\nu t) \\
-u^{xy}\cos(k^o_m\nu t)\cos(k^e_n\nu t) & u^{yy}\cos(k^o_m\nu t)\cos(k^o_n\nu t)
\end{array}\right).
\end{split}
\end{equation}
The differences between the above elements are
only the place of $k^o$ and $k^e$.
\subsection{Elements for zero-choreographic subspace}
To calculate the $ab$ element of
$\braket{zc^+_m|{}^t\!\Delta\mathcal{U}\Delta|zc^+}$,
let $\epsilon^x=(-1)^{\lfloor (n+1)/2\rfloor}$
and $\epsilon^y=-(-1)^{\lfloor (n+1)/2\rfloor}$.
Then
\begin{equation}
\begin{split}
&\left(\begin{array}{ccc}1 & 0 & 0 \\0 & R^{-1/3} & 0 \\0 & 0 & R^{1/3}\end{array}\right)
\!\Delta
\left(\begin{array}{l}
\cos(\ell^a_n \nu t) \\
\cos(\ell^a_n \nu t + 2\pi \epsilon^a_n/3) \\
\cos(\ell^a_n \nu t - 2\pi \epsilon^a_n/3)
\end{array}\right)\\
&=\sqrt{3}\epsilon^a_n
\left(\begin{array}{l}
\sin(\ell^a_n \nu t)\\
\sin(\ell^a_n \nu (t-T/3) - \pi \epsilon^a_n/3)\\
\sin(\ell^a_n \nu (t+T/3)+\pi \epsilon^a_n/3)
\end{array}\right).
\end{split}
\end{equation}
Therefore,
\begin{equation}
\begin{split}
&\braket{zc^+_m|{}^t\!\Delta \mathcal{U} \Delta|zc^+_n}^{ab}\\
=&
\epsilon^a_m \epsilon^b_n
\frac{1}{T}\int_0^T dt\ u^{ab}(t)\Bigg(
\cos\Big((\ell^a_m-\ell^b_n)\nu t\big)
- \cos\big((\ell^a_m+\ell^b_n)\nu t\big)\\
&+\cos\Big((\ell^a_m-\ell^b_n)\nu (t-T/3)-(\epsilon^a_m-\epsilon^b_n)\pi/3\Big)
-\cos\Big((\ell^a_m+\ell^b_n)\nu (t-T/3)-(\epsilon^a_m+\epsilon^b_n)\pi/3\Big)\\
&+\cos\Big((\ell^a_m-\ell^b_n)\nu (t+T/3)+(\epsilon^a_m-\epsilon^b_n)\pi/3\Big)
-\cos\Big((\ell^a_m+\ell^b_n)\nu (t+T/3)+(\epsilon^a_m+\epsilon^b_n)\pi/3\Big)
\Bigg)\\
=&\epsilon^a_m \epsilon^b_n \frac{3}{T}\int_0^T dt\
u(t)L(\ell^a, \ell^b, \epsilon^a, \epsilon^b, m, n).
\end{split}
\end{equation}
The term $L(\ell^a, \ell^b, \epsilon^a, \epsilon^b, m, n)$ turns out to be
\begin{equation}
L(\ell^a, \ell^b, \epsilon^a, \epsilon^b, m, n)
=(-1)^{m+n}\cos\Big( ((-1)^m \ell^a_m -(-1)^n\ell^b_n)\nu t).
\end{equation}
Using
$(-1)^{\lfloor (n+1)/2 \rfloor}(-1)^n
=1,-1,-1,1,1,-1,-1,1,1,\dots=(-1)^{\lfloor n/2 \rfloor}$.
Finally, we get
\begin{equation}
\begin{split}
&\braket{nc^+_m|{}^t\!\Delta \mathcal{U} \Delta| nc^+_n}
=(-1)^{\lfloor m/2 \rfloor + \lfloor n/2 \rfloor} \frac{3}{T}\\
&\times\int_0^T \!dt
\left(\begin{array}{rr}
u^{xx}(t)\cos\Big(((-1)^m\ell^o_m-(-1)^n\ell^o_n)\nu t\Big) &
-u^{xy}(t)\cos\Big(((-1)^m\ell^o_m-(-1)^n\ell^e_n)\nu t\Big) \\
-u^{yx}(t)\cos\Big(((-1)^m\ell^e_m-(-1)^n\ell^o_n)\nu t\Big) &
u^{yy}(t)\cos\Big(((-1)^m\ell^e_m-(-1)^n\ell^e_n)\nu t\Big)\end{array}
\right).
\end{split}
\end{equation}
Exchanging $\ell^o \leftrightarrow \ell^e$,
we get $\braket{nc^-_m|{}^t\!\Delta \mathcal{U} \Delta| nc^-_n}$,
\begin{equation}
\begin{split}
&\braket{nc^-_m|{}^t\!\Delta \mathcal{U} \Delta| nc^-_n}
=(-1)^{\lfloor m/2 \rfloor + \lfloor n/2 \rfloor} \frac{3}{T}\\
&\times\int_0^T \!dt
\left(\begin{array}{rr}
u^{xx}(t)\cos\Big(((-1)^m\ell^e_m-(-1)^n\ell^e_n)\nu t\Big) &
-u^{xy}(t)\cos\Big(((-1)^m\ell^e_m-(-1)^n\ell^o_n)\nu t\Big) \\
-u^{yx}(t)\cos\Big(((-1)^m\ell^o_m-(-1)^n\ell^e_n)\nu t\Big) &
u^{yy}(t)\cos\Big(((-1)^m\ell^o_m-(-1)^n\ell^o_n)\nu t\Big)\end{array}
\right).
\end{split}
\end{equation}
Similarly,
\begin{equation}
\begin{split}
&\left(\begin{array}{ccc}1 & 0 & 0 \\0 & R^{-1/3} & 0 \\0 & 0 & R^{1/3}\end{array}\right)
\!\Delta
\left(\begin{array}{l}
\sin(\ell^a_n \nu t) \\
\sin(\ell^a_n \nu t + 2\pi \epsilon^a_n/3) \\
\sin(\ell^a_n \nu t - 2\pi \epsilon^a_n/3)
\end{array}\right)\\
&=\sqrt{3}\epsilon^a_n
\left(\begin{array}{l}
\cos(\ell^a_n \nu t)\\
-\cos(\ell^a_n \nu (t-T/3) - \pi \epsilon^a_n/3)\\
-\cos(\ell^a_n \nu (t+T/3)+\pi \epsilon^a_n/3)
\end{array}\right).
\end{split}
\end{equation}
Therefore,
\begin{equation}
\begin{split}
&\braket{zs^+_m|{}^t\!\Delta \mathcal{U} \Delta|zs^+_n}^{ab}\\
=&
(-1)^{m+n}
\epsilon^a_m \epsilon^b_n
\frac{1}{T}\int_0^T dt\ u^{ab}(t)\Bigg(
\cos\Big((\ell^a_m-\ell^b_n)\nu t\big)
+ \cos\big((\ell^a_m+\ell^b_n)\nu t\big)\\
&+\cos\Big((\ell^a_m-\ell^b_n)\nu (t-T/3)-(\epsilon^a_m-\epsilon^b_n)\pi/3\Big)
+\cos\Big((\ell^a_m+\ell^b_n)\nu (t-T/3)-(\epsilon^a_m+\epsilon^b_n)\pi/3\Big)\\
&+\cos\Big((\ell^a_m-\ell^b_n)\nu (t+T/3)+(\epsilon^a_m-\epsilon^b_n)\pi/3\Big)
+\cos\Big((\ell^a_m+\ell^b_n)\nu (t+T/3)+(\epsilon^a_m+\epsilon^b_n)\pi/3\Big)
\Bigg)\\
=&(-1)^{m+n}\epsilon^a_m \epsilon^b_n \frac{3}{T}\int_0^T dt\
u(t)L'(\ell^a, \ell^b, \epsilon^a, \epsilon^b, m, n).
\end{split}
\end{equation}
The term $L'(\ell^a, \ell^b, \epsilon^a, \epsilon^b, m, n)$ turns out to be
\begin{equation}
L'(\ell^a, \ell^b, \epsilon^a, \epsilon^b, m, n)
=\cos\Big( ((-1)^m \ell^a_m -(-1)^n\ell^b_n)\nu t).
\end{equation}
Thus, we directly get
\begin{equation}
\begin{split}
\braket{zs^+_m|{}^t\!\Delta\mathcal{U}\Delta|zs^+_n}
&=\braket{zc^+_m|{}^t\!\Delta\mathcal{U}\Delta|zc^+_n},\\
\braket{zs^-_m|{}^t\!\Delta\mathcal{U}\Delta|zs^-_n}
&=\braket{zc^-_m|{}^t\!\Delta\mathcal{U}\Delta|zc^-_n}.
\end{split}
\end{equation}
This is a direct proof of the double degeneracy of $\mathcal{H}$
in $\mathcal{P}_\mathcal{C}'=0$ subspace.
\subsection{Summary for the matrix elements}
Let $\tilde{u}^{ab}(k)$ be
\begin{equation}
\left(\begin{array}{cc}
\tilde{u}^{xx}(k) & \tilde{u}^{xy}(k) \\
\tilde{u}^{yx}(k) & \tilde{u}^{yy}(k)
\end{array}\right)
=\frac{1}{T}\int_0^T \!\!dt\
\left(\begin{array}{cc}
u^{xx}(t) & u^{xy}(t) \\
u^{yx}(t) & u^{yy}(t)
\end{array}\right)
\cos(k\nu t),
\end{equation}
$k=0,1,2,\dots,6N-1$, $N=2^M$.
\subsubsection{Choreographic subspace}
Let $C(k^x, k^y,\epsilon, m,n)$ be
\begin{equation}
C(k^x, k^y,\epsilon, m,n)
=3(-1)^{m+n}\left(\begin{array}{rr}
\tilde{u}^{xx}(k^x+\epsilon k^x) & -\tilde{u}^{xy}(k^x+\epsilon k^y)\\
-\tilde{u}^{yx}(k^y+\epsilon k^x) & \tilde{u}^{yy}(k^y+\epsilon k^y)
\end{array}\right).
\end{equation}
Subspace: $(\mathcal{P}_\mathcal{C}',\mathcal{P}_\mathcal{T}',\mathcal{P}_\mathcal{M}')=(0,1,1)$,\\
name: Choreographic cos$+$,\\
symbol: $cc^+$,\\
range: $m,n=1,2,3,\dots, N$,\\
matrix element:
\begin{equation}
\braket{cc^+_m|{}^t\!\Delta\mathcal{U}\Delta|nc^+_n}
=C(k^o,k^e,-1,m,n)-C(k^o,k^e,1,m,n).
\end{equation}
Subspace: $(\mathcal{P}_\mathcal{C}',\mathcal{P}_\mathcal{T}',\mathcal{P}_\mathcal{M}')=(0,1,-1)$,\\
name: Choreographic cos$-$,\\
symbol: $cc^-$,\\
range: $m,n=1,2,3,\dots, N$,\\
matrix element:
\begin{equation}
\braket{cc^-_m|{}^t\!\Delta\mathcal{U}\Delta|nc^-_n}
=C(k^e,k^o,-1,m,n)-C(k^e,k^o,1,m,n).
\end{equation}
Subspace: $(\mathcal{P}_\mathcal{C}',\mathcal{P}_\mathcal{T}',\mathcal{P}_\mathcal{M}')=(0,-1,1)$,\\
name: Choreographic sin$+$,\\
symbol: $cs^+$,\\
range: $m,n=1,2,3,\dots, N$,\\
matrix element:
\begin{equation}
\braket{cs^+_m|{}^t\!\Delta\mathcal{U}\Delta|ns^+_n}
=C(k^o,k^e,-1,m,n)+C(k^o,k^e,1,m,n).
\end{equation}
Subspace: $(\mathcal{P}_\mathcal{C}',\mathcal{P}_\mathcal{T}',\mathcal{P}_\mathcal{M}')=(0,-1,-1)$,\\
name: Choreographic sin$-$,\\
symbol: $cs^-$,\\
range: $m,n=1,2,3,\dots, N$,\\
matrix element:
\begin{equation}
\braket{cs^-_m|{}^t\!\Delta\mathcal{U}\Delta|ns^-_n}
=C(k^e,k^o,-1,m,n)+C(k^e,k^o,1,m,n).
\end{equation}
\subsubsection{Zero-choreographic subspace}
Let $Z(\ell^x, \ell^y,m,n)$ be
\begin{equation}
\begin{split}
Z(\ell^x, \ell^y,m,n)
=&3(-1)^{\lfloor m/2\rfloor+\lfloor n/2\rfloor}\\
&\times
\left(\begin{array}{rr}
\tilde{u}^{xx}((-1)^m \ell^x_m-(-1)^n \ell^x_n)&
-\tilde{u}^{xy}((-1)^m \ell^x_m-(-1)^n \ell^y_n) \\
-\tilde{u}^{yx}((-1)^m \ell^y_m-(-1)^n \ell^x_n)&
\tilde{u}^{yy}((-1)^m \ell^y_m-(-1)^n \ell^y_n)
\end{array}\right).
\end{split}
\end{equation}
Subspace: $(\mathcal{P}_\mathcal{C}',\mathcal{P}_\mathcal{T}',\mathcal{P}_\mathcal{M}')=(0,\pm 1,1)$,\\
name: Zero-choreographic cos$+$ and sin$+$,\\
symbol: $zc^+$ and $zs^+$,\\
range: $m,n=1,2,3,\dots, 2N$,\\
matrix element:
\begin{equation}
\braket{zc^+_m|{}^t\!\Delta\mathcal{U}\Delta|zc^+_n}
=\braket{zs^+_m|{}^t\!\Delta\mathcal{U}\Delta|zs^+_n}
=Z(\ell^o,\ell^e,m,n).
\end{equation}
Subspace: $(\mathcal{P}_\mathcal{C}',\mathcal{P}_\mathcal{T}',\mathcal{P}_\mathcal{M}')=(0,\pm 1,-1)$,\\
name: Zero-choreographic cos$-$ and sin$-$,\\
symbol: $zc^-$ and $zs^-$,\\
range: $m,n=1,2,3,\dots, 2N$,\\
matrix element:
\begin{equation}
\braket{zc^-_m|{}^t\!\Delta\mathcal{U}\Delta|zc^-_n}
=\braket{zs^-_m|{}^t\!\Delta\mathcal{U}\Delta|zs^-_n}
=Z(\ell^e,\ell^o,m,n).
\end{equation}
.
\section{Discussions}
\subsection{Scaling and coupling dependence of eigenvalues}
Consider the eigenvalue problem $\mathcal{H}\Psi=\lambda \Psi$, namely,
\begin{equation}
\left( -\frac{d^2}{dt^2}+{}^t\!\Delta \mathcal{U} \Delta \right)\Psi = \lambda \Psi
\end{equation}
for extended Newton potential in (\ref{homogeneousPotential}).
The eigenvalues scales $\lambda \to \lambda/\mu^2$ for
the scaling of $t \to \mu t$ and $q \to \mu^{2/(2+\alpha)}q$.
Therefore, it is useful to consider the scale invariant eigenvalue $\tilde{\lambda}$
defined by
\begin{equation}
\tilde{\lambda}=\frac{\lambda}{4\pi^2/T^2}.
\end{equation}
Here, the factor $4\pi^2$ is for later convenience.
Consider a system under a homogeneous potential with coupling constant $g^2>0$,
described by
\begin{equation}
\label{generalizedLagrangian}
L=1/2\sum (dq_\ell/dt)^2 + g^2 \sum V_\alpha(|q_i-q_j|).
\end{equation}
The Lagrangian in (\ref{Lagrangian}) has $g^2=1$.
Then the corresponding eigenvalue problem is
\begin{equation}
\left( -d^2/dt^2+g^2\ {}^t\!\Delta \mathcal{U} \Delta \right)\Psi = \lambda \Psi.
\end{equation}
The eigenvalue for the same period $T$ is the same for $g^2=1$,
because a scale transformation
$t \to t$ and $q \to g^{2/(2+\alpha)} q$ makes $g^2 \to 1$.
Therefore, the scale invariant eigenvalue $\tilde{\lambda}$ is
also invariant for changing coupling constant $g^2>0$.
\subsection{Numerical calculation}
In this subsection, we describe
a detail to calculate the eigenvalues and eigenfunctions
of the Hessian for choreographic sine plus subspace
to which the figure-eight solution belongs.
We introduced a cut-off of the Fourier series
$1, \cos(\nu t), \cos(2\nu t),\dots, \cos(3N\nu t)$
and
$\sin(\nu t), \sin(2\nu t),\dots, \sin(3N\nu t)$
with sufficient large $N=2^M$.
Then, the largest number of $k^o_n$ and $k^e_n$ are
\begin{equation}
\label{maxForKoandKe}
k^o_n=\{1,5,7,\dots, k^o_{N}=3N-1\},\
k^e_n=\{2,4,8,\dots, k^e_{N}=3N-2\}.
\end{equation}
Using the equations
\begin{equation}
\begin{split}
&2\sin(k\nu t)\sin(k'\nu t)=-\cos((k+k')\nu t)+\cos((k-k')\nu t),\\
&2\cos(k\nu t)\cos(k'\nu t)=\cos((k+k')\nu t)+\cos((k-k')\nu t).
\end{split}
\end{equation}
the matrix elements (\ref{uforCCPlus}), (\ref{uforCCMinus}),
(\ref{uforCSPlus}) and (\ref{uforCSMinus})
are expressed by the Fourier integral
\begin{equation}
\label{Fourier}
\tilde{u}(k)=\frac{1}{T}\int_0^T u(t) \cos(k\nu t).
\end{equation}
The largest $k$ to calculate the matrix elements in (\ref{uforCSPlus})
for (\ref{maxForKoandKe})
is $k_\text{max}=2k^o_{N}=6N-2 < 6N-1$.
We calculate the Fourier integral by Fast Fourier Transformation,
\begin{equation}
\tilde{u}(k) =
\operatorname{Re}
\left(\frac{1}{N}\sum_{0 \le s \le 6N-1} u\left(\frac{Ts}{N}\right)
\exp(2\pi i k s/N)
\right).
\end{equation}
By Nyquist-Shannon-Someya sampling theorem,
we have to take the sampling number larger than $2 k_\text{max}$
to calculate $\tilde{u}(k)$ for $0 \le k \le k_\text{max}$ correctly.
Then, the elements (\ref{uforCSPlus}) are
\begin{equation}
\begin{split}
&\braket{cs_m^+|{}^t\!\Delta \mathcal{U} \Delta |cs_n^+}\\
&=(-1)^{m+n}3
\left(\begin{array}{rr}
\tilde{u}_{xx}(k^o_m+k^o_n)+\tilde{u}_{xx}(k^o_m-k^o_n) &
-\tilde{u}_{xy}(k^o_m+k^e_n)-\tilde{u}_{xy}(k^o_m-k^e_n) \\
-\tilde{u}_{xy}(k^e_m+k^o_n)-\tilde{u}_{xy}(k^e_m-k^o_n) &
+\tilde{u}_{yy}(k^e_m+k^e_n)+\tilde{u}_{yy}(k^e_m-k^e_n)
\end{array}\right).
\end{split}
\end{equation}
Then, make the Hessian,
$H_{mn}=\braket{cs_m^+|-d^2/dt^2|cs_n^+}+\braket{cs_m^+|{}^t\!\Delta U \Delta |cs_n^+}$,
and solve the eigenvalue problem $H \Phi = \lambda \Phi$.
Inversely, once we get the eigenvalue $\lambda$ and the eigenfunction
\begin{equation}
\Phi=\left(\begin{array}{c}a_1 \\a_2 \\a_3 \\\vdots \\a_{N}\end{array}\right),
\end{equation}
the eigenfunction $\Psi$ is given by
\begin{equation}
\Psi
= \sum_{1 \le n \le N}cs^+_n a_n
=\left(\begin{array}{c} \delta q(t)\\\delta q(t+T/3) \\\delta q(t-T/3)\end{array}\right),
\end{equation}
where
\begin{equation}
\delta q(t)=\sum_{1 \le n \le N}
\left(\begin{array}{cc}\sin(k^o_n \nu t) & 0 \\0 & \sin(k^e_n \nu t)\end{array}\right)
\left(\begin{array}{c}a_{n}^x \\a_{n}^y\end{array}\right)
=\sum_{1 \le n \le N}
\left(\begin{array}{c}
\sin(k^o_n \nu t)a_{n}^x \\
\sin(k^e_n \nu t)a_{n}^y.
\end{array}\right).
\end{equation}
Each component of the right hand side is a Fourier series.
Using a series $b_k$, defined by
\begin{equation}
b_{k_n}=a_n \mbox{ for } n=1, 2, 3, \dots, N, \mbox{ all other } b_k=0,
\end{equation}
the series is given by
\begin{equation}
\begin{split}
\delta q(t)
&=\sum_{1 \le n \le N}\sin(k_n\nu t) a_n
=\sum_{0 \le k \le 3N-1} \sin(k\nu t) b_k\\
&=-\operatorname{Im} \left(\sum_{0 \le k \le 3N-1} b_k \exp(-2\pi i kt/T)\right).
\end{split}
\end{equation}
We can use Inverse Fast Fourier Transform to calculate the right hand side numerically.
Namely, for discrete time $t=Ts/(3N)$, and $s=0,1,2, \dots, 3N-1$,
\begin{equation}
\delta q\left( \frac{Ts}{3N} \right)
=-\operatorname{Im} \left(\sum_{0 \le k \le 3N-1} b_k \exp(-2\pi i ks/(3N))\right).
\end{equation}
\subsection{Application to the figure-eight solution}
Lower 5 scale invariant eigenvalues $\tilde{\lambda}=\lambda/(4\pi^2/T^2)$
for Newton potential ($\alpha=1$)
are listed in the equation (\ref{eigenValuesForFigureEight}).
\begin{equation}
\label{eigenValuesForFigureEight}
\begin{array}{llllll}
cc+& 1.14719\times10^{-10}& 4.04112& 16.5764& 26.117& 50.469\\
cc-& 1.93391& 7.26091& 19.289& 27.3167& 50.6671\\
cs+& 2.32495& 6.14928& 18.5694& 27.9174& 50.9476\\
cs-& -6.81259\times10^{-10}& 3.54142& 16.4395& 25.7645& 50.272\\
zc+\mbox{ and }zs+& 0.00174304& 0.946257& 5.02738& 8.91162& 12.5666\\
zc-\mbox{ and }zs-& -0.0744809& 1.24605& 4.92153& 9.14293& 11.6371
\end{array}
\end{equation}
Tiny eigenvalues of order $10^{-10}$ in $cc+$ and $cs-$ are
zero eigenvalues that correspond to the conservation of
energy and angular momentum respectively.
This calculation is done with $N=2^9$.
(For definition of $N$, see section \ref{sectionDecomposition}
and figure \ref{blocksOfH}).
\subsection{A question for eigenvalue problem by differential equations}
The eigenvalue problem $\mathcal{H}\Psi = \lambda \Psi$ can be solved
by directly solving the differential equation
(\ref{defOfHessian}) and (\ref{eigenValueProblem}) with periodic boundary condition.
Mathematica provides ``NDEigensystem'' function for this purpose.
This function gives us an easy method to calculate the eigenvalues and
eigenfunctions in a moderate precision.
It is good idea to use this function if you need a moderate precision.
Actually, we used this function twice.
The first time is to get rough estimates of eigenvalues and eigenfunctions at an early stage of our calculations.
The second time is to check our program developed here at the last stage.
In these calculations,
we eliminated the variable $\delta q_2$ by $\delta q_2=-(\delta q_0+\delta q_1)$
to eliminate the trivial subspace.
And we impose periodic boundary condition $\delta q_k(t+T)=\delta q_k(t)$.
Equivalently,
we restricted the function space to zero-centre-of-mass
and periodic. It works fine.
\emph{We have one question}.
Can we restrict the function space more and more in this direct calculation
by imposing suitable boundary conditions?
For example,
we can restrict the function space to choreographic subspace
by imposing the boundary conditions
\begin{equation}
\delta q_1(t)=\delta q_0(t+T/3),\ \delta q_2(t)=\delta q_1(t+T/3),\ \delta q_0(t)=\delta q_2(t+T/3).
\end{equation}
The question is how to realize this boundary condition in numerical calculations,
for example in ``NDEigensystem''?
And how to realize boundary conditions for choreographic cos $+$ in calculations?
A strong motivation for this note
is to classify eigenvalues and eigenvectors in suitable subspace defined by the symmetry.
If we can directly solve the eigenvalue equation in each subspace
with realizing suitable boundary conditions,
this will provide a concise and simple method.
It will be a future investigation.
\section*{Acknowledgements}
We would like to express many thanks to M.~Shibayama
for his work \cite{Shibayama}
that describes a general method to calculate the eigenvalue of the Hessian.
This work was partly supported by JSPS Grant-in-Aid for Scientific Research
17K05146 (HF) and 17K05588(HO).
|
1,116,691,499,231 | arxiv | \section{Introduction}
The Horizontal visibility algorithm \cite{seminalPRE} is a mapping by which an ordered set of $N$ real numbers $\{x_t\}, t=1,...,N$ maps into a graph $\cal G$ with $N$ nodes and adjacency matrix $A_{ij}$. Nodes $i$ and $j$ are connected through an undirected edge ($A_{ij}=A_{ji}=1$) if $x_i$ and $x_j$ have so called horizontal visibility, that is, if every intermediate datum $x_q$ follows $$x_q<\inf\{x_i,x_j\}, \ \forall q\in [i,j]$$
The set of graphs spanned by this mapping are called Horizontal Visibility Graphs (HVGs). These are noncrossing outerplanar graphs with a Hamiltonian path \cite{Gutin, prodinger}, subgraphs of a more general mapping \cite{pnas} that have been recently used in the context of time series analysis and signal processing \cite{review} (see figure \ref{fig_label1} for an illustration). The methodology proceeds by analysing the topological properties of $\cal G$ and, according to that information, characterise the structure of $\{x_t\}$ and its underlying dynamics. Periodic dynamics are retrieved from the mean degree of the associated graphs, and
some recent applications include the description of correlated stochastic and low-dimensional chaotic series \cite{toral}, processes that seem to cluster as HVGs with different exponential degree distributions $P(k)=a \exp(-\lambda k)$ (we recall that the degree distribution describes the probability of a node chosen at random to have degree $k$). The canonical routes to chaos (Feigenbaum, quasiperiodic, and Pomeau-Manneville scenarios) have also been described in graph theoretical terms \cite{chaos, quasi, intermitencia}, and in particular sensibility to initial conditions has been related, via Pesin identity, to the Shannon and the block entropies of the degree distribution of the associated graphs. Both the horizontal and the original version of the mapping are currently extensively used for data analysis purposes (as feature extraction algorithm for feature-based classification) in several disciplines, such as biomedicine \cite{bio} or geophysics \cite{geo} (see \cite{review} for a recent review).\\
\begin{figure}
\centering
\includegraphics[width=1.\columnwidth]{prueba.pdf}
\vspace{-15mm}
\caption{Graphical Illustration of the Horizontal Visibility Algorithm: in the top we plot a sample time series (vertical bars) and in the bottom we represent its associated Horizontal Visibility Graph (HVG), where each node has a certain degree $k$ and the edges are undirected, and the Directed Horizontal Visibility Graph (DHVG), where each node has an ingoing and an outgoing degree, and the edges are directed. The HVG and DHVG are actually the same graph, although in the DHVG the edges are directed and therefore in this latter case the adjacency matrix is not symmetric. The degree sequence of the HVG and DHVG are also represented.}
\label{fig_label1}
\end{figure}
\noindent The fingerprint of the arrow of time and time asymmetries in stationary dynamics can be assessed within this framework by redefining the following node labelling in the transformation: if in the HVG one distinguishes the ingoing degree of a node (where node $i$ has an ingoing edge from $j$ if $j<i$) from its outgoing degree (where node $i$ has an outgoing edge to $j$ if $i<j$), the graph converts into a digraph, so called Directed Horizontal Visibility Graph (DHVG) \cite{irrev}. In a DHVG the ingoing degree distribution $P_\leftarrow(k)$ (the probability that a node chosen at random from $\cal G$ has $k$ ingoing links) and the outgoing degree distribution $P_{\rightarrow}(k)$, defined as the probability that a node chosen at random from $\cal G$ has $k$ outgoing links, are in general different distributions (see figure \ref{fig_label1} for an illustratation). Recent works point that different measures of the distinguishability between $P_{\rightarrow}(k)$ and $P_{\leftarrow}(k)$ \cite{irrev} (amongst other graph properties \cite{irrev2}) can distinguish statistically reversible processes (white noise, stochastic processes with linear correlations, measure preserving chaotic maps) from statistically irreversible stationary processes, such as some dissipative chaotic maps or thermodynamic systems driven out of equilibrium, and are shown to be computationally efficient methods to quantify such asymmetries.\\\
In summary, it appears that the degree distribution of visibility graphs, along with some derived functionals (moments, entropy, etc) carry much information of the signal and the underlying dynamics.\\
\noindent Despite these recent applications, the majority of results using these transformations are numerical and/or heuristic. In this work we focus on the analytical properties of the degree distribution $P(k)$ (and $P_\rightarrow(k)$), when the HVG (or DHVG) is associated to some important classes of dynamical processes. The amount of closed analytical results on these degree distributions is also so far scarce, most of them being found for HVGs which are nontrivial graph theoretical fixed points of some renormalisation group transformation, associated with critical dynamics generated at accumulation points of the canonical routes to chaos \cite{chaos, quasi, intermitencia}. Almost no exact results exist for other nonperiodic dynamical processes, with the exception of uncorrelated random processes \cite{seminalPRE, irrev}. Here we build on previous results and explore in section II how to compute these degree distributions for any given dynamical process with well defined invariant measure. To this end, we propose a general diagrammatic theory similar in spirit to Feynman's approach in quantum theory. Each degree probability $P(k)$ can be computed as a 'perturbative' expansion, where the 'coupling constant' is the number of hidden variables $\alpha$, and for each $\alpha$ a different number of diagrams must me summed up (corrections to the free field). For each $k$, the free field ($\alpha=0$) can be summed up exactly, and diagrams of order $\alpha$ yield a correction of order O($\alpha^{-2}$) in the probability amplitude. In section III we show that the results on uncorrelated processes (for both HVG and DHVG) represent a very special case in this formalism, where all orders of the theory can be summed up exactly, as the $n-$joint distributions completely factorise. We then address in section IV the theory for dynamical processes that fulfill the Markov property and give explicit formulae for a arbitrary diagrams. As case studies we focus on stochastic stationary Markovian processes (Ornstein-Uhlenbeck) in section V and in one dimensional deterministic maps (both chaotic and quasiperiodic) in section VI. For a few terms of the distribution, their diagram expansions can be summed up exactly (up to all orders), but in the general case a convergent perturbative approach should be followed, to calculate each degree probability up to arbitrary precision. In the particular case of chaotic maps, the existence of forbidden patterns \cite{forbidden} drastically decreases the number of allowed diagrams to a finite number, up to a given order, speeding up the convergence of the perturbative analysis.
Then, in section VII we present a general variational approach, by which we can derive analytical results for the entire distribution provided some entropic optimisation hypothesis holds. We show that the entire distributions of chaotic and stochastic Markovian (Ornstein-Uhlenbeck) processes is well approximated if we assume the graphs associated to these processes are maximally entropic. In section VIII we conclude.
\section{A general diagrammatic formalism for degree distributions}
Consider a stationary dynamical process with an underlying invariant density $f(x), \ x \in [a,b]$ (where $a, b \in \mathbb{R}$, and they can be either finite -finite support- or diverge -unbounded support-). $f(x)$ is just a probability density for stationary stochastic processes, or an invariant measure for dynamical systems. Consider also a series of $n$ variables $\{x_0,x_1,...,x_n\}$ extracted from $f(x)$, which can be either a realisation of a stochastic process with underlying density $f(x)$, or a trajectory of some deterministic dynamical system with invariant measure $f(x)$. In both situations, each time series realisation of $n$ variables has a joint probability $f(x_0 \rightsquigarrow x_n)\equiv f(x_0,x_1,x_2,...,x_n)$ (we may call this the propagator). For each realisation of the dynamical process, this ordered set will have an associated HVG/DHVG with a given topology, and in particular with a given degree distribution. As already acknowledged, previous numerical research suggests that such distribution encapsulates and compresses much of the series structure and variability. Consequently, is there any general approach to derive such distributions in a constructive way? The response is positive for large $n$ ($n\rightarrow \infty$), that is, when we consider bi-infinite series and their associated graphs (note, nonetheless, that finite size effects decrease fast with $n$, and therefore whereas theory can only be built in the asymptotic limit, finite size systems converge to the asymptotic solution very fast \cite{seminalPRE}).\\
In what follows we present such a constructive approach. We recall that each datum $x_i$ in the ordered data set is associated with a node with label $i$ in the HVG/DHVG. With a litle abuse of language, from now on we will use $x_i$ to label both the datum and the node (and we will quote as \textit{variable}), although we will make clear when necessary that they represent different mathematical objects.\\
\begin{figure}[h]
\centering
\includegraphics[width=0.50\textwidth]{k4.pdf}
\caption{Set of possible configurations for a seed variable $x_0$ with
$k=4$. Observe that the sign of the subindex in $x_i$ indicates if
the data is located whether at left-hand side of $x_0$ (sign
minus) or at right-hand side. Accordingly, the bounding's variable
subindex directly indicates the amount of data located in that
side. For instance, $C_0$ is the configuration where none of the
$k-2=2$ inner variables are located in the left-hand side of $x_0$, and
therefore the left bounding variable is labelled as $x_{-1}$ and the
right bounding variable is labelled as $x_3$. $C_1$ is the
configuration for which an inner variable is located in the left-hand
side of $x_0$ and another inner variable is located in its right-hand
side. Finally, $C_2$ is the configuration for which both inner
variables are located in the left-hand side of the seed. Notice that an
arbitrary number of hidden variables can be eventually located among
the inner variables, what is schematically represented in the figure as
a row of vertical lines. }\label{k4}
\end{figure}
Consider a datum (node) chosen at random from the bi-infinite sequence $\{x_t\}$, that we label $x_0$ without loss of generality. To calculate the degree distribution $P(k)$ of the HVG (or the outgoing degree distribution $P_\rightarrow(k)$ of the DHVG) is equivalent to calculate the probability that node $x_0$ has degree $k$ (or outdegree $k$) respectively. For each $k$, a different number of variable configurations (relative positions of data at the right hand side and left hand side of $x_0$) are allowed. Each of these configurations can in turn be schematised as a different \textit{diagram} and will contribute with a different correction to the total probability, as will be shown.\\
For illustrative purposes, consider $P(2)$ of an HVG (as HVGs are undirected and connected, by construction $P(1)=0$). The only configuration that allows $x_0$ to have degree $k=2$ requires, by construction, that the variables on the left and right hand side are larger than $x_0$. Label these \textit{bounding variables} $x_{-1}$ and $x_1$ respectively. Accordingly, this unique configuration (diagram) has an associated contribution to $P(2)$
\begin{equation}
P(2)=\int_{x_0}^ b dx_{-1} \int_{a}^ b dx_{0}\int_{x_0}^ b dx_{1}f(x_{-1},x_0,x_1)
\end{equation}
Incidentally, note at this point that, by construction, the HVGs are outerplanar and have a Hamiltonian path, and therefore $P(2)$ is directly related to the probability that a node chosen at random forms a 3-clique (triplet). Therefore equation \ref{cluster} calculates the global clustering coefficient of a HVG.\\
\noindent A similar calculation can be made for DHVGs. Indeed, $P\rightarrow(0)=0$ (the DHVG is connected) and $P\rightarrow(1)$ is equivalent to calculate the probability that $x_1>x_0$:
\begin{equation}
P\rightarrow(1)=\int_{a}^ b dx_{0}\int_{x_0}^ b dx_{1}f(x_0,x_1)
\end{equation}
\begin{figure}[h]
\centering
\includegraphics[width=0.8\columnwidth]{diagrams_HVG.pdf}
\caption{Some diagrammatic contributions to the degree distribution $P(k)$ of a horizontal visibility graph (HVG). In each diagram, the grey node corresponds to the reference datum $x_0$, from which the set of concatenated integrals spans (see the text).}
\label{diagrams_HVG}
\end{figure}
\noindent For $P(k\geq 3)$ (or $P\rightarrow(k)\geq 2$) an arbitrary large number of different contributions should be taken into account. Consider first the HVG: if $x_0$ has degree $k\geq3$, then besides the bounding variables, $k-2$ (visible) \textit{inner} variables should be distributed on the right and left hand side of $x_0$. Due to visibility constraints, there are exactly $k-1$ different possible configurations $\{C_i\}_{i=0..k-2}$, where the index $i$
determines the number of inner variables on the left-hand side of $x_0$
Accordingly, $C_i$ corresponds to the configuration for
which $i$ inner variables are placed at the left-hand side of $x_0$,
and $k-2-i$ inner variables are placed at its right-hand side. Each of
these possible configurations have an associated probability
$p_i\equiv p(C_i)$ that will contribute to $P(k)$ such that
\begin{equation}
P(k)=\sum_{i=0}^{k-2}p_i \label{P_i}.
\end{equation}
In the case of a DHVG, there is only one such configuration $C_0$, such that $P_\rightarrow(k)=p_0 $.\\
\noindent Now, among the inner variables, an arbitrary number of \textit{hidden} variables may appear (see figure \ref{k4} for a graphical illustration of this situation for $P(4)$). In summary, between each pair of inner variables variables $x_i,x_{i+1}$, an arbitrary (eventually infinite) number of hidden variables $\{z_p\}_{p=1}^\infty$ may take place. For each $p$, the different combinations of inner and hidden variables yield different contributions. As we will show, in some cases it is very convenient to represent such set of contributions as a series expansion
\begin{equation}
P(k)=\sum_{\alpha} P^{(\alpha)}(k), \label{pertu}
\end{equation}
where $\alpha$ denotes the number of hidden variables. Up to each order, a set of different configurations can take place. Each of these configurations is indeed a different \textit{diagram}. Following Feynman's approach, the free field theory takes into account the contributions spanned by configurations with no hidden variables ($P^{(0)}(k)$), whereas the interaction field introduces corrections of all orders in the number of hidden variables, which is here the coupling constant. Accordingly, $\alpha=0$ accounts for the diagrams with no hidden variables, $\alpha=1$ account for all the diagrams with one hidden variable, etc.\\
The same formalism can be extended to DHVGs, where for concreteness we focus on the out degree distribution $P_\rightarrow(k)$
\begin{equation}
P_\rightarrow(k)=\sum_{\alpha} P_\rightarrow^{(\alpha)}(k), \label{pertu2}
\end{equation}
In figures \ref{diagrams_HVG} and \ref{diagramsDHVG} we represent some contributing diagrams up to third order corrections for both HVG and DHVGs.\\
\noindent Two general questions arise:\\
(i) Can we explicitely calculate the contribution of any particular diagram contributing to $P(k)$ or $P_\rightarrow(k)$?\\
(ii) What can we say about the convergence properties of the series in (\ref{pertu}, \ref{pertu2})?\\
\noindent Regarding (i), to compute closed form solutions for the entire degree distributions $P(k)$ and $P_\rightarrowtail(k)$ is a hopeless endeavour in the general case, mainly because the n-point propagators $f(x_0 \rightsquigarrow x_n)$ (where $n$ is also arbitrarily long) cannot be calculated explicitely. However, in several particular cases this propagator factorises and therefore some theory can still be constructed. This is for instance the case of an uncorrelated random process with an underlying probability density $f(x)$. For this large class of stochastic processes, the propagator completely factorises
$$f(x_0,x_1,x_2,...,x_n)=f(x_0)f(x_1)f(x_2)\cdots f(x_n).$$
In the next section we show that this simplification is the key aspect that permits us to calculate $P(k)$ and $P_\rightarrow(k)$ in closed form.
\noindent On relation to (ii), note that all expressions of the form \ref{pertu} or \ref{pertu2} should be converging series as they are probabilities. In particular, a diagram in the correction to the free field of order $\alpha$ has $\alpha$ hidden variables. This diagram is considering the possibility that the seed ($x_0$) and the bounding variable ($x_{k-1}$) are separated by a total of $k-1 + \alpha$ intermediate variables. Now, in a previous work \cite{seminalPRE} it was shown that the probability of two variables separated by $n$ intermediate variables are connected $U(n)$ decreases asymptotically as $n^{-2}$ for uncorrelated random processes. For a constant value of the degree $k$, this means that the correction of the diagrams of order $\alpha$ decreases at least as fast as $\alpha^{-2}$, hence the perturbation series is convergent for the uncorrelated case. For other processes this is not rigorously proved, however we will see that processes with fast decaying correlations are likely to also have associated converging perturbation series.
\begin{figure}[h]
\centering
\includegraphics[width=0.8\columnwidth]{diagrams.pdf}
\caption{Some diagrammatic contributions to the degree distribution $P_\rightarrowtail(k)$ of a directed horizontal visibility graph (DHVG). In each diagram, the grey node corresponds to the reference datum $x_0$, from which the set of concatenated integrals spans (see the text)}
\label{diagramsDHVG}
\end{figure}
\section{Uncorrelated processes: exact results for asymptotic distribution}
When the dynamical process under study is a random uncorrelated one, we are in a special case where we don't actually need explicit diagrams to compute the entire degree distribution of both HVG and DHVG, and therefore we don't actually need to follow a perturbative approach such as eqs \ref{pertu} and \ref{pertu2}. We first recall \cite{seminalPRE} the result for HVGs, and further extend this to DHVGs.
\subsection{HVG}
\noindent \textbf{Theorem 1.}\\
Let $X(t)$ a real valued bi-infinite time series created from a random variable
$X$ with probability distribution $f(x)$, with $x \in [a,b]$, and
consider its associated Horizontal Visibility Graph $\cal G$. Then,
\begin{equation}
P(k)=\frac{1}{3}\bigg(\frac{2}{3}\bigg)^{k-2}, \ k=2,3,\dots , \ \forall \ f(x)
\label{uncorr}
\end{equation}
\noindent \textbf{Sketch of the proof.}\\
\noindent The proof proceeds by induction on $k$. Let us begin by computing some easy terms:
\begin{equation}
P(k=2)=\textrm{Prob}(x_{-1},x_1\geq0)
=\int_0^1f(x_0)dx_0\int_{x_0}^1f(x_1)dx_1\int_{x_0}^1f(x_{-1})dx_{-1}.
\label{pk2}
\end{equation}
Now, the cumulative probability distribution function $F(x)$ of
any probability distribution $f(x)$ is defined as
\begin{equation}
F(x)=\int_0^xf(x')dx'\label{cumulative},
\end{equation}
where $dF(x)/dx=f(x)$, $F(0)=0$ and $F(1)=1$. In particular, the
following relation between $f$ and $F$ holds:
\begin{equation}
f(x)F^{n-1}(x)=\frac{1}{n}\frac{dF^n(x)}{dx}.\label{f-F}
\end{equation}
We can accordingly rewrite and compute equation \ref{pk2} as
\begin{equation}
P(k=2)=\int_0^1 f(x_0)[1-F(x_0)]^2dx_0=\frac{1}{3}, \ \forall f(x)
\end{equation}
\noindent In the case $P(k=3)$, two different configurations arise: $C_0$, in which $x_0$
has 2 bounding variables ($x_{-1}$ and $x_2$ respectively) and
a right-hand side inner variable ($x_1$), and the same for $C_1$ but
with the inner variable being place at the left-hand side of the seed:
$$P(k=3)=p(C_0) + p(C_1)\equiv p_0+p_1.$$
\noindent Notice at this point that an arbitrary number $r$ of
hidden variables $n_1,n_2...n_r$ can eventually be located between the
inner data and the bounding variables, and this fact needs to be taken
into account in the probability calculation. The geometrical
restrictions for the $n_j$ hidden variables are $n_j<x_1, \ j=1,...,r$
for $C_0$ and $m_j<x_{-1}, \ j=1,...,s$ for $C_1$. Then,
\begin{eqnarray}
p_0 &=& \textrm{Prob}\bigg((x_{-1},x_2\geq x_0)\cap (x_1<x_0)\cap (\{n_j<x_1\}_{j=1,\dots,r})\bigg), \nonumber \\
p_1 &=& \textrm{Prob}\bigg((x_{-2},x_1\geq x_0)\cap
(x_{-1}<x_0)\cap(\{m_j<x_{-1}\}_{j=1,\dots,s})\bigg).
\label{P_1_k3bis}
\end{eqnarray}
Now, we need to consider every possible hidden variable configuration. In the particular case of an uncorrelated process,
all these contributions can be summed up exactly:
\begin{eqnarray}
&&p_0= \int_0^1f(x_0)dx_0 \int_{x_0}^1f(x_{-1}) dx_{-1}
\int_{x_0}^1 f(x_2)dx_2
\int_0^{x_0}f(x_1)dx_1 + \nonumber\\
&&\sum_{r=1}^\infty \int_0^1f(x_0)dx_0 \int_{x_0}^1
f(x_{-1})dx_{-1} \int_{x_0}^1 f(x_2)dx_2
\int_0^{x_0}f(x_1)dx_1\prod_{j=1}^r\int_0^{x_1}f(n_j)dn_j\nonumber\label{P0k3}
\end{eqnarray}
where the first term corresponds the contribution of a
configuration with no hidden variables and the second sums up the
contributions of $r$ hidden variables. Making use of the properties of
the cumulative distribution $F(x)$ we arrive to
\begin{equation}
p_0=\int_0^1f(x_0)dx_0 \int_{x_0}^1f(x_{-1}) dx_{-1} \int_{x_0}^1
f(x_2)dx_2 \int_0^{x_0}\frac{f(x_1)}{1-F(x_1)}dx_1,
\end{equation}
where we also have made use of the sum of a geometric series. We
can find an identical result for $p_1$, since the last integral on
equation \ref{P0k3} only depends on $x_0$ and consequently the
configuration provided by $C_1$ is symmetrical to the one provided
by $C_0$. We finally have
\begin{equation}
P(k=3)=2p_0=-2\int_0^1f(x_0)(1-F(x_0))^2\ln
(1-F(x_0))dx_0=\frac{2}{9} \ \forall f(x),
\end{equation}
where the last calculation also involves the change of variable
$z=1-F(x)$.\\
\noindent Hitherto, we can deduce that a given configuration $C_i$
contributes to $P(k)$ with a product of integrals according to the
following rules:
\begin{itemize}
\item The seed variable [S] provides a contribution of
$\int_0^1f(x_0)dx_0$. \item Each boundary variable [B] provides a
contribution of $\int_{x_0}^1 f(x)dx$. \item An inner variable [I]
provides a contribution $\int_{x_j}^{x_0} \frac{f(x)dx}{1-F(x)}$.
\end{itemize}
These 'Feynman rules' allow us to schematize in a formal
way the probability associated to each configuration. For instance
in the case $k=2$, $P(k)$ has a single contribution $p_0$
represented by the formal diagram $[B][S][B]$, while for $k=3$,
$P(k)=p_0+p_1$ where $p_0$'s diagram is $[B][S][I][B]$ and $p_1$'s is
$[B][I][S][B]$. It seems quite straightforward to derive a general
expression for $P(k)$, just by applying the preceding rules for
the contribution of each $C_i$. However, there is still a subtle
point to address that becomes evident for the case
$P(k=4)=p_0+p_1+p_2$ (see figure \ref{k4}). While in this case
$C_1$ leads to essentially the same expression as for both
configurations in $k=3$ (and in this sense one only needs to apply
the preceding rules to derive $p_1$), $C_0$ and $C_2$ are
geometrically different configurations. These latter ones are
configurations formed by a seed, two bounding and two
\emph{concatenated} inner variables, and concatenated variables lead to
concatenated integrals. For instance, applying the same formalism
as for $k=3$, one come to the conclusion that for $k=4$,
\begin{equation}
p_0=\int_0^1 f(x_0)dx_0 \int_0^{x_0} \frac{f(x_1)dx_1}{1-F(x_1)}
\int_{x_1}^{x_0} \frac{f(x_2)dx_2}{1-F(x_2)} \int_{x_0}^1
f(x_3)dx_3 \int_{x_0}^1 f(x_{-1})dx_{-1}.
\end{equation}
While for the case $k=3$ every integral only depended on $x_0$
(and consequently we could integrate independently every term
until reaching the dependence on $x_0$), having two concatenated
inner variables on this configuration generates a dependence on the
integrals and hence on the probabilities. For this reason, each
configuration is not equiprobable in the general case, and thus
will not provide the same contribution to the probability $P(k)$
($k=3$ was an exception for symmetry reasons). In order to weight
appropriately the effect of these concatenated contributions, we
can make use of the definition of $p_i$. Since $P(k)$ is formed by
$k-1$ contributions labelled $C_0, C_1...C_{k-2}$ where the index
denotes the number of inner data present at the left-hand side of
the seed, we deduce that in general the $k-2$ inner variables have the
following effective contribution to $P(k)$:
\begin{itemize}
\item $p_0$ has $k-2$ concatenated integrals (right-hand side of
the seed). \item $p_1$ has $k-3$ concatenated integrals
(right-hand side of the seed) and an independent inner data
contribution (left-hand side of the seed). \item $p_2$ has $k-4$
concatenated integrals (right-hand side of the seed) and another 2
concatenated integrals (left-hand side of the seed). \item ...
\item $p_{k-2}$ has $k-2$ concatenated integrals (left-hand side
of the seed).
\end{itemize}
Observe that $p_i$ is symmetric with respect to the seed.\\
\noindent Including this modification in the Feynman rules,
we are now ready to calculate a general expression for $P(k)$.
Formally,
\begin{equation}
P(k)=\sum_{j=0}^{k-2}[S][B]^2[I]_j[I]_{k-2-j} \label{formal},
\end{equation}
where the sum extends to each of the $k-1$ configurations, the
superindex denotes exponentiation and the subindex denotes
concatenation (this latter expression can be straightforwardly
proved by induction on the number of inner variables). The concatenation of $n$ inner variable integrals
$[I]_n$ reads
\begin{equation}
[I]_n=\int_0^{x_0}
\frac{f(x_1)dx_1}{1-F(x_1)}\prod_{j=1}^{n-1}\int_{x_j}^{x_0}\frac{f(x_{j+1})dx_{j+1}}{1-F(x_{j+1})}.\label{recurrencia}
\end{equation}
which can be proved by induction (using the properties of the cumulative distribution and using appropiate change of variables) to reduce to
\begin{equation}
[I]_n=\frac{(-1)^{n}}{n!}\bigg[\ln
\big(1-F(x_0)\big)\bigg]^n.\label{In}
\end{equation}
According to the formal solution \ref{formal} and to equation
\ref{In}, we finally have
\begin{eqnarray}
P(k)&=&\sum_{j=0}^{k-2}\frac{(-1)^{k-2}}{j!(k-2-j)!}\int_0^1f(x_0)[1-F(x_0)]^2[\ln(1-F(x_0))]^{k-2}dx_0\nonumber
\\
&=& 3^{1-k}\sum_{j=0}^{k-2}\frac{(k-2)!}{j!(k-2-j)!} =
\frac{1}{3}\bigg(\frac{2}{3}\bigg)^{k-2}, \ \forall f(x) \ \ \ \ \square \nonumber
\end{eqnarray}
\noindent Surprisingly, we can conclude that for every probability
distribution $f(x)$, the degree distribution $P(k)$ of the
associated horizontal visibility graph
has the same exponential form (note that this result parallels the one for the spectrum of white noise, which is universally flat independently of the underlying probability density $f(x)$ of the noise). This is an exact result for sufficiently long uncorrelated random processes (to avoid finite-size or border effects), that is, we consider bi-infinite series ${x(t)}, t\in \mathbb{Z}$ and the calculations address the asymptotic shape of the degree distribution $P(k)$. However, it should be noted that numerical results for finite size series converge fast to the asymptotic solution.
\subsection{DHVG}
\noindent \textbf{Theorem 2.}\\
Let $\{x_{t}\}_{t=-\infty,\dots,\infty}$ be a bi-infinite sequence of
independent and identically distributed random variables extracted
from a continuous probability density $f(x)$. Then, the out degree distributions of its associated
directed horizontal visibility graph is
\begin{equation}
P_\rightarrow(k)=\bigg(\frac{1}{2}\bigg)^{k},\ k=1,2,3,\dots , \ \forall f(x)
\label{pk}
\end{equation}
\noindent \textbf{Sketch of the proof.}\\
\noindent The proof follows a similar path as for HVG, such that instead of equation \ref{formal} one gets that for a DHVG and for $k\geq1$
$$P_\rightarrow(k)=[S][B][I]_{k-1}=\frac{-1^{k-1}}{(k-1)!}\int_0^1f(x_0)(1-F(x_0))\ln [1-F(x_0)]^{k-1}dx_0=\bigg (\frac{1}{2}\bigg)^{k},\ \forall f(x)$$
where we have used the change of variables $z=1-F(x)$ and the formal solution for the concatenation of $n$ inner variable integrals (eq. \ref{In}) $ \ \square \nonumber $ \\
When variables in $\{x_t\}$ are not uncorrelated anymore, the propagator does not factorise and clean, closed form solutions are more difficult to obtain. In what follows we present a general theory for correlated Markovian dynamics. From now on, for concreteness we focus on the DHVG, although a similar formalism can be extended to HVGs.
\section{Markovian dynamics: a constructive solution}
The second 'easiest' class of dynamical processes are Markovian processes with an integrable invariant measure $f(x)$. For these systems the propagators of the n-joint probabilities factorise into conditional probabilities
$$f(x_0,x_1,x_2,...,x_n)=f(x_0)f(x_1|x_0)f(x_2|x_0)\cdots f(x_n|x_{n-1})$$
Examples include the Ornstein-Uhlenbeck process \cite{kampen} as an example of a stationary stochastic process, the fully chaotic logistic map as an example of an ergodic chaotic map with smooth invariant measure, or irrational rotation as an example of a zero entropy (nonchaotic), ergodic deterministic process. In what follows we show how to treat this special (although very relevant) class of processes, we develop a formal theory to calculate $P_\rightarrowtail(k)$ in terms of diagrammatic expansions, and we apply it to compute the distributions in the aforementioned examples. \\
For the sake of exposition, let us work out a case by case analysis. For the simplest case $P_\rightarrowtail(1)$, the only possible diagram (see figure \ref{diagramsDHVG}) describes the situations where datum $x_1$ bounds $x_0$ ($x_1>x_0$) and therefore
\begin{eqnarray}
P_\rightarrowtail(1) = \int_1^ b f(x_0)dx_0 \int_{x_0}^b f(x_1|x_0)
\label{P1}
\end{eqnarray}
\noindent Second, let us consider $P_\rightarrowtail(2)$. In this case we need to take into account the situations where datum $x_0$ sees $x_1$ and $x_2$, that is: $x_2$ is a bounding variable ($x_2\geqslant x_0$) and an arbitrary number of hidden variables can be placed between $x_1$ and $x_2$. There are an infinite number of diagrams that can be labelled as $[z_i]$, where $i$ determines the number of hidden variables in each diagram. Accordingly,
\begin{eqnarray}
P_\rightarrowtail(2) = \sum_{i=0}^\infty [z_i]\equiv \sum_{\alpha=0}^\infty P_\rightarrowtail^{(\alpha)}(2)
\label{P2}
\end{eqnarray}
where in this case each diagram each correcting order $\alpha$ in the perturbative expansion only includes a single diagram $[z_i]$ that reads
\begin{equation}
[z_0]=\int_{a}^bf(x_0)dx_0 \int_a^{x_0} f(x_1|x_0) dx_1 \int_{x_0}^b f(x_2|x_1)dx_2
\nonumber
\end{equation}
\begin{equation}
[z_1]=\int_{a}^bf(x_0)dx_0 \int_a^{x_0} f(x_1|x_0) dx_1 \int_a^{x_1} f(z_1|x_1)dz_1\int_{x_0}^b f(x_2|z_1)dx_2\nonumber
\end{equation}
\begin{equation}
[z_2]=\int_{a}^bf(x_0)dx_0 \int_a^{x_0} f(x_1|x_0) dx_1 \int_a^{x_1} f(z_1|x_1)dz_1\int_a^{x_1} f(z_2|z_1)dz_2\int_{x_0}^b f(x_2|z_2)dx_2\nonumber
\end{equation}
and for $p>2$,
\begin{equation}
[z_i]=\int_{a}^bf(x_0)dx_0 \int_a^{x_0} f(x_1|x_0) dx_1 \int_a^{x_1} f(z_1|x_1)dz_1 \bigg [\prod_{p=2}^{i-1}\int_a^{x_1} f(z_p|z_{p-1})dz_p\bigg]\int_{x_0}^b f(x_2|z_i)dx_2.
\end{equation}
The third component $P_\rightarrowtail(3)$ is slightly more involved, as there can appear arbitrarily many hidden variables between $x_1$ and $x_2$ and between $x_2$ and $x_3$. Each diagram accounts for a particular combination with $i$ hidden variables between $x_1$ and $x_2$, and $j$ hidden variables between $x_2$ and $x_3$, with the restriction that $x_3$ is a bounding variable and $x_2$ cannot be a bounding variable. There are an infinite number of such contributions, diagrams that can be labelled as $[z_i^{(1)}\star z_j^{(2)}]$:
\begin{equation}
P_\rightarrowtail(3) = \sum_{i=0}^\infty\sum_{j=0}^\infty [z_i^{(1)}\star z_j^{(2)}]\equiv \sum_{\alpha=0}^\infty P_\rightarrowtail^{(\alpha)}(3)
\label{P3}
\end{equation}
where the $[z_i^{(1)}\star z_j^{(2)}]$ diagram, for $i>2$ and $j>2$ yields a correction
\begin{eqnarray}
[z_i^{(1)}\star z_j^{(2)}]&=&\int_{a}^bf(x_0)dx_0 \int_a^{x_0} f(x_1|x_0) dx_1 \int_a^{x_1} f(z_1^{(1)}|x_1)dz_1^{(1)} \bigg [\prod_{p=2}^{i}\int_a^{x_1} f(z_p^{(1)}|z_{p-1}^{(1)})dz_p^{(1)}\bigg]\nonumber \\
&&\ \cdot\int_{x_1}^{x_0} f(x_2|z_i^{(1)})dx_2 \int_a^{x_2} f(z_1^{(2)}|x_2)dz_1^{(2)} \bigg [\prod_{p=2}^{j}\int_a^{x_2} f(z_p^{(2)}|z_{p-1}^{(2)})dz_p^{(2)}\bigg]\int_{x_0}^bf(x_3|z_j^{(2)}) dx_3\nonumber
\end{eqnarray}
As can be seen, in this case more than one diagram contributes to each order in the perturbation expansion. The number of diagrams $N(\alpha, k)$ that contribute to the correction of order $\alpha$ in $P_\rightarrow (k)$ is
\begin{equation}
N(\alpha, k)= {k-1 \choose \alpha}
\end{equation}
\noindent Although explicit integral formulae get more and more involved for larger values of the degree $k$, they can be easily expressed diagrammatically. For instance,
\begin{equation}
P_\rightarrowtail(4) = \sum_{i=0}^\infty\sum_{j=0}^\infty\sum_{k=0}^\infty [z_i^{(1)}\star z_j^{(2)}\star z_k^{(3)}]
\label{P4}
\end{equation}
describes diagrams with an arbitrary amount of hidden variables located between $x_1$ and $x_2$, $x_2$ and $x_3$, or $x_3$ and $x_4$. The general $[z_i^{(1)}\star z_j^{(2)}\star z_k^{(3)}]$ approximant of this expansion (for $i,j,k>2$) reads:
\begin{eqnarray}
[z_i^{(1)}\star z_j^{(2)}\star z_k^{(3)}]&=&
\int_{a}^bf(x_0)dx_0 \int_a^{x_0} f(x_1|x_0) dx_1 \int_a^{x_1} f(z_1^{(1)}|x_1)dz_1^{(1)} \bigg [\prod_{p=2}^{i}\int_a^{x_1} f(z_p^{(1)}|z_{p-1}^{(1)})dz_p^{(1)}\bigg]\nonumber \\
&&\ \cdot\int_{x_1}^{x_0} f(x_2|z_i^{(1)})dx_2 \int_a^{x_2} f(z_1^{(2)}|x_2)dz_1^{(2)} \bigg [\prod_{p=2}^{j}\int_a^{x_2} f(z_p^{(2)}|z_{p-1}^{(2)})dz_p^{(2)}\bigg] \nonumber \\
&&\ \cdot\int_{2}^{x_0} f(x_3|z_j^{(2)}) dx_3 \int_a^{x_3}f(z_1^{(3)}|x_3)dz_1^{(3)}\bigg [\prod_{p=2}^{k}\int_a^{x_3} f(z_p^{(3)}|z_{p-1}^{(3)})dz_p^{(3)}\bigg] \nonumber \\
&&\ \cdot\int_{x_0}^bf(x_4|z_k^{(3)}) dx_4\nonumber
\end{eqnarray}
Finally, by induction we can prove that in the general case the diagrammatic series reads
\begin{equation}
P_\rightarrowtail(k) = \prod_{\Lambda=1}^{k-1} \bigg[\sum_{i{_\Lambda}=0}^\infty\bigg] \bigotimes_{\Lambda=1}^{k-1}[z_{i_\Lambda}^{(\Lambda)}]\bigg (\equiv \sum_{\alpha=0}^{\infty}P_\rightarrow^{(\alpha)}(k)\bigg),
\label{Pk}
\end{equation}
where the general term $$\bigotimes_{\Lambda=1}^{k-1}[z_{i_\Lambda}^{(\Lambda)}]=
[z_{i_1}^{(1)}\star z_{i_2}^{(2)}\dots \star z_{i_{k-1}}^{(k-1)}]$$
is a diagram that introduces the following correction in order $\alpha=\sum_{j=1}^{k-1}i_j$:
\begin{eqnarray}
\bigotimes_{\Lambda=1}^{k-1}[z_{i_\Lambda}^{(\Lambda)}]&=&
\int_{a}^bf(x_0)dx_0 \prod_{\Lambda=1}^{k-1}\bigg\{
\int_a^{x_0} f(x_1|x_0) dx_1 \int_a^{x_1} f(z_1^{(1)}|x_1)dz_1^{(1)} \bigg [\prod_{p=2}^{i}\int_a^{x_1} f(z_p^{(1)}|z_{p-1}^{(1)})dz_p^{(1)}\bigg]\nonumber \\
&&\ \cdot\int_{x_{i-1}}^{x_0} f(x_i|z_{\Lambda}^{(\Lambda)})dx_i \int_a^{x_i} f(z_1^{(\Lambda)}|x_i)dz_1^{(\Lambda)} \bigg [\prod_{p=2}^{\Lambda}\int_a^{x_i} f(z_p^{(\Lambda)}|z_{p-1}^{(\Lambda)})dz_p^{(\Lambda)}\bigg] \bigg\} \int_{x_0}^b f(x_k|z_{\Lambda}^{(\Lambda)})\nonumber
\end{eqnarray}
Concrete evaluation of each diagram depends on the type of dynamics involved (that is, on the explicit invariant density $f(x)$ and propagator $f(x|y)$), much in the same vein as explicit computation of Feynman diagrams depend on the type of field theory involved \cite{ramond}.\\
\section{Stochastic Markovian dynamics: the Ornstein-Uhlenbeck process}
An Ornstein-Uhlenbeck process \cite{kampen} is a stochastic process that describes the velocity of a massive Brownian particle under the influence of friction. It is the only stationary Gaussian Markov process (up to linear transformations). Since it is a Gaussian process, we have
$$f(x)=\frac{\exp(-x^2/2)}{\sqrt{2 \pi}}$$
and the transition probability reads
$$f(x_2|x_1)=\frac{\exp(-(x_2-Kx_1)^2/2(1-K^2))}{\sqrt{2 \pi}(1-K^2)},$$
where $K=\exp(-1/\tau)$ and the correlation function is $C(t)\sim \exp(-t/\tau)$. Variables are Gaussian, therefore $a=-\infty$ and $b=\infty$. Note that in this case we also expect convergent perturbation expansions of the type (\ref{pertu2}), since calculation of long distance visibility $U(n)$ for short-range correlated processes can be approximated, for large $n$, to the uncorrelated case.
\subsection{Exact solutions}
\noindent To begin, consider $P_\rightarrowtail(1)$. Note that this probability is only spanned by a single diagram (no hidden variables), yielding simply
$$P_\rightarrowtail(1)=\int_{-\infty}^{\infty}dx_0f(x_0)\int_{x_0}^{\infty} dx_1f(x_1|x_0)=\frac{1}{2},$$
for $\tau=1.0$.\\
\noindent $P_\rightarrowtail(2)$ is more involved, as it has corrections to the free field solution at all orders (see figure \ref{diagramsDHVG}). In the next section we compute a few terms of this diagrammatic expansion. Here we show that a mathematical trick can be used to compute all orders of $P_\rightarrowtail(2)$ simultaneously, yielding an exact result. The technique is numerical but nonetheless quite sound. We start by rewriting the contribution of each diagram in equation \ref{P2} as a recursive relation
\begin{eqnarray}
P_\rightarrowtail(2)=\int_{-\infty}^{\infty} f(x_0) \sum_{p=0}^\infty I(p|x_0)\nonumber
\label{P+2OU}
\end{eqnarray}
where $I(p|x_0)$ satisfies
\begin{equation}
I(p|x_0)=\int_{-\infty}^{x_0} dx_1f(x_1|x_0) G_p(x_1,x_1,x_0),\label{generalI}
\end{equation}
and $G_p$ satisfies:
\begin{eqnarray}
G_0(x,y,z)&\equiv&\int_z^\infty f(h|y)dh, \\
G_p(x,y,z)&=&\int_{-\infty}^x dh f(h|y)G_{p-1}(x,h,z), \ p\geq1. \label{Ggen}
\end{eqnarray}
(These latter recursions can be straightforwardly proved by induction on the number of hidden variables). The last equations represent a convolution that can be one more time formally rewritten as $G_p=TG_{p-1}$, or $G_p=T^pG_0$,
with an integral operator $T=\int_{-\infty}^x dh f(h|y)$. Accordingly,
\begin{equation}
I(p|x_0)=\int_{-\infty}^{x_0} dx_1f(x_1|x_0) \sum_{p=0}^{\infty}G_p(x_1,x_1,x_0)\equiv\int_{-\infty}^{x_0} dx_1f(x_1|x_0) S(x_1,x_1,x_0).
\label{p2m}
\end{equation}
In the last equation the formal sum $S(x,y,z)$ is defined as
\begin{equation}
S(x,y,z)=\sum_{p=0}^\infty G_p(x,y,z)=\sum_{p=0}^\infty T^p G_0=\frac{1}{1-T}G_0,\label{S1}
\end{equation}
where the convergence of last summation is guaranteed provided the spectral radius $r(T)<1$, that is,
\begin{equation}
\lim_{n\rightarrow\infty} ||T^n||^{1/n}<1,
\end{equation}
where $||T||=\max_{y\in(-\infty,x)} \int_{-\infty}^x dh |f(h|y)|$ is the norm of $T$. This condition is trivially fulfilled as $f(x|y)$ is a
Markov transition probability. Then, equation (\ref{S1}) can be written as $(1-T)S=G_0$, or more concretely
\begin{equation}
S(x,y,z)={G}_0(x,y,z) + \int_{-\infty}^x dh f(h|y) S(x,h,z), \label{SSS}
\end{equation}
which is a Volterra equation of the second kind for $S(x,y,z)$.
Typical one-dimensional Volterra integral equations can be numerically solved applying quadrature formulae for approximate the integral operator \cite{librodeernesto}. The technique can be easily extended in the case that the integral equation involves more than one variable, as it is this case, using a Simpson-type integration scheme to compute the function $S(x,y,z)$. One technical point is that one needs to replace the $-\infty$ limit in the integral by a sufficienly small number $a$. We have found that $a=-10$ is enough for a good convergence of the algorithm. Given a value of $z$ the recursion relation
\begin{eqnarray}
&&S(a,a+n\delta,z)= G_0(a,a+n\delta,z)\nonumber \\
&&S(a+k\delta,a+n\delta,z)= G_0(a,a+n\delta,z)+\delta\sum_{i=0}^{k-1}f(a+i\delta|a+n\delta)S(a+(k-1)\delta,a+i\delta,z)+ O(\delta^2)\nonumber \label{SS}
\end{eqnarray}
where $\delta$ is the step of the algorithm, for $k=0,1,2,\dots$ and $n=0,1,\dots,k$, allows us to compute $S(x,y,z)$ for $y\le x$. We find (for a correlation time $\tau=1.0$)
$$P_\rightarrowtail(2)\approx 0.24,$$
on good agreement with numerical experiments (table \ref{table2}).\\
\subsection{Perturbative expansions}
\noindent In general the infinite set of diagrams that contributes to each probability $P_\rightarrowtail(k)$ cannot be summed up in closed form, and a perturbative approach should be followed. Here we explore the accuracy and convergence of such series expansion (eq. \ref{pertu2}).\\
\noindent $\underline{\alpha=0}.$\\
\noindent Up to zeroth order (free field theory), we count only diagrams with no hidden variables. Accordingly,
the first degree probabilities are
\begin{eqnarray}
P_\rightarrowtail^{(0)}(1)=\int_{-\infty}^{\infty}dx_0f(x_0) \int_{x_0}^{\infty} dx_1f(x_1|x_0)\nonumber \\
P_\rightarrowtail^{(0)}(2)=\int_{-\infty}^{\infty}dx_0f(x_0)\int_{-\infty}^{x_0} dx_1f(x_1|x_0)
\int_{x_0}^{\infty} dx_2f(x_2|x_1)\nonumber \\
P_\rightarrowtail^{(0)}(3)=\int_{-\infty}^{\infty}dx_0f(x_0)\int_{-\infty}^{x_0} dx_1f(x_1|x_0)
\int_{x_1}^{x_0} dx_2f(x_2|x_1)\int_{x_0}^{\infty}dx_3f(x_3|x_2)\nonumber \\
\cdots
\end{eqnarray}
These can be calculated up to arbitrary precision. As an example, for a correlation time $\tau=1.0$ and approximating $(-\infty,\infty)$ by $(-4,4)$ (note that this is a good approximation as the OU process is Gaussian), we find
\begin{eqnarray}
P_\rightarrowtail^{(0)}(1)=0.500\nonumber \\
P_\rightarrowtail^{(0)}(2)=0.151\nonumber \\
P_\rightarrowtail^{(0)}(3)=0.043 \nonumber
\end{eqnarray}
Now, $P_\rightarrowtail^{(0)}(1)$ is exact as long as $P_\rightarrowtail(1)$ does not admit hidden variables.\\\\
\noindent $\underline{\alpha=1}.$\\
\noindent The interaction field couples an arbitrary number of hidden variables, distributed amongst the $k-1$ inner variables. Up to first order (figure \ref{diagramsDHVG}), we further count diagrams with one hidden variable. For instance,
\begin{eqnarray}
P_\rightarrowtail^{(1)}(2)=\int_{-\infty}^{\infty}dx_0f(x_0)\int_{-\infty}^{x_0} dx_1f(x_1|x_0)
\int_{-\infty}^{x_1}f(z_1|x_1)dz_1\int_{x_0}^{\infty} dx_2f(x_2|z_1)\approx 0.0366\nonumber
\end{eqnarray}
Therefore, up to first order, the analytical predictions are
\begin{eqnarray}
P_\rightarrowtail^{\text{th}}(1)=0.5\nonumber \\
P_\rightarrowtail^{\text{th}}(2)=0.19\nonumber
\end{eqnarray}
Similar calculations can be performed for other values of the degree $k$ and for higher order corrections $\alpha \geq 2$, in order to reach arbitrarily accurate estimations. The only restriction we have in this example is the power of our symbolic calculator, as we need to perform symbolic concatenated integrals of Gaussian functions.
\begin{table}
\begin{ruledtabular}
\begin{tabular}{ccccc}
degree $k$ &$P_\rightarrowtail(k)$ (numerics)&$P_\rightarrowtail(k)$ (exact)&$P_\rightarrowtail^{(0)}(k)$ &$P_\rightarrowtail^{(0)}(k)+P_\rightarrowtail^{(1)}(k)$\\
\hline
1&0.50&0.5&0.5&0.5\\
2&0.24&0.24&0.151&0.19\\
3& 0.12&?&0.043&\\
\end{tabular}
\end{ruledtabular}
\caption{\label{table2} Comparison of numerical results and theoretical predictions for the out degree distribution of DHVG associated to an Ornstein-Uhlenbeck process (see the text) with $\tau=1.0$. Numerics are the result of simulations, where a trajectory of $2^{20}$ time steps extracted from the map is mapped to its DHVG and the degree distribution is numerically computed. For $k=1$ perturbation theory at zeroth order is exact due to the restriction of forbidden patterns (see the text). For $k=2$ we can see that up to first order we already reach a reasonably accurate result.}
\end{table}
\noindent To evaluate the accuracy of zeroth and first order in the general case, we make some numerical simulations. We generate time series of $2^{20}$ data extracted from an OU process with the same characteristics, transform the series into a DHVg and compute its degree distribution. A comparison with our theory is shown in table \ref{table2}.
The first term ($k=1$) is exact in our theory and therefore coincides with the numerical experiments. The results of zeroth and first order corrections for $k=2,3$ are still far away from the experiments, however convergence guarantees that such methodology provides an arbitrary accurate result.\\
In the following section we address deterministic maps and show that convergence of the perturbative approach is faster than for stochastic processes.
\section{One dimensional deterministic maps with smooth invariant measure - the chaotic and quasiperiodic cases}
A deterministic map of the form $x_{t+1}=H(x_t)$ also has the Markov property, and therefore the same mathematical framework applies, provided that there exists an (invariant) probability measure $f(x)$ that characterizes the long-run proportion of time spent by the system in the various regions of the phase space. In the case of dissipative chaos, $f(x)$ describes the distribution of visits to different parts of the map's attractor. If such attractor has integer dimension, then the invariant measure is at least some piecewise continuous function and therefore integrable: in that case the general methodology presented in section IV holds. For chaotic maps with fractal attractor, a more general integration theory should be adopted (see the discussion section).\\
As $H$ is a deterministic map, the transition probability $f(x|y)$ (the propagator) reads
$$f(x|y)=\delta(x-H(y)),$$
where $\delta(x)$ is the Dirac generalised function. This means that nested integrals that appear in each diagram only yield a binary ($0/1$) result.
As we will see, this implies that the only effect of these nested integrals is to rescale the range of integration of $x_0$. For instance, in the case of dissipative chaotic maps this rescaling is in turn associated to the fixed point structure of the $n$th iterates $\{H(x), H^{(2)}(x),\cdots,H^{(n)}(x)\}$ (for piecewise linear maps, other criteria should be followed). Let us illustrate this with the calculation of $P_\rightarrowtail(1)$ associated to
a general one dimensional deterministic map $x_{t+1}=H(x_t)$. Recall that $P_\rightarrowtail(1)$ does not allow hidden variables and therefore its computation is only related to a single diagram, whose probability is
$$P_\rightarrowtail(1)=\int_a^b f(x_0)dx_0\int_{x_0}^b \delta (x_1-H(x_0))dx_1.$$
Now, interestingly, the dependent integral vanishes $\forall x_0$ if $H(x_0)<x_0$, and is equal to one otherwise. Therefore, this rescales the integration range of $x_0$
$$[a,b]\Longrightarrow [a',b'],$$
such that for $x_0 \in [a',b']$, $H(x_0)<x_0$. Hence and
$$P_\rightarrowtail(1)=\int_{a'}^{b'} f(x_0)dx_0$$
If the map is unimodal, then $a'=x^*$ and $b'=b$, where $x^*$ is the unstable fixed point of $H$ satisfying $H(x^*)=x^*$. In general, in order to compute each of the diagrams associated to $P_\rightarrowtail(k)$, the $n$-th nested integral rescales the integration range according to the fixed points of $H^{n}(x)$. This property makes the calculation of these diagrams in the case of chaotic dynamics much easier than for stochastic Markovian dynamics. Moreover, the fact that the chaotic trajectories visit the phase space in an orchestrated way introduces forbidden patterns \cite{forbidden} in the temporal order of visits to regions of the phase space. This further introduces 'forbidden diagrams', what will reduce significantly the number of corrections that need to be counted up to each other $\alpha$. On top of that, note that in this case we also expect convergent perturbation expansions of the type (\ref{pertu2}): as the correlation function $<x_t x_{t+\tau}$ of chaotic processes vanishes for distant data ($\tau<<1$), long distance visibility $U(n)$ for chaotic processes can be approximated, for large $n$, to the uncorrelated case. In what follows we focus on two concrete maps: a chaotic map with integer dimension (the fully chaotic logistic map) and a nonchaotic quasiperiodic map.
\subsection{Chaotic dynamics}
For the sake of concreteness, we focus on logistic map $H(x_n)=\mu x_n(1-x_n)$ with parameter $\mu = 4$, where the map is ergodic, the attractor is the whole
interval $[0,1]$ and the invariant measure $f(x)$ corresponds to a Beta distribution
\begin{equation}
f(x)=\frac{1}{\pi \sqrt{x(1-x)}}
\label{rho}
\end{equation}
This is topologically conjugate to the tent and Bernoulli maps. Its fixed points and unstable periodic orbits (fixed points of $H^{(n)}$) are
\begin{eqnarray}
&& H^{(1)}(x)=4x(1-x), \ x^*\equiv S_1=\{0,3/4\}\nonumber \\
&& H^{(2)}(x)=-16x(x-1)(4x^2-4x+1), \ x^*\equiv S_2=\{0, 3/4, \frac{5-\sqrt{5}}{8}, \frac{5+\sqrt{5}}{8}\}\nonumber \\
&& H^{(3)}(x)=-64x(x-1)(4x^2-4x+1)(64x^4-128x^3+80x^2-16x+1), \ x^*\equiv S_3 \label{fp}
\end{eqnarray}
where $S_n, \ n\geq3$ can only be computed numerically according to Abel-Ruffini's theorem (see figure \ref{lm} for a cobweb diagram of several iterates $H^{(n)}(x)$).
\begin{figure}
\centering
\includegraphics[width=0.9\columnwidth]{lm.pdf}
\caption{Cobweb plot of the fully chaotic logistic map $H(x)=4x(1-x)$ and some of its iterates $H^{(n)}(x)$. The specific intervals where $H^{(n)}(x)>(<)x$ depend on the distribution of fixed points $\{x^*\}$, that fulfill $H^{(n)}(x^*)=x^*$.}
\label{lm}
\end{figure}
In what follows we study the diagrammatic expansion for all degree $k$.
To begin, consider $P_\rightarrowtail(1)$, generated by a single diagram
\begin{equation}
P_\rightarrowtail(1)=\int_0^1 f(x_0)dx_0\int_{x_0}^1 \delta (x_1-H(x_0))dx_1.
\end{equation}
The second integral is one for $H(x_0)>x_0$ (and zero otherwise), that is, for $x_0<3/4$, where $x^*=S_1=\{0,3/4\}$ are the fixed points of $H(x)$. Therefore,
\begin{equation}
P_\rightarrowtail(1)=\int_0^{3/4} \frac{dx_0}{\pi\sqrt{x(1-x)}}=\frac{2}{3}
\end{equation}
\noindent In a second step, consider $P_\rightarrowtail(2)$, which is generated by an infinite number of diagrams, each of which with $n$ hidden variables. This again can be written down perturbatively
$$P_\rightarrowtail(2) = \sum_{\alpha=0}^{\infty}P_\rightarrowtail^{(\alpha)}(2).$$ At zeroth order, the diagram reads
\begin{equation}
P_\rightarrowtail^{(0)}(2)= \int_0^1 f(x_0)dx_0\int_{0}^{x_0} \delta (x_1-H(x_0))dx_1\int_{x_0}^1 \delta(x_2-H(x_1))dx_2.
\end{equation}
The last integrals rescale the integration range of $x_0$ according to the solutions of:
\begin{eqnarray}
H(x_0)<x_0\nonumber\\
H^{2}(x_0)>x_0.
\label{GEN1}
\end{eqnarray}
The first condition rescales $x_0 \in [3/4,1]$. For the second condition, we use the fixed points set $S_2$ (see (\ref{fp})), and after a little algebra we find that if we label $p=\frac{5-\sqrt{5}}{8}; q=\frac{5+\sqrt{5}}{8}$, we get
$$H^{2}(x_0)>x_0 \Longleftrightarrow x_0 \in [0,p] \cup [3/4,q].$$
The intersection of both conditions is then $[3/4,(5+\sqrt{5})/8$, hence
\begin{equation}
P_\rightarrowtail^{(0)}(2)= \int_{3/4}^{\frac{5+\sqrt{5}}{8}} \frac{dx_0}{\pi\sqrt{x(1-x)}}=\frac{2}{15}
\end{equation}
Now, the rest of orders require $\alpha>0$ hidden variables. In every such diagram (see figure \ref{diagramsDHVG}), an implicit condition is $x_1<x_0$ and $z_1<x_0$, where $z_1$ is the first hidden variable. However, this is a forbidden 3-pattern: there are no three consecutive points in any orbit of the logistic map forming a strictly decreasing trio \cite{forbidden} Therefore we conclude that the infinite number of diagrams with hidden variables do not further contribute to $P_\rightarrowtail(2)$, and in this special case the probability is exactly solvable
\begin{eqnarray}
&&P_\rightarrow^{(0)}(2)=2/15 \nonumber \\
&&P_\rightarrow^{(\alpha >0 )}(2)=0\nonumber \\
&&\Rightarrow P_\rightarrow(2)=\frac{2}{15}
\end{eqnarray}
\noindent Let us proceed forward with $P_\rightarrowtail(3)$. We will show that the diagrams associated to each perturbation order $\alpha$ can be formally calculated using the fixed point structure of $H^{(n)}(x)$.\\
\noindent $\underline{\alpha=0}$\\
The zeroth order diagram (no hidden variables) contributes with only one diagram such that
$$P_\rightarrowtail^{(0)}(3)=\underbrace{\int_0^1 f(x_0)dx_0}_{I_0}\underbrace{\int_0^{x_0} \delta (x_1-H(x_0))dx_1}_{I_1} \underbrace{\int_{x_1}^{x_0} \delta(x_2-H(x_1))dx_2}_{I_2} \underbrace{\int_{x_0}^1 \delta(x_3-H(x_2))dx_3}_{I_3}.$$
Each integral $I_m$ rescales the integration range in $I_0$ according to the intersection of the solution interval of the following inequalities
\begin{eqnarray}
I_1\rightarrow H(x_0)<x_0\nonumber\\
I_2\rightarrow H^{2}(x_0)<x_0\nonumber\\
I_3\rightarrow H^{3}(x_0)>x_0
\label{GEN2}
\end{eqnarray}
We have:
$$H(x_0)<x_0\Rightarrow x_0 \in [3/4,1].$$
On top of that, the second inequality reduces this range further:
$$H(x_0)<x_0 \cap H^{2}(x_0)<x_0 \Rightarrow x_0 \in [\frac{5+\sqrt{5}}{8},1].$$
Last inequality cannot be solved in an algebraic closed form (the order of the polynomial is larger than five), but the solution can be calculated numerically up to arbitrary precision, finding that the intersection interval is
$$H(x_0)<x_0 \cap H^{2}(x_0)<x_0 \cap H^{3}(x_0)>x_0\Rightarrow x_0 \in [.95, .97],$$
therefore
$$P_\rightarrowtail^{(0)}(3)\approx \int_{0.95}^{0.97} \frac{dx_0}{\pi \sqrt{x(1-x)}}=0.0315$$
\noindent $\underline{\alpha=1}$\\
Note that for $\alpha=1$ (diagrams with one hidden variable), only one of the two possible diagrams is allowed, as the other one is again a forbidden one.
The allowed diagram yields a correction
\begin{eqnarray}
&&P_\rightarrowtail^{(1)}(3)=\underbrace{\int_0^1 f(x_0)dx_0}_{I_0}\underbrace{\int_0^{x_0} \delta (x_1-H(x_0))dx_1}_{I_1} \cdot \nonumber \\ &&\cdot\underbrace{\int_{x_1}^{x_0} \delta(x_2-H^{(2)}(x_0))dx_2}_{I_2} \underbrace{\int_{0}^{x_2} \delta(z_1-H^{(3)}(x_0))dz_1}_{I_3} \underbrace{\int_{x_0}^1 \delta(x_3-H^{(4)}(x_0))dx_3}_{I_4},
\end{eqnarray}
where again each integral $I_m$ contributes with a rescaling of the integration range in $I_0$ according to the solutions of
\begin{eqnarray}
I_1\rightarrow H(x_0)<x_0\nonumber\\
I_2\rightarrow H^{2}(x_0)<x_0\nonumber\\
I_3\rightarrow H^{3}(x_0)<H^{2}(x_0)\nonumber\\
I_4\rightarrow H^{4}(x_0)>x_0
\label{GEN3}
\end{eqnarray}
Proceeding as before, we have:
$$H(x_0)<x_0\Rightarrow x_0 \in [3/4,1].$$
On top of that, the second inequality reduces this range further:
$$H(x_0)<x_0 \cap H^{(2)}(x_0)<x_0 \Rightarrow x_0 \in \bigg[\frac{5+\sqrt{5}}{8},1\bigg],$$
The third inequality reduce it further on:
$$H(x_0)<x_0\ \cap\ H^{(2)}(x_0)<x_0 \ \cap\ H^{(3)}(x_0)< H^{(2)}(x_0) \Rightarrow x_0 \in \bigg[\frac{5+\sqrt{5}}{8},\frac{1}{2}+\frac{\sqrt 3}{4}\bigg]$$
and finally
$$H(x_0)<x_0\ \cap\ H^{(2)}(x_0)<x_0 \ \cap\ H^{(3)}(x_0)< H^{(2)}(x_0) \ H^{4}(x_0)>x_0\Rightarrow x_0 \in [\frac{5+\sqrt{5}}{8}, .925]$$
and therefore the correction of order $\alpha=1$ is
$$P_\rightarrowtail^{(1)}(3)\approx \int_{0.904}^{0.925} \frac{dx_0}{\pi \sqrt{x(1-x)}}=0.024$$
Summarising, up to first order, the predicted probability
$$P_\rightarrowtail(3)\approx P_\rightarrowtail^{(0)}(3)+P_\rightarrowtail^{(1)}(3)=0.0555$$
\noindent $\underline{\alpha>1}$\\
Before addressing the general high order correction $P_\rightarrowtail^{(\alpha>1)}(3)$, we can say a word about forbidden diagrams. Note that the number of diagrams with $\alpha$ hidden variables can be further labelled using a full binary tree. The seed of the tree is a node that corresponds to the diagram with no hidden variables. This node has a total of $k-1$ offsprings, as the first hidden variable can be placed after one of the $k-2$ variables that are not bounding variables. From there, each node has always two children, labelled $L$ (if the new hidden variable is placed to the left) or $R$ (right). Therefore, any diagram can be uniquely labelled. Diagrams with $\alpha$ hidden variables all of them placed after $x_j$ will be labelled as $j$ followed by a string of letters $L$ or $R$ of size $p-1$.\\
On the other hand, note that all the diagrams that are descendants (offsprings, offsprings of offsprings, etc) of a forbidden diagram are, trivially, also forbidden diagrams. Therefore, it is easy to prove that for $k=3$, at order $\alpha>1$, out of the possible $2^{p}$ diagrams $2^{p-1}$ are forbidden. Moreover, forbidden patterns appear at all levels, in a hierarchical way. Accordingly, only one diagram contributes at order $\alpha$, labelled as $R\underbrace{LL \cdots L}_{\alpha-1}$, whose probability reads
$$P_\rightarrowtail^{(\alpha)}(3)=\int_u^v f(x_0)dx_0,$$
where $[u,v]$ is the range of values of $x_0$ which are solution to the set of inequalities
\begin{eqnarray}
H(x_0)<x_0\nonumber\\
H^{(2)}(x_0)<x_0\nonumber\\
H^{(3)}(x_0)<H^{(2)}(x_0)\nonumber\\
H^{(4)}(x_0)<H^{(2)}(x_0)\nonumber\\
\cdots \nonumber\\
H^{(2+\alpha)}(x_0)<H^{(2)}(x_0)\nonumber\\
H^{(3+\alpha)}(x_0)>x_0
\label{GEN4}
\end{eqnarray}
\noindent Abel-Ruffini's theorem precludes closed solutions for $\alpha\geq 0$, however such corrections can be calculated up to arbitrary precision as they are based on univariate algebraic equations. When we try to calculate some approximants, we find
\begin{eqnarray}
P_\rightarrowtail^{(2)}(3)=P_\rightarrowtail^{(3)}(3)=P_\rightarrowtail^{(4)}(3)=0
\end{eqnarray}
as the intersection range for these cases is empty. By induction we can prove that
$$P_\rightarrowtail^{(\alpha>1)}(3)=0,$$ and therefore
the result up to first order is indeed exact:
\begin{eqnarray}
&&P_\rightarrow^{(0)}(3)=0.0315 \nonumber \\
&&P_\rightarrow^{(1)}(3)=0.024 \nonumber \\
&&P_\rightarrow^{(\alpha >1 )}(3)=0\nonumber \\
&&\Rightarrow P_\rightarrow(3)=P_\rightarrow^{(0)}(3)+P_\rightarrow^{(1)}(3)\approx 0.0555
\end{eqnarray}
\noindent In what follows we investigate a general expression for the zeroth order of a general degree $k$. As this has already been calculated for $k\leq3$, let us start with $P_\rightarrowtail(4)$. The zeroth order diagram reads
\begin{eqnarray}
&&P_\rightarrowtail^{(0)}(4)=\underbrace{\int_0^1 f(x_0)dx_0}_{I_0}\underbrace{\int_0^{x_0} \delta (x_1-H(x_0))dx_1}_{I_1} \cdot \nonumber \\ &&\cdot\underbrace{\int_{x_1}^{x_0} \delta(x_2-H^{(2)}(x_0))dx_2}_{I_2} \underbrace{\int_{x_2}^{x_0} \delta(x_3-H^{(3)}(x_0))dx_3}_{I_3} \underbrace{\int_{x_0}^1 \delta(x_4-H^{(4)}(x_0))dx_4}_{I_4}=\nonumber \\
&&\int_u^v f(x_0)dx_0,
\end{eqnarray}
where $[u,v]$ is the range of values of $x_0$ which are solution to the set of inequalities
\begin{eqnarray}
H(x_0)<x_0\nonumber\\
H^{2}(x_0)<x_0\nonumber\\
H^{3}(x_0)<x_0\nonumber\\
H^{4}(x_0)>x_0
\label{GEN5}
\end{eqnarray}
A pattern is now evident and a formal expression for the zeroth order diagram up to arbitrary degree $k$ can be found by induction:
\begin{eqnarray}
P_\rightarrowtail^{(0)}(k)= \int_u^v f(x_0)dx_0, \
\end{eqnarray}
where $[u,v]$ is the range of values of $x_0$ which are solution to the system of inequalities
\begin{eqnarray}
&&H^{i}(x_0)<x_0, i=1,\dots,k-1 \nonumber \\
&&H^{k}(x_0)>x_0
\label{GEN6}
\end{eqnarray}
These inequalities can be solved up to arbitrary numerical precision. Some results include:
\begin{eqnarray}
P_\rightarrowtail^{(0)}(4)=0.03 \nonumber \\
P_\rightarrowtail^{(0)}(5)= 0.02\nonumber \\
P_\rightarrowtail^{(0)}(6)= 0.014\nonumber
\end{eqnarray}
\noindent Note that the analysis performed here is far more general than the case of the logistic map, and indeed the set of inequalities (\ref{GEN1}, \ref{GEN2}, \ref{GEN3}, \ref{GEN4}, \ref{GEN5}, \ref{GEN6}) holds in general provided the map has an invariant, $L_1$-integrable $f(x)$. To evaluate the accuracy and convergence speed of the perturbative expansion in the concrete case of the fully chaotic logistic map, we make some numerical simulations. We generate time series of $2^{20}$ data extracted from a logistic map with the same characteristics, transform the series into a DHVg and compute its degree distribution. A comparison with our theory is shown in table \ref{table1}.
The first three terms $k=1,2,3$ show a perfect agreement with numerics because in those cases the theory was exact. For the rest, we can see that the free field solution (zeroth order) gets more accurate for larger values of $k$.
\begin{table}
\begin{ruledtabular}
\begin{tabular}{cccc}
degree $k$ &$P_\rightarrowtail(k)$ (numerics)&$P_\rightarrowtail^{(0)}(k)$ &$P_\rightarrowtail^{(0)}(k)+P_\rightarrowtail^{(1)}(k)$\\
\hline
1&0.666&0.666&0.666\\
2&0.133&$\frac{2}{15}=0.133$&$\frac{2}{15}$\\
3&0.055&0.0315&0.055\\
4&0.044&0.03&\\
5&0.03&0.02&\\
6&0.02&0.014&\\
\end{tabular}
\end{ruledtabular}
\caption{\label{table1}Comparison of numerical results and theoretical predictions for the out degree distribution of DHVG associated to the fully chaotic logistic map
$x_{t+1}=4x_t(1-x_t)$. Numerics are the result of simulations, where a trajectory of $2^{20}$ time steps extracted from the map is mapped to its DHVG and the degree distribution is numerically computed. For $k=1,2$ perturbation theory at zeroth order is exact due to the restriction of forbidden patterns (see the text). For $k=3$ we need to go up to first order to reach an accurate result.}
\end{table}
\subsection{Quasiperiodic dynamics}
We now address a paradigmatic example of a nonchaotic deterministic dynamics which is nonetheless aperiodic. Consider the Poincare map associated with a flow having two incommensurate frequencies \cite{just}, or simply an irrational rotation
$$H(x)=(x+\omega) \mod 1, $$
where $\omega$ is irrational which without loss of generality we may assume to be smaller than one (otherwise, the map is equivalent to $H(x)=(x+\omega-\lfloor \omega \rfloor) \mod 1 $.) Trajectories are aperiodic and fill up densely the interval $[0,1]$. The map is ergodic and has a uniform invariant measure $U[0,1]$, albeit having zero entropy. In our notation,
$$f(x)=1,$$
therefore the integration rescaling $[a,b]\rightarrow [u,v]$ exposed in the previous section yields a general diagram whose probability correction is simply $v-u$. A cobweb diagram of some iterations of this map is shown in figure \ref{qp}.
The nth-iteration of the map can be simply written as
$$H^{(n)}(x)=(x+n\omega) \mod 1,$$
what yields a simpler set of criteria:
\begin{eqnarray}
H^{(n)}(x)<x &\Longleftrightarrow & x > 1- n\omega\mod 1,\ \forall n \geq 1\nonumber \\
&\Longleftrightarrow & \left\{
\begin{array}{l}
x>1-n\omega, \ n<1/\omega, \\
x>1-n\omega + \lfloor n\omega \rfloor, n>1/\omega %
\end{array}%
\right.
\end{eqnarray}
\begin{figure}
\centering
\includegraphics[width=0.9\columnwidth]{qp.pdf}
\caption{Cobweb plot of the quasiperiodic map $H(x)=x +\omega \mod 1$, where $\omega$ is irrational (in this figure $\omega=\exp(-1)$) and some of its iterates $H^{(n)}(x)$. The specific intervals where $H^{(n)}(x)>(<)x$.}
\label{qp}
\end{figure}
The calculations are conceptually equivalent to the ones in previous section, although the concrete techniques differ as this map does not has fixed points. In what follows we show a summary of the first terms. First,
\begin{equation}
P_\rightarrowtail(1)=\int_0^1 dx_0\int_{x_0}^1 \delta (x_1-H(x_0))dx_1.
\end{equation}
The second integral is $1$ for $H(x_0)>x_0$ (and zero otherwise), that is, for $x_0<1-\omega$. Therefore,
\begin{equation}
P_\rightarrowtail(1)=\int_0^{1-\omega} dx_0=1-\omega
\end{equation}
For $P_\rightarrowtail(2)$ we expand again in a perturbation series
$$P_\rightarrowtail(2) = \sum_{\alpha=0}^{\infty}P_\rightarrowtail^{(\alpha)}(2),$$
whose free field solution $\alpha=0$ reduces to the intersection
$$[1-\omega, 1] \cap [0,1-2\omega]$$
This interval is non null for $\omega>1/2$ and doesn't contribute otherwise (forbidden pattern). For $\omega>1/2$ its contribution is
$$P_\rightarrowtail^{(0)}(2)=\lfloor 2\omega \rfloor - \omega$$
Similarly, the correction of order $\alpha=1$ reduces to the intersection
\begin{eqnarray}
(H(x)< x) \cap (H^{(2)}(x)< H(x)) \cap (H^{(3)}(x)>x)
\end{eqnarray}
which is null for $\omega >1/2$. By iteration, it can be proved that
$$P_\rightarrowtail(2) = \sum_{\alpha=1}^{\infty}P_\rightarrowtail^{(\alpha)}(2)=0, \ \forall \omega$$
Similarly, the free field solution for $P_\rightarrowtail(3)$ reduces to
\begin{eqnarray}
(H(x)< x) \cap (H^{(2)}(x)< x) \cap (H^{(3)}(x)>x).
\end{eqnarray}
The first inequality requires $$x \in [1-\omega, 1], \ \forall \omega.$$ The second one requires
$$x \left\{
\begin{array}{l}
\in [1-2\omega, 1], \ \forall \omega < 1/2, \\
\in [1-2\omega + \lfloor 2\omega \rfloor,1], \ \forall \omega > 1/2 %
\end{array}%
\right.
$$
and the third requires
$$x \left\{
\begin{array}{l}
\in [0, 1-3\omega], \ \forall \omega < 1/3, \\
\in [0,1-3\omega+\lfloor 3\omega \rfloor], \ \forall \omega > 1/3 %
\end{array}%
\right.
$$
So depending on the specific value of $\omega$, this contribution can be a finite or null value. In table \ref{qp} we summarise these theoretical estimations for different values of $\omega$, along with the results of numerical simulations.
\begin{table}
\begin{ruledtabular}
\begin{tabular}{ccccc}
$k$ &$\omega$& $P_\rightarrowtail^{(0)}(k)$ & Numerics\\
\hline
1&$\exp(-1)$&0.632&0.632\\
2&$\exp(-1)$&0&0\\
3&$\exp(-1)$&0.264&0.264\\
1&$\pi-3$&0.858&0.858\\
2&$\pi-3$&0&0\\
3&$\pi-3$&0&0\\
1&$\phi^{-1}$&0.382&0.382\\
2&$\phi^{-1}$&0.382&0.382\\
3&$\phi^{-1}$&0&0.146\\
1&$\sqrt 2-1$&0.5857&0.5857\\
2&$\sqrt 2-1$&0&0\\
3&$\sqrt 2-1$&0.171&0.3431\\
\end{tabular}
\end{ruledtabular}
\caption{\label{tableqp}Comparison of numerical results and theoretical predictions for the out degree distribution of DHVG associated to the irrational rotation
$x_{t+1}=x_t + \omega \mod 1$, where $\omega$ is an irrational number $\in (0,1)$. Numerics are the result of simulations, where a trajectory of $2^{20}$ time steps extracted from the map is transformed into a DHVG and the degree distribution is numerically computed. For $k=1,2$ perturbation theory at zeroth order is exact due to the restriction of forbidden patterns (see the text). For $k=3$ the zeroth order is an accurate approximation of the experimental solution for some values of $\omega$.}
\end{table}
To summarise, in the last sections we have developed a formal theory to compute concrete degree probabilities $P_\rightarrow(k)$ of DHVGs associated to different types of Markovian dynamics. In the next section we take a totally different approach and investigate how to make use of calculus of variations to calculate the whole degree distribution associated to these processes, and for completeness we extend these to both DHVG and HVGs.
\section{Variational techniques}
\noindent Let the Shannon entropy over the degree distribution of an HVG, ${\Phi }\{P(k)\}$ be
$${\Phi}\{P(k)\}=-\sum_{k=2}^{\infty} P(k) \log P(k)$$
On the other hand, periodic series of period $T$ yield HVGs with mean degree \cite{chaos, Gutin}
$$\langle k \rangle_{\text{HVG}}=4\bigg(1-\frac{1}{2T}\bigg),$$
which means that aperiodic series reach the upper bound $\langle k \rangle_{\text{HVG}}=4$, independently of whether the underlying process is deterministic or stochastic. In previous research \cite{chaos} it was shown that HVGs associated to uncorrelated random processes are maximally entropic, in the sense that the degree distributions that maximise $S$, if we require $P(k)$ to be normalised and that $\langle k \rangle_{\text{HVG}}=4$, reduces to equation \ref{uncorr}. In other words, uncorrelated random processes are maximally ${\Phi}_{\text{HVG}}$-entropic processes.\\
\noindent In this section we first show that a similar principle holds for DHVGs and that the DHVGs associated to uncorrelated random processes are maximally entropic.
We then conjecture that stochastic and chaotic processes are also maximally entropic restricted to further requirements: whereas they are also aperiodic
and the mean degree is maximal, they have correlations recorded by $P_\rightarrow(k)$ for small values of the degree. To test this conjecture,
we formulate a similar MaxEnt problem where the values of $P_\rightarrow(s), s=1,2,3$ (calculated exactly in previous sections) are fixed. We also explore this principle for HVGs.
\subsection{Uncorrelated random processes: maximally entropic DHVGs}
First of all note a periodic series of period $T$ (generated from a stochastic or deterministic Markovian dynamical system) maps to a DHVG whose mean in and out degree read
$$\langle k_\leftarrow \rangle_{\text{DHVG}}=\langle k_\rightarrow \rangle_{\text{DHVG}}=2\bigg(1-\frac{1}{2T}\bigg)$$
(the proof trivially follows from the one for the HVG \cite{chaos} and is therefore skipped in this presentation). The maximal mean degree of a DHVG is therefore $\langle k_\leftarrow \rangle_{\text{DHVG}}=\langle k_\rightarrow \rangle_{\text{DHVG}}=2$, reached for aperiodic series.\\
\noindent In order to calculate the degree distribution $P_\rightarrow(k)$ that maximises the entropy of a DHVG ${\Phi}\{P_\rightarrow(k)\}=\sum_{k=1}^{\infty} P_\rightarrow(k) \log P_\rightarrow(k)$, let us define the following Lagrangian
\begin{eqnarray}
{\cal L}_\rightarrow= -\sum_{k=1}^{\infty} P_\rightarrow(k) \log P_\rightarrow(k) -\lambda_0\bigg(\sum_{k=1}^{\infty} P_\rightarrow(k) -1\bigg)- \lambda_1\bigg(\sum_{k=1}^{\infty} k P_\rightarrow(k) - 2\bigg)
\end{eqnarray}
The extremum condition reads
\begin{eqnarray}
\frac{\partial \cal L_\rightarrow}{\partial P_\rightarrow(k)}=0 \nonumber
\end{eqnarray}
whose general solution is an exponential function
$$P_\rightarrow(k)= \exp(-\lambda_0-\lambda_1 k),$$
where $\lambda_0$ and $\lambda_1$ are Lagrange multipliers that can be solved from the constraints. The first constraint is the distribution normalisation
\begin{equation}
\sum_{k=1}^{\infty }{e^{-\lambda _{0}-\lambda _{1}k}}=1,
\label{RR0}
\end{equation}%
which implies the following relation between $\lambda _{0}$ and $\lambda _{1}$
\begin{equation}
e^{\lambda _{0}}=\frac{1}{e^{\lambda_1 - 1}},
\label{RR1}
\end{equation}%
using the sum of a trigonometric series. Now, the second restriction is on the mean degree:
\begin{equation}
\sum_{k=1}^{\infty} ke^{-\lambda_{0}-\lambda_{1}k} =e^{-\lambda_{0}}\sum_{k=1}^{\infty }ke^{-\lambda_{1}k}=2
\label{RR2}
\end{equation}%
Now, notice that
$$ \frac{\partial}{\partial \lambda_1} \bigg( \sum e^{-\lambda_1 k} \bigg)= - \sum k e^{-\lambda_1 k}$$
and therefore differentiating equation \ref{RR0} with \ref{RR1} gives
$$ \sum k e^{-\lambda_1 k}=\frac{e^{\lambda_1}}{(e^{\lambda_1}-1)^2}
$$
which we introduce in equation \ref{RR2} to find
$$\frac{e^{\lambda_1}}{(e^{\lambda_1}-1)^2}=2e^{\lambda_0}.$$
This last expression, together with equation \ref{RR1}, solves $\lambda_0$ and $\lambda_1$
$$\lambda_0=0 ,\ \lambda_1=\log 2.$$
Therefore, the degree distribution that maximizes ${\cal L}_\rightarrow$ is
\begin{equation*}
P(k)=\bigg(\frac{1}{2}\bigg)^k, \ k=1,2,...
\end{equation*}%
which is the result found for uncorrelated random processes (theorem 2). We conclude that the DHVG of these processes are maximally entropic or, equivalently, that uncorrelated random processes are maximally $\Phi_{\text{DHVG}}$-entropic.
\subsection{Ornstein-Uhlenbeck and logistic map I: DHVGS}
\noindent There is numerical evidence \cite{irrev} that suggests that both chaotic and some correlated stochastic processes also have DHVGs with exponentially decaying degree distributions, for $k$ sufficiently large. An heuristic justificaton argues that for long times, chaotic and random uncorrelated processes cannot be distinguished, and the same holds for correlated stochastic processes with a fast decaying correlation function (Ornstein-Uhlenbeck). Therefore we expect that the degree distribution of these processes deviate from the i.i.d. theory for short times, whereas we conjecture that a similar entropic extremization may take place for sufficiently large values. The question of course is to determine what "sufficiently large" means, that is, which is the minimal value of $k$ where the exponential decay starts to be a good approximation. We define an $s$- dependent Lagrangian ${\cal L}_\rightarrow(s)$, such that
\begin{eqnarray}
{\cal L}_\rightarrow(1) &=& -\sum_{k=2}^{\infty} Q_\rightarrow(k) \log Q_\rightarrow(k) -\lambda_0\bigg(\sum_{k=2}^{\infty} Q_\rightarrow(k) -(1-p_1)\bigg)- \nonumber\\
&&- \lambda_1\bigg(\sum_{k=2}^{\infty} kQ_\rightarrow(k) + p_1-2\bigg),
\end{eqnarray}
where $p_1=P_\rightarrow(1)$,
\begin{eqnarray}
{\cal L}_\rightarrow(2) &=& -\sum_{k=3}^{\infty} Q_\rightarrow(k) \log Q_\rightarrow(k) -\lambda_0\bigg(\sum_{k=3}^{\infty} Q_\rightarrow(k) -(1-p_1-p_2)\bigg)- \nonumber\\
&&- \lambda_1\bigg(\sum_{k=3}^{\infty} kQ_\rightarrow(k) + p_1+2p_2-2\bigg),
\end{eqnarray}
where $p_2=P_\rightarrow(2)$, and
\begin{eqnarray}
{\cal L}_\rightarrow(3) &=& -\sum_{k=4}^{\infty} Q_\rightarrow(k) \log Q_\rightarrow(k) -\lambda_0\bigg(\sum_{k=4}^{\infty} Q_\rightarrow(k) -(1-p_1-p_2-p_3)\bigg)- \nonumber\\
&&- \lambda_1\bigg(\sum_{k=4}^{\infty} kQ_\rightarrow(k) + p_1+2p_2+3p_3-2\bigg).
\end{eqnarray}
where $p_3=P_\rightarrow(3)$. From previous sections, we have learned how to compute these terms, either in an exact or in a perturbative way. In this section we only use exact results for these terms. Our approach consists in finding $Q_\rightarrow(k)$ such that
\begin{equation}
P_\rightarrow(k)=\left\{
\begin{array}{l}
p_k, \ k\leq s, \\
Q_\rightarrow(k), \ k > s.%
\end{array}%
\right. \label{pkRG}
\end{equation}%
If these graphs are still maximally entropic, then a MaxEnt argument should predict the correct shape of the full distribution.\\
After a little algebra we come to a system of two equations for the Lagrange multipliers $\{\lambda_0, \lambda_1\}_s$, such that for $s=1$ we have
\begin{eqnarray}
e^{\lambda_0}=\frac{1}{1-p_1} \bigg(\frac{1}{e^{\lambda_1}-1}-e^{-\lambda_1}\bigg)\nonumber \\
e^{\lambda_0}=\frac{1}{2-p_1}\bigg( \frac{e^{\lambda_1}}{(e^{\lambda_1}-1)^2 } - e^{-\lambda_1} \bigg),
\end{eqnarray}
for $s=2$, $\{\lambda_0, \lambda_1\}_2$ fulfills
\begin{eqnarray}
e^{\lambda_0}=\frac{1}{1-p_1-p_2} \bigg(\frac{1}{e^{\lambda_1}-1}-e^{-\lambda_1}-e^{-2\lambda_1}\bigg)\nonumber \\
e^{\lambda_0}=\frac{1}{2-p_1-2p_2}\bigg( \frac{e^{\lambda_1}}{(e^{\lambda_1}-1)^2 } - e^{-\lambda_1} -2e^{-2\lambda_1} \bigg)
\end{eqnarray}
while for $s=3$, $\{\lambda_0, \lambda_1\}_3$ fulfills
\begin{eqnarray}
e^{\lambda_0}=\frac{1}{1-p_1-p_2-p_3} \bigg(\frac{1}{e^{\lambda_1}-1}-e^{-\lambda_1}-e^{-2\lambda_1}e^{-3\lambda_1}\bigg)\nonumber \\
e^{\lambda_0}=\frac{1}{2-p_1-2p_2-3p_3}\bigg( \frac{e^{\lambda_1}}{(e^{\lambda_1}-1)^2 } - e^{-\lambda_1} -2e^{-2\lambda_1} -3e^{-3\lambda_1} \bigg)
\end{eqnarray}
Note again that this formulation can be easily extended for an arbitrary $s$, if needed. Here we will investigate the results for $s=2,3$.\\
\begin{figure}
\centering
\includegraphics[width=0.9\columnwidth]{maxent_caos.pdf}
\caption{Semi log plot of the degree distribution $P_\rightarrow(k)$ of a DHVG associated to a fully chaotic logistic map. Dots are the result of numerical simulations on a time series of $2^{20}$ data, solid lines are the prediction of the variational approach.}
\label{maxent1}
\end{figure}
\noindent Let us start with the fully chaotic logistic map. For ${\cal L}_\rightarrow(2)$, where $p_1=1/3, \ p_2=2/15$, we find
\begin{eqnarray}
&&\lambda_1= \log 2+ \log 5- \log 7 \approx 0.356674944 \nonumber \\
&&\lambda_0= \log \bigg( \frac{1}{1-p_1-p_2} \bigg(\frac{1}{e^{\lambda_1}-1}-e^{-\lambda_1}-e^{-2\lambda_1}\bigg) \bigg)\approx 1.743385883\nonumber
\end{eqnarray}
while for ${\cal L}_\rightarrow(3)$ (where $p_1=1/3, \ p_2=2/15, \ p_3=0.55$), the Lagrange multipliers read $\lambda_1\approx 0.372$, $\lambda_0\approx 1.727$.
In figure \ref{maxent1} we plot the theoretical prediction of $P_\rightarrow(k)$ along with the numerical simulations obtained for a time series of $2^{20}$ data. The agreement is very good for $s=2$ and even better for $s=3$, what confirms that (i) the chaotic process is also maximally ${\Phi}_{\text{DHVG}}$-entropic up to short range restrictions, (ii) short term correlations are captured in $P_\rightarrow(1), P_\rightarrow(2), P_\rightarrow(3)$, and (ii) that a very good theoretical approximation for its full degree distribution is
\begin{equation}
P_\rightarrow(k)=\left\{
\begin{array}{l}
2/3, \ k=1, \\
2/15, \ k=2, \\
0.55, \ k=3, \\
\exp(0.727-0.372 k), \ k \geq 4.%
\end{array}%
\right. \label{pkcaos}
\end{equation}%
\noindent For the Ornstein-Uhlenbeck process we only have exact results for $P_\rightarrow(1)$ and $P_\rightarrow(2)$. For ${\cal L}_\rightarrow(1)$ (where $p_1=1/2$), the Lagrange multipliers are $\lambda_1= \log(2)$, $\lambda_0=0$, that is, the prediction for an uncorrelated process. The prediction clearly improves for ${\cal L}_\rightarrow(2)$, as can be seen in figure \ref{maxent2}, where we plot the theoretical results for $P_\rightarrow(k)$ along with the numerical simulations of the process (series of $2^{20}$ data). This confirms that the Ornstein-Uhlenbeck process is also maximally ${\Phi}_{\text{DHVG}}$-entropic up to short range restrictions. In this case, short term correlations are well captured by $P_\rightarrow(1), P_\rightarrow(2)$. The theoretical degree distribution is in this case
\begin{equation}
P_\rightarrow(k)=\left\{
\begin{array}{l}
1/2, \ k=1, \\
0.24, \ k=2, \\
\exp(-0.733 k), \ k \geq 3.%
\end{array}%
\right. \label{pkcaos}
\end{equation}%
\begin{figure}
\centering
\includegraphics[width=0.9\columnwidth]{maxent_OU.pdf}
\caption{Semi log plot of the degree distribution $P_\rightarrow(k)$ of a DHVG associated to an Ornstein-Uhlenbeck with correlation time $\tau=1.0$. Dots are the result of numerical simulations on a time series of $2^{20}$ data, solid lines are the prediction of the variational approach.}
\label{maxent2}
\end{figure}
\subsection{Ornstein-Uhlenbeck and logistic map II: HVGS}
Previous numerical evidence suggests that these processes also have HVGs with exponentially distributed degree sequences $P(k)\sim \exp(-\lambda k)$, where $\lambda$ distinguish the purely chaotic from the stochastic ones \cite{toral}. In this last section we repeat the same variational arguments for HVGs associated to both Ornstein-Uhlenbeck and fully chaotic logistic map, and find that an entropy maximization principle predicts accurately the degree distribution. In both cases, the Lagrangian reads
\begin{eqnarray}
{\cal L}(s) &=& -\sum_{k=s+1}^{\infty} Q(k) \log Q(k) -\lambda_0\bigg(\sum_{k=s+1}^{\infty} Q(k) -(1-\sum_{i=2}^sp_i)\bigg)- \nonumber\\
&&- \lambda_1\bigg(\sum_{k=s+1}^{\infty} kQ(k) + \sum_{i=2}^sp_i-4\bigg),
\end{eqnarray}
For the logistic map, $P(2)=P(3)=1/3$ \cite{toral}. Accordingly, a MaxEnt principle up to $s=3$ predicts an exponential decay with $\lambda_1=\log(4/3) \approx 0.28$, to be compared with the numerical estimate $\lambda_1^{\text{num}} \approx 0.26$ \cite{toral}. We expect more accurate predictions of the slope of the distribution for higher values of $s$.\\
For the Ornstein-Uhlenbeck with $\tau=1.0$, analytical estimates of the HVG are $P(2)\approx0.3012, \ P(3)\approx 0.23$ \cite{toral}, which yield under a MaxEnt principle an exponentially decaying degree distribution with $\lambda_1 \approx 0.4467$, to be compared with numerics in figure \ref{maxent3}, finding an excellent match between theory and numerical experiment.\\
\noindent Finally, let us observe that the irrational rotation does not fulfill a MaxEnt principle in this context, as expected (correlations do not decay fast for quasiperiodic dynamics).
\begin{figure}
\centering
\includegraphics[width=0.9\columnwidth]{maxent_OU_HVG.pdf}
\caption{Semi log plot of the degree distribution $P(k)$ of a HVG associated to an Ornstein-Uhlenbeck with correlation time $\tau=1.0$. Dots are the result of numerical simulations on a time series of $2^{18}$ data, solid lines are the prediction of the variational approach.}
\label{maxent3}
\end{figure}
\section{Discussion}
In this work we have addressed the degree distribution of horizontal visibility graphs (both undirected and directed). These are classes of graphs constructed from the mapping of dynamics via the so called horizontal visibility algorithm, which are currently widely used for nonlinear time series analysis. In the first part we have developed a diagrammatic theory to analytically compute each component of the out degree distribution. In the case of deterministic dynamical systems, we have found that diagrammatic expansions converge fast due to the presence of forbidden diagrams. A more detailed analysis of these diagrams on relation to ordinal patterns \cite{forbidden} is left for future work.\\
Note that the use of diagrammatic expansions in nonlinear dynamics is an idea first advanced by Percival and Vivaldi \cite{vivaldi}, and later explored by Beck \cite{beck}. Our theory works for dynamical systems that have a $L_1$-integrable invariant measure. Accordingly, dissipative chaotic maps with strange attractors are left out: in those cases, a general measure theory over fractal sets should be followed, this is also an open problem for further research.\\
The theory also applies to stochastic processes with the Markovian property. In this work we dealt with Guassian processes, whose high order diagrammatic corrections are, from a practical point of view, computationally expensive. Further work could explore the saddle-point method to approximate these higher order corrections, to speed up the calculations.\\
In a second part, we have used Jayne's MaxEnt variational approach to obtain estimates of the full degree distributions associated to the chaotic and stochastic cases, finding that these are maximally entropic correlated processes.\\
\noindent To conclude, note that the theory developed in this work can be further extended to other types of Markovian dynamics. In this sense, previous numerical evidence on the degree distribution associated to different types of dynamics
could be analytically addressed via this methodology.
|
1,116,691,499,232 | arxiv | \section{Introduction}
Hybrid quantum systems aim to overcome difficulties in implementing more
complex quantum information processing tasks with individual quantum
systems~\cite{Xiang2013,Kurizki2015} -
a quantum network with solid-state systems as fast
processors, atomic systems as long-lived memories, and optical photons
interfacing spatially separated network nodes is an example~\cite{Kimble2008,Waks2009}.
There, control over the spectral and temporal properties of the exchanged photons is desirable for efficient information transfer
~\cite{Wilk:2007hm,Ritter2012}, but can be hard or impractical to implement.
A common problem in hybrid quantum networks is a mismatch between the
characteristic time scale (or bandwidth) of photons emitted by one node with the optical transition of the receiving node~\cite{Akopian:2011,Ulrich2014,Siyushev2014,Jahn:2015,Meyer:2015}.
Understanding the role of the photon bandwidth in light-matter interaction is therefore important for the further development of hybrid networks.
One realization of an atomic node of a quantum network is a single atom in free space coupled to a strongly focused mode~\cite{Sondermann2013}.
Aside from the potential use in quantum networks, single atoms are also an ideal test bed to study fundamental properties of light-matter interaction.
Following up on recent work on time-resolved scattering of photons with
exponentially rising and decaying profiles~\cite{Leong2016}, we report in this
work on the dependency of single photon scattering on their bandwidth.
\section{Theory}\label{sec:theory}
The interaction between single atoms and light has been studied extensively in semi-classical and fully quantized frameworks~\cite{Bufmmodeheckzlseziek1999,Enk2000,Enk2001,Enk2004,Domokos2002,Imamoglu:2008,Agio:2008,Wrigge2008,Heugel:2009,Leuchs:2009,Tey2009,Wang:2011,Leuchs:2012ps,Trautmann2016}.
Interesting recent results were the complete reflection of
light by a single atom~\cite{Agio:2008,Tey2009} and the complete
absorption of a single photon by reversing the spontaneous emission process from an excited two-level atom into
spherical harmonic modes described many decades ago~\cite{Weisskopf:1930jm,Imamoglu:2008,Leuchs:2009,Wang:2011}.
Practically important cases for traveling photons with a finite
duration have an exponential temporal envelope and consequently a Lorentzian
power spectrum
\begin{equation}\label{eq:lorentzian}
\mathcal{L}\left(\omega\right)\propto
\frac{1}{\left(\omega-\omega_0\right)^2+\Gamma_\textrm{p}^2/4}
\end{equation}
of width $\Gamma_\textrm{p}$ centered at the atomic resonance frequency
$\omega_0$.
For a single photon Fock state with this power spectrum~(Eq.\ref{eq:lorentzian}),
the scattering probability~$\epsilon$ is obtained by solving Eq.~18 and Eq.~21 in Ref.~\cite{Wang:2011},
\begin{equation}\label{eq:tx}
\epsilon =
4\Lambda \left(1-\Lambda \right) \frac{\Gamma_0}{\Gamma_0+\Gamma_\textrm{p}}\,,
\end{equation}
where $\Gamma_0$ is the width of the atomic line of interest and $\Lambda \in [0,1]$ is the spatial overlap between the excitation and the atomic dipole mode~\cite{Sondermann2013,Leong2016}.
Photons of different temporal envelopes can have identical power
spectra -- for example, both exponentially rising and falling temporal envelopes lead
to a Lorentzian power spectrum. In the work presented here, we consider an
exponential decaying envelope $P_\textrm{p}(t)$ for the photon in the time domain,
\begin{equation}\label{eq:photons_decay}
P_\textrm{p}(t) = \Gamma_\textrm{p} \Theta(t) \exp{\left(-\Gamma_\textrm{p} t\right)}\,,
\end{equation}
where $\Theta(t)$ is the Heaviside step function.
For such a photon the probability~$P_\textrm{e}(t)$ of finding a two-level atom in the excited state is given by~\cite{Leong2016,Leuchs:2009}
\begin{equation}\label{eq:P_e_down}
P_{\textrm{e}}(t)=
\frac{4\Lambda \Gamma_0 \Gamma_\textrm{p} }{\left(\Gamma_0-\Gamma_\textrm{p}\right)^2} \Theta(t)\left[ \exp{\left(-\frac{1 }{2}\Gamma_0 t\right)}- \exp{\left(-\frac{1}{2}\Gamma_\textrm{p} t\right)} \right]^2\,,
\end{equation}
with an atomic peak excitation probability
\begin{equation}\label{eq:pemax}
P_\textrm{e,max} = 4 \Lambda \left( \frac{\Gamma_\textrm{p}}{\Gamma_0}\right)^{\frac{\Gamma_0+\Gamma_\textrm{p}}{\Gamma_0-\Gamma_\textrm{p}}}\,.
\end{equation}
According to this model, the highest atomic
peak excitation probability is reached if the bandwidth of the incident photon matches the atomic linewidth, $\Gamma_\textrm{p}=\Gamma_0$,
while the highest scattering probability
is obtained for very narrowband excitation, $\Gamma_\textrm{p} \rightarrow 0$~(Eq.~\ref{eq:tx}) .
\section{Experiment}
To test the model presented in Section~\ref{sec:theory}, we prepare single photon states by heralding on
a time-correlated photon-pair from a parametric process in a cold atomic
ensemble~\cite{Chaneliere:2006,Srivathsan:2013}.
These photons are
focused onto a single atom trapped in a far-off-resonant optical dipole trap.
From the fraction of photons that are scattered out of this focused
excitation mode we can accurately obtain the
the transient atomic excitation~\cite{Piro:2010js,Sandoghdar:2012,Brito:2016,Leong2016}.
\begin{figure}
\centerline{\includegraphics[width=\linewidth]{figure1.eps}}
\caption{\label{fig:setup}
\textbf{(a)}~Optical setup to prepare heralded single photons at 780\,nm which are tightly focused on a single trapped atom.
Four-wave mixing: two co-aligned pump fields (pump1 at 795\,nm and pump2 at 762\,nm) generate herald (776\,nm) and probe (780\,nm) photon pairs in a cold cloud of $^{87}$Rb atoms.
The detection of a herald photon at D$_\textrm{h}$ signals the presence of a single photon in the probe mode.
Single Atom: a $^{87}${Rb} atom is trapped at the focus of a confocal aspheric lens pair~(AL) with an optical dipole trap.
D$_\textrm{h}$, D$_\textrm{f}$: avalanche photodetectors, P: polarizer, F: interference filter, $\lambda$/2, $\lambda$/4: half- and quarter-wave plate, (P)BS: (polarizing) beam splitter, DM: dichroic mirror, AOM: acousto-optic modulator, MOT: magneto-optical trap.
\textbf{(b)}~Level scheme of the FWM process.
\textbf{(c)}~Level scheme of the single $^{87}$Rb atom in the dipole trap.}
\end{figure}
The experimental setup is shown in Fig.~\ref{fig:setup}(a).
A single atom is loaded from a cold ensemble
in a magneto-optical
trap~(MOT) into a far-off-resonant optical dipole trap~\cite{Tey:2008} formed
by the tight focus of a light beam prepared by an aspheric lens
(numerical aperture 0.55). The dipole laser (980\,nm, $42$\,mW, circular
polarization) provides a trap depth of approximately $2$\,mK.
Once trapped, the atom undergoes molasses cooling and is optically pumped into the 5\hflev{S}{1}{2}{2}, $m_F$=-2 state.
A bias magnetic field of $0.7$\,mT is applied along the optical axis.
We obtain heralded single photons from
correlated photon pairs generated by four-wave-mixing (FWM) in a cloud of cold
$^{87}$Rb atoms. This atomic cloud is
repeatedly cooled and refilled by a MOT for 140\,$\mu$s, followed by a photon
pair generation interval of~10\,$\mu$s.
During the photon pair generation, two pump beams with wavelengths 795\,nm and
762\,nm drive a transition from 5\hflev{S}{1}{2}{2} to
5\hflev{D}{3}{2}{3}~[Fig.~\ref{fig:setup}(b)]. A parametric
conversion process along a cascade decay channel generates time-ordered photon pairs.
We collect herald~(776\,nm) and probe~(780\,nm) photons via interference filters into single mode fibers~\cite{Franson:1992cl,Srivathsan:2014jx,Gulati:2014}.
Detecting photons at 776\,nm then heralds single photons at 780\,nm, resonant with the $^{87}$Rb D2 transition.
Due to collective effects in the atomic ensemble, the bandwidth of the probe
photons is broader than the natural linewidth ($\Gamma_0/2\pi=6.07$\,MHz)
of the $5P_{3/2}-5S_{1/2}$ transition~\cite{Jen:2012,Srivathsan:2013}.
We employ these effects to tune the bandwidth $\Gamma_\textrm{p}$ of the probe photons
over a range of 6 to 2~$\Gamma_0$ by controlling
the number and density of atoms in the cold ensemble via the gradient of the magnetic quadrupole field during the MOT phase.
The probe photons are sent to the single atom setup via a 230\,m long optical
fiber. To suppress unheralded photons from the FWM setup, an acousto-optic
modulator~(AOM) acts as a switch
between the photon
source and the single atom that is opened by a heralding event on
detector~D$_\textrm{h}$ for 600\,ns. Optical and electrical delays are set such that the probe photon passes the AOM in the center of this interval.
This AOM also compensates for the 72\,MHz shift of the atomic resonance frequency caused by the bias magnetic field and the dipole trap.
Probe photons are focused onto the atom by the first aspheric lens. The
transmitted probe mode is then collimated again by a second aspheric lens,
subsequently coupled into a single-mode fiber, and sent to the forward
detector~D$_\textrm{f}$. In previous experiments we determined the spatial overlap of the probe mode with the atomic dipole mode to be $\Lambda\approx0.033$~\cite{Leong2016}.
\begin{figure}
\centerline{\includegraphics[width=\linewidth]{figure2.eps}}
\caption{\label{fig:photons}
Coincidence histograms between heralding detector D$_\textrm{h}$ and forward detector D$_\textrm{f}$ for probe photons with five different bandwidths.
The bandwidths, shown in the figure legends, are determined by fitting the histograms to Eq.~\ref{eq:photons_decay}.
Each histogram is normalized to the heralding~efficiency~$\eta_\textrm{f}$.
Error bars are smaller than symbol size (one standard deviation
of propagated Poissonian counting uncertainties).
Detection times are offset
to account for delays introduced by electrical and optical lines.
}
\end{figure}
For each
bandwidth setting, we obtain a coincidence histogram by sorting the detection
times~$t_\textrm{i}$ at~D$_\textrm{f}$ with respect to the
heralding event into $\Delta t=1$\,ns wide time bins.
Figure~\ref{fig:photons} shows reference
histograms~$G_\textrm{0}\left(t_\textrm{i}\right)$ with no trapped atoms, normalized to the heralding efficiency~$\eta_\textrm{f}= \sum_i G_\textrm{0}\left(t_\textrm{i}\right)$ over the time interval $-10\,\textrm{ns}\leq t_{i} \leq 100\,\textrm{ns}$.
The time-ordering of herald and probe photons leads to an asymmetric
exponentially decaying profile, from
which we extract the corresponding photon bandwidth~$\Gamma_\textrm{p}$ by fitting to Eq.~\ref{eq:photons_decay} within the range $2\,\textrm{ns}\leq t_{i} \leq 100$\,ns.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{figure3.eps}
\caption{\label{fig:tx}
Extinction~$\epsilon$ of probe photons with different bandwidth~$\Gamma_\textrm{p}$ (circles). For comparison we include the observed extinction of weak coherent field~(diamond).
Solid line: Eq.~\ref{eq:tx} with $\Lambda=0.033$. Error bars are smaller than symbol size (one standard deviation
of propagated Poissonian counting uncertainties).
}
\end{figure}
When the atom is trapped, we record a corresponding set of probe
histograms~$G(t_\textrm{i})$ in a similar way.
From this, we can determine the extinction via
\begin{equation}
\epsilon= 1 -\sum\limits_i G\left(t_\textrm{i}\right) / \sum\limits_i
G_\textrm{0}\left(t_\textrm{i}\right)
\end{equation}
to characterize the photon-atom interaction. Both summations are performed over the time interval $-10\,\textrm{ns}\leq t_{i} \leq 100\,\textrm{ns}$.
\section{Results}
The results of the extinction measurements for bandwidth values between $6$
to $2\Gamma_0$ are shown in Fig.~\ref{fig:tx}. Consistent with the model
(Eq.~\ref{eq:tx}),
the extinction increases for narrower photon bandwidths.
Since our heralded photon source can not efficiently prepare photons with a
bandwidth below $2\Gamma_0$,
we use 100\,ms long pulses of laser light in a coherent state to simulate photons with $\Gamma_\textrm{p} \rightarrow 0$.
The observed extinction of the weak coherent field~(Fig.~\ref{fig:tx}, diamond) is larger than the extinction of the heralded photons, but deviates considerably from the model.
We attribute
this mostly to linewidth broadening of the atomic transition:
when tuning the frequency of the coherent probe field across the atomic resonance, we observe a linewidth of $2\pi\cdot10.6(6)\,\text{MHz}=1.7(1)\Gamma_0$ in the transmission spectrum.
This broadening can be
caused by intensity fluctuations of the dipole trap field, the atomic motion of the atom, and the probe laser linewidth~\cite{Chin2016}.
The heralded photons from FWM have a larger bandwidth, so their scattering probability is less susceptible to this broadening of the atomic transition.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{figure4.eps}
\caption{\label{fig:pe}
Temporal evolution of the atomic excited population~$P_\textrm{e}(t)$ obtained from the time-resolved changes in transmission detection rate (see Eq.~\ref{eq:P_e_dot}).
Narrowband photons lead to stronger and longer lasting atomic excitation, in agreement with Eq.~\ref{eq:P_e_down}~(solid lines).
Error bars represent one standard deviation of the distributions obtained by a Monte-Carlo method which assumes Poissonian statistics for the detection rates.}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{figure5.eps}
\caption{\label{fig:pe_max}
Atomic peak excitation~$P_\textrm{e, \textrm{max}}(t)$, extracted from the experimental~$P_\textrm{e}(t)$ curves shown in Fig.~\ref{fig:pe}.
Solid line: Eq.~\ref{eq:pemax}.}
\end{figure}
Aside from the scattering probability, we also determine the temporal
evolution of the atomic excited population $P_\textrm{e}(t)$ during the
scattering process.
Any change of the forward detection rate~$
\delta\left(t_\textrm{i}\right)= \left(G_\textrm{0}(t_\textrm{i}) -
G(t_\textrm{i})\right)/\left(\eta_\textrm{f} \Delta t\right)$ is directly related to a change
$\dot{P}_\textrm{e}(t)$ of the atomic population~\cite{Leong2016} via the rate
equation
\begin{equation} \label{eq:P_e_dot}
\dot{P}_\textrm{e}(t) = \delta(t) - (1-\Lambda)\Gamma_0 P_\textrm{e}(t)\,.
\end{equation}
Thus, $P_\textrm{e}(t)$ can be obtained by integrating Eq.~\ref{eq:P_e_dot}
with a relatively low experimental uncertainty. Figure~\ref{fig:pe} shows the
evolution of the resulting $P_\textrm{e}(t)$,
agreeing very well with Eq.~\ref{eq:P_e_down} (solid lines) for narrowband photons,
but exhibit a stronger deviation from the model as the bandwidth is increased.
This deviation
is likely due to the imperfect photon profiles on the sharp rising edge
(Fig.~\ref{fig:photons}) compared to the ideal asymmetric exponentially decaying profile.
The peak excitation probability~$P_\textrm{e,max}$ for each bandwidth in
Fig.~\ref{fig:pe} is shown in Figure~\ref{fig:pe_max}, and is in good agreement with Eq.~\ref{eq:pemax}.
Comparing results for narrowband ($\Gamma_\textrm{p}= 1.96(1)\Gamma_0$) and broadband
($\Gamma_\textrm{p}=6.09(5)\Gamma_0$) photons, we observe an increase in the
peak excitation by a factor~$1.5(2)$.
This relative increase is smaller than the relative increase by a factor of
2.6(4) of the extinction between these two bandwidths, i.e., the atomic peak
excitation has a weaker dependence on the photon bandwidth than the scattering
probability over our accessible bandwidth range.
\section{Discussion and summary}
While often a semiclassical description of light atom-interaction can
explain observations with sufficient accuracy, it is well known that the
photon statistics of the incident light can affect the atomic
excitation dynamics~\cite{Georgiades:1995,Domokos2002,Wang:2011, Strauss:2016}.
Full quantum models of the light-matter interaction predict different scattering probabilities for single photons compared to coherent fields with equal bandwidth and a mean photon number of one~\cite{Domokos2002,Wang:2011}.
As (hybrid) quantum networks in an information processing scenario are likely
to operate by exchanging single photons, testing light-matter interfaces with
single photons -- as opposed to weak coherent fields -- becomes important.
For the experimental parameters in the work presented here ($\Lambda=0.033$,
$\Gamma_\textrm{p}=2\Gamma_0$), the expected difference in scattering
probability is only 0.05\%, and thus within our experimental uncertainties.
However, already a slight improvement of experimental parameters parameters
should make the difference in scattering resolvable, assuming similar
experimental uncertainties.
For $\Lambda=0.1$ and $\Gamma_\textrm{p}=\Gamma_0$, quite within
experimentally reachable range~\cite{Alber2016}, we expect $\epsilon=18.0\%$ for single photon
excitation, but $\epsilon=17.1\%$ for a coherent field.
In summary, we find that the role of the photon bandwidth in the scattering
process with a single atom is well described by a relatively simple excitation
model with a fully quantized field description~\cite{Wang:2011}. Tuning the
photon bandwidth from 6 to 2~$\Gamma_0$, we observe an increase in the
scattering probability as well as in the atomic peak excitation.
Notably, the relative increase in the scattering probability is larger than in the atomic peak excitation.
\section*{Funding}
Ministry of Education in Singapore (AcRF Tier 1);
National Research Foundation, Prime Minister's office (partly under grant No. NRF-CRP12-2013-03);
M. Steiner acknowledges support by the Lee Kuan Yew Postdoctoral Fellowship.
\bibliographystyle{apsrev4-1}
|
1,116,691,499,233 | arxiv | \subsection{Implementation Details}
\label{appendix_imple_details}
For the BlenderBot model \cite{roller-etal-2021-recipes}, we use the distilled version with 2 encoder layers, 12 decoder layers, 1280 hidden dimensions, and 365M parameters. For all our experiments, we implement our code based on an efficient transformer architecture \cite{wolf-etal-2020-transformers}. For PPO experiments, our implementation is adapted from \url{https://https://github.com/lvwerra/trl}. We run all our experiments on a RTX 2080 Ti GPU machine.
For classifiers, we train all models to a maximum of six epochs and choose the best model on corresponding validation set (for both safety and consistency, we observe the best performing model is at epoch 4). For weakly supervised and reinforcement models, we set the maximum number of training steps to 60, where each step consists of 512 randomly sampled data. We run in mini-batch size of 32 for safety and 16 for consistency. We set the learning rate to 2e-4 and optimize with AdamW \cite{loshchilov-etal-2018-decoupled}. We use the adaptive KL controller with the initial coefficient of $0.2$. Even though further training can keep improving actual rewards, we do not keep training due to that the model may diverge too much from the original model so that the generated utterances are not fluent anymore. To prevent unfair comparison to the baselines, we choose the best performing model if the perplexity is within +- 1.5 from the beginning of training. We choose the last epoch if the perplexity is within this range.
\paragraph{Safety Training}
\label{appendix_safety_training}
\begin{figure}[h]
\centering
\includegraphics[width=\columnwidth]{figures/ppo_training.pdf}
\caption{Learning curves for \texttt{Trigger\_PPO}}
\label{fig:trigger_ppo}
\vspace{-1em}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=\columnwidth]{figures/ppo_adv_training.pdf}
\caption{Learning curves for \texttt{Trigger\_PPO\_adv}}
\label{fig:trigger_ppo_adv}
\vspace{-1em}
\end{figure}
Because the perplexity is within the range we set, we use step 60 for evaluation. It takes 106 minutes to train and 5 minutes for evaluation. For weight for in \texttt{Trigger\_PPO\_adv}, we experimented with 0.2, 0.5, and 1.0 where it is slower to converge for weight 0.5 and 1.0. We choose the weight 0.2 in our experiments. For weakly-supervised method, it takes 127 minutes to train the model. Figure \ref{fig:trigger_ppo} shows learning curves for \texttt{Trigger\_PPO} and Figure \ref{fig:trigger_ppo_adv} shows learning curves for \texttt{Trigger\_PPO\_adv} for both the reward of prompt and response. Rewards are the raw logits of the target label.
\paragraph{Consistency Training}
\label{appendix_consistency_training}
\begin{figure}[h]
\centering
\includegraphics[width=\columnwidth]{figures/ppo_contra.pdf}
\caption{Learning curves for \texttt{Trigger\_PPO\_ft}}
\label{fig:trigger_consis}
\vspace{-1em}
\end{figure}
We use step 60 for training data. For fine-tuning on the human-bot data, we train with 100 maximum steps and use step 61 for evaluation. It takes 156 minutes to train and 2 minutes for evaluation. For the weakly-superivsed method, it takes 288 minutes to train. Figure \ref{fig:trigger_consis} shows learning curves for \texttt{Trigger\_PPO\_ft} on the response reward.
\subsection{Human Evaluation}
\label{appendix_human_eval}
Following previous practice in generating adversarial examples and problematic responses \cite{jin-etal-2020-is, nie-etal-2020-i}, we employ three native English speakers for evaluation by first showing them labeled examples from previous research. The annotators need to pass an onboarding test (15 examples each) before the annotation process.
\subsection{Examples}
\label{appendix_examples}
We show parallel examples with safety (Table \ref{table:safety_examples}) exposure tasks comparing different models\footnote{We do not show other lime-picked examples here because for our task, lime-picked represents the conversations that do not expose corresponding problems (i.e. unsafe and inconsistent responses in our experiments). In other words, lime-picked examples will just be normal generated self-chat conversations.}.
\input{tables/safety_examples}
\section{Introduction}
Language models, including dialog models, greatly benefit from training on large amounts of data with the objective of mimicking human generated sentences \cite{radford-etal-2019-language, brown-etal-2020-langauge, zhang-etal-2019-dialogpt, adiwardana-etal-2020-towards, roller-etal-2021-recipes}. However, even with carefully pre-processed training data from online sources, neural dialog models are prone to issues including generic utterances, repetition, contradiction, and lack of safety \cite{li-etal-2016-persona, welleck-etal-2020-neural, li-etal-2020-dont, roller-etal-2021-recipes, xu-etal-2021-beyond}. Compared to modularized dialog systems which are designed to avoid these problems \cite{yu-etal-2019-gunrock, paranjape-etal-2020-neural}, fixing these issues with end-to-end neural models is more challenging, which may hinder real world use of trained models \cite{wolf-etla-2017-why, simonite_2021}. We argue that before solving these problems using simulated data from simplified scenarios, we need to be able to probe the models and expose the problems in a dynamic way.
Even though crucial limitations of neural dialog models are prevalent, they are mostly manually identified and categorized through interactions between model designers and the dialog system during qualitative analysis \cite{roller-etal-2021-recipes}. Recent work proposes asking annotators to converse with dialog models while goading the model into generating problematic responses in a black-box attack setting. Although the data collected in this way can improve the performance of both problem classifiers and model generation, human annotators mostly rely on straightforward and intuitive strategies to collect the dataset, which may only expose superficial problems. For instance, \citet{xu-etal-2020-recipes} instructs crowdworkers to trigger dialog systems into responding with unsafe (offensive or otherwise socially undesirable) utterances, but most of the human messages are either hate speech or controversial statements. Similarly, \citet{nie-etal-2020-i} asks Mechanical Turkers to manually write contradicting dialogs for both humans and bots, or to interact with chatbots,
where a frequent strategy is to ask factual questions intentionally leading to contradiction
(e.g. ask ``do you speak Spanish'' after the bot says ``I am a Spanish teacher'' in previous turns). Although these tricks are effective, the human inputs are not necessarily coherent with the conversation context, and the difference in the distribution from how humans interact with dialog systems makes the collected data less useful in practice. In addition, the data collection procedure is extremely expensive and is not practical for newly trained models. More importantly, systematic problems are still not revealed.
\input{figures/mode_figure}
In this work, we propose to automatically expose problems with neural dialog models in a more systematic setting. Given a conversation context, the goal is to generate a coherent utterance to act as a human \texttt{prompt} through self-chat, which will trigger the dialog system into generating a problematic response. To this end, we propose to learn some trigger hidden states while freezing the original dialog model. We assume that we have some problem classifiers which can be from in-domain or out-of-domain collected data. This is practical because out-of-domain data is relatively easy to collect and we do not require a perfect classifier. Each token can attend to the trigger hidden states when generating the next tokens so that the generated \texttt{prompt} include systematic signals regarding target problems. Specifically, we introduce a weakly-supervised method where we can back-propagate the gradients from the classifier through self-attention and cross-attention. We also introduce a reinforcement learning method that uses classifier results on the model responses as rewards\footnote{Our code is available at \url{https://github.com/DianDYu/trigger}}.
Compared to sentiment neurons \cite{radford-etal-2017-learning}, learning trigger hidden states as a problem switch is a much harder task because our hidden states in a relative shallow model are not trained with a huge amount of clearly distinguished supervised data. Furthermore, exposing more subtle problems (such as contradiction) with a coherent prompt that will indirectly impact on the model response rather than direct conditional text generation is more challenging, similar to probing models in an adversarial attack setting. However, we demonstrate the effectiveness of our proposed methods on automatic problem exposure with the state-of-the-art chatbot Blenderbot \cite{roller-etal-2021-recipes}. We evaluate with two problems: safety and consistency. In addition, we show that the generated examples can help to improve the performance of out-of-domain problem classification as well.
\section{Task Definition}
Given the context of a conversation $c_{k-1} = {x_1, y_1, x_2, y_2, ..., x_{k-1}, y_{k-1}}$, where $x_i, y_i$ represents utterances from each speaker in a turn, we want to generate a \texttt{prompt} $x_k$ while keeping the whole conversation coherent and engaging. The original neural dialog model then considers the whole context ($c_{k-1}; x_k$) to generate a response $y_k$, which is considered as not acceptable to a problem $P$ (e.g. toxic response to the safety problem). Meanwhile, we have some trained classifier $f_P(h(y_k))$ for the problem $P$ which can indicate how likely the contextual representation $h(y_k)$ has the problem.
\section{Methodology}
Since the goal of the task is to expose systematic problems of pre-trained models rather than relying on simple tricks, we generate prompts using the same model in a self-chat paradigm so that when we plug in the generated prompts to the original model we get exactly the same response. Compared to recent work on instructing humans to goad chatbots where annotators have no information of how the models works in a trial-and-error black-box attack manner, we use gradients of the model.
Motivated by recent success in conditional generation without fine-tuning model parameters \cite{li-etal-2021-prefix, yu-etal-2021-attribute}, we propose to learn a trigger prompt hidden states, $h_{trigger}$, while freezing the original dialog model to maintain output distribution and generation quality. Specifically, for an encoder-decoder model (or a language model), we are modeling
\begin{equation}
\label{equ_prompt}
p_\theta(x_k|h_{trigger}, x_{<k}, y_{<k})
\end{equation}
where $h_{trigger}$ is prepended to the beginning of the conversation history and is initialized with the hidden states of the \texttt{bos} (beginning of sentence) token. Before any training, the distribution of $p_\theta(x_k|.)$ will not be modified at all. Once we generate the \texttt{prompt} $x_k$, we use the original model to generate a response as
\begin{equation}
\label{equ_inference}
p_\theta(y_k|x_{<k+1}, y_{<k})
\end{equation}
where $y_k$ may be problematic. A trained classifier $f$ indicates the degree of the problem.
During training, we sample data $(c_{k-1}, x_k, y_k)$ via $c_{k-1} \sim \mathcal{D}$ where $\mathcal{D}$ can be any unlabeled conversation as the context, and $x_k$ and $y_k$ are generated in a two-stage sequence using Equation \ref{equ_prompt} \& \ref{equ_inference}.
In order to optimize $h_{trigger}$ which will boost the level of target problem through attention mechanism~\cite{vaswani-etal-2017-attention}, we propose a weakly-supervised trigger model (Section \ref{model_weakly}) where we backpropagate gradients from the classifier back to the hidden states directly, and a reinforcement learning trigger model (Section \ref{model_rl}) where the classifier results are used as rewards to optimize the hidden states. We illustrate the task and the proposed methods in Figure \ref{fig:illustration}.
\subsection{Weakly-supervised Trigger Model}
\label{model_weakly}
Because the \texttt{prompt} $x_k$ is sampled and detached from the original model, and the classifier operates on the corresponding response $y_k$, we need to connect $h_{trigger}$ with the response. During training, we first generate a prompt $x_k$ and then simulate the attention mechanism by modeling
\begin{equation}
\label{equ_weakly}
p_\theta(y_k|h_{trigger}, x_{<k+1}, y_{<k})
\end{equation}
where compared to generating the actual response using the original dialog model as in Equation~\ref{equ_inference}, $h_{trigger}$ is also used in response generation. We need to apply a classifier $f'$ on the generation hidden states which contains information about $h_{trigger}$ before sampled discrete tokens following \citet{dathathri-etal-2020-plug}\footnote{This classifier $f'$ is only used for training the weakly-supervised model, while a more robust classifier operating on the actual generated response tokens is used for evaluation.}. We can then use cross-entropy loss against the target label as the training signal to optimize $h_{trigger}$. We refer to this model as \texttt{Trigger\_{weakly}}.
Even though this method is relatively straightforward, we note that there are two potential problems. The first one is that because $h_{trigger}$ is considered as one (indirect) input, the optimized hidden states may not necessarily impact on the actual response to lower the loss function. In other words, $h_{trigger}$ is optimized specifically to a loss function regardless of the model output. Another problem is that at inference time when we generate a prompt $x_k$ to get a response using Equation~\ref{equ_inference}, there is mismatch from training so that even with a low training loss, the response generated can be very from that during training. However, as we evaluated empirically, $h_{trigger}$ learned this way is still effective.
In addition, we also experimented with gumbel softmax \cite{maddison-etal-2016-the, jang-etal-2017-categorical} on the generated prompt $x_k$ so that we can input the prompt gumbel vectors which contains information about $h_{trigger}$ using Equation~\ref{equ_inference} during training without the hidden state term. We did not notice a large difference in our preliminary study using an autoregressive language model (GPT-2, \citealp{radford-etal-2019-language}), so we use Equation~\ref{equ_weakly} for optimization with our weakly-supervised trigger model.
\subsection{Reinforcement Trigger Model}
\label{model_rl}
To solve the potential problems with the weakly-supervised trigger model, we leverage reinforcement learning to bypass the challenge in connecting $h\_{trigger}$ with the model response. During training, we use Equation~\ref{equ_prompt} to generate a coherent prompt $x_k$, and send the sampled discrete tokens to Equation~\ref{equ_inference} to get the response $y_k$. Instead of using the hidden states, we input the generated response tokens to the classifier $f$ to get a reward $r(y_k)$, where we use the raw logits of the target label. Following \citet{ziegler-etal-2019-fine}, we add an adaptive KL term to prevent the generated prompt from diverging too far from the original model
\begin{equation}
\label{equ_kl}
KL(x_k) = \beta \log \frac{p_\theta(x_k|h_{trigger}, x_{<k}, y_{<k})}{p_\theta(x_k|x_{<k}, y_{<k})}
\end{equation}
where $\beta$ varies dynamically to achieve a particular value \cite{ziegler-etal-2019-fine}.
The overall reward is thus
\begin{equation}
\label{equ_rl_reward}
R(x_k) = r(y_k) - KL(x_k).
\end{equation}
We optimize $h_{trigger}$ using Proximal Policy Optimization (PPO, \citealp{schulman-etal-2017-proximal}) with the reward $R$ from Equation~\ref{equ_rl_reward}. We refer to this model as \texttt{Trigger\_{PPO}}.
Another potential benefit of using reinforcement learning here in addition to fixing the challenges with weakly-supervised trigger model is that we can tweak the reward function to penalize easy triggers. For instance, human annotators may quickly find that controversial statements and unsafe sentences can goad the bot into generating unsafe responses back \cite{xu-etal-2020-recipes}. If we train $h_{trigger}$ using the original reward function, or use the weakly supervised method where the gradient impacts on both the prompt and the response, then it is very likely that by attending to $h_{trigger}$, the prompt $x_k$ is already problematic. To solve this, we can add a penalty term using the same reward as the response but on the prompt. This encourages the model to only generate acceptable prompts that trigger concerning responses, similar to an adversarial setting. The overall reward is
\begin{equation}
R_{adv}(x_k) = r(y_k) - KL(x_k) - w * r(x_k)
\end{equation}
where $w$ is a weight hyper-parameter to balance between the reward on the prompt and the response. We refer to this model as \texttt{Trigger\_{PPO\_adv}}.
\section{Experiments and Results}
We evaluate our proposed approaches on two problems: safety and consistency by generating prompts that can trigger corresponding problems. In addition, we study whether our generated results can in turn improve the classification performance with out-of-domain data.
For all our experiments, we use the state-of-the-art open-domain chatbot BlenderBot \cite{roller-etal-2021-recipes} as our pre-trained neural dialog model.
The maximum context and response lengths is set to 128 BPE tokens \cite{radford-etal-2019-language}. BlenderBot is pre-trained on Reddit discussions \cite{baumgartner-etal-2020-pushshift} with heuristic filtering and fine-tuned on human-collected clean conversational data including ConvAI2 \cite{zhang-etal-2018-personalizing} and Blended Skill Talk \cite{smith-etal-2020-put}. Because of the fine-tuning data, the chatbot frequently deviates from the current conversation topic and asks simple questions such as ``do you have a pet''. This makes it even harder to generate unsafe and contradictory responses given a coherent prompt.
For decoding, we follow the same procedure as in the original model, except that we use sampling instead of beam-search to increase diversity (which is shown to perform as well as beam search in their paper).
During training, we set a maximum number of training steps with early stopping. To prevent unfair comparison to baselines, instead of selecting the best model based on average reward, we early stop when perplexity diverges too much from the original perplexity. Please see Appendix \ref{appendix_imple_details} for implementation details. We analyze the effect of early stopping in Section \ref{analysis_steps}.
\subsection{Safety}
The safety problem exposure task is to generate coherent prompts where the dialog model will generate unsafe responses given the contexts and prompts. We compare our proposed \texttt{Trigger\_weakly}, \texttt{Trigger\_PPO}, and \texttt{Trigger\_PPO\_adv} with the original model \texttt{BlenderBot}.
\input{tables/safety_results}
\paragraph{Safety Classifier}
We train our safety classifier $f_{safety}$ using data collected from BAD \cite{xu-etal-2020-recipes}. We truncate the conversation history to four utterances from both speakers following the best practice in their paper. We also ignore easy cases where the bot says ``Hey do you want to talk about something else'' from a safety layer during data collection. In addition, we leverage data from BBF \cite{dinan-etal-2019-build} including both single-turn and multi-turn examples. In total, we have a training corpus of 197K examples and we evaluate on the BAD validation set with 12.8K examples. We train the classifier using RoBERTa \cite{liu-etal-2019-roberta}.
The classifier achieves an F1 score of $77.34$ on unsafe examples, which is close to the number reported in \citet{xu-etal-2020-recipes}, so we did not use additional training data and framework
For the classifier used in the weakly-supervised method $f'_{safety}$, we use the same data training a multi-layer perceptron (MLP) on top of frozen BlenderBot hidden states (similar to \citealp{dathathri-etal-2020-plug}). $f'_{safety}$ achieves an F1 score of $69.09$ on unsafe examples.
\paragraph{Training and Evaluation}
During training, we sample contexts of three utterances from the pre-processed BAD training data explained above because BlenderBot can only handle 128 tokens.
For evaluation, we sample contexts of the same length from the BAD validation data. We report the average probability that the response is unsafe and the average probability that the prompt is unsafe using $f_{safety}$, as well as the generated sentence perplexity as automatic evaluation averaged over three random seeds. For human evaluation, we report the percentage of unsafe responses, unsafe prompts, and the language quality of the prompts which indicate both fluency and coherence on an 1 - 5 Likert scale using 150 examples. Details of human evaluation can be found in Appendix \ref{appendix_human_eval}.
\paragraph{Results}
Table \ref{table:safety_results} shows results for the safety exposure task. On the induced responses according to the generated prompts, compared to the baseline model \texttt{BlenderBot}, all our proposed methods substantially increase the chance that the responses are unsafe (with more than 8\% absolute from safety classifier $f_{safety}$, and more than 12\% from human evaluation). This suggests that these methods are effective in exposing safety problems with the pre-trained models. In addition, the relatively low unsafe percentage (9.21\% and 26.32\%) indicates that in general, \texttt{BlenderBot} tends to generate safe responses due to its clean fine-tuning data. Tricking the model into generating unsafe responses is thus very challenging without modifying the model distribution, especially when we want to generate coherent prompts with high language quality.
On the generated prompts, as expected, without any constraint as with \texttt{Trigger\_weakly} and \texttt{Trigger\_PPO}, the model may learn to increase the likelihood of unsafe responses by crafting unsafe prompts, resulting in much higher prompt unsafe probability judged by both the automatic classifier (more than 15\% over the baseline) and human annotation (more than 7\%). However, by adding a penalty to the prompt to reduce its unsafe degree (from 23.68\% to 17.11\% by human evaluation), we can maintain or even outperform unsafe degree in the corresponding responses (26.32\%). Meanwhile, the language quality human annotation results show that penalty on the prompt also helps with maintaining coherence and fluency compared to \texttt{Trigger\_weakly} and \texttt{Trigger\_PPO}.
\subsection{Consistency}
\label{exp_consistency}
The consistency problem exposure task is to generate coherent prompts to trigger responses that contradict their roles in the conversation context. In contrast with safety, since generating inconsistent prompts will not necessarily result in more inconsistent responses, we do not evaluate on \texttt{Trigger\_PPO\_adv}. Instead, we compare \texttt{Trigger\_weakly}, \texttt{Trigger\_PPO}, and the original \texttt{BlenderBot}. We also compare with \texttt{Human\_selected} which picks context-specific prompts that trigger responses labeled as contradictory from multiple sampled pairs in DECODE data collection \cite{nie-etal-2020-adversarial}. It serves as the upper bound for the task.
\paragraph{Consistency Classifier}
We train our consistency classifier $f_{consis}$ using the data collected from DECODE \cite{nie-etal-2020-i}. Because contextual information is crucial for consistency detection,
we do not truncate the context history. The training corpus consists of 27K examples and we evaluate the classifier on the DECODE validation dataset with 4K examples. In order to easily create training signals when optimizing $h_{trigger}$, we train the classifier by concatenating the last response with the context instead of the suggested structured method. Our RoBERTa-based classifier achieves an F1 score of $93.45$ on contradictory utterances, which is close to the results in \citet{nie-etal-2020-i} with additional training data\footnote{Although more accurate classification is beneficial to our model, training more complicated classifiers to achieve only marginal improvements is out of the scope of our work.}.
For the weakly-supervised method, $f'_{consis}$ is similar to $f'_{safety}$, and achieves $86.02$ F1 on contradictory examples.
\paragraph{Training and Evaluation}
During training, because BlenderBot cannot handle longer contexts, we truncate the conversation history to three utterances. We sample examples from the DECODE training data to form $\mathcal{D}$. We note that DECODE training and validation data are collected by asking humans to write utterances for each speaker, which may be different from a chatbot setting. Therefore, for evaluation, we sample contexts from their collected human-bot test set (with 764 examples in total). This set is collected by asking human annotators to interact with multiple chatbots. We report the average probability that the response contradicts with the context using $f_{consis}$ for automatic evaluation. We also report the percentage of contradictory responses for human evaluation on 200 generated examples from three random seeds.
Because the training signal for inconsistency is more delicate compared to other attributes such as sentiment and safety, it may be harder for the model to converge, especially with a diverse set of training examples.
Therefore, we also experiment with training on the human-bot context directly.
It is worth mentioning that even though we train with the same context as for evaluation, the only training signal is from the classifier $f_{consis}$. In other words, none of the models require external information such as real collected prompts and responses with their corresponding gold labels.
More importantly, we select early-stopping based on perplexity instead of cherry-picking the best examples using actual predicted rewards.
Thus it is fair in performance comparison.
Moreover, this is a common practice in the literature \cite{finn-etal-2017-meta}, particularly with reinforcement learning to optimize rewards \cite{wu-etal-2016-google, ziegler-etal-2019-fine}.
\paragraph{Results}
\input{tables/consistency_results}
Table \ref{table:consis_results} summarizes the experiment results on the consistency exposure task. We observe that during training, \texttt{Trigger\_weakly} does not converge so that its performance on the test data (17.86\%) is lower than the baseline. Even though \texttt{Trigger\_PPO} gets higher reward, training is not very stable and its performance on the target data does not increase by a large margin. This suggests that inconsistency signals may not be easily captured to craft corresponding dynamic prompts. When we train the models on human-bot data instead (denoted as \texttt{Trigger\_weakly\_ft} and \texttt{Trigger\_PPO\_ft} respectively), the weakly supervised method still does not converge. However, \texttt{Trigger\_PPO\_ft} learns how to perform the task evaluated by the learning curve (see Appendix \ref{appendix_consistency_training}). We thus do human evaluation on this method. \texttt{Trigger\_PPO\_ft} significantly outperforms the baseline (28.14\% compared to 12.56\%) from human evaluation, suggesting that even with weaker signals, our proposed method is still effective on harder tasks such as inconsistency, which by nature is non-trivial to detect. Lastly, when we compare to the upper bound \texttt{Human\_selected}, which are picked by humans to be inconsistent, we found that human prompts are shorter compared to our generated prompts because of the minimum generation size of 20 suggested by \citet{roller-etal-2021-recipes}. Since the context window of our dialog model is limited, longer prompts indicate less context due to truncation. Given that conversation history is critical in inconsistency, this partially explains the relatively lower performance.
\subsection{Out-of-domain Classification}
\label{exp_classification}
In addition to generating prompts to expose problems of neural dialog models, we examine if the generated prompts and responses can help out-of-domain problem classification. This is critical because due to distribution difference, problem classifiers trained on one domain may not work well on another \cite{gururangan-etal-2020-dont}, especially with problems that are hard to expose. For instance, \citet{nie-etal-2020-i} collect a contradiction dataset by asking humans to generate inconsistent responses, which is a much easier task than tricking dialog models into generating inconsistent utterances within a reasonable interaction budget. They observe a large performance gap between in-domain (human-human) training data and out-of-domain human-bot data. To this end, we experiment with generating out-of-domain problem training data with our proposed methods on the consistency task.
\paragraph{Training and Evaluation}
We use the best performing model from Section \ref{exp_consistency}, \texttt{Trigger\_PPO} to generate prompts and responses for the evaluation human-bot contexts (\texttt{human-bot Trigger\_PPO}). We also generate prompts and responses using the original BlenderBot on the same context (\texttt{human-bot BlenderBot}). For each generated utterance, we use $f_{consis}$ to predict the probability that it contradicts with the conversation history. A threshold ($0.5$ in our experiments) is used to convert the predicted probability to a contradiction label. Then we can train the contradiction detection classifier the same way as explained in Section \ref{exp_consistency}. We compare with the classifier trained on the human-collected DECODE training data. The best model checkpoint to perform evaluation is selected on the DECODE validation data.
\paragraph{Results}
\input{tables/classification_results}
Table \ref{table:classification_results} shows the classification results for contradiction prediction on the human-bot data. Similar to previous findings, even though training on human-human data can achieve a high F1 score on the human-human validation set, it suffers from the out-of-domain distribution. The classifier trained with data generated from the original BlenderBot results in similar performance. In comparison, training with the data generated by \texttt{Trigger\_PPO} significantly outperforms the baseline (67.68 F1 vs. 59.05 F1). This indicates that our proposed method can not only expose model problems, but can help problem detection with new data distribution as well. It can also be used in models that require good domain-specific classification, such as \citet{dathathri-etal-2020-plug}.
\section{Analysis}
In this section, we discuss model performance in depth with qualitative analysis. See Appendix \ref{appendix_examples} for more model comparison parallel examples.
\input{tables/consis_examples}
\paragraph{Training for more steps}
\label{analysis_steps}
As shown in the learning curves in Appendix \ref{appendix_safety_training}, the training reward actually does not saturate when it reaches our set maximum number of steps. In other words, we can expect to see higher rewards with more training steps. However, our PPO model starts to exploit environment quirks to maximize rewards (such as step 80). For instance, for the safety exposure task, the model starts to generate prompts with certain patterns such as ``Then put her ...'' or ``They should ...'', even with \texttt{Trigger\_PPO\_adv} adding the prompt penalty term. For the consistency exposure task, the model starts to use prompts with patterns such as ``I don't understand you ...'', ``How long have you ...'', or ``You are not ...''. The generated prompts still consider context (rather than just generating templates), and can vary from different random seeds. Even though they are more effective in inducing problematic responses, the prompts are less coherent and less diverse, resulting in similar n-grams. This suggests that on the one hand, we may need an additional reward in addition to the relatively straightforward negative penalty.
On the other hand, with more training steps, we may be able to discover more meaningful ``universal triggers'' \cite{wallace-etal-2019-universal} that can trigger target responses regardless of the context.
\paragraph{Weakly-supervised vs. Reinforcement Learning Method}
Although there is a potential discrepancy between training and testing that $h_{trigger}$ may only learn to optimize the classifier $f'_{consis}$ regardless of the actual task for the weakly-supervised method as explained in Section \ref{model_weakly}, we found that in the safety exposure task, it can still increase performance from human annotation. However, this results in much higher unsafe degree for the prompt and lower language quality. Qualitatively, we found that it is more likely to generate unsafe tokens and due to the diverged distribution, the prompts are less grammatical with nonsensical tokens. This suggests that the gradients impact on the prompts more to change the corresponding response attribute, which can also explain its worse performance in tasks where prompt attributes are less dominant to responses such as consistency exposure.
In comparison, the reinforcement learning method does not rely on gradients that flow through both the responses and prompts. Instead, it utilizes rewards through exploration and exploitation so it can be more effective in different tasks.
\paragraph{Exposing more systematic problems}
Previous research mostly exposes superficial problems with easy tricks such as controversial statements and repetitive questions which are unnatural and incoherent. To illustrate, \citet{xu-etal-2020-recipes} show that only 12.9\% responses are offensive if their corresponding prompts are safe. In other words, the vast majority of unsafe responses are induced by unsafe prompts. In comparison, our results show that in the human evaluation test data where 26.32\% responses are unsafe, only 17.11\% prompts are unsafe, indicating that our generated safe prompts are effective in generating problematic responses. Similarly, for consistency, we found that in our preliminary experiments on 100 examples, none of our generated prompts applies easy tricks such as repetitive questions that directly contradicts the context, whereas 15\% of the DECODE human-bot prompts fall in this category. This number is much higher in their collected human-human data with other tricks such as asking numeric questions. In addition, 53\% DECODE-collected prompts contain questions (which are more likely to trigger inconsistent responses in general), whereas 39\% contain questions from our proposed method (close to 43\% in the BlenderBot baseline).
On the other hand, we can find that coherent natural patterns such as ``They should ...'' and ``You are not …'' (rather than easy tricks) are more likely to trigger problematic responses. Together with the evidence that some problem triggers are learnable from our proposed methods above the surface level, we believe that we can expose more systematic problems compared to previous research where human annotators have no direct information to interpret how a natural prompt can trigger corresponding responses.
\section{Related Work}
For our introduced task to expose problems with pre-trained dialog models, the most relevant work is in the fields of controlled generation and adversarial attack. The goal for controlled generation is to generate coherent sentences containing some target attributes, whereas the task for adversarial attack is to craft some examples that can fool some trained classifiers.
\paragraph{Controlled Generation}
Most previous work in controlled text generation involve training or fine-tuning the whole model \cite{ficler-goldberg-2017-controlling, keskar_etal-2019-ctrl, peng-etal-2020-shot, ziegler-etal-2019-fine}. Alternatively, to utilize the high-quality pre-trained language model quality, \citet{dathathri-etal-2020-plug, madotto-etal-2020-plug} propose to perturb token distributions towards a specific attribute with residual adapters \cite{houlsby-etal-2019-parameter}. Recently, \citet{li-etal-2021-prefix, yu-etal-2021-attribute} show that optimizing simple prefix hidden states is effective in controlling pre-trained models, which inspires us to expose problems in neural dialog models by learning $h\_{trigger}$. In terms of applying reinforcement learning to language generation tasks, previous work leverages defined reward functions \cite{wu-etal-2016-google, li-etal-2016-deep, serban-etal-2017-deep} or human preference \cite{ziegler-etal-2019-fine}. All these work targets at generating sentences that contain target attributes. In contrast, our work optimizes prompt generation, which indirectly triggers a pre-trained model generating responses containing target attributes. There is no straightforward way to apply previous techniques to this task.
\paragraph{Adversarial Attacks with Pre-trained Neural Models}
Similar to generating adversarial examples to fool natural language understanding models \cite{zhang-etal-2019-paws, jin-etal-2020-is, li-etal-2020-contextualized, song-etal-2020-universal}, \citet{wallace-etal-2019-universal, sheng-etal-2020-towards} show that some learned discrete nonsensical universal triggers can be used to generate unsafe sentences. On the other hand, \citet{gehman-etal-2020-realtoxicityprompts} finds toxic prompts from naturally occurring sentences. The most similar work to ours is probably directing pre-trained models into generating a list of pre-defined tokens or sentences \cite{he-etal-2018-detecting, liu-etal-2019-say, liu-etal-2020-chat, he-glass-2020-negative}. In comparison, our task needs to generate coherent prompts according to the conversation history. Furthermore, instead of triggering pre-defined egregious responses, our proposed method is more flexible in exposing a wide range of problems such as consistency where crafting the target responses without context in advance is impossible.
\paragraph{Safety and Consistency}
To make machine learning models safe to use especially with language generation, there is a long literature in safety
such as hate speech \cite{zampieri-etal-2020-semeval} and bias \cite{dinan-etal-2019-build, dinan-etal-2020-multi}. Most of these works focus on abusive context detection. On a different line of research, some work introduces conditional generation to reduce toxicity \cite{dathathri-etal-2020-plug, gehman-etal-2020-realtoxicityprompts}.
These techniques mostly requires some toxic classifiers, which as shown in Section \ref{exp_classification}, may not work well for a different model distribution. Recently, \citet{xu-etal-2020-recipes} instructs humans to interact with neural dialog models in an adversarial way in order to induce unsafe responses from chatbots. Although classifiers trained with the introduced dataset are more robust, the collected data is relatively artificial because humans rely on apparent traits such as controversial statement or hate speech, regardless of the semantics and coherence of the conversation.
For consistency, previous work suggests generation grounded by information such as personas \cite{zhang-etal-2018-personalizing} and neural memories \cite{sukhbaatar-etal-2015-end}. In terms of consistency detection, \citet{dziri-etal-2019-evaluating-coherence, welleck-etal-2019-dialogue, li-etal-2020-dont} introduce and suggest using natural language inference to model conversation coherence. Recently, \citet{nie-etal-2020-i} collects a large contradicting human dialog corpus based on a conversational context and show better performance than entailment-based methods. However, as in \citet{xu-etal-2020-recipes}, annotators tend to ask repeating questions to provoke inconsistent answers.
Instead of asking humans to write prompts that may induce problematic responses, which is expensive and unrealistic with newly designed dialog models, we propose to trigger unsafe and inconsistent responses automatically. Our method can expose more systematic errors and is generally applicable to a wide variety of problems with trained neural models.
\section{Conclusion}
In this paper, we propose a weakly-supervised approach and a reinforcement learning approach to automatically expose problems with neural dialog models. Compared to data annotated by humans that rely on simple tricks, our methods can expose more systematic problems with coherent prompts and can help find these problems easily with newly trained neural models. We conduct extensive experiments with a safety exposure task and a consistency exposure task, and show that our proposed methods are effective. In addition, we showed that our method can also be used to generate data to improve performance of out-of-domain problem classifiers. In the future, we plan to extend our methods to other problems such as generic utterances and hallucination \cite{mielke-etal-2020-linguistic}.
\section*{Acknowledgments}
We thank Zhou Yu for early discussion, and the anonymous reviewers for their constructive suggestions. This work was supported by the National Science Foundation under Grant No. 1840191. Any opinions, findings, and conclusions or recommendations expressed are those of the authors and do not necessarily reflect the views of the NSF.
\section*{Ethical Considerations}
Our intended use case is to expose problems with neural (dialog) models where it is impractical to ask human annotators to interact with each new model to find potential errors within a reasonable budget. More importantly, we believe that by finding these problems automatically, we can reveal more systematic problems rather than relying on simple tricks as done in previous data collection. We hope that this can help model developers to find problems with our proposed method so that they can solve them before deploying their trained models. Furthermore, for research on fixing well-known problems in dialog systems, we hope that our work can also help with building more robust methods against realistic adversarial examples. Lastly, even though our methods aim at exposing problems in neural models, they can also be easily modified to avoid such problems and reduce the risk of misuse which we plan to explore more for future work.
|
1,116,691,499,234 | arxiv | \section{Introduction}\label{sec:intro}
The stationary Radiative Transfer Equation (sRTE) models the mesoscopic equilibrium state of particles, such as photons or electrons, propagating across heterogenous media. The density of particles at the equilibrium state results from the interaction of them with the microscopic constituents of the underlying background, by virtue of absorption and scattering. More specifically, denoting by $u(x,\theta)$ the phase-space density of particles, this is, those located at position $x\in{\mathbb R}^d$ and moving in direction $\theta\in{\mathbb S}^{d-1}$, the sRTE is given by
\begin{equation}\label{sRTE}
\theta\cdot\nabla_x u +\lambda u = \mathcal{I}(u)+f,
\end{equation}
where $f(x,\theta)$ represents a source or sink; $\lambda(x)$ measures absorption caused by the medium; and $\mathcal{I}(u)$ controls the rate of collisions between particles and the background. It is commonly defined as an integral operator of the form
\begin{equation*}
\mathcal{I}(u) = \int_{{\mathbb S}^{d-1}}k(x,\theta,\theta')\left(u(x,\theta') - u(x,\theta)\right)d\theta',
\end{equation*}
where the scattering kernel $k(x,\theta',\theta)$ represents the probability of a particle at a location $x$, changing direction from $\theta'$ to $\theta$ due to collisions. It is commonly assumed to be symmetric in the angular variable, this is, $k(x,\theta',\theta)=k(x,\theta,\theta')$.
In specific circumstances, a single or even a few scattering interactions are not enough to considerably change the direction of propagation of traveling particles. However, when collisions occur with high frequency, for example in the propagation of light in biological tissues or in thick atmospheres, their accumulative effect becomes significant and therefore visible at a larger scale. This is the regime of {\em highly forward-peaked scattering}. It intents to model a diffusive transport at a scale where the initial directionality of particles is not completely lost, but frequent enough to produce some mixing effect. It is suitable for the study of electron and photon transport \cite{BL2,LL,OF}, optical imaging and microscopy \cite{CCOCPH,KiKe}, and laser propagation on turbulent media \cite{HaBDH,RoRe}. More details on its derivation and application to inverse problem can be found in \cite{B,BKR}, and we refer the reader to \cite{DL} for a mathematical introduction to the theory of the transport equation.
A standard choice for scattering kernel in this regime is the well-known {\em Henyey--Greenstein} phase function \cite{HG}, defined in 3-dimensional space as
\begin{equation*}
k_{HG}(\theta',\theta) := \frac{1}{\Delta}\cdot \frac{1-g^2}{\left(1+g^2-2g\theta'\cdot\theta\right)^{3/2}}.
\end{equation*}
The parameter $\Delta>0$ is the {\em mean free path}, the mean distance between successive interactions of particles with the underlying medium; while $g\in(-1,1)$ is the {\em anisotropy factor}, representing forward-peaked scattering and determining the {\em mean scattering cosine} associated to $k_{HG}(\theta,\theta')$. The closer $g$ is to 1, the more forward-peaked the scattering is. Physically, the most meaningful distance is the {\em transport mean free path} given by $\Delta^*=\Delta/(1-g)$. It measures the average distance over which particles change directions `significantly' after undergoing a large number of forward-peaked collisions when $g$ is close to $1$.
Its been pointed out in a number of publications that Henyey--Greenstein scattering has small but significant large-angle effect, and this behavior is not accurately captured by standard approximation models such as the (local) Fokker--Planck equation and its respective Fermi--Eyges pencil-beam equation, which are better for numerical implementations and practical purposes. See for instance \cite{BL2, KiKe}. Moreover, the derivation of these simpler approximation models are known to fail for this specific kernel \cite{Po}. Phenomena of this type are not specific to the Henyey--Greenstein kernel and occur also for the Rutherford scattering kernel in electron transport \cite{BL2,LL}.
There has been previous attempts to find better suited approximation model such as the Boltzmann--Fokker--Planck equation and the Generalized Fokker--Planck equation (or Leakeas--Larsen equation) \cite{LL}.
In the highly forward-peaked limit when $g\to 1$, it was shown in \cite{AS} (for the time-evolution case) that (a rescaling of) global solutions to RTE with Henyey-Greenstein scattering cross-section, weakly converge in $L^2$ to functions satisfying a fractional version of Fokker--Planck, where the limiting scattering term involves a singular integral operator which resembles a fractional Laplace-Beltrami operator on the unit sphere. The hypoelliptic property of the associated integro-differential operator was also analyzed.
Similar results have been obtained in \cite{GPR1,GPR2} for the radiative transfer equation with long-range interactions. Analyzing the limiting scaling of this equation, the authors demonstrated the emergence of a Fokker--Planck type operator in the highly forward-peaked limit, with a diffusion component consisting of another singular integral operator whose high frequency behavior equals that of the Laplace-Beltrami operator on the sphere. The first publication provides precise hypoelliptic estimates for such operators.
The main goal of this paper is the study of the propagation of narrow beams in the highly forward-peaked regime. This corresponds to the analysis of the fractional Fokker--Planck equations obtained in \cite{AS}, for particle sources that are highly concentrated in phase-space and with a choice of scales so that $\Delta^*\gg 1$. The latter condition ensures that the beams remain narrow over a domain of interest. In this setting, the magnitude of the diffusion coefficient in the fractional Laplacian is small compared to the spatial dynamic of particles and characterized by a small parameter $\epsilon$. Analogously to what the authors have done in the local case ($s=1$) in \cite{BP}, we provide a higher order approximation to the Fokker--Planck solution by means of pencil-beams, which are solutions to appropriate fractional Fermi pencil-beam equations. We establish error estimates to contrast the accuracy of the approximations for the pencil-beam model and the ballistic (i.e., unscattered) transport solution. We perform this analysis in an adapted 1-Wasserstein sense with a level of accuracy given in terms of powers of the diffusion parameter $\epsilon$. Similarly as in \cite{AS}, we consider a generalized version of the Henyey--Greenstein forward-peaked scattering.
\subsection{Highly forward-peaked radiative transfer for narrow beams}
In space dimension $d\geq 2$,
we say $k(x,\theta',\theta)$ is a {\em Henyey--Greenstein scattering kernel with parameters $(g,s,m)$} whenever
\begin{equation}\label{HG_kernels}
k_g(x,\theta',\theta) := \frac{b(x)}{\Delta}\cdot\frac{(1-g)^m(1+g)^m}{\big((1-g)^2+2g(1-\theta'\cdot\theta)\big)^{\frac{d-1}{2}+s}},\quad(x,\theta',\theta)\in{\mathbb R}^d\times{\mathbb S}^{d-1}\times{\mathbb S}^{d-1},
\end{equation}
$b$ a Lipschitz function, and with $m>0$ so that
\begin{equation}\label{def_m}
\int_{{\mathbb S}^{d-1}}k_g(x,\theta',\theta)d\theta' = O(\Delta^{-1}).
\end{equation}
We recover the standard 3-dimensional kernel when the parameters are $(g,1/2,1)$. For a general fractional exponent $s\in(0,1)$, one easily verifies that in dimension $d=3$, $(g,s,2s)$ satisfies the required scaling for the integral of $b_g$ in \eqref{def_m}.
The collision operator associated to such kernels is given by
\begin{equation*}
\mathcal{I}_g(u) := \int_{{\mathbb S}^{d-1}}k_g(x,\theta,\theta')\left(u(x,\theta') - u(x,\theta)\right)d\theta'.
\end{equation*}
We work in the {\em narrow beam regime for particle transport} defined by the following hypotheses:
\begin{itemize}
\item[i.] The scattering kernel is of Henyey--Greenstein type with parameters $(g,s,m)$.
\item[ii.] The transport mean free path is given by $\Delta^*:=\Delta/(1-g)^m$, where $(1-g)^{m} \ll \Delta$. More precisely, we take $(1-g)^{m} = \epsilon^{2s}\Delta$, for $\epsilon\ll1$.
\item[iii.] The source term $f(x,\theta)$ is highly concentrated near a single point $(x
_0,\theta_0)\in{\mathbb R}^{d}\times{\mathbb S}^{d-1}$.
\end{itemize}
The exponent $2s$ in condition $ii$ is included here for notational convenience, and agrees with the exponent considered in \cite{BP} as we approach the local case $s=1$. Without loss of generality, we assume in most of the article that $(x_0,\theta_0) = (0,N)$ with $N=(0,\dots,0,1)$. We also consider the following modified version of $iii$:
\begin{itemize}
\item[iii'.] $f\in L^\infty_{x,\theta}\cap L^1_{x,\theta}$ is a compactly supported $\delta$-approximation to the identity $\delta_{x}\delta_{N}(\theta)$, i.e., such that $\left|\int f\varphi dxd\theta - \varphi(0,N)\right|\lesssim \delta$ for all $\varphi\in C({\mathbb R}^d\times{\mathbb S}^{d-1})$.
\end{itemize}
\subsection{Fractional Fokker--Planck equation (fFPE)}
As $g\to 1$, a Henyey-Greenstein kernel formally converges to the singular one
\begin{equation}\label{limiting_kernel}
k(x,\theta,\theta') := \frac{2^{m-\frac{d-1}{2}-s}\epsilon^{2s}b(x)}{\big(1-\theta\cdot\theta'\big)^{\frac{d-1}{2}+s}},
\end{equation}
whose link to the fractional Laplacian was noticed in \cite{AS} and established by means of a stereographic transformation $\mathcal{S}:{\mathbb S}^{d-1}\backslash\{S\}\to {\mathbb R}^{d-1}$, with $S=-N$. The unit sphere is considered to be an embedded hyper-surface in ${\mathbb R}^{d}$, where spatial coordinates have been fixed so that $N = (0,\dots,0,1)=-S$. The stereographic coordinates (with respect to $N$) and its associated surface measure are then defined by
$$
v = \mathcal{S}(\theta):= \frac{1}{(1+\theta_d)}(\theta_1,\dots,\theta_{d-1})\quad\text{and}\quad d\theta=\frac{2^{d-1}}{\langle v\rangle^{2(d-1)}}dv,\quad\text{with}\quad \langle v\rangle = (1+|v|^2)^{1/2};
$$
while its inverse transformation is defined as
$$
\theta = \mathcal{S}^{-1}(v) := \left(\frac{2v}{\langle v\rangle^2}, \frac{1-|v|^2}{\langle v\rangle^2}\right).
$$
The identity
\begin{equation*}\label{cos_identity}
1-\theta\cdot\theta' = 2\frac{|v-v'|^2}{\langle v\rangle^2\langle v'\rangle^2},
\end{equation*}
allows us to write the limiting singular integral $\int_{{\mathbb R}^{d-1}}k(x,\theta',\theta)\big(u(\theta')-u(\theta)\big)d\theta'$
in stereographic coordinates as
\begin{equation*}\label{scat_op_fFP}
\int_{{\mathbb R}^{d-1}}\langle v\rangle^{2(d-1)} k_{\mathcal{S}}(x,v',v)\big(u(v')-u(v)\big)dv',
\end{equation*}
where $[u]_\mathcal{S} = u(v)$ stands for the pullback $u(\mathcal{S}^{-1}(v))$, and with kernel such that
\begin{equation*}\label{scat_kernel_fFP}
k_{\mathcal{S}}(x,v',v)dv'dv = \left[k(x,\theta',\theta)d\theta' d\theta\right]_{\mathcal{S}}= \frac{2^{m-2s}\epsilon^{2s} b(x)dv'dv}{|v-v'|^{d-1+2s}\langle v'\rangle^{d-1-2s}\langle v\rangle^{d-1-2s}}.
\end{equation*}
By introducing the following version of the fractional Laplacian on the unit sphere (as in \cite{AS}),
\begin{equation}\label{def:frac_Lap_sphere}
\left[(-\Delta_\theta)^su\right]_{\mathcal{S}} := \langle \cdot\rangle^{d-1+2s}\big(-\Delta_v\big)^sw_{\mathcal{S}},\quad w_{\mathcal{S}} = \frac{[u]_{\mathcal{S}}}{ \langle \cdot\rangle^{d-1-2s}},
\end{equation}
with the Euclidean fractional Laplacian given by the singular integral
\begin{equation*}
(-\Delta_v)^sg(v) := c_{d-1,s}\int_{{\mathbb R}^{d-1}} \frac{f(v) - f(v+z)}{|z|^{d-1+2s}}dz,
\end{equation*}
we see that in the (so far formal) limit $g\to 1$, we encounter the {\em fractional Fokker--Planck equation}
\begin{equation}\label{fFP_diff}
\theta\cdot\nabla_x u +\lambda(x) u +\epsilon^{2s}\sigma(x)\Big((-\Delta_\theta)^su - cu\Big)= f,\quad \text{in }{\mathbb R}^d\times {\mathbb S}^{d-1},
\end{equation}
where $\sigma(x):= c^{-1}_{d-1,s}2^{m-2s}b(x)$ and $c=c_{s,d}>0$, for suitable positive constants $c_{d-1,s},c_{d,s}$ (see appendices A.2 and A.3 in \cite{AS} for more details). A rigorous convergence result is presented in Section \ref{sec:sRTE_fFP} (see also \cite{AS}).
Using this notation, the limiting kernel \eqref{limiting_kernel} takes the form $k(x,\theta,\theta') = \epsilon^{2s}\sigma(x)K(\theta',\theta)$, where we define its angular part and the associated integral operator as
\begin{equation}\label{kernel_K}
K(\theta',\theta) := \frac{c_{d-1,s}2^{s-\frac{d-1}{2}}}{\left(1-\theta\cdot\theta'\right)^{\frac{d-1}{2}+s}},\quad \mathcal{I}_\theta(u):= \int K(\theta',\theta)(u(\theta')-u(\theta))d\theta'.
\end{equation}
It will be occasionally more convenient to work with the integro-differential version
\begin{equation}\label{fFP_id}
\theta\cdot\nabla_x u +\lambda(x) u = \epsilon^{2s}\sigma(x)\mathcal{I}_\theta(u)+f(x,\theta),
\end{equation}
where, in stereographic coordinates, the integral part takes the form
\begin{equation}\label{fFP_kernel}
\begin{aligned}
&[\mathcal{I}_\theta(u)]_\mathcal{S}:= c_{d-1,s} \int_{{\mathbb R}^{d-1}}\big(u(v')-u(v)\big)K_\mathcal{S}(v',v)\langle v\rangle^{2(d-1)}dv',\\
&\text{with}\quad K_\mathcal{S}(v',v):= \frac{\langle v'\rangle^{-(d-1-2s)}\langle v\rangle^{-(d-1-2s)}}{|v-v'|^{d-1+2s}}.
\end{aligned}
\end{equation}
Throughout the paper, we will assume the following conditions on the coefficients of equation \eqref{fFP_id}:
\begin{equation}\label{cond_coeff}
\begin{aligned}
&\text{\em $\sigma$ and $\lambda$ are Lipschitz continuous, and there exist constants}\\
&\text{\em $\sigma_0,\lambda_0>0$, such that $\sigma_0\leq \sigma(x)\leq \sigma_0^{-1}$ and $\lambda_0\leq \lambda(x)\leq \lambda_0^{-1}$.}
\end{aligned}
\end{equation}
\subsection{Fermi pencil-beam approximation}
Similarly as in the original derivation of the Fermi pencil-beam equation (see, for instance, \cite{BL1}), we can derive an approximation to the fractional Fokker--Planck equation \eqref{fFP_diff} by following simple formal computations. Let $u$ be the solution to \eqref{fFP_diff} (or equivalently \eqref{fFP_id}-\eqref{fFP_kernel}) upon which we perform a change of coordinates that we proceed to define. We introduce the {\em stretched coordinate system}, or {\em pencil-beam coordinates}, as
\begin{equation*}
X := ((2\epsilon)^{-1}x',x^d)\quad\text{and}\quad V = \epsilon^{-1}\mathcal{S}(\theta),
\end{equation*}
defined on ${\mathbb R}^{d}\times{\mathbb R}^{d-1}$. In addition, the volume form on ${\mathbb R}^{d}\times{\mathbb S}^{d-1}$ adopts the representation
\begin{equation*}
dxd\theta =\frac{(2\epsilon)^{2(d-1)}}{\langle \epsilon V\rangle^{2(d-1)}}dXdV.
\end{equation*}
We see that in pencil-beam coordinates, the function $U(X,V) = (2\epsilon)^{2(d-1)}u(2\epsilon X', X^d,\mathcal{J}(\epsilon V))$ satisfies the equation
\begin{equation*}
\begin{aligned}
&\left(\frac{V}{\langle\epsilon V\rangle^2}, \frac{1-\epsilon^2|V|^2}{\langle \epsilon V\rangle^2}\right)\cdot \nabla_{X}U + \lambda(2\epsilon X',X^d) U\\
&= \epsilon^{2s}\sigma(2\epsilon X',X^d) \cdot c_{d-1,s} \int_{{\mathbb R}^{d-1}} \left(U(X,V') - U(X,V)\right)K_\mathcal{S}(\epsilon V,\epsilon V')\langle \epsilon V\rangle^{2(d-1)}dV' + F,
\end{aligned}
\end{equation*}
for $F(X,V)=(2\epsilon)^{2(d-1)}f(2\epsilon X',X^d,\mathcal{J}(\epsilon V))$. By ignoring the dependence of $U$ and $F$ with respect to $\epsilon$, we can take the point-wise limit as $\epsilon\to 0$ in order to (formally) deduce the {\em fractional Fermi pencil-beam} equation
\begin{equation}\label{fFpb}
\begin{array}{ll}
\mathcal{P}(U):=\partial_{X^d}U + V\cdot\nabla_{X'} U +\widetilde{\lambda} U + \widetilde{\sigma}(-\Delta_V)^sU= F,& \text{in }{\mathbb R}^d_+\times{\mathbb R}^{d-1},
\end{array}
\end{equation}
where $\widetilde{\sigma}(X^d) = \sigma(0,X^d)$ and $\widetilde{\lambda}(X^d) = \lambda(0,X^d)$. ${\mathbb R}^d_+$ stands for the half-space $X=(X',X^d)\in{\mathbb R}^d$ with $X^d>0$.
We also define the {\em backward (or adjoint) problem} as
\begin{equation}\label{b_fFpb}
\begin{array}{ll}
\mathcal{P}'(W):=-\partial_{X^d}W - V\cdot\nabla_{X'} W +\widetilde{\lambda} U + \widetilde{\sigma}(-\Delta_V)^sW= F,& \text{in }{\mathbb R}^d_+\times{\mathbb R}^{d-1}.
\end{array}
\end{equation}
For more details on the derivation of the Fermi pencil-beam equation we refer the reader to \cite{BL1}.
\subsection{1-Wasserstein distance}
Narrow beam solutions are by construction singular. Approximation errors need to be measured using a metric that adequately captures beam spreading, which is intuitively a form of a `mass' transport. A natural notion to measure the mass transport of probability measures is given by the 1-Wasserstein (or earth-mover) distance. In our setting, while solutions preserve positivity, their total mass may vary. As we have done in, e.g., \cite{BJ, BP} for similar reasons, we consider a generalization of the standard 1-Wasserstein distance.
We denote by $BL_{1,\kappa}$ the set of $\psi\in C({\mathbb R}^d\times{\mathbb S}^{d-1})$ such that $\|\psi\|_\infty\leq 1$ and $\text{\rm Lip\hspace{.1em}}(\psi)\leq\kappa$, for a fixed parameter $\kappa\geq 1$.
Given two finite Borel measures in ${\mathbb R}^d\times{\mathbb S}^{d-1}$, $\mu$ and $\nu$, their 1-Wasserstein distance is defined as
\begin{equation*}
\mathcal{W}^1_\kappa(\mu,\nu) :=\sup\left\{\int \psi (\mu-\nu) : \psi\in BL_{1,\kappa}\right\}.
\end{equation*}
For any $\Omega\subset{\mathbb R}^d$ open and bounded, we can similarly define a 1-Wasserstein distance in the compact region $\bar{\Omega}\times{\mathbb S}^{d-1}$ by restricting the test functions to this set, which we denote by $\mathcal{W}^1_{\bar{\Omega},\kappa}(\mu,\nu)$.
The parameter $\kappa$ defines the spatial scale $\kappa^{-1}$ over which we wish to penalize transport. The uniform bound on $\psi$ allows us to penalize variations in total mass.
\subsection{Main results} \label{subsec:main_results}
Our main result asserts that a suitable transformation of a Fermi pencil-beam solution approximates the solution to the fractional Fokker--Planck equation, at a higher accuracy than the ballistic solution ---the one that completely neglects diffusion (i.e., a solution to \eqref{fFP_diff} for $\epsilon=0$).
\begin{theorem}\label{thm:narrow_beam}
Assume that conditions {\em i}, {\em ii} and {\em iii'} hold with the latter satisfied for $\delta\lesssim \epsilon^{2s}\kappa$. Let $u$ be the solution to the fractional Fokker--Planck equation \eqref{fFP_diff}, $v$ the ballistic solution, and $U$ the solution to the corresponding fractional Fermi pencil-beam equation \eqref{fFpb}. The pencil-beam approximation in the original variables is defined by
\begin{equation*}
\mathfrak{u}(x,\theta) := (2\epsilon)^{-2(d-1)}U((2\epsilon)^{-1}x',x^d,\epsilon^{-1}\mathcal{S}(\theta)).
\end{equation*}
Then, for any $s'\in(0,s)$ in dimension $d\geq 3$, or any $s'\in(2s-1,s)$ in dimension $d=2$, there exist positive constants $M_1=M_1(d,s,s')$, $M_2=M_2(d,s,s')$ and $m=m(d,s)$ (depending also on $\lambda$ and $\sigma$) such that
$$
\mathcal{W}^1_{\kappa}(u,\mathfrak{u})\leq M_1\cdot\epsilon^{2s'}\kappa^{s'},\quad\text{and}\quad m\cdot\min\{\epsilon\kappa,1\}\leq \mathcal{W}^1_{\kappa}(v,\mathfrak{u})\leq M_2\cdot(\epsilon\kappa)^{\min\{2s',1\}},
$$
where $M_1\to \infty$ as $s'\to s$, while $M_2\to\infty$ when $s'\to s$ for $s\leq1/2$, otherwise, $M_2$ is independent of $s'$.
\end{theorem}
The proof of this result is split into Theorems \ref{thm:fFP_vs_pb} and \ref{thm:ball_vs_pb}, which we prove in Sections \ref{subsec:fFpb_approx} and \ref{subsec:ballistic_approx}, respectively.
By linearity, we may generalize the above pencil-beam approximation to sources that do not satisfy condition {\em iii} (and hence {\em iii'}). We construct a general approximation by (continuously) superposing pencil-beams. The details of this definition are postponed to Section \ref{sec:superposition}.
\begin{theorem}[Approximation via continuous superposition]\label{thm:cont_sup}
Under the hypotheses of the previous theorem, except for hypothesis \text{\em iii},
the same conclusion holds for a broad source $f$ satisfying
\begin{equation*}
f\in L^1({\mathbb R}^d\times{\mathbb S}^{d-1}),\quad f\geq 0\quad\text{and}\quad\text{supp}(f)\Subset {\mathbb R}^d\times{\mathbb S}^{d-1},
\end{equation*}
and $\mathfrak{u}(x,\theta) := \int_{{\mathbb R}^d\times{\mathbb S}^{d-1}}f(y,\eta)\mathfrak{u}(x,\theta;y,\eta)dyd\eta$, a continuous superposition of pencil-beams.
\end{theorem}
Our last result concerns the approximation to fractional Fokker--Planck solutions by means of a finite number of pencil-beams.
\begin{theorem}[Approximation via discrete superposition]\label{thm:disc_sup}
Under the hypotheses of the previous theorem, we reach the same conclusions for some $\mathfrak{u}(x,\theta) := \sum_{i=1}^Ia_i\cdot\mathfrak{u}(x,\theta;x_i,\theta_i)$, a discrete superposition of pencil-beams.
\end{theorem}
Throughout the paper, we adopt the notation $a \lesssim b$ to denote an estimate of the form $a\leq Cb$, for a constant that is independent of $\epsilon$ and $\kappa$. We will occasionally use the notation: $Q={\mathbb R}^d\times{\mathbb S}^{d-1}$, $Q_+={\mathbb R}^d_+\times{\mathbb S}^{d-1}$, $\mathcal{Q}={\mathbb R}^d\times{\mathbb R}^{d-1}$, and $\mathcal{Q}_+={\mathbb R}^d_+\times{\mathbb R}^{d-1}$.
The paper is organized as follows. We begin with the study of the sRTE and its Fokker--Planck approximation. Well-posedness results are proven in Section \ref{sec:sRTE_fFP}, as well a convergence result in the 1-Wasserstein framework, along the lines of \cite{AS} for the $L^2$-case. In Section \ref{sec:fFpb}, we show existence and uniqueness of solution to the fractional Fermi pencil-beam equation, and prove some integrability properties of their solutions which will be of great importance in subsequent sections. We then move to the approximation analysis where we define in details the pencil-beam approximation and provide the proof of our main Theorem \ref{thm:narrow_beam}. This is the content of Section \ref{sec:Fpb_approx}. We conclude the article with constructions, based on the narrow beam case, for the superposition of pencil-beams and the proof of Theorems \ref{thm:cont_sup} and \ref{thm:disc_sup} in Section \ref{sec:superposition}.
\section{Stationary Radiative Transfer and fractional Fokker--Planck equations}\label{sec:sRTE_fFP}
\subsection{Well-posedness of sRTE}\label{subsec:sRTE}
The existence and uniqueness of global solutions in $L^2$ is (easily) deduced from the well-posedness in bounded domains \cite{DL}. It follows by an approximation argument after confining equation \eqref{sRTE} to a bounded region $\Omega\times{\mathbb S}^{n-1}$ with smooth boundary, and solving the null Cauchy data problem there. Defining
$$W^2_{\Omega}:=\{u\in L^2(\Omega\times{\mathbb S}^{d-1}\} \;:\; \theta\cdot\nabla_x u\in L^2(\Omega\times{\mathbb S}^{d-1}))\},$$
and similarly $W^2$ when $\Omega={\mathbb R}^d$ in the previous definition,
the standard argument states that the solutions $u_{\Omega}\in W^2_\Omega$ converge to the desired global solution $u$ of \eqref{sRTE} as $\Omega\to {\mathbb R}^d$, and moreover $u\in W^2$. A proof of this can be found in Appendix \ref{appdx:1}.
\begin{theorem}\label{thm:wellposedness_sRTE}
For $f\in L^2(Q)$, there exists a unique solution $u \in W^2$ to the stationary RTE \eqref{sRTE} for an integrable symmetric kernel. Furthermore, if $f\geq 0$ then $u\geq 0$.
\end{theorem}
We need the following two properties of solutions to the sRTE with Henyey--Greenstein kernels. The fist one follows directly from the mass-conservation property: $\mathcal{I}_g[\cdot]$:
\begin{equation}\label{mass_conservation}
\int_{{\mathbb S}^{d-1}} \mathcal{I}_g[u^g](x,\theta)d\theta = \int_{{\mathbb S}^{d-1}} \int_{{\mathbb S}^{d-1}}\big(u^g(\theta')-u^g(\theta)\big)k_g(x,\theta,\theta')d\theta'd\theta=0.
\end{equation}
\begin{lemma}\label{lemma:L1_est}
For a nonnegative source $f\in L^1(Q)$, the solution to \eqref{sRTE}-\eqref{HG_kernels} satisfies the estimate $\|u^g\|_{L^1}\leq \lambda_0^{-1}\|f\|_{L^1}$ for all $g\in(0,1)$.
\end{lemma}
\begin{proof}
This follows by integrating equation \eqref{sRTE} and the fact that for $f\geq 0$, $u^g\geq 0$. Indeed, by integration by parts and \eqref{mass_conservation} we get
$$
\underbrace{\int_Q \theta\cdot\nabla_x u^g dxd\theta}_{=0} +\int_Q \lambda u^gdxd\theta = \underbrace{\int_Q\mathcal{I}_g[u^g]dxd\theta}_{=0}+\int fdxd\theta.
$$
\end{proof}
\begin{lemma}\label{lemma:Linfty_est}
For
$f\in L^\infty(Q)\cap L^2(Q)$
the solution to \eqref{sRTE}-\eqref{HG_kernels} satisfies the estimate $\|u^g\|_\infty\leq \lambda_0^{-1}\|f\|_\infty$ for all $g\in(0,1)$.
\end{lemma}
\begin{proof}
We follow \cite{HJJ}.
Let $M=\|f\|_\infty$ and assume there exists $\alpha>0$ and a bounded set $A\subset{\mathbb R}^n$ with positive measure such that (without loss of generality) $u(x,\theta)>M\lambda_0^{-1}+\alpha$ in $A$.
For any small enough $\delta>0$ we can find a ball $B\subset Q$ such that
$$
\text{meas}(B\cap A)>(1-\delta)\text{meas}(B).
$$
This in particular implies the inequality $\text{meas}(B\backslash A)<\delta\cdot \text{meas}(B)$.
For such ball we take a nonnegative function $h\in C^\infty_c(\bar{B})$ so that $h\lesssim \text{meas}(B)^{-1}$, and
$$\left|\int\Big(\frac{\chi_B}{\text{meas}(B)} - h\Big)dxd\theta\right| = \left|1-\int h dxd\theta\right|<\delta.$$
Let $\varphi\in W^2$, $\varphi\geq 0$, be the unique solution to the backward sRTE,
$$
-\theta\cdot\nabla_x\varphi + \lambda\varphi = \mathcal{I}_g[\varphi] + h,
$$
which according to the previous lemma it satisfies
$$
\int_Q\varphi dxd\theta\leq\lambda_0^{-1}\int_Q hdxd\theta<\lambda_0^{-1}(1+\delta).
$$
Its existence, uniqueness, and regularity follows similarly as in Theorem \ref{thm:wellposedness_sRTE} by means of \cite[Chapter XXI-Proposition 8]{DL}.
Then,
$$
\int f\varphi dxd\theta\leq M\int_Q\varphi dxd\theta \leq M\lambda_0^{-1}(1+\delta).
$$
On the other hand,
\begin{equation}\label{eq1}
\int_Q f\varphi dxd\theta = \int_Q u^ghdxd\theta,
\end{equation}
where we split the integration into two integrals, one over $B\cap A$ and another one over $B\backslash A$. By our initial assumption,
$$
\begin{aligned}
\int_{B\cap A} u^g h dxd\theta&\geq (M\lambda_0^{-1}+\alpha)\left(\int_{B\cap A}\frac{\chi_B}{\text{meas}(B)}dxd\theta - \int_{B\cap A}\Big(\frac{\chi_B}{\text{meas}(B)}-h\Big)dxd\theta\right)\\
&\geq (M\lambda_0^{-1}+\alpha)\left(\frac{\text{meas}(B\cap A)}{\text{meas}(B)} - \delta\right) >(M\lambda_0^{-1}+\alpha)(1-2\delta),
\end{aligned}
$$
and also
$$
\begin{aligned}
\int_{B\backslash A} u^g h dxd\theta \leq \|u^g\|_{L^2}\|h\|_{L^2}
\leq \|u^g\|_{L^2}\|h\|_\infty^{1/2}\text{meas}(B\backslash A)^{1/2}\lesssim \|u^g\|_{L^2}\delta^{1/2}.
\end{aligned}
$$
We then obtain a contradiction by noticing that \eqref{eq1} and previous estimates imply that for some $C>0$, independent of $\delta$,
$$(M\lambda_0^{-1}+\alpha)(1-2\delta) - C\delta^{1/2}\leq M\lambda_0^{-1}(1+\delta),$$
and consequently
$$M\lambda_0^{-1}+\alpha \leq M\lambda_0^{-1} + C\delta^{1/2},$$
which cannot hold if $\delta$ is sufficiently small.
\end{proof}
\subsection{Well-posedness of fFPE}\label{subsec:fFP}
Let's introduce the bilinear form
\begin{equation*}
\mathcal{B}(u,\varphi) := \frac{1}{2}\int_{{\mathbb S}^{d-1}}\int_{{\mathbb S}^{d-1}}K(\theta',\theta)(u(\theta') - u(\theta))(\varphi(\theta') - \varphi(\theta))d\theta' d\theta,
\end{equation*}
for the kernel $K(\theta',\theta)$ as in \eqref{kernel_K}, and define its associated Hilbert space $H^s_{\mathcal{B}}$ given by
\begin{equation*}
H^s_{\mathcal{B}} := \{u\in L^{2}_\theta \;: \; \mathcal{B}(u,u)<+\infty\},\quad \text{for}\quad s\in(0,1),
\end{equation*}
with inner product
\begin{equation*}
\langle u,\varphi\rangle_{H^s_{\mathcal{B}}} := \mathcal{B}(u,\varphi) + \int_{{\mathbb S}^{d-1}}u\varphi d\theta.
\end{equation*}
It is worth mentioning how $H^s_{\mathcal{B}}$ and $H^s({\mathbb R}^{d-1})$ are related to each other when stereographic coordinates are considered. It turns out that
passing to stereographic coordinates, $\|u\|^2_{H^s_{\mathcal{B}}}$ is
equivalent (up to a multiplicative factor) to
\begin{equation*}
\int_{{\mathbb R}^{d-1}}\int_{{\mathbb R}^{d-1}}\frac{(u(v') - u(v))^2}{|v'-v|^{d-1+2s}\langle v'\rangle^{d-1-2s}\langle v\rangle^{d-1-2s}}dv' dv + \int_{{\mathbb R}^{d-1}}\frac{|u(v)|^2}{\langle v\rangle^{2(d-1)}}dv,
\end{equation*}
which yields the inclusion $H^s({\mathbb R}^{d-1})\subset \left[ H^s_\mathcal{B}\right]_{\mathcal{S}}$.
Moreover, one can also verify that
\begin{equation*}
\begin{aligned}
&\int_{{\mathbb R}^{d-1}}\int_{{\mathbb R}^{d-1}}\frac{(u(v') - u(v))^2}{|v'-v|^{d-1+2s}\langle v'\rangle^{d-1-2s}\langle v\rangle^{d-1-2s}}dv'dv \sim \|(-\Delta_v)^{s/2}w_{\mathcal{S}}\|^2_{L^2({\mathbb R}^{d-1})}+\|u\|^2_{L^2({\mathbb S}^{d-1})},
\end{aligned}
\end{equation*}
which implies the following equivalence of norms
\begin{equation}\label{equiv_norms}
\|u\|_{H^s_\mathcal{B}}\sim \|(-\Delta_v)^{s/2}w_{\mathcal{S}}\|_{L^2({\mathbb R}^{d-1})}+\|u\|_{L^2({\mathbb S}^{d-1})} \sim \|(-\Delta_\theta)^{s/2}u\|.
\end{equation}
Let us consider now the Hilbert space $\mathcal{H}^s_\epsilon := L^2_x({\mathbb R}^d;H^s_\mathcal{B})$, equipped with the norm
$$\|f\|^2_{\mathcal{H}^s_\epsilon }:= \|f\|_{L^2_{x,\theta}}^2 + \epsilon^{2s}\int_{{\mathbb R}^d}\|f\|^2_{H^s_\mathcal{B}}dx,$$
where we use the shorthand notation $L^2_{x,\theta}$ for $L^2({\mathbb R}^d\times{\mathbb S}^{d-1})$.
Its dual will be denoted by $(\mathcal{H}^s_\epsilon)'$. Denoting the transport operator $T = \theta\cdot \nabla_x$, we define the solution space for the fractional Fokker--Planck equation as
$$\mathcal{Y}^s_\epsilon := \{f\in \mathcal{H}^s_\epsilon\;:\; Tf\in(\mathcal{H}^s_\epsilon)'\},\quad\text{with norm}\quad \|f\|_{\mathcal{Y}^s_\epsilon}:= \|f\|_{\mathcal{H}^s_\epsilon} + \|Tf\|_{(\mathcal{H}^s_\epsilon)'}.$$
\begin{definition}\label{def:weak_sol}
We say that a function $u\in\mathcal{Y}^s_\epsilon$ is a weak solution of \eqref{fFP_diff} if for all $\varphi\in C^\infty_c({\mathbb R}^d\times{\mathbb S}^{d-1})$, it satisfies
\begin{equation}\label{weak_sol}
\begin{aligned}
&\int -u(\theta \cdot\nabla_x\varphi) + \lambda u\varphi dxd\theta +\epsilon^{2s}\int\sigma(x)\mathcal{B}(u,\varphi)dx = \int f\varphi dxd\theta.
\end{aligned}
\end{equation}
\end{definition}
The well-posedness and some properties of the solutions to the fractional Fokker--Planck equation are summarized next. Additional properties are stated in Theorem \ref{thm:wellposedness_FP2} after we obtain a convergence result for sRTE-solutions in the highly forward-peaked limit.
\begin{theorem}\label{thm:wellposedness_FP}
Let $\sigma,\lambda$ satisfying \eqref{cond_coeff}. For any $f\in L^2_{x,\theta}$ there exists a unique solution $u\in\mathcal{Y}^s_\epsilon$ to \eqref{fFP_diff}. Moreover,
there is $C>0$ such that $\|u\|_{\mathcal{Y}^s_\epsilon}\leq C\|f\|_{L^2_{x,\theta}}$ for all $f\in L^2_{x,\theta}$
\end{theorem}
\begin{proof}
\noindent 1. Following the definition of weak solution, we introduce the bilinear form $a:\mathcal{H}^s_\epsilon\times C^\infty_c\to {\mathbb R}$ so that the left hand side of \eqref{weak_sol} is given by $a(u,\varphi)$, while the right hand defines the linear operator $L(\varphi)$ mapping $L:C^\infty_c\to {\mathbb R}$. The bilinear form becomes a bounded linear operator in $\mathcal{H}^s_\epsilon$ when freezing the second component $\varphi\in C^\infty_c$, and moreover, using integration by parts we get
$$
a(\varphi,\varphi)\geq \min\{\lambda_0,\sigma_0\}\|\varphi\|_{\mathcal{H}^s_\epsilon}^2,\quad\forall \varphi \in C^\infty_c.
$$
Then, applying \cite[Theorem 2.4]{BP} (see also \cite{L}) for the pre-Hilbert space $\mathcal{F} = (C^\infty_c, \|\cdot\|_{\mathcal{H}^s_\epsilon})$ we obtain the existence of a weak solution $u\in \mathcal{H}^s_\epsilon$, which in addition satisfies $u\in \mathcal{Y}^s_\epsilon$, since \eqref{weak_sol} allows to define $Tu$ in the sense of distributions, and $T$ is a bounded linear operator in $\mathcal{H}^s_\epsilon$.
It is not hard to verify that the set $C^\infty_c({\mathbb R}^d\times{\mathbb S}^{d-1})$ is dense in $\mathcal{Y}^s_\epsilon$, and this allows us to obtain the integration by parts formula
\begin{equation}\label{ibp_formula}
\langle u_1,Tu_2\rangle_{\mathcal{H}^s_\epsilon,(\mathcal{H}^s_\epsilon)'} = -\langle u_2,Tu_1\rangle_{\mathcal{H}^s_\epsilon,(\mathcal{H}^s_\epsilon)'}, \quad\forall u_1,u_2\in \mathcal{Y}^s_\epsilon,
\end{equation}
In particular, $\langle u,Tu\rangle_{\mathcal{H}^s_\epsilon,(\mathcal{H}^s_\epsilon)'} = 0$.
The identity \eqref{ibp_formula} is obtained as in \cite{Ba}. More recent proofs of similar density results are presented in \cite{AM,BS}. The density used in this paper is easier to establish as we do not deal with boundaries.
It follows from the weak formulation of the fractional Fokker--Planck equation that
\begin{equation*}
\langle u,Tu\rangle_{\mathcal{H}^s_\epsilon,(\mathcal{H}^s_\epsilon)'} + \int \lambda |u|^2 dxd\theta +\epsilon^{2s}\int\sigma(x)\mathcal{B}(u,u)dx = \int fudxd\theta,
\end{equation*}
and consequently,
$$
\min\{\lambda_0,\sigma_0\}\|u\|_{\mathcal{H}^s_\epsilon}^2\leq \|f\|_{L^2_{x,\theta}}\|u\|_{L^2_{x,\theta}} \leq \frac{1}{2}\delta^{-2}\|f\|^2_{L^2_{x,\theta}}+\frac{1}{2}\delta^{2}\|u\|^2_{L^2_{x,\theta}},
$$
where we choose $\delta>0$ small enough so that we can absorb the last term with the left hand side.
Uniqueness of solutions follows directly from the continuous dependence estimate.
\end{proof}
\subsection{Convergence of sRTE and some properties of fFPE}
In what follows, we take the kernel $k_g(x,\theta,\theta')$ to be in the narrow beam scaling, thus, we assume it satisfies condition $ii$. Then,
\begin{equation}\label{HGk}
k_g(x,\theta,\theta') := \frac{\epsilon^{2s}(1+g)^m b(x)}{\big(1+g^2-2g\theta\cdot\theta'\big)^{\frac{d-1}{2}+s}},
\end{equation}
whose limit as $g\to 1$ is given by \eqref{limiting_kernel}
We next extend the weak-$L^2$ convergence result in \cite{AS} to the topology induced by the Wasserstein distance.
\begin{theorem}\label{thm:approx_fFP}
For any open bounded $\Omega\subset{\mathbb R}^d$ and a nonnegative and integrable function $f$, there exists an increasing sequence $\{g_k\}_k\subset (0,1)$ converging to 1, and $\kappa_0>0$ depending on $\|\lambda\|_{C^1}$ and $\|f\|_{L^1}$, such that
\[
\mathcal{W}^1_{\bar{\Omega},\kappa}(u^{g_k},u)\to 0\quad\text{as}\quad k\to\infty,
\]
for all $\kappa\geq\kappa_0$, and where $u^{g_k}$ are the solutions to \eqref{sRTE} with kernel \eqref{HGk} and parameter $g=g_k$, and $u$ is the solution to the fractional Fokker--Planck equation \eqref{fFP_diff}.
\end{theorem}
\begin{proof}
Let $f$ be a nonnegative integrable function and consider an increasing sequence $\{g_n\}_{n\geq 1}\subset(0,1)$ so that $g_n\to 1$ as $n\to\infty$. We have that $u^{g_n}\geq 0$ for all $n$, and therefore, we can regard $\lambda u^{g_n}/\|f\|_{L^1}$ as a family of Radon probability measures since (as in Lemma \ref{lemma:L1_est})
$$
\int \lambda u^{g_n}dxd\theta = \|f\|_{L^1},\quad\forall n\geq 1.
$$
In the compact set $\bar{\Omega}\times{\mathbb S}^{d-1}$, weak$^*$-compactness for measures (Banach-Alaoglu theorem) yields then the existence of a subsequence $\{u^{g_{n_k}}\}_k$, and another Radon probability measure $\mu$, for which
\begin{equation}\label{weak*_conv}
\frac{1}{\|f\|_{L^1}}\int_{\bar{\Omega}\times{\mathbb S}^{d-1}} \varphi \lambda u^{g_{n_k}}dxd\theta \to \int_{\Omega\times{\mathbb S}^{d-1}} \varphi \mu(x,\theta),\quad\forall \;\varphi\in C_b(\bar{\Omega}\times{\mathbb S}^{d-1}),
\end{equation}
with $C_b(\bar{\Omega}\times{\mathbb S}^{d-1})$ the set of bounded continuous functions in $\bar{\Omega}\times{\mathbb S}^{d-1}$. Writing instead $\psi = \lambda\varphi/\|f\|_{L^1}$, and since $\lambda\in \text{Lip}({\mathbb R}^d\times{\mathbb S}^{d-1})$, then we can find $\kappa_0>0$ such that for all $\kappa\geq \kappa_0$,
\begin{equation*}
\int_{\Omega\times{\mathbb S}^{d-1}} \psi u^{g_{n_k}}dxd\theta \to \int_{\Omega\times{\mathbb S}^{d-1}} \psi \big(\|f\|_{L^1}\lambda^{-1}\mu(x,\theta)\big),\quad\forall \;\psi\in BL_{1,\kappa}(\bar{\Omega}\times{\mathbb S}^{d-1}),
\end{equation*}
Due to uniqueness of the distributional limit, one verifies that $\|f\|_{L^1}\lambda^{-1}\mu = u dxd\theta$, with $u$ solution to \eqref{sRTE} and limiting scattering cross-section \eqref{limiting_kernel} (i.e. the fractional Fokker--Planck equation \eqref{fFP_id}). This follows from similar computations as in \cite{AS}, leading to
$$
\|u^g\|_{L^2_{x,\theta}}+\|\theta\cdot\nabla_xu^g\|_{L^2_{x,\theta}} + \|\mathcal{I}_gu^g\|_{L^2_{x,\theta}}\leq C\|f\|_{L^2_{x,\theta}},
$$
uniformly in $g$, which subsequently imply weak-$L^2_{x,\theta}$ convergence to a limiting function $u$, i.e.
$$
\theta\cdot\nabla_xu^g + \lambda u^g - \mathcal{I}_g[u^g]\;\rightharpoonup\; \theta\cdot\nabla_xu + \lambda u - \mathcal{I}_\theta [u]\quad\text{as}\quad g\to 1.
$$
The previous then implies
\begin{equation*}
\mathcal{W}^{1}_{\bar{\Omega},\kappa}(u^{g_k},u)\to 0\quad\text{as}\quad k\to\infty.
\end{equation*}
\end{proof}
With the aid of the previous convergence theorem, some properties of the radiative transfer solution are inherited by the limiting function. This yields the following additional properties for the solution to the fractional Fokker--Planck equation.
\begin{theorem}\label{thm:wellposedness_FP2}
Let $\sigma,\lambda$ and $f$ as in Theorem \ref{thm:wellposedness_FP}. The unique solution $u$ to the fractional Fokker--Planck equation \eqref{fFP_diff} satisfies the following additional properties:
\begin{enumerate}
\item[1.] (non-negativity) if $f\geq 0$ then $u\geq 0$;
\item[2.] (continuous dependence for bounded sources) $\|u\|_\infty\leq\lambda_0^{-1}\|f\|_\infty$ for all $f\in L^\infty_{x,\theta}\cap L^2_{x,\theta}$.
\end{enumerate}
\end{theorem}
\begin{proof}
\noindent 1. The non-negativity of $u$ for a source term $f\geq 0$ follows from the analogous property of the sRTE in Theorem \ref{thm:wellposedness_sRTE} and the convergence in Theorem \ref{thm:approx_fFP}.\\
\noindent 2. The continuity in the $L^\infty$-norm follows in a similar fashion as Lemma \ref{lemma:Linfty_est} for sRTE. We point out here the main differences in the argument. The exact same computations yield to \eqref{eq1} for a test function $\varphi_g$, solution to the backward sRTE. Instead of estimating directly the right hand side $\int_Qu^g hdxd\theta$, we take the limit as $g\to 1$ and obtain
$$
\int f\varphi dxd\theta = \int uh dxd\theta,
$$
where $\varphi$ is the weak-$L^2_{x,\theta}$ limit of the solutions $\varphi_g$, thus it solves a backward fractional Fokker-Planck equation. It is of course nonnegative and with $L^1_{x,\theta}$-norm bounded by $\lambda_0^{-1}(1+\delta)$.
The proof is then complete by repeating the remaining steps in the proof of Lemma \ref{lemma:Linfty_est}.
\end{proof}
\subsection{Regularity of solutions to fFPR}
The regularity results of this section are based on a local representation of the fractional Fokker--Planck equation, in a neighborhood of an arbitrary $(x_0,\theta_0)\in{\mathbb R}^d\times{\mathbb S}^{d-1}$. The idea is to pseudo-localize the equation in the sense that we can approximate a solution to the Fokker--Planck equation with smooth and compactly supported functions, satisfying a Fokker--Planck equation with a smooth source that might not be compactly supported.
\begin{theorem}[Regularity of solutions]\quad
\begin{enumerate}
\item[1.](Sub-elliptic regularity) For $\sigma,\lambda$ and $f$ as in Theorem \ref{thm:wellposedness_FP}, the unique solution $u$ to the fractional Fokker--Planck equation is a strong solution, and more precisely
$(-\Delta_\theta)^{s}u,\;Tu \in L^2_{loc}({\mathbb R}^d;L^2({\mathbb S}^{d-1}))$;
\item[2.] (Continuity) For $f\in L^\infty_{x,\theta}\cap L^2_{x,\theta}$, the solution $u$ to the fractional Fokker--Planck equation \eqref{fFP_diff} is locally H\"older continuous in ${\mathbb R}^d\times{\mathbb S}^{d-1}$, and consequently, it belongs to $C({\mathbb R}^d\times{\mathbb S}^{d-1})$.
\end{enumerate}
\end{theorem}
\begin{proof}
\noindent 1.
The subelliptic character of fractional order kinetic equation is known to derive, for example, from commutator identities and energy estimates \cite{Bo,A}. In order to adapt these arguments to the setting of the fractional Laplacian on the unit sphere, we pseudo-localize the equation and consider suitable local coordinates that allow us to apply known results.
By a standard localization and mollification technique (see Appendix \ref{appdx:loc_moll}) we can assume without loss of generality that $u$ is a smooth and compactly supported solution to the fractional Fokker--Planck equation \eqref{fFP_diff}, or equivalently \eqref{fFP_id}, with support contained in ${\mathbb R}^n\times{\mathbb S}^{d-1}_+$, and for a right-hand side $f\in C^\infty({\mathbb R}^{d}\times{\mathbb S}^{d-1})\cap L^2({\mathbb R}^{d}\times{\mathbb S}^{d-1})$. By considering beam coordinates, we see that for $\theta(v),\theta'(v')\in {\mathbb S}^{d-1}_+$,
$$
\left(1-\theta\cdot\theta'\right)_{\mathcal{B}} = \frac{\langle v\rangle\langle v'\rangle -v\cdot v' -1}{\langle v\rangle\langle v'\rangle}.
$$
Therefore, multiplying \eqref{fFP_id} by $\langle v\rangle^{-\frac{d}{2}}$ and setting $\widetilde{u}(x,v) = \langle v\rangle^{-\frac{d}{2}-1}u(x,v)$, we obtain the kinetic equation
\begin{equation*}
\partial_{x^d}\tilde{u}+v\cdot\nabla_{x'}\tilde{u} = \int \widetilde{k}(x,v,v')(\tilde{u}(x,v')-\tilde{u}(x,v))dv'+ \widetilde{f}(x,v),
\end{equation*}
for $\widetilde{f} = \sum_{j=0}^3f_j$ with
\begin{equation}\label{func_tf}
\begin{aligned}
&f_0(x,v) = \frac{f(x,v)}{\langle v\rangle^{\frac{d}{2}}},\quad f_1(x,v)=- \lambda(x)\frac{u(x,v)}{\langle v\rangle^{\frac{d}{2}}} \\
&f_2(x,v) = \frac{u(x,v)}{\langle v\rangle^{\frac{d}{2}+1}}\int \widetilde{k}(x,v,v')\left(1 - \frac{\langle v\rangle^{\frac{d+1}{2}}}{\langle v'\rangle^{\frac{d+1}{2}}}\right)dv',\\
&f_3(x,v) = \int \widetilde{k}(x,v,v')\frac{u(x,v')}{\langle v'\rangle^{\frac{d+1}{2}}}\left(\frac{1}{\langle v\rangle^{\frac{1}{2}}} - \frac{1}{\langle v'\rangle^{\frac{1}{2}}}\right)dv',\\
\end{aligned}
\end{equation}
and for a kernel defined by
\begin{equation}\label{kernel_tk}
\widetilde{k}(x,v',v) = 2^{m-\frac{d-1}{2}-s}\epsilon^{2s}\sigma(x)\frac{\langle v\rangle^{s}\langle v'\rangle^{s}}{h(v,v')^{\frac{d-1}{2}+s}},\quad\text{with}\quad h(v,v') = \langle v\rangle\langle v'\rangle -v\cdot v' -1.
\end{equation}
Notice that here, $f(x,v)$ stands for $[f|_{{\mathbb S}^{d-1}_+}]_{\mathcal{B}}$ which is in $L^2({\mathbb R}^d\times{\mathbb R}^{d-1}; \langle v\rangle^{-d}dxdv)$.
We now estimate the $L^2$-norm of $(-\Delta_\theta)^su$ in terms of the kernel $\widetilde{k}(x,v,v')$. By comparing \eqref{fFP_diff} and \eqref{fFP_id}, we have
\begin{equation}\label{fLap_tk}
\begin{aligned}
\|(-\Delta_\theta)^su\|_{L^2_{x,\theta}} &= \|\mathcal{I}_\theta(u) - cu\|_{L^2_{x,\theta}}
\lesssim \left(\int \left([\mathcal{I}_\theta(u)]_{\mathcal{B}} + c[u]_{\mathcal{B}}\right)^2\frac{dv}{\langle v\rangle^{d}}dx\right)^{1/2}\\
&\lesssim \left\| \int \widetilde{k}(x,v,v')(\widetilde{u}(v') - \widetilde{u}(v))dv'\right\|_{L^2_{x,v}} +\left\| u\right\|_{L^2_{x,\theta}}.
\end{aligned}
\end{equation}
The last inequality follows, in particular, from the fact that $\widetilde{f}\in L^2({\mathbb R}^d\times{\mathbb R}^{d-1};dxdv)$, which we prove next. The latter spaces is denoted in this section by $L^2_{x,v}$. Indeed, we directly have that $\|f_0\|_{L^2_{x,v}}\lesssim \|f\|_{L^2_{x,\theta}}$ and $\|f_1\|_{L^2_{x,v}}\lesssim \|u\|_{L^2_{x,\theta}}$, and moreover,
\begin{equation*}
\begin{aligned}
f_2(x,v) &= \frac{u(x,v)}{2\langle v\rangle^{\frac{d}{2}+1}}\int \widetilde{k}(x,v,v+z)\left(2-\frac{\langle v\rangle^{\frac{d+1}{2}}}{\langle v+z\rangle^{\frac{d+1}{2}}}-\frac{\langle v\rangle^{\frac{d+1}{2}}}{\langle v-z\rangle^{\frac{d+1}{2}}}\right)dz\\
&\quad + \frac{u(x,v)}{2\langle v\rangle^{\frac{d}{2}+1}}\int \big(\widetilde{k}(x,v,v-z) - \widetilde{k}(x,v,v+z)\big)\left(1-\frac{\langle v\rangle^{\frac{d+1}{2}}}{\langle v-z\rangle^{\frac{d+1}{2}}}\right)dz;\\
f_3(x,v) &= \int \widetilde{k}(x,v,v+z)\left(\frac{u(v+z)}{\langle v+z\rangle^{\frac{d+1}{2}}} - \frac{u(v)}{\langle v\rangle^{\frac{d+1}{2}}}\right)\left(\frac{1}{\langle v\rangle^{\frac{1}{2}}}-\frac{1}{\langle v+z\rangle^{\frac{1}{2}}}\right)dz\\
&\quad + \frac{u(x,v)}{2\langle v\rangle^{\frac{d+1}{2}}}\int \widetilde{k}(x,v,v+z)\left(\frac{2}{\langle v\rangle^{\frac{1}{2}}}-\frac{1}{\langle v+z\rangle^{\frac{1}{2}}}-\frac{1}{\langle v-z\rangle^{\frac{1}{2}}}\right)dz\\
&\quad + \frac{u(x,v)}{2\langle v\rangle^{\frac{d+1}{2}}}\int \big(\widetilde{k}(x,v,v-z) - \widetilde{k}(x,v,v+z)\big)\left(\frac{1}{\langle v\rangle^{\frac{1}{2}}}-\frac{1}{\langle v-z\rangle^{\frac{1}{2}}}\right)dz\\
\end{aligned}
\end{equation*}
We show in Appendix \ref{appdx:func_h} that for some $\beta\in(0,1)$,
\begin{equation*}
\frac{\beta}{2}|v-v'|^{2}\leq h(v,v')\leq \frac{1}{2}|v-v'|^{2},
\end{equation*}
which then implies the inequality (see \eqref{ineq_h})
\begin{equation*}
\big|\widetilde{k}(x,v,v-z) - \widetilde{k}(x,v,v+z)\big|\lesssim \frac{\langle v\rangle^{2s}}{|z|^{d-1+2s-1}}.
\end{equation*}
The above (and the compact support of $u$) allows us to estimate the $L^2$-norm of $f_2$ and the last two terms in $f_3$, by a constant times $\|u\|_{L^2_{x,\theta}}$. For the remaining term in $f_3$, we notice that
\begin{equation*}
\begin{aligned}
&\left\| \int \widetilde{k}(x,v,v+z)\left(\frac{u(v+z)}{\langle v+z\rangle^{\frac{d+1}{2}}} - \frac{u(v)}{\langle v\rangle^{\frac{d+1}{2}}}\right)\left(\frac{1}{\langle v\rangle^{\frac{1}{2}}}-\frac{1}{\langle v+z\rangle^{\frac{1}{2}}}\right)dz\right\|^2_{L^2_{x,v}}\\
&\lesssim \int \int \widetilde{k}(x,v,v+z)\left(\frac{u(v+z)}{\langle v+z\rangle^{\frac{d+1}{2}}} - \frac{u(v)}{\langle v\rangle^{\frac{d+1}{2}}}\right)^2dzdv\\
&\lesssim \int[\mathcal{I}_\theta(u)]_{\mathcal{B}}udxd\theta + \|u\|_{L^2_{x,\theta}}^2\\
&\lesssim \|(-\Delta_\theta)^{s/2}u\|^2_{L^2_{x,\theta}} + \|u\|_{L^2_{x,\theta}}^2.
\end{aligned}
\end{equation*}
Summarizing, we have shown that
\begin{equation*}
\|\widetilde{f}\|_{L^2_{x,v}}\lesssim \|f\|_{L^2_{x,\theta}}+\|u\|_{L^2_{x,\theta}}+\|(-\Delta_\theta)^{s/2}u\|_{L^2_{x,\theta}}.
\end{equation*}
It then follows from the above and results from \cite{A} (see Appendix \ref{appdx:subellipticity}) that
\begin{equation*}
\|(-\Delta_v)^s\tilde{u}\|_{L^2}+\|(-\Delta_x)^{\frac{s}{1+2s}}\tilde{u}\|_{L^2}\lesssim \|f\|_{L^2} + \|\tilde{u}\|_{L^2} + \|(-\Delta_v)^{s/2}\tilde{u}\|_{L^2},
\end{equation*}
and subsequently (see Remark \ref{remark_D}),
\begin{equation*}
\left\| \int \widetilde{k}(x,v,v')(\tilde{u}(x,v')-\tilde{u}(x,v))dv'\right\|_{L^2_{x,v}}\leq \|f\|_{L^2} + \|\tilde{u}\|_{L^2} + \|(-\Delta_v)^{s/2}\tilde{u}\|_{L^2_{x,v}}.
\end{equation*}
Combining the above, \eqref{fLap_tk}, and the estimates $\|\tilde{u}\|_{L^2_{x,v}}\leq \|u\|_{L^2_{x,\theta}}$ and
\begin{equation*}
\|(-\Delta_v)^{s/2}\tilde{u}\|^2_{L^2_{x,v}}\lesssim \int \int \widetilde{k}(x,v,v+z)
\left(\widetilde{u}(v+z)-\widetilde{u}(v)\right)^2dzdv\lesssim \|(-\Delta_\theta)^{s/2}u\|^2_{L^2_{x,\theta}} + \|u\|_{L^2_{x,\theta}}^2,
\end{equation*}
we finally deduce that
\begin{equation*}
\|(-\Delta_\theta)^su\|^2_{L^2_{x,\theta}}\lesssim \|f\|_{L^2_{x,\theta}} + \|u\|_{L^2_{x,\theta}} + \|(-\Delta_\theta)^{s/2}u\|^2_{L^2_{x,\theta}}.
\end{equation*}
Using equation \eqref{fFP_diff} one also has that $Tu\in L^2({\mathbb R}^d\times {\mathbb S}^{d-1})$ with and analogous estimate.
Lastly, given any bounded set $\Omega\subset{\mathbb R}^d$ and a partition of unity defined on ${\mathbb R}^d\times{\mathbb S}^{d-1}$, the above yields that $(-\Delta_\theta)^s u$ and $Tu$ belong to $L^2(\Omega\times{\mathbb S}^{d-1})$ for any solution to the fractional Fokker--Planck equation \eqref{fFP_diff} with source $f\in L^2({\mathbb R}^d\times{\mathbb S}^{d-1})$.\\
2. Similarly, the continuity of solutions is a consequence of the continuity property satisfied by solutions to fractional order kinetic equations, and more generally satisfied by solutions to some linear transport equations with singular scattering kernels. We again localize the solution in order to use beam coordinates on the sphere.
We assume without loss of generality that $u(\cdot,\cdot)$ is a compactly supported solution to the fractional Fokker--Planck equation \eqref{fFP_diff}, with support in $\mathcal{V}\times{\mathbb S}^{d-1}_+$, for some open bounded set $\mathcal{V}\subset{\mathbb R}^d$, and for a source term $f\in L^2_{x,\theta}\cap L^\infty_{x,\theta}$. By the previous regularity result, it is indeed a strong solution.
By considering beam coordinates, we saw above that $\widetilde{u}(x,v) = \langle v\rangle^{-\frac{d}{2}-1}u(x,v)$ solved
\begin{equation*}
\partial_{x^d}\tilde{u}+v\cdot\nabla_{x'}\tilde{u} = \int \widetilde{k}(x,v,v')(\tilde{u}(x,v')-\tilde{u}(x,v))dv'+ \widetilde{f}(x,v)
\end{equation*}
with $\widetilde{f}$ and $\widetilde{k}$ defined respectively in \eqref{func_tf} and \eqref{kernel_tk}. It turns out that the kernel (see Appendix \ref{appdx:func_h}) satisfies the point-wise inequalities,
\begin{equation*}
\frac{1}{|v-v'|^{d-1+2s}}\lesssim\widetilde{k}(x,v,v')\lesssim \frac{\langle v\rangle^{s}\langle v'\rangle^{s}}{|v-v'|^{d-1+2s}},
\end{equation*}
which directly implies coercivity of the integral operator associated to $\widetilde{k}$. Namely, for every $R\geq1$, there is $C>0$ so that for all $\varphi:{\mathbb R}^{d-1}\to {\mathbb R}$ compactly supported inside the ball $B_{R}$,
\begin{equation*}
\begin{aligned}
&C^{-1}\int_{{\mathbb R}^{d-1}}\int_{{\mathbb R}^{d-1}} \frac{|\varphi(v) -\varphi(v')|^2}{|v-v'|^{d-1+2s}}dvdv' \\
&\leq - \int_{{\mathbb R}^{d-1}}\left(\int_{{\mathbb R}^{d-1}}\widetilde{k}(x,v,v')(\varphi(v')-\varphi(v))dv'\right)\varphi(v)dv + \|\varphi\|^2_{L^2({\mathbb R}^{d-1})}.
\end{aligned}
\end{equation*}
The same inequality helps us to deduce the non-degeneracy condition (in the case $s<1/2$):
\begin{equation*}
\inf_{|e|=1}\int_{B_r(v)}\max((v'-v)\cdot e,0)^2\widetilde{k}(x,v,v')dv'\geq c r^{2-2s},\quad\forall v\in B_R.
\end{equation*}
Finally, we also verify that
\begin{equation*}
\begin{aligned}
&\int_{{\mathbb R}^{d-1}\backslash B_r(v)}\widetilde{k}(x,v,v')dv' \lesssim r^{-2s},\quad \forall r>0 \text{ and }v\in B_R;\\
&\int_{B_R\backslash B_r(v')}\widetilde{k}(x,v,v')dv \lesssim r^{-2s},\quad \forall r>0 \text{ and }v'\in B_R.
\end{aligned}
\end{equation*}
As long as we have $\widetilde{f}\in L^\infty({\mathbb R}^d\times B_R)$, the H\"older regularity result in \cite{IS} guarantees that $\widetilde{u}$ is H\"older continuous in $\mathcal{V}\times{\mathbb R}^{d-1}$, and consequently, $u(x,\theta)$ is continuous in a neighborhood of $(x_0,\theta_0)$.
In order to show that $\widetilde{f}$ is bounded, and since $f,u \in L^\infty_{x,\theta}$ (by virtue of Theorem \ref{thm:wellposedness_FP2}), it remains to deduce the boundedness of $f_2$ and $f_3$ in \eqref{func_tf}. Indeed, denoting $\varphi(v')=\frac{\langle v\rangle^{\frac{d+1}{2}}}{\langle v'\rangle^{\frac{d+1}{2}}}$, we see that
\begin{equation*}
\begin{aligned}
f_2(x,v)&=\int \widetilde{k}(x,v,v')\left(\varphi(v) - \varphi(v')\right)dv'\\
&=\frac{1}{2}\int \widetilde{k}(x,v,v+z)(2\varphi(v) - \varphi(v+z) - \varphi(v-z))dz \\
&\quad+\frac{1}{2}\int (\widetilde{k}(x,v,v-z)-\widetilde{k}(x,v,v+z))(\varphi(v)-\varphi(v-z))dz.
\end{aligned}
\end{equation*}
Moreover, since $\varphi$ is twice-continuously differentiable and using also that $u(x,v)$ is compactly supported, we have that (again from \eqref{ineq_h})
\begin{equation*}
\begin{aligned}
\left|f_2(x,v)\right|&\lesssim |u(x,v)|\left(\int_{|z|<1} \frac{1}{|z|^{d-1-2(1-s)}} dz + \int_{|z|>1} \frac{1}{|z|^{d-1+2s}} dz \right)\lesssim \|u\|_{\infty}.
\end{aligned}
\end{equation*}
We similarly obtain $|f_3(x,v)| \lesssim \|u\|_{\infty}$, which allows us to conclude that $\|\widetilde{f}\|_\infty\lesssim \|f\|_\infty$ and the proof is complete.
\end{proof}
We saw above that solutions to the fFPE can be viewed locally as solutions to evolution integro-differential kinetic equations, and it was this precise local form that allowed us to derive the regularity properties. How the optimal H\"older exponent associated to solutions of \eqref{fFP_diff} depends on the regularity of the source $f$ term (e.g., Lipschitz continuous) remains unclear.
Several recent results related to the more general and/or closely related non-local kinetic models, such as the Boltzmann equation without cut-off, may be found in \cite{IS,IS2,IS3,S}.
It is expected that results along those lines could lead to quantitative approximation estimates in the 1-Wasserstein sense between the (stationary) radiative transfer solution, in the narrow beam regime, and the fractional Fermi pencil-beam solution. We proved above that the radiative transfer solution converged in the highly forward-peaked limit, in a weak sense (and with no known estimates), to the fractional Fokker--Planck solution (Theorem \ref{thm:approx_fFP}), and furthermore, the latter solution can be subsequently approximated by a pencil-beam in the narrow beam regime (Theorem \ref{thm:narrow_beam}, above).
The main obstacle that prevents the application of the regularity results in \cite{IS,IS2,IS3,S} resides in the a-priori regularity assumed on the solutions. The H\"older estimates proved in \cite{IS,S} hold for weak solutions and thus are appropriate for our work. In contrast, the local Schauder estimates and their global extension obtained respectively in \cite{IS2} and \cite{IS3}, assume solutions to be at least classical. A generalization of the Schauder estimates for weak solutions would be necessary to quantify the convergence of radiative transfer to the Fokker-Planck equation, something we do not pursue here.
\section{Analysis of the Fractional Fermi pencil-beam equation}\label{sec:fFpb}
\subsection{Existence of solutions and properties}
Let $F\in C({\mathbb R}_+;\mathcal{S}'({\mathbb R}^{2(n-1)}))$ and $G\in \mathcal{S}'({\mathbb R}^{2(n-1)})$. In this section we will assume $\widetilde{\sigma}(X^n),\widetilde{\lambda}(X^n)\in C(\bar{{\mathbb R}}_+)$. Analogously to the non-fractional case we have an explicit characterization of the solutions to the fractional Fermi pencil-beam equation. However, this time it is explicit only in the Fourier domain.
We denote by $\mathfrak{F}_{X'}[f](\xi,X^d,V) := \int e^{-\text{\rm i} X'\cdot\xi} f(X,V)dX'$ the Fourier transformation of $f$ with respect to $X'$, and similarly we denote by $\mathfrak{F}_V$ the Fourier transform operator with respect to the angular variable $V$. We also write $\mathfrak{F}_{X',V}=\mathfrak{F}_{X'}\mathfrak{F}_{V}$.
\begin{lemma}\label{lemma:solution_fFPB}
For the above choice of parameters there exists a unique solution to
\begin{equation}\label{fFPB}
\left\{\begin{array}{ll}
\partial_{X^n}U + V\cdot \nabla_{X'}U + \widetilde{\sigma}(-\Delta_V)^sU + \widetilde{\lambda} U =F, &X=(X',X^n)\in{\mathbb R}^n_+,\; V\in{\mathbb R}^{n-1}\\
U= G,&(X',V)\in {\mathbb R}^{2(n-1)},\; X^n=0,
\end{array}\right.
\end{equation}
whose Fourier Transform (with respect to transversal and angular variables) is given by
\begin{equation}\label{fFBP_sol}
\begin{aligned}
\mathfrak{F}_{X',V}[U](\xi,X^n,\eta) &= e^{-\int^{X^n}_0\widetilde{\lambda}(r)dr}\mathfrak{F}_{X',V}[G](\xi,\eta+X^n\xi)e^{-\int^{X^n}_0|\eta+(X^n-t)\xi|^{2s}\widetilde{\sigma}(t)dt}\\
&\quad + \int^{X^n}_0e^{-\int^{X^n}_t\widetilde{\lambda}(r)dr}\mathfrak{F}_{X',V}[F](\xi,t,\eta+(X^n-t)\xi)e^{-\int^{X^n}_t|\eta + (X^n-r)\xi|^{2s}\widetilde{\sigma}(r)dr} dt.
\end{aligned}
\end{equation}
\end{lemma}
\begin{proof}
It follows in a similar fashion as in the non-fractional case. We refer the reader to the proof of \cite[Proposition 4.1]{BP}.
\end{proof}
\begin{lemma}\label{lemma:radial_sym} Let $U$ as in the previous lemma with null interior source ($F=0$) and $G=\delta(X)\delta(V)$. Then,
$$\int_{{\mathbb R}^{2(d-1)}} \big(X'\cdot \xi_0\big)U(X',X^d,V)dX'dV=0,\quad\text{for all }\xi_0\in{\mathbb R}^{d-1}\text{ and }X^d>0.$$
$$\int_{{\mathbb R}^{d-1}} (-\Delta_V)^sU(X',X^d,V)dV=0,\quad\text{for all }X'\in{\mathbb R}^{d-1}\text{ and }X^d>0.$$
\end{lemma}
\begin{proof}
Using that integration is equivalent to the zero Fourier coefficient we get
$$
\begin{aligned}
&\int_{{\mathbb R}^{2(d-1)}} \big(X'\cdot \xi_0\big)U(X',X^d,V)dX'dV\\
&\hspace{2em}=\int_{{\mathbb R}^{d-1}} \big(X'\cdot \xi_0\big)\mathfrak{F}_{V}[U](X',X^d,0)dX'\\
&\hspace{2em}=\text{\rm i}\xi_0\cdot \big[\nabla_{\xi}\mathfrak{F}[U]\big](0,X^d,0)\\
&\hspace{2em}=\text{\rm i} e^{-\int^{X^n}_0\widetilde{\lambda}(r)dr}\left(2s\textstyle \int^{X^d}_0(X^d-t)^{2s}\widetilde{\sigma}(t)dt\right)\big(\xi_0\cdot \hat{\xi} \big)|\xi|^{2s-1}e^{-|\xi|^{2s}\int^{X^n}_0(X^n-t)^{2s}\widetilde{\sigma}(t)dt}\Big|_{\xi =0}=0.
\end{aligned}
$$
Similarly,
$$
\begin{aligned}
\int_{{\mathbb R}^{d-1}} (-\Delta_V)^sU(X',X^d,V)dV&=\mathfrak{F}_{V}[(-\Delta_V)^sU](X',X^d,0)\\
&=|\eta|^{2s}\mathfrak{F}[U](X',X^d,\eta)\big|_{\eta=0}\quad =0.
\end{aligned}
$$
\end{proof}
\begin{lemma}\label{lemma:IbyP}
Let $U$ be as in Lemma \ref{lemma:solution_fFPB} with null interior source ($F=0$) and $G=\delta(\cdot-Y',\cdot-W)$. Then, for any $\Phi\in C({\mathbb R}^n_+\times{\mathbb R}^{n-1})\cap H^1({\mathbb R}^n_+;L^2({\mathbb R}^{2(n-1)}))$ we have the following integration by parts formula
$$
\int_{{\mathbb R}^n_+\times{\mathbb R}^{n-1}}U\partial_{X^n}\Phi dXdV = -\int_{{\mathbb R}^n_+\times{\mathbb R}^{n-1}}\big(\partial_{X^n}U\big)\Phi dXdV - \Phi(Y',0,W).
$$
\end{lemma}
\begin{proof}
We use Plancherel's formula to write
$$
\begin{aligned}
&\int_{{\mathbb R}^n_+\times{\mathbb R}^{n-1}}U\partial_{X^n}\Phi dXdV = \int_{{\mathbb R}^{2(n-1)}\times{\mathbb R}_+}\mathfrak{F}_{X',V}[U]\big(\partial_{X^n}\mathfrak{F}_{X',V}[\Phi]\big)dXdV d\xi'd\eta dX^n\\
=&- \int_{{\mathbb R}^{2(n-1)}\times{\mathbb R}_+} \!\!\!\!\!\! \big(\partial_{X^n}\mathfrak{F}_{X',V}[U]\big)\mathfrak{F}_{X',V}[\Phi]dXdV d\xi'd\eta dX^
-\int_{{\mathbb R}^{2(n-1)}}\!\!\!\!\!\mathfrak{F}_{X',V}[U]|_{X^n=0}\mathfrak{F}_{X',V}[\Phi]|_{X^n=0}d\xi d\eta
\\
=&-\int_{{\mathbb R}^n_+\times{\mathbb R}^{n-1}}\big(\partial_{X^n}U\big)\Phi dXd
- \int e^{\text{\rm i} Y'\cdot \xi + \text{\rm i} W\cdot\eta}\mathfrak{F}_{X',V}[\Phi]|_{X^n=0}d\xi d\eta.
\end{aligned}
$$
\end{proof}
\begin{lemma}\label{lemma:FPb_Lip}
Let $F\in C_c(\bar{Q})$ and Lipschitz continuous with respect to $Z=(X',V)$, and $G=0$. Let $U$ be the solution to \eqref{fFPB}. There exists $C>0$ so that $U_{X^n}(Z) = U(X',X^n,V)$ satisfies
$$|U_{X^n}(Z_1) -U_{X^n}(Z_2) |\leq C\text{\rm Lip\hspace{.1em}}(F)|Z_1-Z_2|,$$
for all $Z_1,Z_2\in{\mathbb R}^{2(n-1)}$ and $X^n>0$.
\end{lemma}
\begin{proof}
For arbitrary $(X'_1,X^n,V_1),(X'_2,X^n,V_2)\in {\mathbb R}^{2(n-1)}$ we have
$$
\begin{aligned}
&U(X'_1,X^n,V_1)-U(X'_2,X^n,V_2) \\
&=\int e^{\text{\rm i}(X'_1\cdot \xi)+\text{\rm i}(V_1\cdot \eta)}\mathfrak{F}_{X',V}[U](\xi,X^n,\eta) d\xi d\eta-\int e^{\text{\rm i}(X'_2\cdot \xi)+\text{\rm i}(V_2\cdot \eta)}\mathfrak{F}_{X',V}[U](\xi,X^n,\eta) d\xi d\eta\\
&=\int \int^{X^n}_0e^{-\int^{X^n}_t\widetilde{\lambda}(r)dr}\mathfrak{F}_{V}\mathfrak{F}_{X'}\big[\tau_{X_1' - (X^n-t)(V-V_1)}\tau_{V_1}F-\tau_{X_2' - (X^n-t)(V-V_2)}\tau_{V_2}F\big]\\
&\quad \times \left[e^{-\int^{X^n}_t|\eta + (X^n-r)\xi|^{2s}\widetilde{\sigma}(r)dr} \right]dtd\xi dV\\
&=\int \int^{X^n}_0e^{-\int^{X^n}_t\widetilde{\lambda}(r)dr}\big(\tau_{X_1' - (X^n-t)(V-V_1)}\tau_{V_1}F-\tau_{X_2' - (X^n-t)(V-V_2)}\tau_{V_2}F\big)\\
&\quad \times \mathfrak{F}_{X'}^{-1}\mathfrak{F}_{V}^{-1}\left[e^{-\int^{X^n}_t|\eta + (X^n-r)\xi|^{2s}\widetilde{\sigma}(r)dr} \right]dtdX' dV.
\end{aligned}
$$
Therefore,
$$
\begin{aligned}
\big|U(X'_1,X^n,V_1)-U(X'_2,X^n,V_2) \big| &\leq \text{Lip}(F)\int \int^{X^n}_0|(X'_1-X'_2 - (X^n-t)(V_1-V_2),V_1-V_2)| \\
&\quad \times e^{-\int^{X^n}_t\widetilde{\lambda}(r)dr}\Big|\mathfrak{F}_{X',V}^{-1}\left[e^{-\int^{X^n}_t|\eta + (X^n-r)\xi|^{2s}\widetilde{\sigma}(r)dr} \right]\Big|dtdX' dV\\
&\leq C\text{\rm Lip\hspace{.1em}}(F)\big(|X_1'-X_2'|+|V_1-V_2|\big),
\end{aligned}
$$
for $C>0$, a uniform upper bound of
$$
\int \int^{X^n}_0\big(1+(X^n-t)\big)e^{-\int^{X^n}_t\widetilde{\lambda}(r)dr}\Big|\mathfrak{F}_{X',V}^{-1}\left[e^{-\int^{X^n}_t|\eta + (X^n-r)\xi|^{2s}\widetilde{\sigma}(r)dr} \right]\Big|dtdX' dV<+\infty.
$$
\end{proof}
\subsection{Integrability properties of solutions}\label{subsec:integ_fFpb}
We state some integrability properties for the fundamental solution associated to the fractional Fermi equation that will be used in the following sections. The fractional Fermi equation \eqref{fFPB} is a slight generalization of the fractional Kolmogorov equation (see \S2.4 in \cite{IS})
\begin{equation}\label{Kol_eq}
f_t + v\cdot\nabla_xf + (-\Delta_v)^sf = h,\quad x,v\in{\mathbb R}^{d-1}.
\end{equation}
Lemma \ref{lemma:solution_fFPB} provides us with a fundamental solution (solving \eqref{fFPB} with sources $F=0$ and $G=\delta(X)\delta(V)$) which takes the self-similar form
\begin{equation}\label{self_similar_J}
J(X,V) = c_{d-1}\frac{1}{(X^d)^{d-1+\frac{d-1}{s}}}\mathfrak{J}\left(\frac{X'}{(X^d)^{1+\frac{1}{2s}}},\frac{V}{(X^d)^{\frac{1}{2s}}}\right)\exp\left(-\int_0^{X^d}\widetilde{\lambda}(\tau)d\tau\right),
\end{equation}
for some appropriate constant $c_{d-1}>0$, and with $\mathfrak{J}$ defined via its Fourier transform:
$$
\hat{\mathfrak{J}}(\xi,\eta;X^d) := \exp\left(-\int^1_0|\eta - \tau\xi|^{2s}\widetilde{\sigma}\Big(X^d(1-\tau)\Big)d\tau\right).
$$
In what remains of this section, we abbreviate the exponential factor in the definition of $J$ by writing $\Lambda(X^d):= \exp\left(-\int_0^{X^d}\widetilde{\lambda}(\tau)d\tau\right)$.
The solution to the initial value problem \eqref{fFPB} takes the form
\begin{equation}\label{sol_fFPB}
\begin{aligned}
U(X,V) &= \int_{{\mathbb R}^{d-1}}\int_{{\mathbb R}^{d-1}} G(Y',W)J(X'-Y'-X^dW,X^d,V-W)dWdY'\\
&\quad + \int_0^{X^d}\int_{{\mathbb R}^{d-1}}\int_{{\mathbb R}^{d-1}} F(Y,W)J(X'-Y'-(X^d-Y^d)W,X^d-Y^d,V-W)dWdY.
\end{aligned}
\end{equation}
The next proposition is the analogous to \cite[Proposition 2.1]{IS} for the fractional Fermi equation and we state it without proof.
\begin{proposition}
For $J$ and $\mathfrak{J}$ as above we have:
\begin{itemize}
\item[(1)] The function $\mathfrak{J}$ is $C^\infty$ and decays polynomially at infinity. Moreover, $\mathfrak{J}$ and all its derivatives are integrable in ${\mathbb R}^{2(d-1)}$.
\item[(2)] For every $X^d>0$, $\int_{{\mathbb R}^{2(d-1)}}J(X,V)dX'dV=\Lambda(X^d)$.
\item[(3)] $J\geq 0$ and $\mathfrak{J}\geq 0$.
\item[(4)] For any $X^d>0,$
\begin{equation*}
\begin{aligned}
\|J(\cdot,X^d,\cdot)\|_{L^1({\mathbb R}^{2(d-1)})} &= \Lambda(X^d)\|\mathfrak{J}\|_{L^1({\mathbb R}^{2(d-1)})}\\
\|(-\Delta_V)^{s/2}J(\cdot,X^d,\cdot)\|_{L^1({\mathbb R}^{2(d-1)})} &= (X^d)^{-1/2}\Lambda(X^d)\|(-\Delta_V)^{s/2}\mathfrak{J}\|_{L^1({\mathbb R}^{2(d-1)})}.
\end{aligned}
\end{equation*}
\end{itemize}
\end{proposition}
We use the Riemann-Lebesgue theorem to deduce explicit estimates for the decay at infinity of the kernel $\mathfrak{J}$. Indeed, for any pair of multi-indices $\alpha,\beta$ such that $|\alpha|+|\beta|=d-1$, one verifies that
\begin{equation*}
\Lambda(X^d)^{-1}|\partial^\alpha_{\xi}\partial^\beta_\eta\hat{\mathfrak{J}}(\xi,\eta;X^d)|\lesssim\int_0^1\frac{d\rho}{|\eta+\rho\xi|^{d-1-2s}}e^{-\sigma_0\int^1_0|\eta+\tau\xi|^{2s}d\tau},
\end{equation*}
which is integrable since Fubini's theorem implies the integrability of $(\rho,\eta,\xi)\mapsto \frac{e^{-\sigma_0\int^1_0|\eta+\tau\xi|^{2s}d\tau}}{|\eta+\rho\xi|^{d-1-2s}}$.
This yields (thanks to the Riemann-Lebesgue theorem) that $X'^\alpha V^\beta\mathfrak{J}(X',\cdot,V)\in C_0({\mathbb R}^{2(d-1)})$ and consequently, for some $\gamma>0$,
\begin{equation*}
\begin{aligned}
\Lambda(X^d)^{-1}\mathfrak{J}(X',X^d,V)&\lesssim \frac{1}{(1+|X'|^2+|V'|^2)^{\frac{d-1}{2}+\gamma}}\\
\end{aligned}
\end{equation*}
uniformly in $X^d>0$.
By averaging $\mathfrak{J}$ with respect to either $X'$ or $V$ one deduces that $\gamma\leq s$. This follows from the fact
\begin{equation*}
\begin{aligned}
&\mathfrak{F}_{V\to\eta}\left(\int\mathfrak{J}(X,V)dX'\right) = \exp\left(-\sigma_1(X^d)|\eta|^{2s}\right)\Lambda(X^d)\quad\text{and}\\
&\mathfrak{F}_{X'\to\xi}\left(\int\mathfrak{J}(X,V)dV\right) = \exp\left(-\sigma_2(X^d)|\xi|^{2s}\right)\Lambda(X^d),
\end{aligned}
\end{equation*}
where $\sigma_1(X^d):=\int^1_0\widetilde{\sigma}(\tau X^d)d\tau$ and $\sigma_2(X^d) := \int^1_0\tau^{2s}\widetilde{\sigma}(\tau X^d)d\tau$. Therefore, for some $C,C'>0$,
\begin{equation}\label{decay_avg_J}
\begin{aligned}
&\frac{C^{-1}\Lambda(X^d)}{\left(1+|V|^2\right)^{\frac{1}{2}(d-1 +2s)}}\leq \int\mathfrak{J}(X,V)dX' \leq\frac{C\Lambda(X^d)}{\left(1+|V|^2\right)^{\frac{1}{2}(d-1 +2s)}};\\
&\frac{C'^{-1}\Lambda(X^d)}{\left(1+|X'|^2\right)^{\frac{1}{2}(d-1 +2s)}}\leq \int\mathfrak{J}(X,V)dV' \leq\frac{C'\Lambda(X^d)}{\left(1+|X|^2\right)^{\frac{1}{2}(d-1 +2s)}}.
\end{aligned}
\end{equation}
In a similar fashion we can obtain upper bounds for the decay of the derivatives of $\mathfrak{J}$. For any $\alpha',\beta'$ multi-indices we see that
\begin{equation*}
\Lambda(X^d)^{-1}|\partial^\alpha_{\xi}\partial^\beta_\eta\big(\xi^{\alpha'}\eta^{\beta'}\hat{\mathfrak{J}}(\xi,\eta;X^d)\big)|\lesssim\int_0^1\frac{d\rho}{|\eta+\rho\xi|^{|\alpha|+|\beta|-|\alpha'|-|\beta'|-2s}}e^{-\sigma_0\int^t_0|\eta+\tau\xi|^{2s}d\tau},
\end{equation*}
uniformly in $X^d>0$, and the right hand side is still integrable for $|\alpha|+|\beta|=d-1+|\alpha'|+|\beta'|$ which yields the decay
\begin{equation}\label{decay_der_J}
\big|\partial_{X'}^{\alpha'}\partial_V^{\beta'}\mathfrak{J}(X,V)\big|\lesssim \frac{\Lambda(X^d)}{(1+|X'|^2+|V|^2)^{\frac{d-1 +|\alpha'|+|\beta'|}{2}+\gamma'}},\quad\forall X^d>0,
\end{equation}
for some $0<\gamma'\leq s$.
We then obtain the next.
\begin{lemma}\label{lemma:int_der_J} For any nonnegative real numbers $m,n$, such that $m+n\leq |\alpha|+|\beta|$, with $\alpha,\beta$ multi-indices,
\begin{equation*}
\begin{aligned}
\||X'|^m|V|^n\partial^\alpha_{X'}\partial^\beta_{V}J(X,V)\|_{L^1({\mathbb R}^{2(d-1)})} &= O((X^d)^{\left(1+\frac{1}{2s}\right)(m-|\alpha|) + \frac{1}{2s}(n-|\beta|)}\Lambda(X^d)).
\end{aligned}
\end{equation*}
\end{lemma}
\begin{proof}
We use the self-similar form of $J$ to obtain
\begin{equation*}
\begin{aligned}
&\int_{{\mathbb R}^{2(d-1)}}|X'|^m|V|^n |\partial^\alpha_{X'}\partial^\beta_{V}J(X,V) |dX'dV\\
&=\frac{c_{d-1}\Lambda(X^d)}{(X^d)^{d-1+\frac{d-1}{s}}}\int_{{\mathbb R}^{2(d-1)}}|X'|^m|V|^n \Big|\partial^\alpha_{X'}\partial^\beta_{V}\left(\mathfrak{J}\left(\frac{X'}{(X^d)^{1+\frac{1}{2s}}},\frac{V}{(X^d)^{\frac{1}{2s}}}\right)\right)\Big|dX'dV\\
&=\frac{c_{d-1}\Lambda(X^d)}{(X^d)^{d-1+\frac{d-1}{s}+|\alpha|\left(1+\frac{1}{2s}\right)+|\beta|\frac{1}{2s}}}\int_{{\mathbb R}^{2(d-1)}}|X'|^m|V|^n \Big|(\partial^\alpha_{X'}\partial^\beta_{V}\mathfrak{J})\left(\frac{X'}{(X^d)^{1+\frac{1}{2s}}},\frac{V}{(X^d)^{\frac{1}{2s}}}\right)\Big|dX'dV\\
\end{aligned}
\end{equation*}
By means of a change of variable and the decay estimate \eqref{decay_der_J} we conclude the proof by noticing that
\begin{equation*}
\begin{aligned}
&\int_{{\mathbb R}^{2(d-1)}}|X'|^m|V|^n |\partial^\alpha_{X'}\partial^\beta_{V}J(X,V) |dX'dV\\
&=\frac{c_{d-1}(X^d)^{(d-1)\left(1+\frac{1}{s}\right) + m\left(1+\frac{1}{2s}\right) + n\frac{1}{2s}}}{(X^d)^{d-1+\frac{d-1}{s}+|\alpha|\left(1+\frac{1}{2s}\right)+|\beta|\frac{1}{2s}}}\Lambda(X^d)\int_{{\mathbb R}^{2(d-1)}}|X'|^m|V|^n \Big|\left(\partial^\alpha_{X'}\partial^\beta_{V}\mathfrak{J}\right)\left(X,V\right)\Big|dX'dV\\
&\lesssim (X^d)^{(m-|\alpha|)\left(1+\frac{1}{2s}\right) + (n-|\beta|)\frac{1}{2s}}\Lambda(X^d).
\end{aligned}
\end{equation*}
\end{proof}
\begin{remark}\label{rmk:sharp_decay_J}
We can improve the above estimates for the cases with no differentiation involved. Using the sharp decay estimates for the averages of $\mathfrak{J}$ in \eqref{decay_avg_J}, we obtain that for any $m<2s$,
\begin{equation*}
\begin{aligned}
&\||X'|^m\mathfrak{J}(X,V)\|_{L^1({\mathbb R}^{2(d-1)})}+\||V|^m\mathfrak{J}(X,V)\|_{L^1({\mathbb R}^{2(d-1)})}<\infty,\quad\forall X^d>0.
\end{aligned}
\end{equation*}
This implies
\begin{equation*}
\begin{aligned}
&\||X'|^mJ(X,V)\|_{L^1({\mathbb R}^{2(d-1)})} = O((X^d)^{\left(1+\frac{1}{2s}\right)m}\Lambda(X^d)), &&\text{for }m<2s;\\
&\||V|^nJ(X,V)\|_{L^1({\mathbb R}^{2(d-1)})} = O((X^d)^{\frac{1}{2s}n}\Lambda(X^d)),&& \text{for }n<2s.
\end{aligned}
\end{equation*}
\end{remark}
\begin{remark}
If instead of considering the whole half-space ${\mathbb R}^d_+\times{\mathbb R}^{d-1}$ as domain of integration one restricts the estimates to a compact region, it is possible to obtain estimates for exponents $m\geq 2s$ and $n\geq 2s$. However the appearing constants will grow with the (transversal, i.e. in $X',V$) size of the domain, logarithmically for $m\text{ (or $n$) }=2s$ and polynomially for $m\text{ (or $n$) }>2s$.
\end{remark}
We will need the following decay estimates for the fractional Laplacian of $J$.
\begin{lemma}\label{lemma:int_fLap_J} For any nonnegative real numbers $m,n$, such that $m,n<2s$,
\begin{equation*}
\begin{aligned}
&\||X'|^m(-\Delta_V)^{s}J(X,V)\|_{L^1({\mathbb R}^{2(d-1)})} = O((X^d)^{\left(1+\frac{1}{2s}\right)m-1}\Lambda(X^d));\\
&\||V|^n(-\Delta_V)^{s}J(X,V)\|_{L^1({\mathbb R}^{2(d-1)})} = O((X^d)^{\frac{1}{2s}n -1}\Lambda(X^d)).
\end{aligned}
\end{equation*}
\end{lemma}
\begin{remark}
The constants in the estimates blow up as $m$ or $n$ approaches $2s$.
\end{remark}
\begin{proof}
We do the estimation involving powers of $|V|$, the other case is simpler and follows similarly.
Recall the following singular integral definition of the fractional Laplacian:
$$
(-\Delta_V)^{s}J = \frac{c_{d,s}}{2}\int_{{\mathbb R}^{d-1}}\frac{2J(V)-J(V+z)-J(V-z)}{|z|^{d-1+2s}}dz.
$$
Then,
\begin{equation*}
\begin{aligned}
&\int_{{\mathbb R}^{2(d-1)}} |V|^n |(-\Delta_V)^{s}J(X,V) |dX'dV\\
&\lesssim \int_{{\mathbb R}^{2(d-1)}}|V|^n \int_{{\mathbb R}^{d-1}}\frac{\left| 2J(V)-J(V+z)-J(V-z)\right|}{|z|^{d-1+2s}}dzdX'dV\\
&\lesssim (X^d)^{n/2s -1}\Lambda(X^d)\int_{{\mathbb R}^{2(d-1)}}|V|^n \int_{{\mathbb R}^{d-1}}\frac{\left|2\mathfrak{J}(V)-\mathfrak{J}(V+z)-\mathfrak{J}(V-z)\right|}{|z|^{d-1+2s}}dzdX'dV\\
\end{aligned}
\end{equation*}
We split the integral with respect to $z$ into two integrals for the respective regions $|z|<1$ and $|z|>1$. For the former, we see that
\begin{equation*}
\begin{aligned}
&\int_{{\mathbb R}^{2(d-1)}}|V|^n \int_{|z|<1}\frac{\left|2\mathfrak{J}(V)-\mathfrak{J}(V+z)-\mathfrak{J}(V-z)\right|}{|z|^{d-1+2s}}dzdX'dV\\
&\lesssim \int_{{\mathbb R}^{d-1}} \frac{1}{|z|^{d-1-2(1-s)}}\int^1_0\int_{{\mathbb R}^{2(d-1)}}|V|^n|\nabla_V^2\mathfrak{J}(X,V+rz)|dX'dVdrdz\\
&\lesssim \int_{|z|<1} \frac{1}{|z|^{d-1-2(1-s)}}\int^1_0\int_{{\mathbb R}^{2(d-1)}}|V-rz|^n|\nabla_V^2\mathfrak{J}(X,V)|dX'dVdrdz\\
&\lesssim \left(\int_{|z|<1} \frac{dz}{|z|^{d-1-2(1-s)}}\right)\int_{{\mathbb R}^{2(d-1)}}(|V|^n+1)|\nabla_V^2\mathfrak{J}(X,V)|dX'dV <\infty,
\end{aligned}
\end{equation*}
where the finiteness of the last integral with respect to $(X',V)$ follows from the previous lemma.
On the other hand, for $|z|>1$ and using Remark \ref{rmk:sharp_decay_J} we easily obtain that
\begin{equation*}
\begin{aligned}
&\int_{{\mathbb R}^{2(d-1)}}|V|^n \int_{|z|>1}\frac{\left|2\mathfrak{J}(V)-\mathfrak{J}(V+z)-\mathfrak{J}(V-z)\right|}{|z|^{d-1+2s}}dzdX'dV\\
&\lesssim \left(\int_{|z|>1} \frac{dz}{|z|^{d-1+2s-n}}\right)\int_{{\mathbb R}^{2(d-1)}}|V|^n\mathfrak{J}(X,V)dX'dV <\infty.
\end{aligned}
\end{equation*}
\end{proof}
\begin{remark}\label{rmk:gral_integ}
Similar estimates hold for more general solution to \eqref{fFPB} by means of the representation formula \eqref{sol_fFPB}. We indeed use this in the proof of Lemma \ref{lemma:ord1_approx}.
\end{remark}
\section{Approximation analysis for narrow beams}\label{sec:Fpb_approx}
\subsection{Fractional Fermi pencil-beam approximation}\label{subsec:fFpb_approx}
The next lemma is a crucial step in the proof of the pencil-beam approximation result.
\begin{lemma}\label{lemma:approx_fFpb}
Let $\psi\in C_c(Q)\cap \text{\rm Lip\hspace{.1em}}_{\kappa}(Q)$ so that the backward fractional Fokker-Planck system,
\begin{equation}\label{backward_fFP}
- \theta\cdot\nabla_x\varphi + \lambda\varphi = \mathcal{I}_\theta(\varphi) + \psi.
\end{equation}
has a unique continuous strong solution $\varphi$. Let $U$ be the solution to the fractional Fermi pencil-beam system \eqref{fFPB} with
$$F=0,\quad G=\delta(X')\delta(V),\quad \widetilde{\sigma}=\sigma(0 ,X^d)\quad\text{and}\quad \widetilde{\lambda}=\lambda(0 ,X^d),$$
and extended by zero to $X^d<0$. (i.e. for $X^n>0$, $U$ coincides with the fundamental solution $J$ defined in the previous section). In dimension $d=2$ we take $s'\in(2s-1,s)$ and for $d\geq 3$ we choose $s'\in(0,s)$. Then, there exists $C_{s'}>0$ so that the rescaling
$$\mathfrak{u}(x,\theta) := (2\epsilon)^{-2(d-1)}U((2\epsilon)^{-1}x',x^d,\epsilon^{-1}\mathcal{S}(\theta))$$
satisfies
$$
\left|\int_Q \mathfrak{u}(x,\theta)\psi(x,\theta)dxd\theta -\varphi(0,N)\right|\leq C_{s'}\epsilon^{2s'}\kappa^{s'},
$$
where $N = (0,\dots,0,1)\in{\mathbb S}^{d-1}$ and $C_{s'}\to\infty$ as $s'\to s$.
\end{lemma}
\begin{proof}
We define $\Psi$ and $\Phi$ as the following rescaling of $\psi$ and $\varphi$ respectively:
\begin{equation}\label{rescaling_test_functions}
\varphi(x,\theta) = \Phi((2\epsilon)^{-1}x',x^d,\epsilon^{-1}\mathcal{S}(\theta))\quad\text{and}\quad \psi(x,\theta) = \Psi((2\epsilon)^{-1}x',x^d,\epsilon^{-1}\mathcal{S}(\theta)),
\end{equation}
thus $\|\Phi\|_\infty=\|\varphi\|_\infty\lesssim \|\psi\|_\infty=\|\Psi\|_\infty$.
Then,
$$
\begin{aligned}
\int_{Q}\mathfrak{u}\psi dxd\theta &= \int_{Q}\mathfrak{u}\left(-\theta \cdot\nabla_x\varphi+ \lambda \varphi -\mathcal{I}_\theta(\varphi)\right)dxd\theta \\
&= \int_{Q}\mathfrak{u}\left(-\theta\cdot\nabla_x \varphi +\lambda \varphi +\textstyle\epsilon^{2s}\sigma(x)\Big((-\Delta_\theta)^s\varphi - c_{d,s}\varphi\Big)\right)dxd\theta\\
&=:I_1+I_2+I_3 + I_4.
\end{aligned}
$$
The proof consists in showing that
$$
I_1+I_2+I_3 + I_4 = \Phi(0,0,0) + \int \underbrace{\mathcal{P}(U)}_{=0}\frac{\Phi}{\langle \epsilon V\rangle^{2(d-1)}}dXdV + O(\epsilon^{2s'}\kappa^{s'}),
$$
where we recall $\mathcal{P}$ is the fractional Fermi pencil-beam operator defined in \eqref{fFpb}. \\
\noindent{\it Estimation of $I_1$.} The advection component $I_1=\textstyle -\int_Q\mathfrak{u}\big(\theta\cdot\nabla_x\varphi\big)dxd\theta$ is computed as follows. A change of variables gives us
$$
\begin{aligned}
I_1&= - \int_{{\mathbb R}^{d}_+\times{\mathbb R}^{d-1}} \Big[\Big(\frac{V\cdot \nabla_{X'} \Phi}{1+\epsilon^2|V|^2}\Big)U + \Big(\frac{1-\epsilon^2|V|^2}{1+\epsilon^2|V|^2}\Big)(\partial_{X^d}\Phi)U \Big]\frac{dXdV}{\langle \epsilon V\rangle^{2(d-1)}},
\end{aligned}
$$
where the integration by parts formula from Lemma \ref{lemma:IbyP} and the decay of $U$ at infinity lead to
$$
\begin{aligned}
I_1= &
\int_{{\mathbb R}^{d}_+\times{\mathbb R}^{d-1}} \Big[\Big(\frac{V\cdot \nabla_{X'} U}{1+\epsilon^2|V|^2}\Big)\Phi + \Big(\frac{1-\epsilon^2|V|^2}{1+\epsilon^2|V|^2}\Big)(\partial_{X^d}U)\Phi \Big]\frac{dXdV}{\langle \epsilon V\rangle^{2(d-1)}}\\
&+\Phi(0,0,0)\\
=& \int_{{\mathbb R}^{d}_+\times{\mathbb R}^{d-1}}(V\cdot\nabla_{X'}U+\partial_{X^d}U)\frac{\Phi}{\langle \epsilon V\rangle^{2(d-1)}} dXdV +\Phi(0,0,0)+E_1(\Phi).
\end{aligned}
$$
The error term is given by
$$
\begin{aligned}
E_1 &= -2\epsilon^2\int_{{\mathbb R}^{d}_+\times{\mathbb R}^{d-1}} {\textstyle\frac{|V|^2}{\langle \epsilon V\rangle^{2d}}}\big(V\cdot \nabla_{X'}U +\partial_{X^d}U\big)\Phi dXdV + \epsilon^2\int_{{\mathbb R}^{d}_+\times{\mathbb R}^{d-1}} \textstyle\frac{|V|^2}{\langle \epsilon V\rangle^{2d}}\big(V\cdot \nabla_{X'}U\big)\Phi dXdV.
\end{aligned}
$$
The next simple inequality is used extensively in subsequent estimations in order to reduce the powers of $|V|$ and obtain integrability:
\begin{equation}\label{s'_ineq}
\frac{\epsilon^2|V|^2}{\langle \epsilon V\rangle^{2m}}\leq \epsilon^{2s'}|V|^{2s'}\frac{(\epsilon^2|V|^2)^{1-s'}}{(1+\epsilon^2|V|^2)^{1-s'}}\leq \epsilon^{2s'}|V|^{2s'},\quad\text{for any }s'\in(0,s)\text{ and }m\geq1-s'.
\end{equation}
Then,
$$
\begin{aligned}
|E_1| &\lesssim \epsilon^{2s'}\int_0^\infty\big(\||V|^{2s'}(-\Delta_V)^{s}U\|_{L^1({\mathbb R}^{2(d-1)}}\\
&\hspace{5em}+\||V|^{2s'} V\cdot \nabla_{X'}U\|_{L^1({\mathbb R}^{2(d-1)}}+\||V|^{2s'}U\|_{L^1({\mathbb R}^{2(d-1)}}\big)dX^d,
\end{aligned}
$$
where the integrals on the right hand side are finite according to Lemmas \ref{lemma:int_der_J} and \ref{lemma:int_fLap_J}.\\
\noindent{\it Estimation of $I_2$.} Similarly, for $I_2 = \textstyle \int_{Q}\lambda\mathfrak{u}\varphi dxd\theta$ we obtain
\begin{equation}\label{defJ1}
\begin{aligned}
I_2=& \int_{{\mathbb R}^d_+\times{\mathbb R}^{d-1}}\widetilde{\lambda} U\frac{\Phi}{\langle \epsilon V\rangle^{2(d-1)}} dXdV
+\underbrace{\int_{{\mathbb R}^d_+\times{\mathbb R}^{d-1}} \big(\lambda(\epsilon X',X^d) - \lambda(0 ,X^d)\big)U\frac{\Phi}{\langle \epsilon V\rangle^{2(d-1)}} dXdV}_{=:J_1(\Phi)}\\
=& \int_{{\mathbb R}^d_+\times{\mathbb R}^{d-1}}\widetilde{\lambda} U\frac{\Phi}{\langle \epsilon V\rangle^{2(d-1)}} dXdV
+ J_1(\Phi),
\end{aligned}
\end{equation}
Let us skip for a moment the estimation of $J_1(\Phi)$ and move on to $I_3$ and $I_4$.\\
\noindent{\it Estimation of $I_4$.} We see that
$$
I_4 = - {\textstyle\epsilon^{2s}\sigma(x)c_{d,s}}\int_{Q}\mathfrak{u}\varphi dxd\theta=- {\textstyle\epsilon^{2s}\sigma(x)c_{d,s}}\int_{Q}U\frac{\Phi}{\langle \epsilon V\rangle^{2(d-1)}}dXdV,
$$
thus, from the the explicit dependence on $\epsilon$ we easily verify that $|I_4|\lesssim \epsilon^{2s}\|\psi\|_\infty$.
\noindent{\it Estimation of $I_3$.} Recall $I_3= \textstyle\epsilon^{2s}\sigma(x)\int_{Q}\mathfrak{u}(-\Delta_\theta)^s\varphi dxd\theta$. We drop for a moment the dependence in $X'$ and by abusing notation write $\sigma$ instead of $\sigma(2\epsilon X',X^d)$. We have
$$
\begin{aligned}
I_3 = & \int_{{\mathbb R}^d_+\times{\mathbb R}^{d-1}}\int_{{\mathbb R}^{d-1}}\frac{U(V)}{\langle \epsilon V\rangle^{d-1-2s}}\Big(\frac{\Phi(V)}{\langle \epsilon V\rangle^{d-1-2s}}-\frac{\Phi(V')}{\langle \epsilon V'\rangle^{d-1-2s}}\Big)\frac{\sigma\;dV'dXdV}{2^{2s}|V-V'|^{d-1+2s}}\\
= & \int_{{\mathbb R}^d_+\times{\mathbb R}^{d-1}}\int_{{\mathbb R}^{d-1}}\frac{\Phi(V)}{\langle \epsilon V\rangle^{d-1-2s}}\Big(\frac{U(V)}{\langle \epsilon V\rangle^{d-1-2s}}-\frac{U(V')}{\langle \epsilon V'\rangle^{d-1-2s}}\Big)\frac{\sigma\;dV'dXdV}{2^{2s}|V-V'|^{d-1+2s}}\\
=& \frac{1}{2^{1+2s}} \int_{{\mathbb R}^d_+\times{\mathbb R}^{d-1}}\int_{{\mathbb R}^{d-1}}\frac{\Phi(V)}{\langle \epsilon V\rangle^{d-1-2s}}\Big(\frac{2U(V)}{\langle \epsilon V\rangle^{d-1-2s}}-\frac{U(V+z)}{\langle \epsilon(V+z)\rangle^{d-1-2s}}-\frac{U(V-z)}{\langle \epsilon(V-z)\rangle^{d-1-2s}}\Big)\frac{\sigma\;dzdXdV}{|z|^{d-1+2s}}\\
=& \frac{1}{2^{1+2s}} \int_{{\mathbb R}^d_+\times{\mathbb R}^{d-1}}\int_{{\mathbb R}^{d-1}}\frac{\Phi(V)}{\langle \epsilon V\rangle^{2(d-1-2s)}}\frac{\big(2U(V)-U(V+z)-U(X,V-z)\big)}{|z|^{d-1+2s}}\sigma dzdXdV\\
&+\frac{1}{2^{1+2s}} \int_{{\mathbb R}^d_+\times{\mathbb R}^{d-1}}\int_{{\mathbb R}^{d-1}}\frac{\Phi(V)}{\langle \epsilon V\rangle^{d-1-2s}}\left(\frac{1}{\langle \epsilon V\rangle^{d-1-2s}}-\frac{1}{\langle \epsilon(V+z)\rangle^{d-1-2s}}\right)\frac{U(V+z)}{|z|^{d-1+2s}}\sigma\;dzdXdV\\
&+\frac{1}{2^{1+2s}} \int_{{\mathbb R}^d_+\times{\mathbb R}^{d-1}}\int_{{\mathbb R}^{d-1}}\frac{\Phi(V)}{\langle \epsilon V\rangle^{d-1-2s}}\left(\frac{1}{\langle \epsilon V\rangle^{d-1-2s}}-\frac{1}{\langle \epsilon(V-z)\rangle^{d-1-2s}}\right)\frac{U(V-z)}{|z|^{d-1+2s}}\sigma\;dzdXdV\\
=:& I_{3,1}+I_{3,2}+I_{3,3}.
\end{aligned}
$$
The first integral gives
\begin{equation}\label{I_31}
I_{3,1} = \int \widetilde{\sigma}(X^d)\frac{\Phi}{\langle \epsilon V\rangle^{2(d-1)}}(-\Delta_V)^sU dXdV + E_{3}(\Phi) + J_2(\Phi),
\end{equation}
with error terms
$$
\begin{aligned}
E_3 =& \int_{{\mathbb R}^d_+\times{\mathbb R}^{d-1}}\frac{\Phi}{\langle \epsilon V\rangle^{2(d-1)}} (-\Delta_V)^sU\left(\langle \epsilon V\rangle^{4s}-1\right)\frac{1}{2^{2s}}\sigma(2\epsilon X',X^d)\;dXdV,
\end{aligned}
$$
and
\begin{equation}\label{defJ2}
\begin{aligned}
J_2 =& \frac{1}{2^{2s}}\int_{{\mathbb R}^d_+\times{\mathbb R}^{2(d-1)}}\big(\sigma(2\epsilon X',X^d)-\sigma(0,X^d)\big)\frac{\Phi}{\langle \epsilon V\rangle^{2(d-1)}}(-\Delta_V)^sU dX dV.
\end{aligned}
\end{equation}
On the other hand,
$$
\begin{aligned}
I_{3,2}+I_{3,3}&= \frac{1}{2^{1+2s}} \int_{{\mathbb R}^d_+\times{\mathbb R}^{d-1}}\int_{{\mathbb R}^{d-1}}\sigma(2\epsilon X',X^d)\frac{\Phi(X,V)}{\langle \epsilon V\rangle^{d-1-2s}}\\
&\hspace{3em} \times\left(\frac{2}{\langle \epsilon V\rangle^{d-1-2s}}-\frac{1}{\langle \epsilon(V+z)\rangle^{d-1-2s}}-\frac{1}{\langle \epsilon(V-z)\rangle^{d-1-2s}}\right)\frac{U(V+z)}{|z|^{d-1+2s}}\;dzdXdV\\
&\quad +\frac{1}{2^{1+2s}} \int_{{\mathbb R}^d_+\times{\mathbb R}^{d-1}}\int_{{\mathbb R}^{d-1}}\sigma(2\epsilon X',X^d)\frac{\Phi(X,V)}{\langle \epsilon V\rangle^{d-1-2s}}\\
&\hspace{3em}\times \left(\frac{1}{\langle \epsilon V\rangle^{d-1-2s}}-\frac{1}{\langle \epsilon(V-z)\rangle^{d-1-2s}}\right)\frac{(U(V-z) - U(V+z))}{|z|^{d-1+2s}}\;dzdXdV.
\end{aligned}
$$
By denoting $h(z) = \frac{1}{\langle \epsilon(V+z)\rangle^{d-1-2s}}$, the above simplifies to
$$
\begin{aligned}
&I_{3,2}+I_{3,3} \\
&= \frac{1}{2^{1+2s}} \int_{{\mathbb R}^d_+\times{\mathbb R}^{d-1}}\int_{{\mathbb R}^{d-1}}\sigma(2\epsilon X',X^d)\frac{\Phi(X,V)}{\langle \epsilon V\rangle^{d-1-2s}} \frac{\left(2h(0)-h(z)-h(-z)\right)}{|z|^{d-1+2s}}U(V+z)\;dzdXdV\\
&\quad +\frac{1}{2^{1+2s}} \int_{{\mathbb R}^d_+\times{\mathbb R}^{d-1}}\int_{{\mathbb R}^{d-1}}\sigma(2\epsilon X',X^d)\frac{\Phi(X,V)}{\langle \epsilon V\rangle^{d-1-2s}}\left(h(0)-h(-z)\right)\frac{(U(V-z) - U(V+z))}{|z|^{d-1+2s}}\;dzdXdV.
\end{aligned}
$$
To continue with the estimation we split the each integrals into two, for the respective regions $|z|>1$ and $|z|\leq 1$. For the former we use that
$$
\nabla_zh(z)=\frac{(d-1-2s)\epsilon^2(V+z)}{\langle \epsilon(V+z)\rangle^{d+1-2s}}$$
therefore
\begin{equation*}\label{der1_h}
|h(0)-h(z)|\leq C\epsilon^2|z|(|V\pm z|+|z|), \quad\forall |z|>1,\, V\in{\mathbb R}^{d-1},
\end{equation*}
which directly implies
$$
|2h(0)-h(z)-h(-z)|\leq|h(0)-h(z)|+|h(0)-h(-z)|\leq C\epsilon^2|z|(|V+z|+|z|).
$$
Since $|h|\leq1$ we then have that for any $s'\in(0,s)$,
$$|h(0)-h(z)|^{s'+(1-s')}\leq C|h(0)-h(z)|^{s'}\leq C\epsilon^{2s'}(|V\pm z|^{2s'}+|z|^{2s'}),$$
and similarly for $|2h(0)-h(z)-h(-z)|$.
The above gives us that on the region $|z|>1$,
$$
|I_{3,2}+I_{3,3}|\leq C\epsilon^{2s'}\left(\textstyle \int_{|z|>1}\frac{1}{|z|^{d-1+2(s-s')}}dz\right)\|\sigma\|_\infty\|\Phi\|_{\infty}\|(1+|V|)U\|_{L^1}\leq C\epsilon^{2s'}\|\psi\|_\infty.
$$
For the second part, $|z|\leq 1$, we compute
$$
\nabla_z^2h(z) = \frac{(d-1-2s)\epsilon^2 I}{\langle \epsilon(V+z)\rangle^{d+1-2s}} + \frac{(d-1-2s)(d+1-2s)\epsilon^4(V+z)^t(V+z)}{\langle \epsilon(V+z)\rangle^{d+3-2s}},
$$
thus, from the inequality $\frac{\epsilon^2|V+z|^2}{\langle \epsilon(V+z)\rangle^{d+3-2s}}\leq 1$ it follows
\begin{equation}\label{der2_h}
|2h(0)-h(z)-h(-z)|\leq C\epsilon^2|z|^2,\quad\forall |z|\leq 1,\, V\in{\mathbb R}^{d-1},
\end{equation}
and this leads to the estimate
$$
\begin{aligned}
& \int_{{\mathbb R}^d_+\times{\mathbb R}^{d-1}}\int_{{\mathbb R}^{d-1}}\sigma(2\epsilon X',X^d)\frac{\Phi(X,V)}{\langle \epsilon V\rangle^{d-1-2s}} \frac{\left(2h(0)-h(z)-h(-z)\right)}{|z|^{d-1+2s}}U(V+z)\;dzdXdV\\
&\quad \leq C\epsilon^2\|\sigma\|_\infty\|\Phi\|_\infty\|U\|_{L^1}\left(\textstyle \int_{|z|<1}\frac{1}{|z|^{d-1 - 2(1-s)}}dz\right)\leq C\epsilon^2\|\psi\|_\infty.
\end{aligned}
$$
On the other hand, we notice that
$$
h(0)-h(-z) = \int^1_0z\cdot\nabla h(tz)dt = (d-1-2s)\epsilon^2 \int^1_0\frac{z\cdot(V-tz)}{\langle \epsilon(V-tz)\rangle^{d+1-2s}}dt.
$$
Thus
$$
\begin{aligned}
&\int_{{\mathbb R}^d_+\times{\mathbb R}^{d-1}}\int_{{\mathbb R}^{d-1}}\sigma(2\epsilon X',X^d)\frac{\Phi(X,V)}{\langle \epsilon V\rangle^{d-1-2s}}\left(h(0)-h(-z)\right)\frac{(U(V-z) - U(V+z))}{|z|^{d-1+2s}}\;dzdXdV\\
&\quad = (d-1-2s)\epsilon^2 \int\int^1_0\sigma(2\epsilon X',X^d)\frac{\Phi(X,V)}{\langle \epsilon V\rangle^{d-1-2s}}\frac{z\cdot(V-tz)}{\langle \epsilon(V-tz)\rangle^{d+1-2s}}\frac{(U(V-z) - U(V+z))}{|z|^{d-1+2s}}\;dtdzdXdV\\
&\quad = C\epsilon^2 \int\int^1_0\frac{\sigma(2\epsilon X',X^d)\Phi(X,V)}{\langle \epsilon V\rangle^{d-1-2s}\langle \epsilon(V-tz)\rangle^{d+1-2s}}\frac{z\cdot(U(V-z)V - U(V+z)V)}{|z|^{d-1+2s}}\;dtdzdXdV+ O(\epsilon^2),
\end{aligned}
$$
where the error terms is given by
$$
\begin{aligned}
&C\epsilon^2 \int\frac{\sigma(2\epsilon X',X^d)\Phi(X,V)}{\langle \epsilon V\rangle^{d-1-2s}}\left(\int^1_0\frac{tdt}{\langle \epsilon(V-tz)\rangle^{d+1-2s}}\right)\frac{(U(V-z) - U(V+z))}{|z|^{d-1-2(1-s)}}\;dzdXdV \\
&\hspace{3em}\leq C\epsilon^2\|\sigma\|_\infty\|\Phi\|_\infty\|U\|_{L^1}\Big(\textstyle\int_{|z|<1}\frac{1}{|z|^{d-1-2(1-s)}}dz\Big).
\end{aligned}
$$
For the remaining integral we see that
$$
z\cdot(U(V-z)V - U(V+z)V) = \big(U(V-z)z\cdot(V-z) - U(V+z)z\cdot(V+z)\big) + |z|^2\big(U(V-z)+U(V+z)\big).
$$
Thus, for any $\hat{z} = z/|z|$, with $0<|z|<1$, by defining the function
$$
g_{\hat{z}}(V) = (\hat{z}\cdot V)U(V),
$$
we have that
$$
\begin{aligned}
&\int\int^1_0\frac{\sigma(2\epsilon X',X^d)\Phi(X,V)}{\langle \epsilon V\rangle^{d-1-2s}\langle \epsilon(V-tz)\rangle^{d+1-2s}}\frac{z\cdot(U(V-z)V - U(V+z)V)}{|z|^{d-1+2s}}\;dtdzdXdV\\
&\hspace{3em} = \int\int^1_0\frac{\sigma(2\epsilon X',X^d)\Phi(X,V)}{\langle \epsilon V\rangle^{d-1-2s}\langle \epsilon(V-tz)\rangle^{d+1-2s}}\left(\frac{g_{\hat{z}}(V-z) - g_{\hat{z}}(V+z)}{|z|}\right)\frac{1}{|z|^{d-1-2(1-s)}}\;dtdzdXdV\\
&\hspace{4em} + \int\int^1_0\frac{\sigma(2\epsilon X',X^d)\Phi(X,V)}{\langle \epsilon V\rangle^{d-1-2s}\langle \epsilon(V-tz)\rangle^{d+1-2s}}\left(U(V-z)+U(V+z)\right)\frac{1}{|z|^{d-1-2(1-s)}}\;dtdzdXdV,
\end{aligned}
$$
and hence, denoting by $D^{z}_Vg_{\hat{z}}:=\frac{g_{\hat{z}}(V+z) - g_{\hat{z}}(V)}{|z|}$ the difference quotient of $g_{\hat{z}}(V)$, the integral is bounded by
$$
\begin{aligned}
& C\|\sigma\|_\infty\|\Phi\|_\infty\left(\int \left(\int |D^{z}_Vg_{\hat{z}}(V)| dVdX\right)\frac{1}{|z|^{d-1-2(1-s)}}dz + \|U\|_{L^1}\Big(\textstyle\int_{|z|<1}\frac{1}{|z|^{d-1-2(1-s)}}dz\Big)\right)\\
&\quad \leq C\|\sigma\|_\infty\|\Phi\|_\infty\left(\sup_{\hat{w}\in {\mathbb S}^{d-2}}\Big(\sup_{|z|<1}\|D^{z}_Vg_{\hat{w}}(V)\|_{L^1}\Big) + \|U\|_{L^1}\right)\Big(\textstyle\int_{|z|<1}\frac{1}{|z|^{d-1-2(1-s)}}dz\Big)\\
&\quad\leq C\|\sigma\|_\infty\|\Phi\|_\infty\left(\||V|\nabla_V U\|_{L^1} + \|U\|_{L^1}\right),
\end{aligned}
$$
where the last inequality follows from \cite[\S 5.8.2]{Evans}, while the boundedness of the $L^1$-norms of $U$ are due to Lemma \ref{lemma:int_der_J}. This concludes the proof that $|I_{3,2} + I_{3,3}|\leq C\epsilon^{2s'}\|\psi\|_\infty$.
We still need to estimate the error terms of $I_{3,1}$ (defined in \eqref{I_31}).
By simply noticing that $|\langle \epsilon V\rangle^{4s}-1|\lesssim \epsilon^2|V|^2\langle \epsilon V\rangle^{2(2s-1)}$, then
$$
|E_3|\lesssim \epsilon^2\|\sigma\|_\infty \int_{{\mathbb R}^d_+\times{\mathbb R}^{d-1}}\frac{|\Phi||V|^2}{\langle \epsilon V\rangle^{2(d-2s)}} |(-\Delta_V)^sU|\;dXdV.
$$
Therefore, we get an $O(\epsilon^{2s'})$ upper bound by means of the inequality \eqref{s'_ineq} with $m=d-2s$
and the integrability Lemma \ref{lemma:int_fLap_J}.
Notice that for \eqref{s'_ineq} to hold we need $d-2s\geq1-s'$, or equivalently $s'\geq2s+1-d$. For $d=2$ this translates into $s'\geq 2s-1$, while for $d\geq 3$ it is always satisfied. This explains the hypothesis imposed in the statement of the theorem.
We deduce $|E_3|\lesssim\epsilon^{2s'}\|\psi\|_\infty$ with constant depending on $\int_{{\mathbb R}_+}\||V|^{2s'}(-\Delta_V)^sU\|_{L^1({\mathbb R}^{2(d-1)})}dX^d$.\\
\noindent{\it Estimation of $J_1$ and $J_2$.} Let us now estimate $J_1$ and $J_2$ (defined respectively in \eqref{defJ1} and \eqref{defJ2}).
We write $J_1$ and $J_2$ as follows,
\begin{equation*}
\begin{aligned}
J_1 =& \frac{1}{2}\int_{{\mathbb R}^d_+\times{\mathbb R}^{d-1}} \big(\lambda(\epsilon X',X^d) - \lambda(0 ,X^d)\big)U(X,V)\Phi(X,V)dXdV + O(\epsilon^{2s'}).
\end{aligned}
\end{equation*}
with a reminder depending on $\||V|^{2s'}U(X,V)\|_{L^1}$,
and
$$
\begin{aligned}
J_2 =& \frac{1}{2^{2s}}\int_{{\mathbb R}^d_+\times{\mathbb R}^{d-1}}\big(\sigma(2\epsilon X',X^d)-\sigma(0,X^d)\big)\Phi(X,V)\big[(-\Delta_V)^sU(X,V)\big]dV dX + O(\epsilon^{2s'}),
\end{aligned}
$$
with a reminder depending on $ \||V|^{2s'}(-\Delta_V)^sU(X,V)\|_{L^1}$. To proceed we need the next lemma whose proof is postponed until the end of this section.
\begin{lemma}[Sub-optimal approximation for the adjoint equation] \label{lemma:ord1_approx}
Let $\varphi(x,\theta) = \Phi((2\epsilon)^{-1}x',x^d,\epsilon^{-1}\mathcal{S}(\theta))$ and $\psi(x,\theta) = \Psi((2\epsilon)^{-1}x',x^d,\epsilon^{-1}\mathcal{S}(\theta))$ with $\varphi$ solution to the backward fractional Fokker--Planck equation
$$-\theta\cdot\nabla_x\varphi + \lambda\varphi = \mathcal{I}_\theta[\varphi] + \psi,\quad \psi \in C_c(Q).$$
Let $W$ be the unique solution to the backward fractional Fermi pencil-beam equation \eqref{b_fFpb} with source $\Psi$. Then, for any $f$ absolutely integrable function in ${\mathbb R}^d_+\times{\mathbb R}^{d-1}$ and for any $s'\in (0,s)$, there exists $C>0$ (depending on $\sigma,\lambda$ and $\|f\|_{L^1}$) such that
\begin{equation*}
\Big|\int_{{\mathbb R}^d_+\times{\mathbb R}^{d-1}} f\Phi dXdV - \int_{{\mathbb R}^d_+\times{\mathbb R}^{d-1}} f WdXdV \Big|\leq C\epsilon^{\min\{1,2s'\}}\|\Psi\|_\infty.
\end{equation*}
with $C=C_{s'}$ blowing up to infinity as $s'\to s$ for $s<1/2$, and independent of $s'$ otherwise.
\end{lemma}
In view of the previous lemma we get
\begin{equation*}
\begin{aligned}
J_1 =& \frac{1}{2}\int_{{\mathbb R}^d_+\times{\mathbb R}^{d-1}} \big(\lambda(\epsilon X',X^d) - \lambda(0 ,X^d)\big) W(X,V)U(X,V)dXdV + O(\epsilon^{2s'}),
\end{aligned}
\end{equation*}
and
\begin{equation*}
\begin{aligned}
J_2 =& \frac{1}{2^{2s}} \int_{{\mathbb R}^d_+\times{\mathbb R}^{d-1}}\big(\sigma(2\epsilon X',X^d)-\sigma(0,X^d)\big)W(X,V)\big[(-\Delta_V)^sU(X,V)\big]dV dX + O(\epsilon^{2s'}),
\end{aligned}
\end{equation*}
where we used the Lipschitz property of $\lambda$ and $\sigma$ to obtain the $O(\epsilon^2)$-error when interchanging $\Phi$ with $W$. In order to utilize the Lipschitz continuity of $W$ we employ Lemma \ref{lemma:radial_sym}. We first split the computations into two cases. For $s\leq1/2$ we do not need Lemma \ref{lemma:radial_sym} and simply obtain
\begin{equation*}
\begin{aligned}
|J_1| \lesssim& \; \epsilon^{2s'}\|\lambda\|_{C^1}^{2s'}\|\lambda\|_\infty^{1-2s'}\|\psi\|_\infty^{1-2s'}\||X|^{2s'}U(X,V)\|_{L^1} + C\epsilon^{2s'},
\end{aligned}
\end{equation*}
for some $s'\in(0,s)$. Otherwise, for $s\in(1/2,1)$, the lemma allows us to write
\begin{equation*}
\begin{aligned}
J_1 =& \frac{\epsilon}{2}\int_{{\mathbb R}^d_+\times{\mathbb R}^{d-1}} \nabla_{X'}\lambda(0,X^n)\cdot X' \left(W(X,V) - W(0,V)\right)U(X,V)dXdV + O(\epsilon^{2s'}).
\end{aligned}
\end{equation*}
from which we deduce
\begin{equation*}
|J_1|\lesssim \epsilon^{1+r}\|\psi\|_\infty^{1-r}\kappa^{r}\|\lambda\|_{C^1}\||X'|^{1+r}U(X,V)\|_{L^1} + C\epsilon^{2s'},
\end{equation*}
for $r=2s'-1$ and $s'\in(1/2,s)$. In both cases, we end up with an estimate of the form $|J_1|\lesssim \epsilon^{2s'}\kappa^{s'}$.
Regarding $J_2$, Lemma \ref{lemma:radial_sym} yields
\begin{equation*}
\begin{aligned}
J_2 =& \frac{1}{2^{2s}}\int_{{\mathbb R}^d_+\times{\mathbb R}^{d-1}}\big(\sigma(2\epsilon X',X^d)-\sigma(0,X^d)\big)\\
&\hspace{5em}\times\big(W(X,V) - W(X,0)\big)\big[(-\Delta_V)^sU(X,V)\big]dV dX + O(\epsilon^{2s'}).
\end{aligned}
\end{equation*}
Similarly as above, we reduce the exponent by interpolating upper bounds and obtain
\begin{equation*}
\begin{aligned}
|J_2| \leq \frac{1}{2^{2s}}\|\sigma\|_{\infty}^{1-s'}\|\psi\|_{\infty}^{1-s'}& \int_{{\mathbb R}^d_+\times{\mathbb R}^{d-1}}\big|\sigma(2\epsilon X',X^d)-\sigma(0,X^d)\big|^{s'}\\
&\times\big|W(X,V) - W(X,0)\big|^{s'}\big|(-\Delta_V)^sU(X,V)\big|dV dX + O(\epsilon^{2s'}),
\end{aligned}
\end{equation*}
which then gives rise to
$$
\begin{aligned}
|J_2| &\leq C\epsilon^{2s'}\|\sigma\|_{\infty}^{1-s'}\|\sigma\|^{s'}_{C^1}\|\psi\|_{\infty}^{1-s'}\kappa^{s'} \int_{{\mathbb R}^d_+\times{\mathbb R}^{d-1}}|X'|^{s'}|V|^{s'}\big|(-\Delta_V)^sU(X,V)\big|dV dX + O(\epsilon^{2s'}),
\end{aligned}
$$
and consequently $|J_2| \leq C\epsilon^{2s'}\kappa^{s'}$. The positive constant depends on
\begin{equation*}
\begin{aligned}
\||X'|^{2s'}(-\Delta_V)^sU(X,V)\|_{L^1}\quad\text{and}\quad \||V|^{2s'}(-\Delta_V)^sU(X,V)\|_{L^1}.
\end{aligned}
\end{equation*}
which are finite thanks to Lemma \ref{lemma:int_fLap_J}.
\end{proof}
\begin{theorem}\label{thm:fFP_vs_pb}
Let $\delta\lesssim \epsilon^2\kappa$ and $f\in L^\infty_{x,\theta}\cap L^1_{x,\theta}$ be a compactly supported $\delta$-approximation to the identity $\delta_{0}(x)\delta_{N}(\theta)$, this is $\left|\int f\varphi dxd\theta - \varphi(0,N)\right|\lesssim \delta$ for all $\varphi\in C(Q)$. We let $u(x,\theta)$ be the solution to the fractional Fokker--Planck equation \eqref{fFP_diff} (or equivalently \eqref{fFP_id}-\eqref{fFP_kernel}).
For any $s'\in(2s-1,s)$ in dimension $d=2$, and $s'\in(0,s)$ for $d\geq 3$, there exists a constant $C=C(s',d,\lambda,\sigma)>0$, blowing up as $s'\to s$, such that
$$
\mathcal{W}^1_{\kappa}(\mathfrak{u},u)\leq C\epsilon^{2s'}\kappa^{s'}.
$$
\end{theorem}
\begin{proof}
For an arbitrary $\psi\in C_c(Q)\cap \text{\rm Lip\hspace{.1em}}_{\kappa}(Q)$ let $\varphi$ be the unique solution to \eqref{backward_fFP}. We see that
\begin{equation*}
\begin{aligned}
\left| \int (\mathfrak{u}-u)\psi dxd\theta \right|&=\left| \int \mathfrak{u}\psi dxd\theta - \int f\varphi dxd\theta\right|\\
&\leq \left| \int \mathfrak{u}\psi dxd\theta -\varphi(0,N)\right|+\left|\varphi(0,N) - \int f\varphi dxd\theta\right|.
\end{aligned}
\end{equation*}
The proof follows from the hypothesis on the source $f$ and Lemma \ref{lemma:approx_fFpb}.
\end{proof}
\begin{proof}[Proof of Lemma \ref{lemma:ord1_approx}]
We let $\text{\bf U}$ be the solution to the inhomogeneous fractional Fermi pencil-beam equation $\mathcal{P}(\text{\bf U}) = f$ with boundary conditions $\text{\bf U}|_{X^n=0}=0$, and let $\text{\bf u}$ be its rescaling
\begin{equation*}
\text{\bf u}(x,\theta) = (2\epsilon)^{-2(d-1)}\text{\bf U}((2\epsilon)^{-1}x',x^d,\epsilon^{-1}\mathcal{S}(\theta)).
\end{equation*}
One verifies that
$$
\int_{\mathcal{Q}_+} f WdXdV = \int_{\mathcal{Q}_+} \text{\bf U} \Psi dXdV,
$$
and hence a change variables leads to
$$
\int_{\mathcal{Q}_+} f WdXdV = \int_{Q_+} \text{\bf u} \psi dxd\theta + R(\epsilon^{2s'}),
$$
with the reminder term given explicitly by
$$
R(\epsilon^{2s'})=\int_{\mathcal{Q}} \text{\bf U}\Psi\left(1-\langle \epsilon V\rangle^{-(d-1)}\right)dXdV \leq \epsilon^{2s'}\big(\|\text{\bf U}\|_{L^1({\mathbb R}^d\times B_1)} + \||V|^{2s'}\text{\bf U}\|\big)_{L^1({\mathbb R}^d\times B_1^c)}\|\Psi\|_\infty.
$$
The proof then reduces to showing that
\begin{equation*}
\Big|\int_{Q_+} \text{\bf u}\psi dxd\theta - \int_{\mathcal{Q}_+} f \Phi dXdV \Big|\leq C\epsilon^{\min\{1,2s'\}}\|\Psi\|_\infty,
\end{equation*}
for any $s'\in(0,1)$, with the constant depending on this choice.
We proceed as in the proof of Lemma \ref{lemma:approx_fFpb} using the following decomposition
$$
\begin{aligned}
\int_{Q_+}\text{\bf u}\psi dxd\theta &= \int_{\mathcal{Q}_+}\text{\bf u}\left(-\theta \cdot\nabla_x\varphi+ \lambda \varphi -\mathcal{I}_\theta(\varphi)\right)dxd\theta \\
&=:I_1+I_2+I_3 + I_4,
\end{aligned}
$$
where the term $I_j$ are defined as in Lemma \ref{lemma:approx_fFpb} with $U$ replaced by $\text{\bf U}$. The objective is to show that
\begin{equation}\label{sum_Ij}
\begin{aligned}
I_1+I_2+I_3 + I_4 &= \int_{{\mathbb R}^{d-1}\times{\mathbb R}^{d-1}}\underbrace{\text{\bf U}(X',0,V)}_0\frac{\Phi(X',0,V)}{\langle \epsilon V\rangle^{2(d-1)}}dX'dV\\
&\hspace{10em} + \int_{\mathcal{Q}_+} \underbrace{\mathcal{P}(\text{\bf U})}_{=f}\Phi dXdV + O(\epsilon^{\min\{1,2s'\}}).
\end{aligned}
\end{equation}
Most of the estimates performed in the proof of Lemma \ref{lemma:approx_fFpb} remain identical, although with upper bounds now depending on various integrals of $\text{\bf U}$ (which are guaranteed to be finite by virtue of Lemma \ref{lemma:int_der_J} and the convolution formula for pencil-beam solutions \eqref{sol_fFPB}). These are subsequently bounded by the $L^1$-norm of $f$ (see \S\ref{subsec:integ_fFpb} and Remark \ref{rmk:gral_integ}).
The main difference is that we are interested here in obtaining a sub-optimal accuracy for the error, namely $O(\epsilon^{\min\{1,2s'\}})$, so that Lipschitz-continuity of $\sigma$ and $\lambda$ is enough to deduce \eqref{sum_Ij}.
\end{proof}
\subsection{Ballistic approximation}\label{subsec:ballistic_approx}
We denote by $v$ the solution to the {\em ballistic transport equation}
\begin{equation}\label{ballistic_eq}
\theta\cdot\nabla_x v +\lambda(x) v = f(x,\theta),
\end{equation}
which, for a source $f\in L^1({\mathbb R}^{d}\times{\mathbb S}^{d-1})$, has the explicit form
\begin{equation}\label{ballistic_sol}
v(x,\theta) =\mathcal{L}[f](x,\theta):= \int^{\infty}_0 e^{-\int^t_0\lambda(x-s\theta)ds}f(x-t\theta,\theta)dt.
\end{equation}
For a point source $f = \delta_0(x)\delta_{N}(\theta)$ the solution is defined in the distributional sense as
\begin{equation}\label{ballistic_sol_dist}
\langle v,\psi\rangle = \int^\infty_0e^{-\int^t_0\lambda(\vec{0},s)ds}\psi(tN,N)dt,\quad \forall \psi \in C({\mathbb R}^{d}\times{\mathbb S}^{d-1}),
\end{equation}
where $\vec{0}$ stands for the origin in ${\mathbb R}^{d-1}$.
We now compare the order of approximation between the ballistic solution and the pencil-beam approximation introduced in the previous section. We obtain a lower and upper bound for their 1-Wasserstein distance for different choices of parameters.
\begin{theorem}\label{thm:ball_vs_pb}
There exists a constant $c>0$ (independent of $\epsilon$ and $\kappa$) such that
$$
c\epsilon\kappa\leq \mathcal{W}^1_{\kappa}(v,\mathfrak{u})\quad \text{for}\quad1\leq\kappa\leq\epsilon^{-1}\quad\text{and} \quad c\leq \mathcal{W}^1_{\kappa}(v,\mathfrak{u})\quad \text{for}\quad \kappa>\epsilon^{-1}
$$
Moreover, there exist constants $C_{s'}>0$ and $C_{s}>0$, with the former blowing up to infinity as $s'\in(0,s)$ approaches $s$, and the latter blowing up to infinity as $s \searrow\frac{1}{2}$, such that
\begin{equation*}
\mathcal{W}^1_{\kappa}(v,\mathfrak{u})\leq C_{s'}(\epsilon\kappa)^{2s'}\quad \text{if}\quad s\in(0,1/2]\quad\text{and}\quad \mathcal{W}^1_{\kappa}(v,\mathfrak{u})\leq C_s\epsilon\kappa\quad \text{if}\quad s\in(1/2,1).
\end{equation*}
\end{theorem}
\begin{proof}
\noindent{\em Upper bound.} Let $\psi$ be an arbitrary test function in $BL_{1,\kappa}({\mathbb R}^{d}\times{\mathbb S}^{d-1})$ and $\Psi$ its rescaling according to \eqref{rescaling_test_functions}, which is Lipschitz with constant $O(\epsilon\kappa)$. For $s'\in(0,s)$ we have that
\begin{equation*}
\begin{aligned}
\langle v-\mathfrak{u},\psi\rangle &= \int^\infty_0e^{-\int^t_0\lambda(\vec{0},s)ds}\Psi(\vec{0},t,\vec{0})dt -\int_0^\infty\int_{{\mathbb R}^{2(d-1)}}U(X',t,V)\Psi(X',t,V) \frac{dX'dVdt}{\langle \epsilon V\rangle^{2(d-1)}} \\
&= \int^\infty_0\left(e^{-\int^t_0\lambda(\vec{0},s)ds}\Psi(\vec{0},t,\vec{0}) -\int_0^\infty\int_{{\mathbb R}^{2(d-1)}}U(X',t,V)\Psi(X',t,V) dX'dV\right)dt + O(\epsilon^{2s'}).
\end{aligned}
\end{equation*}
The expression inside parentheses is equivalent to
\begin{equation*}
\int_{{\mathbb R}^{2(d-1)}}J(X',t,V)\left(\Psi(\vec{0},t,\vec{0}) -
\Psi(X',t,V) \right)dX'dV,
\end{equation*}
with $J$ the fundamental solution to the fractional Fermi pencil-beam of Section \ref{subsec:integ_fFpb}.
There are two scenarios depending on the values of the exponent $s$. For $s>1/2$, we take $s'=1/2$ above and get
\begin{equation*}
\langle v-\mathfrak{u},\psi\rangle\lesssim \epsilon\kappa\int^\infty_0 e^{-\int^t_0\lambda(\vec{0},r)dr}\||(X',V)|J(X',t,V)\|_{L^1({\mathbb R}^{2(d-1)})}dt=O(\epsilon\kappa).
\end{equation*}
On the other hand, for $s\leq1/2$, we interpolate the Lipschitz and the $L^\infty$ bound of the difference $\Psi(\vec{0},t,\vec{0}) -
\Psi(X',t,V)$ to get
\begin{equation*}
\langle v-\mathfrak{u},\psi\rangle\lesssim (\epsilon\kappa)^{2s'}\int^\infty_0 e^{-\int^t_0\lambda(\vec{0},r)dr}\||(X',V)|^{2s'}J(X',t,V)\|_{L^1({\mathbb R}^{2(d-1)})}dt=O((\epsilon\kappa)^{2s'}).
\end{equation*}
for any $s'\in(0,s)$ with the constant in the estimate blowing up as $s'$ approaches $s$.
The desired estimates follow after taking supremum with respect to $\psi$.\\
\noindent{\em Lower bound.} Let us choose a specific test function: $\psi(x,\theta)=\psi_1(x')= e^{-\kappa|x'|}$.
When passing to stretched coordinate we denote $\Psi_1(X') = e^{-2\epsilon\kappa|X'|}$
and therefore $\Psi(X,V)=\Psi_1(X')$. The Fourier Transform of $\Psi_1$ is given by
$$\hat{\Psi}_1(\xi):=\mathfrak{F}_{X'\to\xi}[\Psi_1](\xi) = c_{d}\frac{\epsilon\kappa}{\left(4\pi^2\epsilon^2\kappa^2+|\xi|^2\right)^{\frac{d}{2}}},$$
for some constant $c_d>0$.
We find that
\begin{equation}\label{ball_test}
\begin{aligned}
\langle v,\psi\rangle &= \int^\infty_0e^{-\int^t_0\lambda(\vec{0},s)ds}\Psi(\vec{0},t,\vec{0})dt = \int^\infty_0e^{-\int^t_0\lambda(\vec{0},s)ds}\int_{{\mathbb R}^{d-1}}\hat{\Psi}_1(\xi)d\xi dt\\
&=c_d\epsilon\kappa\int^\infty_0e^{-\int^t_0\lambda(\vec{0},s)ds}\int_{{\mathbb R}^{d-1}}\frac{1}{\left(4\pi^2\epsilon^2\kappa^2+|\xi|^2\right)^{\frac{d}{2}}}d\xi dt.
\end{aligned}
\end{equation}
On the other hand,
$$
\begin{aligned}
\langle \mathfrak{u},\psi\rangle &= \int_0^\infty\int_{{\mathbb R}^{2(d-1)}}U(X',t,V)\Psi(X',t,V) \frac{dX'dVdt}{\langle \epsilon V\rangle^{2(d-1)}} \\
&\leq \int_0^\infty\int_{{\mathbb R}^{2(d-1)}}U(X',t,V)\Psi_1(X') dX'dVdt.
\end{aligned}
$$
Then, the integration with respect to $V$ and the Plancherel's identity yield
\begin{equation}\label{pb_test}
\begin{aligned}
\langle \mathfrak{u},\psi\rangle&\leq \int_0^\infty\int_{{\mathbb R}^{d-1}}\hat{U}(\xi,t,0)\hat{\Psi}_1(\xi,t) d\xi dt\\% + O(\epsilon^{2s'})\\
&\leq c_d\epsilon\kappa\int_0^\infty e^{-\int^t_0\lambda(\vec{0},s)ds}\int_{{\mathbb R}^{d-1}}\frac{e^{-\tau(t)|\xi|^{2s}}}{\left(4\pi^2\epsilon^2\kappa^2+|\xi|^2\right)^{\frac{d}{2}}}d\xi dt
\end{aligned}
\end{equation}
where $\tau(t) := \int^t_0r^{2s}\widetilde{\sigma}(r)dr$ ($\approx t^{2s+1}$).
Let us compare this to \eqref{ball_test}. We get
$$
\begin{aligned}
\langle v-\mathfrak{u},\psi\rangle \geq c_d\epsilon\kappa \int_0^\infty e^{-\int^t_0\lambda(\vec{0},s)ds}\int_{{\mathbb R}^{d-1}}\frac{\big(1-e^{-\tau(t)|\xi|^{2s}}\big)}{\left(4\pi^2\epsilon^2\kappa^2+|\xi|^2\right)^{\frac{d}{2}}}d\xi dt > 0
\end{aligned}
$$
To simplify notation, we consider a generic constant $c>0$ independent of $\epsilon$ and $\kappa$ so that once passing to spherical coordinates we get
$$
\begin{aligned}
\langle v-\mathfrak{u},\psi\rangle &\geq c\epsilon\kappa\int_0^\infty e^{-\int^t_0\lambda(\vec{0},s)ds}\int_{0}^\infty \frac{\big(1-e^{-\tau(t)r^{2s}}\big)}{\left(4\pi^2\epsilon^2\kappa^2+r^2\right)^{\frac{d}{2}}}r^{d-2}drdt
\end{aligned}
$$
We now analyze the integral $I(t) := \epsilon\kappa\int_{0}^\infty \frac{\big(1-e^{-\tau(t)r^{2s}}\big)}{\left(4\pi^2\epsilon^2\kappa^2+r^2\right)^{\frac{d}{2}}}r^{d-2}dr$. The term inside the parentheses is bounded from below by
$$
\begin{aligned}
1-e^{-\tau(t)r^{2s}}&=2s\tau(t)\int^r_0\rho^{2s-1}e^{-\tau(t)\rho^{2s}}d\rho \\
&\geq 2s\tau(t)e^{-\tau(t)r^{2s}}\int^r_0\rho^{2s-1}d\rho\\
&= \tau(t)e^{-\tau(t)r^{2s}}r^{2s}
\end{aligned}
$$
which leads to
$$
I(t) \geq \epsilon\kappa\tau(t)\int_{0}^\infty \frac{e^{-\tau(t)r^{2s}}r^{d-2+2s}}{\left(4\pi^2\epsilon^2\kappa^2+r^2\right)^{\frac{d}{2}}}dr.
$$
We then get
$$
I(t) \geq \epsilon\kappa\tau(t)e^{-\tau(t)}\int_{0}^1 \frac{r^{d-2(1-s)}}{\left(4\pi^2(\epsilon\kappa)^2+1\right)^{\frac{d}{2}}}dr\geq c\epsilon\kappa\left(\frac{1}{\left((2\pi\epsilon\kappa)^2+1\right)^{\frac{d}{2}}}\right)\tau(t)e^{-\tau(t)}.
$$
Consequently, under the resolution restriction $\kappa\leq \epsilon^{-1}$, we obtain the estimate
$$
\langle v-\mathfrak{u},\psi\rangle\geq c\epsilon\kappa\int^\infty_0\tau(t)e^{-\tau(t)}e^{-\int^t_0\lambda(\vec{0},r)dr}dt
$$
Let us now consider the case $\kappa>\epsilon^{-1}$. We can use a different lower bound, namely
$$
\begin{aligned}
\epsilon\kappa\int_{0}^\infty \frac{e^{-\tau(t)r^{2s}}r^{d-2+2s}}{\left(4\pi^2\epsilon^2\kappa^2+r^2\right)^{\frac{d}{2}}}dr
&\geq \frac{\epsilon\kappa e^{-\tau(t)(\epsilon\kappa)^{2s}}}{(\epsilon\kappa)^{d}\left(4\pi^2+1\right)^{\frac{d}{2}}}\int_{0}^{\epsilon\kappa} r^{d-2+2s}dr\geq c(\epsilon\kappa)^{2s}e^{-\tau(t)(\epsilon\kappa)^{2s}},
\end{aligned}
$$
thus
$$
\langle v-\mathfrak{u},\psi\rangle\geq c(\epsilon\kappa)^{2s}\int^\infty_0\tau(t)e^{-\tau(t)(\epsilon\kappa)^{2s}}e^{-\int^t_0\lambda(\vec{0},s)ds}dt \gtrsim (\epsilon\kappa)^{2s}\int^\infty_0t^{2s+1}e^{-\alpha t^{2s+1}(\epsilon\kappa)^{2s}}e^{-\beta t}dt
$$
for some $\alpha,\beta>0$ depending on $\|\sigma\|_\infty$ and $\|\lambda\|_{\infty}$. Performing the change of variables $\rho = t(\epsilon\kappa)^{2s/(2s+1)}$, we get
\begin{equation*}
\begin{aligned}
(\epsilon\kappa)^{2s}\int^\infty_0t^{2s+1}e^{-\alpha t^{2s+1}(\epsilon\kappa)^{2s}}e^{-\beta t}dt &\geq (\epsilon\kappa)^{-\frac{2s}{2s+1}}\int^{\infty}_0\rho^{2s+1}e^{-\alpha \rho^{2s+1}-\beta\rho}d\rho.
\end{aligned}
\end{equation*}
which is a worse lower bound since $\epsilon\kappa>(\epsilon\kappa)^{-\frac{2s}{2s+1}}$ for all $s\in(0,1)$ when $\epsilon\kappa>1$.
Since the requirement on the Lipschitz constant for $\psi$ is $\text{Lip}(\psi)\leq \kappa$ (with $\kappa>\epsilon^{-1}$), by taking $\psi_1(x) = e^{-\epsilon^{-1}|x'|}$ and repeating previous computations we arrive to the stronger lower bound $\langle v-\mathfrak{u},\psi\rangle = O(1).$
\end{proof}
\section{Approximation via superposition of pencil-beams}\label{sec:superposition}
Let $f\in L^1(Q)$ be such that
\begin{equation*}
f\geq 0\quad\text{and}\quad\text{supp}(f)\Subset {\mathbb R}^d\times{\mathbb S}^{d-1}.
\end{equation*}
We denote by $\{U_{y,\eta}(X,V)\}_{y,\eta}$ a continuous family of pencil-beams indexed by the parameters $(y,\eta)$, so that $U_{y,\eta}$ is the solution to the fractional Fermi pencil-beam equation \eqref{fFpb} with initial condition $G=\delta(X')\delta(V - \epsilon^{-1}\mathcal{S}(\eta))$, extended by zero for $X^d<0$ (the source $F$ is always taken to be null), and for diffusion and absorption coefficients
\begin{equation*}
\widetilde{\sigma}(X^d) = \sigma(y+X^d\eta)\quad\text{and}\quad \widetilde{\lambda}(X^d) = \lambda(y+X^d\eta).
\end{equation*}
The construction of the approximation via pencil beams is based on the superposition of the $U_{y,\eta}$'s. We define their respective affine transformations as
\begin{equation*}
\mathfrak{u}(\cdot,\cdot ; y,\eta) := (2\epsilon)^{-2(d-1)}U_{y,\eta}\circ T^\epsilon_y,\quad (y,\eta)\in \text{\rm supp\hspace{.1em}}(f),
\end{equation*}
where each $T^\epsilon_{y,\eta}:{\mathbb R}^d\times{\mathbb S}^{d-1}\ni (x,\theta)\mapsto (X,V)\in {\mathbb R}^d\times{\mathbb R}^{d-1}$ encodes an affine transformation of the spatial coordinates plus a rescaled stereographic projection of the angular variables. They are given by
\begin{equation*}
T^\epsilon_{y,\eta}(x,\theta) := \left((2\epsilon)^{-1}\Pi_{\eta^\perp}(x-y), \eta\cdot(x-y),\epsilon^{-1}\mathcal{S}_{\eta}(\theta)\right),
\end{equation*}
where $\Pi_{\eta^\perp}x := (\text{Id}-\eta\eta^T)x$ is the orthogonal projection of $x$ onto the subspace $\eta^\perp$, while $S_{\eta}$ denotes the stereographic projection of the unit sphere that takes $\eta$ to the origin in ${\mathbb R}^{d-1}$ (defined similarly as in Section \ref{sec:intro}).
The approximation via pencil-beams is then defined as
\begin{equation}\label{def:general_approx}
\mathfrak{u}(x,\theta) := \int_{Q}f(y,\eta)\mathfrak{u}(x,\theta;y,\eta)dyd\eta,
\end{equation}
which acts in the distributional sense according to the rule
\begin{equation*}
\langle\mathfrak{u},\phi\rangle := \int_{Q}\int_Qf(y,\eta)\mathfrak{u}(x,\theta;y,\eta)\phi(x,\theta)dxd\theta dyd\eta, \quad \phi\in C_0(Q).
\end{equation*}
We give once again the statement of the approximations results for the superposition of beams, as formulated in Section \ref{subsec:main_results}. These results follows almost directly from the narrow beam case.
\begin{theorem*}[Theorem \ref{thm:cont_sup}]
The same conclusion of Theorem \ref{thm:narrow_beam} holds for a source $f$ satisfying
\begin{equation*}
f\in L^1(Q),\quad f\geq 0\quad\text{and}\quad\text{supp}(f)\Subset {\mathbb R}^d\times{\mathbb S}^{d-1}.
\end{equation*}
and $\mathfrak{u}$ given by the continuous superposition of pencil-beam in \eqref{def:general_approx}.
\end{theorem*}
\begin{proof}
Let $\psi$ be an arbitrary test function in $BL_{1,\kappa}({\mathbb R}^{d}\times{\mathbb S}^{d-1})$ and $\varphi$ the solution to the backward Fokker--Planck system. We see that
\begin{equation*}
\begin{aligned}
\int_Q\psi(x,\theta)\left(u-\mathfrak{u}\right)dxd\theta &= \int_{Q}f(y,\eta)\varphi(y,\eta)dyd\eta\\
&\quad - \int_{Q}f(y,\eta)\left(\int_Q\mathfrak{u}(x,\theta;y,\eta)\psi(x,\theta)dxd\theta \right)dyd\eta.
\end{aligned}
\end{equation*}
Therefore, the estimate for the 1-Wasserstein distance between the Fokker--Planck solution and the superposition of pencil-beams follows by noticing that
\begin{equation*}
\left|\int_Q\mathfrak{u}(x,\theta;y,\eta)\psi(x,\theta)dxd\theta - \varphi(y,\eta)\right|\lesssim \epsilon^{2s'}\kappa^{s'}
\end{equation*}
as a consequence of Lemma \ref{lemma:approx_fFpb}, and for a constant uniform with respect to $(y,\eta)$ but depending on $\|f\|_{L^1}$, $\|\lambda\|_{C^1}$ and $\|\sigma\|_{C^1}$.
Regarding the upper and lower bounds of the distance between the ballistic transport solution (i.e. $v(x,\theta)$) and the pencil-beam approximation, we again use the explicit expression of $v$ given in \eqref{ballistic_sol}, and see that
\begin{equation*}
\begin{aligned}
\int_Q\psi(x,\theta)\left(v-\mathfrak{u}\right)dxd\theta &= \int_Q\int^{\infty}_0 e^{-\int^t_0\lambda(y-s\eta)ds}f(y-t\eta,\eta)\psi(y,\eta)dtdyd\eta\\
&\quad - \int_{Q}f(y,\eta)\left(\int_Q\mathfrak{u}(x,\theta;y,\eta)\psi(x,\theta)dxd\theta \right)dyd\eta\\
&= \int_Q f(y,\eta)\int^{\infty}_0e^{-\int^t_0\lambda(y+s\eta)ds}\psi(y+t\eta,\eta)dtdyd\eta\\
&\quad - \int_{Q}f(y,\eta)\left(\int_Q\mathfrak{u}(x,\theta;y,\eta)\psi(x,\theta)dxd\theta \right)dyd\eta.
\end{aligned}
\end{equation*}
Therefore, the upper and lower bounds arise from the analysis of the difference
\begin{equation}\label{diff_lb}
\int^{\infty}_0e^{-\int^t_0\lambda(y+s\eta)ds}\psi(y+t\eta,\eta)dt - \int_Q\mathfrak{u}(x,\theta;y,\eta)\psi(x,\theta)dxd\theta.
\end{equation}
For each pair $(y,\eta)$, a coordinates rotation plus passing to stretched coordinates allow us to simplify the second integral and get
\begin{equation*}
\int_0^\infty\int_{{\mathbb R}^{2(d-1)}}U_{y,\eta}(X',t,V)\Psi(X',t,V) \frac{dX'dVdt}{\langle \epsilon V\rangle^{2(d-1)}},
\end{equation*}
for $\Psi$ defined by the relation $\varphi(x,\theta) = \Psi\circ T^\epsilon_{y,\eta}(x,\theta)$.
Following the proof of Theorem \ref{thm:ball_vs_pb} it is not hard to realize that
\begin{equation*}
\int_0^\infty\int_{{\mathbb R}^{2(d-1)}}U_{y,\eta}(X',t,V)\Psi(X',t,V) \frac{dX'dVdt}{\langle \epsilon V\rangle^{2(d-1)}} = \int^\infty_0e^{-\int^t_0\lambda(y+s\eta)ds}\Psi(\vec{0},t,\vec{0})dt + O((\epsilon\kappa)^{2s'}),
\end{equation*}
where we take $s'\in(0,s)$ for $s\in(0,1/2]$ and $s'=1/2$ for $s\in(1/2,1)$, and with the remainder independent of the values of $y$ and $\eta$. By noticing that
\begin{equation*}
\Psi(\vec{0},t,\vec{0}) = \psi(y+t\eta,\eta),\footnotemark
\end{equation*}
\footnotetext{If $t=\eta\cdot(x-y)$, then $0 = (\text{Id} -\eta\eta^T)(x-y) = (x-y) - t\eta$, this is, $x=y+t\eta$.}
the upper bound follows directly.
The same computations performed in the proof of Theorem \ref{thm:ball_vs_pb} to obtain a lower bound can be carried out here, in order to bound from below the differences in \eqref{diff_lb} uniformly with respect to $y$ and $\eta$.
\end{proof}
We end this article by stating the analogous result in the case of a discrete superposition of pencil-beams. Given a spread out source $f$ as in the previous theorem, which we assume to be bounded, we can approximate it in the 1-Wasserstein sense as a finite sum of delta sources. Indeed, there exists a simple function $g = \sum_{i=1}^Ia_i\chi_{R_i}$, $a_i\geq 0$ and with $\chi_{R_i}$ the characteristic function for the open set $R_i\subset Q$, which without loss of generality can be taken of diameter $|R_i| \leq \epsilon^2$, so that $\int|f-g|d xd\theta <\epsilon^2\kappa$. We set $\widetilde{a}_i = a_i\text{meas}(R_i)$, with $\text{meas}(R_i)$ denoting the measure of $R_i$ with respect to $dxd\theta$. Then,
\begin{equation*}
\mathcal{W}^1_\kappa(f,\widetilde{f})\lesssim \epsilon^{2}\kappa,\quad\widetilde{f}(x,\theta):= \sum_{i=1}^I \widetilde{a}_i\delta_{x_i}(x)\delta_{\theta_i}(\theta),
\end{equation*}
since for any $\psi\in BL_{1,\kappa}$,
\begin{equation*}
\begin{aligned}
\left|\int \psi(f-\widetilde{f})dxd\theta\right|&\leq \left|\int \psi(f-g)dxd\theta\right|+\left|\int \psi(g-\widetilde{f})dxd\theta\right|\\
&\leq \|\psi\|_\infty\epsilon^2\kappa + \sum^I_{i=1}a_i\left|\int_{R_i}\left(\psi(x,\theta) - \psi(x_i,\theta_i)\right)dxd\theta\right|\\
&\lesssim \epsilon^2\kappa + \sum_{i=1}^Ia_i\text{meas}(R_i)|R_i|\kappa = O(\epsilon^2\kappa).
\end{aligned}
\end{equation*}
Considering the same affine transformations defined previously, we set
\begin{equation}\label{def:disc_approx}
\mathfrak{u}(x,\theta) := \sum_{i=1}^I\widetilde{a}_i\cdot\mathfrak{u}(x,\theta;x_i,\theta_i),
\end{equation}
a discrete superposition of pencil beams, where
\begin{equation*}
\mathfrak{u}(\cdot,\cdot ; x_i,\theta_i) := (2\epsilon)^{-2(d-1)}U_{x_i,\theta_i}\circ T^\epsilon_{x_i,\theta_i},
\end{equation*}
with the $U_{x_i,\theta_i}$'s defined as in the continuous case. Following the same steps as in the previous proof, we deduce the following result:
\begin{theorem*}[Theorem \ref{thm:disc_sup}]
The same conclusion of the Theorem \ref{thm:narrow_beam} holds for a source $f$ satisfying
\begin{equation*}
f\in L^1(Q)\cap L^\infty(Q),\quad f\geq 0\quad\text{and}\quad\text{supp}(f)\Subset {\mathbb R}^d\times{\mathbb S}^{d-1}.
\end{equation*}
and $\mathfrak{u}$ given by the discrete superposition of pencil-beam in \eqref{def:disc_approx}.
\end{theorem*}
\bigskip
\section*{Acknowledgment}
This research was partially supported by the Office of Naval Research, Grant N00014-17-1-2096 and by the National Science Foundation, Grant DMS-1908736.
\bigskip
\begin{appendix}
\section{Proof of Theorem \ref{thm:wellposedness_sRTE}}\label{appdx:1}
Let $f\in L^2({\mathbb R}^d\times{\mathbb S}^{d-1})$ and $u_\Omega \in W^2_\Omega$ the unique solution to
\begin{equation}\label{sRTE_bc}
\left\{\begin{array}{ll}
\theta\cdot\nabla_x u +\lambda u = \mathcal{I}(u)+f,& \text{in }\Omega\times{\mathbb S}^{d-1},\\
u=0,&\text{on }\Gamma_-,
\end{array}\right.
\end{equation}
which is non-negative whenever $f\geq 0$. Since the scattering kernel is integrable and assumed to be symmetric with respect to its angular entries, the existence and uniqueness of $u_\Omega$ follows from \cite[Chapter XXI-Theorem 4]{DL} provided $\lambda\geq \lambda_0>0$. The non-negativity property can be derived from the proof of Proposition 6 in the same reference.
Moreover, by zero-continuing $u_{\Omega}$ to the whole space, and regardless of $\Omega$, we have the estimates
$$\|u_{\Omega}\|_{L^2({\mathbb R}^{d}\times{\mathbb S}^{n-1})}\leq \|f\|_{L^2({\mathbb R}^n\times{\mathbb S}^{d-1})},$$
thus, the Banach-Alaoglu theorem gives us a limit function $u\in L^2({\mathbb R}^d\times{\mathbb S}^{d-1})$ and a convergent sequence $\{u_{\Omega_{k}}\}_k$, such that $u_{\Omega_k}\to u$ weakly in $L^2$. Moreover, the limit satisfies the same inequality which then gives us uniqueness of solutions. It remains to show that $u\in W^2$ and it satisfies \eqref{sRTE}.
Without loss of generality, we assume $\{\Omega_k\}_k$ is an increasing sequence of open bounded subsets of ${\mathbb R}^d$. Let $\Omega_{k_0}$ be an arbitrary element. For any $\varphi\in C_c^\infty({\mathbb R}^d\times{\mathbb S}^{d-1})$ there exist $k_0>0$ so that $\varphi\in C_c^\infty(\Omega_{k_0}\times{\mathbb S}^{d-1})$, and for $k> k_0$ we have that
$$
\begin{aligned}
&\int_{{\mathbb R}^d\times{\mathbb S}^{d-1}}\big((-\theta\cdot\nabla_x\varphi)u_{\Omega_k} +\lambda\varphi u_{\Omega_k} \big)dxd\theta\\
&\hspace{2em}= \int_{{\mathbb R}^d\times{\mathbb S}^{d-1}}\int_{{\mathbb S}^{d-1}}k(x,\theta',\theta)(u_{\Omega_k}(x,\theta') - u_{\Omega_k}(x,\theta))\varphi(x,\theta)d\theta'dxd\theta + \int_{{\mathbb R}^d\times{\mathbb S}^{d-1}} f\varphi dxd\theta.
\end{aligned}
$$
Therefore, we can take the limit as $k\to\infty$ and obtain
$$
\begin{aligned}
&\int_{{\mathbb R}^d\times{\mathbb S}^{d-1}}\big((-\theta\cdot\nabla_x\varphi)u +\lambda\varphi u \big)dxd\theta\\
&\hspace{2em}= \int_{{\mathbb R}^d\times{\mathbb S}^{d-1}}\int_{{\mathbb S}^{d-1}}k(x,\theta,\theta')(u(x,\theta) - u(x,\theta'))\varphi(x,\theta)d\theta'dxd\theta + \int_{{\mathbb R}^d\times{\mathbb S}^{d-1}} f\varphi dxd\theta.
\end{aligned}
$$
On the other hand, using \eqref{sRTE_bc} we see that $\|(\theta\cdot\nabla_xu_{\Omega_k})\mathds{1}_{\Omega_k}\|_{L^2({\mathbb R}^d\times{\mathbb S}^{d-1})}\leq C_{\lambda,b}\|f\|_{L^2({\mathbb R}^d\times{\mathbb S}^{d-1})}$, thus passing to a subsequence we deduce that $u\in W^2$ and $u$ is a strong solution of \eqref{sRTE}.
We directly deduce that $u\geq 0$ for $f\geq 0$.
\section{Localization and mollification}\label{appdx:loc_moll}
We consider stereographic coordinates on the unit sphere and smooth and compactly supported mollification kernels $\xi(x)$ and $\eta(v)$. The localization and mollification technique is a standard procedure for local operators and we thus focus on the singular integral term in \eqref{fFP_id}. For a fixed pair $(y,v'')\in {\mathbb R}^d\times{\mathbb R}^{d-1}$, we consider the test function $\varphi(x,\theta)$ defined via stereographic coordinates as $[\varphi]_{\mathcal{S}}(x,v) = \chi(y,v'')\xi^\delta(y-x)\eta^\delta(v''-v)\langle v\rangle^{2(d-1)}$ with $\chi$ a smooth compactly supported function, and
$$\xi^\delta(x) = \delta^{-d}\xi(\delta^{-d}x)\quad\text{and}\quad\eta^\delta(x) = \delta^{-(d-1)}\eta(\delta^{-(d-1)}x),$$
with $\delta>0$ small, and $\xi$ and $\eta$ smooth, nonnegative functions, with compact support around their respective origins, and such that $\int\xi(x) dx=\int\eta(v) dv=1$.
In the weak formulation of Fokker--Planck \eqref{weak_sol}, the non-local part corresponds to the integral $\int\sigma(x)\mathcal{B}(u,\varphi)dx$, which in stereographic coordinates rewrites as
\begin{equation*}
\begin{aligned}
&\int\sigma(x)\mathcal{B}(u,\varphi)dx\\
&=\frac{1}{2}\int\int \sigma(x)\int K(\theta',\theta)(u(x,\theta') - u(x,\theta))(\varphi(x,\theta')-\varphi(x,\theta))d\theta'd\theta dx\\
&=\int\int \sigma(x)(-\Delta_v)^{s/2}\left(\frac{u(x,v)}{\langle v\rangle^{d-1-2s}}\right)\cdot (-\Delta_v)^{s/2}\left(\frac{\varphi(x,v)}{\langle v\rangle^{d-1-2s}}\right)dvdx \\
&\quad - c\int\int \sigma(x)u(x,v)\varphi(x,v)\langle v\rangle^{-2(d-1)}dvdx.
\end{aligned}
\end{equation*}
We use the notation $w(x,v)=\frac{u(x,v)}{\langle v\rangle^{d-1-2s}}$. Then, for the test function introduced above, we have that
\begin{equation*}
\begin{aligned}
&\int\sigma(x)\mathcal{B}(u,\varphi)dx\\
&=\chi(y,v'')\int\int \sigma(x)\xi^\delta(y-x)((-\Delta_v)^{s/2}w)(x,v)\int \frac{\eta^\delta(v''-v)\langle v\rangle^{d-1+2s}-\eta^\delta(v''-v')\langle v'\rangle^{d-1+2s}}{|v-v'|^{d-1+s}}dv'dv dx \\
&\quad+ c\chi(y,v'')\int\int\sigma(x)u(x,v)\xi^\delta(y-x)\eta^\delta(v''-v)dxdv\\
&=\chi(y,v'')\langle v''\rangle^{d-1+2s}\int\int \sigma(x)\xi^\delta(y-x)((-\Delta_v)^{s/2}w)(x,v)((-\Delta_v)^{s/2}\eta )(v''-v)dv dx \\
&\quad- \chi(y,v'')\int\int \sigma(x)\xi^\delta(y-x)((-\Delta_v)^{s/2}w)(x,v)\\
&\quad\times \int \frac{\eta^\delta(v''-v)\left(\langle v\rangle^{d-1+2s} - \langle v''\rangle^{d-1+2s}\right)-\eta^\delta(v''-v')\left(\langle v'\rangle^{d-1+2s}-\langle v''\rangle^{d-1+2s}\right)}{|v-v'|^{d-1+s}}dv'dvdx\\
&\quad+ c(\chi\cdot(\xi^\delta\otimes\eta^\delta)*(\sigma u))(y,v'')\\
&=\sigma(y)\chi(y,v'')\langle v''\rangle^{d-1+2s}\int\int \xi^\delta(y-x)((-\Delta_v)^{s/2}w)(x,v)((-\Delta_v)^{s/2}\eta )(v''-v)dv dx \\
&\quad+\chi(y,v'')\langle v''\rangle^{d-1+2s}\int\int (\sigma(x)-\sigma(y))\xi^\delta(y-x)((-\Delta_v)^{s/2}w)(x,v)((-\Delta_v)^{s/2}\eta )(v''-v)dv dx \\
&\quad- \chi(y,v'')\int\int \sigma(x)\xi^\delta(y-x)((-\Delta_v)^{s/2}w)(x,v)\\
&\quad\times \int \frac{\eta^\delta(v''-v')\left(\langle v'\rangle^{d-1+2s} - \langle v''\rangle^{d-1+2s}\right)-\eta^\delta(v''-v)\left(\langle v\rangle^{d-1+2s}-\langle v''\rangle^{d-1+2s}\right)}{|v-v'|^{d-1+s}}dv'dvdx\\
&\quad+ c(\chi\cdot(\xi^\delta\otimes\eta^\delta)*(\sigma u))(y,v'').
\end{aligned}
\end{equation*}
Using the Fourier definition of the fractional (Euclidean) Laplacian, we see that
$$
(-\Delta_v)^{s}(f*g) = \mathfrak{F}^{-1}(|k|^{s}\mathfrak{F}(f)\cdot |k|^{s}\mathfrak{F}(g)) = \big((-\Delta_v)^{s}f\big)*\big((-\Delta_v)^{s}g\big).
$$
Thus, the first term on the right-hand side is equivalent to
\begin{equation*}
\sigma(y)\langle v''\rangle^{d-1+2s}\chi(y,v'')(-\Delta_v)^{s}\left(\xi^\delta\otimes\eta^\delta*w\right)(y,v'').
\end{equation*}
However, we would like to obtain an expression for $\langle v\rangle^{-(d-1-2s)}(\chi\cdot(\xi^\delta\otimes\eta^\delta*u))$, which involves computing the commutators between convolution and multiplication by $\langle v\rangle^{-(d-1-2s)}$, and between the fractional Laplacian and multiplication by $\chi$. We have
\begin{equation*}
\eta*(fg) = f(\eta*g) + \int\eta(v')g(v-v')(f(v-v')-f(v))dv'
\end{equation*}
and also that
\begin{equation*}
(-\Delta_v)^s(fg) = f(-\Delta_v)^s(g) + g(-\Delta_v)^s(f).
\end{equation*}
Then, denoting $\widetilde{u}(x,v)=\chi(x,v)\cdot (\xi^\delta\otimes\eta^\delta*u
)(x,v)$ and $\widetilde{w}(x,v) = \frac{\widetilde{u}(x,v)}{\langle v\rangle^{d-1-2s}}$, we see that
\begin{equation*}
\begin{aligned}
&\chi(y,v'')(-\Delta_v)^{s}\left(\xi^\delta\otimes\eta^\delta*w\right)(y,v'')\\
&= ((-\Delta_v)^{s}\widetilde{w})(y,v'') - \frac{(\xi^\delta\otimes\eta^\delta*u
)(y,v'')}{\langle v''\rangle^{d-1-2s}}((-\Delta_v)^{s}\chi)(y,v'')\\
&\quad - \int \frac{\left(\frac{(\xi^\delta\otimes\eta^\delta*u
)(y,v)}{\langle v\rangle^{d-1-2s}}-\frac{(\xi^\delta\otimes\eta^\delta*u
)(y,v'')}{\langle v''\rangle^{d-1-2s}}\right)\left(\chi(y,v)-\chi(y,v'')\right)}{|v-v''|^{d-1+2s}}dv\\
&\quad +\chi(y,v'')\left((-\Delta_v)^{s}\int\int \xi^\delta(y-x)\eta^\delta(v-v')u(x,v')\left(\frac{1}{\langle v'\rangle^{d-1-2s}} - \frac{1}{\langle v\rangle^{d-1-2s}}\right)dv'dx \right)(y,v'').
\end{aligned}
\end{equation*}
Bringing the above computations together, we conclude that
\begin{equation*}
\int\sigma(x)\mathcal{B}(u,\varphi)dx = \sigma(y)[(-\Delta_\theta)^s(\widetilde{u})](y,v'') + h(y,v''),
\end{equation*}
with an error $h(y,v'')$ given by
\begin{equation*}
\begin{aligned}
&h(y,v'') \\
&= - \sigma(y)\langle v''\rangle^{d-1+2s}\frac{(\xi^\delta\otimes\eta^\delta*u
)(x,v'')}{\langle v''\rangle^{d-1-2s}}((-\Delta_v)^{s}\chi)(y,v'')\\
&\quad - \sigma(y)\langle v''\rangle^{d-1+2s}\int \frac{\left(\frac{(\xi^\delta\otimes\eta^\delta*u
)(x,v)}{\langle v\rangle^{d-1-2s}}-\frac{(\xi^\delta\otimes\eta^\delta*u
)(x,v'')}{\langle v''\rangle^{d-1-2s}}\right)\left(\chi(y,v)-\chi(y,v'')\right)}{|v-v''|^{d-1+2s}}dv\\
&\quad +\sigma(y)\langle v''\rangle^{d-1+2s}\chi(y,v'')\left((-\Delta_v)^{s}\int\int \xi^\delta(y-x)\eta^\delta(v-v')u(x,v')\zeta_1(v';v)dv'dx \right)(y,v'')\\
&\quad+\chi(y,v'')\langle v''\rangle^{d-1+2s}\int\int (\sigma(x)-\sigma(y))\xi^\delta(y-x)((-\Delta_v)^{s/2}w)(x,v)((-\Delta_v)^{s/2}\eta^\delta )(v''-v)dv dx \\
&\quad- \chi(y,v'')\int\int \sigma(x)\xi^\delta(y-x)((-\Delta_v)^{s/2}w)(x,v)\int \frac{\eta^\delta(v''-v')\zeta_2(v',v'')-\eta^\delta(v''-v)\zeta_2(v,v'')}{|v-v'|^{d-1+s}}dv'dvdx\\
&\quad+ c(\chi\cdot(\xi^\delta\otimes\eta^\delta)*(\sigma u))(y,v'')\\
&=:\sum_{i=1}^6I_i,
\end{aligned}
\end{equation*}
where $\zeta_1(v';v)=\frac{1}{\langle v'\rangle^{d-1-2s}} - \frac{1}{\langle v\rangle^{d-1-2s}}$ and $\zeta_2(v,v'') = \langle v\rangle^{d-1+2s}-\langle v''\rangle^{d-1+2s}$.
We will show that $h\in L^2({\mathbb R}^d\times{\mathbb R}^{d-1};\langle v\rangle^{-2(d-1)}dxdv)$, with norm bounded by the sum of $\|u\|_{L^2}$ and $\|(-\Delta_\theta)^{s/2}u\|_{L^2}$, times a constant factors that is independent of the mollifying parameter $\delta$. We recall that integrating over the unit sphere translates into integration on ${\mathbb R}^{d-1}$ for the measure $\langle v\rangle^{-2(d-1)}dv$. Our main tool throughout the next computations is Young's inequality.
$\bullet$ We start with $I_6$, whose upper bounds is obtained by a simple application of Young's inequality:
\begin{equation*}
\|I_6\|_{L^2({\mathbb R}^{d}\times{\mathbb S}^{d-1})}\lesssim \|\langle v\rangle^{-(d-1)}\chi \cdot\xi^\delta\otimes\eta^\delta*\sigma u\|_{L^2({\mathbb R}^{d}\times{\mathbb R}^{d-1})}\lesssim \|\xi^\delta\otimes\eta^\delta\|_{L^1({\mathbb R}^{d}\times{\mathbb R}^{d-1})}\|u\|_{L^2({\mathbb R}^{d}\times{\mathbb S}^{d-1})}.
\end{equation*}
$\bullet$ Let's recall the equivalence of norms in \eqref{equiv_norms}. For $I_5$, we have that
\begin{equation*}
\|I_5\|_{L^2({\mathbb R}^{d}\times{\mathbb S}^{d-1})} \lesssim C\|(-\Delta_\theta)^{s/2}u\|_{L^2({\mathbb R}^{d}\times{\mathbb S}^{d-1})},
\end{equation*}
with a constant depending on
\begin{equation}\label{Young_1}
\begin{aligned}
&\sup_{v''\in \sup(\chi)}\left\|\langle v''-\cdot\rangle^{d-1-2s}\int \frac{\eta^\delta(v')\zeta_2(v''-v',v'')-\eta^\delta(\cdot)\zeta_2(v''-\cdot,v'')}{|\cdot-v'|^{d-1+s}}dv'\right\|_{L^1({\mathbb R}^{d-1})};\\
&\sup_{v\in {\mathbb R}^{d-1}}\left\|\langle v\rangle^{d-1-2s}\int \frac{\eta^\delta(\cdot-v')\zeta_2(v',\cdot)-\eta^\delta(\cdot-v)\zeta_2(v,\cdot)}{|v-v'|^{d-1+s}}dv'\right\|_{L^1(\text{\rm supp\hspace{.1em}}(\chi))}.
\end{aligned}
\end{equation}
Let us verify that the previous quantities are finite and remain bounded as $\delta\to 0$. After rescaling the integration variables, the first integral takes the form
\begin{equation*}
\int \langle v''-\delta v\rangle^{d-1-2s}\left|\int \frac{\zeta_3(v+z,v'')-\zeta_3(v,v'')}{\delta^s|z|^{d-1+s}}dz\right|dv,
\end{equation*}
where $\zeta_3(v,v'')=\eta(v)\zeta_2(v''-\delta v,v'')=\eta(v)(\langle v''-\delta v\rangle^{d-1+2s} - \langle v''\rangle^{d-1+2s})$. For $|z|<1$,
\begin{equation*}
\begin{aligned}
&|\zeta_3(v+z,v'')-\zeta_3(v,v'')|\\
&\leq |z|\int^1_0|\nabla \zeta_3(v+tz,v'')|dt\\
&\leq \delta|z|\int^1_0\int^1_0|\nabla\eta(v+tz)||(v''-\tau\delta (v+tz))\cdot (v+tz)|\langle v''-\tau\delta (v+tz)\rangle^{d-3+2s}d\tau dt \\
&\quad + \delta |z|\int^1_0|\eta(v+tz)||v+tz|\langle v''-\delta (v+z)\rangle^{d-3+2s}dt.
\end{aligned}
\end{equation*}
Therefore, since $v''\in\text{\rm supp\hspace{.1em}}(\chi)$ and $\eta$ is compactly supported, we have
\begin{equation*}
\begin{aligned}
&\int \langle v''-\delta v\rangle^{d-1-2s}\int_{|z|<1} \frac{\zeta_3(v+z,v'')-\zeta_3(v,v'')}{\delta^s|z|^{d-1+s}}dzdv\\
&\leq \delta^{1-s} \int\int_{|z|<1}\int^1_0\int^1_0 \frac{|\nabla\eta(v+tz)||v+tz|\langle v''-\tau\delta (v+tz)\rangle^{d-2+2s}\langle v''-\delta v\rangle^{d-1-2s}}{|z|^{d-1-(1-s)}}d\tau dtdzdv\\
&\quad + \delta^{1-s} \int\int_{|z|<1} \int^1_0\frac{|\eta(v+tz)||v+tz|\langle v''-\delta (v+z)\rangle^{d-3+2s}\langle v''-\delta v\rangle^{d-1-2s}}{|z|^{d-1-(1-s)}}dt dzdv\\
&\lesssim \delta^{1-s}\int_{|z|<1} \frac{1}{|z|^{d-1-(1-s)}}dz.
\end{aligned}
\end{equation*}
For $|z|\geq 1$ we instead have that
\begin{equation*}
\begin{aligned}
&|\zeta_3(v+z,v'')-\zeta_3(v,v'')|\\
&\leq \delta\eta(v+z)\int^1_0|(v''-t\delta(v+z))\cdot(v+z)|\langle v''-t\delta(v+z)\rangle^{d-3+2s}dt\\
&\quad + \delta\eta(v)\int^1_0|(v''-t\delta v)\cdot v|\langle v''-t\delta v\rangle^{d-3+2s}dt.
\end{aligned}
\end{equation*}
Thus, using again the compactness of the support of $\eta$, for all $v''\in\text{\rm supp\hspace{.1em}}(\chi)$,
\begin{equation*}
\begin{aligned}
&\int \langle v''-\delta v\rangle^{d-1-2s}\int_{|z|\geq1} \frac{\zeta_3(v+z,v'')-\zeta_3(v,v'')}{\delta^s|z|^{d-1+s}}dzdv\\
&\leq \delta^{1-s}\int\int_{|z|\geq 1}\int^1_0 \frac{\eta(v+z)|(v''-t\delta(v+z))\cdot(v+z)|\langle v''-t\delta(v+z)\rangle^{d-3+2s}\langle v''-\delta v\rangle^{d-1-2s}}{|z|^{d-1+s}}dtdzdv\\
&\quad + \delta^{1-s}\int\int_{|z|\geq 1}\int^1_0\frac{\eta(v)|(v''-t\delta v)\cdot v|\langle v''-t\delta v\rangle^{d-3+2s}\langle v''-\delta v\rangle^{d-1-2s}}{|z|^{d-1+s}}dtdzdv\\
&\lesssim \delta^{1-s}\int_{|z|\geq1} \frac{1}{|z|^{d-1+s}}dz.
\end{aligned}
\end{equation*}
Regarding the second quantity in \eqref{Young_1} we see that after a rescaling of all the variables $v,v'$ and $v''$ we get
\begin{equation*}
\int_{\delta\text{\rm supp\hspace{.1em}}(\chi)}\langle \delta v\rangle^{d-1-2s}\left|\int \frac{\eta(v''-v')\zeta_2(\delta v',\delta v'')-\eta(v''-v)\zeta_2(\delta v,\delta v'')}{\delta^s|v-v'|^{d-1+s}}dv'\right|dv''.
\end{equation*}
We enlarge the domain of integration for $v''$ to a fixed ball $B_r$ containing $\text{\rm supp\hspace{.1em}}(\chi)$. Since $\eta$ has compact support, we can find $R>0$ large enough so that $\eta(v''-v')=\eta(v''-v) = 0$ whenever $|v'|,|v|>R$.
Following similar computations as above we deduce the following upper bound for the region $|v|,|v'|<R$:
\begin{equation*}
\begin{aligned}
&\int_{B_r}\langle \delta v\rangle^{d-1-2s}\left|\int_{|v'|<R} \frac{\eta(v''-v')\zeta_2(\delta v',\delta v'')-\eta(v''-v)\zeta_2(\delta v,\delta v'')}{\delta^s|v-v'|^{d-1+s}}dv'\right|dv''\\
&\lesssim \delta^{1-s}\int_{|z|<2R} \frac{1}{|z|^{d-1-(1-s)}}dz.
\end{aligned}
\end{equation*}
In the case of $|v|<R$ and $|v'|>R$, we obtain an upper bound, up to a constant factor, given by
\begin{equation*}
\begin{aligned}
&\int_{B_r}\left|\int_{R/2<|v'|<2R} \frac{\eta(v''-v')\zeta_2(\delta v',\delta v'')-\eta(v''-v)\zeta_2(\delta v,\delta v'')}{\delta^s|v-v'|^{d-1+s}}dv'\right|dv''\\
&+\int_{B_r}\left|\int_{|v'|>2R} \frac{\zeta_2(\delta v,\delta v'')}{\delta^s|v-v'|^{d-1+s}}dv'\right|dv''\\
&\lesssim \delta^{1-s}\left(\int_{|z|<R}\frac{1}{|z|^{d-1-(1-s)}}dz+\int_{|z|>2R}\frac{1}{|z|^{d-1+s}}dz\right),
\end{aligned}
\end{equation*}
and similarly, for $|v|>R/2$ and $|v'|<R$ we get the upper bounds
\begin{equation*}
\begin{aligned}
&\int_{B_r}\left|\int_{|v'|<R} \frac{\eta(v''-v')\zeta_2(\delta v',\delta v'')-\eta(v''-v)\zeta_2(\delta v,\delta v'')}{\delta^s|v-v'|^{d-1+s}}dv'\right|dv''\\
&\lesssim \delta^{1-s}\int_{|z|<3R}\frac{1}{|z|^{d-1-(1-s)}}dz,
\end{aligned}
\end{equation*}
if $R/2<|v|<2R$, and for $|v|>2R$,
\begin{equation*}
\begin{aligned}
&\int_{B_r}\langle \delta v\rangle^{d-1-2s}\left|\int_{|v'|<R} \frac{\eta(v''-v')\zeta_2(\delta v',\delta v'')}{\delta^s|v-v'|^{d-1+s}}dv'\right|dv''\\
&\lesssim \delta^{1-s}\frac{\langle \delta v\rangle^{d-1-2s}}{(|v|-|R|)^{d-1+s}} \;\lesssim\;\delta^{1-s}.
\end{aligned}
\end{equation*}
$\bullet$ We analyze now the $L^2$-norm of $I_4$. Again by Young's inequality we obtain the estimate
\begin{equation*}
\|I_4\|_{L^2({\mathbb R}^{d}\times{\mathbb S}^{d-1})} \lesssim C\|(-\Delta_\theta)^{s/2}u\|_{L^2({\mathbb R}^{d}\times{\mathbb S}^{d-1})},
\end{equation*}
with a constant depending on
\begin{equation}\label{Young_2}
\begin{aligned}
&\sup_{(y,v'')\in \text{\rm supp\hspace{.1em}}(\chi)}\left\|(\sigma(\cdot)-\sigma(y))\xi^\delta(y-\cdot)((-\Delta_v)^{s/2}\eta^\delta)(v''-\cdot)\right\|_{L^1({\mathbb R}^d\times{\mathbb R}^{d-1})};\\
&\sup_{(x,v)\in {\mathbb R}^d\times {\mathbb R}^{d-1}}\left\|(\sigma(x)-\sigma(\cdot))\xi^\delta(\cdot-x)((-\Delta_v)^{s/2}\eta^\delta)(\cdot-v)\right\|_{L^1(\text{\rm supp\hspace{.1em}}(\chi))}.
\end{aligned}
\end{equation}
Both terms above are $O(\delta^{1-s})$, since for all $x,y\in {\mathbb R}^{d}$,
\begin{equation*}
\|(\sigma(\cdot)-\sigma(y))\xi^\delta(y-\cdot)\|_{L^1({\mathbb R}^d)}, \|(\sigma(x)-\sigma(\cdot))\xi^\delta(\cdot-x)\|_{L^1({\mathbb R}^d)}\lesssim \delta
\end{equation*}
and for all $v\in{\mathbb R}^{d-1}$ and $v''\in\text{\rm supp\hspace{.1em}}(\chi)$,
\begin{equation*}
\|((-\Delta_v)^{s/2}\eta^\delta)(v''-\cdot)\|_{L^1({\mathbb R}^{d-1})}, \|((-\Delta_v)^{s/2}\eta^\delta)(\cdot-v)\|_{L^1(\text{\rm supp\hspace{.1em}}(\chi))}\lesssim \delta^{-s}.
\end{equation*}
$\bullet$ We write $I_3$ as follows
\begin{equation*}
\begin{aligned}
I_3 &= \sigma(y)\langle v''\rangle^{d-1+2s}\chi(y,v'')\int \xi^\delta(y-x) \frac{u(x,v')}{\langle v'\rangle^{d-1}}\\
&\quad \times \int\frac{\eta^\delta(v-v')\left(\langle v'\rangle^{2s}-\frac{\langle v'\rangle^{d-1}}{\langle v\rangle^{d-1-2s}}\right)-\eta^\delta(v''-v')\left(\langle v'\rangle^{2s}-\frac{\langle v'\rangle^{d-1}}{\langle v''\rangle^{d-1-2s}}\right)}{|v-v''|^{d-1+2s}}dvdv'dx.
\end{aligned}
\end{equation*}
Thus, we obtain the estimate
\begin{equation*}
\|I_3\|_{L^2({\mathbb R}^{d}\times{\mathbb S}^{d-1})}\leq C\|u\|_{L^2({\mathbb R}^{d}\times{\mathbb S}^{d-1})},
\end{equation*}
for a constant depending on
\begin{equation}\label{Young_3}
\begin{aligned}
&\sup_{v''\in \text{\rm supp\hspace{.1em}}(\chi)}\left\| \int\frac{\eta^\delta(v-\cdot)\left(\langle \cdot\rangle^{2s}-\frac{\langle \cdot \rangle^{d-1}}{\langle v\rangle^{d-1-2s}}\right)-\eta^\delta(v''-\cdot)\left(\langle \cdot\rangle^{2s}-\frac{\langle \cdot\rangle^{d-1}}{\langle v''\rangle^{d-1-2s}}\right)}{|v-v''|^{d-1+2s}}dv\right\|_{L^1({\mathbb R}^{d-1})};\\
&\sup_{v'\in {\mathbb R}^{d-1}}\left\| \int\frac{\eta^\delta(v-v')\left(\langle v'\rangle^{2s}-\frac{\langle v'\rangle^{d-1}}{\langle v\rangle^{d-1-2s}}\right)-\eta^\delta(v''-v')\left(\langle v'\rangle^{2s}-\frac{\langle v'\rangle^{d-1}}{\langle \cdot\rangle^{d-1-2s}}\right)}{|v-\cdot|^{d-1+2s}}dv \right\|_{L^1(\text{\rm supp\hspace{.1em}}(\chi))}.
\end{aligned}
\end{equation}
By rescaling $v,v'$ and $v''$ and after a change of variables the integrand takes the form
\begin{equation*}
\begin{aligned}
&\int\frac{\eta^\delta(v''+z-v')\left(\langle v'\rangle^{2s}-\frac{\langle v'\rangle^{d-1}}{\langle v''+z\rangle^{d-1-2s}}\right)-\eta^\delta(v''-v')\left(\langle v'\rangle^{2s}-\frac{\langle v'\rangle^{d-1}}{\langle v''\rangle^{d-1-2s}}\right)}{|z|^{d-1+2s}}dz\\
& = \frac{1}{2}\int\frac{\zeta_4(v''+z,v')+\zeta_4(v''-z,v')-2\zeta_4(v'',v')}{\delta^{2s}|z|^{d-1+2s}}dz
\end{aligned}
\end{equation*}
for $\zeta_4(v,v') = \eta(v-v')\left(\langle \delta v'\rangle^{2s}-\frac{\langle \delta v'\rangle^{d-1}}{\langle \delta v\rangle^{d-1-2s}}\right)$. We also notice that for $v''\in B_r$ (see definition above), since $\eta$ has compact support, the integrand is compactly supported as a function of $z$ and $v'$. Therefore, we set $R>0$ so that such support is contained $|z|,|v'|<R$. We have that
\begin{equation*}
\begin{aligned}
\left|\zeta_4(v''+z,v')+\zeta_4(v''-z,v')-2\zeta_4(v'',v')\right| \leq \|\nabla^2 \zeta_4\|_\infty|z|^2
\end{aligned}
\end{equation*}
and the desired estimates follow after showing that $\|\nabla^2 \zeta_4\|_\infty\lesssim \delta^2$. With this in mind, we first observe that
\begin{equation*}
\left|\langle \delta v'\rangle^{2s}-\frac{\langle \delta v'\rangle^{d-1}}{\langle \delta v\rangle^{d-1-2s}}\right|\lesssim \delta^2\int^1_0\frac{|v+t(v'-v)|}{\langle \delta(v+t(v'-v))\rangle^{d+1-2s}}|v-v'|dt\lesssim \delta^2,
\end{equation*}
in the support of the integrand. We also have that on the same support,
\begin{equation*}
\left|\nabla_v\left(\langle \delta v'\rangle^{2s}-\frac{\langle \delta v'\rangle^{d-1}}{\langle \delta v\rangle^{d-1-2s}}\right)\right|,\left|\nabla^2_v\left(\langle \delta v'\rangle^{2s}-\frac{\langle \delta v'\rangle^{d-1}}{\langle \delta v\rangle^{d-1-2s}}\right)\right|\lesssim \delta^2.
\end{equation*}
$\bullet$ For $I_2$ we see that for $|v''|>R$ with $R>0$ large enough so the support of $\chi$ is contained in the ball $B_{R/2}\times B_{R/2}$, then
\begin{equation*}
\begin{aligned}
|I_2| &\lesssim \langle v''\rangle^{d-1+2s}\int \frac{\left(\frac{(\xi^\delta\otimes\eta^\delta*u
)(y,v)}{\langle v\rangle^{d-1-2s}}-\frac{(\xi^\delta\otimes\eta^\delta*u
)(y,v'')}{\langle v''\rangle^{d-1-2s}}\right)\chi(y,v)}{(|v''|-R/2)^{d-1+2s}}dv\\
&\lesssim \int_{B_{R/2}}\left(\frac{(\xi^\delta\otimes\eta^\delta*u
)(y,v)}{\langle v\rangle^{d-1-2s}}-\frac{(\xi^\delta\otimes\eta^\delta*u
)(y,v'')}{\langle v''\rangle^{d-1-2s}}\right)dv\\
&\lesssim \int_{B_{R/2}}|(\xi^\delta\otimes\eta^\delta*u
)(y,v)|dv + |(\xi^\delta\otimes\eta^\delta*u
)(y,v'')|,
\end{aligned}
\end{equation*}
and consequently,
\begin{equation*}
\begin{aligned}
&\int_{{\mathbb R}^d\times B_R^c} |I_2(y,v'')|^2\frac{dydv''}{\langle v''\rangle^{2(d-1)}}\\
&\lesssim \int_{B_{R/2}\times B_R^c} \left|\int_{B_{R/2}}|(\xi^\delta\otimes\eta^\delta*u
)(y,v)|dv + |(\xi^\delta\otimes\eta^\delta*u
)(y,v'')|\right|^2\frac{dydv''}{\langle v''\rangle^{2(d-1)}
\lesssim \|u\|_{L^2({\mathbb R}^d\times{\mathbb S}^{d-1})}.
\end{aligned}
\end{equation*}
On the other hand, noticing that
\begin{equation*}
\begin{aligned}
\frac{(\eta^\delta\otimes\eta^\delta*u)}{\langle v\rangle^{d-1-2s}}
&= \int \eta^\delta(v')w(v-v')dv
+ \int \eta^\delta(v-v')u(v')\left(\frac{1}{\langle v\rangle^{d-1-2s}} - \frac{1}{\langle v'\rangle^{d-1-2s}}\right)dv',
\end{aligned}
\end{equation*}
and denoting $\zeta_5(v,v')=\eta^\delta(v-v')\left(\frac{1}{\langle v\rangle^{d-1-2s}} - \frac{1}{\langle v'\rangle^{d-1-2s}}\right)$, we have that
\begin{equation*}
\begin{aligned}
&\int_{{\mathbb R}^d\times B_{R}} |I_2(y,v'')|^2\frac{dydv''}{\langle v''\rangle^{2(d-1)}}\\
&\lesssim \int_{B_{R/2}\times B_{R}}\langle v''\rangle^{4s}\left|\int \frac{\left(\frac{(\xi^\delta\otimes\eta^\delta*u
)( y, v)}{\langle v\rangle^{d-1-2s}}-\frac{(\xi^\delta\otimes\eta^\delta*u
)( y,v'')}{\langle v''\rangle^{d-1-2s}}\right)(\chi( y,v) - \chi( y,v''))}{|v-v''|^{d-1+2s}}dv\right|^2dydv''\\
&\lesssim \int_{B_{R/2}\times B_{R}}\langle v''\rangle^{4s}\left|\int \frac{\left((\xi^\delta\otimes\eta^\delta*w
)( y, v)-(\xi^\delta\otimes\eta^\delta*w
)( y,v'')\right)(\chi( v) - \chi( v''))}{|v-v''|^{d-1+2s}}dv\right|^2dydv''\\
&\quad + \int_{B_{R/2}\times B_{R}}\langle v''\rangle^{4s}\left|\int (\chi( y,v) - \chi(y, v''))\right.\\
&\hspace{3em} \times\left.\frac{ \left(\int \xi^\delta(y-x)\zeta_5(v,v')u(x,v')dxdv' - \int \xi^\delta(y-x)\zeta_5(v'',v')u(x,v')dxdv' \right)}{|v-v''|^{d-1+2s}}dv\right|^2dydv''\\
&\lesssim \|(-\Delta_v)^{s/2}(\xi^\delta\otimes\eta^\delta*w
)\|_{L^2}^2\\
&\quad + \int_{B_{R/2}\times B_{R}}\left|\int\xi^\delta(y-x)\left( \int\int \frac{ \left|\zeta_5(v,v')- \zeta_5(v'',v')\right|}{|v-v''|^{d-1+2s-1}}|u(x,v')|dv' dv\right)dx\right|^2dydv''\\
&\lesssim \|\xi^\delta\otimes\eta^\delta\|_{L^1}^2\|(-\Delta_v)^{s/2}w
\|_{L^2}^2\\
&\quad + \int_{B_{R/2}\times B_{R}}\left|\int\int \frac{ \left|\zeta_5(v,v')- \zeta_5(v'',v')\right|}{|v-v''|^{d-1+2s-1}}|u(y,v')|dv' dv\right|^2dydv''\\
&\lesssim \|(-\Delta_v)^{s/2}w\|_{L^2}^2\\
&\quad + \int_{B_{R/2}\times B_{R}}\left|\int\left(\int \frac{ \left|\zeta_5(v,v')- \zeta_5(v'',v')\right|}{|v-v''|^{d-1+2s-1}}\langle v'\rangle^{d-1}dv\right) \frac{|u(y,v')|}{\langle v'\rangle^{d-1}}dv' \right|^2dydv''\\
&\lesssim \|(-\Delta_\theta)^{s/2}u\|_{L^2}^2 + C\|u\|_{L^2({\mathbb R}^{d}\times{\mathbb S}^{d-1})},
\end{aligned}
\end{equation*}
with $C$ depending on both:
\begin{equation*}
\begin{aligned}
&\sup_{v''\in B_R}\int \int \frac{ \left|\zeta_5(v,v')- \zeta_5(v'',v')\right|}{|v-v''|^{d-1+2s-1}}\langle v'\rangle^{d-1}dv dv';\\
&\sup_{v'\in {\mathbb R}^{d-1}}\int_{B_R} \int \frac{ \left|\zeta_5(v,v')- \zeta_5(v'',v')\right|}{|v-v''|^{d-1+2s-1}}\langle v'\rangle^{d-1}dv dv''.
\end{aligned}
\end{equation*}
We need to make sure that these quantities do not diverge as $\delta\to 0$. Indeed, rescaling all three variables $v,v'$ and $v''$, we get
\begin{equation*}
\begin{aligned}
&\int \int \frac{ \left|\zeta_5(v,v')- \zeta_5(v'',v')\right|}{|v-v''|^{d-1+2s-1}}\langle v'\rangle^{d-1}dv dv'\\
&=\int \int \frac{ \left|\eta(v-v')\left(\frac{1}{\langle \delta v\rangle^{d-1-2s}} - \frac{1}{\langle \delta v'\rangle^{d-1-2s}}\right)- \eta(v''-v')\left(\frac{1}{\langle \delta v''\rangle^{d-1-2s}} - \frac{1}{\langle \delta v'\rangle^{d-1-2s}}\right)\right|}{\delta^{2s-1}|v-v''|^{d-1+2s-1}}\langle \delta v'\rangle^{d-1}dv dv'.
\end{aligned}
\end{equation*}
The numerator above is bounded by
\begin{equation*}
\begin{aligned}
&|v-v''|\int^1_0|\nabla\eta(v''+t(v-v'')-v')|\left|\frac{1}{\langle \delta (v''+t(v-v''))\rangle^{d-1-2s}} - \frac{1}{\langle \delta v'\rangle^{d-1-2s}}\right|\langle \delta v'\rangle^{d-1}dt\\
&+\delta|v-v''|\int^1_0|\eta(v''+t(v-v'')-v')|\frac{\delta\left|v''+t(v-v'')\right|}{\langle \delta(v''+t(v-v''))\rangle^{d+1-2s} }\langle \delta v'\rangle^{d-1}dt\\
&\lesssim \delta|v-v''|\int^1_0 (|\nabla\eta(v''+t(v-v'')-v')| + |\eta(v''+t(v-v'')-v')| )dt,
\end{aligned}
\end{equation*}
which implies that the quantities above are $O(\delta^{2(1-s)})$.
We conclude this section with the estimation of $\|I_1\|_{L^2}$. We see that
\begin{equation*}
\begin{aligned}
\|I_1\|_{L^2}^2&\lesssim \int\left(\frac{(\xi^\delta\otimes\eta^\delta*u
)(y,v'')}{\langle v''\rangle^{d-1}}\frac{((-\Delta_v)^{s}\chi)(y,v'')}{\langle v''\rangle^{d-1-4s}}\right)^2dydv''\\
&\lesssim\left\|\frac{(\xi^\delta\otimes\eta^\delta*u
)(y,v'')}{\langle v''\rangle^{d-1}}\right\|_{L^2}^2\left\|\frac{((-\Delta_v)^{s}\chi)(y,v'')}{\langle v''\rangle^{d-1-4s}}\right\|_{\infty}^2.
\end{aligned}
\end{equation*}
The first term on the left is estimated by noticing that
\begin{equation*}
\begin{aligned}
\left\|\frac{(\xi^\delta\otimes\eta^\delta*u
)(y,v'')}{\langle v''\rangle^{d-1}}\right\|_{L^2}^2&= \left\|\int \frac{\xi^\delta(y-x)\eta^\delta(v''-v)\langle v\rangle^{d-1}}{\langle v''\rangle^{d-1}}\frac{u(x,v)}{\langle v\rangle^{d-1}} dxdv\right\|_{L^2}^2 \leq C \|u\|_{L^2({\mathbb R}^d\times{\mathbb S}^{d-1})}^2,
\end{aligned}
\end{equation*}
for a constant depending on
\begin{equation*}
\sup_{v''\in{\mathbb R}^{d-1}}\int \frac{\eta^\delta(v''-v)\langle v\rangle^{d-1}}{\langle v''\rangle^{d-1}}dv
=\sup_{v''\in{\mathbb R}^{d-1}}\int \frac{\eta(v''-v)\langle \delta v\rangle^{d-1}}{\langle \delta v''\rangle^{d-1}}dv \lesssim \|\eta\|_{L^1};
\end{equation*}
and similarly
\begin{equation*}
\sup_{v\in{\mathbb R}^{d-1}}\int \frac{\eta^\delta(v''-v)\langle v\rangle^{d-1}}{\langle v''\rangle^{d-1}}dv''
\lesssim \|\eta\|_{L^1}.
\end{equation*}
On the other hand,
\begin{equation*}
\begin{aligned}
\left|\frac{((-\Delta_v)^{s}\chi)(y,v'')}{\langle v''\rangle^{d-1-4s}}\right|& = \frac{1}{\langle v''\rangle^{d-1-4s}} \int \frac{|\chi(y,v''+z) + \chi(y,v''-z) - 2\chi(y,v'')|}{|z|^{d-1+2s}}dz\\
&\leq \left(\sup_{v'',z\in{\mathbb R}^{d-1}}\frac{|\nabla^2_z\chi(y,v''+z)|}{\langle v''\rangle^{d-1-4s}}\right)\int\frac{1}{|z|^{d-1-2(1-s)}}dz \\
&\quad + \left(\sup_{v'',z\in{\mathbb R}^{d-1}}\frac{|\chi(y,v''+z)|}{\langle v''\rangle^{d-1-4s}}\right)\int\frac{1}{|z|^{d-1+2s}}dz.
\end{aligned}
\end{equation*}
Therefore, $\left\|\frac{((-\Delta_v)^{s}\chi)(\cdot)}{\langle \cdot\rangle^{d-1-4s}}\right\|_{\infty}<\infty$.
\section{Estimates for $h(v,v')$}\label{appdx:func_h}
We recast $h(v,v')$ as
\begin{equation}\label{ident_h}
\begin{aligned}
h(v,v')=\langle v\rangle\langle v'\rangle - v\cdot v' -1 &= \langle v\rangle\langle v'\rangle - \frac{1}{2}(|v|^2 + |v'|^2) + \frac{1}{2}|v-v'|^2 -1\\
&=\frac{1}{2}|v-v'|^2 - \frac{1}{2}\left(\langle v\rangle - \langle v'\rangle\right)^2.
\end{aligned}
\end{equation}
This directly gives
\begin{equation}\label{ineq1_h}
h(v,v')\leq \frac{1}{2}|v-v'|^2.
\end{equation}
On the other hand, denoting $p(v') = \langle v'\rangle $, one realizes that it satisfies
\begin{equation*}
|\nabla p(v')| \leq \frac{|v'|}{\langle v'\rangle}=\sqrt{1-\frac{1}{\langle v'\rangle^2}}.
\end{equation*}
Therefore,
\begin{equation*}
|p(v') - p(v)| \leq \left(\int^1_0\sqrt{1-\frac{1}{\langle v+t(v'-v)\rangle^2}}dt\right)|v'-v|.
\end{equation*}
If $\rho =|v-v'| >|v| $ we see that
\begin{equation*}
\begin{aligned}
&\int^1_0\sqrt{1-\frac{1}{\langle v+t(v'-v)\rangle^2}}dt \leq \int^1_0\sqrt{1-\frac{1}{1+2(1+t^2)\rho^2}}dt\\
\leq& \frac{\sqrt{2}\rho}{\sqrt{1+2\rho^2}}\int^1_0\sqrt{1+t^2}d
<\left.\left(\frac{t\sqrt{1+t^2}}{2} + \frac{1}{2}\ln\left(t+\sqrt{1+t^2}\right)\right)\right|_{0}^1
=\frac{1}{\sqrt{2}}\left(1-\frac{1}{\sqrt{2}}\ln(1+\sqrt{2})\right),
\end{aligned}
\end{equation*}
which is strictly positive (this follows from the inequality $\ln(1+x)<x$ for $x>0$) and smaller than 1. This implies
\begin{equation*}
|p(v') - p(v)| \leq \frac{1}{\sqrt{2}}\left(1-\frac{1}{\sqrt{2}}\ln(1+\sqrt{2})\right)|v'-v|.
\end{equation*}
Similarly, if now we set $\rho = |v|>|v'-v|$, the same computations leads us to conclude that there is $c\in(0,1)$ such that
\begin{equation*}
|p(v') - p(v)| \leq c|v-v'|,\quad\forall v,v'\in{\mathbb R}^{d-1}.
\end{equation*}
Then, from the above, \eqref{ident_h}, and \eqref{ineq1_h}, we conclude that
\begin{equation}\label{final_ineq_h}
\frac{\beta}{2}|v-v'|^2\leq h(v,v')\leq \frac{1}{2}|v-v'|^2,
\end{equation}
for $\beta=\left(1-\frac{1}{2}\left(1-\frac{1}{\sqrt{2}}\ln(1+\sqrt{2})\right)^2\right)\in(0,1)$.
In addition, $h(v,v+z)$ satisfies the following inequality for $|v|<R$ and $|z|<1$:
\begin{equation}\label{ineq_h}
\begin{aligned}
&\left|\frac{1}{h(v,v+z)^{\alpha}} - \frac{1}{h(v,v-z)^{\alpha}}\right|\\
&\lesssim
\frac{\left|h(v,v+z)^{\alpha-1}+ h(v,v-z)^{\alpha}\right|}{h(v,v+z)^{\alpha}h(v,v-z)^{\alpha-1}}\left| \left(\langle v\rangle - \langle v+z\rangle\right)^2-\left(\langle v\rangle - \langle v-z\rangle\right)^2\right|\\
&\lesssim \frac{1}{|z|^{2(\alpha+1)}}\left| \left(\langle v\rangle - \langle v+z\rangle\right)^2-\left(\langle v\rangle - \langle v-z\rangle\right)^2\right|\\
&\lesssim \frac{1}{|z|^{d-1+2s+2}}\left|\langle v+z\rangle- \langle v-z\rangle\right|\left| 2\langle v\rangle - \langle v+z\rangle - \langle v-z\rangle \right
\ \lesssim \frac{1}{|z|^{d-1+2s-1}},
\end{aligned}
\end{equation}
for a constant depending on $R$.
\section{Subelliptic estimates}\label{appdx:subellipticity}
In this appendix we use results from \cite{A} to obtain subelliptic regularity for a kinetic equation with singular scattering kernel of fractional order. We consider
\begin{equation}\label{kinetic_eq}
\partial_tf+v\cdot\nabla_xf = \int_{{\mathbb R}^{d}}k(t,x,v,z)(f(v+z) - f(v))dz + g,\quad t\in{\mathbb R},\;x\in{\mathbb R}^d, v\in{\mathbb R}^d,
\end{equation}
for a kernel given by
\begin{equation}\label{kinetic_kernel}
k(t,x,v,z)=a(t,x)\frac{\langle v\rangle^s\langle v+z\rangle^s}{h(v,v+z)^{\frac{d-1}{2}+s}},\quad \text{with}\quad h(v,v+z) = \frac{1}{2}|z|^2-\frac{1}{2}(\langle v\rangle - \langle v+z\rangle)^2,
\end{equation}
and with $a(t,x)$ a continuous and positive function, bounded from below by a positive constant.
We prove the following:
\begin{theorem}\label{thm:ext_Alexandre}
Let $f$ be a smooth solution to \eqref{kinetic_eq}-\eqref{kinetic_kernel} with compact support contained in a bounded set $\Omega\subset{\mathbb R}\times{\mathbb R}^d\times B_\rho$ for some small $\rho>0$. Then,
\begin{equation*}
\|(-\Delta_v)^{s}f\|_{L^2} + \|(-\Delta_x)^{\frac{s}{1+2s}}f\|_{L^2}\leq C_{k,\Omega}\left(\|g\|_{L^2}+\|f\|_{L^2}+ \|(-\Delta_v)^{s/2}f\|_{L^2}\right).
\end{equation*}
\begin{proof} We rewrite equation \eqref{kinetic_eq} in the following form
\begin{equation}\label{kinetic_eq2}
\partial_t+v\cdot\nabla_xf + \widetilde{a}(t,x,v)(-\Delta_v)^sf= \widetilde{g},
\end{equation}
with $\widetilde{a}(t,x,v) = \left(\frac{2}{\beta}\right)^{\frac{d-1}{2}+s}\langle v\rangle^{2s}a(t,x)$ and, denoting $\alpha = \frac{d-1}{2}+s$, a source term given by
\begin{equation*}
\begin{aligned}
\widetilde{g}&= g +\langle v\rangle^{s}a(t,x)\int \frac{(f(v+z) - f(v))(\langle v+z\rangle^s - \langle v\rangle^s)}{h(v,v+z)^{\alpha}}dz \\
&\quad+ \langle v\rangle^{2s}a(t,x)\int (f(v+z) - f(v))\left(\frac{1}{h(v,v+z)^{\alpha}} - \frac{1}{(\frac{\beta}{2}|z|^2)^{\alpha}}\right)dz.
\end{aligned}
\end{equation*}
Furthermore, since $f(t,x,v)$ is compactly supported, we can assume without loss of generality that $\widetilde{a}$ is of the form $b^2(t,x,v)\chi(t,x,v)+a_-$, for $b^2(t,x,v)=(\widetilde{a}(t,x,v)-\widetilde{a}_-)$, $\chi$ smooth and compactly supported, and $\widetilde{a}_-=\inf_{t,x,v}\widetilde{a}(t,x,v)>0$ constant. This can be seen by following an application of Jensen's inequality when integrating over the region $|v|>R\gg \rho$. In fact,
\begin{equation*}
\begin{aligned}
\int_{|v|>R}|\langle v\rangle^{2s}(-\Delta_v)^sf |^2dv &= \int_{|v|>R}\langle v\rangle^{4s}\left(\int_{|z|<\rho}\frac{|f(z)|}{|v-z|^{d-1+2s}}dz\right)^2dv\\
&\lesssim \int_{|z|<\rho}|f(z)|^2\left(\int_{|v|>R}\frac{\langle v\rangle^{4s}}{(|v|-\rho)^{2(d-1)+4s}}dv\right)dz
\ \lesssim \|f\|^2_{L^2}.
\end{aligned}
\end{equation*}
We then take a cut-off function $\chi$, with compact support satisfying $\Omega\subset\text{\rm supp\hspace{.1em}}(\chi)\subset{\mathbb R}\times{\mathbb R}^{d-1}\times B_{R}$, and write
\begin{equation*}
\begin{aligned}
\widetilde{a}(t,x,v)(-\Delta_v)^sf &= (\widetilde{a}(t,x,v)-\widetilde{a}_-)(-\Delta_v)^sf +\widetilde{a}_-(-\Delta_v)^sf \\
&= (b^2(t,x,v)\chi(t,x,v)+\widetilde{a}_-)\int\frac{(f(v)-f(v+z))}{|z|^{d-1+2s}}dz\\
&\quad + b^2(t,x,v)(1- \chi(t,x,v))\int\frac{(f(v)-f(v+z))}{|z|^{d-1+2s}}dz.
\end{aligned}
\end{equation*}
Therefore, $f$ satisfies the equation
\begin{equation*}
\partial_t+v\cdot\nabla_xf + (b^2\chi +\widetilde{a}_-)(-\Delta_v)^sf= \widetilde{G},
\end{equation*}
for a source $\widetilde{G}$ given by
\begin{equation*}
G = \widetilde{g} + b^2(t,x,v)(1- \chi(t,x,v))\int\frac{(f(v)-f(v+z))}{|z|^{d-1+2s}}dz,
\end{equation*}
which we claim it satisfies
\begin{equation*}
\|G\|_{L^2}\lesssim \|g\|_{L^2}+\|f\|_{L^2}+\|(-\Delta_v)^{s/2}f\|_{L^2}+\rho \|(-\Delta_v)^{s}f\|_{L^2}.
\end{equation*}
For this, it only remains to estimate the norm of the two integral terms in the definition of $\widetilde{g}$. Let us write $\widetilde{g} = g_0 + g_1+g_2$ with $g_0 = g$.
For the second summand, we see that in the region $|v|>R\gg \rho$, we repeat some computations above in order to get
\begin{equation*}
\begin{aligned}
\int_{|v|>R}|g_1(v)|^2dv &=\int_{|v|>R}\left(\langle v\rangle^sa(t,x)\int\frac{|f(v+z)|(\langle v+z\rangle^s - \langle v\rangle^s)}{|z|^{d-1+2s}}dz\right)^2dv\\
&\lesssim \int_{|z|<\rho}|f(z)|^2\left(\int_{|v|>R}\frac{\langle v\rangle^{4s}}{(|v|-\rho)^{2(d-1)+4s}}dv\right)dz
\ \lesssim \ \|f\|^2_{L^2}.
\end{aligned}
\end{equation*}
For $|v|<R$, the estimate follows by noticing that $g_1(v)$ satisfies
\begin{equation*}
\begin{aligned}
&\left| g_1(v)\right| \lesssim \int_{|z|<1}\frac{|f(v+z)-f(v)|}{|z|^{d-1+2s-1}}dz + \int_{|z|>1}\frac{|f(v+z)-f(v)|}{|z|^{d-1+2s}}dz.
\end{aligned}
\end{equation*}
Therefore, using H\"older's inequality, and Jensen's inequalities for the finite measures $|z|^{-(d-1+2s)}dz$ in $|z|>1$, we obtain
\begin{equation*}
\begin{aligned}
\int_{|v|<R}\left| g_1(v)\right|^2dv &\lesssim \int_{|v|<R}\left(\int_{|z|<1}\frac{|f(v+z)-f(v)|}{|z|^{d-1+2s-1}}dz\right)^2dv + \int_{|v|<R}\left(\int_{|z|>1}\frac{|f(v+z)-f(v)|}{|z|^{d-1+2s}}dz\right)^2dv\\
&\lesssim \int_{|v|<R}\left(\int_{|z|<1}\frac{|f(v+z)-f(v)|^2}{|z|^{d-1+2s}}dz\right)\left(\int_{|z|<1}\frac{1}{|z|^{d-1-(1-s)}}\right)dv \\
&\quad + \int_{|v|<R}\int_{|z|>1}\frac{|f(v+z)-f(v)|^2}{|z|^{d-1+2s}}dzdv
\ \lesssim \ \|(-\Delta_v)^{s/2}f\|^2_{L^2}+\|f\|^2_{L^2}.
\end{aligned}
\end{equation*}
We now estimate $g_2(v)$, which we rewrite as
\begin{equation}\label{g2}
\begin{aligned}
g_2(v)=&\frac{\langle v\rangle^{2s}a(t,x)}{2}\int (f(v+z) + f(v-z)- 2f(v))\left(\frac{1}{h(v,v+z)^{\alpha}} - \frac{1}{(\frac{1}{2}|z|^2)^{\alpha}}\right)dz\\
&-\frac{\langle v\rangle^{2s}a(t,x)}{2}\int (f(v+z) - f(v))\left(\frac{1}{h(v,v+z)^{\alpha}} - \frac{1}{h(v,v-z)^{\alpha}}\right)dz.
\end{aligned}
\end{equation}
We then split the integration in two and write $g_2(v) = g_2^{>}(v)+g_2^{<}(v)$, with each summand associated to the respective region of integration $|z|>1$ and $|z|<1$. The integrals in $g_2^{>}(v)$ are easily bounded by $\|f\|_{L^2}$ and it thus remains to analyze $g_2^{<}(v)$.
We next decomposes $g_2^{<}(v) = g_{2,1}^{<}(v)+g_{2,2}^{<}(v)$ following \eqref{g2}. We notice that for $g_{2,2}^{<}(v)$ we can repeat the trick of restricting the estimate to $|v|<R$ for some $R\gg \rho$, since otherwise, the compact support of $f$ allows to estimate the $L^2$-norm of those integrals by $\|f\|_{L^2}$. Then, the remaining part (i.e. for $|v|<R$) is bounded by a constant factor times
\begin{equation*}
\begin{aligned}
&\int_{|z|<1} |f(v+z) - f(v)|\left|\frac{1}{h(v,v+z)^{\alpha}} - \frac{1}{h(v,v-z)^{\alpha}}\right|dz.
\end{aligned}
\end{equation*}
It follows directly from \eqref{ineq_h} that
\begin{equation*}
|g_{2,2}^{<}(v)|\lesssim \int_{|z|<1}\frac{|f(v+z)-f(v)|}{|z|^{d-1+2s-1}}dz,\quad \text{for all }|v|<R,
\end{equation*}
and consequently
\begin{equation*}
\|g_{2,2}^{<}(v)\|_{L^2}\lesssim \|(-\Delta_v)^{s/2}f\|_{L^2}+\|f\|_{L^2}.
\end{equation*}
Finally for $g_{2,1}^{<}(v)$, we take a smooth cutoff $\chi$ supported inside $\{|v|<2\rho\}$, and $\chi=1$ in $\{|v|\leq \rho\}$, and split the coefficient $\langle v\rangle^{2s}a(t,x) = a_1(t,x,v) + a_2(t,x,v)$, with $a_1(t,x,v)=\chi(v)\langle v\rangle^{2s}a(t,x)$. When estimating the norm of $g_{2,1}^{<}(v)$, we obtain an integral associated to each coefficient $a_1$ and $a_2$. For the latter, the associated integral is easily bounded by $C_\rho\|f\|_{L^2}$ as we have done before (now taking $R=2\rho$), for a constant that increases as $\rho \to 0$. For $a_1$ we have that
\begin{equation*}
\begin{aligned}
&\|\chi(v)g_{2,1}^{<}(v)\|_{L^2}^2 \\
&= \int\left(a_1(t,x,v)\int_{|z|<1} (f(v+z) + f(v-z)- 2f(v))\left(\frac{1}{h(v,v+z)^{\alpha}} - \frac{1}{(\frac{1}{2}|z|^2)^{\alpha}}\right)dz\right)^2dv\\
&=\int\left(\int_{|z|<1} \hat{f}(\eta)(e^{i\eta\cdot z} + e^{-i\eta\cdot z}- 2)\int e^{-i\eta\cdot v}a_1(t,x,v)\left(\frac{1}{h(v,v+z)^{\alpha}} - \frac{1}{(\frac{1}{2}|z|^2)^{\alpha}}\right)dvdz\right)^2d\eta.
\end{aligned}
\end{equation*}
We first notice that due to the compact support of $a_1(t,x,v)$ as a function of $v$, which is contained in the ball $|v|<2\rho$, we have that
\begin{equation*}
\left|\int e^{-i\eta\cdot v}a_1(t,x,v)\left(\frac{1}{h(v,v+z)^{\alpha}} - \frac{1}{(\frac{1}{2}|z|^2)^{\alpha}}\right)dv\right|\lesssim \frac{\rho^2}{|z|^{2\alpha}}.
\end{equation*}
This follows from the fact that the above difference inside the parentheses is bounded (up to a constant factor) by $|v|^2/|z|^{2\alpha}$.
We split the integration in $\|\chi(v)g_{2,2}^{<}(v)\|_{L^2}^2$ into several regions. For $|\eta|<1$, we simply estimate
\begin{equation*}
|e^{i\eta\cdot z} + e^{-i\eta\cdot z}- 2|\lesssim |\eta|^2|z|^2\leq |z|^2,
\end{equation*}
which allows us to bound the integral on this region by a constant times $\|f\|_{L^2}^2$. For $1<|\eta|<1/|z|$, we use the the inequality $|e^{i\eta\cdot z} + e^{-i\eta\cdot z}- 2|\lesssim |\eta|^2|z|^2$, and
\begin{equation*}
|\eta|^2\int_{|z|<1/|\eta|}\frac{1}{|z|^{2\alpha-2}}dz\lesssim |\eta|^{2s}.
\end{equation*}
Finally, for $|\eta|>1/|z|$ and $\delta\in(0,2s)$, we have that $|z|^{-2\alpha}\leq |\eta|^{2s-\delta}/|z|^{d-1+\delta}$ and
\begin{equation*}
|\eta|^{2s-\delta}\int_{1/|\eta|<|z|<1}\frac{1}{|z|^{d-1+\delta}}dz\lesssim |\eta|^{2s-\delta}\frac{1}{t^\delta}\Big|^{t=1}_{t=1/|\eta|}\lesssim |\eta|^{2s}.
\end{equation*}
With the aids of the previous inequalities and Jensen's inequality, we obtain
\begin{equation*}
\begin{aligned}
\|\chi(v)g_{2,1}^{<}(v)\|_{L^2}^2&\lesssim \rho^2\int |\hat{f}(\eta)|^2(1 + |\eta|^{2s})d\eta\lesssim\rho^2\left(\|f\|^2_{L^2}+\|(-\Delta_v)^{s}f\|^2_{L^2}\right).
\end{aligned}
\end{equation*}
In summary, we have deduced the estimate
\begin{equation*}
\|g_2(v)\|_{L^2}\lesssim C_\rho \|f\|_{L^2} + \|(-\Delta_v)^{s/2}f\|_{L^2}+\rho\|(-\Delta_v)^sf\|_{L^2}.
\end{equation*}
We then apply Theorem 1.2 from \cite{A} and obtain
\begin{equation*}
\|(-\Delta_v)^{s}f\|_{L^2} + \|(-\Delta_x)^{\frac{s}{1+2s}}f\|_{L^2}\leq C_k\left(\|\widetilde{G}\|_{L^2}+\|f\|_{L^2}\right),
\end{equation*}
where if $f$ is assumed to have a sufficiently small support, then we can absorb the term $\rho \|(-\Delta_v)^{s}f\|_{L^2}$ in the upper bound with the left hand side and conclude that
\begin{equation*}
\|(-\Delta_v)^{s}f\|_{L^2} + \|(-\Delta_x)^{\frac{s}{1+2s}}f\|_{L^2}\leq C_{k,\rho}\left(\|g\|_{L^2}+\|f\|_{L^2}+\|(-\Delta_v)^{s/2}f\|_{L^2}\right).
\end{equation*}
\end{proof}
\begin{remark}\label{remark_D}
From the previous theorem and its proof we also obtain that
\begin{equation*}
\left\|\int_{{\mathbb R}^{d}}k(t,x,v,z)(f(v+z) - f(v))dz\right\|_{L^2}\lesssim \|f\|_{L^2}+ \|(-\Delta_v)^{s/2}f\|_{L^2}+\|(-\Delta_v)^{s}f\|_{L^2}.
\end{equation*}
\end{remark}
\end{theorem}
\end{appendix}
|
1,116,691,499,235 | arxiv | \section{Copy-paste in detail}
Copy and pasted method introduced in Sec.~PUT REF TO SECTION is a new flexible approach to construct light-cones for arbitrary observers respect to a particular source in an N-body simulation. These method solve the problem that arbitrary lines-of-sight are likely to lie outside of the limits of the simulation.
The program allows three different initializations for the reference system observer cluster/box:observer aligned with the cluster, observer aligned with the center of the box, random line-of-sight. In the first case there will not be any intrinsic tilt since observer and cluster are align, while in the other two cases due to the lack of alignment there will be an intrinsic tilt between the cluster and the line of sight of the observer. This intrinsic tilt means
\section{Summary and conclusions}
This paper addresses the fundamental question of the role of small-scale environment ($\rm <1\,Mpc$) in triggering accretion events onto the supermassive black holes at the nuclear regions of galaxies. We tackle this issue by developing a flexible semi-empirical model that populates the dark-matter halos of cosmological N-body simulations with AGN using observational relations on the incidence of accretion events among galaxies. This zero-order assumption of the model is that the probability of accretion events are independent of the parent halo masses of galaxies, i.e. agnostic to their small-scale environment. Moreover, the observationally derived AGN incidence probabilities adopted by the model are representative of the field galaxy population, i.e. those outside massive groups or clusters. This model is used to predict the fraction of AGN in massive clusters of galaxies at different redshifts and compare against observational results by carefully taking into account observational selections effects. Any differences between model predictions and observations would point to an environmental dependence of AGN triggering mechanisms. Our main findings are
\begin{enumerate}
\item the X-ray AGN fraction in massive clusters are larger than the model prediction at $z\sim1.25$. This points to a strong environmental dependence of AGN triggering at high redshift. Black hole accretion events are promoted in massive halos relative to the model expectation, which in turn represents the field expectation.
\item the model predictions are consistent with the observed AGN fractions of interemediate redshift ($z\sim0.75$) clusters of galaxies. At this redshift it appears that massive halos do not promote or suppress AGN activity relative to the field predictions of the model.
\item at low redshift, $z\sim0.2$, the model overpredicts the fraction of $L_X(\rm 2-10\,keV) >10^{43} \, erg \, s^{-1}$ AGN in clusters compared to observations. This suggests a suppression of AGN activity in clusters relative to the field expectation of the model at $z\sim0.2$.
\item overall the points above suggest a differential redshift evolution of the AGN fraction in clusters relative to the field predictions of our semi-empirical model.
\end{enumerate}
\noindent The observed trends above may be related to the increasing gas content of galaxies with increasing redshift, coupled with mechanisms such as galaxy interactions or ram-pressure. Both of these processes under certain conditions could promote AGN activity among galaxies that fall onto massive clusters at higher redshift.
\section{Discussion}\label{sec:discussion}
In this work we study the fraction of AGN in massive clusters with a semi-empirical modelling technique that allows the generation of realistic AGN mock catalogues. Dark matter haloes from cosmological simulations are seeded with galaxies using abundance matching techniques \citep[][]{behroozi19_um}. On top of these galaxies accretion events by supermassive black holes are painted. The latter step is based on state-of-the-art specific accretion rate distributions derived from observations \citep[][]{age17_sar,aird18_sar}. The zero-order assumption of the semi-empirical model is that the incidence of accretion events in galaxies is independent of environment, i.e. galaxies are seeded with AGN using the same empirical relations independent of the mass of the parent halo. We refer to the predictions of the model as "field expectation" for the reasons explained in Section~\ref{sec:agn_galax_conec}. This methodology reproduces by construction the halo mass function, stellar mass function and the X-ray luminosity function as demonstrated in Figure~\ref{fig:cube}.
\begin{figure*}
\centering
\includegraphics[width=1\textwidth]{pictures/discus/smf_cube_clustSat.pdf}
\caption{Stellar mass function normalized to the total stellar mass of the sample. Each panel correspond to one of the redshifts $z=0.2, 0.75$ and 1.25. The blue curve is for all galaxies in the corresponding \textsc{UniverseMachine} box and therefore represents the field stellar mass function. The orange color is the stellar mass function of the satellites of the selected clusters (see Section~\ref{sec:selec_eff} for more details). The shaded orange region correspond to the Poisson noise uncertainty.}
\label{fig:smf_field_clust}
\end{figure*}
\subsection{AGN fractions in high redshift clusters}
A striking result in Figure~\ref{fig:fractions} is the higher observed fraction of AGN in massive clusters at $z=1.25$ compared to the model predictions. The largest discrepancy of nearly 1\,dex is for powerful AGN with $L_X(\rm 2-10\,keV) > 10^{44} \, erg \, s^{-1}$. An enhanced fraction is also observed for moderate luminosity AGN, $L_X(\rm 2-10\,keV) > 10^{43} \, erg \, s^{-1}$, but the fact that the observations can only place a lower limit does not allow firm conclusions on the amplitude of the effect. These findings can be attributed to systematic differences between field and cluster either in the AGN specific accretion rate distributions or the stellar mass function of the galaxy population. The former option could be interpreted as evidence for environmental dependence of the AGN triggering efficiency. The latter would indicate differences in the galaxy populations as a function of position on the cosmic web.
Observations indicate that the shape of the total (i.e. independent of galaxy type or SFR) stellar mass function of galaxies does not strongly depend on environment out to $z\approx1.5$ \citep[e.g.][]{Vulcani2013, Nantais2016}. This behaviour is also reproduced in the \textsc{UniverseMachine} semi-empirical model. This is demonstrated in Figure \ref{fig:smf_field_clust} that compares the (average) stellar mass function of mock galaxies in the same massive clusters used in our analysis with that of all galaxies in the \textsc{UniverseMachine} box. The mass functions in Figure \ref{fig:smf_field_clust} are normalised so that their integral yields the same stellar mass density. This allows direct comparison of the mass function shapes, which are remarkably similar between massive clusters and the full box. The latter is dominated by field galaxies (i.e. not associated with massive halos) and by construction reproduces the observed stellar mass function at different redshifts estimated using extragalactic survey fields \citep[e.g.][]{muzzinsmf2013,ilbert13_smf,Moustakas_2013}. The construction of AGN mocks using the seeding process described in section \ref{sec:agn_galax_conec} is primarily sensitive to the shape of the galaxy mass function. The evidence above therefore suggests that the difference between observations and semi-empirical model predictions in Figure \ref{fig:fractions} cannot be attributed to systematic variations of the total stellar mass function with environment.
We acknowledge differences between field and cluster mass functions for star-forming and quiescent galaxies \citep[e.g.][]{Vulcani2013, Nantais2016,papovich18_smf,vanderBurg20}, in the sense that dense environments host a larger fraction of quenched galaxies. Nevertheless, such variations are second order effect in our empirical AGN-seeding model. This is shown in Figure \ref{fig:fractions}, where the predictions using the \cite{aird18_sar} specific accretion rate distribution model that includes star-formation dependence and those based on the \cite{age17_sar} specific accretion rate distribution (no SFR-dependence) are similar at $z=1.25$.
The higher fraction of AGN in $z\ga1$ clusters compared to the field expectation in Figure \ref{fig:fractions} contradicts the results of \cite{martini13}, who report similar fractions. The field AGN fraction of \cite{martini13} is estimated from annular regions centered on the clusters with inner and outer radii of 2 and 6\,arcmin respectively. These regions may include filaments, infalling groups and generally dense structures associated with the nodes of the cosmic web where the cluster is found. They may therefore not be entirely representative of the true field. Relevant to this point are recent results by \cite{koulouridis19} who studied the radial distribution of massive galaxy clusters at $z\approx1$. They report a statistically significant overdensity of X-ray selected AGN within the infall cluster region at a project distance of $\sim2-2.5 \,R_{500}$ relative to the cluster center. This interval lies within the region used by \cite{martini13} to determine their field AGN fractions. The enhanced number of AGN found by \cite{koulouridis19} could bias high the field AGN fractions estimated by \cite{martini13}. Our findings are consistent with the higher fraction of infrared selected AGN in massive clusters of galaxies at $z\ga1$ reported by \cite{alberts16}. We caution nevertheless that the selection function of that sample is very different from that of \cite{martini13}. \cite{alberts16} identifies AGN by fitting model templates to the multi-wavelength SEDs of mid-infrared selected cluster members.
\subsection{Reproducing high-$z$ AGN fractions in clusters}
It is interesting to speculate on specific accretion rate distributions that reproduce the \cite{martini13} AGN fractions in massive clusters of galaxies at $z\approx1.25$ shown in Figure \ref{fig:fractions}. A distribution is required that produces more luminous AGN compared to the current model. There are clearly many functional forms that could achieve that. For simplicity we approach this problem by assuming a Gaussian for the specific accretion rate distribution with parameters (mean, scatter) free to vary. We caution that this problem has a broad range of non-unique solutions. This is ultimately related to the limited observational constraints that are not sufficient to break degeneracies among model parameters. There are essentially only two data points, one of which an upper limit, on the cluster AGN fraction at $z\approx1.25$ shown in Figure \ref{fig:fractions}. It is nevertheless, instructive to explore parameter combinations (mean, scatter) that yield AGN fractions consistent with the observations. In practice we assume a Gaussian specific accretion rate distribution with a given mean/scatter that is independent of stellar mass or SFR. This is applied to \textsc{UniverseMachine} galaxies (i.e. similar to Section \ref{sec:agn_galax_conec}) to produce light-cones of clusters (Section \ref{sec:lc}) on which selection effects are applied (Section \ref{sec:selec_eff}). The resulting AGN factions are then required to be within the $1\,\sigma$ uncertainty of the \cite{martini13} $L_X>\rm 10^{44}\, erg \, s^{-1}$ data point or larger than the lower limit for $L_X>\rm 10^{43}\, erg \, s^{-1}$ AGN in Figure \ref{fig:fractions}. The general trend is that broader distributions (larger scatter) require lower means $\lambda_{sBHAR}$ to reproduce the data. We also find that Gaussians with mean of $\lambda_{sBHAR} \ga10^{-1}$ produce too many luminous AGN for any scatter value and are therefore not allowed by the observations. Similarly a mean value of $\lambda_{sBHAR}\la10^{-3}$ produce too few luminous AGN for any scatter value and are also rejected.
Figure~\ref{fig:gauss_models} (left panel) shows a selected number of Gaussian specific accretion rate distributions which produce AGN fractions consistent with the observations. These distributions have more power at intermediate specific accretion rates ($10^{-2}\la \lambda_{sBHAR} \la 10^{-1}$) compared to the \cite{age17_sar} and \cite{aird18_sar} models that represent the field AGN specific accretion rate distributions. The Gaussian specific accretion rate models plotted in Figure~\ref{fig:gauss_models} make different predictions on the number of cluster AGN as a function of accretion luminosity. This is demonstrated in the middle panel of Figure~\ref{fig:gauss_models}, which plots the predicted cluster AGN X-ray luminosity functions for the different Gaussian specific accretion rate distribution models. The cluster XLF prediction is different in both shape and normalisation from the field one, also plotted in Figure~\ref{fig:gauss_models}. Future observations that provide a broad luminosity baseline and sufficient AGN number statistics can help constrain the models by e.g. directly measuring the XLF as a function of environment. We have also confirmed that even if massive clusters in the \textsc{UniverseMachine} box are assigned the higher normalisation XLFs shown in Figure~\ref{fig:gauss_models}, the total AGN XLF averaged across environments is consistent with observations. This is because massive clusters are rare as a result of the shape of the Halo Mass Function and therefore have a minor contribution to the mean XLF of the \textsc{UniverseMachine} box.
Finally, the difference between field and cluster specific accretion rates plotted in the left panel of Figure~\ref{fig:gauss_models} can be the result of varying black hole Eddington ratio distributions and/or duty cycles as a function of environment. These are the physical quantities that convolve to yield the specific accretion rate distribution \citep[e.g.][]{Shankar2020,allevato21}. The analysis presented in this paper cannot identify which of the two quantities (Eddington ratio or duty cycle) is primary driving the differences in the AGN fraction between field and cluster environments. Further work is needed to address this issue and associate the observed differences to physical quantities directly related to the accretion flow onto the supermassive black hole.
\subsection{Evolution of AGN incidence in cluster vs. field
Contrary to the results at $z=1.25$ discussed above, Figure \ref{fig:fractions} shows that at lower redshift, $z=0.75$, the fraction of AGN in massive clusters is consistent with the model predictions (i.e. field expectation, see Section~\ref{sec:agn_galax_conec}) within the observational data uncertainties. This conclusion does not strongly depend on the details of the semi-empirical modelling, e.g. which specific accretion rate distribution is adopted for seeding galaxies with AGN or the type of observational selection effects (mass vs. apparent magnitude cut) applied to the mock sample. At even lower redshift, $z=0.2$ our analysis tentatively suggests a paucity of AGN in cluster galaxies compared to the field. This is mainly driven by the higher (factor 2--3) fraction of AGN with $L_X ( \rm 2 - 10 \, keV ) > 10^{43} \, erg \,s^{-1}$ predicted by the model compared to the observations. For more luminous AGN, $L_X ( \rm 2 - 10 \, keV ) > 10^{44} \, erg \,s^{-1}$, no firm conclusions can be made because the observations only provide an upper limit to the AGN fraction in clusters. We also caution that at this luminosity cut the semi-empirical model that uses the \cite{age17_sar} specific accretion rate distribution is biased high. In this model flavor a large fraction ($\approx10$\%; see Section \ref{sec:selec_eff}) of the massive central galaxies is assigned powerful AGN. This fraction is much higher than the observed incidence of luminous AGN ($\approx 1-3$\%) among the Brightest Cluster Galaxies \cite{yiang18_bcgs}. This discrepancy is ultimately related to the stellar mass dependence of the \cite{age17_sar} empirical specific accretion rates. A stronger such dependence exists in the \cite{aird18_sar} specific accretion rate distribution. As a result this model predicts much lower $L_X ( \rm 2 - 10 \, keV ) > 10^{44} \, erg \,s^{-1}$ AGN fractions in $z=0.2$ clusters.
\begin{figure*}
\centering
\includegraphics[width=1\textwidth]{pictures/discus/gauss_models.pdf}
\caption{Comparison of Gaussian and the observationally-derived (see Section~\ref{sec:agn_galax_conec}) specific accretion rate distribution models for the cluster AGN sample described in Section~\ref{sec:selec_eff} at $z=1.25$. The left panel shows the specific accretion rate distributions that describe the probability of a galaxy hosting an AGN with specific accretion rate $\lambda_{\rm{sBHAR}}$ at $1<z<1.5$ and stellar masses $10^{10.5}<M_\star/M_\odot<10^{11}$. The observationally derived models are the purple \citep{aird18_sar} and green \citep{age17_sar} curves. Shaded region indicate 68\% confidence intervals. The yellow, light brown and dark brown curves correspond to the Gaussian distributions (see text for details) with different means and dispersions as indicated in the legend. The middle panel compares the X-ray luminosity function for the cluster sample, predicted by the specific accretion rate distribution models (lines) with observations \citep[orange points;][]{age17_sar}. The different lines indicate the observationally derived (representative of the field population) specific accretion rate models \citep[solid, ][green and purple respectively, ]{age17_sar,aird18_sar} and the Gaussian specific accretion-rate models, proposed to match the AGN fraction in clusters at this redshift (dashed, with different colors indicating different Gaussian parameters as indicated in the legend). The right panel shows the predicted AGN fraction in clusters at $z=1.25$ following the selection effects as explained in Section~\ref{sec:selec_eff}. The stars (red or blue) are the observationally measured AGN fractions of \citet{martini09, martini13}. Different colours indicate different luminosity cut. Red is for AGN with $L_X (\rm 2-10\, keV)> 10^{43}\,erg\,s^{-1}$ and blue corresponds to $L_{X} \rm (2-10\, keV)> 10^{44}\,erg\,s^{-1}$. The predictions from the observationally derived specicific accretion-rate models in Figure~\ref{fig:fractions} are shown here with horizontal dashed-doted \citep[][]{age17_sar} and solid lines \citep[][]{aird18_sar}. The circle, diamond and plus markers correspond to the predictions of the three Gaussian specific accretion-rate models shown in the left panel with parameters as indicated in the legend.}
\label{fig:gauss_models}
\end{figure*}
Overall the evidence above points to a differential redshift evolution of the incidence of AGN in clusters relative to the field. Massive structures at $z\ga1$ are found to be more efficient in triggering accretion events onto supermassive black holes compared to less dense regions. Such an environmental dependence is not present at $z\approx0.75$ and possibly inverses at low redshift, $z\approx0.2$, in the sense that clusters likely become less active regions for black hole growth compared to the field.
\cite{Eastman2007} also find evidence for differential evolution of the AGN population as a function of environment. They compared the fraction of X-ray selected AGN in cluster of galaxies between $z\approx0.2$ and 0.6. Their analysis suggests an accelerated evolution of the AGN population in dense environments compared to the field between these redshifts. X-ray observations of individual proto-clusters at higher redshift $z\ga2$ find that AGN are more common among galaxies in these environments compare to the field \citep[][]{lehmer09,dNorth10, krishnan17}. These trends suggests that the probability of a galaxy hosting an accretion event depends on its small-scale environment, $\rm <1\,Mpc$, in agreement with our findings.
It is also interesting that observational studies of the Halo Occupation Distribution (HOD) of AGN are broadly consistent with this picture. At low-redshift, $z\la0.2$, the HOD of X-ray AGN is proposed to have satellite fractions that increase with halo mass, albeit less rapidly than the galaxy population \citep[][]{miyaji11,allevato12}. This points to a decreasing fraction of AGN relative to galaxies with increasing halo mass, similar to our conclusions for low redshift clusters. At higher redshift however, studies of the quasi-stellar object projected correlation function find that the quasi-stellar object satellite fractions increase with halo mass similar to galaxies \citep[][]{richardson12,shen13}. The evidence above is therefore consistent with a picture whereby the fraction of AGN in massive clusters evolves with redshift.
\subsection{Physical interpretation}
Next we explore different physical mechanisms that could be responsible for the trends discussed in the previous sections. In that respect it is interesting that the redshift evolution of the fraction of AGN in clusters relative to the field bears similarities to the star-formation properties of cluster member galaxies as a function of cosmic time. In the local Universe clusters are dominated by quiescent galaxies and their integrated SFRs are significantly lower than the field expectation \citep[e.g.][]{chung2011}. This changes however toward higher redshift. The total cluster SFR normalised to halo mass increases with lookback time and possibly catches up with the field at $z>1$ \cite[e.g.][]{Webb2013, Popesso2015, alberts16}. This is also accompanied by an inversion of the SFR vs. density relation \citep{Butcher_Oelmer1984}, whereby cluster cores at $z\ga1$ host large fractions of actively star-forming galaxy populations \cite[e.g.][]{Elbaz2007, alberts16}. The trends above could be explained by the increasing fraction of the cold gas content of galaxies toward higher redshift \citep[e.g.][]{Tacconi2010, Saintonge2013, Santini2014, Gobat2020} and the increasing fraction of galaxy interactions in overdense regions compared to the field. However, it is still unclear if cluster member galaxies at $z\ga1$ are as gas rich as their field counterparts. Stacking analysis at millimeter wavelengths using ALMA continuum observations find that star-forming galaxies in massive clusters at $z\ga1$ are on average significantly more deficient in molecular gas relative to the field \citep{Alberts2022}. This is at odds with CO emission-line observations of galaxies in clusters that typically detect molecular gas fractions comparable or higher than the field \citep[][]{Noble2019, Williams2022}. Despite this discrepancy the general picture emerging from these studies is that star-formation in massive clusters is associated with infalling galaxies during their first passage through the dense region that either manage to retain/replenish their molecular gas \citep[e.g.][]{kotecha22, vulcani18} or have been largely stripped but continue to consume any remaining gas by forming stars before being rapidly quenched \citep[e.g. ``delay then rapid'' scenario][]{wetzel13} .
It is therefore possible that AGN in clusters are also associated with galaxies that fall for the first time into the deep potential well of the overdensity. Relevant to this point is the discovery of an excess of X-ray AGN at the outskirts (2--3\,$R_{500}$) of massive clusters ($M_{500} > 10^{14}\, M_\odot$) at $z\approx1$ \citep{koulouridis19}. This finding points to a direct link between the growth of supermassive black holes and the infall region of massive structures in the cosmic web. Therefore the strong evolution of the AGN fraction in clusters may be linked to the increasing molecular gas content with increasing redshift of the galaxies that fall into massive halos. It is then interesting to speculate on the physical mechanisms that could lead to the {\it differential} evolution of the AGN fraction in clusters relative to the field{\bf, possible processes include:}
\begin{enumerate}
\item Galaxy interactions. These are expected to be more common in dense environments and the outskirts of clusters. Moreover numerical simulations show that the merging rate of dark matter halos as well as the accretion rate onto them increases with redshift \citep[e.g.][]{Gottloeber2001, Fakhouri2008, McBride2009}. Therefore the flow of galaxies from the cosmic web onto massive halos is expected to be higher at earlier times therefore leading to enhanced galaxy interaction rates. The Semi-Analytic Model of \cite{Gatti2016} suggests that galaxy interactions dominate relatively luminous AGN $L_X\ga10^{43} \rm \, erg \, s^{-1}$ in massive halos ($\ga 10^{14}\,M_\odot$) and low/intermediate redshift, $z\la1$. Instead, internal processes, such as disk instabilities, become important for lower luminosity AGN in dense environments. At much higher redshift, $z>2$, the relative contribution of galaxy interactions and internal processes in triggering AGN inverses, with disk instabilities dominating in massive halos. This is because of the higher gas and disk fraction of galaxies at these higher redshift \citep{Gatti2016}.
\item Ram-pressure. Interaction between the Intra-Cluster Medium (ICM) and the cold gas of the galaxy, can make the latter lose angular momentum (without stripping it), hence facilitating its flow to the nuclear regions where it can accrete onto the supermassive black hole. Simulations have shown that this is possible, if the density of the ICM is not very dense (e.g. in the outskirts of clusters) \citep[e.g.][]{marshall18, Ricarte2020}. Observations of galaxies in local clusters with morphological evidence for ongoing ram-pressure stripping indeed reveal a high incidence ($\sim27-51\%$) of AGN optical emission-line signatures \citep{peluso22}. It is further expected that the ICM is more tenuous toward higher redshift and in the cluster outskirts. This is the regime where the physical conditions are more favorable for the ram-pressure to have a positive impact on AGN triggering. Such a scenario could lead to the observed trend of increasing AGN fractions in clusters relative to the field with increasing redshift. Clusters at $z\approx0.2$ have dense ICMs and the ram-pressure is sufficiently strong to strip galaxies off their gas even at the outskirts \citep[e.g.][]{zinger18, arthur19} and hence, perhaps suppress powerful AGN. Instead, at higher redshift the conditions may be appropriate for the ram-pressure to have a positive effect and boost the numbers of luminous AGN. The radial distribution of AGN within massive halos could provide further constraints this scenario \citep[e.g.][]{marshall18}.
\end{enumerate}
In addition to the processes above, simulations also highlight the potential of cosmic filaments in channeling matter and streams of cold gas deep into the potential well of massive overdensities in the Universe \citep[e.g.][]{Keres2005, Dekel2006, Keres2009}, particularly at high redshift, $z\ga1$. Galaxies that fall into clusters along such filaments are shielded from the hot ICM and can therefore retain at least part of their gas reservoirs even close to the cluster core \citep{kotecha22}. This delays the quenching of galaxies by modulating ram-pressure stripping or strangulation and allows the formation of new stars in dense cluster environments. It can be expected that the same process also promotes black hole growth in the galaxies that accrete onto massive halos through filaments. Furthemore \cite{kotecha22} argue that this effect is more pronounced at high redshifts $z=1-3$ since at this epoch the cold flow filaments are expected to have a higher temperature contrast relative to the ICM and cluster halos are typically less massive. Such an evolution pattern is consistent with the higher fraction of AGN in massive clusters at $z\ga1$ compared to lower redshift. Additional processes (e.g. interactions, positive impact of ram-pressure) need to be invoked to explain the difference between the AGN fractions in cluster and field at $z\ga1$.
\section{Introduction}
It is now widely accepted that supermassive black holes are found at the centres of most, if not all, galaxies in the local Universe \citep[][]{Kormendy_Ho2013}. These compact objects are believed to have grown their masses throughout cosmic time via accretion of material from their surroundings \citep[e.g.][]{Soltan1982, Marconi2004, Merloni_Heinz2008}. Such accretion events generate large amounts of energy that can be detected as radiation across the electromagnetic spectrum. The astrophysical sources associated with such events are dubbed Active Galactic Nuclei \citep[AGN, ][]{Padovani2017}. Observational campaigns in the last 20 years aiming at detecting and characterising large samples of AGN have painted a comprehensive picture of the cosmological evolution of this population and have provided a quantitative description of the accretion history of the Universe out to high redshift \citep[e.g.][]{Ueda2014, aird15_xlf, Brandt_Alexander2015}. What remains challenging to understand however, are the physical processes that trigger accretion events onto supermassive black holes and therefore drive the observed black hole growth as a function of redshift. The different supermassive black hole fueling mechanisms proposed in the literature can broadly be grouped into external ({\it ex-situ}) and internal ({\it in-situ}) in nature. The latter are related to the secular evolution of galaxies, e.g. disk instabilities \citep[e.g.][]{Hopkins_Hernquist2006, Gatti2016}, the creation of bars \citep{Cisternas2015}, stellar winds \citep{Ciotti_Ostriker2007, Kauffmann_Heckman2009} or the biased collapse of the baryons in the inner region of the halo \citep[e.g.][]{Lapi2011, Lapi2018}, and could lead to gas inflows toward the central regions of the galaxy and feeding of the central black hole. {\it Ex-situ} processes are those that act on a galaxy from its environment. They include for example, galaxy interactions \citep{DiMatteo2005, Gatti2016}, cold gas inflows \citep{Bournaud2012, DeGraf2017}, or cooling flows in massive clusters \citep[e.g.][]{Fabian1994}. The balance between {\it ex-situ} and {\it in-situ} supermassive black hole fuelling processes likely depends, among others, on redshift, position or the cosmic web and intrinsic galaxy properties, such as gas content and structural parameters.
One approach for exploring the relative importance of the diverse mechanisms above in modulating the growth of supermassive black holes is to study the incidence of AGN in galaxies as a function of e.g. their morphology, star-formation rate (SFR) or position on the cosmic web \citep{Brandt_Alexander2015}. Such investigations can shed light on the conditions that promote or suppress accretion events onto the supermassive black holes of galaxies and make inferences on the physics at play. Environmental studies in particular, i.e. how AGN populate galaxy groups, filaments, clusters and field, could provide information on the balance between {\it in-situ} and {\it ex-situ} process for activating supermassive black holes. This potential have motivated observational studies to characterise AGN populations in different environments. At low redshift ($z\approx0.1$) there is evidence that the fraction of AGN in high density regions is lower compared to the field \citep[e.g.][]{Kauffmann2004, Popesso_Biviano2006, Koulouridis2010, Lopes2017, Mishra_Dai2020}. This may indicate the decreasing incidence of mergers in massive clusters \citep{Popesso_Biviano2006} and/or the impact of processes that strip the gas reservoirs of galaxies and hence, lead to the suppression of their nuclear activity. There are also claims that the AGN radial distribution is skewed to the cluster outskirts relative to the general galaxy population \citep[e.g][]{deSouza2016, Lopes2017}. This finding coupled with suggestions that cluster AGN show high velocity dispersion \citep[e.g][]{Haines2012, Pimbblet2013, Lopes2017} points to a link between accretion events and galaxies that fall onto high density regions from larger scales. Contrary to the findings above there are also observational studies that claim little or no dependence of the AGN fraction on environment at low redshift \citep[e.g.][]{Miller2003, Haggard2010, Pimbblet2013}. At least part of the discrepancy is likely related to selection effects. These include the accretion luminosity threshold adopted in the various studies \citep{Kauffmann2004, Pimbblet2013}, differences between field and cluster environments in the properties of the overall galaxy population (e.g. SFR, morphology) used to determine fractions \citep[e.g.][]{vonderLinden2010, Lopes2017, Man2019} or the methods adopted for selecting AGN (e.g. optical emission lines, X-ray emission, mid-infrared colours).
Outside the local Universe ($z\ga0.1$) there is evidence that the group/cluster environments become more active in terms of black hole growth. The fraction of AGN in such dense regions increases with increasing redshift \citep{martini09, martini13, Bufanda2017} at a rate that appears to be faster than the field AGN evolution \citep{Eastman2007}. At redshift $z\ga1$ the fraction of AGN in clusters is at least as high as the field expectation \citep{martini13, alberts16} suggesting efficient triggering of accretion events. This is possibly associated with the higher incidence of interactions in these environments \citep[e.g.][]{alberts16} and/or the larger cold gas content of galaxies at earlier times \citep[e.g.][]{Tacconi2010} combined with the impact of the ram-pressure experienced by galaxies as they fall into the cluster potential well \citep[][]{Poggianti2017, Ricarte2020}. The evidence above emphasises the role of environment for understanding AGN triggering mechanisms and underlines the need to better constrain the redshift at which the cluster AGN fractions are on par with the field or even exceed it \citep[][]{alberts16}.
In this work we revisit the incidence of AGN in massive clusters of galaxies out to $z\approx1.25$ by developing a semi-empirical modeling approach to interpret observational results from the literature \citep{martini09,martini13}. The feature of our modeling methodology is control over systematics and observational selection effects. We use observationally motivated relations to populate massive dark matter halos extracted from N-body cosmological simulations with AGN and galaxies. These are then used to mimic observations of clusters of galaxies by including in a realistic manner the relevant selection effects, such as cluster membership definition, flux or luminosity cuts, etc. These mocks are then compared with real observations to make inferences on the evolution of the AGN fraction in clusters relative to the field expectation. Section~\ref{sec:obs} presents the observations and selection bias that we attempt to reproduce. Section~\ref{sec:methods} describes the generation of the mock catalogues and the implementation of the different selection effects into the simulations. The comparison of the semi-empirical model predictions with the observations is presented in Section~\ref{sec:results}. Finally, Section~\ref{sec:discussion} discusses the results in the context of AGN triggering mechanisms. We adopt a flat $\Lambda$CDM cosmology with parameters $\Omega_{\rm{m}}$ = 0.307, $\Omega_{\rm{\Lambda}}$= 0.693, $h = 0.678$ consistent with the Planck results \citep{planck16}.
\section*{Acknowledgements}
The authors thank the anonymous referee for their careful reading and their comments of the paper. We are grateful as well to \'Angel Ruiz Camu\~nas for his helpful comments and
advises. This project has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No 860744. AL is partly supported by the PRIN MIUR 2017 prot. 20173ML3WW 002 ‘Opening the ALMA window on the cosmic evolution of gas, stars, and massive black holes’. SB acknowledges the project PGC2018-097585-B-C22, MINECO/FEDER, UE of the Spanish Ministerio de Economia, Industria y Competitividad. This research made use of Astropy,\footnote{http://www.astropy.org} a community-developed core Python package for Astronomy \citep{astropy:2013, astropy:2018}; \texttt{Colossus}\footnote{https://bdiemer.bitbucket.io/colossus/index.html} \citep{diemer18_colossus}, \texttt{Numpy}\footnote{https://numpy.org/} \citep{numpy2011}, \texttt{Scipy}\footnote{https://scipy.org/} \citep{2020SciPy-NMeth}, \texttt{Matplotlib}\footnote{https://matplotlib.org/} \citep{matplotlib2007} and \texttt{HaloMod}\footnote{\url{https://pypi.org/project/halomod/}} \citep{Murray2013, Murray2021}. For the purpose of open access, the authors have applied a CC-BY public copyright license to any author accepted manuscript version arising. \\
\section*{Data Availability}
The data products and relevant code to reproduce the results of this paper are available at \url{https://github.com/IvanMuro/agn_frac_data_release}
\bibliographystyle{mnras}
\section{Methodology}\label{sec:methods}
This section describes our approach for generating mock observations of galaxies and AGN in massive structures of the cosmic web. The starting point of our method are cosmological N-body simulations \citep[e.g. ][]{millennium2006, bolshoi2011, MultiDark2016} that describe the formation and evolution of dark matter halos in the Universe under the influence of gravity. These are coupled with empirical relations that associate dark matter halos with galaxies (galaxy-halo connection). Accretion events associated to supermassive black holes are then painted on top of those galaxies using recent observational results on the incidence of AGN in galaxies (AGN-galaxy connection). The implicit assumption of this latter step is that the probability of galaxies hosting an accretion event does not depend on environment, i.e., halo mass. Light-cones are then generated to mimic real observations of AGNs and galaxies on the cosmic web. These steps above are described in detail in the following sections.
\subsection{Galaxy-halo connection: (Sub-)Halo Abundance matching techniques}
It is well established that the main sites of galaxy formation in the Universe are halos of dark matter. These provide the necessary gravitational potential for the various baryonic physical processes to act and form the luminous structures (i.e. galaxies) we observe. Among the different methods proposed in the literature for associating galaxies (i.e. luminous baryonic matter) with dark matter halos, the semi-empirical approach of abundance matching offers a number of advantages. With relatively small number of parameters, this approach can successfully reproduce observed properties of galaxies such as their stellar masses or SFRs. In the basic implementation of abundance matching it is assumed that most massive halos are associated with the most massive galaxies. This approach yields a relation between dark matter mass and stellar mass as a function of redshift that is in reasonable agreement with observational results \citep[e.g. occupation number, two point correlation function or cross bias, see ][]{vale04_am,kravtsov04_am,tasitsiomi04_am}. Recent implementations of abundance matching techniques include an increasing level of complexity in the way halos are associated with baryonic mass and galaxies. For example, the halo mass vs. stellar mass relation is parameterized by analytic functions allowing for intrinsic scatter \citep[e.g. ][] {behroozi10_scatter, moster10_scatter}, baryonic process such as star-formation in galaxies are modeled using information on the accretion/merger history of halos, diverse observational results (e.g. stellar mass functions, galaxy clustering properties) are used to tune the various model parameters and produce realistic mock galaxy catalogues out to high redshift \citep{behroozi19_um, moster18_emerge}.
In this work we use the \textsc{UniverseMachine} data release 1\footnote{\url{https://www.peterbehroozi.com/data.html}} \citep[][]{behroozi19_um} implemented for the MultiDark PLanck2 \citep[MDPL2, ][]{MultiDark2016} dark matter N-body simulation. We choose to use the MDPL2 because it is one of the largest volume, high resolution and public cosmological simulations. It has a box size of 1\,000 Mpc/$h$, a mass resolution of $1.5\times10^9\,M_\odot/h$ and 3\,840$^3$ ($\sim$57$\cdot$10$^9$) particles. Individual dark matter halos in the MDPL2 are identified using \textsc{Rockstar} \citep{behroozi13_rockstar}. This is a state-of-the-art halo finder that uses both the 6-dimensional phase-space distribution of dark matter particles and temporal information to identify bound structures, i.e. dark matter halos.
\textsc{Rockstar} is efficient in detecting and measuring the properties of both the largest collapsed structures (parent haloes) and sub-structures within them (satellites haloes). The evolution of haloes through cosmic time is tracked in the form of merger trees computed by the code \textsc{Consistent Trees} \citep{behroozi13_consist_trees}. In this work we consider only dark matter halos with at least 100 times the MDPL2 mass resolution, i.e. $M_{\rm peak}>1.5\times10^{11}\,M_\odot/h$. This limit ensures that the inferred properties of dark matter halos, such as their position and total mass are not affected by the finite resolution of the simulations.
\textsc{UniverseMachine} assigns stellar masses (and hence galaxies) to dark matter halos by parameterizing the star formation history (SFH) of individual halos. The SFR in a halo is assumed to be a function of the depth of the halo's potential well, its assembly history and cosmic time. The maximum circular velocity, $v_{\rm max}$, is used as a proxy of the depth of the potential well. The $v_{\rm max}$, corresponds to the circular velocity of the halo when it reaches its historical maximum mass ($M_{\rm peak}$ parameter in the MDPL2 catalogues). The halo assembly history is parametrized by the $v_{\rm max}$ variations ($\Delta v_{\rm max}$) across cosmic time. \textsc{UniverseMachine} therefore assumes a parametric analytic function SFR$(v_{\rm max}, \Delta v_{\rm max}, z)$ to determine the SFR for each halo across cosmic time. Integrating the SFR along the assembly and merger history of a galaxy it is then possible to determine the corresponding stellar mass. The parameter space of SFR$(v_{\rm max}, \Delta v_{\rm max}, z)$ function is explored in an iterative manner by estimating at each step observables (stellar mass functions, UV luminosity functions, the UV–stellar mass relation, specific and cosmic SFRs, galaxy quenched fractions, galaxy auto-correlation functions and the quenched fraction of central galaxies as a function of environmental density) and comparing them with observations at different redshfits. A Monte Carlo Markov Chain (MCMC) approach is used to sample the model parameter space and yield posteriors for the model parameters.
The end product of \textsc{UniverseMachine} are catalogues of dark matter halos, each of which is assigned a galaxy stellar mass and a SFR. By construction the galaxy population is consistent with observations, including the stellar mass function at different redshift, the evolution of the SFR density of the Universe and the Main Sequence of star-formation. In the following we use the ``observed'' \textsc{UniverseMachine} values for the stellar mass and SFR of mock galaxies. These are estimated by adding systematic errors (also free parameters in \textsc{UniverseMachine}) to the ``true'' values to account for observational effects (e.g. Eddington bias). We note however, that our final results and conclusions are not sensitive to this choice. For dark matter haloes we use virial values as defined by \cite{bryan98_deltavir} for mass and radius.
\subsection{AGN-galaxy connection: specific accretion rate distributions} \label{sec:agn_galax_conec}
The assignment of AGN to the \textsc{UniverseMachine} galaxies is also based on empirical relations that associate the probability of a supermassive black hole accretion event to the properties of its host. The relevant observable is the specific accretion rate, $\lambda_{\rm{sBHAR}} \propto L_X({\rm 2-10\,keV}) /M_\star$. In this definition $L_X(\rm 2-10\, keV)$ is the AGN X-ray luminosity in the $2-10$\,keV band and $M_\star$ is the stellar mass of the parent galaxy. The specific accretion rate provides an estimate of how much X-ray luminosity is emitted by the AGN per unit stellar mass of the host galaxy. In this work we choose to scale the ratio $L_X({\rm 2-10\, keV})/M_\star$ as
\begin{equation}
\lambda_{\rm{sBHAR}} = \frac{\rm{k}_{\rm{bol}}}{1.26 \times 10^{38} \times 0.002} \cdot \left(\frac{L_X(2-10\,\rm{keV})}{\rm{erg}\,\rm{s}^{-1}}\right) \cdot \left( \frac{M_\odot}{M_\star}\right).
\label{eq:sar_def}
\end{equation}
\noindent The above equation assumes a Margorrian-type relation between stellar and black hole mass $\rm{M}_{BH} = 0.002 \cdot M_\star$ \citep[][]{marconi03}, an AGN bolometric correction $\rm{k}_{\rm{bol}} = L_{\rm{bol}} / L_X(2-10\,\rm{keV}) = 25$ \citep[][]{elvis94} and the Eddington luminosity of the black hole $1.26\times10^{38}\,\rm{erg\,s^{-1}}$. The scaling factors in Equation~(\ref{eq:sar_def}) make $\lambda_{\rm{sBHAR}}$ resemble an Eddington ratio, i.e. the AGN bolometric luminosity normalised to the Eddington luminosity of the black hole. It is emphasised that the multiplicative constants in Equation~(\ref{eq:sar_def}) do not affect our analysis and the assignment of AGN luminosities to \textsc{UniverseMachine} galaxies.
Large multi-wavelength observational programs have enabled the estimation of stellar masses, X-ray luminosities and hence $\lambda_{\rm{sBHAR}}$ for large samples of AGN \citep{aird12_primus, bongiorno12_cosmos, schulze15, age17_sar}. These observations made possible the determination of the fraction of galaxies at fixed stellar mass that host an accretion event with specific accretion rate $\lambda_{\rm{sBHAR}}$. These fractions can then be turned into specific accretion rate probability distribution functions, P($\lambda_{\rm{sBHAR}}$), which describe the probability of an accretion event with parameter $\lambda_{\rm{sBHAR}}$ in a galaxy. Recent observational studies have measured the specific accretion rate distribution as a function of redshift and host galaxy properties such as stellar mass and SFR \citep[][]{aird18_sar,age17_sar}. In this work we use these two independent estimates of the specific accretion rate distribution.
\cite{age17_sar} combined a number of extragalactic X-ray survey fields with multi-wavelength data to construct a non-parametric model of the specific accretion rate distribution. Their methodology required that the convolution of the P($\lambda_{\rm{sBHAR}}$) with the galaxy stellar mass function yields the observed number of X-ray AGN in bins of luminosity, redshift and stellar mass. \cite{aird18_sar} started with a sample of near-infrared selected galaxies for which stellar masses and SFRs were estimated. X-ray observations were then used to extract the X-ray photons at the positions of individual galaxies. These were then fed into a flexible Bayesian mixture model to determine in a non-parametric manner the corresponding specific accretion rate distribution of star-forming and quiescent galaxies as a function of stellar mass and redshift. Despite differences in the methodology the \cite{age17_sar} and \cite{aird18_sar} constraints on the specific accretion rate distribution are in good agreement \citep[see][]{age17_sar}. Both \cite{age17_sar} and \cite{aird18_sar} measured P$(\lambda_{\rm{sBHAR}})$ as a function of redshift out to $z\approx3$. Figure~\ref{fig:sar} graphically shows examples of the specific accretion rate distributions used in our analysis.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{pictures/methodology/sar_mod.pdf}
\caption{Specific accretion rate distributions that describe the probability of a galaxy hosting an AGN with specific accretion rate $\lambda_{\rm sBHAR}$. The shaded regions correspond to different observational measurements of P($\lambda_{\rm sBHAR}$). The purple colour shows the \citet{aird18_sar} result, where different line styles indicate different galaxy types: star-forming galaxies (solid line) or passive (dash-dotted). The green colour shows the \citet{age17_sar} constraints on the specific accretion are distribution. All curves correspond to the redshift interval $z=0.5-1$ and stellar mass interval $M_\star=10^{10.5}-10^{11}M_\odot$. The extent of the shaded regions correspond to the 68\% confidence interval around the median (bold curves).}
\label{fig:sar}
\end{figure}
Using these distributions we associate AGN X-ray luminosities to galaxies in the \textsc{UniverseMachine} catalogues. This process is done in a probabilistic approach. For each mock galaxy with stellar mass $M_\star$ and redshift $z$ the corresponding specific accretion rate distributions from either \cite{age17_sar} or \cite{aird18_sar} are sampled to draw random $\lambda_{\rm{sBHAR}}$. These are then used to assign X-ray luminosities to individual galaxies by inverting Equation~(\ref{eq:sar_def}). \cite{aird18_sar} provides separate P$(\lambda_{\rm{sBHAR}})$ for star-forming and passive galaxies. In this case we split the \textsc{UniverseMachine} galaxies into these two classes using the relation of \cite{aird18_sar}
\begin{equation}
\log \rm{SFR}_{\rm{cut}} = -8.9 + 0.76 \log \left(\frac{M_\star}{M_\odot}\right) + 2.95 \left( 1 + \it{z} \right).
\label{eq:sfr_cut}
\end{equation}
\noindent At fixed stellar mass and redshift galaxies with star-formation rate above and below $\rm{SFR}_{\rm{cut}}$ are considered star-forming and passive respectively.
The extragalactic survey fields used by \cite{age17_sar} and \cite{aird18_sar} are dominated by low density regions of the cosmic web. Groups and cluster of galaxies, although present in these samples, are subdominant simply because of the form of the halo mass function and the relevantly small FOV of most of the survey fields used. The derived specific accretion rate distributions are therefore representative of the field galaxy population, i.e. those outside massive groups or clusters of galaxies. For this reason in what follows we refer to the predictions of the model as ``field expectation''. The adopted specific accretion rate distributions are agnostic to the parent halo mass of individual galaxies, hence the the zero-order assumption of the model that the incidence of AGN in galaxies is independent of environment.
The final products of the AGN seeding process are MDPL2 cosmological boxes at fixed redshift with galaxies (from \textsc{UniverseMachine}) and AGNs (from random sampling of the $\lambda_{\rm{sBHAR}}$ distribution). Figure~\ref{fig:cube} graphically demonstrates that our semi-empirical methodology by construction reproduces the halo mass function of dark matter haloes, the stellar mass function of galaxies and the AGN X-ray luminosity function. It is also demonstrated that this approach produces AGN mocks with the large-scale clustering ($\ga 1$\,Mpc) consistent with observations \citep{age19_mock, aird21_mock}. Moreover, the AGN duty cycle, defined as the probability of galaxies above a given stellar mass hosting an AGN above a given accretion luminosity, are inherent in the derivation of specific accretion rate distributions above. As a result our AGN and galaxy mocks are consistent with independently derived determinations of the AGN duty cycles \citep[e.g.][]{Goulding2014}. Put differently, the stellar mass function of the mock AGN host galaxies at fixed X-ray luminosity threshold is consistent with observational constraints \citep{Georgakakis2011, age17_sar}.
It is noted that the above methodology for seeding galaxies and halos with AGN is similar to that proposed by \cite{Shankar2020} and \cite{allevato21} for generating mock AGN samples based on the semi-empirical approach. In these studies satellites and central galaxies/AGN can be treated separately by changing their relative duty cycles. In our approach there is no distinction between the two, i.e. it is assumed that both central and satellites are described by the same duty cycle.
\begin{figure*}
\includegraphics[width=1.75\columnwidth]{pictures/methodology/cubes_rast_2d.pdf}\\
\caption{Graphical workflow of the the semi-empirical modeling to construct mock AGN catalogues based on dark matter N-body simulations. The top panels correspond to a 10 Mpc/$h$ slice of a box from MDPL2 cosmological simulation with 1\,000 Mpc/$h$ side size at the snapshot $z=0.75$. The dots represent the positions of dark matter haloes with masses $M_{\rm peak}>10^{12}\,M_\odot/h$ (top left panel), galaxies within these dark matter haloes (top middle panel) and AGN within the same dark matter haloes (top right panel). Only AGN with $L_X\,(2-10\,\rm{keV})>10^{42}\,\rm{erg\,s^{-1}}$ using the specific accretion distribution described in \citet[]{age17_sar}. The construction of AGN mocks proceeds from left to right in this figure. Dark matter halos (black dots in the top left panel) are populated with galaxies (blue points in the top middle panel), using abundance matching \citep[\textsc{UniverseMachine}, ][]{behroozi19_um}. These galaxies are seeded with accretion events following the observationally derived distributions of these events \citep[e.g.][]{age17_sar,aird18_sar}. The key feature of this approach is the reproduction by construction of the predictions and observables shown in the lower set of panels. Lower left panel: halo mass function predicted by theoretical models in orange (HaloMod, \url{https://pypi.org/project/halomod/}) and simulations in blue \citep[MDPL2,][]{MultiDark2016}. Lower center panel: stellar mass function where circles represent observations \citep[][]{muzzinsmf2013,ilbert13_smf} and the dashed curve the semi-empirical model prediction. Lower right panel: X-ray luminosity function where circles represent observations \protect\cite[][]{age17_sar} and the curves represent the two independently derived models using the specific accretion rate distributions of \protect\citet[][ green dot-dashed]{age17_sar} and \protect\citet[][ purple dashed]{aird18_sar}}.
\label{fig:cube}
\end{figure*}
\subsection{Light-cones}\label{sec:lc}
The comparison of the predictions from the simulations with the observations is following the principles of forward modeling. For that the \textsc{UniverseMachine} boxes need to be first projected onto the sky plane to mimic real observations. We assume a box with a XYZ Cartesian coordinate system. This is offset along the Z-axis by the co-moving distance $D_c(z)$ at redshift $z$. The observer is placed at $\rm{Z}=0$\,Mpc$/h$ and on the XY plane. The line-of-sight angle of the observer relative to every galaxy in the box is then estimated. This angle can be split into a right ascension and declination on the unit sphere. The redshift of each object corresponds to its co-moving distance from the observer. The finite FOV of real observations can also be imposed by defining a sight-line from the observer to a given light-cone direction, estimating the angular distances of all mock galaxies relative to this direction and then rejecting the ones with angular distances larger than the adopted FOV.
For the analysis presented in this paper we construct light-cones in the vicinity of clusters. We consider three redshifts $z=0.2, 0.75$ and 1.25, which correspond to the mean redshifts of the cluster samples presented by \cite{martini09, martini13}. We select \textsc{UniverseMachine} boxes with scale factor\footnote{Defined as $a=1/(1+z)$.} 0.4505, 0.5747 and 0.8376 that approximately correspond to each of the redshifts above. For a given box a massive dark matter halo is selected (see Section \ref{sec:selec_eff} for details) and is placed at a co-moving distance $D_c(z_c)$ from the observer, where $z_c$ is the box redshift, i.e. one of 0.2, 0.75 and 1.25. The light-cone to the cluster is then constructed with an opening angle that is defined by the user (see next section for details). The end product are dark matter haloes, mock galaxies and AGN projected on the sky that mimic real observations.
\subsection{Selection effects}\label{sec:selec_eff}
This section describes how observational selection effects are implemented into our simulations to allow comparison with the results of \cite{martini09,martini13} on the fraction of AGN in galaxy clusters. The characteristics of the observations we attempt to mimic can be grouped into three broad categories that relate to the richness/mass of the cluster sample, the galaxy/AGN cluster membership criteria and the apparent brightness or stellar mass of the galaxy/AGN sample. Below we discuss each of them in detail.
\subsubsection{Cluster sample}\label{sec:clust_sample}
We define the cluster sample by adopting a minimum virial mass threshold. \cite{martini09} provide velocity dispersion for their cluster sample as a measure of their masses. However, the \textsc{UniverseMachine} dataset does not include velocity dispersion information and therefore a mapping is required between this parameter and halo mass. For the latter we adopt the analytical relations presented by \cite{munari13_disp_hm_rela} based on N-body simulations. This allows us to associate individual \textsc{UniverseMachine} halos with a velocity dispersion and then threshold on this quantity to mimic the \cite{martini09} cluster sample selection. We choose the lower halo mass limit in such a way that our parent cluster sample reproduces the median velocity dispersion of the \cite{martini09} sample. The adopted virial halo mass limits are $5.7\cdot10^{14}$ M$_\odot/h$ and $3.6\cdot10^{14}$ M$_\odot/h$ for $z=0.2$ and 0.75 respectively. For the high-redshift clusters of \cite{martini13} there is only scattered information on their halo masses. Literature results suggest masses of few times $10^{14}\,M_\odot$, based on dynamical measurements or estimates from X-ray luminosities. For our high-redshift simulations ($z=1.25$) we therefore select halos with virial mass $>3\cdot10^{14}\,M_\odot/\it{h}$ to mimic the cluster sample of \cite{martini13}. Our results and conclusions are not sensitive to this threshold.
At the mass limits above there are 388, 157 and 18 parent halos in the MDLP2/\textsc{UniverseMachine} boxes at redshifts 0.2, 0.75 and 1.25 respectively. These numbers exclude halos close to the box edges, whose volume as defined in observations (see text below for more details), intersects the box boundaries. The rapid decrease in the number of clusters in each sample is because of the strong evolution of the halo mass function with redshift. These clusters are then projected onto the sky as described in Section \ref{sec:lc}. We choose to place the observer at the same (X,Y) position as the cluster with respect to the reference system of the box. This results in light-cones centered on the each of the selected massive halos, with a line-of-sight perpendicular to the (X,Y)-plane of the box. We define the FOV in terms of the virial radius of the cluster.
\subsubsection{Cluster membership}
Although in the simulation box the satellite galaxies associated with a given parent halo are known, we prefer to follow an observational-motivated approach for defining cluster membership based on the projected and radial distances relative to the cluster center. Cluster member candidates are selected to lie within the projected $R_{\rm vir}$ radius of the corresponding halo. This is similar to the selection of \cite{martini09,martini13}. We choose to use $R_{\rm vir}$ instead of $R_{200}$ since they are similar, but the last is not present in \textsc{UniverseMachine} dataset. We checked that this assumption does not affect our final results and conclusions.
The radial distance of clusters member candidates relative to the cluster centre is measured in redshift space as in real observations. Cluster members are those mock galaxies or AGN with redshift difference to the cluster $\Delta z\cdot c \leq 3\sigma_v(1 + z)$, where $z$ is the redshift of the cluster (fixed to be one of the redshifts of interest, i.e. $z=0.2$, 0.75 or 1.25), $c$ represents the speed of light and $\sigma_v$ is the velocity dispersion of the cluster determined using \cite{munari13_disp_hm_rela}. This definition corresponds to the selection criteria adopted by \cite{martini09} for defining cluster members and it is used for the clusters at redshifts $z=0.2$ and 0.75. In the case of \cite{martini13} 3$\sigma_v$ is fixed to $2\,000\, {\rm km/s}$ for all the clusters. This restriction is also adopted in our simulation for the clusters at redshift $z=1.25$. This condition defines the volume of the cluster in terms of its velocity dispersion. Figure~\ref{fig:fov_cuts} shows an example of a simulated observation and demonstrates the impact of selection effects.
\begin{figure}
\centering
\includegraphics[width=.49\textwidth]{pictures/methodology/fov_example_cuts_ang.pdf}
\caption{The light-cone of a massive cluster in \textsc{UniverseMachine} at redshift $z=1.25$ and halo mass $M_{\rm{vir}}=10^{14.51}M_\odot/h$. Large circle marks the virial radius of the cluster, all symbols correspond to \textsc{UniverseMachine} galaxies within the light-cone. Different symbols and colors demonstrate selection effects as described in Section~\ref{sec:selec_eff}. Purple crosses, red empty squares and blue diamonds indicate galaxies that are excluded from the sample because of the selection effects as indicated in the legend. Green circles and orange triangles indicate the final sample of galaxies and AGNs with $L_X(2-10\,\rm keV)>10^{43}$ erg/s after applying all the selection effects.}
\label{fig:fov_cuts}
\end{figure}
\subsubsection{Galaxy/AGN sample selection}\label{sec:sample_select}
\begin{table}
\centering
\caption{Selection effects adopted for defining the mock cluster galaxy and AGN samples.}
\label{tab:sel_effects_values}
\begin{tabular}{c|c|c|c|c|c}
$z_{\rm box}$ & $M_{\rm vir,lim}$ & $m_{\rm lim}$ & band & $M_{\star, \rm lim}$ & L$_{\rm X, lim}$ \\
(adim.)& ($M_\odot/h$) & (mag) & (adim.) & ($M_\odot$) & (erg\,s$^{-1}$)\\
(1) & (2) & (3) & (4) & (5) & (6) \\\hline
1.25 & $3\cdot10^{14}$ & 18.5 & IRAC1 & 2.2$\cdot$10$^{10}$ & $10^{43}$ and $10^{44}$ \\%\cline{1-4}
0.75 & 3.6$\cdot10^{14}$ & 23.3 & R band & 2.2$\cdot$10$^{10}$ & $10^{43}$ and $10^{44}$\\%\cline{1-4}
0.2 & 5.7$\cdot10^{14}$ & 19.1 & R band & 2.2$\cdot$10$^{10}$ & $10^{43}$ and $10^{44}$\\ \hline \hline
\end{tabular}
\begin{list}{}{}
\item (1) Redshift of each cluster corresponding to those of \citet[][]{martini09,martini13}, (2) Minimum dark matter halo mass (virial) adopted to define the parent cluster sample at different redshifts, (3) minimum magnitude threshold adopted to define galaxy/AGN cluster members at different redshifts, (4) Photometric band used to define the magnitude threshold, (5) Stellar mass limit used to select galaxy/AGN cluster members at different redshifts and (6) X-ray luminosity limit adopted to define the AGN sample at different redshifts.
\end{list}
\end{table}
In observations AGN and/or galaxy samples are typically selected above a given apparent magnitude limit. In \cite{martini09,martini13} for example, this is set relative to the knee of the optical and/or mid-infrared luminosity function of galaxies at the corresponding cluster redshift (see Section~\ref{sec:obs}). In simulations however, like the semi-empirical model described in this work, galaxies are defined by their intrinsic properties, such as stellar mass, SFR and accretion luminosity. Associating these physical properties to apparent magnitudes requires assumptions on e.g. the SFH of galaxies, the spectral energy distribution (SED) of stellar populations or the shape and normalization of the dust attenuation curve. Assigning SEDs to simulated galaxies is therefore far from trivial and inevitably requires additional modelling steps \citep[e.g.][]{ueda20,pearl21_remapSFR}
Our baseline model/observation comparison avoids these additional steps. Instead we make the simplifying assumption that the knee of the observed galaxy optical or mid-infrared luminosity function traces the knee of the underlying galaxy stellar mass function, $M^*_\star$. This allows us to translate the $R$-band and $3.6\,\rm \mu$m apparent magnitude limits of $0.4\,L^*$ adopted by \citet[see Section \ref{sec:obs}]{martini09,martini13} to stellar mass cuts. Mock galaxies are selected to be more massive than $\log\,M^*_\star/M_\odot - 0.4\,\rm dex$. The break of the mass function is fixed to $\log\,M^*_\star=10^{10.7}\,M_\odot$ independent of redshift based on the parametrisation of \cite{ilbert13_smf}. Although the \cite{ilbert13_smf} study refers to field galaxies, observations show that the shape of stellar mass function is similar in massive clusters \citep[e.g.][]{Vulcani2013, Nantais2016}. The translation of the $0.4\,L^*$ apparent magnitude limit to a $\log\,M^*_\star/M_\odot-0.4\,\rm dex$ threshold implies the same average mass-to-light ratio for galaxies. This approximation is justifiable in the case of the high redshift cluster sample ($z\approx1.25$) of \cite{martini13}, where galaxies are selected in the $IRAC$ $3.6\,\mu$m band. At the mean cluster redshift, this wavelength roughly corresponds to rest-fame near-infrared ($\approx$ 1.6$\,\mu$m), where the mass-to-light ratio is not a strong function of the galaxy stellar population. We acknowledge that in the low-redshift ($z=0.2$ and 0.75) cluster sample of \cite{martini09}, galaxies are selected in the $R$-band, where variations of the mass-to-light ratio as a function of the star-formation rate are important. For this sample the approximation of a constant mass-to-light ratio is rough and should be taken with caution.
We further address the limitations above by assigning apparent magnitudes to mock galaxies in the light-cones following the methodology described in \cite{ueda20}. \textsc{UniverseMachine} galaxies are assumed to be described by exponentially declining SFH. The parameters of the SFH model are constrained to reproduce the \textsc{UniverseMachine} stellar masses and instantaneous SFRs of the galaxies at their assigned redshifts in the light-cone. The \cite{BruzualCharlot03} stellar library and the \cite{chabrier03} initial mass function are used to synthesize stellar populations for the adopted SFH. The SEDs of star-forming galaxies are extincted by dust assuming the \cite{calcetti2000} law and $E(B - V)$ = 0.4 mag. The magnitudes of passive galaxies are not extincted by dust. This empirical model is shown to reproduce reasonably well the distribution of apparent magnitudes of galaxies in the COSMOS field \citep{muzzinsmf2013}.
The assigned apparent magnitudes are sensitive to the instantaneous SFR of mock galaxies. The empirical model of \cite{ueda20} assumes that star-forming galaxies are on the Main Sequence of star-formation \citep[][]{schreiber+15}, while quiescent galaxies lie $1-1.5$\,dex below it depending on redshift. This offset for the quenched galaxies is empirically determined to reproduce the observed magnitude distribution of passive galaxies as a function of redshift in the COSMOS field. The \textsc{UniverseMachine} star-forming galaxies are also constrained to follow the Main Sequence of star-formation at different redshifts and are therefore consistent with the assumptions of the empirical SED model of \cite{ueda20}. In contrast, the SFR distribution of quenched galaxies is not as well constrained by observations \citep[see discussion by][]{pearl21_remapSFR}. Passive galaxies in \textsc{UniverseMachine} are assigned specific star-formation rates (sSFR) that are drawn from a non-evolving log-normal distribution with mean $\log \rm{sSFR}/\rm{yr}=-11.8$ and scatter 0.36\,dex, motivated by observations in the local Universe \citep[][]{behroozi15}. This SFR is 2\,dex below the main sequence of star-formation at low redshift and therefore inconsistent with the assumptions of the empirical SED model of \cite{ueda20}. This difference in sSFR results in passive galaxies with too faint apparent magnitudes for the empirical model of \cite{ueda20}. In this work we therefore adopt the definition of quenched galaxies given by Equation~\ref{eq:sfr_cut}
but then re-scale the corresponding sSFR according to the \cite{ueda20} prescriptions (i.e. 1$-$1.5\,dex below the main should that be $M_R^*(z=0)$ sequence) before assigning them magnitudes.
Once apparent magnitudes are assigned to mock galaxies in the light-cone, we apply cuts similar to those adopted by \cite{martini09,martini13}, i.e. one magnitude fainter than the break of the luminosity function in the $R$ (low- and medium-redshift cluster samples, $z=0.2$ and 0.75) and 3.6$\,\mu$m (high-redshift cluster samples; $z=1.25$) bands. The magnitude limits for the different samples are summarized in Table~\ref{tab:sel_effects_values}. For the $R$-band in particular, following \cite{martini09} we assume that the knee of the luminosity function evolves with redshift as $M_R(z)=M_R(z=0)+z$ with $M_R^*(z=0)=-21.92$ \citep[][]{christlein03_break_lf}. This absolute magnitude is converted to apparent magnitude in the $R$-band assuming an elliptical galaxy SED \citep[][]{ilbert09_photoRed} for the K-corrections. The estimated knee of the luminosity function in apparent magnitudes is $m_R^*=17.1$, 22.3\,mag at $z= 0.2$ and 0.75 respectively. The limits listed in Table \ref{tab:sel_effects_values} are one magnitude fainter than the apparent magnitude of knee of the optical luminosity function at the relevant redshift. For the 3.6\,$\mu$m band the knee of the luminosity function is assumed to 17.5 mag (apparent magnitude) at $z=1.25$ \citep[][]{mancone10}. Mock galaxies and AGNs in the simulated light-cones are selected to be brighter than the magnitudes limits listed in Table \ref{tab:sel_effects_values}. For the mock AGN we further apply the X-ray luminosity limits of \cite{martini09,martini13} to define two samples with $L_X(\rm 2-10\,keV) > 10^{43}\,\rm{erg \,s^{-1}}$ and $\rm >10^{44}\,\rm{erg \,s^{-1}}$.
In the analysis above we ignore the contribution of AGN light to the observed magnitude of galaxies. Modeling this component requires knowledge of the obscuration properties of the AGN. This latter parameter has the strongest impact on the observed spectral energy distribution of these sources. The specific accretion rate distribution used in this work to seed galaxies with AGN luminosities do not account for the AGN obscuration and therefore cannot be used to model this effect \citep[see also][]{ueda20}. We simplify this problem by ignoring the AGN contribution to the emission of the galaxy in the optical/mid-infrared. This simplification is equivalent to the assumption that mock AGN are completely obscured and hence subdominant relative to the stellar emission of galaxies. We will discuss the impact of this assumption on the results and conclusions in the next sections.
\section{Introduction}
The journal \textit{Monthly Notices of the Royal Astronomical Society} (MNRAS) encourages authors to prepare their papers using \LaTeX.
The style file \verb'mnras.cls' can be used to approximate the final appearance of the journal, and provides numerous features to simplify the preparation of papers.
This document, \verb'mnras_guide.tex', provides guidance on using that style file and the features it enables.
This is not a general guide on how to use \LaTeX, of which many excellent examples already exist.
We particularly recommend \textit{Wikibooks \LaTeX}\footnote{\url{https://en.wikibooks.org/wiki/LaTeX}}, a collaborative online textbook which is of use to both beginners and experts.
Alternatively there are several other online resources, and most academic libraries also hold suitable beginner's guides.
For guidance on the contents of papers, journal style, and how to submit a paper, see the MNRAS Instructions to Authors\footnote{\label{foot:itas}\url{http://www.oxfordjournals.org/our_journals/mnras/for_authors/}}.
Only technical issues with the \LaTeX\ class are considered here.
\section{Obtaining and installing the MNRAS package}
Some \LaTeX\ distributions come with the MNRAS package by default.
If yours does not, you can either install it using your distribution's package manager, or download it from the Comprehensive \TeX\ Archive Network\footnote{\url{http://www.ctan.org/tex-archive/macros/latex/contrib/mnras}} (CTAN).
The files can either be installed permanently by placing them in the appropriate directory (consult the documentation for your \LaTeX\ distribution), or used temporarily by placing them in the working directory for your paper.
To use the MNRAS package, simply specify \verb'mnras' as the document class at the start of a \verb'.tex' file:
\begin{verbatim}
\documentclass{mnras}
\end{verbatim}
Then compile \LaTeX\ (and if necessary \bibtex) in the usual way.
\section{Preparing and submitting a paper}
We recommend that you start with a copy of the \texttt{mnras\_template.tex} file.
Rename the file, update the information on the title page, and then work on the text of your paper.
Guidelines for content, style etc. are given in the instructions to authors on the journal's website$^{\ref{foot:itas}}$.
Note that this document does not follow all the aspects of MNRAS journal style (e.g. it has a table of contents).
If a paper is accepted, it is professionally typeset and copyedited by the publishers.
It is therefore likely that minor changes to presentation will occur.
For this reason, we ask authors to ignore minor details such as slightly long lines, extra blank spaces, or misplaced figures, because these details will be dealt with during the production process.
Papers must be submitted electronically via the online submission system; paper submissions are not permitted.
For full guidance on how to submit a paper, see the instructions to authors.
\section{Class options}
\label{sec:options}
There are several options which can be added to the document class line like this:
\begin{verbatim}
\documentclass[option1,option2]{mnras}
\end{verbatim}
The available options are:
\begin{itemize}
\item \verb'letters' -- used for papers in the journal's Letters section.
\item \verb'onecolumn' -- single column, instead of the default two columns. This should be used {\it only} if necessary for the display of numerous very long equations.
\item \verb'doublespacing' -- text has double line spacing. Please don't submit papers in this format.
\item \verb'referee' -- \textit{(deprecated)} single column, double spaced, larger text, bigger margins. Please don't submit papers in this format.
\item \verb'galley' -- \textit{(deprecated)} no running headers, no attempt to align the bottom of columns.
\item \verb'landscape' -- \textit{(deprecated)} sets the whole document on landscape paper.
\item \verb"usenatbib" -- \textit{(all papers should use this)} this uses Patrick Daly's \verb"natbib.sty" package for citations.
\item \verb"usegraphicx" -- \textit{(most papers will need this)} includes the \verb'graphicx' package, for inclusion of figures and images.
\item \verb'useAMS' -- adds support for upright Greek characters \verb'\upi', \verb'\umu' and \verb'\upartial' ($\upi$, $\umu$ and $\upartial$). Only these three are included, if you require other symbols you will need to include the \verb'amsmath' or \verb'amsymb' packages (see section~\ref{sec:packages}).
\item \verb"usedcolumn" -- includes the package \verb"dcolumn", which includes two new types of column alignment for use in tables.
\end{itemize}
Some of these options are deprecated and retained for backwards compatibility only.
Others are used in almost all papers, but again are retained as options to ensure that papers written decades ago will continue to compile without problems.
If you want to include any other packages, see section~\ref{sec:packages}.
\section{Title page}
If you are using \texttt{mnras\_template.tex} the necessary code for generating the title page, headers and footers is already present.
Simply edit the title, author list, institutions, abstract and keywords as described below.
\subsection{Title}
There are two forms of the title: the full version used on the first page, and a short version which is used in the header of other odd-numbered pages (the `running head').
Enter them with \verb'\title[]{}' like this:
\begin{verbatim}
\title[Running head]{Full title of the paper}
\end{verbatim}
The full title can be multiple lines (use \verb'\\' to start a new line) and may be as long as necessary, although we encourage authors to use concise titles. The running head must be $\le~45$ characters on a single line.
See appendix~\ref{sec:advanced} for more complicated examples.
\subsection{Authors and institutions}
Like the title, there are two forms of author list: the full version which appears on the title page, and a short form which appears in the header of the even-numbered pages. Enter them using the \verb'\author[]{}' command.
If the author list is more than one line long, start a new line using \verb'\newauthor'. Use \verb'\\' to start the institution list. Affiliations for each author should be indicated with a superscript number, and correspond to the list of institutions below the author list.
For example, if I were to write a paper with two coauthors at another institution, one of whom also works at a third location:
\begin{verbatim}
\author[K. T. Smith et al.]{
Keith T. Smith,$^{1}$
A. N. Other,$^{2}$
and Third Author$^{2,3}$
\\
$^{1}$Affiliation 1\\
$^{2}$Affiliation 2\\
$^{3}$Affiliation 3}
\end{verbatim}
Affiliations should be in the format `Department, Institution, Street Address, City and Postal Code, Country'.
Email addresses can be inserted with the \verb'\thanks{}' command which adds a title page footnote.
If you want to list more than one email, put them all in the same \verb'\thanks' and use \verb'\footnotemark[]' to refer to the same footnote multiple times.
Present addresses (if different to those where the work was performed) can also be added with a \verb'\thanks' command.
\subsection{Abstract and keywords}
The abstract is entered in an \verb'abstract' environment:
\begin{verbatim}
\begin{abstract}
The abstract of the paper.
\end{abstract}
\end{verbatim}
\noindent Note that there is a word limit on the length of abstracts.
For the current word limit, see the journal instructions to authors$^{\ref{foot:itas}}$.
Immediately following the abstract, a set of keywords is entered in a \verb'keywords' environment:
\begin{verbatim}
\begin{keywords}
keyword 1 -- keyword 2 -- keyword 3
\end{keywords}
\end{verbatim}
\noindent There is a list of permitted keywords, which is agreed between all the major astronomy journals and revised every few years.
Do \emph{not} make up new keywords!
For the current list of allowed keywords, see the journal's instructions to authors$^{\ref{foot:itas}}$.
\section{Sections and lists}
Sections and lists are generally the same as in the standard \LaTeX\ classes.
\subsection{Sections}
\label{sec:sections}
Sections are entered in the usual way, using \verb'\section{}' and its variants. It is possible to nest up to four section levels:
\begin{verbatim}
\section{Main section}
\subsection{Subsection}
\subsubsection{Subsubsection}
\paragraph{Lowest level section}
\end{verbatim}
\noindent The other \LaTeX\ sectioning commands \verb'\part', \verb'\chapter' and \verb'\subparagraph{}' are deprecated and should not be used.
Some sections are not numbered as part of journal style (e.g. the Acknowledgements).
To insert an unnumbered section use the `starred' version of the command: \verb'\section*{}'.
See appendix~\ref{sec:advanced} for more complicated examples.
\subsection{Lists}
Two forms of lists can be used in MNRAS -- numbered and unnumbered.
For a numbered list, use the \verb'enumerate' environment:
\begin{verbatim}
\begin{enumerate}
\item First item
\item Second item
\item etc.
\end{enumerate}
\end{verbatim}
\noindent which produces
\begin{enumerate}
\item First item
\item Second item
\item etc.
\end{enumerate}
Note that the list uses lowercase Roman numerals, rather than the \LaTeX\ default Arabic numerals.
For an unnumbered list, use the \verb'description' environment without the optional argument:
\begin{verbatim}
\begin{description}
\item First item
\item Second item
\item etc.
\end{description}
\end{verbatim}
\noindent which produces
\begin{description}
\item First item
\item Second item
\item etc.
\end{description}
Bulleted lists using the \verb'itemize' environment should not be used in MNRAS; it is retained for backwards compatibility only.
\section{Mathematics and symbols}
The MNRAS class mostly adopts standard \LaTeX\ handling of mathematics, which is briefly summarised here.
See also section~\ref{sec:packages} for packages that support more advanced mathematics.
Mathematics can be inserted into the running text using the syntax \verb'$1+1=2$', which produces $1+1=2$.
Use this only for short expressions or when referring to mathematical quantities; equations should be entered as described below.
\subsection{Equations}
Equations should be entered using the \verb'equation' environment, which automatically numbers them:
\begin{verbatim}
\begin{equation}
a^2=b^2+c^2
\end{equation}
\end{verbatim}
\noindent which produces
\begin{equation}
a^2=b^2+c^2
\end{equation}
By default, the equations are numbered sequentially throughout the whole paper. If a paper has a large number of equations, it may be better to number them by section (2.1, 2.2 etc.). To do this, add the command \verb'\numberwithin{equation}{section}' to the preamble.
It is also possible to produce un-numbered equations by using the \LaTeX\ built-in \verb'\['\textellipsis\verb'\]' and \verb'$$'\textellipsis\verb'$$' commands; however MNRAS requires that all equations are numbered, so these commands should be avoided.
\subsection{Special symbols}
\begin{table}
\caption{Additional commands for special symbols commonly used in astronomy. These can be used anywhere.}
\label{tab:anysymbols}
\begin{tabular*}{\columnwidth}{@{}l@{\hspace*{50pt}}l@{\hspace*{50pt}}l@{}}
\hline
Command & Output & Meaning\\
\hline
\verb'\sun' & \sun & Sun, solar\\[2pt]
\verb'\earth' & \earth & Earth, terrestrial\\[2pt]
\verb'\micron' & \micron & microns\\[2pt]
\verb'\degr' & \degr & degrees\\[2pt]
\verb'\arcmin' & \arcmin & arcminutes\\[2pt]
\verb'\arcsec' & \arcsec & arcseconds\\[2pt]
\verb'\fdg' & \fdg & fraction of a degree\\[2pt]
\verb'\farcm' & \farcm & fraction of an arcminute\\[2pt]
\verb'\farcs' & \farcs & fraction of an arcsecond\\[2pt]
\verb'\fd' & \fd & fraction of a day\\[2pt]
\verb'\fh' & \fh & fraction of an hour\\[2pt]
\verb'\fm' & \fm & fraction of a minute\\[2pt]
\verb'\fs' & \fs & fraction of a second\\[2pt]
\verb'\fp' & \fp & fraction of a period\\[2pt]
\verb'\diameter' & \diameter & diameter\\[2pt]
\verb'\sq' & \sq & square, Q.E.D.\\[2pt]
\hline
\end{tabular*}
\end{table}
\begin{table}
\caption{Additional commands for mathematical symbols. These can only be used in maths mode.}
\label{tab:mathssymbols}
\begin{tabular*}{\columnwidth}{l@{\hspace*{40pt}}l@{\hspace*{40pt}}l}
\hline
Command & Output & Meaning\\
\hline
\verb'\upi' & $\upi$ & upright pi\\[2pt]
\verb'\umu' & $\umu$ & upright mu\\[2pt]
\verb'\upartial' & $\upartial$ & upright partial derivative\\[2pt]
\verb'\lid' & $\lid$ & less than or equal to\\[2pt]
\verb'\gid' & $\gid$ & greater than or equal to\\[2pt]
\verb'\la' & $\la$ & less than of order\\[2pt]
\verb'\ga' & $\ga$ & greater than of order\\[2pt]
\verb'\loa' & $\loa$ & less than approximately\\[2pt]
\verb'\goa' & $\goa$ & greater than approximately\\[2pt]
\verb'\cor' & $\cor$ & corresponds to\\[2pt]
\verb'\sol' & $\sol$ & similar to or less than\\[2pt]
\verb'\sog' & $\sog$ & similar to or greater than\\[2pt]
\verb'\lse' & $\lse$ & less than or homotopic to \\[2pt]
\verb'\gse' & $\gse$ & greater than or homotopic to\\[2pt]
\verb'\getsto' & $\getsto$ & from over to\\[2pt]
\verb'\grole' & $\grole$ & greater over less\\[2pt]
\verb'\leogr' & $\leogr$ & less over greater\\
\hline
\end{tabular*}
\end{table}
Some additional symbols of common use in astronomy have been added in the MNRAS class. These are shown in tables~\ref{tab:anysymbols}--\ref{tab:mathssymbols}. The command names are -- as far as possible -- the same as those used in other major astronomy journals.
Many other mathematical symbols are also available, either built into \LaTeX\ or via additional packages. If you want to insert a specific symbol but don't know the \LaTeX\ command, we recommend using the Detexify website\footnote{\url{http://detexify.kirelabs.org}}.
Sometimes font or coding limitations mean a symbol may not get smaller when used in sub- or superscripts, and will therefore be displayed at the wrong size. There is no need to worry about this as it will be corrected by the typesetter during production.
To produce bold symbols in mathematics, use \verb'\bmath' for simple variables, and the \verb'bm' package for more complex symbols (see section~\ref{sec:packages}). Vectors are set in bold italic, using \verb'\mathbfit{}'.
For matrices, use \verb'\mathbfss{}' to produce a bold sans-serif font e.g. \mathbfss{H}; this works even outside maths mode, but not all symbols are available (e.g. Greek). For $\nabla$ (del, used in gradients, divergence etc.) use \verb'$\nabla$'.
\subsection{Ions}
A new \verb'\ion{}{}' command has been added to the class file, for the correct typesetting of ionisation states.
For example, to typeset singly ionised calcium use \verb'\ion{Ca}{ii}', which produces \ion{Ca}{ii}.
\section{Figures and tables}
\label{sec:fig_table}
Figures and tables (collectively called `floats') are mostly the same as built into \LaTeX.
\subsection{Basic examples}
\begin{figure}
\includegraphics[width=\columnwidth]{example}
\caption{An example figure.}
\label{fig:example}
\end{figure}
Figures are inserted in the usual way using a \verb'figure' environment and \verb'\includegraphics'. The example Figure~\ref{fig:example} was generated using the code:
\begin{verbatim}
\begin{figure}
\includegraphics[width=\columnwidth]{example}
\caption{An example figure.}
\label{fig:example}
\end{figure}
\end{verbatim}
\begin{table}
\caption{An example table.}
\label{tab:example}
\begin{tabular}{lcc}
\hline
Star & Mass & Luminosity\\
& $M_{\sun}$ & $L_{\sun}$\\
\hline
Sun & 1.00 & 1.00\\
$\alpha$~Cen~A & 1.10 & 1.52\\
$\epsilon$~Eri & 0.82 & 0.34\\
\hline
\end{tabular}
\end{table}
The example Table~\ref{tab:example} was generated using the code:
\begin{verbatim}
\begin{table}
\caption{An example table.}
\label{tab:example}
\begin{tabular}{lcc}
\hline
Star & Mass & Luminosity\\
& $M_{\sun}$ & $L_{\sun}$\\
\hline
Sun & 1.00 & 1.00\\
$\alpha$~Cen~A & 1.10 & 1.52\\
$\epsilon$~Eri & 0.82 & 0.34\\
\hline
\end{tabular}
\end{table}
\end{verbatim}
\subsection{Captions and placement}
Captions go \emph{above} tables but \emph{below} figures, as in the examples above.
The \LaTeX\ float placement commands \verb'[htbp]' are intentionally disabled.
Layout of figures and tables will be adjusted by the publisher during the production process, so authors should not concern themselves with placement to avoid disappointment and wasted effort.
Simply place the \LaTeX\ code close to where the figure or table is first mentioned in the text and leave exact placement to the publishers.
By default a figure or table will occupy one column of the page.
To produce a wider version which covers both columns, use the \verb'figure*' or \verb'table*' environment.
If a figure or table is too long to fit on a single page it can be split it into several parts.
Create an additional figure or table which uses \verb'\contcaption{}' instead of \verb'\caption{}'.
This will automatically correct the numbering and add `\emph{continued}' at the start of the caption.
\begin{table}
\contcaption{A table continued from the previous one.}
\label{tab:continued}
\begin{tabular}{lcc}
\hline
Star & Mass & Luminosity\\
& $M_{\sun}$ & $L_{\sun}$\\
\hline
$\tau$~Cet & 0.78 & 0.52\\
$\delta$~Pav & 0.99 & 1.22\\
$\sigma$~Dra & 0.87 & 0.43\\
\hline
\end{tabular}
\end{table}
Table~\ref{tab:continued} was generated using the code:
\begin{verbatim}
\begin{table}
\contcaption{A table continued from the previous one.}
\label{tab:continued}
\begin{tabular}{lcc}
\hline
Star & Mass & Luminosity\\
& $M_{\sun}$ & $L_{\sun}$\\
\hline
$\tau$~Cet & 0.78 & 0.52\\
$\delta$~Pav & 0.99 & 1.22\\
$\sigma$~Dra & 0.87 & 0.43\\
\hline
\end{tabular}
\end{table}
\end{verbatim}
To produce a landscape figure or table, use the \verb'pdflscape' package and the \verb'landscape' environment.
The landscape Table~\ref{tab:landscape} was produced using the code:
\begin{verbatim}
\begin{landscape}
\begin{table}
\caption{An example landscape table.}
\label{tab:landscape}
\begin{tabular}{cccccccccc}
\hline
Header & Header & ...\\
Unit & Unit & ...\\
\hline
Data & Data & ...\\
Data & Data & ...\\
...\\
\hline
\end{tabular}
\end{table}
\end{landscape}
\end{verbatim}
Unfortunately this method will force a page break before the table appears.
More complicated solutions are possible, but authors shouldn't worry about this.
\begin{landscape}
\begin{table}
\caption{An example landscape table.}
\label{tab:landscape}
\begin{tabular}{cccccccccc}
\hline
Header & Header & Header & Header & Header & Header & Header & Header & Header & Header\\
Unit & Unit & Unit & Unit & Unit & Unit & Unit & Unit & Unit & Unit \\
\hline
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
\hline
\end{tabular}
\end{table}
\end{landscape}
\section{References and citations}
\subsection{Cross-referencing}
The usual \LaTeX\ commands \verb'\label{}' and \verb'\ref{}' can be used for cross-referencing within the same paper.
We recommend that you use these whenever relevant, rather than writing out the section or figure numbers explicitly.
This ensures that cross-references are updated whenever the numbering changes (e.g. during revision) and provides clickable links (if available in your compiler).
It is best to give each section, figure and table a logical label.
For example, Table~\ref{tab:mathssymbols} has the label \verb'tab:mathssymbols', whilst section~\ref{sec:packages} has the label \verb'sec:packages'.
Add the label \emph{after} the section or caption command, as in the examples in sections~\ref{sec:sections} and \ref{sec:fig_table}.
Enter the cross-reference with a non-breaking space between the type of object and the number, like this: \verb'see Figure~\ref{fig:example}'.
The \verb'\autoref{}' command can be used to automatically fill out the type of object, saving on typing.
It also causes the link to cover the whole phrase rather than just the number, but for that reason is only suitable for single cross-references rather than ranges.
For example, \verb'\autoref{tab:journal_abbr}' produces \autoref{tab:journal_abbr}.
\subsection{Citations}
\label{sec:cite}
MNRAS uses the Harvard -- author (year) -- citation style, e.g. \citet{author2013}.
This is implemented in \LaTeX\ via the \verb'natbib' package, which in turn is included via the \verb'usenatbib' package option (see section~\ref{sec:options}), which should be used in all papers.
Each entry in the reference list has a `key' (see section~\ref{sec:ref_list}) which is used to generate citations.
There are two basic \verb'natbib' commands:
\begin{description}
\item \verb'\citet{key}' produces an in-text citation: \citet{author2013}
\item \verb'\citep{key}' produces a bracketed (parenthetical) citation: \citep{author2013}
\end{description}
Citations will include clickable links to the relevant entry in the reference list, if supported by your \LaTeX\ compiler.
\defcitealias{smith2014}{Paper~I}
\begin{table*}
\caption{Common citation commands, provided by the \texttt{natbib} package.}
\label{tab:natbib}
\begin{tabular}{lll}
\hline
Command & Ouput & Note\\
\hline
\verb'\citet{key}' & \citet{smith2014} & \\
\verb'\citep{key}' & \citep{smith2014} & \\
\verb'\citep{key,key2}' & \citep{smith2014,jones2015} & Multiple papers\\
\verb'\citet[table 4]{key}' & \citet[table 4]{smith2014} & \\
\verb'\citep[see][figure 7]{key}' & \citep[see][figure 7]{smith2014} & \\
\verb'\citealt{key}' & \citealt{smith2014} & For use with manual brackets\\
\verb'\citeauthor{key}' & \citeauthor{smith2014} & If already cited in close proximity\\
\verb'\defcitealias{key}{Paper~I}' & & Define an alias (doesn't work in floats)\\
\verb'\citetalias{key}' & \citetalias{smith2014} & \\
\verb'\citepalias{key}' & \citepalias{smith2014} & \\
\hline
\end{tabular}
\end{table*}
There are a number of other \verb'natbib' commands which can be used for more complicated citations.
The most commonly used ones are listed in Table~\ref{tab:natbib}.
For full guidance on their use, consult the \verb'natbib' documentation\footnote{\url{http://www.ctan.org/pkg/natbib}}.
If a reference has several authors, \verb'natbib' will automatically use `et al.' if there are more than two authors. However, if a paper has exactly three authors, MNRAS style is to list all three on the first citation and use `et al.' thereafter. If you are using \bibtex\ (see section~\ref{sec:ref_list}) then this is handled automatically. If not, the \verb'\citet*{}' and \verb'\citep*{}' commands can be used at the first citation to include all of the authors.
\subsection{The list of references}
\label{sec:ref_list}
It is possible to enter references manually using the usual \LaTeX\ commands, but we strongly encourage authors to use \bibtex\ instead.
\bibtex\ ensures that the reference list is updated automatically as references are added or removed from the paper, puts them in the correct format, saves on typing, and the same reference file can be used for many different papers -- saving time hunting down reference details.
An MNRAS \bibtex\ style file, \verb'mnras.bst', is distributed as part of this package.
The rest of this section will assume you are using \bibtex.
References are entered into a separate \verb'.bib' file in standard \bibtex\ formatting.
This can be done manually, or there are several software packages which make editing the \verb'.bib' file much easier.
We particularly recommend \textsc{JabRef}\footnote{\url{http://jabref.sourceforge.net/}}, which works on all major operating systems.
\bibtex\ entries can be obtained from the NASA Astrophysics Data System\footnote{\label{foot:ads}\url{http://adsabs.harvard.edu}} (ADS) by clicking on `Bibtex entry for this abstract' on any entry.
Simply copy this into your \verb'.bib' file or into the `BibTeX source' tab in \textsc{JabRef}.
Each entry in the \verb'.bib' file must specify a unique `key' to identify the paper, the format of which is up to the author.
Simply cite it in the usual way, as described in section~\ref{sec:cite}, using the specified key.
Compile the paper as usual, but add an extra step to run the \texttt{bibtex} command.
Consult the documentation for your compiler or latex distribution.
Correct formatting of the reference list will be handled by \bibtex\ in almost all cases, provided that the correct information was entered into the \verb'.bib' file.
Note that ADS entries are not always correct, particularly for older papers and conference proceedings, so may need to be edited.
If in doubt, or if you are producing the reference list manually, see the MNRAS instructions to authors$^{\ref{foot:itas}}$ for the current guidelines on how to format the list of references.
\section{Appendices and online material}
To start an appendix, simply place the \verb'
\section{Observations}\label{sec:obs}
This work uses the observational measurements of the fraction of X-ray AGN in galaxy clusters presented by \cite{martini09,martini13}. Typical halo masses of these clusters are few times $10^{14}\,M_\odot$ (see Section~\ref{sec:clust_sample} for more details). In this section we describe the most salient details of these observations and the corresponding data analysis. Of particular interest to our work are the (i) the definition of cluster membership in the observations and (ii) the magnitude/flux limits that are used to define the galaxy and AGN samples. The inferred AGN fractions strongly depend on these selection effects and it is therefore important to reproduce them in the simulations before comparing with the observations.
\cite{martini09} used a sample of 32 massive galaxy clusters out to redshift $z=1.3$ with available \textit{Chandra} X-ray observations. Their low redshift sub-sample consists of 17 clusters at $z<0.4$ (mean redshift $\bar{z}=0.19$). These 17 clusters include the 10 presented in \cite{martini06} and 7 additional ones selected from the \textit{Chandra} archive to be the nearest most massive clusters with virial radius that fits within the \textit{Chandra} field-of-view (FOV). The high redshift sub-sample numbers 15 clusters in the redshift interval $z=0.4-1.3$ (average redshift $\bar{z}=0.72$). Cluster member candidates are selected within the projected $R_{200}$\footnote{The virial radius of a cluster is defined to be the distance from the cluster center where the local density is 200 times the mean density of the Universe.} radius of each cluster in the sample. The number of AGN and galaxy cluster members is determined to the apparent $R$-band magnitude limit $m_{R}^{*}(z)+1$, where $m_{R}^{*}(z)$ is the break of the $R$-band luminosity function at the cluster redshift. The latter is estimated assuming that the absolute magnitude break of the luminosity function evolves as $M_R^*(z) = M_R^*(z=0)-z$ with $M_R^*(z=0)$ from \cite{christlein03_break_lf} and early type galaxy spectral energy distribution for the K-correction. For clusters with a high redshift identification completeness the number of galaxy members is estimated by counting sources with $R$-band magnitude brighter than $m_R^*(z)+1$ and redshift difference ($\Delta z$) relative to the mean cluster redshift ($z$), $\Delta z \cdot c < 3\,\sigma_v (1+z)$, where $\sigma_v$ is the cluster velocity dispersion and $c$ the speed of light. For clusters with limited spectroscopic redshift follow-ups (mostly high-redshift sub-sample) the number of galaxy members is estimated using the cluster-richness vs velocity dispersion relation of \cite{becker07_richnes_sigma}. This empirical relation is calibrated to yield the number of early-type galaxy cluster members that are more luminous than 0.4\,$L^*$ (i.e. equivalent to $m_R^*(z)+1$) within the $R_{200}$ radius. AGN cluster members are also selected to have apparent magnitude brighter than $m_R^*(z)+1$ and redshifts that are consistent with $\Delta z\cdot c < 3\,\sigma_v(1+z)$, i.e. similar to galaxies. The observed number of X-ray AGN cluster members is also corrected for the spectroscopic completeness of each cluster (typically $>60$\% for AGN). The depth of the {\it Chandra} X-ray observations means that AGN samples are complete to hard-band luminosities $L_X(\rm 2-10\,keV) >10^{43}\, erg\,s^{-1}$. Less luminous X-ray sources suffer incompleteness because of the sensitivity of the \textit{Chandra} observations and are not used for the estimation of AGN fractions.
The \cite{martini13} cluster sample is composed of 13 of the most statistically significant extended X-ray sources detected in the {\it Chandra} survey of the Bootes field with spectroscopic identifications in the redshift interval $z= 1-1.5$ \citep{Eisen08_clust_selec}. Cluster member candidates, AGN or galaxies, are selected to lie within the projected $R_{200}$ radius and have {\it Spitzer} $3.6\,\mu \rm m$-band apparent magnitude brighter than $m_{3.6}^*(z)+1$. The quantity $m_{3.6}^*(z)$ is the break of the $3.6\,\mu \rm m$ luminosity function at redshift $z$ adopted from \cite{mancone10}. Both spectroscopic and photometric redshifts are used to determine cluster membership. Sources with spectroscopic redshift within $\Delta z\cdot c<\pm2\,000 (1+z)\,\rm km\,s^{-1}$ (i.e. similar to \citealt{martini09} fixing $3\sigma_v=2\,000\,\rm km\,s^{-1}$) off the cluster redshift are assumed to be members. In the case of photometric redshift estimates this condition is modified so that at least 30\% of the photometric redshift probability density function is required to lie within the above redshift interval. All X-ray selected AGN cluster member candidates in the sample of \cite{martini13} have spectroscopic redshifts. The AGN sample is complete to the X-ray luminosity $L_X(\rm 2-10\,keV) =10^{44}\, erg \, s^{-1}$. For less luminous systems, $L_X(\rm 2-10\,keV) >10^{43}\, erg \, s^{-1}$, \cite{martini13} provide lower limits for the AGN cluster fraction.
\section{Results}\label{sec:results}
\begin{figure*}
\centering
\includegraphics[width=1\textwidth]{pictures/results/general_fractions.pdf}
\caption{Fraction of AGN relative to galaxies above a stellar mass limit as a function of stellar mass. The curves correspond to the predictions of the semi-empirical model described in Section \ref{sec:agn_galax_conec}. Different colours correspond to AGN more luminous than $L_X(\rm 2-10\,keV)>10^{42}\,erg\,s^{-1}$ (green), $\rm >10^{43}\,erg\,s^{-1}$ (red) and $\rm >10^{44}\,erg\,s^{-1}$ (blue). Different line styles indicate different specific accretion rate distribution models adopted to generate the AGN mocks (see Section~\ref{sec:agn_galax_conec}). The dashed-dotted lines are for the \citet{age17_sar} and the solid lines correspond to the \citet{aird18_sar} specific accretion-rate distributions. Each panels correspond to redshifts from left to right of $z=0.2$, 0.75 and 1.25.}
\label{fig:frac_SM}
\end{figure*}
\subsection{Impact of selection effects on the incidence of AGN}\label{sec:results_impact_select_eff}
We first explore the sensitivity of the estimate AGN fractions to observational selection effects. Figure \ref{fig:frac_SM} plots the AGN duty cycle of our semi-empirical model as a function of host galaxy stellar mass limit. The former quantity is defined as the fraction of mock AGN above a given X-ray luminosity threshold among galaxies more massive than a given stellar mass limit (x-axis of Figure \ref{fig:frac_SM}). In this exercise all galaxies in a given \textsc{UniverseMachine} box are used independent of halo mass. We iterate that the AGN fractions plotted in Figure \ref{fig:frac_SM} are consistent with independent observations that use the stellar mass function of galaxies and AGN hosts to infer duty cycles \citep[see][]{age17_sar}. Figure \ref{fig:frac_SM} shows that the AGN duty cycle is sensitive to both the X-ray and the host galaxy stellar mass thresholds. There is a general trend of increasing AGN fraction with decreasing X-ray luminosity. This is because less luminous AGN are more common than high accretion luminosity events (i.e. the X-ray luminosity function). Also the AGN duty cycle at fixed luminosity drops with decreasing stellar mass below about $10^{11}\,M_{\odot}$. This is because in our implementation lower stellar mass hosts require higher $\lambda_{\rm{sBHAR}}$ to produce AGN above a given luminosity cut. However, higher specific accretion rates are less likely. This is demonstrated by the form of the specific accretion rate distribution plotted in Figure \ref{fig:sar}, which strongly decreases with increasing $\lambda_{\rm{sBHAR}}$. For stellar masses $\ga 10^{11}\,M_{\odot}$ the duty cycle curves of Figure~\ref{fig:frac_SM} either increase, decrease or remain nearly flat with increasing stellar mass. This is related to the stellar mass dependence of the specific accretion-rate distributions of \cite{age17_sar} and \cite{aird18_sar} used in our analysis. It is also worth noting that the differences between the two specific accretion rate distributions models above are stronger for the lowest redshift panel ($z=0.2$). This is related to the small sample of low redshift AGN in the \cite{age17_sar} and \cite{aird18_sar} studies. In any case, the important point of Figure~\ref{fig:frac_SM} is that AGN fractions are sensitive to the choice of the stellar mass and X-ray luminosity thresholds, i.e. the observational selections effects. This emphasises the importance of carefully treating this issue, as described in Sections \ref{sec:sample_select}.
For completeness we also show in Figure \ref{fig:frac_HM} how the fraction of AGN varies with parent halo mass. \textsc{UniverseMachine} galaxies (central or satellites) above a given stellar mass threshold are grouped by parent halo mass. At a fixed halo mass the fraction of galaxies that host AGN above a given accretion luminosity limit is estimated and plotted. As expected the resulting curves are nearly flat with halo mass since the explicit assumption of the semi-empirical model construction is that the probability of a galaxy hosting an accretion event is agnostic to halo mass. Nevertheless, the strong correlation between halo and stellar mass as well as the X-ray luminosity cut imprint systematic trends onto the curves of Figure~\ref{fig:frac_HM}. These are manifested for example, by the increasing AGN fraction toward lower halo masses. This is more pronounced for the curves with a high stellar mass cut, $\log M_{\star}/M_{\odot} > 10.8$. This is because this threshold essentially removes a large number of lower mass galaxies, which are typically found in low mass halos. Additionally the form of the specific accretion rate distributions dictates that more massive galaxies are more likely to host AGN above a fixed accretion luminosity threshold. The net effect is the observed increase in the AGN fraction toward the low halo mass end in the case of the higher stellar mass threshold in Figure~\ref{fig:frac_HM}.
\subsection{X-ray AGN fractions in clusters}
\begin{figure*}
\centering
\includegraphics[width=1\textwidth]{pictures/results/general_fractions_HOD_hm_smFilter_central.pdf}
\caption{Fraction of AGN in galaxies as a function of parent halo mass. The curves correspond to the predictions of the semi-empirical model described in Section \ref{sec:agn_galax_conec} using the specific accretion rate distribution from \citet{aird18_sar}. Different colours correspond to AGN more luminous than $L_X(\rm 2-10\,keV)>10^{42}\,erg\,s^{-1}$ (green), $\rm >10^{43}\,erg\,s^{-1}$ (red) and $\rm >10^{44}\,erg\,s^{-1}$ (blue). Different line styles indicate different stellar mass limit cuts M$_{\star,lim}$>10$^{10.3}$ M$_\odot$ (solid line) and M$_{\star,lim}$>10$^{10.8}$ M$_\odot$ (dashed-dotted line). Each panel corresponds to redshifts from left to right of $z=0.2$, 0.75 and 1.25.}
\label{fig:frac_HM}
\end{figure*}
Having demonstrated the strong impact of selection effects on the calculation of AGN fractions, we next turn to the comparison of the our model predictions with the observed fractions of AGN in massive galaxy clusters. Figure~\ref{fig:fractions} plots the fraction of AGN among cluster member galaxies as a function of redshift. The observational results of \cite{martini09,martini13} at mean redshifts $z = 0.2$, 0.75 and 1.25 are compared with the predictions of our semi-empirical model using either the \cite{aird18_sar} or the \cite{age17_sar} specific accretion rate distributions. The model predictions for the mass- and magnitude-limited samples are presented in different panels. Cluster AGN fractions are estimated for two luminosity thresholds, $L_X(\rm 2-10\,keV)>10^{43}\,\rm{erg\,s^{-1}}$ and $>10^{44}\,\rm{erg\,s^{-1}}$ indicated by different colors. The uncertainties assigned to the model fractions are determined using bootstrap resampling. For each cluster a total of 10 AGN realisations are generated by repeating the seeding process of Section \ref{sec:agn_galax_conec} 10 times (re-seeded samples). For a given cluster these 10 realisations differ in their AGN populations because of the stochastic nature of the seeding process. This results in an extended cluster sample that is 10 times larger than the original, i.e. 3880, 1570, 180 for $z=0.2$, 0.75 and 1.25 respectively. At fixed redshift clusters are drawn with replacement from the extended sample to generate a total of 100 sub-samples which are used to determine the 68\% confidence interval around the mean. These confidence intervals are the errobars of the model predictions plotted in Figure~\ref{fig:fractions}. They represent the uncertainty of the mean expected fraction of AGN per cluster.
The different model flavors broadly yield consistent results at fixed redshift and X-ray luminosity threshold. At the lowest redshift bin however, differences are apparent between the AGN fractions at $L_X(\rm 2-10\,keV)>10^{44}\,erg\,s^{-1}$ for the models using the \cite{aird18_sar} and \cite{age17_sar} specific accretion rate distributions.
These differences are ultimately linked to the observationally measured specific accretion rate distributions and their dependence on the stellar mass of the AGN hosts. The probability of high mass galaxies hosting an accretion event is lower in the \cite{aird18_sar} specific accretion rate distributions compared to the \cite{age17_sar} ones. As a result the AGN fractions predicted by the model flavor that uses the \cite{age17_sar} specific accretion rate distributions are higher. This is primarily because of luminous AGN assigned to the central galaxies of the clusters. About 42 out 388 ($\sim$11\%) of the mock clusters have a central galaxy with assigned AGN luminosity $L_X(\rm 2-10\,keV)>10^{44}\,\rm{erg\,s^{-1}}$. Such a high incidence AGN is inconsistent with observational constraints on the X-ray properties of Brightest Cluster Galaxies in local massive clusters \citep[][]{yiang18_bcgs}. This is a limitation of the \cite{age17_sar} observationally determined specific accretion rate distributions.
The fractions predicted by the semi-empirical model flavor with the apparent magnitude selection effects (right panel in Figure~\ref{fig:fractions}) does not include the contribution of AGN emission to the mock galaxy SED. In the case of unobscured AGN this contribution may dominate over the host galaxy stellar component in the optical/mid-infrared bands, particularly at high accretion luminosities ($L_X \rm \ga10^{44}\, erg \, s^{-1}$). The estimated model fractions may therefore be underestimated, because a higher fraction of mock AGN would be brighter than the adopted apparent magnitude cut, if their contribution to the mock galaxy SED was modeled. We estimate nevertheless that this effect is small. We modify the methodology of Section~\ref{sec:selec_eff} by including all AGN cluster members above the luminosity limits $L_X(\rm 2-10\,keV)>10^{43}\,\rm{erg\,s^{-1}}$ or $>10^{44}\,\rm{erg\,s^{-1}}$ in the calculation of the corresponding AGN cluster fractions, irrespective of the apparent magnitude of their host galaxy. This is equivalent to assuming that all of the mock AGN are unobscured and therefore their apparent magnitudes dominate that of their hosts. This approach nevertheless increases the estimated fraction at any redshift by $\la 0.1$\,dex and therefore does not impact our results and conclusions.
\begin{table}
\caption{Fraction of X-ray AGN relative to galaxies in simulated massive clusters for different redshifts.}
\centering
\begin{tabular}{c|c|c|c|c|c}
\multirow{2}{*}{z} & \multirow{2}{*}{L$_{\rm X,lim}$} & \multicolumn{2}{c}{f$_{\rm G+17}\cdot 10^{-2}$} & \multicolumn{2}{c}{f$_{\rm A+18}\cdot 10^{-2}$} \\\cline{3-6}
& & SM & Mag & SM & Mag \\
(1) & (2) & \multicolumn{2}{c}{(3)} & \multicolumn{2}{c}{(4)} \\\hline
\multirow{2}{*}{1.25} & 10$^{43}$ & 2.48$^{+0.10}_{-0.13}$ & 2.43$^{+0.01}_{-0.12}$ & 3.21$^{+0.02}_{-0.17}$ & 3.08$^{+0.02}_{-0.16}$ \\
& 10$^{44}$ & 0.23 $^{+0.04}_{-0.04}$ & 0.22$^{+0.04}_{-0.04}$ & 0.28$^{+0.04}_{-0.04}$ & 0.27 $^{+0.04}_{-0.04}$ \\ \hline
\multirow{2}{*}{0.75} & 10$^{43}$& 1.56 $^{+0.03}_{-0.03}$ & 1.04$^{+0.020}_{-0.021}$ & 1.21$^{+0.03}_{-0.03}$ & 0.635$^{+0.020}_{-0.016}$ \\
& 10$^{44}$& 0.096$^{+0.007}_{-0.007}$ & 0.063$^{+0.005}_{-0.005}$ & 0.062$^{+0.006}_{-0.006}$ & 0.038$^{+0.004}_{-0.004}$ \\ \hline
\multirow{2}{*}{0.2} & 10$^{43}$ & 0.300$^{+0.007}_{-0.007}$ & 0.266$^{+0.006}_{-0.006}$ & 0.328$^{+0.008}_{-0.008}$ & 0.208$^{+0.006}_{-0.006}$ \\
& 10$^{44}$ & 0.084$^{+0.004}_{-0.004}$ & 0.072$^{+0.003}_{-0.003}$ & 0.036$^{+0.003}_{-0.003}$ & 0.0263$^{+0.0020}_{-0.0020}$ \\ \hline
\end{tabular}
\begin{list}{}{}
\item (1) Redshift, (2) X-ray luminosity threshold on the $2-10$ keV band, adopted for the AGN sample, in units of erg/s, (3) fractions corresponding to the \cite{age17_sar} model, for the stellar mass (SM) and magnitude (Mag) selected samples, (4) fractions corresponding to \cite{aird18_sar} model, for the stellar mass (SM) and magnitude (Mag) selected samples.
\end{list}
\label{tab:fractions}
\end{table}
\begin{figure*}
\centering
\includegraphics[width=.8\textwidth]{pictures/results/agn_fractions_line_sm_mag_final_sfUM_qAGE_ang.pdf}
\caption{Evolution of the X-ray AGN fractions in massive clusters. The observations (stars) and semi-empirical model predictions (lines and shaded regions) are plotted at redshifts $z = 0.2, 0.75$ and 1.25. In both panels the stars (red or blue) are the observationally measured AGN fractions of \citet[][see Section \protect\ref{sec:obs}]{martini13}. Different colours indicate different luminosity cuts. Red is for AGN with $L_X (\rm 2-10\, keV)> 10^{43}\,erg\,s^{-1}$ and blue corresponds to $L_{X} \rm (2-10\, keV)> 10^{44}\,erg\,s^{-1}$. For the shake of clarity the blue stars are shifted by -0.12 in the x-axis direction. The errorbars correspond to the 1$\sigma$ error. The lower limit at $z=1.25$ is due to incompleteness and the upper limit at $z=0.2$ corresponds to the 3$\sigma$ confidence level. In both panels the lines and corresponding shaded regions represent the predictions of the semi-empirical model on the AGN fraction under different assumptions on the selection effects. On the left panel the simulated galaxy/AGN samples are selected above the stellar mass limit listed in Table \ref{tab:sel_effects_values}. The right panel corresponds to mock AGN/galaxy samples selected above the apparent magnitude thresholds listed in Table \ref{tab:sel_effects_values} (see Section \ref{sec:selec_eff} for more details). Lines of different colour indicate different luminosity cuts as explained above. The blue lines should therefore be compared with the blue stars and the same for the red lines/symbols. Different line styles indicate different specific accretion rate distribution models adopted to generate the AGN mocks (see Section~\ref{sec:agn_galax_conec}). The dashed-dotted lines are for the \citet{age17_sar} and the solid lines correspond to the \citet{aird18_sar} specific accretion-rate distributions. The shaded regions within which lines are embedded correspond to the 68\% confidence intervals of the mean value calculated using the bootstrapping technique described in the text.}
\label{fig:fractions}
\end{figure*} |
1,116,691,499,236 | arxiv |
\section{Introduction}\label{introduction}
About every sixth death in the world is cancer-related, with colorectal cancer (CRC) being the most dominant globally~\cite{haggar2009colorectal}. During a colonoscopy, a tiny video camera connected to the end of an endoscope creates a video signal of the internal mucosa of the patient's colon. The video data is shown on a monitor for real-time diagnosis by the endoscopists. Various randomized clinical examinations and prospective cohort investigations have testified that conventional colonoscopy with polypectomy lowered the incidence of CRC significantly by 40-90\% and decreased mortality~\cite{rex2002rationale}. Therefore, the requirement of colonoscopy continues to increase. Current European Union guidelines also recommend such screening for those above 50 years old~\cite{von2012european}.
The main contribution of the paper are follows.
\begin{enumerate}
\item We develop our \ac{GAN} to generate synthetic images to be used in the training dataset to deal with the challenges related to dataset problem.
\item The experimental results on the Kvasir-dataset shows that the proposed model outperforms its competitors and also forms new baseline for the publicly available dataset.
\end{enumerate}
\section{Related Works}\label{related work}
Polyp segmentation has been an active area of research. One of the significant challenges in the field was the availability of an open-access dataset for experimentation and benchmarking. Jha et al.~\cite{jha2020kvasir} approached the problem of the lack of data insufficiency by annotating the polyp images with the help of a team of medical experts, generating ground truth information and bounding box information and made it publicly available to research community. For benchmarking on the dataset, they used Fuzzy C-mean Clustering and ResUNet model. They obtained a dice coefficient of 0.787763 and \ac{mIoU} of $0.777$ on the test dataset with ResUNet~\cite{zhang2018road}.
There have been several individual work and challenges been done to reach the goal of developing an automated poly detection system for clinical implementation. For the medical image segmentation problem, usually, an encoder-decoder network like U-Net~\cite{ronneberger2015u} is mostly famous. Jha et al.~\cite{jha2019resunet++} proposed ResUNet++~\cite{jha2019resunet++,} to improve the results on the Kvasir-SEG dataset. ResUNet++ is an encoder-decoder network which takes the advantage of ResUNet~\cite{zhang2018road}, deep residual learning~\cite{he2016deep}, squeeze and excitation network~\cite{hu2018squeeze}, Atrous spatial pyramid pooling~\cite{he2015spatial} and UNet~\cite{ronneberger2015u}. They have compared their results with U-Net and ResUNet and showed the improvement over both the baselines. Recently, they have proposed DoubleUNet~\cite{jha2020doubleu} for medical image segmentation which gave state-of-the-art results with four publicly available medical image segmentation dataset.
Moreover, there are other promising works in the field of colonoscopy.......
The detection and classifying of cancers in colonoscopy videos has been researched for several years~\cite{c5, c6}. However, there is still room for improvement since the accuracy is not at the level of expert endoscopists. A research paper from the European Journal of Nuclear Medicine and Molecular Imaging (EJNMMI) Research found that a CNN model is not significantly better than a doctor or a classical algorithm when it comes to detecting tumors~\cite{c1}. However, it is much simpler to set up and is less discriminative than a traditional algorithm. There is still potential to increase the performance and to look into alternative ways of training machine learning models.
Overall, there have been several works done in the field of polyp segmentation. However, for a two-class problem, there are still chances for improving the models to reach the goal of better or close to endoscopist level detection. Therefore, in order to full fill the gap, we do more experiments on this problem to explore the possibility of building a better-automated system that could possibly be useful in the future.
\section{Proposed Method}\label{methods}
Our work looks at using conventional neural networks (CNN) to classify images into eight different findings located in the GI tract. The findings include both lesions and anatomical landmarks, as both are important to detect during a routine investigation of the GI tract. The eighth categories are polyp, ulcerative colitis, esophagitis Z-line, pylorus, cecum, dyed-lifted polyps, and dyed-resection margins. The developed model should find and categorize each image into the correct category. To supplement the CNN model we created a generative adversarial network (GAN). The GAN is used to generate synthetic data which may be used by the CNN to become more general and capable of predicting unseen data samples. The goal with this paper is to make as good as possible model given the data set of 8000 images divided between eight categories (1000 images for each category). Due to the relatively low number of images per class, deep neural networks have a tendency to easily overfit on the data. Therefor, we decided to split the data into three separate parts. The largest of the set was used for training. The next two parts were quite smaller, and was used for testing and validation. We wanted data that my algorithm never had the opportunity to use before in order to validate it’s performance. This will result in a more real to life performance of the algorithm since it will have to perform on images it has never seen before.
In the following few sections, we describe our pipeline for prepossessing the data, developing the CNN-based model, and developing a GAN to generate synthetic data samples.
\section{Experiments}\label{experiments}
In this section, we present datasets, evaluation metrics, experiment setup and configuration, and data augmentation techniques used in all the experiments.
There were in all two types of deep leaning models used in this project. The first one is a convolutional neural network (CNN). These algorithms are often used for images classification and are quite well suited for that task. An important thing to note is that CNN’s are quite flexible. They can be used on all types of data that is fitted into a grid. It is unique in that it tries to mimic how humans think. This makes it a fascinating algorithm to explore further and more desirable algorithm to test compared to traditional algorithms such as random forest.
\subsection{Dataset}
We have used dataset from KvasirV2~\cite{pogorelov2017kvasir}. We have used 4 classes out of 8 class. They are ......
\subsection{Data pre-possessing}
We developed a pre-processing pipeline that does a couple of different things depending on if this is the first or later import of data into the model. It can ether perform a strait import or it can load already existing files into the pipeline.
To get a better understanding of how the navigation box affects performance, we decided to first to run a model with the box information present, and then one more time where the box information was blacked out.
\subsection{Implementation Details}
All the experiments were implemented in .... using Keras and Tensorflow as backend. In our experiment, we divided the dataset into 80:10:10, where 80\% of the dataset was used for training, 10\% was used for evaluation and rest for testing. During the training, we downscale the image to .... to mention the aspect ratio. The size of the images were .......We used ......loss function, ...optimizer. The batch size was kept to ...... The \ac{GAN} models was trained upto ...epoch and the learning rate was set to ...... We have used early stopping .....Our implemntation details can be found at GitHub repository.
\subsection{Evaluation Metrics}
For the evaluation of our model, we have chosen Sørensen–dice coefficient (DSC), \ac{mIoU}, Precision, and Recall, Specificity, Accuracy and \ac{FPS}.
\begin{equation}{\label{eq:1}}
\text{Recall}~(REC) = \frac{TP} {TP + FN}
\end{equation}
\begin{equation}{\label{eq:2}}
\text{Specificity}~(SPEC) = \frac{TN} {TN + FP}
\end{equation}
\begin{equation}{\label{eq:3}}
\text{Precision}~(PREC) =\frac{TP} {TP + FP}
\end{equation}
\begin{equation}{\label{eq:4}}
\text{Accuracy}~(ACC) ={\frac{TP + TN} {TP + TN + FP + FN}}
\end{equation}
\begin{equation}{\label{eq:7}}
\text {Frame Per Second}~(FPS) = {\frac{1} {sec /frame}}
\end{equation}
In the above equations, TP, FP, TN, FN represents true positives, false positives, true negatives, and false negatives, respectively, for the classification outputs.
\subsection{Data Augmentation}
\section{Results}
\label{results}
In this section, we will present our approach and compare the proposed methods with the state-of-the art methods. Moreover, we demonstrate that the efficiency of the proposed method.
\subsection{Comparison with the recent works}
\begin{table}[!t]
\caption{Result on the KvasirV2 dataset}
\def1.2{1.2}
\begin{tabular}{|l|l|l|l|l|l|}
\hline
Method & Recall & Precision & Accuracy & MCC\\ \hline
GAN & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ \hline
ResUNet++\cite{jha2019resunet++} & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ \hline
....... & 0.0000 & 0.0000 &0.0000 & 0.0000 \\ \hline
\end{tabular}
\label{tab:result}
\end{table}
\section{Discussion}\label{discussion}
\subsection{Synthetic vs Real data}
An aspect of note in this paper is the relation between what features are seen as important and what is not important during the feature detection face. A way to spot important features is trough the creation of a GAN-model. If the GAN could detect the same features as was important for the CNN and then replicate them in a new data set field with fake images. The gold would then be to use the fake images to train the CNN model, and then use the real data to validate the models performance. A success would lead to a reduced need for data gathering during initial construction of the model.
\subsubsection{The quality of the synthetic data}
Since this is the first time I have used and generated data with a GAN model the data did not turn out that great. There are several problems specifically the regeneration of colors. The color regeneration consisted constantly of wrong colors. And jet, the features are actually, quite similar. If the GAN generated image is tough of as a compressed feature maps of the organelles, it is quite possible to see the connections between the real image and the synthetic.
\input{figures/esophagitis_real_vs_fake}
As seen on Figure~\ref{fig:esophagitis} the feature of text on the side is kept in the synthetic image. It is not comprehensive, but the fact that it is text on the left hand side is considered a feature for this category. This is something that should be left out of the data in order to make a more generalized model. This might be a factor that makes the model perform so well on category two both in the color and gray scale precision. Not all the images in this category has this text information and that is reflected in the synthetic data as well.
\subsubsection{Using the synthetic data for predictions}
The model producing synthetic data performed really badly. So badly that I considered not even trying to train based on the images. And jet, the result is interesting even thou it is quite simply outperformed by both the gray scale and the standard color model by a wide margin.
\input{tables/color_model_synthetic_data}
The Table~\ref{tab:color_model_with_synthetic_data} show the performance for the model. 0 to 3 represent 4 of the categories. These were the only categories I had time to create synthetic data for. Every category has a recall, precision and a F1 score associated. At the bottom is an aggregate for the overall performance of the model including a weighted average. The model was only trained with synthetic data, and then tested with real data.
There are some instigates which in intern could be apply to the real images. It found all images in category 0 as seen on the models recall score, however about 3/4 of the images it guessed was category 0 tuned out to be other categories giving it a F1-score of just 0.42. We can also see that the model got a perfect precision score on category 2, however it only got a 0.18 on recall giving it a F1 score of 0.31. This means that the model only guessed category 2 when it had a specific feature or very high confidence.
Another important thing to bring up when it comes to this model was the fact that it overfitted extremely quickly. This lead me to the realization that these images contain to few features. The missing features might have been found given more training time to the GAN and by creating a higher quality image at the end of the process.
\subsubsection{Valuable lesson learned form the GAN}
\input{figures/cecum_fake}
There are some valuable insides that can be learned from making the GAN algorithm. Firstly, what features the GAN recognized tended to be quite prevalent in the category. For instance, I had realized that a lot of the images had a green box with text down in its left corner, however it was first when I trained the GAN on category normal cecum that I realized how important it would be to remove this particular feature. Yes, many images in the category esophagitis had text on the left hand side, but this feature was not as prevalent as the green box in normal-cecum. This insight mad me run my code one more time, but this time with the removing of the green box.
Figure~\ref{fig:fakececum} is a sample of just some of the many GAN generated images in the category normal cecum which contained a feature that is quite frankly the green box.
\subsection{Gray vs Color}
There are several differences in the categorical data when it comes to the quality of the images. Some have high resolution images with grate color saturation. Others categories contain only lower resolution images. One problem that might occur in such circumstances is the outcome becomes overfitted to the specific data set. Color data is important and contains a lot of information. But how much of the final quality of the model and its performance relays on the color data, and are all categories equally reliant.
Two models were made where the only difference was the change in the color data. Both trained for seven epochs, and both had the same amount of data for training, testing and validation. The images used is scaled down to a 64x64 size. The expectation is that the color images will perform better, but how much better will be the important factor. And which categories rely the most on color data.
\input{tables/color_model_performance}
\input{tables/greyscale_model_performance}
The most important metric to look at is the F1 score. Overall the color image model got a 0.84 (Table~\ref{tab:gray_scale_model_performance}) and the gray scale got 0.63 (Table~\ref{tab:gray_scale_model_performance}). The color model performed significantly better based only on this information. However, the interesting inside is not that the color algorithm performed better, but which categories relay the most on color data and are these categories so reliant on the cooler data as to render them useless on other datasets?
Category 0 dyed lifted polyps is quite connected to the color data. It is a polyp that is appearing as having a blue color in all the images. The F1 score on color is 0.84 while the gray scale is 0.38. We can from this conclude that the category is quite connected to the color data. The gray scale model is performing better than random guessing (that would be an around \verb|1/8 = 12.
The other category that is significantly worse is category 6 polyps. The model that uses colors got a F1 score of 0.77 and the gray scale got 0.4. Category 0 and category 6 performance might actually be related. The images are quite close to one and other when the color information is removed. Both of the categories also have a low precision score which tells us the ratio between correct guesses divided by the total amount of guesses. Both have about 1/3 correct guesses which means that about 2/3 of the guesses are wrong in both cases.
Looking at the performance metrics in Tensorbord does not make the outcome any better for the gray scale model. The metric that might be most important to look at when it comes to training is the loss. Figure~\ref{fig:tensorBoard_showing_loos} shows the validation loss for the two models. The light blue is the gray scale model and the dark blue is the color model.
In figure \ref{fig:tensorBoard_showing_loos} the curve for the gray scale model is going up, this is often the case when the model is overfitted to the data. This means that the model used would probably be at its best and would not likely perform any better if it would have been run for more epochs. This point is exemplified further by looking at its accuracy over time.
In figure \ref{fig:tensorBoard_showing_accuracy}, we can see the models validation accuracy. This parametric has stagnated which in turn is an indicator that the model is no longer leaning anything new from the data. There is no clear answer to the question if the model will perform well or not on another datasets, however one thing is clear and that is the fact that the model needs as much data as it can get and that includes the color values.
\subsection{Color vs Gray scale without green text box on the side of some images}
Color information has proven to be quite important for the model, but is this because of the green box with information that exist in some of the category images. In order to check the performance for the model more thoroughly I decided to extend my prepossessing pipeline by adding a function called \verb|remove_green_box()|. The function uses OpenCV in order to check for a green rectangle. If it is detected it will generate a black box and draw that on top of the images. This is not optimal since a lot of the images now will have a black box there in stead, however, we should be able to see a change in predictions if the model wa susing information inside the box in order to make its predictions.
To check if the performance now declined and by have much I decided to run the code one more time, both in color and in gray scale to see if it had an impact on the two different models.
\input{tables/color_model_info_box_removed}
\input{tables/greyscale_model_info_box_removed}
What came out of this was actually quite fascinating. The categories most effected by the green box was the same categories that tended to be problematic for the gray scale model as seen in table \ref{tab:grayscale_model_performance_with_removed_info_box}. The color model got a high precision score on category 0, however it did get a really low recall of 0.46. All data for the color model is in table \ref{tab:color_model_performance_with_removed_info_box}. This means that the model needed to be quite confident before it guessed category 0. The opposite is true for category 6, it has a lower then average precision and recall score. This means that it guessed a lot wrong and that it did get few of the total correct as well. It seems that the original color model actually was looking at the box in order to get information about the pictures. However it is hard to say anything definitively.
Overall the color model performed quite similar, with or without the green boxes. However the model without the green box information might actually be better, even thou it got a lower overall F1 score. Since this model is not restricted by information only accessible in some particular cases. This in turn should make the second color model better suited for other data sets.
The gray scale model actually performed a little better than the one with the box on the side. This might be because the ration between signal and noise is reduced, meaning it is now less distracting information in the image. However, the gray scale model performed quite badly compared to the color model. Therefore the color information is still vital for the models overall performance.
\section{Conclusion}
\label{conclusion}
In this paper, we have proposed a novel method for polyp image classification using generative adversarial network. The basic idea behind using \ac{GAN} was to generate more synthetic images and feed it into the network so that the model can classify the polyp with high accuracy. Our results demonstrated the efficiency of the proposed model with the sensitvity of ... and precision of ....which outperforms previous approaches.
We believe that the use of GAN algorithms will increase over the next coming years, and their applications will differ quite significantly. It is possible to already see a future where a researcher can start out with a small data set and use that in order to build a GAN which in turn can create a greater data set. But the application of GAN will influence everything from art, science, news and even law. For this specific use case it is worth noting that looking at the images generated by the GAN can be used to understand how a model sees a particular category. If the produced image contains a lot of metadata this tells us that the CNN model might actually be quite reliant on that information and not actually finding features in the images directly. The conclusion when it comes to synthetic data is not then that it is useless to train CNN models. But rather that it takes time and precision to make a high quality performing GAN. For now, the better performing model is the one that is trained on real data.
In future, we will extend our model for multi-instance segmentation and multi-instance detection problem. We will also apply more effort into applying the optimum hyperparamters to achieve high performance and will apply post-processing technique to improve the baseline results.
\bibliographystyle{ieeetr}
\section{Introduction}\label{introduction}
The goal of \emph{Medico automatic polyp segmentation challenge} the benchmarking of polyp segmentation algorithms on new test images for automatic polyp segmentation that can detect and mask out polyps (including irregular, small or flat polyps) with high accuracy. The main goal of the challenge is to benchmark different computer vision and machine learning algorithms on the same dataset that could promote to build novel methods which could be potentially useful in clinical settings. Moreover, we emphasize on robustness and generalization of the methods to solve the limitations related to data availability and method comparison. The detailed challenge description can be found here \url{https://multimediaeval.github.io/editions/2020/tasks/medico/}.
After three years of organizing the Medico Multimedia Task~\cite{riegler2017multimedia, pogorelov2018medico, hicks2019acm}, we present the fourth iteration in the series. With a focus on assessing human semen quality last year~\cite{hicks2019acm}, this year we build on the 2017~\cite{riegler2017multimedia} and 2018~\cite{pogorelov2018medico} challenges of automatically detecting anomalies in video and image data from the GI tract. We introduce a new task for automatic polyp \textit{segmentation}. In the prior \gls{gi} challenges, we classified the images into various classes. We are now interested in identifying each pixel of the lesions from the provided polyp images in this challenge.
The task is important because \gls{crc} is the third most leading cause of cancer and fourth most prevailing strain in terms of cancer incidence globally~\cite{bray2018global}. Regular screening through colonoscopy is a prerequisite for early cancer detection and prevention of \gls{crc}. Regardless of the achievement of colonoscopy examinations, the estimated polyp miss rate is still around 20\%~\cite{kaminski2010quality}, and there are large inter-observer variabilities~\cite{mahmud2015computer}. An automated \gls{cadx} system detecting and highlighting polyps could be of great help to improve the average endoscopist performance.
In recent years, \glspl{cnn} have advanced medical image segmentation algorithms. However, it is essential to understand the strengths and weaknesses of the different approaches via performance comparison on a common dataset. There are a large number of available studies on automatic polyp segmentation~\cite{fan2020pranet,guo2020polyp,jha2019resunet++,mahmud2020polypsegnet,jha2020doubleu,guo2019giana,wang2018development,jharealtime}. However, most of the conducted studies are performed on a restricted dataset which makes it difficult for benchmarking, algorithm development and reproducible results. Our challenge is utilizing the publicly available Kvasir-SEG dataset~\cite{jha2020kvasir}. The entire Kvasir-SEG dataset is used for training and an additional and unseen test dataset for benchmarking the algorithms.
In summary, the Medico 2020 challenge can support building future systems and foster open, comparable and reproducible results where \textit{the objective of the task is to find efficient solutions automatic polyp segmentation}, both in terms of pixel-wise accuracy and processing speed.
For the clinical translation of technologies, it is essential to design methods on multi-centered and multi-modal datasets. We have recently released several gastrointestinal endoscopy~\cite{pogorelov2017kvasir,pogorelov2017nerthus,borgli2020hyperkvasir}, wireless capsule endoscopy~\cite{smedsrud2020kvasir}, endoscopic instrument ~\cite{jhakvasirinstrument}, and polyp datasets~\cite{jha2020kvasir}. Thus, we have put in significant effort to address the challenges related to lack of public available datasets in the field of \gls{gi} endoscopy.
\begin{figure} [!t]
\centering
\includegraphics[height=1.6cm]{figures/10_image.jpg}
\includegraphics[height=1.6cm]{figures/3_image.jpg}
\includegraphics[height=1.6cm]{figures/26.jpg}
\includegraphics[height=1.6cm]{figures/11image.jpg}
\includegraphics[height=1.6cm]{figures/3mask.jpg}
\includegraphics[height=1.6cm]{figures/10_mask.jpg}
\includegraphics[height=1.6cm]{figures/26mask.jpg}
\includegraphics[height=1.6cm]{figures/11mask.jpg}
\caption{Polyps and corresponding masks from Kvasir-SEG}
\label{fig:images}
\end{figure}
\section{Dataset}\label{dataset}
The Kvaris-SEG~\cite{jha2020kvasir} training dataset can be downloaded from \url{https://datasets.simula.no/kvasir-seg/}. It contains 1,000 polyp images and their corresponding ground truth mask as shown in Figure~\ref{fig:images}. The dataset was collected from real routine clinical examinations at B{\ae}rum Hospital in Norway by expert gastroenterologists. The resolution of images varies from $332 \times 487$ to $1920 \times 1072$ pixels. Some of the images contain a green thumbnail in the lower-left corner of the images showing the scope position marking from the ScopeGuide (Olympus) (see Figure~\ref{fig:testimages}). We annotate another separate dataset consisting of 160 new polyp images and use the resulting dataset as the test set to benchmark the participants' approaches. Figure~\ref{fig:testimages} shows some examples of test images used in the challenge.
\begin{figure} [!t]
\centering
\includegraphics[height=1.6cm]{figures/test_images/1.jpg}
\includegraphics[height=1.6cm]{figures/test_images/2.jpg}
\includegraphics[height=1.6cm]{figures/test_images/3.jpg}
\includegraphics[height=1.6cm]{figures/test_images/4.jpg}
\includegraphics[height=1.6cm]{figures/test_images/5.jpg}
\includegraphics[height=1.6cm]{figures/test_images/6.jpg}
\includegraphics[height=1.6cm]{figures/test_images/7.jpg}
\includegraphics[height=1.6cm]{figures/test_images/8.jpg}
\caption{Examples polyps from the test images}
\label{fig:testimages}
\end{figure}
\section{Task Description}
The participants are invited to submit their solutions for the two following tasks: segmentation and efficiency (speed).
\subsection{The automatic polyp segmentation task}
This task invites participants to develop new algorithms for segmentation of polyps. The main focus is to develop an efficient system in terms of diagnostic ability and processing speed and accurately segment the maximum polyp area in a frame from the provided colonoscopic images.
There are several ways to evaluate the segmentation accuracy. The most commonly used metrics by the wider medical imaging community are the correct \textbf{Dice similarity coefficient (DSC)} or overlap index, and the \textbf{mean Intersection over Union (mIoU)}, also known as the Jaccard index. In clinical applications, the gastroenterologists are interested in pixel-wise detail information extraction from the potential lesions. The metrics such as DSC and mIoU are used to compare the pixel-wise similarity between the predicted segmentation maps and the original ground truth of the lesions.
The DSC is a metric for comparison of the similarities between two given samples. If \textit{tp}, \textit{tn}, \textit{fp}, and \textit{fn} represent the number of true positive, true negative, false positive and false negative per-pixel predictions for an image, respectively, then the DSC is given as
\begin{equation*}\label{eq:dice}
\text{DSC} = \frac{2 \cdot tp} {2 \cdot tp + fp + fn}
\end{equation*}
Furthermore, the IoU is then defined as the ratio of intersection of two metrics over a union of two corresponding metrics. The mean IoU computes IoU of each semantic class of an image and calculate the mean over each classes. The IoU is defined as:
\begin{equation*}\label{eq:iou}
\text{IoU} = \frac{tp} {tp + fp + fn}
\end{equation*}
Moreover, in the polyp image segmentation task (i.e., a binary segmentation task), \textbf{precision} (positive predictive value) shows over-segmentation, and \textbf{recall} (true positive rate) shows under-segmentation. Over-segmentation means that the predicted image covers more area than the ground truth in some part of the frame. The under-segmentation implies that the algorithm has predicted less polyp content in some portion of the image compared to its corresponding ground truth. We also encourage participants to calculate precision and recall, and these are given by:
\begin{equation*}
\text{Precision} =\frac{tp} {tp + fp}
\end{equation*}
\begin{equation*}
\text{Recall} = \frac{tp} {tp + fn}.
\end{equation*}
The main metric for evaluation and ranking of the teams is \textbf{mIoU}. There is a direct correlation between mIoU and DSC. Therefore, we have only used one metric. If the teams have the same mIoU values, then the teams will be further evaluated on the basis of the higher value of the DSC. For the evaluation, we ask the participants to submit the predicted masks in a zip file. The resolution of the predicted masks must be equal to the test images.
\subsection{The algorithm speed efficiency task}
Real-time polyp detection is required for live patient examinations in the clinic. It can gain gastroenterologist attention to the region of interest. Thus, we also ask participants to participate in the efficiency task. The algorithm efficiency task is similar to the previous task, but it puts a stronger emphasis on the algorithm's speed in terms of frames-per-second.
Submissions for this task will be evaluated based on both the algorithm's speed and segmentation performance. The segmentation performance (the segmentation accuracy) will be measured using the same \textbf{mIoU} metric as described above for the first task, whereas speed will be measured by \textbf{frames-per-second (FPS)} according to the following formula:
\begin{equation*}\label{fps}
FPS = {\frac{\#frames} {sec}}
\end{equation*}
For this task, we require participants to submit their proposed algorithm as part of a Docker image so that we can evaluate it on our hardware. We evaluate the performance of the algorithm on the Nvidia GeForce GTX 1080 system. For the team ranking, we set a certain mIoU as threshold for considering it as a valid efficient segmentation solution and rank according to the FPS.
\vspace{-2mm}
\section{Discussion and Outlook}\label{Discussion}
Currently, there is a growing interest in the development of \gls{cadx} systems that could act as a second observer and digital assistant for the endoscopists. Algorithmic benchmarking is an efficient approach to analyze the results of different methods. A comparison of different approaches can help us to identify challenging cases in the data. We then can discriminate the image frames into simple, moderate, and challenging images. Later on, we can target to develop models on the challenging images that are usually missed out during a routine examination to design better \gls{cadx} systems. We hope that this approach would help us to design better performing algorithms/models that may increase the efficiency of the health system.
\bibliographystyle{ACM-Reference-Format}
\section{Evaluation Methodology}
|
1,116,691,499,237 | arxiv | \section{Introduction}
\label{sec:intro}
Let $(X,d)$ be a metric space, and consider a subset
$T=\{t_\xi:\xi\in\Xi\}$ of $X$ that is parametrized by a bounded subset
$\Xi$ of $\mathbb{R}^d$. Roughly speaking, we are interested in the
following question: can $T$ be viewed as a \emph{finite-dimensional}
subset of $X$? It is certainly tempting to think so, as the parameter
set $\Xi$ is finite-dimensional. This idea is easily made precise if the
induced metric $d_T(\xi,\xi')=d(t_\xi,t_{\xi'})$ on $\Xi$ is comparable
to a norm on $\mathbb{R}^d$, so that $T$ inherits the Euclidean
geometry. However, there are natural examples whose geometry is highly
non-Euclidean, so that the conclusion is far from obvious. The aim of
this paper is to investigate in detail such a problem that arises from
applications in statistics.
To set the stage for the problem that we will consider, let us recall
some metric notions of dimension. For a subset $T$ of
a metric space $(X,d)$, the \emph{covering number} $N(T,\varepsilon)$ is
the smallest cardinality of a covering of $T$ by
$\varepsilon$-balls \cite{KT59}:
$$
N(T,\varepsilon) = \inf\bigg\{
n:\exists~x_i\in X,~i=1,\ldots,n\mbox{ s.t.\ }
T\subseteq\bigcup_{i=1}^n B(x_i,\varepsilon)
\bigg\},
$$
where $B(x,\varepsilon)=\{x'\in X:d(x,x')\le\varepsilon\}$. The covering
number, or equivalently the metric entropy
$\log N(T,\varepsilon)$, quantifies the capacity of the set $T$,
and its scaling in $\varepsilon$ is closely connected to
dimension. Indeed, let $|\cdot|$ be a norm on $\mathbb{R}^d$, so that
$(\mathbb{R}^d,|\cdot|)$ is a finite-dimensional Banach space. A standard
estimate \cite[Lemma 4.14]{Mas07} gives
$$
N(B(t,\delta),\varepsilon) \le \bigg(
\frac{3\delta}{\varepsilon}
\bigg)^d
$$
for any $\varepsilon\le\delta$, where
$B(t,\delta)=\{x\in\mathbb{R}^d:|x-t|\le\delta\}$.
This estimate has two trivial
consequences: first, for any bounded $T\subset
(\mathbb{R}^d,|\cdot|)$, there is a constant $C_1$ so that
\begin{equation}
\label{eq:globalent}
N(T,\varepsilon) \le \bigg(\frac{C_1}{\varepsilon}\bigg)^d
\end{equation}
for all $\varepsilon$ sufficiently small. On the other hand, if we fix
a distinguished point $t_0\in T$, there is a constant $C_2$ such that
for all $\varepsilon/\delta$ sufficiently small
\begin{equation}
\label{eq:localent}
N(T\cap B(t_0,\delta),\varepsilon) \le
\bigg(\frac{C_2\delta}{\varepsilon}\bigg)^d.
\end{equation}
Either (\ref{eq:globalent}) or (\ref{eq:localent}) may be used as a
notion of finite-dimensionality for a set $T$ in a general metric space
$(X,d)$: a set satisfying the \emph{global} entropy
bound (\ref{eq:globalent}) has finite Kolmogorov dimension
$\log N(T,\varepsilon)/\log(1/\varepsilon)\lesssim d$, while a
set satisfying the \emph{local} entropy bound (\ref{eq:localent}) has finite
local\footnote{
The doubling (Assouad) dimension of a set $T$ is defined as
the supremum of the local doubling dimension
$\sup_{\varepsilon}\log N(T\cap B(t_0,2\varepsilon),\varepsilon)$
with respect to $t_0$
\cite{Ass83,Hei01}. For the purposes of this paper, we will
consider mainly the local version of this concept where
the point $t_0$ is fixed.
} doubling dimension
$\log N(T\cap B(t_0,2\varepsilon),\varepsilon)
\lesssim d$.
Clearly (\ref{eq:localent}) implies
(\ref{eq:globalent}), but not conversely.
Now consider a parametrized set $T=\{t_\xi:\xi\in\Xi\}$ in a metric
space $(X,d)$, where $\Xi$ is a bounded subset of $\mathbb{R}^d$, and
let $|\cdot|$ be a norm on $\mathbb{R}^d$.
As $(\Xi,|\cdot|)$ is finite-dimensional in either sense (\ref{eq:globalent})
or (\ref{eq:localent}), these properties are inherited by $T$ provided that
the metric $d$ is comparable to $|\cdot|$.
Indeed, if we have a H\"older-type upper bound
$d(t_\xi,t_{\xi'})\le C|\xi-\xi'|^\alpha$, then $T$ satisfies the
global entropy bound (\ref{eq:globalent}); if we have in addition the
lower bound $d(t_\xi,t_{\xi_0})\ge c|\xi-\xi_0|^\alpha$, we obtain
the local entropy bound (\ref{eq:localent}) with $t_0=t_{\xi_0}$.\footnote{
If $d(t_\xi,t_{\xi'})\le C|\xi-\xi'|^\alpha$,
then any covering of $\Xi$ by balls of radius
$(\varepsilon/C)^{1/\alpha}$ yields a covering
of $T$ by $\varepsilon$-balls, so that
$N(T,\varepsilon)\le N(\Xi,(\varepsilon/C)^{1/\alpha}) \le
(C'/\varepsilon)^{d/\alpha}$. If also
$d(t_\xi,t_{\xi_0})\ge c|\xi-\xi_0|^\alpha$, then
$\{\xi\in\Xi:d(t_\xi,t_{\xi_0})\le\delta\}\subseteq
\Xi\cap B(\xi_0,(\delta/c)^{1/\alpha})$, so
$N(T\cap B(t_{\xi_0},\delta),\varepsilon)\le
(C''\delta/\varepsilon)^{d/\alpha}$.
}
The upper bound is easily obtained in many
cases of interest, so that finite-dimensionality in the sense
(\ref{eq:globalent}) is not too problematic.
The lower bound is much more delicate, however.
In its absence, finite-dimensionality in the sense (\ref{eq:localent})
is far from obvious.
We will investigate these issues in the context of a prototypical
example, to be described presently, that is of significant independent
interest. Fix a probability density $f_0$ on $\mathbb{R}^d$
(that is, $f_0\ge 0$ and $\int f_0\,dx=1$), and consider the class
$$
\mathcal{M}_q = \bigg\{ x\mapsto
\sum_{i=1}^q\pi_if_0(x-\theta_i):
\pi_i\ge 0,~
\sum_{i=1}^q \pi_i = 1,~
\theta_i \in \Theta
\bigg\}
$$
of convex combinations of $q$ translates of $f_0$, where $\Theta$ is a
bounded subset of $\mathbb{R}^d$. Such densities appear in numerous
statistical applications, where they are frequently known as \emph{location
mixtures}. $\mathcal{M}_q$ is a subset of the space $\mathcal{M}$ of all
probability densities on $\mathbb{R}^d$, endowed with a suitable metric
$d$.
$\mathcal{M}_q$ is parametrized by the finite-dimensional subset
$\Xi_q=\Delta_{q-1}\times\Theta^q$ of $\mathbb{R}^{qd+q-1}$, where
$\Delta_{q-1}$ is the $q$-simplex. Natural metrics $d$ satisfy a
H\"older-type upper bound with respect to a norm on $\Xi_q$ (e.g., step
2 in the proof of Theorem \ref{thm:mainmixtures} below). However, the
corresponding lower bound is impossible to obtain.
\begin{example}
\label{ex:geom}
We will write $f_\theta(x)= f_0(x-\theta)$ for simplicity.
Fix $\theta^\star\in\Theta$ and let $f^\star=f_{\theta^\star}$.
Then $f^\star\in\mathcal{M}_2$, but $f^\star$ is not uniquely
represented by a parameter in $\Xi_2$:
\begin{align*}
&\{(\pi,\theta)\in\Xi_2:
d(\pi_1f_{\theta_1}+\pi_2f_{\theta_2},f^\star)=0\}
={}\\
&\{\pi\in\Delta_1,\theta_1=\theta_2=\theta^\star\}\cup
\{\pi_1=0,\theta_1\in\Theta,\theta_2=\theta^\star\}\cup
\{\pi_1=1,\theta_1=\theta^\star,\theta_2\in\Theta\}.
\end{align*}
Clearly $d$ cannot be lower bounded by any norm on $\Xi_2$,
as such a bound would necessarily imply that
$\{(\pi,\theta)\in\Xi_2:
d(\pi_1f_{\theta_1}+\pi_2f_{\theta_2},f^\star)=0\}$
consists of a single point. Thus the above approach to
(\ref{eq:localent}) is useless here.
\end{example}
\begin{figure}
\centering
\includegraphics[width=.85\textwidth]{hball}
\vskip-.4cm
\caption{\small
Let $f_\theta(x)=e^{-2(x-\theta)^2}$, $f^\star = f_{0.5}$,
$\mathcal{M}_2=\{pf_{\theta_1}+
(1-p)f_{\theta_2}:p,\theta_1,\theta_2\in[0,1]\}$.
The plots illustrate
(a) the set of parameters $(p,\theta_1,\theta_2)$ corresponding
to the Hellinger ball $\{f\in\mathcal{M}_2:h(f,f^\star)\le 0.05\}$; and
(b) the parameter set
$\{(p,\theta_1,\theta_2):N(p,\theta_1,\theta_2)\le 0.05\}$ with
$N(p,\theta_1,\theta_2) =
|p(\theta_1-0.5)+(1-p)(\theta_2-0.5)|
+\tfrac{1}{2}p(\theta_1-0.5)^2
+\tfrac{1}{2}(1-p)(\theta_2-0.5)^2$. The two plots are related by
the local geometry
Theorem \ref{thm:localgeom}, which yields
$c^\star N(p,\theta_1,\theta_2)\le h(pf_{\theta_1}+(1-p)f_{\theta_2},f^\star)
\le C^\star N(p,\theta_1,\theta_2)$.
}
\label{fig:hball}
\end{figure}
The phenomenon illustrated in this example can be stated more generally.
For $f^\star\in\mathcal{M}_{q^\star}$ such that $q^\star<q$ (note that
$f^\star\in\mathcal{M}_q$ as $\mathcal{M}_{q}\subset\mathcal{M}_{q+1}$
for all $q$), the subset of parameters $\Xi_q(\delta)\subset\Xi_q$
corresponding to the ball
$\mathcal{M}_q(\delta)=\{f\in\mathcal{M}_q:d(f,f^\star)\le\delta\}$
behaves nothing at all like a ball in a finite-dimensional Banach space
(see Figure \ref{fig:hball}(a)): indeed, the diameter of $\Xi_q(\delta)$
is even bounded away from zero as $\delta\downarrow 0$. There is
therefore no hope to deduce a local entropy bound of the form
(\ref{eq:localent}) for $N(\mathcal{M}_q(\delta),\varepsilon)$ directly
from the corresponding bound in $\mathbb{R}^{qd+q-1}\supset\Xi_q$.
This provides a vivid illustration of the difficulty of
establishing local entropy bounds in geometrically irregular settings.
Nevertheless, we will be able to obtain local entropy bounds for the
mixture classes $\mathcal{M}_q$ in section \ref{sec:mixtures} below.
For concreteness, we endow $\mathcal{M}_q$ with the
Hellinger metric $h(f,g)=\|\sqrt{f}-\sqrt{g}\|_{L^2}$, which is the
relevant metric for statistical applications \cite[ch.\ 7]{vdG00},
\cite{Mas07} (however, our results are easily adapted to other commonly
used probability metrics---the total variation metric $d_{\rm
TV}(f,g)=\|f-g\|_{L^1}$, for example---using almost identical proofs).
The main result, Theorem \ref{thm:localmixtures}, provides an explicit
bound of the form (\ref{eq:localent}) for $\mathcal{M}_q$ under suitable
smoothness assumptions on $f_0$.
The fundamental challenge that we face in the proof is to develop a
sharp quantitative understanding of the local geometry of mixtures
(illustrated in Figure \ref{fig:hball}). The key result that we prove in
this direction is Theorem \ref{thm:localgeom}, which forms the central
contribution of this paper. As this result is rather technical,
we postpone its description to section \ref{sec:mixturesGeo} below.
However, an important consequence of this result is as follows:
given a mixture $f^\star =
\sum_{i=1}^{q^\star}\pi_i^\star f_{\theta_i^\star}$, one can choose
sufficiently small neighborhoods $A_1,\ldots,A_{q^\star}$ of
$\theta_1,\ldots,\theta_{q^\star}$, respectively, such that for any
$q\ge 1$ and
mixture $f=\sum_{i=1}^q\pi_if_{\theta_i}$, the Hellinger metric
$h(f,f^\star)$ is of the same order as
$$
\sum_{\theta_j\in A_0}\pi_j +\sum_{i=1}^{q^\star}
\Bigg\{
\Bigg|
\sum_{\theta_j\in A_i}\pi_j
-\pi_i^\star\Bigg|
\\ \mbox{}
+
\Bigg\|
\sum_{\theta_j\in A_i}\pi_j(\theta_j-\theta_i^\star)
\Bigg\|
+ \frac{1}{2}
\sum_{\theta_j\in A_i}\pi_j\|\theta_j-\theta_i^\star\|^2
\Bigg\}
$$
(here $A_0=\mathbb{R}^d\backslash (A_1\cup\cdots\cup A_{q^\star})$).
This pseudodistance controls precisely the set of parameters in $\Xi_q$
with density close to $f^\star$, see Figure \ref{fig:hball} for an
example.
Let us emphasize that while the local geometry theorem relates the
Hellinger metric on $\mathcal{M}_q$ to a pseudodistance on $\Xi_q$,
the latter is not a norm or even a metric. It is therefore still not
possible to control the local entropy of $\mathcal{M}_q$ as in the case
where the metric is comparable to a norm on $\Xi_q$. Instead, we deduce
the local entropy bound in two steps. First, we observe that the local
geometry theorem allows us to obtain a \emph{global} entropy bound of
the form (\ref{eq:globalent}) for the class of weighted densities
$$
\mathcal{D}_q=\bigg\{
\frac{\sqrt{f/f^\star}-1}{h(f,f^\star)}:
f\in\mathcal{M}_q,~f\ne f^\star
\bigg\},
$$
as the above pseudodistance controls the coefficients
in the Taylor expansion of $f$. This is accomplished in Theorem
\ref{thm:mainmixtures}. The global entropy bound for
$\mathcal{D}_q$ now yields a local entropy bound for $\mathcal{M}_q$
using a slicing procedure. The latter is not specific to
mixtures, and will be developed first in a general setting in section
\ref{sec:entropy}.
Beside their intrinsic interest, the results in this paper are of direct
relevance to statistical applications. Many problems in statistics and
probability make use of estimates on the metric entropy of classes of
densities: metric entropy controls the rate of convergence of uniform
limit theorems in probability, and is therefore of central importance in
the design and analysis of statistical estimators
\cite{VW96,vdG00,Mas07}. Such applications frequently require a
slightly stronger notion of metric entropy known as bracketing entropy,
which we will consider throughout this paper; see section
\ref{sec:entropy}. In infinite-dimensional situations, the global
entropy is chiefly of interest: global entropy estimates for various
classes of probability densities can be found in
\cite{VW96,vdG00,Mas07,BGL07,GLW10}. However, in finite-dimensional
settings, global entropy bounds are known to yield sub-optimal results,
and here local entropy bounds are essential to obtain optimal
convergence rates of estimators \cite[\S 7.5]{vdG00}. In
the case of mixtures, the difficulty of obtaining local entropy bounds
was noted, e.g., in \cite{GW00,MM11}. Applications of the results
in this paper are given in \cite{GasHanOrder,GasRouHMM}.
\section{From global entropy to local entropy}
\label{sec:entropy}
The classical notion of covering numbers $N(T,\varepsilon)$ was defined
in the introduction. We will consider throughout this paper a somewhat
finer notion of covering by brackets (order intervals) rather than by
balls. In this section, we will work in the general setting of normed
vector lattices (normed Riesz spaces, see \cite{AB06}).
\begin{definition}
Let $(X,\|\cdot\|)$ be a normed vector lattice. For any subset
$T\subseteq X$ and $\varepsilon>0$, the bracketing number
$N_{[]}(T,\varepsilon)$ is defined as
$$
N_{[]}(T,\varepsilon) = \inf\bigg\{
n:\exists~l_i,u_i\in X,~\|u_i-l_i\|\le\varepsilon,~
i=1,\ldots,n
\mbox{ s.t.\ }
T\subseteq\bigcup_{i=1}^n [l_i,u_i]
\bigg\},
$$
where $[l,u]=\{x\in X:l\le x\le u\}$.
\end{definition}
Note that as $[l,u]\subset B(l,\|u-l\|)$, it is
evident that $N(T,\varepsilon)\le N_{[]}(T,\varepsilon)$ for any
$T\subseteq X$ and $\varepsilon>0$. Bounds on the bracketing number
therefore imply bounds on the covering number, but not conversely.
The finer covering by brackets is essential in many
probabilistic and statistical applications \cite{VW96,vdG00,Mas07}.
Let $(X,\|\cdot\|)$ be a normed vector lattice, and let us fix a subset
$T\subseteq X$ and a distinguished point $t_0\in T$. Our general aim is to
obtain an estimate on the local covering (or bracketing) number
$N(T\cap B(t_0,\delta),\varepsilon)$ that is polynomial in
$\delta/\varepsilon$. As is explained in the introduction, such
estimates can be much more difficult to obtain than the corresponding
estimates on the global covering number $N(T,\varepsilon)$ that
are polynomial in $1/\varepsilon$. Unfortunately, the latter is strictly
weaker than the former.
Nonetheless, global covering estimates can be useful. For any $t\ne t_0$,
define
$$
d_t = \frac{t-t_0}{\|t-t_0\|},\qquad\quad
D_0 = \{d_t:t\in T,~t\ne t_0\}.
$$
The main message of this section is that a \emph{local} covering
estimate for $T$ can be obtained from a \emph{global} covering estimate
for the weighted class $D_0\subseteq X$. As global entropy estimates can
be much easier to obtain than local entropy estimates, this provides a
useful approach to obtaining local entropy bounds for geometrically
complex classes. We state a precise result for bracketing numbers
as will be needed in the sequel; a trivial modification of the proof
yields a version for covering numbers. In the next section, this result
will be applied in the context of mixtures.
\begin{theorem}
\label{thm:localglobal}
Let $(X,\|\cdot\|)$ be a normed vector lattice. Fix $T\subseteq X$
and $t_0\in T$, and let $D_0$ be as above.
Suppose that there exist $q,C_0\ge 1$ and $\varepsilon_0>0$ such that
$$
N_{[]}(D_0,\varepsilon)\le
\bigg(\frac{C_0}{\varepsilon}\bigg)^q
\quad\mbox{for every }\varepsilon\le\varepsilon_0.
$$
Choose any $d\in X$ such that $|d_t|\le d$ for all $t\in T$, $t\ne t_0$.
Then
$$
N_{[]}(T\cap B(t_0,\delta),\rho) \le
\bigg(
\frac{8C\delta}{\rho}
\bigg)^{q+1}
$$
for all $\delta,\rho>0$ such that $\rho/\delta < 4 \wedge 2\|d\|$,
where $C = C_0(1 \vee \|d\|/4\varepsilon_0)$.
\end{theorem}
\begin{remark}
Theorem \ref{thm:localglobal} requires an upper bound $d\in X$ on
$|D_0|$, that is, $D_0$ must be order-bounded. But the assumptions
of the Theorem already require that $N_{[]}(D_0,\varepsilon_0)<\infty$,
which is easily seen to imply order-boundedness of $D_0$. The latter
therefore does not need to be added as a separate assumption.
\end{remark}
\begin{remark}
In Theorem \ref{thm:localglobal}, a global covering bound for $D_0$ of order
$(1/\varepsilon)^{q}$ gives a local covering bound for $T$ of order
$(\delta/\varepsilon)^{q+1}$. It is instructive to note that this
polynomial scaling cannot be improved. Indeed, let $T$ be the unit
(Euclidean) ball in $\mathbb{R}^{q+1}$, and let $t_0=0$.
Then $D_0$ is the unit sphere in $\mathbb{R}^{q+1}$ and therefore
has Kolmogorov dimension $q$, but the covering number of
$B(0,\delta)$ is of order $(\delta/\varepsilon)^{q+1}$.
The same conclusion holds also for bracketing (rather than covering)
numbers.
\end{remark}
\begin{remark}
A natural question is whether a converse to the above results can be
obtained. In general, however, this is not possible: the class $D_0$
can be much richer than the original class $T$, as the following simple
example illustrates. Let $(X,\|\cdot\|)$ be an infinite-dimensional Hilbert
lattice and let $(e_k)_{k\ge 1}$ be an orthonormal basis. Let
$T=\{2^{-k}e_k:k\ge 1\}\cup\{0\}$ and $t_0=0$. Then
$N_{[]}(T\cap B(t_0,2^{-r}),2^{-k}) \le k-r+1$ for $k\ge r$, so
$N_{[]}(T\cap B(t_0,\delta),\varepsilon) \le
\log_2(8\delta/\varepsilon) \le (8\delta/\varepsilon)^{3/2}$
for all $\varepsilon/\delta\le 1$.
But here we have $D_0 = \{e_k:k\ge 1\}$, so
$N_{[]}(D_0,\varepsilon)\ge N(D_0,\varepsilon)=\infty$ for $\varepsilon>0$ small enough.
\end{remark}
We now proceed to the proof of Theorem \ref{thm:localglobal}. The main
idea of the proof is to partition the set $T\cap B(t_0,\delta)$ into
shells $\{t\in T:r^{-n}\delta\le\|t-t_0\|\le r^{-n+1}\delta\}$ for a
suitable choice of $r>0$. The bracketing number of each shell is then
controlled by that of the normalized class $D_0$ at scale $\sim
r^n\rho/\delta$. Such a slicing procedure is commonly used in the
reverse direction in the theory of weighted empirical processes
(see, e.g., \cite[sec.\ 5.3]{vdG00}). Here we apply this
idea directly to the bracketing numbers.
\begin{proof}[Proof of Theorem \ref{thm:localglobal}]
The assumption implies that
$$
N_{[]}(D_0,\varepsilon)\le
\bigg(\frac{C_0}{\varepsilon\wedge\varepsilon_0}\bigg)^q
\quad\mbox{for every }\varepsilon>0.
$$
If $\varepsilon<\|d\|/4$, then
$$
\frac{\varepsilon}{\varepsilon\wedge\varepsilon_0}
\le 1\vee \frac{\|d\|}{4\varepsilon_0}.
$$
We therefore have
$$
N_{[]}(D_0,\varepsilon)\le
\bigg(\frac{C}{\varepsilon}\bigg)^q
\quad\mbox{for every }\varepsilon<\|d\|/4,
$$
where $C$ is as defined in the Theorem. This estimate will be used below.
Fix $\varepsilon,\delta>0$ and let
$N=N_{[]}(D_0,\varepsilon)$. Then there exist
$l_1,u_1,\ldots,l_N,u_N\in X$ such that $\|u_i-l_i\|\le\varepsilon$
for all $i=1,\ldots,N$, and for every $t\in T$, $t\ne t_0$ there is an
$1\le i\le N$ such that
$l_i\le d_t\le u_i$. Choose $t\in T$ such that $r^{-n}\delta\le
\|t-t_0\|\le r^{-n+1}\delta$ (with $r>1$ to be chosen later).
Then there exists $1\le i\le N$ so that
$$
(r^{-n}l_i\wedge r^{-n+1}l_i)\,\delta + t_0 \le
t \le
(r^{-n}u_i\vee r^{-n+1}u_i)\,\delta + t_0.
$$
Note that
\begin{align*}
\|u_i\,r^{-n}\delta-l_i\,r^{-n}\delta\| &\le
r^{-n}\delta\varepsilon,\\
\|u_i\,r^{-n+1}\delta-l_i\,r^{-n+1}\delta\| &\le
r^{-n+1}\delta\varepsilon,\\
\|u_i\,r^{-n+1}\delta-l_i\,r^{-n}\delta\| &\le
(r-1)r^{-n}\delta + r^{-n+1}\delta\varepsilon,\\
\|u_i\,r^{-n}\delta-l_i\,r^{-n+1}\delta\| &\le
(r-1)r^{-n}\delta + r^{-n+1}\delta\varepsilon,
\end{align*}
where the latter two estimates follow from
$l_i\le d_t\le u_i$, $\|d_t\|=1$, and
\begin{align*}
(u_i-l_i)\,r^{-n}\delta &\le
u_i\,r^{-n+1}\delta-l_i\,r^{-n}\delta
- d_t\,(r-1)r^{-n}\delta
\le
(u_i-l_i)\,r^{-n+1}\delta, \\
(u_i-l_i)\,r^{-n}\delta &\le
u_i\,r^{-n}\delta-l_i\,r^{-n+1}\delta
+ d_t\,(r-1)r^{-n}\delta
\le
(u_i-l_i)\,r^{-n+1}\delta.
\end{align*}
As $|a\vee b-c\wedge d|\le |a-c|+|a-d|+|b-c|+|b-d|$, we can estimate
$$
\|(r^{-n}u_i\vee r^{-n+1}u_i)\,\delta-
(r^{-n}l_i\wedge r^{-n+1}l_i)\,\delta\| \le
2(r-1)r^{-n}\delta + 4r^{-n+1}\delta\varepsilon.
$$
Therefore, we have shown that
$$
N_{[]}(\{t\in T:r^{-n}\delta\le
\|t-t_0\|\le r^{-n+1}\delta\},
2(r-1)r^{-n}\delta + 4r^{-n+1}\delta\varepsilon) \le
N_{[]}(D_0,\varepsilon)
$$
for arbitrary $\varepsilon,\delta>0$, $r>1$, $n\in\mathbb{N}$.
In particular,
$$
N_{[]}(\{t\in T:r^{-n}\delta\le
\|t-t_0\|\le r^{-n+1}\delta\},\rho) \le
N_{[]}(D_0,\tfrac{1}{4}r^{n-1}\rho/\delta -
\tfrac{1}{2}(1-1/r))
$$
for every $\delta>0$, $r>1$, $n\in\mathbb{N}$,
$\rho>2(r-1)r^{-n}\delta$.
Choose an envelope $d\in X$ such that $|d_t|\le d$ for all $t\in T$,
$t\ne t_0$. Evidently
$$
t_0-r^{-n}\delta\, d\le t \le t_0+r^{-n}\delta\,d
$$
for all $t\in T$ such that $\|t-t_0\|\le r^{-n}\delta$. Therefore
$$
N_{[]}(\{t\in T:\|t-t_0\|\le r^{-
\lceil H\rceil}\delta\},
2r^{-H}\delta\|d\|)=1
$$
for all $\delta>0$, $r>1$, $H>0$. Thus we can estimate
\begin{align*}
&N_{[]}(T\cap B(t_0,\delta),
2r^{-H}\delta\|d\|) \\
&\quad\le
1+\sum_{n=1}^{\lceil H\rceil}
N_{[]}(\{t\in T:r^{-n}\delta\le
\|t-t_0\|\le r^{-n+1}\delta\},2r^{-H}\delta\|d\|) \\
&\quad\le 1+\sum_{n=1}^{\lceil H\rceil}
N_{[]}(D_0,\{r^{n-H-1}\|d\|-
(1-1/r)\}/2)
\end{align*}
whenever $\delta>0$, $r>1$, $H>0$ such that $\|d\| >
(1-1/r)r^{H}$. In particular,
$$
N_{[]}(T\cap B(t_0,\delta),
2r^{-H}\delta\|d\|)
\le 1+\sum_{n=1}^{\lceil H\rceil}
N_{[]}(D_0,r^{n-H-1}\|d\|/4)
$$
whenever $\delta>0$, $r>1$, $H>0$ such that $\|d\| \ge
2(1-1/r)r^{H}$, where we have used that the bracketing number is
a nonincreasing function of the bracket size.
Now recall that
$$
N_{[]}(D_0,\varepsilon)\le
\bigg(\frac{C}{\varepsilon}\bigg)^q
\quad\mbox{for every }0<\varepsilon<\|d\|/4,
$$
where $q,C\ge 1$. Thus
$$
N_{[]}(T\cap B(t_0,\delta),
2r^{-H}\delta\|d\|) \le 1+
\sum_{n=1}^{\lceil H\rceil}r^{-(n-1)q}
\bigg(
\frac{8C}{2r^{-H}\|d\|}
\bigg)^q
$$
whenever $\delta>0$, $r>1$, $H>0$ such that $\|d\| \ge
2(1-1/r)r^{H}$. But
$$
\sum_{n=1}^{\lceil H\rceil}r^{-(n-1)q} \le
\frac{1}{1-1/r^q} \le
\frac{1}{1-1/r} \le
\frac{\|d\|}{2(1-1/r)r^{H}}
\frac{4C}{2r^{-H}\|d\|}
$$
as $r>1$ and $q,C\ge 1$. We can therefore estimate
$$
N_{[]}(T\cap B(t_0,\delta),
2r^{-H}\delta\|d\|) \le
\frac{\|d\|}{2(1-1/r)r^{H}}
\bigg(
\frac{8C}{2r^{-H}\|d\|}
\bigg)^{q+1}
$$
whenever $\delta>0$, $r>1$, $H>0$ such that $\|d\| \ge
2(1-1/r)r^{H}$.
We now fix $\delta,\rho>0$ such that $\rho/\delta < 4\wedge 2\|d\|$,
and choose
$$
r = \frac{4}{4-\rho/\delta},\qquad
H = \frac{\log(2\|d\|\delta/\rho)}{\log r}.
$$
Clearly $r>1$ and $H>0$. Moreover, note that our choice of $r$ and $H$
implies that $\|d\|=2(1-1/r)r^H$ and $\rho=2r^{-H}\delta\|d\|$. We
have therefore shown that
$$
N_{[]}(T\cap B(t_0,\delta),
\rho) \le
\bigg(
\frac{8C\delta}{\rho}
\bigg)^{q+1}
$$
for all $\delta,\rho>0$ such that $\rho/\delta < 4\wedge 2\|d\|$.
\end{proof}
\section{The local entropy of mixtures}
\label{sec:mixtures}
\subsection{Definitions and main results}
\label{sec:mixturesTh}
Let $\mu$ be the Lebesgue measure on $\mathbb{R}^d$. We
fix a positive probability density $f_0$ with respect to $\mu$
($f_0>0$ and $\int f_0 d\mu=1$),
and consider mixtures (finite convex combinations) of densities
in the class
$$
\{
f_\theta:\theta\in\mathbb{R}^d
\},\qquad
f_\theta(x) = f_0(x-\theta)\quad\forall\,x\in\mathbb{R}^d.
$$
In everything that follows we fix a nondegenerate mixture $f^\star$
of the form
$$
f^\star=\sum_{i=1}^{q^\star}\pi_i^\star f_{\theta_i^\star}.
$$
Nondegenerate means that $\pi_i^\star>0$ for all $i$, and
$\theta_i^\star\ne\theta_j^\star$ for all $i\ne j$.
Let $\Theta\subset\mathbb{R}^d$ be a bounded parameter set such that
$\{\theta^\star_i:i=1,\ldots,q^\star\}\subseteq\Theta$, and denote
its diameter by $2T$ (that is, $\Theta$ is included in some closed
Euclidean ball of radius $T$). We consider for $q\ge 1$ the
family of $q$-mixtures
$$
\mathcal{M}_q = \bigg\{\sum_{i=1}^q\pi_if_{\theta_i}:
\pi_i\ge 0,~\sum_{i=1}^q\pi_i=1,~\theta_i\in\Theta\bigg\}.
$$
The goal of this section is to obtain a local entropy bound for
$\mathcal{M}_q$ at the point $f^\star$, where $\mathcal{M}_q$ is
endowed with the Hellinger metric
$$
h(f,g) = \bigg[\int \big(\textstyle{\sqrt{f}-\sqrt{g}}
\big)^2d\mu\bigg]^{1/2},
\qquad f,g\in\mathcal{M}_q.
$$
That is, we seek bounds on quantities such as
$N_h(\{f\in\mathcal{M}_q:h(f,f^\star)\le \varepsilon\},\delta)$, where
$N_h$ denotes the covering number in the metric space
$(\mathcal{M}_q,h)$ (i.e., covering by Hellinger balls). In fact, we
prove a stronger bound of bracketing type. Our choice of the Hellinger
metric and the particular form of the bracketing number to be considered
is directly motivated by statistical applications \cite[ch.\ 7]{vdG00},
\cite[\S 7.4]{Mas07}; see \cite{GasHanOrder,GasRouHMM} for
statistical applications of the results below. We will adhere to this
setting for concreteness, though other metrics may similarly be
considered.
In the sequel, we denote by $\|\cdot\|_p$ the $L^p(f^\star
d\mu)$-norm, that is, $\|g\|_p^p = \int |g|^p f^\star d\mu$. Note that
the Hellinger metric can be written as $h(f,g) =
\|\sqrt{f/f^\star}-\sqrt{g/f^\star}\|_2$. To obtain covering bounds
for $\mathcal{M}_q$ in the Hellinger metric, we can therefore apply
the results of section \ref{sec:entropy} for the case where
$(X,\|\cdot\|)$ is the Banach lattice $(L^2(f^\star d\mu),\|\cdot\|_2)$,
$T=\{\sqrt{f/f^\star}:f\in\mathcal{M}_q\}$, and $t_0=1$. Indeed,
it is easily seen that\footnote{
It is an artefact of our definitions that
the centers of the balls that define the minimal cover of cardinality
$N(\mathcal{H}_q(\varepsilon),\delta)$ need not lie in the
set $\{\sqrt{f/f^\star}:f\in\mathcal{M}_q\}$, while the centers
of the balls in the minimal cover associated to
$N_h(\{f\in\mathcal{M}_q:h(f,f^\star)\le \varepsilon\},\delta)$
must lie in $\mathcal{M}_q$. This accounts for the additional
factor $2$ in the inequality
$N_h(\{f\in\mathcal{M}_q:h(f,f^\star)\le \varepsilon\},2\delta)\le
N(\mathcal{H}_q(\varepsilon),\delta)$.
}
$$
N_h(\{f\in\mathcal{M}_q:h(f,f^\star)\le \varepsilon\},2\delta) \le
N(\mathcal{H}_q(\varepsilon),\delta) \le
N_{[]}(\mathcal{H}_q(\varepsilon),\delta),
$$
where we have defined
$$
\mathcal{H}_q(\varepsilon)=
\{\sqrt{f/f^\star}:f\in\mathcal{M}_q,~
\|\sqrt{f/f^\star}-1\|_2\le\varepsilon\}
\subset L^2(f^\star d\mu).
$$
Our aim is to obtain a polynomial bound for the bracketing number
$N_{[]}(\mathcal{H}_q(\varepsilon),\delta)$. To this end,
we will apply Theorem \ref{thm:localglobal} to the weighted class
$\mathcal{D}_q$ defined by
$$
\mathcal{D}_q=\{d_f:f\in\mathcal{M}_q,~f\ne f^\star\},
\qquad\quad
d_f = \frac{\sqrt{f/f^\star}-1}{\|\sqrt{f/f^\star}-1\|_2}.
$$
The essential difficulty is now to control the global entropy of
$\mathcal{D}_q$.
The following notation will be used throughout:
\begin{align*}
H_{0}(x)&=\sup_{\theta\in\Theta}f_{\theta}(x)/f^{\star}(x),\\
H_{1}(x)&=\sup_{\theta\in\Theta}
\max_{i=1,\ldots,d}|
\partial f_\theta(x) / \partial\theta^i|/f^{\star}(x),\\ H_{2}(x)&=\sup_{\theta\in\Theta}
\max_{i,j=1,\ldots,d}|
\partial^{2} f_\theta(x) / \partial\theta^i\partial\theta^j|/f^{\star}(x),
\\ H_{3}(x)&=
\sup_{\theta\in\Theta}\max_{i,j,k=1,\ldots,d}|
\partial^{3} f_\theta(x) / \partial\theta^i\partial\theta^j\partial\theta^k|/f^{\star}(x)
\end{align*}
when $f_0$ is sufficiently differentiable,
$\mathcal{M}=\bigcup_{q\ge 1}\mathcal{M}_q$, and
$\mathcal{D}=\bigcup_{q\ge 1}\mathcal{D}_q$.
\begin{aspta}
The following hold:
\begin{enumerate}
\item $f_0\in C^3$ and $f_0(x)$, $(\partial f_0/\partial\theta^i)(x)$ vanish as
$\|x\|\to\infty$.
\item $H_{k}\in L^{4}(f^{\star}d\mu)$ for $k=0,1,2$ and $H_{3}\in L^{2}(f^{\star}d\mu)$.
\end{enumerate}
\end{aspta}
We can now state our main result, whose proof is given in section
\ref{sec:proofmainmixtures}.
\begin{theorem}
\label{thm:mainmixtures}
Suppose that Assumption A holds. Then there exist constants
$C^\star$ and $\delta^\star$, which depend on $d$, $q^\star$ and $f^\star$
but not on $\Theta$, $q$ or $\delta$, such that
$$
N_{[]}(\mathcal{D}_{q},\delta) \le
\bigg(
\frac{
C^\star (T\vee 1)^{1/3}
(\|H_0\|_4^4\vee\|H_1\|_4^4\vee\|H_2\|_4^4\vee\|H_3\|_2^2)
}{\delta}
\bigg)^{10(d+1)q}
$$
for all $q\ge q^\star$, $\delta\le\delta^\star$.
Moreover, there is a function $D\in L^4(f^\star d\mu)$ with
$$
\|D\|_4 \le K^\star (\|H_0\|_4\vee\|H_1\|_4\vee\|H_2\|_4),
$$
where $K^\star$ depends only on $d$ and $f^\star$, such that
$|d|\le D$ for all $d\in\mathcal{D}$.
\end{theorem}
\begin{remark}
Assumption A is essentially a smoothness assumption on $f_0$. Some
sort of smoothness is certainly needed for a result such as
Theorem \ref{thm:mainmixtures} to hold: see \cite[\S 3]{Ciu02} for
a counterexample in the non-smooth case.
\end{remark}
Combining Theorems
\ref{thm:localglobal} and \ref{thm:mainmixtures}, we immediately obtain
a local entropy bound.
\begin{theorem}
\label{thm:localmixtures}
Suppose that Assumption A holds. Then
$$
N_{[]}(\mathcal{H}_{q}(\varepsilon),\delta) \le
\bigg(
\frac{C_\Theta\,\varepsilon}{\delta}
\bigg)^{10(d+1)q+1}
$$
for all $q\ge q^\star$ and $\delta/\varepsilon\le 1$, where
$$
C_\Theta = L^\star\, (T\vee 1)^{1/3} \,
(\|H_0\|_4^4\vee\|H_1\|_4^4\vee\|H_2\|_4^4\vee\|H_3\|_2^2)^{5/4}
$$
and $L^\star$ is a constant that depends only on $d$, $q^\star$ and $f^\star$.
\end{theorem}
To illustrate these results, let us consider the important case of
Gaussian location mixtures, which are widely used in applications
(see, e.g., \cite{GW00,GvdV01,MM11}).
\begin{example}[Gaussian mixtures]
\label{ex:gaussian}
Consider mixtures of standard Gaussian densities
$f_0(x)=(2\pi)^{-d/2}e^{-\|x\|^2/2}$, and let
$\Theta(T) = \{\theta\in\mathbb{R}^d:\|\theta\|\le T\}$. Fix a nondegenerate
mixture $f^\star$, and define $T^\star=
\max_{i=1,\ldots,q^\star}\|\theta_i^\star\|$. Denote by
$\mathcal{H}_q(\varepsilon,T)$ the Hellinger ball associated to the
parameter set $\Theta(T)$. Then
$$
N_{[]}(\mathcal{H}_{q}(\varepsilon,T),\delta) \le
\bigg(
\frac{C_1^\star e^{C_2^\star T^2}\varepsilon}{\delta}
\bigg)^{10(d+1)q+1}
$$
for all $q\ge q^\star$, $T\ge T^\star$, and $\delta/\varepsilon\le 1$,
where $C_1^\star,C_2^\star$ are constants that depend on $d$, $q^\star$ and
$f^\star$ only. To prove this, it evidently suffices to show that
Assumption A holds and that $\|H_k\|_4$ for $k=0,1,2$ and $\|H_3\|_2$
are of order $e^{CT^2}$. These facts are readily verified
by a straightforward computation.
\end{example}
Let us emphasize a key feature of Theorems \ref{thm:mainmixtures} and
\ref{thm:localmixtures}: the dependence of the entropy bounds on the
order $q$ and on the parameter set $\Theta$ is explicit (see, e.g.,
Example \ref{ex:gaussian}). In particular, we find that for every
$f^\star$, the local doubling dimension of $\mathcal{M}_q$ at $f^\star$
is of the same order as the dimension of the natural parameter set for
mixtures $\Delta_{q-1}\times\Theta^q$, which answers the basic question
posed in the introduction. Obtaining this explicit dependence, which is
important in applications \cite{GasHanOrder}, is one of the main
technical challenges of the proof. In order to show only
that $N_{[]}(\mathcal{H}_{q}(\varepsilon),\delta)$ is polynomial
in $\varepsilon/\delta$ without explicit control of the order, the proof
could be simplified and substantially generalized---see Remark
\ref{rem:mixother} below for some discussion.
In contrast to the dependence on $q$ and $\Theta$, however, the proofs of
Theorems \ref{thm:mainmixtures} and \ref{thm:localmixtures} do not
provide any control of the dependence of the constants on $f^\star$. In
particular, while we can control the local doubling dimension of
$\mathcal{M}_q$ at $f^\star$ in terms of $q$, we do not know whether the
dependence on $f^\star$ can be eliminated.
\begin{remark}
We have not optimized the constants in
Theorem \ref{thm:mainmixtures} and Theorem \ref{thm:localmixtures}.
In particular, the constant $10$ in the exponent can likely be improved.
On the other hand, it is unclear whether the dependence on the diameter of
$\Theta$ is optimal. Indeed, if one is only interested in
global entropy $N_{[]}(\mathcal{H}_q,\delta)$ where
$\mathcal{H}_q=\{\sqrt{f/f^\star}:f\in\mathcal{M}_q\}$, then it can be
read off from the proof of Theorem \ref{thm:mainmixtures} that the constants
in the entropy bound depend on $\|H_0\|_1$ and $\|H_1\|_1$ only, which are
easily seen to scale polynomially in $T$ due to the translation invariance
of the Lebesgue measure. Therefore, for example in the case of Gaussian
mixtures, one can obtain a \emph{global} entropy bound which scales only
polynomially as a function of $T$, whereas the above \emph{local} entropy
bound scales as $e^{CT^2}$. The behavior of local entropies is much more
delicate than that of global entropies, however, and we do not know whether
it is possible to obtain a local entropy bound that scales polynomially in $T$
for the Hellinger metric. On the other hand, if $\mathcal{M}_q$ is endowed
with the total variation metric $d_{\rm TV}(f,g)=\int |f-g|d\mu$ rather than
the Hellinger metric, then an easy modification of our proof
yields a local entropy bound that depends only on
$\|H_i\|_1$ ($i=0,\ldots,3$), and therefore scales polynomially in $T$.
In this case the scaling matches that of the
global entropy, and is therefore optimal.
\end{remark}
\begin{remark}
\label{rem:mixother}
The problems that we address in this section could be investigated
in a more general setting. Let $\mathcal{F}=\{f_\theta:\theta\in\Theta\}$
be a given family of probability densities (where
$\Theta$ is a bounded subset of $\mathbb{R}^d$), and define
$$
\mathcal{M}_q = \bigg\{
\sum_{i=1}^q\pi_if_{\theta_i}:\pi_i\ge 0,~
\sum_{i=1}^q\pi_i=1,~\theta_i\in\Theta
\bigg\}.
$$
The case that we have considered corresponds to the choice
$\mathcal{F}=\{f_0(\,\cdot\,-\theta):\theta\in\Theta\}$,
but in principle any parametrized family $\mathcal{F}$ may be considered.
Remarkably, most of the proof of Theorem \ref{thm:mainmixtures} does not
rely at all on the specific choice of $\mathcal{F}$, so that very
similar techniques may be used to study more general mixtures. The only
point where the structure of $\mathcal{F}$ has been used is in the local
geometry Theorem \ref{thm:localgeom} below, whose proof (using Fourier
methods) relies on the specific form of location mixtures. We believe
that essentially the same result holds more generally, but a different
method of proof would likely be needed.
The proof of Theorem \ref{thm:localgeom} below is rather technical: the
difficulty lies in the fact that the result holds uniformly in the order
$q$. This is necessary in order to obtain bounds in Theorems
\ref{thm:mainmixtures} and \ref{thm:localmixtures} that depend
explicitly on $q$. If the explicit dependence on $q$ is not needed,
then our proof of Theorem \ref{thm:localgeom} can be simplified and
adapted to hold for much more general classes $\mathcal{F}$, see
\cite{GasRouHMM}.
Finally, we note that $\mathcal{M}=\bigcup_q\mathcal{M}_q$ is simply the
convex hull of $\mathcal{F}$. The problem of estimating the metric
entropy of convex hulls has been widely studied
\cite{CKP99,Gao01,Gao04,GW00,GvdV01}. In general, however, the convex
hull is infinite-dimensional, so that this problem is quite distinct
from the problems we have considered.
\end{remark}
\begin{remark}
Weighted entropy bounds as in Theorem \ref{thm:mainmixtures} are of
independent interest. A qualitative version of this bound (without
uniform control in $q$ and $T$) was assumed in \cite{DCG99},
which provided inspiration for the present effort. However, in
\cite[Prop.\ 3.1]{DCG99}, it is assumed without justification that one
can choose a multiplicative rather than additive remainder term in a
Taylor expansion. The requisite justification is provided (in a much
more precise form) by the local geometry theorem to be described
presently. Developing a precise understanding of the local geometry
of mixtures is the fundamental challenge to be surmounted in our
setting, and our
local geometry result therefore constitutes the central contribution
of this paper.
\end{remark}
\subsection{The local geometry of mixtures}
\label{sec:mixturesGeo}
At the heart of the proof of Theorem \ref{thm:mainmixtures} lies a
result on the local geometry of location mixtures, Theorem
\ref{thm:localgeom} below. Before we can develop this result, we
must introduce some notation.
Define the Euclidean balls $B(\theta,\varepsilon)=\{\theta'\in\mathbb{R}^d:
\|\theta-\theta'\|<\varepsilon\}$, denote by $\langle u,v\rangle$ the
inner product of two vectors $u,v\in\mathbb{R}^d$, and denote by
$\langle A,u\rangle= \{\langle\theta,u\rangle:\theta\in A\}\subseteq\mathbb{R}$
the inner product of a set $A\subseteq\mathbb{R}^d$ with a vector
$u\in\mathbb{R}^d$.
\begin{lemma}
\label{lem:bubble}
It is possible to choose a bounded convex neighborhood $A_i$ of
$\theta_i^\star$ for every $i=1,\ldots,q^\star$ such that, for some
linearly independent family $u_1,\ldots,u_d\in\mathbb{R}^d$,
the sets $\{\langle A_i,u_j\rangle:i=1,\ldots,q^\star\}$ are disjoint
for every $j=1,\ldots,d$.
\end{lemma}
\begin{proof}
We first claim that one can choose linearly independent
$u_1,\ldots,u_d$ such that $|\{\langle\theta_i^\star,u_j\rangle:
i=1,\ldots,q^\star\}|=q^\star$ for every $j=1,\ldots,d$. Indeed,
note that the set
$\{u\in\mathbb{R}^d:|\{\langle\theta_i^\star,u\rangle:
i=1,\ldots,q^\star\}|<q^\star\}$ is a finite union of $(d-1)$-dimensional
hyperplanes, which has Lebesgue measure zero. Therefore, if we draw
a rotation matrix $T$ at random from the Haar measure on
$\mathrm{SO}(d)$, and let $u_i=Te_i$ for all $i=1,\ldots,d$
where $\{e_1,\ldots,e_d\}$ is the standard Euclidean basis in
$\mathbb{R}^d$, then the desired property will hold with unit
probability. To complete the proof, it suffices to choose
$A_i = B(\theta_i^\star,\varepsilon/4)$ with
$\varepsilon = \min_k\min_{i\ne j}|\langle\theta_i^\star-
\theta_j^\star,u_k\rangle|$.
\end{proof}
We now fix once and for all a family of neighborhoods
$A_1,\ldots,A_{q^\star}$ as in Lemma \ref{lem:bubble}. The precise
choice of these sets only affects the constants in the proofs below and
is therefore irrelevant to our final result; we only presume that
$A_1,\ldots,A_{q^\star}$ remain fixed throughout the proofs. Let us
also define $A_0=\mathbb{R}^d\backslash(A_1\cup\cdots\cup A_{q^\star})$.
Then $\{A_0,\ldots,A_{q^\star}\}$ partitions the parameter set
$\mathbb{R}^d$ in such a way that each bounded element $A_i$,
$i=1,\ldots,q^\star$ contains precisely one component of the mixture
$f^\star$, while the unbounded element $A_0$ contains no components of
$f^\star$.
Let us define for each finite measure $\lambda$ on
$\mathbb{R}^d$ the function
$$
f_{\lambda}(x) = \int f_\theta(x)\,\lambda(d\theta).
$$
We also define the derivatives $D_1f_\theta(x)\in\mathbb{R}^d$ and
$D_2f_\theta(x)\in\mathbb{R}^{d\times d}$ as
$$
[D_1f_\theta(x)]_{i} = \frac{\partial}{\partial\theta^i}f_\theta(x),
\qquad
[D_2f_\theta(x)]_{ij} = \frac{\partial^2}{\partial\theta^i
\partial\theta^j}f_\theta(x).
$$
Denote by $\mathfrak{P}(A)$ the space of probability measures supported
on $A\subseteq\mathbb{R}^d$, and denote by $M_+^d$ the family of all
$d\times d$ positive semidefinite (symmetric) matrices.
\begin{definition}
Let us write
\begin{multline*}
\mathfrak{D} = \{(\eta,\beta,\rho,\tau,\nu):
\eta_1,\ldots,\eta_{q^\star}\in\mathbb{R},~
\beta_1,\ldots,\beta_{q^\star}\in\mathbb{R}^d,~
\rho_1,\ldots,\rho_{q^\star}\in M_+^d,\\
\tau_0,\ldots,\tau_{q^\star}\ge 0,~
\nu_0\in\mathfrak{P}(A_0),\ldots,
\nu_{q^\star}\in\mathfrak{P}(A_{q^\star})\}.
\end{multline*}
Then we define for each $(\eta,\beta,\rho,\tau,\nu)\in\mathfrak{D}$
the function
$$
\ell(\eta,\beta,\rho,\tau,\nu) =
\tau_0\frac{f_{\nu_0}}{f^\star}+
\sum_{i=1}^{q^\star}\bigg\{
\eta_i\frac{f_{\theta_i^\star}}{f^\star}
+\beta_i^*\frac{D_1f_{\theta_i^\star}}{f^\star}
+\mathrm{Tr}\bigg[\rho_i\frac{D_2f_{\theta_i^\star}}{f^\star}
\bigg]
+\tau_i\frac{f_{\nu_i}}{f^\star}
\bigg\},
$$
and the nonnegative quantity
\begin{multline*}
N(\eta,\beta,\rho,\tau,\nu) = \tau_0+
\sum_{i=1}^{q^\star}|\eta_i+\tau_i| +
\sum_{i=1}^{q^\star}
\bigg\|\beta_i+\tau_i\int(\theta-\theta_i^\star)
\,\nu_i(d\theta)\bigg\| +
\mbox{} \\
\sum_{i=1}^{q^\star} \mathrm{Tr}[\rho_i]
+ \sum_{i=1}^{q^\star}
\frac{\tau_i}{2}\int\|\theta-\theta_i^\star\|^2\nu_i(d\theta).
\end{multline*}
\end{definition}
We now formulate the key result on the local geometry of the mixture
class $\mathcal{M}$.
\begin{theorem}
\label{thm:localgeom}
Suppose that
\begin{enumerate}
\item $f_0\in C^2$ and $f_0(x)$, $D_1f_0(x)$ vanish as $\|x\|\to\infty$.
\item $\|[D_1f_0]_i/f^\star\|_1<\infty$ and $\|[D_2f_0]_{ij}/f^\star\|_1<\infty$
for all $i,j=1,\ldots,d$.
\end{enumerate}
Then there exists a constant $c^\star>0$ such that
$$
\|\ell(\eta,\beta,\rho,\tau,\nu)\|_1 \ge
c^\star\,N(\eta,\beta,\rho,\tau,\nu)\quad
\mbox{for all }(\eta,\beta,\rho,\tau,\nu)\in\mathfrak{D}.
$$
\emph{[}The constant $c^\star$ may depend on $f^\star$ and
$A_1,\ldots,A_{q^\star}$ but not on $\eta,\beta,\rho,\tau,\nu$.\emph{]}
\end{theorem}
Before we turn to the proof, let us introduce a notion that is familiar
in quantum mechanics. If $(\Omega,\Sigma)$ is a measurable space,
call the map $\lambda:\Sigma\to\mathbb{R}^{d\times d}$ a \emph{state}\footnote{
Our terminology is in analogy with the notion of a
state on the $C^*$-algebra $\mathbb{C}^{d\times d}\otimes
C_{\mathbb{C}}(\Omega)$, where $\Omega$ is a compact metric space and
$C_{\mathbb{C}}(\Omega)$ is the algebra of complex-valued continuous
functions on $\Omega$. Such states can be
represented by the complex-valued counterpart of our definition.
} if
\begin{enumerate}
\item $A\mapsto[\lambda(A)]_{ij}$ is a signed measure for
every $i,j=1,\ldots,d$;
\item $\lambda(A)$ is a nonnegative symmetric matrix for every $A\in\Sigma$;
\item $\mathrm{Tr}[\lambda(\Omega)]=1$.
\end{enumerate}
It is easily seen that for any unit vector $\xi\in\mathbb{R}^d$, the
map $A\mapsto\langle\xi,\lambda(A)\xi\rangle$ is a sub-probability measure.
Moreover, if $\xi_1,\ldots,\xi_d\in\mathbb{R}^d$ are linearly independent,
there must be at least one $\xi_i$ such that
$\langle\xi_i,\lambda(\Omega)\xi_i\rangle>0$. Finally, let
$B\subset\mathbb{R}^d$ be a compact set and let $(\lambda_n)_{n\ge 0}$
be a sequence of states on $B$. Then there exists a subsequence along
which $\lambda_n$ converges weakly to some state $\lambda$ on $B$
in the sense that $\int \mathrm{Tr}[M(\theta)\lambda_n(d\theta)]
\to \int \mathrm{Tr}[M(\theta)\lambda(d\theta)]$
for every continuous function $M:B\to\mathbb{R}^{d\times d}$.
To see this, it suffices to note that we may extract a subsequence
such that all matrix elements $[\lambda_n]_{ij}$ converge weakly
to a signed measure by the compactness of $B$, and it is evident
that the limit must again define a state.
\begin{proof}[Proof of Theorem \ref{thm:localgeom}]
Suppose that the conclusion of the theorem does not hold. Then there must
exist a sequence of coefficients
$(\eta^{n},\beta^{n},\rho^{n},\tau^{n},\nu^{n})\in\mathfrak{D}$ with
$$
\frac{\|\ell(\eta^{n},\beta^{n},\rho^{n},\tau^{n},\nu^{n})\|_1}{
N(\eta^{n},\beta^{n},\rho^{n},\tau^{n},\nu^{n})}
\xrightarrow{n\to\infty}0.
$$
Let us fix such a sequence throughout the proof.
Applying Taylor's theorem to
$u\mapsto f_{\theta_i^\star+u(\theta-\theta_i^\star)}$, we can write
for $i=1,\ldots,q^\star$
\begin{align*}
& \eta_{i}^{n}\frac{f_{\theta_{i}^\star}}{f^\star}
+ \beta_{i}^{n*} \frac{D_{1}f_{\theta_{i}^\star}}{f^\star}
+ \mathrm{Tr}\bigg[\rho_{i}^{n}
\frac{D_{2}f_{\theta_{i}^\star}}{f^\star}
\bigg]
+ \tau_{i}^{n} \frac{f_{\nu_{i}^{n}}}{f^\star}
\\ &\mbox{}=
\left(\eta_{i}^{n}+\tau_{i}^{n}\right)
\frac{f_{\theta_{i}^\star}}{f^\star}
+ \bigg(\beta_{i}^{n}+\tau_{i}^{n} \int (\theta-\theta_{i}^\star)
\,\nu_{i}^{n}(d\theta)\bigg)^*
\frac{D_{1}f_{\theta_{i}^\star}}{f^\star}
+ \mathrm{Tr}\bigg[\rho_{i}^{n}
\frac{D_{2}f_{\theta_{i}^\star}}{f^\star}
\bigg]
\\ &\quad\mbox{}
+ \frac{\tau_{i}^{n}}{2}\int \|\theta-\theta_{i}^\star\|^{2}\,
\nu_{i}^{n}(d\theta) \int \mathrm{Tr}\bigg[
\bigg\{\int_{0}^{1}
\frac{D_{2}f_{\theta_i^\star+u(\theta-\theta_{i}^\star)}}{f^\star}\,
2(1-u)\,du\bigg\} \,\lambda_{i}^{n}(d\theta)\bigg]
\end{align*}
where $\lambda_{i}^{n}$ is the state on $A_i$ defined by
$$
\int \mathrm{Tr}[M(\theta)\,\lambda_{i}^{n}(d\theta)] =
\frac{\int \mathrm{Tr}[M(\theta)\,(\theta-\theta_{i}^\star)
(\theta-\theta_{i}^\star)^*]\,
\nu_{i}^{n}(d\theta)}{
\int\|\theta-\theta_{i}^\star\|^{2}\,\nu_{i}^{n}(d\theta)}
$$
(it is clearly no loss of generality to assume that $\nu_i^n$ has
no mass at $\theta_i^\star$ for any $i,n$, so that everything is well
defined). We now define the coefficients
$$
\displaylines{
a_{i}^{n} = \frac{\eta_{i}^{n} + \tau_{i}^{n}}{
N(\eta^{n},\beta^{n},\rho^{n},\tau^{n},\nu^{n})},\qquad
b_{i}^{n} = \frac{\beta_{i}^{n} +
\tau_{i}^{n}\int(\theta-\theta_{i}^\star)\,\nu_{i}^{n}(d\theta)}{
N(\eta^{n},\beta^{n},\rho^{n},\tau^{n},\nu^{n})},
\cr
c_{i}^{n} = \frac{\rho_{i}^{n}}{
N(\eta^{n},\beta^{n},\rho^{n},\tau^{n},\nu^{n})},\qquad
d_{i}^{n} = \frac{\frac{\tau_{i}^{n}}{2}
\int\|\theta-\theta_{i}^\star\|^{2}\,\nu_{i}^{n}(d\theta)}{
N(\eta^{n},\beta^{n},\rho^{n},\tau^{n},\nu^{n})}
}
$$
for $i=1,\ldots,q^\star$, and
$$
a_0^n = \frac{\tau_{0}^{n}}{
N(\eta^{n},\beta^{n},\rho^{n},\tau^{n},\nu^{n})}.
$$
Note that
$$
|a_0^n| + \sum_{i=1}^{q^\star}\left\{|a_{i}^{n}|+\|b_{i}^{n}\|
+\mathrm{Tr}[c_{i}^{n}]+|d_{i}^{n}|\right\}=1
$$
for all $n$. We may therefore extract a subsequence such that:
\begin{enumerate}
\item There exist $a_i\in\mathbb{R}$, $b_i\in\mathbb{R}^d$,
$c_i\in M_+^d$, and $a_0,d_i\ge 0$ (for $i=1,\ldots,q^\star$) with
$|a_0|+\sum_{i=1}^{q^\star}\left\{|a_{i}|+\|b_{i}\|+\mathrm{Tr}[c_{i}]
+|d_{i}|\right\}=1$, such that $a_0^n\to a_0$ and $a_i^n\to a_i$,
$b_i^n\to b_i$, $c_i^n\to c_i$, $d_i^n\to d_i$ as $n\to\infty$
for all $i=1,\ldots,q^\star$.
\item There exists a sub-probability measure $\nu_0$ supported on
$A_0$, such that $\nu_0^n$ converges vaguely to $\nu_0$ as $n\to\infty$.
\item There exist states $\lambda_i$ supported on
$\mathop{\mathrm{cl}}A_i$ for $i=1,\ldots,q^\star$, such that $\lambda_i^n$
converges weakly to $\lambda_i$ as $n\to\infty$ for every $i=1,\ldots,q^\star$.
\end{enumerate}
The functions
$\ell(\eta^{n},\beta^{n},\rho^{n},\tau^{n},\nu^{n}) /
N(\eta^{n},\beta^{n},\rho^{n},\tau^{n},\nu^{n})$ converge
pointwise along this subsequence to the function $h/f^\star$ defined by
\begin{multline*}
h = a_0\,f_{\nu_0} +
\sum_{i=1}^{q^\star}\bigg\{
a_i\,f_{\theta_{i}^\star}
+ b_i^*\,D_{1}f_{\theta_{i}^\star}
+ \mathrm{Tr}[c_i\,D_{2}f_{\theta_{i}^\star}] \\ \mbox{}
+ d_i\int \mathrm{Tr}\bigg[\bigg\{\int_{0}^{1}
D_{2}f_{\theta_i^\star+u(\theta-\theta_{i}^\star)}\,
2(1-u)\,du\bigg\} \,\lambda_i(d\theta)\bigg]
\, \bigg\}.
\end{multline*}
But as $\|\ell(\eta^{n},\beta^{n},\rho^{n},\tau^{n},\nu^{n})\|_1 /
N(\eta^{n},\beta^{n},\rho^{n},\tau^{n},\nu^{n})\to 0$, we have
$\|h/f^\star\|_1=0$ by Fatou's lemma. As $f^\star$ is strictly positive,
we must have $h\equiv 0$.
To proceed, we need the following lemma.
\begin{lemma}
\label{lem:flem}
The Fourier transform $F[h](s):=\int e^{\mathrm{i}\langle x,s\rangle}h(x)dx$
is given by
\begin{multline*}
F[h](s) = F[f_0](s)\, \bigg[
a_0\int e^{\mathrm{i}\langle\theta,s\rangle}\,\nu_0(d\theta) +
\sum_{i=1}^{q^\star}\bigg\{
a_i\,e^{\mathrm{i}\langle\theta_{i}^\star,s\rangle}
+ \mathrm{i}\langle b_i,s\rangle\,e^{\mathrm{i}\langle\theta_{i}^\star,s\rangle}
\mbox{} \\
- \langle s,c_i s\rangle\,e^{\mathrm{i}\langle\theta_{i}^\star,s\rangle}
- d_i\,e^{\mathrm{i}\langle\theta_{i}^\star,s\rangle}
\int \phi(\mathrm{i}\langle\theta-\theta_{i}^\star,s\rangle)\,
\langle s,\lambda_i(d\theta)s\rangle
\bigg\}\bigg]
\end{multline*}
for all $s\in\mathbb{R}^d$.
Here we defined the function $\phi(u) = 2(e^u-u-1)/u^2$.
\end{lemma}
\begin{proof}
The $a_i,b_i,c_i$ terms are easily computed using integration by parts.
It remains to compute the Fourier transform of the function
$$
[\Xi_i(x)]_{jk} =
\int \bigg\{\int_{0}^{1}
[D_{2}f_{\theta_i^\star+u(\theta-\theta_{i}^\star)}(x)]_{jk}\,
2(1-u)\,du\bigg\} \,[\lambda_i(d\theta)]_{kj}.
$$
We begin by noting that
\begin{multline*}
\int \int \int_{0}^{1}
|[D_{2}f_{\theta_i^\star+u(\theta-\theta_{i}^\star)}(x)]_{jk}|\,
2(1-u)\,du\,dx\,|[\lambda_i]_{kj}|(d\theta) = \\
\|[\lambda_i]_{kj}\|_{\rm TV}\int |[D_2f_0(x)]_{jk}|\,dx < \infty.
\end{multline*}
We may therefore apply Fubini's theorem, giving
\begin{align*}
F[[\Xi_i]_{jk}](s) &=
- F[f_0](s)\,
s_j s_k\, e^{\mathrm{i}\langle\theta_i^\star,s\rangle}
\int\bigg\{\int_{0}^{1}
e^{\mathrm{i} u\langle\theta-\theta_{i}^\star,s\rangle}
2(1-u) du\bigg\} [\lambda_i(d\theta)]_{kj} \\ \mbox{}
&=
- F[f_0](s)\,
s_j s_k\,e^{\mathrm{i}\langle\theta_i^\star,s\rangle}
\int \phi(\mathrm{i}\langle\theta-\theta_i^\star,s\rangle)\,
[\lambda_i(d\theta)]_{kj},
\end{align*}
where we have computed the inner integral using integration by parts.
\end{proof}
Let $u_1,\ldots,u_d\in\mathbb{R}^d$ be a linearly independent family
satisfying the condition of Lemma \ref{lem:bubble}.
As $F[h](s)=0$ for all $s\in\mathbb{R}^d$, we obtain
$$
\Phi^\ell(\mathrm{i} t) :=
a_0\,\Phi_0^\ell(\mathrm{i} t)+
\sum_{i=1}^{q^\star}
e^{\mathrm{i} t\langle\theta_{i}^\star,u_\ell\rangle}
\big\{
a_i + \mathrm{i} t \langle b_i,u_\ell\rangle
- t^2 \langle u_\ell,c_i u_\ell\rangle
- d_i\, t^2\, \Phi_i^\ell(\mathrm{i} t)
\big\} = 0
$$
for all $\ell=1,\ldots,d$ and $t\in[-\iota,\iota]\subset\mathbb{R}$
for some $\iota>0$, where we defined
$$
\Phi_i^\ell(\mathrm{i} t) =
\int \phi(\mathrm{i} t\langle\theta-\theta_{i}^\star,u_\ell\rangle)\,
\langle u_\ell,\lambda_i(d\theta)u_\ell\rangle
$$
for $i=1,\ldots,q^\star$, and
$$
\Phi_0^\ell(\mathrm{i} t) =
\int e^{\mathrm{i} t\langle\theta,u_\ell\rangle}\,\nu_0(d\theta).
$$
Indeed, it suffices to note that $F[f_0](0)=1$ and that $s\mapsto
F[f_0](s)$ is continuous, so that this claim follows from Lemma
\ref{lem:flem} and the fact that $F[f_0](s)$ is nonvanishing in a
sufficiently small neighborhood of the origin.
As all $\lambda_i$ have compact support, it is easily seen that for every
$i=1,\ldots,q^\star$, the function $\Phi_i^\ell(z)$ is defined for all
$z\in\mathbb{C}$ by a convergent power series. The function
$\Psi^\ell(\mathrm{i} t):=\Phi^\ell(\mathrm{i} t)-a_0\,\Phi_0^\ell(\mathrm{i} t)$ is
therefore an entire function with $|\Psi^\ell(z)|\le k_1e^{k_2|z|}$ for
some $k_1,k_2>0$ and all $z\in\mathbb{C}$. But as $\Phi^\ell(\mathrm{i} t)=0$
for $t\in[-\iota,\iota]$, it follows from \cite{Luk70}, Theorem 7.2.2
that $a_0\,\Phi_0^\ell(\mathrm{i} t)$ is the Fourier transform of a finite
measure with compact support. Thus we may assume without loss of
generality that the law of $\langle\theta,u_\ell\rangle$ under the
sub-probability $\nu_0$ is compactly supported for every
$\ell=1,\ldots,d$, so by linear independence $\nu_0$ must be compactly
supported. Therefore, the function $\Phi^\ell(z)$ is defined for all
$z\in\mathbb{C}$ by a convergent power series. But as $\Phi^\ell(z)$
vanishes for $z\in\mathrm{i}[-\iota,\iota]$, we must have $\Phi^\ell(z)=0$ for
all $z\in\mathbb{C}$, and in particular
\begin{equation}
\label{eq:laplace}
\Phi^\ell(t) =
a_0\,\Phi_0^\ell(t)+
\sum_{i=1}^{q^\star}
e^{t\langle\theta_{i}^\star,u_\ell\rangle}
\big\{
a_i + t \langle b_i,u_\ell\rangle
+ t^2 \langle u_\ell,c_i u_\ell\rangle + d_i\, t^2\, \Phi_i^\ell(t)
\big\} = 0
\end{equation}
for all $t\in\mathbb{R}$ and $\ell=1,\ldots,d$.
In the remainder of the proof, we argue that (\ref{eq:laplace}) can not
hold, thus completing the proof by contradiction.
At the heart of our proof is an inductive argument. Recall that
by construction, the projections $\{\langle A_i,u_\ell\rangle:
i=1,\ldots,q^\star\}$ are disjoint open intervals in $\mathbb{R}$
for every $\ell=1,\ldots,d$.
We can therefore relabel them in increasing order: that is, define
$(\ell1),\ldots,(\ell q^\star)\in\{1,\ldots,q^\star\}$ so that
$\langle \theta_{(\ell1)}^\star,u_\ell\rangle <
\langle \theta_{(\ell2)}^\star,u_\ell\rangle < \cdots <
\langle \theta_{(\ell q^\star)}^\star,u_\ell\rangle$. The following
key result provides the inductive step in our proof.
\begin{prop}
\label{prop:finduction}
Fix $\ell\in\{1,\ldots,d\}$, and define
$$
\tilde\Phi_0^\ell(t) :=
a_0\,\Phi_0^\ell(t)+\sum_{i=1}^{q^\star}
a_i\,e^{t\langle\theta_{i}^\star,u_\ell\rangle}.
$$
Suppose that for some $j\in\{1,\ldots,q^\star\}$ we have
$\Phi^{\ell,j}(t)=0$ for all $t\in\mathbb{R}$, where
$$
\Phi^{\ell,j}(t):=
\tilde\Phi_0^\ell(t)+
\sum_{i=1}^{j}
e^{t\langle\theta_{(\ell i)}^\star,u_\ell\rangle}
\big\{
t \langle b_{(\ell i)},u_\ell\rangle
+ t^2 \langle u_\ell,c_{(\ell i)} u_\ell\rangle
+ d_{(\ell i)}\, t^2\, \Phi_{(\ell i)}^\ell(t)
\big\}.
$$
Then
$d_{(\ell j)}\langle u_\ell,\lambda_{(\ell j)}(\mathbb{R}^d)u_\ell\rangle=0$,
$\langle u_\ell,c_{(\ell j)} u_\ell\rangle=0$, and
$\langle b_{(\ell j)},u_\ell\rangle=0$.
\end{prop}
\begin{proof}
Let us write for simplicity $\theta_i^\ell =
\langle\theta_i^\star,u_\ell\rangle$, and denote by $\lambda_i^\ell$
and $\nu_0^\ell$ the finite measures on $\mathbb{R}$ defined such that
$\int f(x)\lambda_i^\ell(dx) = \int f(\langle\theta,u_\ell\rangle) \langle
u_\ell,\lambda_i(d\theta)u_\ell\rangle$ and $\int f(x)\nu_0^\ell(dx) =
\int f(\langle\theta,u_\ell\rangle) \nu_0(d\theta)$, respectively. For
notational convenience, we will assume in the following that $(\ell
i)=i$ and $\nu_0^\ell(\{\theta_i^\ell\})=0$ for all
$i=1,\ldots,q^\star$. This entails no loss of generality: the former
can always be attained by relabeling of the points $\theta_i^\star$,
while $\tilde\Phi_0^\ell$ is unchanged if we replace $\nu_0^\ell$ and
$a_i$ by
$\nu_0^\ell(\,\cdot\,\cap\mathbb{R}\backslash\{\theta_1^\ell,\ldots,
\theta_{q^\star}^\ell\})$ and $a_i +
a_0\,\nu_0^\ell(\{\theta_i^\ell\})$, respectively. Note that
$$
\langle A_i,u_\ell\rangle =
\mbox{}]\theta_i^{\ell-}, \theta_i^{\ell+}[\mbox{},
\quad\mbox{where}\quad
\theta_i^{\ell-}<\theta_i^\ell<
\theta_i^{\ell+}<\theta_{i+1}^{\ell-}\quad\mbox{for all }i
$$
by our assumptions ($\langle A_i,u_\ell\rangle$ must be an interval
as $A_i$ is convex).
\textbf{Step 1}. We claim that the following hold:
$$
a_i = 0\mbox{ for all }i\ge j+1\quad\mbox{and}\quad
a_0\,\nu_0^\ell([\theta_{j+1}^\ell,\infty[\mbox{})=0.
$$
Indeed, suppose this is not the case. Then it is easily seen that
$$
\liminf_{t\to\infty} \frac{|\tilde\Phi_0^\ell(t)|}{
e^{t\theta_{j+1}^\ell}}>0,
$$
where we have used that $\nu_0^\ell$ has no mass at
$\{\theta_1^\ell,\ldots,\theta_{q^\star}^\ell\}$. On the other hand,
as $\phi$ is positive and increasing and as $\lambda_i$ is supported
on $\mathop{\mathrm{cl}}A_i$, we can estimate
$$
0\le
\frac{t^2\,e^{t\theta_i^\ell}\,\Phi_i^\ell(t)}{e^{t\theta_{j+1}^\ell}}
\le t^2\,e^{-t(\theta_{j+1}^\ell-\theta_i^\ell)}\,
\phi(t\{\theta_j^{\ell+}-\theta_i^\ell\})\,
\lambda_i^\ell(\mathbb{R})
\xrightarrow{t\to\infty}0
$$
for $i=1,\ldots,j$. But then we must have
$$
0 =
\liminf_{t\to\infty} \frac{|\Phi^{\ell,j}(t)|}{
e^{t\theta_{j+1}^\ell}}>0,
$$
which yields the desired contradiction.
\textbf{Step 2}. We claim that the following hold:
$$
d_j\lambda_j^\ell([\theta_j^\ell,\infty[\mbox{})=0,
\quad \langle u_\ell,c_j u_\ell\rangle = 0,\quad\mbox{and}\quad
a_0\,\nu_0^\ell([\theta_j^\ell,\infty[\mbox{})=0.
$$
Indeed, suppose this is not the case. As $\nu_0^\ell(\{\theta_j^\ell\})=0$,
we can choose $\varepsilon>0$ such that
$\nu_0^\ell([\theta_j^\ell+\varepsilon,\infty[\mbox{})\ge
\nu_0^\ell([\theta_j^\ell,\infty[\mbox{})/2$. As $a_0,d_j\ge 0$, and using
that $\phi$ is positive and increasing with $\phi(0)=1$ and that
$e^{\varepsilon t}\ge (\varepsilon t)^2/2$ for $t\ge 0$,
we can estimate
\begin{multline*}
a_0\,\Phi_0^\ell(t)+ e^{t\theta_j^\ell}\big\{
t^2\langle u_\ell,c_j u_\ell\rangle+
d_j\,t^2\,\Phi_j^\ell(t)\big\} \ge \mbox{}\\
t^2\,e^{t\theta_j^\ell}\,\bigg\{
\frac{\varepsilon^2}{4}\,
a_0\,\nu_0^\ell([\theta_j^\ell,\infty[\mbox{})
+ \langle u_\ell,c_j u_\ell\rangle +
d_j\,\lambda_j^\ell([\theta_j^\ell,\infty[\mbox{})\bigg\} > 0
\end{multline*}
for all $t\ge 0$. On the other hand, it is easily seen that
$$
\frac{1}{t^2\,e^{t\theta_j^\ell}}
\left[
\sum_{i=1}^{j}
e^{t\theta_i^\ell}
\big\{
a_i
+ t \langle b_i,u_\ell\rangle \big\}
+
\sum_{i=1}^{j-1}
e^{t\theta_i^\ell}
\big\{
t^2 \langle u_\ell,c_{i} u_\ell\rangle
+ d_i\, t^2\, \Phi_i^\ell(t)
\big\}\right] \xrightarrow{t\to\infty} 0.
$$
But this would imply that
$$
0 = \lim_{t\to\infty}
\frac{\Phi^{\ell,j}(t)}{a_0\,\Phi_0^\ell(t)+ e^{t\theta_j^\ell}\{
t^2\langle u_\ell,c_j u_\ell\rangle+
d_j\,t^2\,\Phi_j^\ell(t)\}} = 1,
$$
which yields the desired contradiction.
\textbf{Step 3}. We claim that the following hold:
$$
d_j\,\lambda_j^\ell([\theta_j^{\ell-},\theta_j^\ell[\mbox{})=0
\quad\mbox{and}\quad
a_0\,\nu_0^\ell([\theta_j^{\ell-},\theta_j^\ell[\mbox{})=0.
$$
Indeed, suppose this is not the case. We can compute
\begin{multline*}
0 = \frac{d^2}{dt^2}\left(
\frac{\Phi^{\ell,j}(t)}{e^{t\theta_j^\ell}}\right)
= d_j\int e^{t(\theta-\theta_j^\ell)}\,
\lambda_j^\ell(d\theta)
+ a_0 \int e^{t(\theta-\theta_j^\ell)}\,
(\theta-\theta_j^\ell)^2\,\nu_0^\ell(d\theta) \\
\mbox{} +
\sum_{i=1}^{j-1}
\frac{d^2}{dt^2}\,
e^{-t(\theta_j^\ell-\theta_i^\ell)}\big\{
a_i + t\langle b_i,u_\ell\rangle
+ t^2\langle u_\ell,c_i u_\ell\rangle
+ d_i\,t^2\,\Phi_i^\ell(t)\big\},
\end{multline*}
where the derivative and integral may be exchanged by \cite{Wil91},
Appendix A16. We now note that as $a_0,d_j\ge 0$, we can estimate
for $t\ge 0$
\begin{multline*}
d_j\int e^{t(\theta-\theta_j^\ell)}\,
\lambda_j^\ell(d\theta)
+ a_0 \int e^{t(\theta-\theta_j^\ell)}\,
(\theta-\theta_j^\ell)^2\,\nu_0^\ell(d\theta) \ge \mbox{}\\
e^{t(\theta_j^{\ell-}-\theta_j^\ell)}
\Bigg\{
d_j\,\lambda_j^\ell([\theta_j^{\ell-},\theta_j^\ell[\mbox{})
+
a_0\int_{[\theta_j^{\ell-},\theta_j^\ell[}
(\theta-\theta_j^\ell)^2\,\nu_0^\ell(d\theta)
\Bigg\}>0.
\end{multline*}
On the other hand, as $(e^x-1)/x$ is positive and increasing, we obtain
for $t\ge 0$
\begin{align*}
& e^{-t(\theta_j^{\ell-}-\theta_j^\ell)}
\left|\frac{d^2}{dt^2}\,
e^{-t(\theta_j^\ell-\theta_i^\ell)}\,t^2\,\Phi_i^\ell(t)\right|
\\ &\quad \mbox{}
= e^{-t(\theta_j^{\ell-}-\theta_j^\ell)}
\times e^{-t(\theta_j^\ell-\theta_i^\ell)} \times
\Bigg|
(\theta_j^\ell-\theta_i^\ell)^2
\int t^2\phi(t\{\theta-\theta_i^\ell\})\,\lambda_i^\ell(d\theta)
\\ &\qquad\qquad \mbox{}
-2(\theta_j^\ell-\theta_i^\ell)
\int
\frac{e^{t(\theta-\theta_i^\ell)}-1}{\theta-\theta_i^\ell}
\,\lambda_i^\ell(d\theta)
+ \int e^{t(\theta-\theta_i^\ell)}\,\lambda_i^\ell(d\theta)
\Bigg|
\\ &\quad \mbox{}
\le e^{-t(\theta_j^{\ell-}-\theta_i^\ell)}
\Bigg\{
(\theta_j^\ell-\theta_i^\ell)^2\,t^2\,
\phi(t\{\theta_i^{\ell+}-\theta_i^\ell\})
\\ &\qquad\qquad\qquad\qquad\qquad \mbox{}
+ 2\,(\theta_j^\ell-\theta_i^\ell)\,
\frac{e^{t(\theta_i^{\ell+}-\theta_i^\ell)}-1}{
\theta_i^{\ell+}-\theta_i^\ell}
+ e^{t(\theta_i^{\ell+}-\theta_i^\ell)}
\Bigg\}
\,\lambda_i^\ell(\mathbb{R}),
\end{align*}
which converges to zero as $t\to\infty$ for every $i<j$. It follows that
$$
0 = \lim_{t\to\infty}\frac{\frac{d^2}{dt^2}\left(
\Phi^{\ell,j}(t)/e^{t\theta_j^\ell}\right)}{
d_j\int e^{t(\theta-\theta_j^\ell)}\,
\lambda_j^\ell(d\theta)
+ a_0 \int e^{t(\theta-\theta_j^\ell)}\,
(\theta-\theta_j^\ell)^2\,\nu_0^\ell(d\theta)}
=1,
$$
which yields the desired contradiction.
\textbf{Step 4}. Recall that $\lambda_j^\ell$ is supported on
$[\theta_j^{\ell-},\theta_j^{\ell+}]$ by construction. We have
therefore established in the previous steps that the following hold:
$$
d_j\langle u_\ell,\lambda_j(\mathbb{R}^d)u_\ell\rangle=
\langle u_\ell,c_j u_\ell\rangle=
a_0\,\nu_0^\ell([\theta_j^{\ell-},\infty[\mbox{})=0,\qquad
a_i=0\mbox{ for }i>j.
$$
It is therefore easily seen that
$$
0 = \lim_{t\to\infty}\frac{\Phi^{\ell,j}(t)}{t\,e^{t\theta_j^\ell}}
= \langle b_j,u_\ell\rangle.
$$
Thus the proof is complete.
\end{proof}
We can now perform the induction by starting from (\ref{eq:laplace}) and
applying Proposition \ref{prop:finduction} repeatedly. This yields
$d_j\langle u_\ell,\lambda_j(\mathbb{R}^d)u_\ell\rangle =
\langle u_\ell,c_j u_\ell\rangle = \langle b_j,u_\ell\rangle = 0$
for all $j=1,\ldots,q^\star$ and $\ell=1,\ldots,d$. As $u_1,\ldots,u_d$
are linearly independent and $c_j\in M_+^d$, this implies
that $b_j=0$, $c_j=0$ and $d_j=0$ for all $j=1,\ldots,q^\star$, so that
$$
a_0\int e^{\mathrm{i}\langle\theta,s\rangle}\,\nu_0(d\theta) +
\sum_{i=1}^{q^\star}a_i\,e^{\mathrm{i}\langle\theta_i^\star,s\rangle}
= 0
$$
for all $s\in\mathbb{R}^d$ (this follows as above by Lemma \ref{lem:flem},
$h\equiv 0$, $F[f_0](s)\ne 0$ for $s$ in a neighborhood of the origin, and
using analyticity). But
by the uniqueness of Fourier transforms, this implies that the signed
measure $a_0\,\nu_0+\sum_{i=1}^{q^\star} a_i\,\delta_{\{\theta_i^\star\}}$
has no mass. As $\nu_0$ is supported on $A_0$, this implies that
$a_j=0$ for all $j=1,\ldots,q^\star$.
We have therefore shown that $a_i,b_i,c_i,d_i=0$ for all
$i=1,\ldots,q^\star$. But recall that
$|a_0|+\sum_{i=1}^{q^\star}\{|a_i|+\|b_i\|+\mathrm{Tr}[c_i]+|d_i|\}=1$,
so that evidently $a_0=1$.
To complete the proof, it remains to note that
$$
\int \frac{\ell(\eta^{n},\beta^{n},\rho^{n},\tau^{n},\nu^{n})}{
N(\eta^{n},\beta^{n},\rho^{n},\tau^{n},\nu^{n})}\,f^\star d\mu
= \sum_{i=0}^{q^\star} a_i^n
\xrightarrow{n\to\infty} 1.
$$
But this is impossible, as
$$
\left\|
\frac{\ell(\eta^{n},\beta^{n},\rho^{n},\tau^{n},\nu^{n})}{
N(\eta^{n},\beta^{n},\rho^{n},\tau^{n},\nu^{n})}\right\|_1
\xrightarrow{n\to\infty}0
$$
by construction. Thus we have the desired contradiction.
\end{proof}
\subsection{Proof of Theorem \ref{thm:mainmixtures}}
\label{sec:proofmainmixtures}
The proof of Theorem \ref{thm:mainmixtures} consists of a sequence of
approximations, which we develop in the form of lemmas. \emph{Throughout
this section, we always presume that Assumption A holds.}
We begin by establishing the existence of an envelope function.
\begin{lemma}
\label{lem:senvelope}
Define $S = (H_0+H_1+H_2)\,d/c^\star$.
Then $S\in L^4(f^\star d\mu)$, and
$$
\frac{|f/f^\star-1|}{\|f/f^\star-1\|_1} \le S
\quad\mbox{for all }f\in\mathcal{M}.
$$
\end{lemma}
\begin{proof}
That $S\in L^4(f^\star d\mu)$ follows directly from Assumption A.
To proceed, let $f\in\mathcal{M}_q$, so that we can write $f=\sum_{i=1}^q
\pi_i f_{\theta_i}$. Then
$$
\frac{f - f^\star}{f^\star} =
\sum_{j:\theta_j\in A_0}\pi_j \frac{f_{\theta_j}}{f^\star}+
\sum_{i=1}^{q^\star}\Bigg\{
\Bigg(
\sum_{j:\theta_j\in A_i}\pi_j
-\pi_i^\star\Bigg)\frac{f_{\theta_i^\star}}{f^\star}
+ \sum_{j:\theta_j\in A_i}
\pi_j\,
\frac{f_{\theta_j}-f_{\theta_i^\star}}{f^\star}
\Bigg\}.
$$
Taylor expansion gives
\begin{multline*}
f_{\theta_j}(x)-f_{\theta_i^\star}(x)=(\theta_j-\theta_i^\star)^*
D_1f_{\theta_i^\star}(x)+ \\
\frac{1}{2}\int_{0}^{1}(\theta_j-\theta_i^\star)^*D_2f_{\theta_i^\star+u(\theta_j-\theta_i^\star)}(x)\,(\theta_j-\theta_i^\star)\, 2(1-u)\,du.
\end{multline*}
Using Assumption A, we find that
\begin{multline*}
\left|\frac{f - f^\star}{f^\star}\right|\le
\Bigg[ \sum_{j:\theta_j\in A_0}\pi_j +\sum_{i=1}^{q^\star}\Bigg\{
\Bigg|
\sum_{j:\theta_j\in A_i}\pi_j
-\pi_i^\star\Bigg| +
\Bigg\|
\sum_{j:\theta_j\in A_i}\pi_j(\theta_j-\theta_i^\star)
\Bigg\|
\\ \mbox{}
+ \frac{1}{2}
\sum_{j:\theta_j\in A_i}\pi_j\|\theta_j-\theta_i^\star\|^2
\Bigg\}\Bigg]
\,(H_0+H_1+H_2)\,d.
\end{multline*}
On the other hand, Theorem \ref{thm:localgeom} gives
\begin{multline*}
\left\|\frac{f - f^\star}{f^\star}\right\|_1\ge
c^\star
\Bigg[ \sum_{j:\theta_j\in A_0}\pi_j +\sum_{i=1}^{q^\star}
\Bigg\{
\Bigg|
\sum_{j:\theta_j\in A_i}\pi_j
-\pi_i^\star\Bigg|
\\ \mbox{}
+
\Bigg\|
\sum_{j:\theta_j\in A_i}\pi_j(\theta_j-\theta_i^\star)
\Bigg\|
+ \frac{1}{2}
\sum_{j:\theta_j\in A_i}\pi_j\|\theta_j-\theta_i^\star\|^2
\Bigg\}\Bigg].
\end{multline*}
The proof follows directly.
\end{proof}
\begin{cor}
\label{cor:denvelope}
$|d|\le D$ for all $d\in\mathcal{D}$, where $D=2S\in L^4(f^\star d\mu)$.
\end{cor}
\begin{proof}
Using $\|f-f^\star\|_{\rm TV}\le 2h(f,f^\star)$ and
$|\sqrt{x}-1|\le |x-1|$, we find
$$
|d_f| = \frac{|\sqrt{f/f^\star}-1|}{h(f,f^\star)} \le
\frac{|f/f^\star-1|}{\frac{1}{2}\|f/f^\star-1\|_1} \le
2S,
$$
where we have used Lemma \ref{lem:senvelope}.
\end{proof}
Next, we prove that the Hellinger normalized densities
$d_f$ can be approximated by chi-square normalized densities
for small $h(f,f^\star)$.
\begin{lemma}
\label{lem:chi2}
For any $f\in\mathcal{M}$, we have
$$
\left|
\frac{\sqrt{f/f^\star}-1}{h(f,f^\star)} -
\frac{f/f^\star-1}{\sqrt{\chi^2(f||f^\star)}}
\right| \le
\{4\|S\|_4^2 S+2S^2\}\,h(f,f^\star),
$$
where we have defined the chi-square divergence
$\chi^2(f||f^\star)=\|f/f^\star-1\|_2^2$.
\end{lemma}
\begin{proof}
Let us define the function $R$ as
$$
\sqrt{\frac{f}{f^\star}}-1 = \frac{1}{2}\left\{
\frac{f-f^\star}{f^\star}+R
\right\}.
$$
Then we have
\begin{multline*}
\frac{\sqrt{f/f^\star}-1}{h(f,f^\star)} -
\frac{f/f^\star-1}{\sqrt{\chi^2(f||f^\star)}} =
\frac{f/f^\star-1+R}{\|f/f^\star-1+R\|_2}-
\frac{f/f^\star-1}{\|f/f^\star-1\|_2} = \mbox{}\\
\frac{(f/f^\star-1+R)\{\|f/f^\star-1\|_2-\|f/f^\star-1+R\|_2\}
+R\|f/f^\star-1+R\|_2
}{\|f/f^\star-1+R\|_2\,\|f/f^\star-1\|_2},
\end{multline*}
so that by the reverse triangle inequality and Corollary
\ref{cor:denvelope}
$$
\left|
\frac{\sqrt{f/f^\star}-1}{h(f,f^\star)} -
\frac{f/f^\star-1}{\sqrt{\chi^2(f||f^\star)}}\right|
\le
\frac{2\|R\|_2S+|R|}{\|f/f^\star-1\|_2}.
$$
Now note that $R = -(\sqrt{f/f^\star}-1)^2\ge -(f/f^\star-1)^2$.
Therefore, by Lemma \ref{lem:senvelope},
$$
|R| \le \left(\frac{f-f^\star}{f^\star}\right)^2
\le S^2\left\|\frac{f-f^\star}{f^\star}\right\|_1^2
\le S^2\left\|\frac{f-f^\star}{f^\star}\right\|_1
\left\|\frac{f-f^\star}{f^\star}\right\|_2.
$$
The proof is easily completed using $\|f-f^\star\|_{\rm TV}\le
2h(f,f^\star)$.
\end{proof}
Finally, we need one further approximation step.
\begin{lemma}
\label{lem:deriv}
Let $q\in\mathbb{N}$ and $\alpha>0$. Then for every $f\in\mathcal{M}_q$
such that $h(f,f^\star)\le\alpha$, it is possible to choose coefficients
$\eta_i\in\mathbb{R}$,
$\beta_{i}\in\mathbb{R}^{d}$, $\rho_i\in M_+^d$ for $i=1,\ldots,q^{\star}$,
and $\gamma_i\ge 0$, $\theta_i\in\Theta$ for $i=1,\ldots, q$,
such that $\sum_{i=1}^{q^\star}\mathrm{rank}[\rho_i]\le q\wedge dq^\star$,
$$
\displaylines{
\sum_{i=1}^{q^\star}|\eta_i|\le
\frac{1}{c^\star}+\frac{1}{\sqrt{c^\star\alpha}},
\qquad
\sum_{i=1}^{q^\star}\|\beta_i\|\le
\frac{1}{c^\star}+\frac{2T}{\sqrt{c^\star\alpha}},
\cr
\sum_{i=1}^{q^\star}\mathrm{Tr}[\rho_{i}]\le \frac{1}{c^\star},
\qquad
\sum_{j=1}^q|\gamma_j|\le \frac{1}{\sqrt{c^\star\alpha}\wedge c^\star},
}
$$
and
$$
\left|
\frac{f/f^\star-1}{\sqrt{\chi^2(f||f^\star)}} -
\ell
\right|\le \frac{d^{3/2}\sqrt{2}}{3(c^\star)^{5/4}}\,
\{\|H_3\|_2\, S + H_3\}\,\alpha^{1/4},
$$
where we have defined
$$
\ell = \sum_{i=1}^{q^\star}\bigg\{
\eta_i\frac{f_{\theta_i^\star}}{f^\star}
+\beta_i^*\frac{D_1f_{\theta_i^\star}}{f^\star}
+\mathrm{Tr}\bigg[\rho_i
\frac{D_2f_{\theta_i^\star}}{f^\star}\bigg]
\bigg\}
+ \sum_{j=1}^q\gamma_j\frac{f_{\theta_j}}{f^\star}.
$$
\end{lemma}
\begin{proof}
As $f\in\mathcal{M}_q$, we can write $f=\sum_{j=1}^q\pi_jf_{\theta_j}$.
Note that by Theorem \ref{thm:localgeom}
$$
h(f,f^\star)
\ge \frac{c^\star}{4} \sum_{i=1}^{q^\star}
\sum_{j:\theta_j\in A_i}\pi_j\|\theta_j-\theta_i^\star\|^2.
$$
Therefore, $h(f,f^\star)\le\alpha$ implies
$\pi_j\|\theta_j-\theta_i^\star\|^2\le 4\alpha/c^\star$ for
$\theta_j\in A_i$. In particular, whenever $\theta_j\in A_i$, either
$\pi_j\le
2\sqrt{\alpha/c^\star}$ or $\|\theta_j-\theta_i^\star\|^2\le
2\sqrt{\alpha/c^\star}$. Define
$$
J=\bigcup_{i=1,\ldots,q^\star}\left\{j:\theta_j\in A_i,~
\|\theta_j-\theta_i^\star\|^2\le 2\sqrt{\alpha/c^\star}\right\}.
$$
Taylor expansion gives
$$
f_{\theta_j}(x)-f_{\theta_i^\star}(x)=(\theta_j-\theta_i^\star)^*
D_1f_{\theta_i^\star}(x)
+\frac{1}{2}(\theta_j-\theta_i^\star)^*D_2f_{\theta_i^\star}(x)
\,(\theta_j-\theta_i^\star)
+ R_{ji}(x),
$$
where $|R_{ji}| \le \frac{1}{6}d^{3/2}\|\theta_j-\theta_i^\star\|^3H_3$.
We can therefore write
$$
\frac{f-f^\star}{f^\star} = L +
\sum_{i=1}^{q^\star}\sum_{j\in J:\theta_j\in A_i}
\pi_j R_{ji},
$$
where we have defined
\begin{multline*}
L = \sum_{i=1}^{q^\star}\Bigg\{
\Bigg(\sum_{j\in J:\theta_j\in A_i}\pi_j-\pi_i^\star\Bigg)
\frac{f_{\theta_i^\star}}{f^\star} +
\sum_{j\in J:\theta_j\in A_i}\pi_j(\theta_j-\theta_i^\star)^*\,
\frac{D_1f_{\theta_i^\star}}{f^\star} \\
\mbox{} + \frac{1}{2}
\sum_{j\in J:\theta_j\in A_i}\pi_j(\theta_j-\theta_i^\star)^*\,
\frac{D_2f_{\theta_i^\star}}{f^\star}(\theta_j-\theta_i^\star)\Bigg\}
+ \sum_{j\not\in J}\pi_j\,\frac{f_{\theta_j}}{f^\star}.
\end{multline*}
Now note that
\begin{align*}
\left|
\frac{f/f^\star-1}{\sqrt{\chi^2(f||f^\star)}} -
\frac{L}{\|L\|_2}
\right| &\le
\frac{|f/f^\star-1|}{\|f/f^\star-1\|_2}
\frac{\|f/f^\star-1-L\|_2}{\|L\|_2}
+ \frac{|f/f^\star-1-L|}{\|L\|_2} \\
&\le
\frac{\|f/f^\star-1-L\|_2\,S+|f/f^\star-1-L|}{\|L\|_2},
\end{align*}
where we have used Lemma \ref{lem:senvelope}.
By Theorem \ref{thm:localgeom}, we obtain
$$
\|L\|_2 \ge \|L\|_1 \ge
\frac{c^\star}{2}\sum_{i=1}^{q^\star}
\sum_{j\in J:\theta_j\in A_i}\pi_j\|\theta_j-\theta_i^\star\|^2.
$$
Therefore, we can estimate
$$
\frac{|f/f^\star-1-L|}{\|L\|_2} \le
\frac{d^{3/2}H_3}{3c^\star}
\frac{\sum_{i=1}^{q^\star}\sum_{j\in J:\theta_j\in A_i}
\pi_j\|\theta_j-\theta_i^\star\|^3}{
\sum_{i=1}^{q^\star}
\sum_{j\in J:\theta_j\in A_i}\pi_j\|\theta_j-\theta_i^\star\|^2} \le
\left(\frac{4\alpha}{c^\star}\right)^{1/4}
\frac{d^{3/2}H_3}{3c^\star}
$$
where we have used the definition of $J$. Setting $\ell=L/\|L\|_2$,
we obtain
$$
\left|
\frac{f/f^\star-1}{\sqrt{\chi^2(f||f^\star)}} -
\ell
\right| \le
\frac{d^{3/2}\sqrt{2}}{3(c^\star)^{5/4}}\,
\{\|H_3\|_2\, S + H_3\}\,\alpha^{1/4}.
$$
It remains to show that for our choice of $\ell=L/\|L\|_2$, the coefficients
$\eta,\beta,\rho,\gamma$ in the statement of the lemma satisfy the
desired bounds. These coefficients are
$$
\displaylines{
\eta_i=\frac{1}{\|L\|_2}\Bigg(
\sum_{j\in J:\theta_j\in A_i}\pi_j-\pi_i^\star\Bigg),\qquad
\beta_i=\frac{1}{\|L\|_2}\sum_{j\in J:\theta_j\in A_i}
\pi_j(\theta_j-\theta_i^\star),
\cr
\rho_{i}=\frac{1}{2\|L\|_2}
\sum_{j\in J:\theta_j\in A_i}
\pi_j(\theta_j-\theta_i^\star)
(\theta_j-\theta_i^\star)^*,
\qquad
\gamma_j=\frac{\pi_j\mathbf{1}_{j\not\in J}}{\|L\|_2}.
}
$$
Clearly $\mathrm{rank}[\rho_i]\le \#\{j:\theta_j\in A_i\}\wedge d$, so
$\sum_{i=1}^{q^\star}\mathrm{rank}[\rho_i]\le q\wedge dq^\star$.
Moreover,
\begin{multline*}
\|L\|_2 \ge c^\star\Bigg[\sum_{j:\theta_j\in A_0}\pi_j +
\sum_{i=1}^{q^\star}
\Bigg\{
\Bigg|\sum_{j:\theta_j\in A_i}\pi_j-\pi_i^\star\Bigg| \\
+\Bigg\|
\sum_{j:\theta_j\in A_i}\pi_j(\theta_j-\theta_i^\star)\Bigg\|
+ \frac{1}{2}
\sum_{j:\theta_j\in A_i}\pi_j\|\theta_j-\theta_i^\star\|^2\Bigg\}
\Bigg]
\end{multline*}
by Theorem \ref{thm:localgeom}. It follows that
$\sum_{i=1}^{q^\star}\mathrm{Tr}[\rho_{i}]\le 1/c^\star$.
Now note that for $j\not\in J$ such that $\theta_j\in A_i$, we have
$\|\theta_j-\theta_i^\star\|^2>2\sqrt{\alpha/c^\star}$ by construction.
Therefore
$$
\|L\|_2
\ge c^\star\Bigg[
\sum_{j\not\in J:\theta_j\in A_0}\pi_j +
\frac{1}{2}\sum_{i=1}^{q^\star}
\sum_{j\not\in J:\theta_j\in A_i}
\pi_j\|\theta_j-\theta_i^\star\|^2
\Bigg]
\ge ( \sqrt{c^\star\alpha} \wedge c^\star )\,\sum_{j\not\in J}\pi_j.
$$
It follows that $\sum_{j=1}^q|\gamma_j|\le 1/ ( \sqrt{c^\star\alpha} \wedge c^\star )$.
Next, we note that
$$
\sum_{i=1}^{q^\star}
\Bigg|\sum_{j\in J:\theta_j\in A_i}\pi_j-\pi_i^\star\Bigg|
\le
\sum_{i=1}^{q^\star}
\Bigg|\sum_{j:\theta_j\in A_i}\pi_j-\pi_i^\star\Bigg| +
\sum_{j\not\in J:\theta_j\not\in A_0}\pi_j.
$$
Therefore $\sum_{i=1}^{q^\star}|\eta_i|\le
1/c^\star+1/\sqrt{c^\star\alpha}$. Finally, note that
$$
\sum_{i=1}^{q^\star}
\Bigg\|\sum_{j\in J:\theta_j\in A_i}
\pi_j(\theta_j-\theta_i^\star)\Bigg\| \le
\sum_{i=1}^{q^\star}
\Bigg\|\sum_{j:\theta_j\in A_i}
\pi_j(\theta_j-\theta_i^\star)\Bigg\| +
2T\sum_{j\not\in J:\theta_j\not\in A_0}\pi_j.
$$
Therefore $\sum_{i=1}^{q^\star}\|\beta_i\|\le
1/c^\star+2T/\sqrt{c^\star\alpha}$. The proof is complete.
\end{proof}
We can now complete the proof of Theorem \ref{thm:mainmixtures}.
\begin{proof}[Proof of Theorem \ref{thm:mainmixtures}]
Let $\alpha>0$ be a constant to be chosen later on, and
$$
\mathcal{D}_{q,\alpha}=\{d_f:f\in\mathcal{M}_q,~
f\ne f^\star,~
h(f,f^\star)\le\alpha\}.
$$
Then clearly
$$
N_{[]}(\mathcal{D}_q,\delta) \le
N_{[]}(\mathcal{D}_{q,\alpha},\delta) +
N_{[]}(\mathcal{D}_q\backslash\mathcal{D}_{q,\alpha},\delta).
$$
We will estimate each term separately.
\textbf{Step 1} (\emph{the first term}).
Define
$$
\mathbb{M}_{q}=
\{(m_{1},\ldots,m_{q^\star})\in
\mathbb{Z}_+^{q^\star}:m_{1}+\cdots+m_{q^\star}=q
\wedge dq^\star\}.
$$
For every $m\in\mathbb{M}_q$, we define the family of functions
\begin{multline*}
\mathcal{L}_{q,m,\alpha}= \Bigg\{
\sum_{i=1}^{q^\star}\bigg\{
\eta_i\frac{f_{\theta_i^\star}}{f^\star}
+\beta_i^*\frac{D_1f_{\theta_i^\star}}{f^\star}
+\sum_{j=1}^{m_{i}}\rho_{ij}^*\frac{D_2f_{\theta_i^\star}}{f^\star}\rho_{ij}
\bigg\}
+ \sum_{j=1}^{q}\gamma_j\frac{f_{\theta_j}}{f^{\star}}:\mbox{}\\
(\eta,\beta,\rho,\gamma,\theta)\in\mathfrak{I}_{q,m,\alpha}\Bigg\},
\end{multline*}
where
\begin{multline*}
\mathfrak{I}_{q,m,\alpha}=\Bigg\{(\eta,\beta,\rho,\gamma,\theta)\in
\mathbb{R}^{q^\star}\times (\mathbb{R}^d)^{q^\star}\times
(\mathbb{R}^d)^{m_{1}}\times\cdots\times (\mathbb{R}^d)^{m_{q^\star}}
\times \mathbb{R}^q\times\Theta^q: \mbox{}\\
\sum_{i=1}^{q^\star}|\eta_i|\le
\frac{1}{c^\star}+\frac{1}{\sqrt{c^\star\alpha}},
\qquad
\sum_{i=1}^{q^\star}\|\beta_i\|\le
\frac{1}{c^\star}+\frac{2T}{\sqrt{c^\star\alpha}},
\mbox{}\\
\sum_{i=1}^{q^\star}\sum_{j=1}^{m_{i}}\|\rho_{ij}\|^2\le \frac{1}{c^\star},
\qquad
\sum_{j=1}^q|\gamma_j|\le \frac{1}{\sqrt{c^\star\alpha}\wedge c^\star}
\Bigg\}.
\end{multline*}
Define the family of functions
$$
\mathcal{L}_{q,\alpha}= \bigcup_{m\in\mathbb{M}_{q}}\mathcal{L}_{q,m,\alpha}
$$
From Lemmas \ref{lem:chi2} and \ref{lem:deriv}, we find that for
any function $d\in\mathcal{D}_{q,\alpha}$, there exists a function
$\ell\in \mathcal{L}_{q,\alpha}$ such that (here we use that
$h(f,f^\star)\le\sqrt{2}$ for any $f$)
$$
|d-\ell| \le
\{4\|S\|_4^2 S+2S^2\}\,(\alpha\wedge\sqrt{2})
+ \frac{d^{3/2}\sqrt{2}}{3(c^\star)^{5/4}}\,
\{\|H_3\|_2\, S + H_3\}\,\alpha^{1/4}.
$$
Using $\alpha\wedge\sqrt{2}\le 2^{3/8}\alpha^{1/4}$ for all $\alpha>0$, we
can estimate
$$
|d-\ell| \le \alpha^{1/4}\,U,\qquad
U=\left( \frac{1+\|H_3\|_2}{(c^\star)^{5/4}}+
8\|S\|_4^2 + 4\right)d^{3/2}\,\{S+S^2+H_3\},
$$
where $U\in L^2(f^\star d\mu)$ by Assumption A.
Now note that if $m_1\le\ell\le m_2$ for some functions $m_1,m_2$ with
$\|m_2-m_1\|_2\le\varepsilon$, then $m_1-\alpha^{1/4}\,U\le d\le
m_2+\alpha^{1/4}\,U$ with $\|(m_2+\alpha^{1/4}\,U)
-(m_1-\alpha^{1/4}\,U)\|_2\le \varepsilon+2\alpha^{1/4}\|U\|_2$.
Therefore
$$
N_{[]}(\mathcal{D}_{q,\alpha},
\varepsilon+2\alpha^{1/4}\|U\|_2) \le
N_{[]}(\mathcal{L}_{q,\alpha},\varepsilon) \le
\sum_{m\in\mathbb{M}_{q}}N_{[]}(\mathcal{L}_{q,m,\alpha},\varepsilon)
\quad
\mbox{for }\varepsilon>0.
$$
Of course, we will ultimately choose $\varepsilon,\alpha$ such that
$\varepsilon+2\alpha^{1/4}\|U\|_2=\delta$.
We proceed to estimate the bracketing number
$N_{[]}(\mathcal{L}_{q,m,\alpha},\varepsilon)$. To this end, let
$\ell,\ell'\in\mathcal{L}_{q,m,\alpha}$, where $\ell$ is defined by the
parameters $(\eta,\beta,\rho,\gamma,\theta)\in\mathfrak{I}_{q,m,\alpha}$ and
$\ell'$ is defined by the parameters
$(\eta',\beta',\rho',\gamma',\theta')\in\mathfrak{I}_{q,m,\alpha}$. Note that
$$
\sum_{i=1}^{q^\star}\sum_{j=1}^{m_i}
\Bigg|
\rho_{ij}^*\frac{D_2f_{\theta_i^\star}}{f^\star}\rho_{ij}-
(\rho_{ij}')^*\frac{D_2f_{\theta_i^\star}}{f^\star}\rho_{ij}'
\Bigg|
\le
\frac{2d}{\sqrt{c^\star}}\,H_2
\sum_{i=1}^{q^\star}\sum_{j=1}^{m_i}
\,\|\rho_{ij}-\rho_{ij}'\|.
$$
We can therefore estimate
\begin{multline*}
|\ell-\ell'| \le
H_0\sum_{i=1}^{q^\star}|\eta_i-\eta_i'| +
H_1\sqrt{d}\sum_{i=1}^{q^\star}\|\beta_i-\beta_i'\| +
H_0\sum_{j=1}^q|\gamma_j-\gamma_j'| +
\\
\frac{\sqrt{d}}{\sqrt{c^\star\alpha}\wedge c^\star}
\,H_1 \max_{j=1,\ldots,q}\|\theta_j-\theta_j'\| +
\frac{2d\sqrt{dq^\star}}{\sqrt{c^\star}}\,H_2\,
\Bigg[
\sum_{i=1}^{q^\star}\sum_{j=1}^{m_i}
\,\|\rho_{ij}-\rho_{ij}'\|^2\Bigg]^{1/2}.
\end{multline*}
where we have used that $|f_{\theta}-f_{\theta'}|/f^\star \le
\|\theta-\theta'\|\,H_1 \sqrt{d}$ by Taylor expansion.
Therefore, writing $V=(H_0+H_1+H_2)\,d\sqrt{dq^\star}$, we have
$$
|\ell-\ell'| \le V\,
\tnorm{(\eta,\beta,\rho,\gamma,\theta)-
(\eta',\beta',\rho',\gamma',\theta')}_{q,m,\alpha},
$$
where $\tnorm{\cdot}_{q,m,\alpha}$ is the norm on
$\mathbb{R}^{(1+d)q^\star+ d(q\wedge dq^\star)+(1+d)q}$ defined by
\begin{multline*}
\tnorm{(\eta,\beta,\rho,\gamma,\theta)}_{q,m,\alpha} =
\sum_{i=1}^{q^\star}|\eta_i|
+ \sum_{i=1}^{q^\star}\|\beta_i\|
+ \sum_{j=1}^q|\gamma_j| \\
+
\frac{1}{\sqrt{c^\star\alpha}\wedge c^\star}
\max_{j=1,\ldots,q}\|\theta_j\|
+
\frac{2}{\sqrt{c^\star}}\,
\Bigg[
\sum_{i=1}^{q^\star}\sum_{j=1}^{m_i}
\,\|\rho_{ij}\|^2\Bigg]^{1/2}.
\end{multline*}
Note that if $\tnorm{(\eta,\beta,\rho,\gamma,\theta)-
(\eta',\beta',\rho',\gamma',\theta')}_{q,m,\alpha}\le \varepsilon'$,
then we obtain a bracket $\ell'-\varepsilon'V\le \ell \le
\ell'+\varepsilon'V$ of size
$\|(\ell'+\varepsilon'V)-(\ell'-\varepsilon'V)\|_2=
2\varepsilon'\|V\|_2$. Thus
$$
N_{[]}(\mathcal{L}_{q,m,\alpha},\varepsilon)
\le N(\mathfrak{I}_{q,m,\alpha},\tnorm{\cdot}_{q,m,\alpha},
\varepsilon/2\|V\|_2)\quad
\mbox{for }\varepsilon>0,
$$
where $N(\mathfrak{I}_{q,m,\alpha},\tnorm{\cdot}_{q,m,\alpha},\varepsilon')$
denotes the covering number of $\mathfrak{I}_{q,m,\alpha}$ with respect to the
$\tnorm{\cdot}_{q,m,\alpha}$-norm.
But note that, by construction, $\mathfrak{I}_{q,m,\alpha}$ is included
in a $\tnorm{\cdot}_{q,m,\alpha}$-ball of radius not exceeding
$(6+3T)/(\sqrt{c^\star\alpha}\wedge c^\star)$. Therefore, using the
standard fact that the covering number of the $r$-ball
$B(r)=\{x\in B:\tnorm{x}\le r\}$ in any $n$-dimensional normed space
$(B,\tnorm{\cdot})$ satisfies $N(B(r),\tnorm{\cdot},\varepsilon)\le
(\frac{2r+\varepsilon}{\varepsilon})^n$, we obtain
$$
N_{[]}(\mathcal{L}_{q,m,\alpha},\varepsilon) \le
\left(\frac{4\|V\|_2(6+3T)/(\sqrt{c^\star\alpha}\wedge c^\star)
+\varepsilon}{\varepsilon}
\right)^{(1+d)q^\star+ d(q\wedge dq^\star)+(1+d)q}.
$$
In particular, if $\varepsilon\le 1$ and $\alpha\le c^\star$, then
$$
N_{[]}(\mathcal{L}_{q,m,\alpha},\varepsilon) \le
\left(\frac{(24+12T)\|V\|_2/\sqrt{c^\star}
+\sqrt{c^\star}}{\varepsilon\sqrt{\alpha}}
\right)^{3(d+1)q}.
$$
Finally, note that the cardinality of $\mathbb{M}_q$ can be
estimated as
$$
\#\mathbb{M}_q =
{q^\star + q\wedge dq^\star -1 \choose q\wedge dq^\star}
\le
2^{2q},
$$
where we have used that ${n\choose k}\le 2^n$ and
$q\ge q^\star$. We therefore obtain
\begin{align*}
N_{[]}(\mathcal{D}_{q,\alpha},\delta) &\le
\sum_{m\in\mathbb{M}_{q}}
N_{[]}(\mathcal{L}_{q,m,\alpha},
\delta-2\alpha^{1/4}\|U\|_2) \\
&\le
\left(\frac{24(2+T)\|V\|_2/\sqrt{c^\star}
+\sqrt{c^\star}}{(\delta-2\alpha^{1/4}\|U\|_2)\sqrt{\alpha}}
\right)^{3(d+1)q}
\end{align*}
whenever $\delta\le 1$ and $\alpha\le (\delta/2\|U\|_2)^4\wedge c^\star$.
\textbf{Step 2} (\emph{the second term}).
For $f,f'\in\mathcal{M}_q$
with $h(f,f^\star)>\alpha$ and $h(f',f^\star)>\alpha$,
\begin{align*}
|d_f-d_{f'}| &=
\frac{|
(\sqrt{f/f^\star}-1)(h(f',f^\star)-h(f,f^\star))+
(\sqrt{f/f^\star}-\sqrt{f'/f^\star})h(f,f^\star)|}{h(f,f^\star)h(f',f^\star)}
\\ &\le
\frac{
|\sqrt{f/f^\star}-1|}{h(f,f^\star)}
\frac{\|\sqrt{f'/f^\star}-\sqrt{f/f^\star}\|_2}
{h(f',f^\star)}
+
\frac{|\sqrt{f/f^\star}-\sqrt{f'/f^\star}|}{h(f',f^\star)}
\\ &\le
\|\sqrt{f'/f^\star}-\sqrt{f/f^\star}\|_2\,
\frac{2S}{\alpha}
+
\frac{|\sqrt{f/f^\star}-\sqrt{f'/f^\star}|}{\alpha},
\end{align*}
where we have used Corollary \ref{cor:denvelope}.
Now note that
$$
\big|\sqrt{a}-\sqrt{b}\big|^2 \le
\big|\sqrt{a}-\sqrt{b}\big|\left(\sqrt{a}+\sqrt{b}\right)
=|a-b|
$$
for any $a,b\ge 0$. We can therefore estimate
$$
|d_f-d_{f'}| \le
\frac{
\|(f-f')/f^\star\|_1^{1/2}2S +
|(f-f')/f^\star|^{1/2}}{\alpha},
$$
Now note that if we write
$f=\sum_{i=1}^q\pi_if_{\theta_i}$ and
$f'=\sum_{i=1}^q\pi_i'f_{\theta_i'}$, then
$$
\left|\frac{f-f'}{f^\star}\right| \le
H_0\sum_{i=1}^q|\pi_i-\pi_i'| +
H_1\sqrt{d}\max_{i=1,\ldots,q}\|\theta_i-\theta_i'\|.
$$
Defining
$$
W = \|H_0+H_1\sqrt{d}\|_1^{1/2}2S +
(H_0+H_1\sqrt{d})^{1/2},
$$
we obtain
$$
|d_f-d_{f'}| \le
\frac{W}{\alpha}\,\tnorm{(\pi,\theta)-(\pi',\theta')}_q^{1/2},
\qquad
\tnorm{(\pi,\theta)}_q=\sum_{i=1}^q|\pi_i| +
\max_{i=1,\ldots,q}\|\theta_i\|
$$
(clearly $\tnorm{\cdot}_q$ defines a norm on $\mathbb{R}^{(d+1)q}$).
Now note that if $\tnorm{(\pi,\theta)-(\pi',\theta')}_q\le\varepsilon$,
then we obtain a bracket $d_{f'}-\varepsilon^{1/2}W/\alpha\le
d_f\le d_{f'}+\varepsilon^{1/2}W/\alpha$ of size
$\|(d_{f'}+\varepsilon^{1/2}W/\alpha)-(d_{f'}-\varepsilon^{1/2}W/\alpha)\|_2
= 2\varepsilon^{1/2}\|W\|_2/\alpha$. Therefore
$$
N_{[]}(\mathcal{D}_q\backslash\mathcal{D}_{q,\alpha},\delta)
\le
N(\Delta_q\times\Theta^q,
\tnorm{\cdot}_q,\alpha^2\delta^2/4\|W\|_2^2),
$$
where we have defined the simplex
$\Delta_q=\{\pi\in\mathbb{R}^q_+:\sum_{i=1}^q\pi_i=1\}$.
We can now estimate the quantity on the right hand side of this expression
as before, giving
$$
N_{[]}(\mathcal{D}_q\backslash\mathcal{D}_{q,\alpha},\delta)
\le
\left(
\frac{8(1+T)\|W\|_2^2 +(c^\star)^4}{\alpha^2\delta^2}
\right)^{(d+1)q}
$$
for $\delta\le 1$ and $\alpha\le c^\star$.
\textbf{End of proof.} Choose $\alpha=(\delta/4\|U\|_2)^4$. Collecting
the various estimates above, we find that for
$\delta\le 1\wedge 4(c^\star)^{1/4}$ (as $\|U\|_2\ge\|S\|_1\ge 1$
by Lemma \ref{lem:senvelope})
\begin{align*}
N_{[]}(\mathcal{D}_{q},\delta)
&\le
\left(\frac{768(2+T)\|U\|_2^2\|V\|_2/\sqrt{c^\star}
+32\|U\|_2^2\sqrt{c^\star}}{\delta^3}
\right)^{3(d+1)q} \\
&\qquad\quad\mbox{}+
\left(
\frac{4^{10}(1+T)\|U\|_2^{8}\|W\|_2^2
+ 4^{8}\|U\|_2^{8}(c^\star)^4}{\delta^{10}}
\right)^{(d+1)q} \\
&\le
\left(
\frac{
c_0^\star\,(T\vee 1)^{1/3}\,
(\|U\|_2\vee\|V\|_2\vee\|W\|_2)
}{\delta}
\right)^{10(d+1)q}
\end{align*}
where $c_0^\star$ is a constant depends only on $c^\star$. It follows that
$$
N_{[]}(\mathcal{D}_{q},\delta) \le
\left(
\frac{
C^\star (T\vee 1)^{1/3}
(\|H_0\|_4^4\vee\|H_1\|_4^4\vee\|H_2\|_4^4\vee\|H_3\|_2^2)
}{\delta}
\right)^{10(d+1)q}
$$
for all $\delta\le\delta^\star$, where $C^\star$ and $\delta^\star$
are constants that depend only on $c^\star$, $d$, and $q^\star$.
This establishes the estimate given in the statement of the Theorem.
The proof of the second half of the Theorem follows from
Corollary \ref{cor:denvelope} and $\|H_0\|_4\ge 1$.
\end{proof}
\textbf{Acknowledgments.}
The authors thank Jean Bretagnolle for providing an
enlightening counterexample that guided some of our proofs, and
the anonymous referee for very helpful comments that have improved
the presentation.
\bibliographystyle{amsplain}
|
1,116,691,499,238 | arxiv | \section{Introduction}
Extragalactic planetary nebulae (PNe) are detectable at great distances, via their strong [\ion{O}{iii}] 5007 \AA{} emission, with surveys extending out as far as the Coma Cluster, $\sim$ 100 Mpc, where PNe were detected within the intracluster medium \citep{gerhard_detection_2005}. Their ionised [\ion{O}{iii}] 5007 \AA{} emission is unresolved at such distances, and when embedded within the bright stellar continuum of the galaxy only a careful modelling of the stellar continuum allows one to measure the PNe [\ion{O}{iii}] flux.
This is why traditionally the study of extragalactic PNe is focused in the halo region of galaxies, where the stellar background is minimal.
Previous studies have applied a range of observation techniques for the task of detecting extragalactic PNe: photometric observations using either 'on-off' band imaging or narrow-band filters of regions outside the galaxy central core \citep{mcmillan_planetary_1993, feldmeier_calibrating_2007, herrmann_planetary_2008}, while counter-dispersed slit-less spectroscopy was employed for the Planetary Nebulae Spectrograph (PN.S) instrument, used in \citet{douglas_pns_2007} and \citet{coccato_kinematic_2009}, and later in the extended PN.S (ePN.S) in \citet{pulsoni_extended_2018}.
However, these techniques fall short when exploring the central regions of galaxy, where spectroscopy is instead needed to uncover the presence of PNe amidst the stellar background and diffuse ionised-gas emission.
Integral-field spectroscopy is ideal in this respect, in particular when using instruments such as MUSE \citep[Multi Unit Spectroscopic Explorer][]{bacon_muse_2010} which, with its large field-of-view allows one to efficiently map the PNe populations of nearby galaxies \citep[e.g. ][]{kreckel_revised_2017}
An example of the growing spatial coverage that single observations can now achieve, across a single galaxy, would be M31. Thanks to its relatively close proximity, the PNe are detectable across the entire galaxy \citep{ciardullo_planetary_1989, merrett_deep_2006, pastorello_planetary_2013, jacoby_survey_2013, martin_sitelle_2018, bhattacharya_survey_2019-1, bhattacharya_survey_2019}.
Over the past 32 years, one of the chief motivations for detecting extragalactic PNe has been to use them as tracer for the stellar kinematics in the faint outskirts of galaxies, where direct spectroscopic measurement are prohibitive. Indeed, numerous studies have shown how PNe can be used in this way to probe the structure of galaxy stellar halos and to measure their dark matter content\citep{romanowsky_dearth_2003,merrett_deep_2006, douglas_pns_2007, coccato_kinematic_2009, kafle_need_2018, martin_sitelle_2018, longobardi_kinematics_2018, pulsoni_extended_2018, bhattacharya_survey_2019-1}.
At the same time, research has also shown that the distribution of PNe absolute magnitudes in [\ion{O}{iii}] 5007 \AA{}, $\mathrm{M_{5007}}$, appears to be nearly invariant between galaxies, with a noticeable cut-off at the bright end \citep{jacoby_planetary_1989-1, ciardullo_planetary_1989, jacoby_planetary_1989, ciardullo_planetary_1989-1}. Notable discrepancies around the invariant nature of the PNLF arise when observing beyond 2 magnitudes from the cut-off, with evidence of dips appearing towards the faint end of the PNLF within the Large and Small Magellanic Clouds (LMC and SMC, respectively), as found by \citet{jacoby_survey_2002} and \citet{hernandez-martinez_emission_2009}. As such, a generalised analytical planetary nebulae luminosity function (PNLF) was first proposed by \citet{ciardullo_planetary_1989}, who used the PNe observed in M31 as Cepheid Variable distance estimates to calibrate the PNLF bright-end absolute magnitude [\ion{O}{iii}] 5007 \AA{}, $\mathrm{M_{5007}^{*}}$ at a value of $-4.48$ mag.
More recently, a value of $\mathrm{M_{5007}^{*}} = -4.53 \pm 0.06$ mag has been decided on for galaxies with stellar populations that are more metal rich than the LMC \citep{ciardullo_planetary_2012}, with the stated census that $\mathrm{M_{5007}^{*}}$ does fade in smaller, more metal-poor galaxies.
In regards to extra-galactic distance determinations, the PNLF method has proven itself a reliable and unique tool, in particular given that PNe are expected to reside within all types of galaxies.
Supernovae type Ia (SNIa) measurements are indeed serendipitous by nature, whereas using surface brightness fluctuations (SBF) and Cepheid variable stars is restricted mostly only to early- and late-type galaxies, respectively.
Previous works found evidence of a systematic difference between distances derived from the PNLF and SBF \citep{ciardullo_comparison_1993}, although more recently both \citet{kreckel_revised_2017} and \citet{hartke_halo_2017}
found that their PNe-based distances, to NGC628 and M49 respectively, are in good agreement with SBF measurements.
As for the application of the PNLF method to the Fornax cluster, beyond our previous work \citep[][hereafter S20]{spriggs_fornax_2020}, there are recorded efforts using the aforementioned photometry and slit-less spectroscopy approaches \citep{mcmillan_planetary_1993, teodorescu_planetary_2005, feldmeier_calibrating_2007}. Each one reporting good agreement to the respective distances measurements available to their dates. Now though, with more detailed SBF, SNIa, and other distance estimators, we see that the Fornax cluster may be further away than the initial estimates ranging between 16 Mpc and 19 Mpc.
Here, we continue the detection work of S20, and carry out a census of PNe populations within the central, middle and halo regions of the remaining early-type galaxies (ETGs) of the Fornax3D (F3D) survey \citep{sarzi_fornax3d_2018}. We introduce our PNLF modelling methodology, improving upon the method of S20. We then report independent distance estimates for each galaxy, and compare our findings with other results and techniques from the literature.
This paper is organised as follows. In Sect.~2, we introduce the F3D data set, along with reduction steps. Sect.~3 reviews the process of PNe detection, filtering, and cataloguing, as introduced in S20, then develops on these practises through the introduction of our PNLF modelling approach and how we tested the robustness and reliability of this method. The PNe catalogue is introduced in Sect.~4, detailing the contents, with notes on objects of unique interest. This is accompanied by the PNLF distances and how they stack up against distance measurements from other methods. We also briefly discuss the 3D structure of the Fornax cluster, as prescribed by our PNLF distances. Finally, we wrap up our findings and draw some future perspectives in Sect.~5
\section{Data}
Our work uses the MUSE data of the F3D survey, in the Wide Field Mode (WFM) so as to reach a spatial sampling of 0.2\arcsec $\times$ 0.2\arcsec\ across a $1\arcmin \times 1\arcmin$ field of view (FOV). The wavelength coverage that MUSE spans 4650--9300 \AA{}, with a spectral sampling of 1.25 \AA{} $\textrm{pixel}^{-1}$. The Fornax3D survey included multiple MUSE pointings (between two and three) to cover extended objects, and to ensure in particular that observations for ETGs reached down to the same limiting surface brightness of $\mu_B=25\ \textrm{mag} \ \textrm{arcsec}^{-2}$. The survey ensured that the central pointings would be observed under good seeing conditions (for a FWHM $\leq$ 0.8\arcsec) precisely to enable a study of point sources such as PNe and globular clusters \citep[e.g.\ ][]{fahrion_constraining_2019} as well as of galactic nuclei. In the outer regions the main interest was to characterise the properties of stellar population in the faint outskirts of galaxies, which is why these regions were observed under more lenient seeing conditions but over longer exposure times.
The MUSE data was reduced using the MUSE pipeline \citep{weilbacher_design_2012, weilbacher_muse-drp_2016}, within the ESOREFLEX \citep{freudling_automated_2013} environment, as reported in \citet{sarzi_fornax3d_2018} and \citet{iodice_fornax_2019}. The reduction phase deals with key steps such as telluric correction, sky-subtraction and absolute flux calibration. All datacubes have been flux calibrated and this flux calibration has been further verified using HST images, as in \citet{sarzi_fornax3d_2018} and S20.
The Fornax3D data taken from different pointings were aligned and presented as a single mosaic in \citet{iodice_fornax_2019}. For the purposes of detecting PNe it is important to account for the different image quality of these different observations, and therefore here we analyse separately the combined observations from each individual pointing.
\subsection{Residual cube preparation}
Our PNe detection method is based on residual data, that is, reduced MUSE spectra that have first been analysed by the Galaxy IFU Spectroscopy Tool pipeline \footnote{\href{https://abittner.gitlab.io/thegistpipeline}{https://abittner.gitlab.io/thegistpipeline}} \citep[GIST,][]{bittner_gist_2019} and finalised by removing the stellar continuum. The GIST pipeline employs both pPXF \citep{cappellari_parametric_2004,cappellari_improving_2017} and PyGandALF (see GIST documentation): the first handles spaxel-by-spaxel modelling of the stellar continuum, using the full MILES stars library of templates \citep{vazdekis_miuscat_2012,vazdekis_uv-extended_2016} while the second focuses on the nebular emissions. The results of the GIST pipeline efforts are then utilised to produce a residual data cube after subtracting the modelled stellar continuum, which therefore still contains the emission lines. We then later re-fit the residual cube for [\ion{O}{iii}] emission only, imposing a fixed line-width on the emission lines. This was found in S20 to produce a better map in [\ion{O}{iii}] 5007 \AA{} emissions, which led to a greater number of PNe detected.
\section{Methods}
\subsection{Compiling the PNe catalogue}
In order to detect and characterise PNe in our sample galaxies using the MUSE data we adopt the methodology that is described in full in S20\footnote{Code hosted on Github: \href{https://github.com/tspriggs}{https://github.com/tspriggs} \citep{thomas_spriggs_tspriggsmuse_pne_fitting_2021}} and which consists of the following steps:
Firstly, we carry out a spaxel-by-spaxel fit to the [\ion{O}{iii}] 4959,5007 \AA{} emission line doublet in the residual cube, imposing a fixed intrinsic line-width of $30\,\rm km\,s^{-1}$ typical of unresolved PNe in order to better highlight the presence of such unresolved [\ion{O}{iii}] sources. Previously identified regions with diffuse nebular emission or with significant and systematic stellar-continuum fit residuals are masked.
Secondly, we extract MUSE residual minicubes around each candidate PNe source 1.8\arcsec$\times$1.8\arcsec wide and over a 100 \AA\ wavelength range. This is followed by the evaluation of the imaging point-spread function (PSF). This is achieved through the simultaneous fit of the residual minicubes for the four best-detected PNe candidate sources, in terms of their central value for the A/rN ratio, using the 3D cube-fitting process described in S20.
Thirdly, we fit for the PNe candidate total [\ion{O}{iii}] flux and kinematics using the 3D cube-fitting process of S20 while now holding the spatial distribution of the [\ion{O}{iii}] flux to that of the previously derived PSF. following this, we conduct an evaluation of the minicube fit quality and point-source detection. Candidate PNe with central A/rN ratio less than three are deemed undetected and are discarded, as are sources whose minicube fits returned $\chi^2$ values above the 3$\sigma$ limit for the $\chi^2$ statistics (for the given number of degree of freedom).
Finally, we conduct a spectral characterisation and PNe impostor identification process. After extracting PSF-weighted aperture spectra around each PNe candidate sources (from the original MUSE cube) we used GIST's GandALF fits along to extract H$\beta$, H$\alpha$, [\ion{N}{ii}]\,6583 and [\ion{S}{ii}]\,6716,6731 doublet line measurements, subsequently identifying PNe impostor sources from supernovae remnants or compact \ion{H}{ii} regions using diagnostic diagrams as seen in S20. This is accompanied by the identification of any kinematic interlopers among the remaining PNe candidate. PNe sources with exceedingly high or low velocities (beyond a 3 $\sigma$ limit) compared to the stellar line-of-sight velocity distribution (LOSVD) along the PNe direction are not considered to belong to the galaxy under consideration.
Following these steps, we finally arrive at a catalogue of 1350 unique PNe across 21 ETGs in the Fornax cluster, located either within the central, middle or halo pointings of the F3D survey.
For these objects, we list their spatial coordinates, line-of-sight velocities, as well as their measured fluxes, and magnitudes with appropriate errors, in the measured [\ion{O}{iii}] 5007\AA{} emission lines.
The catalogue also includes PNe impostor sources and PNe interlopers, as well as a handful of PNe objects that were considered to be over-luminous (see, e.g. the case of FCC~167 in S20) and which will also be excluded by our following PNLF analysis.
\subsection{PNLF fitting and distance estimation}
In S20 we followed the simplest approach for estimating the distance to a galaxy based on the brightest observed PNe. In this case the PNLF distance modulus is simply $\mu_\mathrm{PNLF} = \mathrm{m_{5007,\rm min}} - \mathrm{M_{5007}^{*}}$ where $\mathrm{m_{5007,\rm min}}$ is the [\ion{O}{iii}] 5007 \AA{} magnitude of the brightest PN and $\mathrm{M_{5007}^{*}}$ is the M31 Cepheid distance calibrated absolute magnitude of the PNLF bright cut off.
This approach is relatively sound in the case of galaxies presenting a significant number of PNe (as in FCC~167 and FCC~219, the objects studied in S20) but will clearly lead to biased results when dealing with few PNe. Indeed in this case we are more likely to miss the rarest objects at the bright cut-off of the PNLF, thus deducing an overestimated distance.
Due to a large range in the number of PNe observed across our 21 galaxies, we decided to implement a distance estimation technique based on the entire observed PNLF, taking full advantage of our knowledge for the PNe-detection completeness function in each object in order to exploit the whole range of observed $\mathrm{m_{5007}}$ values. We follow the completeness correcting method of S20, assessing the level of incompleteness of our samples over the observed FOV. In short though, we calculate the completeness of detection for a given apparent magnitude ($m_{5007}$) from the fraction of galaxy stellar light within the MUSE FOV, where a PNe of that particular magnitude can be detected. This procedure is also informed by the PSF of the observation, along with the signal to noise cut of 3 $\times$ A/rN. This is also the method applied, and described, within \citet{galan-de_anta_fornax_2021}.
Then, for each galaxy we explore the parameter space of $\mu_\mathrm{PNLF}$, evaluating for a value that would lead to a completeness-corrected PNLF model through simply minimising the Kolmogorov-Smirnov statistic D. To achieve this, we utilise the Python based Scipy package's 'KS\_1samp' function, passing the supremum value (D) to the scalar minimiser of LMfit \citep{newville_lmfit_2014, newville_lmfitlmfit-py_2019}, applying the Nelder-Mead optimisation method \citep{nelder_simplex_1965}.
In terms of model PNLF, we used the analytic form provided by \citet{longobardi_planetary_2013}
\begin{equation}
N(M) = c_{1} \ e^{ \ c_2 \ M_{5007}} \ \big[1 - e^{ \ 3( M^{*}_{5007} - \ M_{5007} )}\big],
\label{eq:PNLF}
\end{equation}
where $c_{1}$ is a normalisation factor, $c_{2}$ defines the functional form of the PNLF, $\mathrm{M_{5007}^{*}}$ is the bright-end cutoff value first and finally, $\mathrm{M_{5007}}$ is the absolute magnitude of the PNe in [\ion{O}{iii}] 5007\AA{}. In that follows we adopt the most recently calibrated values of $\mathrm{M_{5007}^{*}} = -4.53 \pm 0.06$ for the bright-end cut-off \citep{ciardullo_planetary_2012} and hold $c_{2}$ to the standard value of 0.307 derived by \citet{ciardullo_planetary_1989}.
We note that systematic variations in $\mathrm{M_{5007}^{*}}$ have been reported at very low stellar metallicities, below [M/H] < -0.5 \citep{bhattacharya_survey_2021}, but since only one of our objects (FCC\,119, see Tab. 1C in \citet{iodice_fornax3d_2019}) would appear to fall in this regime, such potential systematic effect would not greatly impact our overall results.
Appendix A shows that in principle our methodology can estimate the $c_{2}$ parameter but also that our current data do not provide strong evidence for departing from the assumed $c_{2} = 0.307$ value. Since the KS-statistics is independent of normalisation, we effectively ignore $c_{1}$ while looking for the best $\mu_\mathrm{PNLF}$ value through our KS-statistic minimisation. Instead, we finally adjust the normalisation of our best completeness-corrected model for the observed PNLF by simply integrating it and matching the result to observed number of PNe.
In this case, availing of the entire PNe sample allowsfor tighter constraints on any estimates for the total number of expected PNe by integrating the PNLF down to some magnitude limit, as this depends on the original Poisson uncertainty on the actual number of PNe used to constrain the PNLF model in first place.
\placefigsimsDMtwoobjects
\placefigsimsDM
\placefigerrorcomparison
To evaluate our uncertainties around our best $\mu_\mathrm{PNLF}$ values we resort to simulations. Namely, starting from the best-model PNLF (at the best $\mu_\mathrm{PNLF}$ and accounting for the completeness), we sample such a parent distribution $N$ times to produce a $N$ synthetic observed PNLF, which we then re-fit through our KS-statistic minimisation routine to obtain $N$ best $\mu_\mathrm{PNLF}$ values. The $16^{\rm th}$ and $84^{\rm th}$ percentile of such a $\mu_\mathrm{PNLF}$ distribution provides our $\pm 1\sigma$ uncertainties on the measured $\mu_\mathrm{PNLF}$. Typically, $N=100$ deliver stable results.
In sampling the best-model PNLF we account for Poisson statistics. For this, each time we sample the parent PNLF model we use a varying number drawn from a Poisson distribution with expectation value equal to the observed number of PNe. Accounting for Poisson statistics is necessary considering how the formation of PNe is itself a stochastic process. Put it simply, the observed number of PNe is likely to be different even if we were hypothetically to observe in the same way two identical galaxies at the same distance, which in turn would lead to slightly different PNLF distance estimates.
To further account for observational errors in our magnitude measurements, we perturb our original data according to our measured errors and add in quadrature to the error budget the standard deviation for the $\mu_\mathrm{PNLF}$ distribution that is obtained in this way.
Both accounting for Poisson statistics and observational errors contribute very little to the error budget, around 3\% an 2\%, respectively. Indeed, as will be shown in the next section, how well $\mu_\mathrm{PNLF}$ can be estimated depends primarily on the number of PNe and on depth of our detection completeness.
Finally, we note that \cite{ciardullo_planetary_2012} report a formal error of 0.06 mag for $\mathrm{M_{5007}^{*}}$, which we also add in quadrature to our PNLF errors.
\subsection{PNLF simulations}
Simulations act as a useful diagnostic tool to understand how our PNLF modelling handles different PNe samples sizes ($N_\mathrm{{PNe \, sim}}$), especially when dealing with small PNe numbers where we wish to understand whether and to which extent our solution may be biased.
The simulations are based on a finely sampled intrinsic PNLF model that assumes the standard $c_\mathrm{2 \, in}$ = 0.307 value, which is then shifted taking a distance modulus of $\mu_\mathrm{PNLF \, in}$ = 31.45 mag and finally corrected for incompleteness. From such a parent PNLF profile we then draw $N_\mathrm{{PNe \, sim}}$ apparent $m_\mathrm{{5007\,sim}}$ values to create a large set of synthetic PNLF, with $m_\mathrm{{5007\,sim}}$ varying between 5 and 200 in order to cover the range of recorded $N_\mathrm{{PNe}}$ seen in our central pointings (Table \ref{tab:galaxy_info}). Each synthetic PNLF is then passed to our PNLF-fitting algorithm for parameter estimation.
Assuming the PNLF given in Eq.~\ref{eq:PNLF} and while only fitting for $\mu_\mathrm{PNLF}$, Fig.~\ref{fig:simulations_dM_FCC193_FCC147} shows how this parameter is recovered with increasing precision with increasing PNe sample size, without incurring in any significant bias even at low $N_\mathrm{{PNe \, sim}}$ numbers. The top panel of Fig.~\ref{fig:simulations_dM_FCC193_FCC147} is based on simulations that adopt the completeness profile observed in FCC~193, where our data allow one to detect PNe some 2.5 magnitudes below the PNLF bright cut-off, whereas the lower panel shows the case of FCC~147 where only probe the PNLF 1.5 magnitudes below this limit. In both instances the scatter in the recovered $\mu_\mathrm{PNLF}$ scales as $1/\sqrt{N_\mathrm{{PNe \, sim}}}$, but at a given number of PNe the accuracy in recovering $\mu_\mathrm{PNLF}$ would actually decrease for deeper the observations.
Accounting within our simulation for the entire range of completeness functions observed across our sample (Fig.~\ref{fig:simulations_dM}) produces an average trend with $N_\mathrm{{PNe \, sim}}$ for the scatter in the recovered $\mu_\mathrm{PNLF}$ that captures well the errors in our actual $\mu_\mathrm{PNLF}$ measurements for our 21 objects (Fig.~\ref{fig:sim_data_error_comparison}).
Overall, this exercise gives us confidence in our $\mu_\mathrm{PNLF}$ estimates and errors, even in objects with less than 20 PNe.
\section{Results and discussion}
In this section, we first review the catalogue of the PNe found within the ETG population of the F3D survey. Then, moving onto the results of applying our PNLF fitting method (see Sect.~3) to each galaxy. We then discuss the accuracy of the PNLF as a distance indicator, when compared to SBF measurements.
\subsection{The F3D PNe catalogue}
We present a catalogue of PNe detected across 21 galaxies, reporting a total of 1350 unique PNe. Though the PNe of each galaxy are our main interest, each galaxy's catalogue contains sources highlighted as impostors, following the criteria of S20. Such impostors are labelled, as described in Section~3.1, following the criteria of S20: supernova remnants (SNR), HII regions, and over-luminous sources (of which we find two in total; one each in FCC\,167 and FCC\,276). Only the sources in our catalogues that have the ID of 'PN' are used for the PNLF modelling and distance estimate efforts.
A sample of our catalogued PNe, from FCC\,083, can be found in appendix A, where the two regions covered in the F3D survey are labelled as: 'center' and 'halo'. The rest of the catalogue is stored as digital records, uploaded to the VizieR On-line Data Catalog. The table contains a 'Source ID' column that follows the same naming convention introduced in S20, using the F3D label used at the start. The RA and Dec are included for future source cross-matching. '$m_{5007}$' is the apparent magnitude of the [\ion{O}{iii}] 5007 \AA{} emission. 'A/rN' is the amplitude-to-residual noise ratio, essentially a signal-to-noise measure. 'LOSV (km s$^{-1}$)' is the reported observed line-of-sight velocity of the source, derived from the wavelength position of the [\ion{O}{iii}] 5007 \AA{} emission line. The 'label' column states each source's assigned identity: PN, SNR (supernova remnant), HII (compact HII region), or OvLu (over luminous source). Finally, an index column is included for easier source comparison, highlighting sources that appear in the overlap of pointings: center C-00, middle M-00 and halo H-00.
All the galaxies in Table~\ref{tab:galaxy_info}, are shown in Figs.~\ref{fig:all_PNLF} and \ref{fig:F3D_FOV}. Every galaxy's FOV is shown via their signal-to-noise map in [\ion{O}{iii}] 5007 \AA{}, with catalogued PNe circled. Alongside the FOV, we present the modelled PNLF of the catalogued sources. If there is halo or middle data present, they are also presented in different colours, as explained in the legends of each plot. The blue histograms show the distribution of detected PNe $m_{5007}$ values, while the black line indicates a scaled form of the \citet{jacoby_planetary_1989} form of the PNLF. We then scale this PNLF to form the incompleteness-corrected PNLF, shown by dashed lines (colour dependant on pointing location). The blue shaded regions surrounding the C89 PNLF line, as well as the dashed incompleteness-corrected profile, show the 1 $\sigma$ confidence limit. After reviewing the literature for works that may contain matching PNe, the only other surveys that contained such matches (for FCC\,167 and FCC219 by \citet{feldmeier_calibrating_2007} and \citet{mcmillan_planetary_1993} respectively) have already been discussed and compared with our samples in S20. We are then confident that we are cataloguing sources never before detected.
\placefigvelocitycomparison
\subsection{Dynamical tracers}
The PNe observed within the central regions of a galaxy can be assumed to be travelling with the same systemic velocity of the galaxy. To this end, we can verify this assumption by taking the median of the LOSV values, as measured from the 3D fitting code, using the mean wavelength position to measure each PNe's individual LOSV. Fig.~\ref{fig:velocity_comparison} shows the comparison of our median LOSV values, to those given by \citet{iodice_fornax3d_2019}, who present the heliocentric systemic velocity of each galaxy, as measured from the stars in the central regions. In agreement with our original assumption, we see that the PNe median LOSV values and \citet{iodice_fornax3d_2019} systemic velocities are in very good agreement.
\placegalaxytable
\subsection{PNLF distances}
Our deep MUSE data allow one to evaluate the accuracy and reliability of the standard candle nature of the PNLF, as an independent distance measurement technique. Using the PNe catalogue that we have produced we can now derive independent PNLF distance estimates for each of the 21 ETGs targeted by the F3D project as well as an overall average distance to the Fornax cluster, which is estimated to reside $\sim$ 20 Mpc away \citep{tonry_sbf_2001,blakeslee_acs_2009, tully_cosmicflows-2_2013, de_grijs_clustering_2020-1}.
Table \ref{tab:galaxy_info} contains the result from each galaxy, along with the halo and middle regions, where available. It lists the number of detected PNe ($N_\mathrm{PNe}$), as well as the number of expected PNe within 2.5 magnitudes of the PNLF cut-off ($N_{2.5}$). Errors on $N_{2.5}$ are derived on the basis of the Poisson uncertainties associated to the actual observed number of PNe $N_\mathrm{PNe}$, after rescaling. Here, we also present our best-fit values of $\mu_\mathrm{{PNLF}}$, and distance (in Mpc), as measured from the PNe populations.
Table~\ref{tab:galaxy_info} also includes revised distance estimates for the FCC~167 and FCC~219 ($\mu_\mathrm{PNLF}$ = 31.30 $\pm$ 0.07 mag and 31.48 $\pm$ 0.10 mag, respectively) that are consistent with the values reported in S20 (31.24 $\pm$ 0.11 mag and 31.42 $\pm$ 0.10 mag), owing to the fact that in first place these objects have a fair number of PNe. For FCC~219, we also note the agreement between PNLF and TRGB method, where \citet{hoyt_carnegie_2021} report a distance of 31.36 $\pm$ 0.04 $\pm$ 0.05 mag, which is within the uncertainty of our proposed 31.46$^{+0.11}_{-0.12}$.
In the case of FCC\,167, \citet{roth_towards_2021} have recently used the same MUSE data set to benchmark their refined 'on-off' band imaging PNe detection methodology, finding a smaller distance for FCC~167, with $\mu_\mathrm{PNLF} = 31.10 \, \pm \, 0.04$ mag.
Although this comparison highlights a radial trend for the difference between our respective $m_{5007}$ measurements, possibly due to some unaccounted stellar-population effect (which we limit through a detailed spectral fitting), the origin for the distance offset appears to lie with the brightest PNe in the central pointing FCC~167, which all reside in low-stellar background regions.
In this regime the simulations of S20 show that our $m_{5007}$ measurements are rather robust, leaving room only for an offset in absolute flux calibration to explain the reported difference in the inferred distance. In this respect we avail of HST images to check the calibration of the MUSE data \citep[see][for details]{sarzi_fornax3d_2018}, which in the case of FCC~167 only required a 3\% re-scaling towards lower fluxes (adding 0.03 mag). For what follows, we note that across our sample such a re-scaling averages out, indicating no systematic offset.
\placefigSBFminPNLF
\placefigTensionSBFminPNLF
To validate our PNLF distance estimates, we started by comparing our individual to measurements to the SBF distances, which exist for all but one of our sample galaxies (FCC~161). Using the latest SBF values from either \citet{blakeslee_acs_2009} or \citet{blakeslee_surface_2010} (based on HST-ACS in the F850LP and F814W pass-band, respectively), Fig.~\ref{fig:PNLF_minus_SBF} shows how overall our PNLF distances agree well with their SBF counterparts, with a median difference of $\Delta \mu = \mu_\mathrm{{PNLF}} - \mu_\mathrm{{SBF}} = 0.003$ mag and a standard deviation of 0.21 mag.
In comparison, \citet{ciardullo_planetary_2012} reported a median $\Delta \mu = -0.33$ mag corresponding to SBF distances being on average larger than PNLF estimates.
Taking a step further, in Fig.~\ref{fig:tension_PNLF_minus_SBF} we show the distribution for the statistical tension between the PNLF and SBF distance measurements, dividing the $\Delta \mu$ values by the quadratic sum of their respective errors. Figure~\ref{fig:tension_PNLF_minus_SBF} further shows how our PNLF values agree on average with the SBF distances, with a large majority of cases showing less than a 1$\sigma$ tension.
How our PNLF distance measurements compare to the SBF estimates of \citet{blakeslee_acs_2009} can be further appreciated in Fig.~\ref{fig:Dist_comp_Bl}, where 15 of the 20 common objects indeed are consistent within the errors, as would be expected when adopting 1$\sigma$ errors.
Of the galaxies covered here, we note the presence of three edge-on galaxies: FCC\,153, FCC\,170 and FCC\,177. Both FCC\,170, and FCC\,177 measured $\mu_{\mathrm{PNLF}}$ show distinguished offsets from their SBF distance counterparts (Fig.~\ref{fig:Dist_comp_Bl}). This would agree with both \citet{blakeslee_distances_1999} and \citet{mei_acs_2007}, where the former reported differences between SBF distances and other distance methods for these objects, whereas the latter noted that SBF measurements for edge-on galaxies poses a greater challenge, compared to measurements with face-on galaxies.
In fact, \citet{ciardullo_planetary_2012} also remarks that the two galaxies that show the largest disagreement between SBF and PNLF distance measurements in their sample are edge-on.
The disagreement between these two distance methods is believed, in part, to arise from the steeper gradient in stellar light along the line of sight of an edge-on galaxy, compared to the smoother stellar profile of a face-on galaxy.
\citet{iodice_fornax3d_2019} find evidence of a potentially ongoing interaction between FCC\,143 and FCC\,147, from the Fornax Deep Survey \citep[FDS,][]{iodice_fornax_2019} images. The \citet{blakeslee_acs_2009} SBF distances indicate a separation of $\sim$ 0.3 Mpc. We are at least able to confirm the distance of FCC\,147 with a excellent agreement with the SBF measurement. However, the presence of only a handful of PNe within FCC\,143 means that our distance estimate is no where near as reliable compared to that of FCC\,147.
\placefigDistBl
\placefigDistCF
\subsection{The distance to the Fornax cluster}
Taking the weighted average of the PNLF derived distances, we estimate a distance to the Fornax cluster of 19.86 $\pm$ 0.32 Mpc.
This is in remarkably good agreement, with similar uncertainties, with both 's SBF mean distance of 20.0 $\pm$ 0.3 $\pm$ 1.4 Mpc \citep{blakeslee_acs_2009} and recommended mean distance of 19.1$^{+1.4}_{-1.2}$ Mpc, from Cepheid distances, SBF, and Tip of the Red Giant Branch (TRGB) methods \citep{de_grijs_clustering_2020-1}.
Our reported average distance corresponds to a distance modulus of $\mu_\mathrm{PNLF}$ = 31.49 $\pm$ 0.04 mag and is based on all 21 ETGs listed in Table~\ref{tab:galaxy_info}, whereas \citet{blakeslee_acs_2009} report a median distance modulus to Fornax of $\mu_\mathrm{SBF}$ = 31.51 $\pm$ 0.09 mag based on over 43 objects they imaged with HST.
To further gauge the reliability of our PNLF derived distances, Fig.~\ref{fig:Dist_comp_CF3} compares our values with those in the CosmicFlows-3 catalogue of \citet{tully_cosmicflows-3_2016}. This catalogue contains statistically derived mean values obtained combining published distances derived using Supernova Type Ia, the Fundamental Plane (FP), Globular Cluster Luminosity Function (GCLF), and the TRGB. As such measurements come with their own errors and potential biases, the values of CosmicFlows-3 catalogue show larger overall uncertainties compared to the SBF measurements of \citet{blakeslee_acs_2009}. Our PNLF values agree within the errors with the CosmicFlows-3 distances for all but two objects, providing further confidence also in the case of FCC~161 that was not targeted by \citet{blakeslee_acs_2009}. This is reflected also by the agreement between our PNLF average distance to Fornax and the value inferred using the CosmicFlows-3 value for our sample galaxies, which stands at 19.4 $\pm$ 0.4 Mpc ($\mu_\mathrm{PNLF}$= 31.44 $\pm$ 0.04 mag).
\placefigtangRmag
\placefigstructure
\subsection{Fornax cluster spatial structure}
Our PNLF distance measurements allows us to map out the ETG population of the Fornax cluster, comparing our independently derived distances within the cluster, and more specifically, to its brightest cluster galaxy (BCG); namely NGC\,1399 (FCC\,213). Here, we treat NGC\,1399 as the central point of the cluster, with our ETGs situated around it. Before mapping the spatial distribution of our sample galaxies within the cluster, we first review the current distance estimates for NGC\,1399.
\citet{blakeslee_acs_2009} reported a distance to NGC\,1399 of 20.9 $\pm$ 0.9 Mpc, though this was updated to 21.1 $\pm$ 0.7 Mpc in \citet{blakeslee_surface_2010}. A few years earlier and always based on SBF, \citet{tonry_sbf_2001} measured a distance of 19.95 $\pm$ 1.47 Mpc. The CosmicFlows-3 survey reports a statistical mean distance from multiple sources of 22.08 $\pm$ 0.12 Mpc \citep{tully_cosmicflows-2_2013}.
MUSE data for NGC\,1399 were collected both during the F3D survey, covering two external regions, and over the course of previous campaigns for the central regions (Prog. ID. 296.B-5054(A)). Upon reviewing these observations, which were taken under less than ideal seeing conditions compared to our other targets, we identified seven potential PNe in one of the F3D halo pointings. Initial estimates from converting the apparent magnitude of brightest PNe directly to the cut-off position produced an initial distance guess of $\sim$ 22 Mpc. Although this may agree with the results of \citet{tully_cosmicflows-2_2013}, the large uncertainties that would come with modelling such a sparse PNLF ($\sim$0.45 mag, see Fig.~2) would not allow us to replace the previous other estimates for the distance to NGC~1399 nor provide a robust reference to help with the present discussion.
Though BCGs are expected to be close to the barycentre of a cluster, it is not always the case that they reside at the geometric centre of the cluster. NGC~1399 resides deep in the potential well of the cluster as traced by the X-ray emission of its hot intracluster gas, although the cluster itself comprises both of a main virialised region around NGC~1399 and of a secondary, offset group of in-falling objects around NGC~1316.
In this respect, it is interesting to note how the \citet{blakeslee_acs_2009,blakeslee_surface_2010} distance measurements for NGC~1399 would seeming place it not only behind the majority of our sample galaxies located within the virial radius of the Fornax cluster (according both to their and our PNLF measurements) but also beyond their own average distance to the cluster based also on objects further out.
On the other hand, although the \citet{tonry_sbf_2001} distance for NGC~1399 comes with a larger error, it is closer to the average cluster distances presented in \citet{blakeslee_surface_2010}, \citet{tully_cosmicflows-2_2013}, and \citet{de_grijs_clustering_2020-1} as well as our own estimate for Fornax cluster distance.
It would therefore seem a more indicative distance to NGC~1399 and the barycentre of the Fornax cluster.
Another expectation for the structure of a galaxy cluster is mass segregation as a result of dynamical friction, in particular for less massive clusters where tidal stripping cannot act as an obstacle to mass segregation as it does in the most massive ones \citep[see, e.g.][]{kim_yzics_2020}.
To start investigating whether that happened in Fornax, in Fig.~\ref{fig:tangential_dist_RMag} we show the tangential separation of each galaxy from NGC\,1399 compared to their $r$-band absolute magnitude \citep{iodice_fornax_2019}, used as proxy for their mass. As projected on the sky, the majority of galaxies reside within 1 degree separation, with the uncertainty in their PNLF distance decreasing with brighter $r$-band magnitudes. To a first order, the more luminous and massive galaxies appear closer to the expected cluster centre (i.e. NGC\,1399), with less luminous objects being more spread out in projected sky separation.
To more accurately depict the 3D structure of the Fornax cluster, we compare the spatial distances between each galaxy to NGC\,1399, against their absolute $r$-band magnitude.
Considering both the distance estimate to NGC~1399 of \citet{blakeslee_surface_2010} and \citet{tonry_sbf_2001}, in Fig.~\ref{fig:fornax_structure_comp} we produce three different scenarios for the spatial distribution of the ETGs covered here.
The first panel (left), compares the distances taken from \citet{blakeslee_acs_2009, blakeslee_surface_2010} to their own distance of 21.1 Mpc for NGC\,1399 \citep{blakeslee_surface_2010}. The galaxies are scattered between 0.5 and 2.1 Mpc, with little indication for a trend with absolute $r$-band magnitude.
The second panel (middle) compares the SBF distances of \citet{blakeslee_acs_2009} against the NGC\,1399 distance of 19.95 Mpc \citep{tonry_sbf_2001}. In contrast to the first panel, we find that the majority of galaxies are concentrated within 1 Mpc of NGC\,1399.
The final panel (right) compares our PNLF distances to the \citet{tonry_sbf_2001} NGC\,1399 distance. Here, we find that, again, the majority of galaxies reside within 1.5 Mpc, though we also note some galaxies showing a separation of $\sim$ 2.5 Mpc. Although in this case we note that the brightest object is closer to NGC1399, overall from these comparisons we do not find clear evidence of mass segregation within the virial radius of the Fornax Cluster, in particular when we consider the large uncertainties in projected PNLF distance, which would correspond to similarly wide errors in the 3D separation.
\subsection{Planetary nebulae sample sizes across the cluster}
The accuracy of our PNLF distance measurement depends primarily on the PNe sample size, which in turn for a simple and relatively old stellar population is expected to simply scale with the luminosity of that population \citep{renzini_global_1986}.
For our sample of ETGs this is confirmed by trend shown in Fig.~\ref{fig:N_PNe_vs_r_mag}, between the $r$-band magnitude corresponding to the stellar flux encompassed by our single F3D pointings (without accounting regions excluded in our PNe analysis) and the expected number of PNe within 2.5 magnitudes from the bright cut-off of the PNLF, as inferred from our best match to the observed PNLF in these same pointings. Much of the scatter in the observed trend can be explained by the varying imaging quality underlying our PNLF measurements, whereby a broader PSF lead to less PNe being detected and to a larger uncertainty in the inferred intrinsic number of PNe. Other factors may also be at work, however, such a dependency with stellar metallicity for the specific number of PNe \citep[][]{buzzoni_planetary_2006} leading to more PNe rich population in metal regions, as would be probed better by our middle or halo pointing in the case of large objects.
Following the efforts of \cite{martin-navarro_fornax_2019}, Martin-Navarro et al. (2021, in prep) now presents stellar-population parameter maps for the entire, which provides a strong basis to further investigate the link between PNe and their parent populations in a future paper.
\placefigRmagNPNe
\section{Conclusions}
Extending the work of \citet{spriggs_fornax_2020} to the full sample of ETGs targeted by the Fornax3D survey we have presented a catalogue of 1350 unique PNe sources across 21 different objects. The catalogue includes the PNe positions, their [\ion{O}{iii}] 5007 \AA{} flux and LOSV, with the PNe population of each galaxy being catalogued in standardised tables, available online: Tables\,\ref{tab:FCC083_center} to A.36 .
Using the luminosity function observed in each galaxy (limited to the central pointing for larger objects) we have have derived independent distance estimates by matching such PNLFs across their entire magnitude range thanks to a careful treatment of the PNe detection incompleteness. Using simulations we have checked the behaviour of our estimated distance errors and demonstrated that our PNLF distance estimates remain unbiased even when based on a low number of PNe. Furthermore, although the present data do not allow one to set meaningful constraints on the shape of the PNLF, as characterised by the generalised form introduced by \citet{longobardi_planetary_2013}, our methods are capable of detecting possible variations for PNe sample sizes not too far from the one we presently obtained.
Our PNLF distance estimate are generally consistent within the errors with distances derived using SBF \citep[][]{blakeslee_acs_2009,blakeslee_surface_2010} or through the combination of various methods other than PNLF \citep[][]{tully_cosmicflows-2_2013,tully_cosmicflows-3_2016}. In the case of SBF, on average our PNLF distance modulii closely match the SBF values, thus appearing to largely reduce previous tensions between these two methodologies.
With PNLF distances for 21 ETGs within the virial radius of the Fornax cluster we estimated a weighted average distance to Fornax of 19.86 $\pm$ 0.32 Mpc. This is consistent within the errors with the distance to this cluster derived from SBF values for all 43 ETGs observed \citet[][20 Mpc]{blakeslee_acs_2009} across the whole cluster and the one recommended by \citet[][19.1 Mpc]{de_grijs_clustering_2020-1} combining a variety of methods.
We also investigated the spatial distribution of the ETGs of the F3D survey, in particular in relation to the BCG NGC~1399. Though we do not produce our own distance to this BCG, we review different cluster structure scenarios that arise from the NGC\,1399 SBF distances of \citet{blakeslee_acs_2009,blakeslee_surface_2010} and \citet{tonry_sbf_2001}, corresponding to 21.1 Mpc and 19.95 Mpc, respectively.
Assuming the \citet{blakeslee_surface_2010} value, NGC~1399 would be behind most of the objects we observe and $\sim$1 Mpc further than the average SBF distance provided by \citet[][]{blakeslee_acs_2009} for the Fornax cluster.
On the other hand, that the SBF distance to NGC\,1399 of \citet{tonry_sbf_2001} agrees better with \citet[][]{blakeslee_acs_2009} distance to Fornax and the average PNLF distance provided by our objects inside the virial radius. This is consistent with the expectation that NGC~1399 is well centred on the potential well of the Fornax cluster, as traced also by the distribution of the hot intracluster medium.
Mass segregation should be facilitated in a relatively modest cluster such as Fornax, but we found no strong
evidence for it, independent of the assumed distance to NGC~1399 and whether we used SBF of PNLF distances. More accurate distance estimates will be needed to further explore the 3D structure of Fornax and other nearby clusters.
The present catalogue will be of great values for future investigation of the central PNe populations of ETGs, in particular in relation to the properties of their parent stellar population.
The MUSE data at hand will allow us to measure stellar age, metallicity, $\alpha$-element abundance \citep{martin-navarro_fornax_2019}, and the star-formation history \citep{pinna_fornax_2019-1, pinna_fornax_2019} of the stellar population in our sample galaxies in the same regions where our PNe are detected. Using this information, we explore in particular how the luminosity specific number of PNe $\alpha_{2.5}$ derived here will vary across the sample as a function of of these stellar population properties, in addition to measurements for the near and far-UV continuum, testing the results of previous comparisons \citep[e.g.][]{buzzoni_planetary_2006}.
To conclude, we note the exciting prospects performing adaptive-optics (AO) observations with MUSE observations for the purpose of exploring the PNe population of nearby galaxies.
Indeed, if present seeing-limited MUSE data allow one to detect PNe and measure PNLF distances out to a distance of 20 Mpc, finding over a hundred of PNe in the most massive objects and thus measuring distances within a 5\% accuracy \citep[see also S20][]{roth_towards_2021}, AO-assisted observations have the potential to at least double the reachable distance with the PNLF method.
This can be understood considering that whereas the background noise in the MUSE observations is not set to vary dramatically for objects further out (owing to how surface brightness is conserved with distance), AO-observations delivering a 0.4\arcsec wide point-spread function instead of the 0.8\arcsec width (typical for good-seeing conditions at Paranal) will lead to a four-fold increase in the central [\ion{O}{iii}] 5007 \AA{} peak flux and spectral amplitude, thus compensating for the flux attenuation observed for PNe twice as further away.
By the same token, AO-observations of objects as far away as Virgo or Fornax will allow one to reach $-1.5$ mag further down from the PNLF bright cut-off, making it possible to further test the universality of the PNLF and better understand its origin.
\begin{acknowledgements}
We would like to thank the referee for their constructive responses, which helped to improved the content and clarity of this manuscript. Based on observations collected at the European Southern Observatory under ESO pro-gramme 296.B-5054(A). TS thanks R. Jackson, S. Kaviraj and A. Bittner for their help and ideas that aided in the construction and presentation of this paper. This work was supported by Science and Technology Facilities Council [grant number ST/R504786/1]. GvdV acknowledges funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme under grant agreement No 724857 (Consolidator Grant ArcheoDyn). RMcD acknowledges financial support as a recipient of an Australian Research Council Future Fellowship (project number FT150100333). J.~F-B and I.~M-N acknowledge support through the RAVET project by the grant PID2019-107427GB-C32 from the Spanish Ministry of Science, Innovation and Universities (MCIU), and through the IAC project TRACES which is partially supported through the state budget and the regional budget of the Consejer\'ia de Econom\'ia, Industria, Comercio y Conocimiento of the Canary Islands Autonomous Community. FP acknowledges support from grant PID2019-107427GB-C32 from The Spanish Ministry of Science and Innovation. EMC acknowledges support by Padua University grants DOR1885254/18, DOR1935272/19, and DOR2013080/20 and by MIUR grant PRIN 2017 20173ML3WW\_001.
This project made use of the following software packages: LMfit \citep{newville_lmfit_2014,newville_lmfitlmfit-py_2019}, Astropy, a community-developed core Python package for Astronomy \citep{robitaille_astropy_2013}, scipy \citep{eric_jones_and_travis_oliphant_and_pearu_peterson_and_et_al_scipy_2001}, Numpy \citep{van_der_walt_numpy_2011}, matplotlib \citep{hunter_matplotlib_2007} and Pandas \citep{mckinney_data_2010}.
\end{acknowledgements}
\bibliographystyle{aa}
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1,116,691,499,239 | arxiv | \section{Introduction}
The ability to predict and optimize the behavior of complex socio-technical systems is one of this century's main scientific themes. It is becoming clearer and clearer how the behavior of large-scale infrastructural, social, economic, and financial networks can have a huge societal impact, e.g., enabling or limiting access to essential services, such as mobility and energy, influencing the outcome of elections, or destabilizing entire economies. A central aspect of such network systems is the role played by interconnections in amplifying and propagating shocks through cascading mechanisms that may increase the fragility of a system \cite{Allen.Gale:2000,Buldyrev.ea:2010,NAS-2012,Como.ea:2013b}. The term \emph{systemic risk} refers to the possibility that even small shocks localized in a limited part of the network can propagate and amplify through cascading mechanisms, thus possibly achieving a significant global impact \cite{Haldane.May:2011,Savla.ea:2014,Acemoglu2015a,Acemoglu2015}.
In this realm, a central problem is to find adequate models for networks, that are sufficiently elaborate to incorporate such propagation phenomena, yet simple enough to allow for mathematical tractability. Simple contagion models such as epidemic contact models prove inadequate as based on purely pairwise interactions. A more complex model taking into account cumulative neighborhood effects is the linear threshold model \cite{Adler:91,Watts:2002,Rossi.ea:2019} whose applicability is however limited by the fact that states of the nodes are described by pure binary variables simply expressing whether the node has been affected by the shock. In most of the applicative contexts where the network represents a physical infrastructure or an interconnected financial system, this simplification is way too strong. Indeed, in these cases, the cascading mechanism is rather triggered by a process naturally working with continuous variables such as, e.g., power flows in electric grids, traffic volumes in transportation systems, or assets values and payments in financial networks.
In this paper, we undertake a fundamental study of a model, originally proposed in the seminal work \cite{Eisenberg.Noe:2001}
to study systemic risk in networks of financial institutions interconnected by mutual obligations.
It is one of the simplest continuous models where the shock propagation phenomena are caused by a saturation nonlinearity of a positive linear static system.
Precisely, we consider the following fixed point equation
\begin{equation}\label{eq1}
x_i=\min \left\{ \max \left\{ \sum_{j=1}^n x_jP_{ji}+c_i, 0 \right\} , w_i \right\}\,,\qquad i=1,\ldots,n\,,
\end{equation}
where $P$ is a row-stochastic or sub-stochastic $n\times n$ matrix and $w \in \mathbb{R}^n$ is a nonnegative vector that jointly describe a \emph{network} and where $c \in \mathbb{R}^n$ is an \emph{exogenous flow} vector.
Equation \eqref{eq1} can be more compactly rewritten as
\begin{equation}\label{eq:nonlinear-system}
x=S_0^w(P'x+c)\,.
\end{equation}
where $S_0^w$ denotes the vector saturation function
\begin{equation}\label{eqd}
\left(S_0^w(x)\right)_i = \min \{ \max \{ x_i, 0\}, w_i\}, \qquad i=1,\ldots,n\,,
\end{equation}
We shall refer to vectors $x$ that are solutions of \eqref{eq:nonlinear-system} as \emph{equilibria} of the network $(P,w)$ with exogenous flow $c$.
While standard arguments guarantee existence of such network equilibria, their uniqueness and dependence on the exogenous flows prove to be more delicate issues that will be the object of this paper.
In financial networks \cite{Eisenberg.Noe:2001}, the entries of the vector $w$ represent the obligations of the various institutions (say banks), those of the exogenous flow $c$ represent the balance between assets possessed by the entities and their obligations towards institutions outside the network, while $P$ describes the way obligations of an entity are split among the others and describe the way the financial system is interconnected.
An equilibrium $x$ represents, in this context, a set of payments that clear the network in a consistent way. Previous works including \cite{Liu2010,Acemoglu2015a,Glasserman2015,Ren2016} have analyzed conditions for uniqueness of the clearing payment $x$ and studied its dependence on the exogenous flow vector $c$. Their main focus is to understand the extent to which a shock hitting the value of the assets of a single node $i$ (perturbation of $c_i$) reflects on the entire network and leads to possible cascade effects. More precisely, they define the concept of a default node as one for which the quantity $\sum_jP_{ji}x_j+c_i$ (representing the liquidity of the entity $i$) is below the value of its obligation $w_i$ and distinguishing in partial or total default if, respectively, $\sum_jP_{ji}x_j+c_i>0$ or not. Although simple, this framework turns out to be very useful for analyzing how losses propagate through the financial system. Considering the importance of this application, we will discuss the financial model more accurately in Section \ref{sec:financial}.
Network equilibria defined as fixed points of equation \eqref{eq1} can also be seen as the Nash equilibria of a properly defined quadratic games ---as illustrated in Section \ref{sec:game}--- as well as the equilibrium points of a dynamical flow network with exogenous flows and capacity constraints that is briefly discussed in Section \ref{sec:compartmental} and whose dynamical behavior has been analyzed in detail in \cite{Massai.ea:IFAC2020}.
The present paper presents a systematic study of the equilibria described by equation (\ref{eq:nonlinear-system}) and gives four fundamental contributions:
\begin{enumerate}
\item[(i)] We prove that the set of nodes in partial or full default (in the financial jargon discussed above) is invariant through the set of all equilibria.
\item[(ii)] We analyze the structure of the solutions with respect to the topological properties of the graph. In particular we show how to effectively construct all equilibria starting from anyone of them.
\item[(iii)] We prove a general result that provides necessary and sufficient conditions for uniqueness of the equilibrium, which subsumes the other theorems assessing uniqueness present in literature. In particular, in the case of a strongly connected network, conditions for uniqueness can be easily checked a priori without doing any preliminary computation.
\item[(iv)] We show that the equilibrium exhibits a jump discontinuity with respect to the exogenous flow vector $c$ when this is crossing certain linear sub-manifolds where uniqueness of equilibrium is lost. This is of particular interest in the financial context, as it shows that even small shocks affecting the values of the assets of few nodes, can trigger catastrophic aggregated loss to the system and cause the default of several agents. We provide an analytical description of the discontinuity set and we quantify the size of the jump for an aggregated loss function.
%
%
%
\end{enumerate}\smallskip
The rest of this paper is organized as follows. The remainder of this Introduction is devoted to a discussion of some related literature as well as a brief explanation of the notational conventions to be followed throughout the paper. Section \ref{sec:motivations} is devoted to present three motivating examples for the model considered in the paper. Section \ref{structure} is devoted to establishing a number of preliminary results on the structure of the equilibria. Uniqueness results as well a general expression describing all solutions in non uniqueness cases is presented in Section \ref{uniqueness}. Section \ref{discontinuous} is devoted to the analysis of jump discontinuities in the equilibrium with respect to variation of the exogenous flow vector. The papers ends with Section \ref{conclusions} dedicated to draw some conclusions and open problems.
\subsection{Related Literature}
The seminal work by Eisenberg and Noe \cite{Eisenberg.Noe:2001} triggered the development of a large literature on similar models. Ren et al. \cite{Ren2016} explore several conditions on the
uniqueness of the clearing payment vector. It is shown that the clearing payment vector is, in general, not unique, but rather a certain kind of net value of a bank is. Liu and Statum \cite{Liu2010} use linear programming to provide a sensitivity analysis of Eisenberg and Noe model with respect to certain parameters. Elsinger et al. \cite{Elsinger2006} use the Eisenberg and Noe model together with standard tools from modern risk management to assess systemic financial stability of the Austrian banking system; they find that financial contagion is rare but can nonetheless wipe out a major part of the banking system. Cifuentes et al. \cite{Cifuentes2005} extend the original Eisenberg and Noe model by considering fire sales; Rogers et al \cite{Rogers2013} include bankruptcy costs using a recovery function that drops discontinuously at the default boundary and then decreases linearly with the amount of assets available. Elsinger and Helmut \cite{Elsinger2009} enrich the original model by considering cross-holdings that lead to additional spillover effects when a shock hits the network.
Glasserman and Young \cite{Glasserman2015} estimate the extent to which interconnections increase expected losses and defaults under a wide range of shock distributions and they provide bounds on the potential magnitude of network effects on contagion and loss amplification; they also provide an extensive survey on the financial networks topic in \cite{Peyton2002}. Acemoglu et al. \cite{Acemoglu2015} develop a unified framework for the study of how network interactions can function as a mechanism for propagation and amplification of microeconomic shocks while in \cite{Acemoglu2015a} they provide an analysis of a particular case of the Eisenberg and Noe model and they prove some rigorous results about the resilience of different network topologies depending on the shock magnitude.
\subsection{Notation}
We briefly explain the notation to be used throughout this paper.
We denote vectors with lower case and matrices with upper case. Sets are denoted with calligraphic letters. A subscript associated to vectors, for instance $v_{\mathcal A}$, represents the sub-vector that is the restriction of $v\in\mathbb R^n$ over the indexes contained in the set $\mathcal A \subseteq \{1,2, \dots, n\}$. The same notation holds for matrices: $P_{\mathcal A \mathcal B}$ represents the sub-matrix of $P$ obtained by considering rows and columns associated to the indexes contained in sets $\mathcal A$ and $\mathcal B$, respectively. We indicate with $\mathbbm{1}$ the all-$1$ vector (independently from its dimension). We consider on $\mathbb{R}^n$ the partial order given by entry-wise ordering: if $x,y\in\mathbb{R}^n$, $ x\leq y$ if $x_i\leq y_i$ for every $i$ and we use the strict inequality symbol $<$ if moreover the two vectors are different. Throughout the paper the ordering of vectors will be meant in this sense.
A non-negative matrix $P \in \mathbb{R}_{+}^{n \times n}$ is called \emph{ (row) sub-stochastic} if the sum of the elements in each row never exceeds $1$, namely $P\mathbbm{1} \le \mathbbm{1}$. Notice that in the literature it is often assumed that sub-stochastic matrices have the additional property that for at least one row there is strict inequality. Here we prefer not to follow this convention and in this way our class of sub-stochastic matrices contains also the \emph{stochastic} matrices that are those for which $P\mathbbm{1}=\mathbbm{1}$.
\section{Three applications}\label{sec:motivations}
In this section, we describe three motivating applications. We start in Section \ref{sec:financial} by presenting a model of financial networks generalizing the one first considered in \cite{Eisenberg.Noe:2001}. We then provide an interpretation in terms of Nash equilibria of a certain quadratic game, as explained in Section \ref{sec:game}. Finally, in Section \ref{sec:compartmental} we discuss a dynamical flow network model with fixed routing and capacity constraints whose rest points coincide with the equilibria.
\subsection{Payment equilibria in financial networks}\label{sec:financial}
We consider a set $\mathcal V=\{1,\dots , n\}$ of financial entities (e.g., banks, broke dealers,...) interconnected by internal and external obligations that are specified by a non-negative matrix $W\in\mathbb{R}^{n\times n}$ and three vectors $a, \,b,\, u \in\mathbb{R}^n_+$ that have the following interpretations:
\begin{itemize}
\item $W_{ij} \ge 0$ is the
liability of node $i$ to node $j$;
\item $a_i$ is the total value of assets and credits of $i$
from external entities;
\item $b_i$ is the total liability
of node $i$ to external non-financial entities;
\item $u_i$ is the total liability
of node $i$ to external financial entities.
\end{itemize}
The quantity $v_i=\sum_j W_{ji}-\sum_j W_{ij}+a_i-b_i-u_i$ is the net worth of node $i$. If the condition $v_i\geq 0$ is verified for every $i\in\mathcal V$, it means that each node is fully liable and in principle capable to pay back all its liabilities to the nodes in the network as well the external ones. In case when instead some nodes do not satisfy the condition $v_i\geq 0$, namely they are not fully liable, it is necessary to determine a consistent set of payments among the various nodes.
Put $w_i=\sum_j W_{ij}+u_i$ and
$$P_{ij}=\left\{\begin{array}{ll}W_{ij}/w_i\quad &\text{if}\, w_i>0\\ 0\quad &\text{otherwise}\end{array}\right.$$
We define by $X_{ij}$ the payment from node $i$ to node $j$ and by $X_{i{\rm o}}$ the payment from node $i$ to external financial entities. Assuming that liabilities to non-financial entities have a higher seniority and that all other payments (including those to external financial entities) should be proportional to the corresponding liabilities, a consistent set of payments among the nodes has to satisfy the relations
\begin{equation}\label{payments}
\begin{array}{rcl}X_{ij} &=& \min \left\{ P_{ij} \max \left\{ \sum_{k} X_{ki}+a_i-b_i,0 \right\}, W_{ij}\right\}\\[5pt]
X_{i{\rm o}} &=& \min \left\{ u_i/w_i \max \left\{ \sum_{k} X_{ki}+a_i-b_i,0 \right\}, u_i\right\}\end{array}
\end{equation}
Let $x_i=\sum_{j }X_{ij}+X_{i{\rm o}}$ be the total payment of node $i$ to the financial entities. Summing the relations in (\ref{payments}) and using the fact that $W_{ij}=w_iP_{ij}$, we obtain
\begin{equation}\label{eq.a0}
x_i=\min \left\{ \max \left\{ \sum_{k } X_{ki}+a_i-b_i,0 \right\}, w_i\right\}
\end{equation}
so that, $X_{ij}=x_iP_{ij}$. Relation (\ref{eq.a0}) can thus be rewritten as
\begin{equation}\label{eq.a1}
x_i=\min \left\{ \max \left\{ \sum_{k } x_kP_{ki}+a_i-b_i,0 \right\}, w_i\right\}
\end{equation}
This set of relations is equivalent to (\ref{payments}). Indeed, if the vector $x$ solves (\ref{eq.a1}), then $X_{ij}=x_iP_{ij}$ solves (\ref{payments}).
This coincides with (\ref{eq1}) with exogenous flow $c=a-b$.
Notice that the matrix $P$ is sub-stochastic in its strict sense (i.e., at least one row does not sum to $1$) when either there exist nodes with a positive liability towards external financial entities, or nodes with no financial liabilities.
In this financial setting, it is often considered the case when we start from a fully liable configuration, that is $v_i\geq 0$ for all $i$, leading to a solution $x$ of (\ref{eq.a1}) such that $x_i\geq w_i$ for all $i$. We then assume that the outside assets suffer a shock $\epsilon\in \mathbb{R}^n_+$ so that their values reduce to $a-\epsilon$ possibly making some of the $v_i$'s negative. The study of the amount of nodes in default $x_i< w_i$ has a function of the shock $\epsilon$ is one of the key issues.
\subsection{A quadratic network game} \label{sec:game}
Consider a quadratic network game with player set $\mathcal V=\{1,2,\ldots,n\}$ where each player $i$ chooses her action $X_{i\cdot}=(X_{i0},X_{i1},\ldots,X_{in})\in\mathbb{R}_+^{n+1}$ so as to maximize her utility
\begin{equation}\label{eq:utility}u_i(X)=-\sum_{\substack{j\in\{0,\ldots,n\}:\\W_{ij}>0}}\frac{W_{ij}}2\left(\frac{X_{ij}}{W_{ij}}-1\right)^2\,,\end{equation}
under the following constraints:
\begin{equation}\label{budget-constraint} \sum_{j=0}^nX_{ij}\le\sum_{j=1}^nX_{ji}+c_i\,.\end{equation}
Above, $W\in\mathbb{R}_+^{n\times n}$ is a nonnegative zero-diagonal matrix such that $w_i=\sum_{j=0}^nW_{ij}>0$ for $1\le i\le n$,
$c\in\mathbb{R}^n$ is a given vector, and $X\in\mathbb{R}_+^{(n-1)\times n}$ denotes the configuration of all players' actions.
Note that the best response of player $i$ necessarily assigns $X_{ij}=0$ for every $j=0,\ldots,n$ whenever $\sum_{j=1}^nX_{ji}+c_i\le0$ and
$X_{ij}=W_{ij}$ for every $j=0,\ldots,n$ whenever $\sum_{j=1}^nX_{ji}+c_i\ge w_i$.
On the other hand, when $0<\sum_{j=1}^nX_{ji}+c_i< w_i$, then the constraint \eqref{budget-constraint} is met with equality by the best response of a player $i$ and this can then be determined by setting to $0$ the partial derivatives of the Lagrangian
$$\sum_{\substack{j\in\{0,\ldots,n\}:\\W_{ij}>0}}\frac{W_{ij}}2\left(\frac{X_{ij}}{W_{ij}}-1\right)^2+\lambda\left(\sum_{j=0}^nX_{ij}-\sum_{j=1}^nX_{ji}-c_i\right)\,.$$
We thus get
$
X_{ij}=P_{ij}\left(\sum_{k=1}^nX_{ki}+c_i\right)\,,\qquad 0\le j\le n\,,$$
where $$P_{ij}=\frac{W_{ij}}{w_i}\,,\qquad 0\le i\le n\,,\qquad1\le j\le n\,.$$ Putting together our findings in the three cases, the best response of a player $i=1,\ldots,n$ reads
$$X_{ij}=P_{ij}S_0^{w_i}\left(\sum_{k=1}^nX_{ki}+c_i\right)\,,\qquad 0\le j\le n\,.$$
The Nash equilibria of this game are those $X^*\in\mathbb{R}_+^{n\times(n+1)}$ with entries $X^*_{ij}=x_iP_{ij}$, where $x=X^*\mathbbm{1}$ is a solution of \eqref{eq:nonlinear-system}.
\subsection{Dynamical flow networks with finite capacities and losses}\label{sec:compartmental}
Consider a dynamical flow network consisting of finitely many cells $i=1,2,\ldots,n$, each containing a quantity $x_i(t)$ of the same commodity at time $t\ge0$. The state of the system is described by the vector $x(t)=(x_i(t))_{1\le i\le n}$ and evolves in continuous time as the cells can exchange flow both among themselves and with the external environment. The network flow dynamics is then described by the system of ordinary differential equations \begin{equation}\label{flow-dynamics}\dot x_i=f_i(x)\,,\qquad i=1,\ldots,n\,,\end{equation}
where $f_i(x)$ represents the total instantaneous net flow (inflow minus outflow) on the cell $i$. The form of the instantaneous flow $f_i(x)$ follows from the considerations below:
\begin{itemize}
\item The instantaneous outflow from cell $i$ is exactly $x_i$. A constant fraction $P_{ij}\ge0$ of the quantity of commodity $x_i$ flows directly towards another cell $j\ne i$ in the network, while the remaining part $(1-\sum_jP_{ij})x_i$ leaves the network directly.
\item The instantaneous inflow to cell $i$ is a quantity between $0$ and $w_i$
\item Besides the inflow from other cells, each cell $i$ possibly receives a constant instantaneous exogenous inflow $a_i\ge0$ from outside the network
while a constant instantaneous outflow $b_i\ge0$ is drained, if available, directly from cell $i$ towards the external environment. The part possibly exceeding $w_i$ is lost.
\end{itemize}
This leads to the following form for the total instantaneous net flow:
\begin{equation}\label{flow-exact}f_i(x)=S_0^{w_i}\left(\sum\nolimits_jP_{ji}x_j+c_i\right)-x_i\,,\qquad 1\le i\le n\,,\end{equation}
where $c_i=a_i-b_i$. It is then clear that the rest points of the dynamical flow network \eqref{flow-dynamics}--\eqref{flow-exact} coincide with the equilibria of the network $(P,w)$ with exogenous flow $c$. In fact, as proven in \cite{Massai.ea:IFAC2020},
every solution of the dynamical flow network \eqref{flow-dynamics}--\eqref{flow-exact} with initial state $x(0)$ converges to some network equilibrium in the limit as $t$ grows large.
\section{Structural properties of the equilibria}\label{structure}
In this section, we assume we have fixed a sub-stochastic matrix $P\in\mathbb{R}_+^{n\times n}$, a nonnegative vector $w\in\mathbb{R}^n_+$, and a vector $c\in\mathbb{R}^n$, and we study the equilibria of the network $(P,w)$ with exogenous flow $c$, namely the solutions of the fixed-point equation \eqref{eq:nonlinear-system}. We denote by $$\mathcal X=\left\{x\in\mathbb{R}^n:\,\eqref{eq:nonlinear-system}\right\}$$ the set of such equilibria. It is useful to introduce the complete lattice
\begin{equation}\label{lattice}\mathcal L_0^w=\{x\in\mathbb{R}^n:\,0\le x\le w\}\,,\end{equation}
and note that the vector saturation function $S_0^w : \mathbb{R}^n \to \mathcal L_0^w$. This implies that necessarily $\mathcal X\subseteq \mathcal L_0^w$.
Existence of such equilibria directly follows from Brower's fixed point theorem, since the lattice $\mathcal L_0^w$ is a nonempty, convex, and compact set, and $x\mapsto S_0^w(P'x+c)$ maps $\mathcal L_0^w$ in itself with continuity. Hence, we always have that the equilibrium set $\mathcal X$ is nonempty. On the other hand, equilibria are not unique in general: e.g., in the simple network with $n=2$, $P_{ij}=|i-j|$ for $i,j=1,2$, $w=\mathbbm{1}$, and $c=0$, the equilibria are all the vectors $x=(t,t)$ for $0\le t\le1$.
In the analysis of the structure of $\mathcal X$, an important role will be played by the graph underlying the matrix $P$, namely the directed graph $\mathcal G_P=(\mathcal V,\mathcal E)$ with node set $\mathcal V=\{1,2,\ldots,n\}$, and link set $\mathcal E=\{(i,j)\in\mathcal V\times\mathcal V:\,P_{ij}>0\}$.
The contribution of this rest of this section is threefold. First, we prove that among the equilibria there is always a minimal one and a maximal one. Moreover, we show that such minimal and maximal equilibria both depend monotonically on the exogenous flow vector $c$ and that they can be computed by a natural iterative distributed algorithm. Finally, we get to the main result of this section introducing a fundamental partition of the node set into three subsets and prove that such partition is invariant with respect to the entire set of equilibria. Such results will be instrumental to all derivations in the next sections.
\subsection{Minimal and maximal equilibria}
We start with the following simple result.
\begin{lemma}\label{lemma:monotone}
Let $w\in\mathbb{R}_+^n$ be a nonnegative vector and let $P\in\mathbb{R}_+^{n\times n}$ be a nonnegative matrix. Then, the map
\begin{equation}\label{eq:monotone-map}(x,c)\mapsto S_0^w(P'x+c)\end{equation}
is continuous and monotone nondecreasing from $\mathbb{R}^n\times\mathbb{R}^n$ to the complete lattice $\mathcal L_0^w\subseteq\mathbb{R}^n$.
\end{lemma}
\proof
Continuity follows immediately from the fact that \eqref{eq:monotone-map} is the composition of the saturation function $S_0^w$ with the linear map $(x,c)\mapsto P'x+c$. Similarly, monotonicity is implied by the fact that both those functions
preserve the natural partial ordering of $\mathbb{R}^n$ (this is true for the linear map beacuse $P$ is a nonnegative matrix).
\qed
We now consider the iterative algorithm
\begin{equation}\label{eq:iterative-algorithm}x(t+1)=S_0^w(P'x(t)+c)\,,\qquad t\ge0\,,\end{equation}
whose fixed points coincide with the solutions of \eqref{eq:nonlinear-system}.
\begin{proposition}\label{proposition:c} Let $\mathcal X$ be the set of equilibria of the network $(P,w)$ with exogenous flow vector $c\in\mathbb{R}^n$. Let $x(t)$ be the sequence determined by (\ref{eq:iterative-algorithm}) with initial condition $x(0)=x_0$. Then,
\begin{enumerate}
\item[(i)] The set $\mathcal X$ admits a minimal element $\underline x$ and a maximal element $\overline x$ (in the sense of the usual entry-wise ordering in $\mathbb{R}^n$);
\item[(ii)] If $x_0=0$, then $x(t)\to\underline x$ for $t\to +\infty$;
\item[(iii)] If $x_0=\mathbbm{1}$, then $x(t)\to\overline x$ for $t\to +\infty$.
\end{enumerate}
Indicating with $\underline x(c)$ and $\overline x(c)$ the minimal and maximal equilibrium as a function of the exogenous flow vector $c$, we have that
\begin{enumerate}
\item[(iv)] $c\mapsto\underline x(c)$ and $c\mapsto\overline x(c)$ are monotone nondecreasing maps from $\mathbb{R}^n$ to $\mathcal L^w_0$.
\end{enumerate}
\end{proposition}
\proof
For simplicity of notation, throughout the proof, we put $\Phi(x)=S_0^w(P'x+c)$.
Assume first that $x_0=0$. In this case the sequence $x(t)$ determined by (\ref{eq:iterative-algorithm}) is non-decreasing and this can be shown with an induction argument on $t\ge0$. Indeed, on the one hand, since $\Phi(0)\in\mathcal L_0^w$, we have that $x(0)=0\le\Phi(0)=x(1)$. On the other hand, since $\Phi(x)$
is monotone by Lemma \ref{lemma:monotone}, we have that, if $x(t)\le x(t+1)$ for some $t\ge0$, then $x(t+2)=\Phi(x(t+1))\ge\Phi(x(t))=x(t+1)$. Being non-decreasing and bounded, the sequence $x(t)$ admits limit $\underline x$. Since $\Phi$ is continuous, $\underline x=\Phi(\underline x)$ is necessarily a solution of \eqref{eq:nonlinear-system}.
We now prove by contradiction that $\underline x$ is minimal in $\mathcal X$. If not, let $x^*\in\mathcal X$ be such that $x^*<\underline x$.
and let $t^*=\sup\{t\ge0:\,x(t)\le x^*\}$. Notice that, since $x(0)=0\le x^*$ and $\lim x(t)=\underline x$,
$t^*$ is necessarily finite and such that $x(t^*)\le x^*$. Monotonicity of $\Phi$ implies that $x(t^*+1)=\Phi(x(t^*))\le\Phi(x^*)=x^*$, thus contradicting the way $t^*$ was defined. Therefore, $\underline x$ is minimal in $\mathcal X$.
This proves (ii) and the first part of (i).
A completely parallel argument shows that, with initial condition $x(0)=w$ the iterations \eqref{eq:iterative-algorithm} converge to a maximal solution $\overline x$ of \eqref{eq:nonlinear-system}. This proves (iii) and the second part of (i).
We now prove (iv). Consider two vectors $c_1,c_2\in\mathbb{R}^n$ such that $c_1\le c_2$ and let $x_1(t)$ and $x_2(t)$ the two sequences determined by the iterative algorithm (\ref{eq:iterative-algorithm}) with, respectively, $c=c_1$ and $c=c_2$, and with initial condition $x_1(0)=x_2(0)=0$. Using (ii), we obtain that the two sequences converge, respectively, to the minimal solutions $\underline x(c_1)$ and $\underline x( c_2)$. Using Lemma \ref{lemma:monotone}
we can see that
$$x_1(t)\leq x_2(t)\;\Rightarrow\; x_1(t+1)=S_0^w(P'x_1(t)+c_1)\le S_0^w(P'x_2(t)+c_2)= x_2(t+1)\,.$$
Since $x_1(0)=x_2(0)$, by induction we have that $x_1(t)\le x_2(t)$ for all $t$. This yields $\underline x(c_1)\le\underline x(c_2)$. We have proven that $\underline x(c)$ is monotone nondecreasing. The same property for the maximal solution $\overline x(c)$ follows by an equivalent argument.
\qed
\begin{remark}
Observe that the iteration \eqref{eq:iterative-algorithm} can be implemented as a distributed algorithm, whereby at each time $t\ge0$ each node $i\in\mathcal V$ using only the current states $x_j(t)$ of its in-neighbors $\{j\in\mathcal V:\,P_{ji}>0\}$ update its state according to $x_i(t+1)=S_0^{w_i}(\sum_j P_{ji}x_i(t)+c_i)$. The complexity of each iteration of \eqref{eq:iterative-algorithm} is therefore of the order of the number of links in the network.
\end{remark}
\subsection{A fundamental partition}
We now show that, regardless of whether the system \eqref{eq:nonlinear-system} admits a unique solution or multiple ones, the node set $\mathcal V$ can be partitioned in three subsets such that every equilibrium $x$ has entries saturated to $w$, unsaturated, and, respectively, saturated to $0$ on these subsets. Remarkably, such partition can be proven to be equilibrium-invariant, a fact that that will play a key role in the future analysis.
For every equilibrium $x\in\mathcal X$, we proceed by partitioning the node set $\mathcal V$ as
\begin{equation}\label{eq:node-partition}\mathcal V=\mathcal V^x_- \cup \mathcal V^x_+ \cup \mathcal V^x_0\,,\end{equation}
where:
\begin{itemize}
\item $\mathcal V^x_+=\l\{i\in\mathcal V:\,c_i+\sum_{k \ne i} P_{ki}x_k>w_i\r\}$ is the set of \emph{surplus} nodes;
\item $\mathcal V^x_0=\l\{i\in\mathcal V:\,0 \le c_i+\sum_{k \ne i} P_{ki}x_k \le w_i\r\}$ is the set of \emph{exposed} nodes;
\item $\mathcal V^x_-=\l\{i\in\mathcal V:\,c_i+\sum_{k \ne i} P_{ki}x_k<0\r\}$ is the set of \emph{deficit} nodes.
\end{itemize}
By the way these sets have been defined, it follows that
\begin{equation}\label{eq:deficit-surplus}\begin{array}{ll}x_i=0\,,\quad\; & {\rm if}\, i\in\mathcal V^x_-\\
x_i=w_i\,,\quad\; & {\rm if}\, i\in\mathcal V^x_+ \\
x_i=c_i+\sum_{j \ne i} P_{ji}x_j \quad\; & {\rm if}\, i\in\mathcal V^x_0\,.\end{array}\end{equation}
The following is the key result of this section and will be instrumental to all our future derivations.
\begin{proposition}\label{proposition:partition}The partition \eqref{eq:node-partition} is invariant over all equilibria $x\in\mathcal X$.
\end{proposition}
\proof
We consider the maximal equilibrium $\overline{x}$ and any another equilibrium $x$ and we show that they share the same node partition \eqref{eq:node-partition}. To begin with, notice that, since $\overline x\ge x$, we have $\mathcal V_{\overline{x}}^+ \supseteq \mathcal V^x_+ $ and $ \mathcal V_{\overline{x}}^- \subseteq \mathcal V^x_-$. Let us split nodes in five different classes, $\mathcal C_1, \mathcal C_2, \mathcal C_3, \mathcal C_4, \mathcal C_5$, and write any vector $y\in\mathbb{R}^\mathcal V$ in a block form accordingly $y=(y_1, y_2,y_3,y_4, y_5)$
For simplicity of notation indicate $Q_{ij}:=(P')_{\mathcal C_i\mathcal C_j}$.
The five classes of nodes correspond to the four possible cases in which $\overline{x}$ and $x$ can differ and are precisely defined below:
\begin{itemize}
\item $\mathcal C_1 = \mathcal V^x_+$ is the set of nodes that are surplus for both equilibria; it holds $w_1=x_1^+=x_1$;
\item $\mathcal C_2=\mathcal V_{\overline{x}}^+ \backslash \mathcal V^x_+$ is the set of nodes that are surplus for $\overline{x}$ but not for $x$; it holds
$w_2=\overline{x}_2<\sum_{k =1 }^4 Q_{2k}\overline{x}_k+c_2$ and $x_2 \ge \sum_{k =1 }^4 Q_{2k}x_k+c_2$;
\item $\mathcal C_3 = \mathcal V^x_0 \cap \mathcal V_{\overline{x}}^0$ is the set of nodes that are exposed for both equilibria; it holds $\overline{x}_3=\sum_{k =1 }^4 Q_{3k}\overline{x}_k+c_3$ and $x_3=\sum_{k =1 }^4 Q_{3k}x_k+c_3$;
\item $\mathcal C_4 = \mathcal V_{\overline{x}}^0 \backslash \mathcal V^x_0$ is the set of nodes that are exposed for $\overline{x}$ and deficit for $x$; it holds $\overline{x}_4=\sum_{k =1 }^4 Q_{4k}\overline{x}_k+c_4$ and $x_4=0>\sum_{k =1 }^4 Q_{4k}x_k+c_4$;
\item $\mathcal C_5 = \mathcal V_{\overline{x}}^-$ is the set of nodes that are deficit for both equilibria; it holds $\overline{x}_5=x_5=0$.
\end{itemize}
For our claim to hold, we need to prove that $\mathcal C_2=\mathcal C_4=\emptyset$.
We put $\Delta x=\overline{x}-x \ge 0$ and we notice that, for classes $\mathcal C_1$ and $\mathcal C_5$ we have that $\Delta x_1=\Delta x_5 =0$. For the remaining blocks we can write:
\begin{equation}
\begin{cases}\label{key-1}
\Delta x_2 = \overline{x}_2-x_2 < \sum_{k =1 }^4 Q_{2k}\overline{x}_k+c_2 - \sum_{k =1 }^4 P_{k2}^{'}x_k-c_2= \sum_{k =2 }^4 Q_{2k}\Delta x_k \\[12pt]
\Delta x_3 = \overline{x}_3-x_3 = \sum_{k =1 }^4 Q_{3k}\overline{x}_k+c_3 - \sum_{k =1 }^4 P_{k3}^{'}x_k-c_3= \sum_{k =2 }^4 Q_{3k}\Delta x_k \\[12pt]
\Delta x_4 =\overline{x}_4-x_4 < \sum_{k =1 }^4 Q_{4k}\overline{x}_k+c_4 - \sum_{k =1 }^4 P_{k4}^{'}x_k-c_4= \sum_{k =2 }^4 Q_{4k}\Delta x_k
\end{cases}
\end{equation}
Since the matrix
\begin{equation}\label{dxx}
Q=\begin{bmatrix}
Q_{22}& Q_{23} & Q_{24} \\
Q_{32}& Q_{33}& Q_{34} \\
Q_{42}& Q_{43}& Q_{44}
\end{bmatrix}
\end{equation}
is a sub-matrix of $P'$, necessarily its transpose $Q'$ is sub-stochastic. By contradiction, if $\mathcal C_2\cup \mathcal C_4\neq\emptyset$, using (\ref{key-1}), the fact that $\Delta x_2+\Delta x_4>0$, and the fact that $\mathbbm{1}'Q\leq \mathbbm{1}'$, we obtain
$$\mathbbm{1}'\Delta x_2+\mathbbm{1}'\Delta x_3+\mathbbm{1}'\Delta x_4<\mathbbm{1}'Q('\Delta x_2+\Delta x_3+\Delta x_4)\le
\mathbbm{1}'\Delta x_2+\mathbbm{1}'\Delta x_3+\mathbbm{1}'\Delta x_4\,.$$
This contradiction implies that, necessarily, $\mathcal C_2=\mathcal C_4=\emptyset$, thus proving the result.
\qed
We gather some immediate consequences of Proposition \ref{proposition:partition} in the following result.
\begin{corollary}\label{corollary:partition}
For every network $(P,w)$ and exogenous flow $c$, there exists a partition of the node set
\begin{equation}\label{eq:partition-unique}\mathcal V=\mathcal V_+\cup \mathcal V_0\cup \mathcal V_-\,,\end{equation}
such that
\begin{enumerate}
\item[(i)] for every equilibrium $x\in\mathcal X$
\begin{equation}\label{not0uniqueness2} x_{\mathcal V_-}=0\,,\qquad x_{\mathcal V_0}=P_{\mathcal V_0\mathcal V_0}x_{\mathcal V_0}+c_{\mathcal V_0}\,,\qquad x_{\mathcal V_+}=w_{\mathcal V_+}\,;\end{equation}
\item[(ii)] for every two equilibria $x^{(1)},x^{(2)}\in\mathcal X$,
\begin{equation}\label{not0uniqueness} x^{(1)}_{\mathcal V_-} =x^{(2)}_{\mathcal V_-}\,,\quad x^{(1)}_{\mathcal V_+} =x^{(2)}_{\mathcal V_+}\,.\end{equation}
\end{enumerate}
\end{corollary}
Corollary \ref{corollary:partition} implies that the check for uniqueness can always be restricted to those entries of the equilibria that are in $\mathcal V_0$ and that such entries of the equilibria solve a linear system of equations. However, the outstanding difficulty in the analysis of the equilibrium set $\mathcal X$ is that the partition \eqref{eq:partition-unique} is not known a priori. This will be dealt with in the following section.
\section{Uniqueness of the equilibria}\label{uniqueness}
In this section, we derive necessary and sufficient conditions for uniqueness of equilibria in a network $(P,w)$ with exogenous flow $c$. We shall first consider two fundamental special cases ---when the matrix $P$ is sub-stochastic and out-connected (Section \ref{sec:outconnected}) and, respectively, when $P$ is stochastic and irreducible (Section \ref{sec:irreducible})--- and then build on them in order to state and prove the general result (Section \ref{sec:general}).
\subsection{The out-connected case}\label{sec:outconnected}
Consider a sub-stochastic matrix $P\in\mathbb{R}^{n\times n}$ and let $$\mathcal O=\left\{i \in \mathcal V \:: \ \sum\nolimits_j P_{ij} <1 \right\}$$ be the set of the rows of $P$ that sum up to less than $1$. The matrix $P$ is said to be \emph{out-connected} if the set $\mathcal O$ is nonempty and globally reachable in the graph $\mathcal G_P$ (i.e., from every node there is a path to a node in $\mathcal O$). The spectrum of any out-connected sub-stochastic matrix is contained in the open unitary disk centered in $0$, as stated in the following result, whose proof can be found, e.g., in \cite{Como.Fagnani:2016}.
\begin{lemma}\label{lemma:spectral-radius}
Let $P\in\mathbb{R}_+^{n\times n}$ be an out-connected sub-stochastic matrix. Then, $P$ has spectral radius strictly smaller than $1$.
\end{lemma}
By combining Proposition \ref{proposition:partition} and Lemma \ref{lemma:spectral-radius}, we can prove
the following first uniqueness result.
\begin{proposition}\label{prop:uniquenss-outconnected}
For a network $(P,w)$ with $P$ sub-stochastic out-connected and for any exogenous flows $c$, the equilibrium is unique.
\end{proposition}
\proof
Let $x^{(1)}, x^{(2)}\in\mathcal X$ and put $y=x^{(1)}- x^{(2)}$. We know from point (ii) of Corollary \ref{corollary:partition} that $y_i=0$ for every $i\in \mathcal V_-\cup\mathcal V_+$. The proof is finished if $\mathcal V_0=\emptyset$. Otherwise, let
$z\in\mathbb{R}^{\mathcal V_0}$ and $Q\in\mathbb{R}^{\mathcal V_0\times\mathcal V_0}$ be the restrictions of $y$ to $\mathcal V_0$ and of $P$ to $\mathcal V_0\times\mathcal V_0$, respectively. It then follows from point (i) of Corollary \ref{corollary:partition} that
$z$ satisfies the equation $z=Q'z$. By Lemma \ref{lemma:spectral-radius}, the matrix $(I-Q)$ is invertible and thus $z=0$. We thus have that $x^{(1)}=x^{(2)}$ and this completes the proof.
\qed
\subsection{The irreducible stochastic case}\label{sec:irreducible}
We now study the case when $P$ is an irreducible stochastic matrix (i.e., the associated graph $\mathcal G_P$ is strongly connected). The following result gathers standard algebraic properties of irreducible stochastic matrices that will be instrumental in the following derivations.
\begin{lemma}\label{lemma:irreducible-stochastic}
Let $P\in\mathbb{R}^{n\times n}$ be an irreducible stochastic matrix. Then:
\begin{enumerate}
\item[(i)] the matrix $I-P$ has rank $n-1$ and there exists a unique invariant probability vector $\pi=P'\pi$. Moreover, such invariant probability vector has positive entries $\pi_i>0$ for every $i\in\mathcal V$;
\item[(ii)] for every zero-sum vector $c\in\mathbb{R}^n$, the serie
\begin{equation}\label{eq:nu-def} \nu=\frac12\sum_{k\ge0}\l(\frac{I+P'}2\r)^kc\end{equation}
is convergent and its limit $\nu$ satisfies $\nu=P'\nu+c$;
\item[(iii)] any square matrix $Q$ obtained as the restriction of $P$ to a proper subset of nodes $\mathcal U\subseteq\mathcal V$ has spectral radius smaller than $1$.
\end{enumerate}
\end{lemma}
\proof (i) This is quite standard, c.f., e.g., \cite{Norris}. \\
(ii) Let $\overline P=(I+P)/2$. Then, $\overline P$ is stochastic irreducible and aperiodic so that the matrix series
$$H=\frac12\sum_{k\ge0}\left(\overline P^k-\mathbbm{1}\pi'\right)$$
is convergent. Observe that, for any zero-sum vector $c$, we have $$H'c=\lim_{t\to+\infty}\frac12\sum_{k=0}^t\left((\overline P')^k-\pi\mathbbm{1}'\right)c
=\lim_{t\to+\infty}\frac12\sum_{k=0}^t(\overline P')^kc=\nu\,.$$
Observe that
$$\frac12 H+\frac12HP=H\overline P=\frac12\sum_{k\ge0}\left(\overline P^{k+1}-\mathbbm{1}\pi'P\right)=\frac12\sum_{k\ge1}\left(\overline P^{k}-\mathbbm{1}\pi'\right)=H-\frac12I+\frac12\mathbbm{1}\pi'\,,$$
so that $H=HP+I-\mathbbm{1}\pi'$, which in turn implies that
$$\nu=H'c=P'H'c+c-\pi\mathbbm{1}'c=\nu+c\,,$$
where the last equality follows since $\mathbbm{1}'c=0$. This completes the proof. \\
(iii) It is sufficient to apply Lemma \ref{lemma:spectral-radius}, upon noticing that the square matrix $Q$ obtained as the restriction of an irreducible stochastic matrix $P$ to any proper subset of nodes $\mathcal U\subseteq\mathcal V$ is an out-connected sub-stochastic matrix.
\qed
We can now state and prove the following result giving, for irreducible networks, an explicit characterization of the condition of non-uniqueness as well a representation of the set of equilibria.
\begin{proposition}\label{proposition:uniqueness-irreducible}
For a network $(P,w)$ with $P$ stochastic irreducible, and for any exogenous flow $c$, the set of equilibria $\mathcal X$ contains more than one elements if and only if
$\sum_ic_i=0$ and
\begin{equation}\label{eq:-min<min} \min _{i} \l\{\frac{\nu_{i}}{\pi_{i}}\r\}+\min _{i}\left\{ \frac{w_{i}-\nu_{i}}{\pi_{i}}\right\}>0\,,\end{equation}
where $\nu$ and $\pi$ are defined as in Lemma \ref{lemma:irreducible-stochastic}. Moreover, in this case, the set of equilibria is given by
\begin{equation}\label{eq:segment}\mathcal X=\left\{x=\nu+\alpha\pi:\,-\min _{i} \l\{\frac{\nu_{i}}{\pi_{i}}\r\}\le\alpha\le\min _{i}\left\{ \frac{w_{i}-\nu_{i}}{\pi_{i}}\right\}\r\}\,.\end{equation}
\end{proposition}
\proof
We first analyze the solution on $\mathbb{R}^n$ of the non-saturated linear system
\begin{equation}\label{eq:linear-system}x=P'x+c\,.\end{equation}
Since $P$ is in this case row-stochastic, summing up all the entries of both sides of \eqref{eq:linear-system} gives $$\mathbbm{1}'x=\mathbbm{1}'P'x+\mathbbm{1}'c=\mathbbm{1}'x+\mathbbm{1}'c$$ so that, for solutions to exist, it must hold that $\sum_ic_i=\mathbbm{1}'c=0$.
On the other hand, if condition $\sum_ic_i=0$ is satisfied, since $P$ is irreducible, Lemma \ref{lemma:irreducible-stochastic} ensures that $1-P$ has rank $n-1$ and that the set of solutions of \eqref{eq:linear-system} coincides with the line
\begin{equation}\label{line}\mathcal H=\{x=\nu+\alpha\pi:\,\alpha\in\mathbb{R}\}\,,\end{equation} with the vectors $\nu$ and $\pi$ defined as in Lemma \ref{lemma:irreducible-stochastic} (ii) and (i), respectively.
Notice that solutions of the linear system (\ref{eq:linear-system}) that stay in the complete lattice are automatically equilibria. In other words, $$\mathcal H \cap \mathcal L_0^w\subseteq \mathcal X$$
Observe, moreover, that $\mathcal H\cap\mathcal L_0^w$ coincides with the right-hand side of \eqref{eq:segment} and that condition \eqref{eq:-min<min} is equivalent to saying that $\mathcal H\cap\mathcal L_0^w$ is a segment of strictly positive length.
We are now ready to prove the statements of the theorem.
\begin{enumerate}
\item[(i)]
Suppose first that there are multiple equilibria, namely $|\mathcal X|>1$.
Consider the node set partition \eqref{eq:partition-unique} that is common to all solutions of system \eqref{eq:nonlinear-system} by Proposition \ref{proposition:partition}. If $\mathcal V_-\cup\mathcal V_+\neq\emptyset$, since $\mathcal V_0$ is a proper subset of $\mathcal V$, Lemma \ref{lemma:irreducible-stochastic} (iii) guarantees that the restriction $Q$ of $P$ to $\mathcal V_0\times\mathcal V_0$ has spectral radius smaller than $1$. Arguing exactly as in the proof of Proposition \ref{prop:uniquenss-outconnected}, we then deduce that $|\mathcal X|=1$. This contradicts the assumption made. Therefore, necessarily $\mathcal V=\mathcal V_0$. In this case, it follows from point (i) of Corollary \ref{corollary:partition} that all equilibria are solutions of (\ref{eq:linear-system}), namely $\mathcal H \cap \mathcal L_0^w= \mathcal X$.
By previous considerations, since this set is not empty, the condition $\mathbbm{1}'c=0$ must hold. Moreover, $|\mathcal X|>1$ implies that $\mathcal H \cap \mathcal L_0^w$ must be a segment of positive length that, as previously observed, is equivalent to condition \eqref{eq:-min<min}.
\item[(ii)] Suppose now that $\sum_ic_i=0$ and that condition \eqref{eq:-min<min} holds true. Then previous considerations imply that $\mathcal H \cap \mathcal L_0^w\subseteq\mathcal X$ is a segment of positive length. Non-uniqueness of equilibria is thus proven.
\item[(iii)] Finally notice that if any of the two equivalent conditions hold, then, from proof (i) we have that $\mathcal H \cap \mathcal L_0^w= \mathcal X$ and this is equivalent to representation (\ref{eq:segment}).
\end{enumerate}
This completes the proof.
\qed
The result above has a simple geometric interpretation in part already exploited in the proof. Assuming that $\sum_ic_i=0$, the line $\mathcal H$ defined in (\ref{line}) is the set of solutions of the non-saturated linear system (\ref{eq:linear-system}). The non-uniqueness condition (\ref{eq:-min<min}) is simply the condition that this line intersects the interior part of the lattice $\mathcal L_0^w$ and the set of equilibria in this case is the segment obtained by this intersection. The minimal and maximal equilibria are the boundary points of this interval. We notice that the arguments used in the proof also show that in the case of non-uniqueness, necessarily all nodes must be exposed nodes, namely $\mathcal V=\mathcal V_0$.
\begin{testexample}\label{example:small-network} Consider the network $(P,w)$ where
%
$$P=\begin{bmatrix}
0 & 0.75 & 0.25 \\
0 & 0 & 1\\
0.3 & 0.7 & 0
\end{bmatrix}\,,\qquad w=\begin{bmatrix}5\\ 3\\ 2\end{bmatrix}\,.$$
Notice that the matrix $P$ is stochastic and irreducible. The associated graph $\mathcal G_P$
is depicted in Figure \ref{fig:example-small-network}.
\begin{fixedfigure}
\begin{tikzpicture}[thick,scale=0.7, every node/.style={transform shape}]
\draw (-3.5,-1) ellipse (0.5 and 0.5);
\draw (0.5,-1) ellipse (0.5 and 0.5);
\draw [-triangle 60] plot[smooth, tension=.7] coordinates {(0.5,-0.5) (0.5,0.5) (-1,1)};
\draw [-triangle 60] plot[smooth, tension=.7] coordinates {(-3.5,-1.5) (-1.5,-2) (0.5,-1.5)};
\node at (-3.5,-1) {1};
\node at (0.5,-1) {2};
\node at (-1.5,1) {3};
\draw (-1.5,1) ellipse (0.5 and 0.5);
\draw plot[smooth, tension=.7] coordinates {(-1,1)};
\draw [-triangle 60] plot[smooth, tension=.7] coordinates {(-3.5,-0.5) (-3.5,0.5) (-2,1)};
\draw [-triangle 60] plot[smooth, tension=.7] coordinates {(-1.5,0.5) (-2,-0.5) (-3,-1)};
\draw [-triangle 60] plot[smooth, tension=.7] coordinates {(-1.5,0.5) (-1,-0.5) (0,-1)};
\node at (-1.5,-2.5) {$ 0.75 $};
\node at (-3.5,1) {$ 0.25 $};
\node at (0.5,1) {$ 1 $};
\node at (-2.5,-0.5) {$ 0.3 $};
\node at (-0.5,-0.5) {$ 0.7 $};
\end{tikzpicture}
\caption{The network of Example \ref{example:small-network}}
\label{f0N2}
\end{fixedfigure}
We analyze uniqueness for two possible exogenous flows $$ c^{(1)}= [-1, 1, 0]'\,\qquad c^{(2)}= [-2, 2, 0]'\,.$$
Notice first that $\sum_ic^{(1)}_i=\sum_ic^{(2)}_i=0$. We now check condition (\ref{eq:-min<min}).
Indicating with $\nu^i$, for $i=1,2$, the vector defined in (\ref{eq:nu-def}) for $c=c^{\circ}$ and with $\pi$ the invariant probability of $P$, a direct computation shows that
\begin{equation}\label{key}
\begin{array}{l}\displaystyle\min _{i} \l\{\frac{\nu^1_{i}}{\pi_{i}}\r\}+\min _{i}\left\{ \frac{w_{i}-\nu^1_{i}}{\pi_{i}}\right\} \approx 1.60>0\\[10pt]
\displaystyle\min _{i} \l\{\frac{\nu^2_{i}}{\pi_{i}}\r\}+\min _{i}\left\{\frac{w_{i}-\nu^2_{i}}{\pi_{i}}\right\} \approx -6.41 <0
\end{array}
\end{equation}
By Proposition \ref{proposition:uniqueness-irreducible} we deduce that for the flow $ c^{(1)}$ there are multiple equilibria, while for the flow $c^{(2)}$ the equilibrium is unique.
The structure of the equilibria is shown in Figure \ref{f3dd}. Notice how in the first case the line $\mathcal H$ intersects the open part of the lattice, while in the second is external. The unique equilibrium in this last case is a single point necessarily on the boundary of the lattice as some of its entries are saturated either to $0$ or to $w_i$.
\begin{fixedfigure}
\subfloat[The network $(P,w)$ with exogenous flow $c^{(1)}$ admits multiple equilibria (the red dots and the red segment).]{\begin{tikzpicture}[thick,scale=0.8, x=0.75pt,y=0.75pt,yscale=-1,xscale=1]
\draw [color={rgb, 255:red, 69; green, 127; blue, 6 } ,draw opacity=1 ][fill={rgb, 255:red, 126; green, 211; blue, 33 } ,fill opacity=0.38 ] (277.53,221.02) -- (208.41,161.26) -- (209.5,84) -- (309.6,69.45) -- (378.72,129.21) -- (377.63,206.47) -- cycle ; \draw [color={rgb, 255:red, 69; green, 127; blue, 6 } ,draw opacity=1 ] (209.5,84) -- (278.62,143.76) -- (277.53,221.02) ; \draw [color={rgb, 255:red, 69; green, 127; blue, 6 } ,draw opacity=1 ] (278.62,143.76) -- (378.72,129.21) ;
\draw (159.3,211.2) -- (160.3,70.2) (155.37,201.17) -- (163.37,201.23)(155.44,191.17) -- (163.44,191.23)(155.51,181.17) -- (163.51,181.23)(155.58,171.17) -- (163.58,171.23)(155.65,161.17) -- (163.65,161.23)(155.73,151.17) -- (163.73,151.23)(155.8,141.17) -- (163.8,141.23)(155.87,131.17) -- (163.87,131.23)(155.94,121.17) -- (163.94,121.23)(156.01,111.17) -- (164.01,111.23)(156.08,101.17) -- (164.08,101.23)(156.15,91.17) -- (164.15,91.23)(156.22,81.17) -- (164.22,81.23)(156.29,71.18) -- (164.29,71.23) ;
\draw (159.3,211.2) -- (228.3,279.2) (169.23,215.37) -- (163.61,221.07)(176.35,222.39) -- (170.74,228.09)(183.48,229.41) -- (177.86,235.11)(190.6,236.43) -- (184.98,242.13)(197.72,243.45) -- (192.1,249.15)(204.84,250.47) -- (199.23,256.16)(211.97,257.49) -- (206.35,263.18)(219.09,264.51) -- (213.47,270.2)(226.21,271.52) -- (220.59,277.22) ;
\draw (228.3,279.2) -- (443.5,250) (237.67,273.89) -- (238.75,281.82)(247.58,272.55) -- (248.66,280.47)(257.49,271.2) -- (258.57,279.13)(267.4,269.86) -- (268.47,277.79)(277.31,268.51) -- (278.38,276.44)(287.22,267.17) -- (288.29,275.1)(297.13,265.82) -- (298.2,273.75)(307.04,264.48) -- (308.11,272.41)(316.94,263.14) -- (318.02,271.06)(326.85,261.79) -- (327.93,269.72)(336.76,260.45) -- (337.84,268.37)(346.67,259.1) -- (347.75,267.03)(356.58,257.76) -- (357.66,265.68)(366.49,256.41) -- (367.57,264.34)(376.4,255.07) -- (377.48,263)(386.31,253.72) -- (387.38,261.65)(396.22,252.38) -- (397.29,260.31)(406.13,251.03) -- (407.2,258.96)(416.04,249.69) -- (417.11,257.62)(425.95,248.35) -- (427.02,256.27)(435.86,247) -- (436.93,254.93) ;
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\draw [color={rgb, 255:red, 0; green, 0; blue, 0 } ,draw opacity=1 ][fill={rgb, 255:red, 255; green, 0; blue, 0 } ,fill opacity=1 ] (298.75,183.57) .. controls (298.75,181.78) and (300.21,180.32) .. (302,180.32) .. controls (303.79,180.32) and (305.25,181.78) .. (305.25,183.57) .. controls (305.25,185.37) and (303.79,186.82) .. (302,186.82) .. controls (300.21,186.82) and (298.75,185.37) .. (298.75,183.57) -- cycle ;
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\draw (120,146) node {$x_{3}$};
\draw (165,268) node {$x_{1}$};
\draw (335,301) node {$x_{2}$};
\draw (147,79) node {$6$};
\draw (146,104) node {$4$};
\draw (146,129) node {$2$};
\draw (146,153) node {$0$};
\draw (140,179) node {$-2$};
\draw (139,203) node {$-4$};
\draw (161,238) node {$4$};
\draw (195,267) node {$2$};
\draw (333,280) node {$2$};
\draw (391,272) node {$0$};
\draw (274,287) node {$4$};
\draw (218,290) node {$0$};
\draw (258,144) node {$\overline{x}$};
\draw (315,169) node {$\underline{x}$};
\draw (397,212) node [color={rgb, 255:red, 74; green, 144; blue, 226 } ,opacity=1 ] {$\mathcal{H}$};
\draw (299,107) node [color={rgb, 255:red, 69; green, 125; blue, 3 } ,opacity=1 ] {$\mathcal L_0^w$};
\end{tikzpicture}}\quad
\subfloat[The network $(P,w)$ with exogenous flow $c^{(2)}$ admits a unique equilibrium (the red dot).]{\begin{tikzpicture}[thick,scale=0.78, x=0.75pt,y=0.75pt,yscale=-1,xscale=1]
\draw [color={rgb, 255:red, 69; green, 127; blue, 6 } ,draw opacity=1 ][fill={rgb, 255:red, 126; green, 211; blue, 33 } ,fill opacity=0.38 ] (277.53,221.02) -- (208.41,161.26) -- (209.5,84) -- (309.6,69.45) -- (378.72,129.21) -- (377.63,206.47) -- cycle ; \draw [color={rgb, 255:red, 69; green, 127; blue, 6 } ,draw opacity=1 ] (209.5,84) -- (278.62,143.76) -- (277.53,221.02) ; \draw [color={rgb, 255:red, 69; green, 127; blue, 6 } ,draw opacity=1 ] (278.62,143.76) -- (378.72,129.21) ;
\draw (159.3,211.2) -- (160.3,70.2) (155.37,201.17) -- (163.37,201.23)(155.44,191.17) -- (163.44,191.23)(155.51,181.17) -- (163.51,181.23)(155.58,171.17) -- (163.58,171.23)(155.65,161.17) -- (163.65,161.23)(155.73,151.17) -- (163.73,151.23)(155.8,141.17) -- (163.8,141.23)(155.87,131.17) -- (163.87,131.23)(155.94,121.17) -- (163.94,121.23)(156.01,111.17) -- (164.01,111.23)(156.08,101.17) -- (164.08,101.23)(156.15,91.17) -- (164.15,91.23)(156.22,81.17) -- (164.22,81.23)(156.29,71.18) -- (164.29,71.23) ;
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\draw (228.3,279.2) -- (443.5,250) (237.67,273.89) -- (238.75,281.82)(247.58,272.55) -- (248.66,280.47)(257.49,271.2) -- (258.57,279.13)(267.4,269.86) -- (268.47,277.79)(277.31,268.51) -- (278.38,276.44)(287.22,267.17) -- (288.29,275.1)(297.13,265.82) -- (298.2,273.75)(307.04,264.48) -- (308.11,272.41)(316.94,263.14) -- (318.02,271.06)(326.85,261.79) -- (327.93,269.72)(336.76,260.45) -- (337.84,268.37)(346.67,259.1) -- (347.75,267.03)(356.58,257.76) -- (357.66,265.68)(366.49,256.41) -- (367.57,264.34)(376.4,255.07) -- (377.48,263)(386.31,253.72) -- (387.38,261.65)(396.22,252.38) -- (397.29,260.31)(406.13,251.03) -- (407.2,258.96)(416.04,249.69) -- (417.11,257.62)(425.95,248.35) -- (427.02,256.27)(435.86,247) -- (436.93,254.93) ;
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\draw (165,268) node {$x_{1}$};
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\draw (333,280) node {$2$};
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\draw (274,287) node {$4$};
\draw (218,290) node {$0$};
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\draw (295,110) node [color={rgb, 255:red, 69; green, 125; blue, 3 } ,opacity=1 ] {$\mathcal L_0^w$};
\draw (293,154) node {$x$};
\end{tikzpicture}
}
\caption{Sets of equilibria for the network.}
\label{f3dd}
\end{fixedfigure}
\end{testexample}
\subsection{Uniqueness in the general case} \label{sec:general}
We now study uniqueness of equilibria for general networks $(P,w)$, where $P$ is an arbitrary sub-stochastic matrix and $w$ an arbitrary nonnegative vector.
In order to proceed, we need to introduce some further graph-theoretic notions. Consider the graph $\mathcal G_P=(\mathcal V,\mathcal E)$ associated with $P$ that, we recall, has node set $\mathcal V=\{1,2,\ldots,n\}$ and link set $\mathcal E=\{(i,j)\in\mathcal V\times\mathcal V:\,P_{ij}>0\}$.
A \emph{trapping set} is any nonempty subset of nodes $\mathcal S \subseteq \mathcal V$ with no links pointing out of it, i.e., such that $P_{ij}=0$ for every $i\in\mathcal S$ and $j\in\mathcal V\setminus\mathcal S$. A trapping set is referred to as \emph{irreducible} if it does not strictly contain any other trapping set. A \emph{sink} node $j$ is a node such that $\{j\}$ is a trapping set (and, as such, it is necessarily irreducible). The \emph{transient} part $\mathcal T \subseteq \mathcal V$ of a graph is the subset of nodes that do not belong to any irreducible trapping set. The node set $\mathcal V$ can be partitioned as the disjoint union \begin{equation}\label{V-partition}\mathcal V = \mathcal T \cup \mathcal S_1\cup \mathcal S_2\cup\ldots\cup \mathcal S_m\,,\end{equation} of the transient part $\mathcal T$ and of the distinct irreducible trapping sets $\mathcal S_l$ for $1\le l\le m$, with $0\le m\le n$. Observe that partition \eqref{V-partition} is determined by the matrix $P$ only. In the special case when $\mathcal G_P$ is strongly connected, the whole node set
$\mathcal V$ is itself an irreducible trapping set, so that in this case $\mathcal T=\emptyset$, $m=1$, and $\mathcal S_1=\mathcal V$. See Figure \ref{fig:trapping} for an example of a graph with a nonempty transient part $\mathcal T$ and a single trapping set $\mathcal S_1$.
\begin{fixedfigure}
\centering
\begin{tikzpicture}[thick, x=0.75pt,y=0.75pt,yscale=-1,xscale=1]
\draw [line width=0.75] (147,156) .. controls (147,148.27) and (153.27,142) .. (161,142) .. controls (168.73,142) and (175,148.27) .. (175,156) .. controls (175,163.73) and (168.73,170) .. (161,170) .. controls (153.27,170) and (147,163.73) .. (147,156) -- cycle ;
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\draw [line width=0.75] (342,169) -- (315,169) ;
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\draw [line width=0.75] (299,155) -- (298.06,124) ;
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\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][fill={rgb, 255:red, 74; green, 144; blue, 226 } ,fill opacity=0.12 ][line width=2.25] (260,108.8) .. controls (260,97.31) and (269.31,88) .. (280.8,88) -- (359.2,88) .. controls (370.69,88) and (380,97.31) .. (380,108.8) -- (380,171.2) .. controls (380,182.69) and (370.69,192) .. (359.2,192) -- (280.8,192) .. controls (269.31,192) and (260,182.69) .. (260,171.2) -- cycle ;
\draw [color={rgb, 255:red, 245; green, 166; blue, 35 } ,draw opacity=1 ][fill={rgb, 255:red, 245; green, 166; blue, 35 } ,fill opacity=0.12 ][line width=2.25] (136.5,108.5) .. controls (136.5,97.01) and (145.81,87.7) .. (157.3,87.7) -- (222.7,87.7) .. controls (234.19,87.7) and (243.5,97.01) .. (243.5,108.5) -- (243.5,170.9) .. controls (243.5,182.39) and (234.19,191.7) .. (222.7,191.7) -- (157.3,191.7) .. controls (145.81,191.7) and (136.5,182.39) .. (136.5,170.9) -- cycle ;
\draw (123,99) node [color={rgb, 255:red, 245; green, 166; blue, 35 } ,opacity=1 ] {$\mathcal{T}$};
\draw (396,97) node [color={rgb, 255:red, 74; green, 144; blue, 226 } ,opacity=1 ] {$\mathcal{S}_1$};
\end{tikzpicture}
\caption{\label{fig:trapping}A graph ${\mathcal{G}}$ containing an irreducible trapping set $\mathcal S_1$ and a nonempty transient part $\mathcal T$. }
\label{f2}
\end{fixedfigure}
The following result gives a complete analysis of the structure of the equilibria for generic networks. We recall the standing notation assumption: if $x$ is a vector and $P$ is a matrix, $x_{\mathcal A}$, $P_{\mathcal A\mathcal B}$ denote restrictions to, respectively, the index sets $\mathcal A$ or $\mathcal A\times\mathcal B$.
\begin{theorem}\label{theorem:general} Consider a network $(P,w)$, where $P$ is any sub-stochastic matrix, and an exogenous flow $c$.
Consider the partition \eqref{V-partition} of the node set into its transient part and its irreducible trapping sets of the graph $\mathcal G_P$.
Then, the equilibria $x\in\mathcal X$ split according to such partition
$$x=(x_{\mathcal T}, x_{\mathcal S_1}, \dots , x_{\mathcal S_m})$$
in such a way that
\begin{enumerate}
\item[(i)] their projection $x_{\mathcal T}$ on the transient part $\mathcal T$ is unique;
\item[(ii)] their projection $x_{\mathcal S_l}$ on a trapping set $\mathcal S_l$ that is out-connected or such that
\begin{equation}\label{eq:unique}\sum_{j\in\mathcal S_l}\left(c_j+\sum_{i\in\mathcal T}P_{ij}x_i\right)\ne0\,,\end{equation}
is unique;
\item[(iii)] their projection $x_{\mathcal S_l}$ on a trapping set $\mathcal S_l$ that is not out-connected and such that
\begin{equation}\label{eq:necessary-nonunique}\sum_{j\in\mathcal S_l}\left(c_j+\sum_{i\in\mathcal T}P_{ij}x_i\right)=0\,,\end{equation}
is non-unique if and only if
\begin{equation}\label{eq:min<min-3}\min_i \l\{\frac{\nu^{(l)}_i}{\pi^{(l)}_i}\r\}+\min_i\l\{\frac{w_i-\nu^{(l)}_i}{\pi^{(l)}_i}\r\}> 0\,,\end{equation}
where
\begin{itemize}
\item $\pi^{(l)}=(P_{\mathcal S_l\mathcal S_l})'\pi^{(l)}$ is the unique invariant probability vector of the block $P_{\mathcal S_l\mathcal S_l}$;
\item $\nu^{(l)}=\frac12\sum_{k\ge0}\l(\frac{I+(P_{\mathcal S_l\mathcal S_l})'}2\r)^k(c_{\mathcal S_l}+(P_{\mathcal T\mathcal S_l})'x_{\mathcal T})$
\end{itemize}
Moreover, in this case, the projection $x_{\mathcal S_l}$ of any equilibrium satisfies
\begin{equation}\label{eq:segment-2}x_{\mathcal S_l}= \nu^{(l)}+\alpha\pi^{(l)}\,,\qquad -\min _{i\in\mathcal S_l} \l\{\frac{\nu^{(l)}_{i}}{\pi^{(l)}_{i}}\r\}\le\alpha\le\min _{i\in\mathcal S_l}\left\{ \frac{w_{i}-\nu^{(l)}_{i}}{\pi^{(l)}_{i}}\right\}\,.\end{equation}
\end{enumerate}
\end{theorem}
\proof
First, we observe that, by the way the transient part $\mathcal T$ is defined, the projection of any equilibrium $x$ on $\mathcal T$ satisfies
$$x_{\mathcal T}= S_0^{w_{\mathcal T}}\left((P_{\mathcal T\mathcal T})'x_\mathcal T+c_{\mathcal T}\right)\,,$$
The fact that $\mathcal T$ is the transient part implies that the matrix $P_{\mathcal T\mathcal T}$ is sub-stochastic and out-connected, so that we can readily apply Proposition \ref{prop:uniquenss-outconnected} to prove claim (i).
We now study separately the behavior on each trapping set $\mathcal S_l$, for $l=1,\ldots,m$ observing that the relative projection of the equilibrium $x$ satisfies
$$x_{\mathcal S_l}= S_0^{w_{\mathcal S_l}}\left((P_{\mathcal S_l\mathcal S_l})'x_{\mathcal S_l}+(P_{\mathcal T\mathcal S_l})'x_\mathcal T+c_{\mathcal S_l}\right)\,,$$
Claims (ii) and (iii) then follow by applying Proposition \ref{proposition:uniqueness-irreducible}.
\qed
\section{Continuity and the lack thereof}\label{discontinuous}
In this section, we study how, for a given network $(P,w)$, the set of equilibria depends on the exogenous flow vector $c$. This analysis is crucial to study the way exogenous shocks affect the network equilibrium. Indeed, recall that in considered financial network model, we have that $c=a-b-\epsilon$ where $a$ and $b$ represent the vector of assets values and liabilities towards external entities and $\epsilon$ is the vector of shock affecting the value of the assets.
Thus, the resilience of the system with respect to shocks is in the end determined by the way solutions depend on the parameter vector $c$.
In the following we will consider a fixed network $(P,w)$ and a varying flow vector $c$ and use the notation
$$\mathcal X(c),\quad \overline x(c),\quad \underline x(c)$$
to denote, respectively, the set of equilibria, the maximal and the minimal equilibria of $(P,w)$ with exogenous flow $c$.
Moreover, let
\begin{equation}\label{eq:UM-def}\mathcal U=\left\{c\in \mathbb{R}^n:\,|\mathcal X(c)|=1\right\}\,,\qquad \mathcal M=\mathbb{R}^n\setminus\mathcal U\,,\end{equation}
be the subsets of exogenous flow vectors for which there is a unique equilibrium and, respectively, there are multiple equilibria.
For exogenous flow vectors $c\in\mathcal U$, we shall also use the notation $$x(c)=\underline x(c)=\overline x(c)$$ for the unique equilibrium.
The following result gives a complete picture of the behavior of the set of equilibria $\mathcal X(c)$ as a function of $c$. It shows that
the set $\mathcal M$ where the equilibrium is not unique is contained in the union of a finite number of linear manifolds of co-dimension $1$ in $\mathbb{R}^n$, that outside of it the equilibrium $x(c)$ is a piecewise continuous function of the exogenous flow $c$ that undergoes jump discontinuities when $c$ crosses the non-uniqueness set $\mathcal M$.
\begin{theorem}\label{theo:continuity} Let $(P,w)$ be a network.
Let $\mathcal U$ and $\mathcal M$ be defined as in \eqref{eq:UM-def} and let $m$ be number of irreducible trapping sets in $\mathcal G_P$.
Then,
\begin{enumerate}
\item[(i)] the non-uniqueness set $\mathcal M$ is the union of at most $m$ linear sub-manifolds of co-dimension $1$ in $\mathbb{R}^n$;
\item[(ii)] the map $c\mapsto x(c)$ is continuous on the uniqueness set $\mathcal U$;
\item[(iii)] for every exogenous flow $c^*\in\mathcal M$,
$$ \liminf\limits_{{\substack{c\in\mathcal U\\ c\to c^*}}} x(c)=\underline x(c^*)\,,\qquad \limsup\limits_{\substack{c\in\mathcal U\\ c\to c^*}}x(c)=\overline x(c^*)\,.$$
\end{enumerate}
\end{theorem}
\proof (i) Consider the partition \eqref{V-partition} of the node set of the graph $\mathcal G_P$ into its transient part and its irreducible trapping sets.
It then follows from Theorem \ref{theorem:general}(iii) that an exogenous flow $c\in\mathcal M$ must satisfy equation \eqref{eq:necessary-nonunique} for some irreducible trapping set $\mathcal S_l$.
Each such condition determines a linear sub-manifold of co-dimension $1$ in $\mathbb{R}^n$. Therefore, we can conclude that the set $\mathcal M$ of those exogenous flow vectors $c$ for which the equilibria are not unique is the union of at most $m$ linear sub-manifolds of co-dimension $1$ in $\mathbb{R}^n$.
(ii) and (iii): Consider first any exogenous flow $c^*\in\mathbb{R}^n$ and a sequence $\left(c{(k)}\right)_{k\ge1}$ in $\mathbb{R}^n$ such that
\begin{equation}\label{sequence}c{(k)}\stackrel{k\to+\infty}{\longrightarrow} c^*\,,\qquad\underline x(c{(k)})\stackrel{k\to+\infty}{\longrightarrow} x^*\,.\end{equation} Since $$\underline x(c(k))=S_0^w(P'\underline x(c(k))+c(k))\,,$$
for all $k\ge1$, passing to the limit in both sides of the above we obtain by continuity that
$$x^*=S_0^w(P'x^*+c^*)\,,$$ i.e., $x^*\in\mathcal X(c^*)$, so that in particular
\begin{equation}\label{limit}\underline x(c^*)\leq x^*\leq \overline x(c^*)\,.\end{equation}
The arbitrariness of the sequence $(c(k))_k$ satisfying \eqref{sequence} and \eqref{limit} imply that
\begin{equation}\label{liminf} \liminf\limits_{{\substack{c\in\mathcal U\\ c\to c^*}}} x(c)\geq\underline x(c^*)\end{equation}
Consider now $c(k)=c^*-1/k\mathbbm{1}$. A continuity argument shows that for those trapping sets for which condition \eqref{eq:unique} holds true with $c=c^*$, it continues to hold true with $c=c(k)$ for sufficiently large $k$. On the other hand, for those trapping sets for which condition \eqref{eq:necessary-nonunique} holds true for $c=c^*$, condition \eqref{eq:unique} holds true instead for $c=c(k)$ and sufficiently large $k$. This says that, independently from $c^*$, $c(k)\in\mathcal U$ for sufficiently large $k$. It then follows from Proposition \ref{proposition:c} (iv) that, since $c(k)\stackrel{k\to+\infty}{\longrightarrow} c^*$ and $c(k)\le c^*$ for every $k\ge1$,
$$ x(c(k))=\underline x(c(k))c(k)\stackrel{k\to+\infty}{\longrightarrow} x^*\leq \underline x(c^*)\,.$$
By combining the above with (\ref{liminf}) we deduce that
\begin{equation}\label{liminfeq} \liminf\limits_{{\substack{c\in\mathcal U\\ c\to c^*}}} x(c)=\underline x(c^*)\end{equation}
A completely analogous argument allows one to prove that
\begin{equation}\label{limsupeq} \limsup\limits_{{\substack{c\in\mathcal U\\ c\to c^*}}} x(c)=\overline x(c^*)\end{equation}
Now, in case that $c^*\in\mathcal U$, we have that $\underline x(c^*)=\overline x(c^*)$ and then relations (\ref{liminfeq}) and (\ref{limsupeq}) together yield (ii). In case instead that $c^*\in\mathcal M$,
relations (\ref{liminfeq}) and (\ref{limsupeq}) together prove (iii).
\qed\medskip
We finally characterize the size of the maximum jump $\overline x(c^*)-\underline x(c^*)$.
\begin{corollary}\label{coro:jump} Let $(P,w)$ be a network.
Consider the partition \eqref{V-partition} of the node set into its transient part $\mathcal T$ and its irreducible trapping sets $\mathcal S_l$ for $l=1,\ldots, m$. For each $l$, let $\pi^{(l)}$ be the invariant probability vector of the block $P_{\mathcal S_l\mathcal S_l}$. Then, the maximal norm of a jump discontinuity in the equilibrium is given by
\begin{equation}\label{eq:max-jump}\max_{c\in\mathbb{R}^n}\|\overline x(c)-\underline x(c)\|_p^p=\sum_{1\le l\le m}\left(\min_{i\in\mathcal S_l}\frac{w_i}{\pi_i^{(l)}}\right)^p\|\pi^{(l)}\|_p^p\,,\end{equation}
\end{corollary}
\proof
First observe that, for every $c\in\mathbb{R}^n$, Theorem \ref{theorem:general} guarantees that there exist scalars $\alpha_l$ such that
\begin{equation}\label{eq:coro-1}(\overline x(c))_{\mathcal T}-(\underline x(c))_{\mathcal T}=0\,,\qquad (\overline x(c))_{\mathcal S_l}-(\underline x(c))_{\mathcal S_l}=\alpha_l\pi^{(l)}\,,\end{equation}
for $1\le l\le m$. Now, observe that
$0\le\overline x_{i}(c)-\underline x_{i}(c)\le w_i$ for every $i\in\mathcal V$, so that
$0\le\alpha_l\pi_i^{(l)}\le w_i$ for every $i\in\mathcal S_l$, which in turn implies that
\begin{equation}\label{eq:coro-2}0\le\alpha_l\le\min_{i\in\mathcal S_l}\frac{w_i}{\pi^{(l)}_i}\,,\end{equation}
for $1\le l\le m$. It then follows from (\ref{eq:coro-1}) and \eqref{eq:coro-2} that
$$\|\overline x(c)-\underline x(c)\|_p^p\le\sum_{1\le l\le m}\left(\min_{i\in\mathcal S_l}\frac{w_i}{\pi_i^{(l)}}\right)^p\|\pi^{(l)}\|_p^p\,,$$
for every $c\in\mathbb{R}^n$.
It remains to prove equality in \eqref{eq:max-jump}. For this, consider $c=0$ and notice that, since $0\in\mathcal X(0)$, then $\underline x(0)=0$. Notice now that on each irreducible trapping set $\mathcal S_l$, the non-uniqueness conditions (\ref{eq:necessary-nonunique}) and (\ref{eq:min<min-3}) of Theorem \ref{theorem:general} hold true (since $\nu^{(l)}=0$) and it follows from expression (\ref{eq:segment-2}) that
$$\left(\overline x(0)\right)_{\mathcal S_l}=\min_{i\in\mathcal S_l}\frac{w_i}{\pi^{(l)}_i}\pi^{(l)}$$
Therefore,
$$\|\overline x(0)-\underline x(0)\|_p^p=\sum_{1\le l\le m}\left(\min_{i\in\mathcal S_l}\frac{w_i}{\pi_i^{(l)}}\right)^p\|\pi^{(l)}\|_p^p\,,$$
thus completing the proof.
\qed
We conclude this section with the following example.
\begin{testexample}\label{example:loss}
In this example we will show how the jump discontinuity that occurs when the exogenous flow $c$ crosses the critical set $\mathcal M$ can be interpreted in the context of financial networks. For sake of simplicity, we will consider a strongly connected graph, but the analysis can be carried on in the general case.
To measure the aggregated effect of a shock on a network, it is useful to introduce a risk measure known as \emph{systemic loss} as defined in \cite{Peyton2002}. Let $c^{\circ}$ an arbitrary exogenous flow for which all nodes in the financial network are fully liable, i.e., such that $x(c^{\circ})=w$, and let $c\le c^{\circ}$ be the exogenous flow after a shock. Then, define the systemic loss as the aggregated difference between the net worth of the nodes before the shock, i.e., $v^{\circ}=P'w+c^{\circ}-w$, and their net worth after the shock, i.e. $v=P'x(c)+c-w$.
Observe that
\begin{equation}
l\left(c^{\circ}, c\right) =\mathbbm{1}^{\prime}\left(v^{\circ}-v\right)=\mathbbm{1}^{\prime}\left(P^{\prime} w+c^{\circ}-w-\left(P^{\prime} x(c)+c-w\right)\right) =\mathbbm{1}^{\prime}c^{\circ}-\mathbbm{1}'c+\mathbbm{1}'w-\mathbbm{1}'x(c)\,.
\end{equation}
In the rightmost side of the above, the term $(\mathbbm{1}'c^{\circ}-\mathbbm{1}'c)$ represents the direct loss inflicted by the shock, while $(\mathbbm{1}'w-\mathbbm{1}'x(c))$, called the shortfall term, represents the indirect loss triggered by reduced payments.
Let us now compute the size of the jump of the loss function when the exogenous flow $c$ crosses the critical region $\mathcal M$. By using the representation of solutions derived in Proposition \ref{proposition:uniqueness-irreducible}, we obtain that for every $c^*\in\mathcal M$, the jump in the loss is given by
\begin{equation}\label{eq:exdisc}
\begin{aligned}
\Delta l\left(c^{*}\right) &=\liminf _{c \in \mathcal{U} \atop c \rightarrow c^{*}} l\left(c^{\circ}, c\right)-\limsup _{c \in \mathcal{U} \atop c \rightarrow c^{*}} l\left(c^{\circ}, c\right)
=\mathbbm{1}^{\prime}\left(\overline{x}\left(c^{*}\right)-\underline{x}\left(c^{*}\right)\right)
\\[2ex] &=\mathbbm{1}^{\prime}\left(\nu+\min _{i}\left\{\frac{w_{i}-\nu_{i}}{\pi_{i}}\right\} \pi-\nu+\min _{i}\left\{\frac{\nu_{i}}{\pi_{i}}\right\} \pi\right) \\[2ex] &=\min _{i}\left\{\frac{\nu_{i}}{\pi_{i}}\right\}+\min _{i}\left\{\frac{w_{i}-\nu_{i}}{\pi_{i}}\right\}\,. \end{aligned}
\end{equation}
Let us consider again the network of Example \ref{example:small-network} (Figure \ref{f0N2}) and compute how the loss function varies when an exogenous shock of magnitude $\epsilon \in [0,14]$ hits the network. Specifically, let us consider an initial vector $c^{\circ}=[5, 2, 2]'$ and an exogenous shock $c$ such that: \begin{equation*}\label{key}
c=c^{\circ}-\epsilon q, \quad \quad q= \begin{bmatrix}
0.07\\
0.59 \\
0.34
\end{bmatrix}, \quad \epsilon \in [0,14].
\end{equation*}
Notice that we have decided to perturb the vector $c^{\circ}$ along the particular direction $q$; such $q$ can be interpreted as a normalized vector collecting the sensibilities of nodes with respect to the shock magnitude $\epsilon$.
The jump discontinuity must happen when uniqueness conditions fail to hold, which, by Proposition \ref{proposition:uniqueness-irreducible}, means that first of all it must be $\mathbbm{1}'c=0 \implies \epsilon=9$, which gives us a candidate critical vector $c^*=[4.4, -3.3, -1.1]'$. For such $c^*$, a direct computation shows that $\min _{i}\left\{\frac{\nu_{i}}{\pi_{i}}\right\}+\min _{i}\left\{\frac{w_{i}-\nu_{i}}{\pi_{i}}\right\} \approx 14.6>0$, which implies that $c^* \in \mathcal M$. Let us plot the loss function and the solution vector $x$ as functions of $\epsilon$ in Fig. \ref{lossdisc}.
\begin{fixedfigure}
\centering
\subfloat[Loss as a function of $\epsilon$. We can see the jump discontinuity at $\epsilon=9$. ]
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} \quad
\subfloat[Solution vector as a function of $\epsilon$. We can see the jump discontinuity at $\epsilon=9$, which brings node 1 and 3 to default.]
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\draw (337,238.89) node {$\epsilon $};
\draw (66,77.89) node {$x( \epsilon )$};
\draw (91,248.49) node [scale=0.7] {$0$};
\draw (230,254) node [scale=0.7,color={rgb, 255:red, 255; green, 0; blue, 0 } ,opacity=1 ] {$9$};
\draw (293,74.9) node [scale=0.8,color={rgb, 255:red, 255; green, 0; blue, 0 } ,opacity=1 ] [align=left] {Node failure};
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\draw (141,141.93) node [scale=0.8] [align=left] {\textcolor[rgb]{0.25,0.46,0.02}{Node 2}};
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\end{tikzpicture}
}
\caption{ test}
\label{lossdisc}
\end{fixedfigure}
Figure \ref{lossdisc} (a) shows how the loss function varies piece-wise linearly until $\epsilon=9$, where it undergoes the jump discontinuity. The size of the jump can be computed using \eqref{eq:exdisc}
\begin{equation}\label{key}
\Delta l(c^*)=\min _{i}\left\{\frac{\nu_{i}}{\pi_{i}}\right\}+\min _{i}\left\{\frac{w_{i}-\nu_{i}}{\pi_{i}}\right\} \approx 14.6.
\end{equation}
It is particularly useful also to plot the solution vector $x$ as a function of $\epsilon$ (Figure \ref{lossdisc} (b)); we can notice that all nodes are solvent for $\epsilon < 6.5$ while for $\epsilon \approx 6.5$ node 2 goes bankrupt as its outflow falls below $w_2=3$. As the shock magnitude increases, we reach the discontinuity point at $\epsilon=9$ where the network suffers a dramatic crisis as node 1 and 3 suddenly default. Notice in particular how node 3 goes from fully solvent ($x_3=w_3$) to completely insolvent ($x_3=0$) around $\epsilon=9$.
\end{testexample}
\section{Conclusions}\label{conclusions}
This paper has analyzed network equilibria modeled as the solutions of a linear fixed point equation with a saturation inequality. Necessary and sufficient conditions for uniqueness and a general expression describing all such equilibria for a general network have been proved. Finally, the dependence of the network equilibria on the exogenous flows in the network has been studied highlighting the existence of jump discontinuities. This model was first considered to determining clearing payments in the context of networked financial institutions interconnected by obligations and it is one of the simplest continuous model where shock propagation phenomena and cascading failure effects may occur. Our results contribute to an in-depth analysis of such phenomena.
The understanding of the extent to which the network topology determines the structure of the solutions as well the possibility of these cascading effects to occur is still not sufficiently understood. As a future project, we aim at studying this for random networks with prescribed degree distributions.
|
1,116,691,499,240 | arxiv | \section{Introduction}
The analysis of time series of meteorological data has an increasing interest in many fields. Their analysis is interesting in order to have a better understanding of the atmospheric phenomena, to determine the climate of a geographical data or to predict the occurrence of extreme events.
In this work, motivated by our real data example, we are interested in the modelling of a bivariate time series of wind speed and direction.
The analysis of wind time series have been carried out by means of different approaches, for example mixture type models (\cite{Ailliot2012}), harmonic analyses (\cite{belu2013}), ARMA-GARCH models (\cite{Lojowska2010}) or stochastic differential equations (\cite{Zarate2013}). Among the others, the hidden Markov model (HMM), a class of mixture model for time series, proved to be well suited to model wind data, see for example
\cite{Martin1999}, \cite{Holzmann2006}, \cite{Ailliot2012}, \cite{Bulla2012}, \cite{lagona2015} or \cite{mastrantonio2015}.
In the HMM, the data-generative distribution is expressed
as a mixture, with components that belong, generally, to an ``easy tractable'' distribution (called \emph{emission distribution} or \emph{regime-specific density}). The belonging to a component of the mixture, called also \emph{regime} or \emph{state}, depends on an underlying and unobservable discrete valued Markov process.
The wind direction is a circular variable, i.e. a variable that represents an angle or a point over the unit circle, while the speed is a linear (or inline) one. Hence, if a time series of wind direction and speed is modelled through an HMM, the emission distribution must be defined over a mixed circular-linear domain. Generally conditional independence between the circular and linear variables is assumed, but exceptions exist (see for example \cite{Bulla2012,mastrantonio2015}).
Although HMMs for wind data have been proposed in the literature, here, due to particular features of our data, a direct application of the models previously proposed is not possible. The measured linear and circular variables are interval-censored (\cite{Lindsey1998}) and they are recorded as discrete variables; more precisely the wind speed is recorded as integer knots and the direction is measured on a discrete scale with 36 equally spaced values over the unit circle. Furthermore, the direction is strictly related to the wind speed as when the latter is too small, the direction is not recorded and, due to the low instrument sensitivity, a recorded value of 0 or 1 wind speed is not reliable.
To take into account all these features, we introduce a new bivariate distribution to be used as the regime-specific density in the HMM. The model is estimated under a non-parametric Bayesian framework, using the hierarchical Dirichlet process-HMM (HDP-HMM) of \cite{Teh2006} and the modification introduced by \cite{fox2011}, namely the sticky HDP-HMM (sHPD-HMM). The sHPD-HMM allows us to estimate, as a model parameter, the number of regimes occupied by the time series. The model is applied to simulated and real data examples.
The paper is organized as follows. In Section \ref{sec:motivo} we describe the motivating example while in Section \ref{sec:dist} we introduce the circular-linear distribution by first formalizing the marginal distribution of the linear component, Section \ref{sec:lin}, and then the one of the circular, conditioned to the linear one, Section \ref{sec:circ}. Section \ref{sec:model} is devoted to the model specification while Section \ref{sec:prior} gives implementation details. Section \ref{sec:ex} contains the simulated examples (Section \ref{sec:sim}) and the real data one (Section \ref{sec:real}). The paper ends with a discussion (Section \ref{sec:disc}).
\section{Motivating example}\label{sec:motivo}
In last decades the Italian cost suffers an intensification of erosion, in particular in South Italy, along the coastline on the Tyrrhenian side, with manmade coastal structures. This is obviously a result of both natural and anthropogenic causes. The study of wind time series, recorded along the cost, is of a great importance since the wind generates waves, a significant factor of coastal geomorphology.
The data are recorded with an anemometer, located on the rocky cape of Capo Palinuro, in the town of Centola, province of Salerno, South Italy. The meteorological station of Capo Palinuro is one of the coastal stations managed by the Meteorological Service of the Military Italian Air Force. The instrument is placed away from obstacles, 10 meters above ground.
The data are
provided by the National Center of Aeronautical Meteorology and Climatology (C.N.M.C.A.), special office of the Meteorological Service of the Italian Air Force.
In this work we focus on the year 2010, that is a particular one. While the station, for decades, has recorded winds blowing between North and North-West or South and South-East octants, in 2010 the prevailing winds (the directions with the higher frequency) are the ones blowing from North-East and South-West, which is almost the opposite of the general trend. Then, it is important to understand and characterize in a greater details, i.e. through an HMM, the wind distribution of this year.
\subsection{Data description}
The linear variable is interval-censored, i.e. it is recorded with an error of 1 knot, and let $Y^*$ be the recorded wind speed, if $Y^*=c$, with $c \in \mathbb{Z}^+$, the real (continuous) variable has value $\geq \min\{c-0.5,0 \}$ but $<c+0.5$. Further uncertainty is added to the values below 2 knots as the instrument sensitivity does not allow reliable recordings.
Also the wind direction is interval-censored and it is measured in degrees
on a discrete scale with 36 distinct values. If the recorded circular variable, $X$, assumes value $c$, with $c \in \{ \frac{2 \pi}{36}j\}_{j=0}^{35}$\footnote{The direction recorded by the instrument assumes values in $(0,10,20,\dots, 350)$ but we rescale it to $[0,2 \pi)$. }, the real direction has value $\geq c-\frac{1}{2}\frac{2 \pi}{36}$ but $<c+\frac{1}{2}\frac{2 \pi}{36}$.
If the (continuous) wind speed is too low, i.e. close to 0, the instrument could not be able to record $X$ and we have a missing recording.
This missing gives information about the non-reliable linear recording, i.e. they are \emph{informative} and \emph{non-ignorable}, (\cite{RUBIN1976}) and then
the process that generates the missing observation must be taken into account in the model specification.
Due to the instrument malfunction, we have 2920 observations with, respectively, 3 and 2 non-informative missing in the circular and linear variables. $y_t^*$ assumes value 0 494 times while
$x_t $ is equal to $ \{\emptyset \}$ 328 times. Figure \ref{fig:obs} shows
barplots of the circular and linear data.
\begin{figure}[t!]
\centering
{\subfigure[Wind speed]{\includegraphics[scale=0.4]{OBSlin}}}
{\subfigure[Wind direction]{\includegraphics[scale=0.4]{OBScirc}}}
\caption{Linear (a) and circular (b) barplot of the variable used in the real data example.} \label{fig:obs}
\end{figure}
\section{The model } \label{sec:dist}
Before the formalization of the HMM (Section \ref{sec:model}), we introduce the circular-linear distribution we use in the model specification.
\subsection{The linear distribution} \label{sec:lin}
We code
the information carried by the recorded linear variable ($Y^*$) into two new variables. One representing the ``true'' discrete wind speed with support $\mathbb{Z}^+$, $Y$, and a Bernoulli variable $W$, that assumes value 0 when $Y^*<2$ and 1 otherwise. The marginal distribution of $Y$ is setted to be Poisson with parameter $\lambda_y$: $Y\sim P(\lambda)$. Between $Y$ and $Y^*$ the following relation exists:
\begin{equation} \label{eqy11}
\Big\{
\begin{array}{lccc}
Y = Y^* & \mbox{ } & \mbox{if }Y^*\geq 2, \\
Y \in \{0,1\} & \mbox{ } & \mbox{if }Y^*<2.
\end{array}
\end{equation}
\\
Let $W$ be a binary random variable such that
\begin{align}
W=1 & \mbox{ if } Y^*\ge 2,\label{eq:w1}\\
W=0 & \mbox{ if } Y^* <2 ,\label{eq:w2}
\end{align}
with
\begin{equation}\label{eq:ber}
W \sim Bern(1-e^{-\lambda_{y}}(1+\lambda_{y})),
\end{equation}
where $Bern(\cdot)$ indicates the Bernoulli distribution. We define the distribution of $Y|W, \lambda_y$ as follows:
\begin{equation} \label{eq:y1}
P(y|w=1, \lambda_y)=
\frac{\lambda_y^{y}e^{-\lambda_y}}{y!} \frac{1}{ 1-e^{-\lambda_{y}}(1+\lambda_{y}) } I(y \in {\mathbb{Z}^+\backslash \{0,1\}})
\end{equation}
and
\begin{equation} \label{eq:y2}
P(y|w=0, \lambda_y)=
\left(\frac{\lambda_{y}e^{-\lambda_y}}{e^{-\lambda_{y}}(1+\lambda_{y})}\right)^y \left(1-\frac{\lambda_{y}e^{-\lambda_y}}{e^{-\lambda_{y}}(1+\lambda_{y})}\right)^{1-y} I(y \in {\{0,1\}})
\end{equation}
where $I(y \in {A})$ is the indicator function.
\subsection{The circular distribution} \label{sec:circ}
To model properly the non-ignorable missing value of $X$, we specify the domain of the circular variable as a mixed one, composed of a discrete circle $\mathbb{D} = \{\frac{2 \pi}{36}j\}_{j=0}^{l-1}$, with $36$ equally spaced values, and the empty set, $\{ \emptyset\}$. The direction assumes value $\{ \emptyset\}$ if the measure is a missing non-ignorable and
then the distribution of the circular variable is an hurdle one. The hurdle distribution, first introduced by \cite{Mullahy1986}, assumes that the observation, in this case the wind direction, comes from two data-generative processes. With probability $\nu^*$ the observation belongs to the hurdle, i.e. the empty set, otherwise it belongs to the other portion of the mixed domain, i.e. $\mathbb{D}$, and a distribution over $\mathbb{D}$ must be chosen.
Since the circular variable can assume value $\{ \emptyset\}$ only if the discrete wind speed is 0, the probability $\nu^*$ must depend on $Y$. Indeed $(X=\{\emptyset\}) \Rightarrow (Y=0)$ but it is not true that $(Y=0)\Rightarrow (X=\{\emptyset\})$, as $X$ may be measured when a light wind is blowing (below 1 knot). We specify the hurdle probability as $\nu^*= \nu I(y=0)$ and, as required, $\nu^*\neq 0$ only if $Y=0$.\\
In the literature several distributions for discrete circular variables have been proposed, see for example the wrapped Binomial (\cite{Girija2014}) or the wrapped Weibull (\cite{Sarma2011}). \cite{mastrantonio2015f} proved that for most of them,
the inference they allow strongly depends on the choice of the reference system origin and orientation, while a ``proper'' distribution must be independent on these choices, for details see \cite{mastrantonio2015f}. As distribution over $\mathbb{D}$, we choose the invariant wrapped Poisson (IWP), proposed by \cite{mastrantonio2015f}, since it is a proper and flexible distribution for discrete circular data that is easy to implement in a Bayesian framework.
The density of $X$ is
\begin{equation} \label{eq:hiwp1}
P(x|y, \lambda_{x},\eta,\xi) = \left( \nu^* \right)^{I(x=\emptyset)} \left( (1-\nu^* ) P(x|x \in \mathbb{D},\lambda_{x},\eta,\xi) \right)^{I(x\in\mathbb{D})}.
\end{equation}
where $P(x|x \in \mathbb{D},\lambda_{x},\eta,\xi)$ is the pmf of the IWP:
\begin{equation} \label{eq:unw3}
P(x|x \in \mathbb{D}, \lambda_{x},\eta, \xi)= \sum_{k=0 }^{\infty} \frac{\lambda^{(\eta \theta- \xi ) \text{ mod } (2 \pi) 36/(2 \pi)+ k36 }e^{-\lambda}}{((\eta \theta- \xi )\text{ mod } (2 \pi) 36/(2 \pi)+ k36)!}, \, \xi \in \mathbb{D},\, \eta \in \{-1,1\}.
\end{equation}
\eqref{eq:unw3} is obtained by wrapping the density of a linear transformation of a Poisson distributed variable. More in details, we first obtain the density of $Q^* = \eta\left( Q \frac{2 \pi}{36} +\xi \right)$, where $Q \sim P(\lambda_x)$, that is
\begin{equation} \label{eq:yty}
\frac{\lambda^{(\eta q^*- \xi )36/(2 \pi) }e^{-\lambda}}{((\eta q^*- \xi )36/(2 \pi))!},
\end{equation}
then we find the distribution of the associated wrapped variable $X = Q^* \text{ mod } 2 \pi$, that is \eqref{eq:unw3}. Note that the transformation $X = Q^* \text{ mod } 2 \pi$ wraps the linear variable $Q^*$, and its density, around the discrete circle $\mathbb{D}$. $k$, i.e. \emph{winding number}, identify in \eqref{eq:unw3} the $k^{th}$ portion of the domain of $Q^*$ that is wrapped around the circle.
We can compute in closed form the directional mean and concentration of the IWP (\cite{mastrantonio2015f}), i.e. the circular counterparts of linear mean and concentration:
\begin{align}
\mu =& \eta\xi+\lambda \sin\left(\eta \frac{2 \pi}{36}\right), \label{eq:cricmean}\\
c =& e^{-\lambda \left(1- \cos \left(\frac{2 \pi}{36}\right)\right)}.\label{eq:cricconc}
\end{align}
Equations \eqref{eq:cricmean} and \eqref{eq:cricconc}, as the mean and concentration for the inline variables, are useful statistics to describe the circular variable behaviour.
It is not easy to work directly with equation \eqref{eq:unw3}, since it involves an infinite sum. When a wrapped distribution is used (\cite{coles98,Jona2013,mastrantonio2015c}), a standard approach is to introduce the latent random variable $K$, and to work with the joint density of $(X,K)$ that is the summand in \eqref{eq:unw3} and does not require the infinite sum evaluation:
\begin{equation} \label{eq:aa}
P(x,k|x \in \mathbb{D}, \lambda_{x},\eta, \xi)= \frac{\lambda^{(\eta \theta- \xi ) \text{ mod } (2 \pi) 36/(2 \pi)+ kl }e^{-\lambda}}{((\eta \theta- \xi )\text{ mod } (2 \pi) 36/(2 \pi)+ kl)!}.
\end{equation}
The winding number is needed also to define an efficient sampling scheme for the IWP parameters, see \cite{mastrantonio2015f} and Section \ref{sec:prior}. \\
The augmented hurdle type density is
\begin{equation} \label{eq:hiwp2}
P(x,k|y, \lambda_{x},\eta,\xi) = \left( \nu^* \right)^{I(x=\emptyset)} \left( (1-\nu^* ) P(x,k|x \in \mathbb{D},\lambda_{x},\eta,\xi) \right)^{I(x\in\mathbb{D})}.
\end{equation}
We write $X,K,Y,W|\lambda_{x},\eta, \xi, \nu,\lambda_{y} \sim HiwpP_{\mathbb{D}}(\lambda_x, \eta, \xi,\nu,\lambda_y)$ if $W|\lambda_y $ is distributed as \eqref{eq:ber}, $Y|W, \lambda_y$ as in \eqref{eq:y1} and \eqref{eq:y2} and $X,K|Y, \lambda_{x},\eta,\xi$
as in \eqref{eq:hiwp2}.
\subsection{The HMM specification} \label{sec:model}
Following the formalization introduced in Sections \ref{sec:lin} and \ref{sec:circ}, we proceed by modeling the four-variate time series $\{ \mathbf{x}, \mathbf{k}, \mathbf{y},\mathbf{w} \}$, where $\mathbf{x}= \{ x_t\}_{t=1}^T$, $\mathbf{k}=\{k_t\}_{t=1}^T$, $\mathbf{y}= \{y_t\}_{t=1}^T$ and $\mathbf{w}= \{w_t\}_{t=1}^T$.
We time-cluster the data with a non-parametric Bayesian HMM, namely the sHDP-HMM, that allows us to group the time series in homogeneous regimes, as the standard HMM, but it does not need to assume known the number of regimes $R$ that can be estimated along with the other model parameters. \\
The belonging to a regime is coded via a discrete random variable $z_t$; if $z_t=r$, at time $t$ the system is in regime $r$. In the sHDP-HMM is assumed that $z_t \in \mathbb{Z}^+ \backslash \{0 \} $, i.e. the number of regimes is potentially infinite. Indeed since $T$ is finite, the elements of the time series $\mathbf{z}=\{z_t \in \mathbb{Z}^+ \backslash \{0 \} \}_{t=1}^T$ will assume only a finite number of states ($R$), i.e. the number of non-empty regimes.
The sHDP-HMM is a hierarchical model where, let $\boldsymbol{\psi}_r=\{\lambda_{x,r}, \eta_r, \xi_r, \nu_r,\lambda_{y,r}\}$ be the vector of parameters, the first stage is
\begin{align}
P( \mathbf{x}, \mathbf{k}, \mathbf{y}, \mathbf{w}|\mathbf{z}, \{\boldsymbol{\psi}_{r}\}_{r=1}^{\infty}) =& \prod_{t =1}^T \prod_{r=1}^{\infty} \left[ P(x_t,k_t,y_t,w_t|\boldsymbol{\psi}_{r}) \right]^{I({r=z_t})}, \label{eq:hmm1}\\
X_t,K_t,Y_t,W_t|\boldsymbol{\psi}_r \sim & HiwpP_{\mathbb{D}}(\lambda_{x,r}, \eta_k, \xi_k,\nu_k,\lambda_{y,r}), \label{eq:hmm12}
\end{align}
At the second level of the hierarchy, $\mathbf{z}$ is assumed to follow a first order Markov process, i.e. $P(z_t|z_{t-1},z_{t-2},\dots , z_{1})= P(z_t|z_{t-1})$, with
\begin{equation} \label{eq:hmm2}
P(z_t | z_{t-1} , \boldsymbol{\pi}_{z_{t-1}}) = \pi_{z_{t-1}z_t}.
\end{equation}
and $\boldsymbol{\pi}_j= \{ \pi_{ji} \}_{i=1}^{\infty}$ is the $j^{th}$ row of the transition matrix, i.e. the matrix that rules the probabilities to move from one state to another. The initial state $z_0$, and the associated vector of probabilities, cannot be estimated consistently since we have no observations at time 0 (for details see \cite{cappe2005}) and, without loss of generality, we set $z_0=1$. The equations \eqref{eq:hmm1}, \eqref{eq:hmm12} and \eqref{eq:hmm2} define an HMM with an infinite number of states, i.e. $z_t \in \mathbb{Z}^+\backslash \{0\}$, infinite number of parameters, $\{\boldsymbol{\psi}_{r}\}_{r=1}^{\infty}$, and vectors of probabilities of infinite length, $\boldsymbol{\pi}_j$s.
The sHDP-HMM specification is concluded assuming the following:
\begin{align}
\boldsymbol{\pi}_{{r}} | \rho,\gamma,\{\beta_j \}_{j=1}^{\infty},\boldsymbol{\psi}_{r}& \sim DP\left(\gamma, \sum_{j=1}^{\infty} \left( (1-\rho) \beta_j+\rho I(r=j) \right) \right), \, \rho\in [0,1], \gamma\in \mathbb{R}^+, \label{eq:distpi}\\
\beta_r &= \beta_r^* \prod_{j=1}^{r-1}(1-\beta_j^*) \\
\beta_r^* |\tau&\sim B(1,\tau), \, \tau \in \mathbb{R}^+\\
\boldsymbol{\psi}_{r}| H &\sim H , \label{eq:H}
\end{align}
where $\beta_r>0, \,r=1,2,\dots, \infty $ and $\sum_{r=1}^{\infty} \beta_{r}=1$, i.e. $\{\beta_r \}_{r=1}^{\infty}$ is a vector of probabilities, $B(\cdot,\cdot)$ is the Beta distribution, $DP(\cdot)$ indicates the Dirichlet process while $H$ is a distribution over $\boldsymbol{\psi}_r$ that acts as a prior for the model parameters.
Without loss of generality, let suppose that the first $R$ states are the non-empty ones, and let $\boldsymbol{\pi}_r^*=\left( \pi_{r1},\dots , \pi_{rR}, \sum_{j=R+1}^{\infty}\pi_{rj} \right)$. The definition of the Dirichlet process (see for example \cite{fox2011}) implies that \eqref{eq:distpi} can be written as
\begin{align}
&\boldsymbol{\pi}_r^* |\rho,\gamma,\{\beta_j \}_{j=1}^{\infty},\boldsymbol{\psi}_{r}\sim \\
&Dir\left(\gamma((1-\rho)\beta_{1}+\rho I(r=1)) ,\dots,\gamma((1-\rho)\beta_{R}+\rho I(r=R) ) , \gamma(1-\rho)\sum_{j=R+1}^{\infty}\beta_j \right) \label{eq:sw},
\end{align}
where $Dir(\cdot)$ is the Dirichlet distribution.
From \eqref{eq:sw} it is clear that, for $r,j=1,\dots,R$, $E(\pi_{rj}) = (1-\rho)\beta_r+\rho I(r=j)$, $Var(\pi_{rj}) = \frac{ ((1-\rho)\beta_{j}+\rho I(r=j)) (1- (1-\rho)\beta_{j}-\rho I(r=j)) }{\gamma+1}$. The vector $\{\beta_r \}_{r=1}^{\infty}$ and $\rho$ rules the mean value of the $\boldsymbol{\pi}_r^*$ (and then also the one of $\boldsymbol{\pi}_r$) and $\rho $ is an additional weight added to the auto-transition probability (or self-transition probability) $\pi_{rr}$. $\rho$ is needed otherwise the HMM tends to create redundant states (\cite{teh2010}).
$\gamma$ is directly proportional to the precision (the inverse of the variance) of $\pi_{rj}$. The parameters $\rho$, $\gamma$ and $\tau$ rule the number of non empty regimes, $R$, and as they increase, $R$ decreases (see \cite{fox2011}).
For a more detailed explanation of the properties and interpretation of the sHDP-HMM we refer the reader to \cite{fox2011}.
\subsection{Prior distributions and implementation details} \label{sec:prior}
To specify the prior distribution over $\boldsymbol{\psi}_r$, $H$, we follow two standard advices. The first is to have prior distributions that make the model parameters easy to update in the Markov chain Monte Carlo (MCMC) algorithm, i.e. within a Gibbs step. The second advice is to use prior distributions that allow an easy learning from the data and a robust estimation of the posterior distribution and it is usually achieved using weakly informative priors
In the simulated and real data examples (Section \ref{sec:ex}), we will use the same set of priors. Specifically $\lambda_{y,r}\sim G(a_y,b_y)I(0,c_y)$, where $G(\cdot,\cdot)I(0,\cdot)$ indicates the truncated gamma distribution expressed in terms of shape and rate that, under specific set of parameters, can resemble the uniform distribution over $[0,c_y]$. Due to the standard conjugacy between the Poisson and the gamma, the prior on $\lambda_{y,r}$ leads to a truncated gamma full conditional. We use a uniform distribution for $\nu_r$ that leads to a beta full conditional.
For the IWP parameters we follows the work of \cite{mastrantonio2015f}. They show that an efficient MCMC algorithm can be defined if we set an upper bound on the range of $\lambda_{x,r}$, i.e. $\lambda_{x,r}\in [0,\lambda_{x,max}]$, we use a truncated gamma distribution for $\lambda_{x,r}$ and we fix the maximum value that $k_t$s can assume. The value $\lambda_{x,max}$ is chosen such that the IWP evaluated with $\lambda_{x,r}=\lambda_{x,max}$ is indistinguishable from a circular discrete uniform. \cite{mastrantonio2015f} propose to find the value $\lambda_{x,max}$ by using the wrapped normal approximation of the IWP and the truncation strategy of \cite{Jona2013}. Once we have the maximum value of $\lambda$, we find the maximum value of the $k_t$s that is needed to obtain a reasonable approximation of the IWP, i.e. $k_{max}= \lceil \frac{3 \sqrt{\lambda_{max}}}{l}+\frac{\lambda_{max}}{l}-\frac{1}{2} \rceil$, where $\lceil \cdot \rceil$ is the ceiling operator. Then following \cite{mastrantonio2015f}, regardless on the prior distributions for $\eta_{r}$ and $\xi_r$, we can define and efficient Gibbs sampler. For $\eta_r$ and $\xi_r$ we use uniform distributions in their respective domains. \\
$R$ depends on $\rho$, $\gamma$ and $\tau$ and then we choose to threat them as random quantities, with non informative prior for each of them, so to be non informative on the distribution of $R$. Following \cite{fox2011}, the following priors, $\rho \in U(0,1)$, $\gamma \sim G(a_{\rho},b_{\rho})$ and $\tau \sim G(a_{\tau},b_{\tau})$, lead to full conditionals easy to simulate. The MCMC update of the regime indicator variables is done with the beam sampler (\cite{VanGael2008}).\\
With regard to the time series $\mathbf{y}$, we can have two types of missing observations; when $y_t^*=0,1$, i.e. an informative missing since we know that the value of $y_t$ is either 0 or 1, and when $y_t^*$ is missing due to the malfunction of the anemometer, i.e. a non informative missing. In both cases the missing observations are estimated during the model fitting. If $y_t$ is missing and $y_t^*=0$ then $w_t=0$ and the full conditional of $y_t$ is $Bern\left( \frac{\lambda_{y,r}e^{-\lambda_{y,r}}}{e^{-\lambda_{{y,r}}}(1+\lambda_{y,r})} \right)$ if $x_t \in \mathbb{D}$, see equation \eqref{eq:yty}, while $y_t=0$ if $x = \{\emptyset\}$. If both $y_t$ and $y_t^*$ are missing, then $w_t$ is missing as well and, if $x = \{ \emptyset \}$ we have that $y_t=0$ and $w_t=0$, otherwise if $x \in \mathbb{D}$, we can simulate from the joint full conditional of $(y_t,w_t)$ by first simulate $y_t$ from a Poisson with parameter $\lambda_{y,r}$, and then we set $w_t=1$ if $y_t \geq 2$ and 0 otherwise.
The non informative missing observation in the time series $\mathbf{x}$, due to a malfunction of the station, can be easily simulated, i.e. if $y_t \neq 0$ we simulate a value from the IWP, while if $y_t =0$ we simulate $w_t$ from a $Bern(\nu_r)$ and, if $w_t=1$ then $x_t = \{ \emptyset \}$ while if $w_t=0$ we simulate a value from the IWP.
\begin{figure}[t!]
\centering
{\subfigure[Linear densities]{\includegraphics[scale=0.4]{ExLin1}}}
{\subfigure[Circular densities]{\includegraphics[scale=0.4]{ExCirc1}}}
\caption{Simulate example: densities used to generate the example 1. The solid line is the first regime, the dashed the second and the dotted the third } \label{fig:Ex1}
\end{figure}
\begin{figure}[t!]
\centering
{\subfigure[Linear densities]{\includegraphics[scale=0.4]{ExLin2}}}
{\subfigure[Circular densities]{\includegraphics[scale=0.4]{ExCirc2}}}
\caption{Simulate example: densities used to generate the example 2. The solid line is the first regime, the dashed the second and the dotted the third } \label{fig:Ex2}
\end{figure}
\begin{figure}[t!]
\centering
{\subfigure[Linear densities]{\includegraphics[scale=0.4]{ExLin3}}}
{\subfigure[Circular densities]{\includegraphics[scale=0.4]{ExCirc3}}}
\caption{Simulate example: densities used to generate the example 3. The solid line is the first regime, the dashed the second and the dotted the third } \label{fig:Ex3}
\end{figure}
\begin{figure}[t!]
\centering
{\subfigure[Linear densities]{\includegraphics[scale=0.4]{ExLin4}}}
{\subfigure[Circular densities]{\includegraphics[scale=0.4]{ExCirc4}}}
\caption{Simulate example: densities used to generate the example 4. The solid line is the first regime, the dashed the second and the dotted the third } \label{fig:Ex4}
\end{figure}
\begin{table}[t!]
\centering
\begin{tabular}{c|ccc}
& Regime 1 & Regime 2 & Regime 3 \\ \hline \hline
$\hat{\lambda}_{y,r}$ & 1.056 & 10.059 & 30.103 \\
CI & (0.981 1.138)& (9.860 10.266)& (29.771 30.467) \\
$\hat{\lambda}_{x,r}$ & 4.887 & 1.024& 5.055\\
CI & (4.740 5.049)& (0.961 1.093)& (4.917 5.190) \\
$\hat{\nu}_r$ & 0.102 & 0.001& 0.002\\
CI & (0.078 0.124)& (0.000 0.004)& (0.000 0.005) \\ \hline
\end{tabular}
\caption{Simulated example: posterior estimates $(\hat{})$ and 95\% credible intervals (CI) for $\lambda_{y,r}$, $\lambda_{x,r}$ and $\nu_{r}$: example 1.} \label{tab:sim1}
\end{table}
\begin{table}[t!]
\centering
\begin{tabular}{c|ccc}
& Regime 1 & Regime 2 & Regime 3 \\ \hline \hline
$\hat{\lambda}_{y,r}$ & 1.010 & 5.130& 10.249 \\
CI & (0.930 1.093)& (4.954 5.307)& (9.993 10.518) \\
$\hat{\lambda}_{x,r}$ & 4.931 & 2.075& 4.941\\
CI & (4.765 5.100)& (1.978 2.173)& (4.767 5.106) \\
$\hat{\nu}_r$ & 0.094 &0.001 & 0.001\\
CI & (0.083 0 0.106)& (0.000 0.004)& (0.000 0.004) \\ \hline
\end{tabular}
\caption{Simulated example: posterior estimates $(\hat{})$ and 95\% credible intervals (CI) for $\lambda_{y,r}$, $\lambda_{x,r}$ and $\nu_{r}$: example 2.} \label{tab:sim2}
\end{table}
\begin{table}[t!]
\centering
\begin{tabular}{c|ccc}
& Regime 1 & Regime 2 & Regime 3 \\ \hline \hline
$\hat{\lambda}_{y,r}$ & 1.015 & 10.029& 30.419\\
CI & (0.940 1.089)& (9.817 10.244)& (30.075 30.776) \\
$\hat{\lambda}_{x,r}$ & 243.205 & 1.016& 5.041 \\
CI & (170.534 303.098)& (0.949 1.083)& (4.896 5.189) \\
$\hat{\nu}_r$ & 0.094 &0.001 & 0.001\\
CI & (0.083 0.105)& (0.000 0.004)& (0.000 0.004) \\ \hline
\end{tabular}
\caption{Simulated example: posterior estimates $(\hat{})$ and 95\% credible intervals (CI) for $\lambda_{y,r}$, $\lambda_{x,r}$ and $\nu_{r}$: example 3.} \label{tab:sim3}
\end{table}
\begin{table}[t!]
\centering
\begin{tabular}{c|ccc}
& Regime 1 & Regime 2 & Regime 3 \\ \hline \hline
$\hat{\lambda}_{y,r}$ & 0.954 & 4.907& 10.117\\
CI & (0.881 1.030)& (4.701 5.111)& (9.868 10.379) \\
$\hat{\lambda}_{x,r}$ & 256.186 &2.927 & 4.919 \\
CI & (183.894 301.445)& (2.811 3.050)& (4.759 5.080) \\
$\hat{\nu}_r$ & 0.094 & 0.001& 0.001\\
CI & (0.083 0.107)& (0.000 0.004)& (0.000 0.004) \\ \hline
\end{tabular}
\caption{Simulated example: posterior estimates $(\hat{})$ and 95\% credible intervals (CI) for $\lambda_{y,r}$, $\lambda_{x,r}$ and $\nu_{r}$: example 4.} \label{tab:sim4}
\end{table}
\begin{table}[t!]
\centering
\begin{tabular}{cc|ccc}
&&&Destination \\
&& Regime 1&Regime 2&Regime 3 \\
\hline
&Regime 1& 0.771 & 0.101&0.128 \\
&& (0.743 0.812)& (0.083 0.121)& (0.098 0.150)\\
Origin & Regime 2& 0.108 & 0.788 & 0.104\\
& & (0.090 0.128)& (0.763 0.813)& (0.085 0.122)\\
& Regime 3& 0.095 &0.107 & 0.798\\
& & (0.079 0.113)& (0.090 0.126)& (0.774 0.820)\\
\hline
\hline
\end{tabular}
\caption{Simulated example: posterior mean estimates and 95 \% credible intervals for the transition probability matrix: example 1.} \label{tab:sim5}
\end{table}
\begin{table}[t!]
\centering
\begin{tabular}{cc|ccc}
&&&Destination \\
&& Regime 1&Regime 2&Regime 3 \\
\hline
& Regime 1& 0.772 & 0.121& 0.107\\
& & (0.742 0.809)& (0.097 0.147)& (0.086 0.132)\\
Origin & Regime 2& 0.099 &0.790 &0.110 \\
& & (0.081 0.119)& (0.762 0.817)& (0.090 0.135)\\
& Regime 3& 0.097& 0.132 & 0.791\\
& & (0.077 0.120)& (0.094 0.164)& (0.746 0.820)\\
\hline
\hline
\end{tabular}
\caption{Simulated example: posterior mean estimates and 95 \% credible intervals for the transition probability matrix: example 2. } \label{tab:sim6}
\end{table}
\begin{table}[t!]
\centering
\begin{tabular}{cc|ccc}
&&&Destination \\
&& Regime 1&Regime 2&Regime 3 \\
\hline
& Regime 1&0.777 & 0.108& 0.114\\
& & (0.752 0.802)& (0.090 0.128)& (0.095 0.134)\\
Origin & Regime 2& 0.122& 0.774&0.104 \\
& & (0.092 0.142)& (0.749 0.800)& (0.085 0.123)\\
& Regime 3& 0.107&0.102 & 0.791\\
& & (0.090 0.126)& (0.085 0.121)& (0.766 0.813)\\
\hline
\hline
\end{tabular}
\caption{Simulated example: posterior mean estimates and 95 \% credible intervals for the transition probability matrix: example 3. } \label{tab:sim7}
\end{table}
\begin{table}[t!]
\centering
\begin{tabular}{cc|ccc}
&&&Destination \\
&& Regime 1&Regime 2&Regime 3 \\
\hline
& Regime 1& 0.786&0.112 & 0.101 \\
& & (0.760 0.812)& (0.091 0.136)& (0.082 0.123)\\
Origin & Regime 2& 0.112 &0.799 & 0.090\\
& & (0.091 0.133)& (0.770 0.826)& (0.069 0.113)\\
& Regime 3& 0.104 & 0.095&0.800 \\
& & (0.084 0.126)& (0.073 0.121)& (0.771 0.828)\\
\hline
\hline
\end{tabular}
\caption{Simulated example: posterior mean estimates and 95 \% credible intervals for the transition probability matrix: example 4. } \label{tab:sim8}
\end{table}
\section{Application} \label{sec:ex}
In this Section, the HMM is applied to simulated and real data examples.
\subsection{Simulated examples} \label{sec:sim}
We simulate 4 datasets with 3 regimes ($R=3$), from the model described in Section \ref{sec:model}.
The parameters are chosen so that in two examples (examples 1 and 3) the marginal circular distributions are slightly overlapping (Figures \ref{fig:Ex1} (b) and \ref{fig:Ex3} (b)) as well as the linear ones (Figures \ref{fig:Ex1} (a) and \ref{fig:Ex3} (a)), and in two examples (examples 2 and 4) the overlapping involves larger portions of the distributions (Figures \ref{fig:Ex2} and \ref{fig:Ex4}). In the examples 3 and 4, one of the marginal circular distribution is a discrete uniform. In all the examples we set $T=3000$ and $\mathbb{D}= \left \{ \frac{2\pi}{36}j \right\}_{j=0}^{35}$. The transition matrix has diagonal elements equal to 0.8 while the off-diagonal ones are 0.1. \\
We use the following set of parameters:
\begin{itemize}
\item Example 1:
\begin{equation}
\boldsymbol{\lambda}_y= \left[
\begin{array}{c}
1 \\10 \\30
\end{array}
\right],\,
\boldsymbol{\lambda}_x= \left[
\begin{array}{c}
5 \\1 \\5
\end{array}
\right],\,
\boldsymbol{\eta}= \left[
\begin{array}{c}
-1 \\1 \\1
\end{array}
\right],\,
\boldsymbol{\xi}= \left[
\begin{array}{c}
5\frac{2\pi}{36} \\15\frac{2\pi}{36} \\0
\end{array}
\right],\,
\boldsymbol{\nu}= \left[
\begin{array}{c}
0.1 \\0 \\0
\end{array}
\right].
\end{equation}
\item Example 2:
\begin{equation}
\boldsymbol{\lambda}_y= \left[
\begin{array}{c}
1 \\5 \\10
\end{array}
\right],\,
\boldsymbol{\lambda}_x= \left[
\begin{array}{c}
5 \\1 \\5
\end{array}
\right],\,
\boldsymbol{\eta}= \left[
\begin{array}{c}
-1 \\1 \\1
\end{array}
\right],\,
\boldsymbol{\xi}= \left[
\begin{array}{c}
10\frac{2\pi}{36} \\15\frac{2\pi}{36} \\10\frac{2\pi}{36}
\end{array}
\right],\,
\boldsymbol{\nu}= \left[
\begin{array}{c}
0.1 \\0 \\0
\end{array}
\right].
\end{equation}
\item Example 3:
\begin{equation}
\boldsymbol{\lambda}_y= \left[
\begin{array}{c}
1 \\10 \\30
\end{array}
\right],\,
\boldsymbol{\lambda}_x= \left[
\begin{array}{c}
300 \\1 \\5
\end{array}
\right],\,
\boldsymbol{\eta}= \left[
\begin{array}{c}
-1 \\1 \\1
\end{array}
\right],\,
\boldsymbol{\xi}= \left[
\begin{array}{c}
5\frac{2\pi}{36} \\15\frac{2\pi}{36} \\0
\end{array}
\right],\,
\boldsymbol{\nu}= \left[
\begin{array}{c}
0.1 \\0 \\0
\end{array}
\right].
\end{equation}
\item Example 4;
\begin{equation}
\boldsymbol{\lambda}_y= \left[
\begin{array}{c}
1 \\5 \\10
\end{array}
\right],\,
\boldsymbol{\lambda}_x= \left[
\begin{array}{c}
300 \\1 \\5
\end{array}
\right],\,
\boldsymbol{\eta}= \left[
\begin{array}{c}
-1 \\1 \\1
\end{array}
\right],\,
\boldsymbol{\xi}= \left[
\begin{array}{c}
10\frac{2\pi}{36} \\15\frac{2\pi}{36} \\10\frac{2\pi}{36}
\end{array}
\right],\,
\boldsymbol{\nu}= \left[
\begin{array}{c}
0.1 \\0 \\0
\end{array}
\right].
\end{equation}
\end{itemize}
The example 4 is particularly challenging since there is a strong overlap between the circular distributions (see Figure \ref{fig:Ex4}). \\
Here, and in the real data example of Section \ref{sec:real}, we run models estimations using a MCMCs with 100000 iterations, a burnin of 50000 and keep for inference one observation every 10 samples, i.e. for posterior estimates we use 5000 samples. Since in the simulated examples and in the real data one, we never observed a value of the linear variable greater than 50, then as prior for $\lambda_{y,r}$, the mean of the linear variable, we use $G(1, 0.00005)I(0, 50)$.
Following \cite{mastrantonio2015f}, as priors for $\lambda_{x,r}$ we use $G(1, 0.00005)I(0, 500)$. To conclude the priors specification we choose
$\gamma \sim G(1,0.1)$, $\tau \sim G(1,0.1)$, that are standard weak informative distributions, and discrete uniform for $\nu_k$ and $\xi_k$.\\
In all four examples the posterior distribution of $R$ is concentrated over 3 (regimes).
The parameters $\lambda_{y,r}$s, $\nu_{r}$s and the transition matrices
are correctly estimated\footnote{ A parameter is considered correctly estimated if the 95\% credible interval (CI) contains the value used to simulate the data.}, see Tables from \ref{tab:sim1} to \ref{tab:sim8}, and the marginal posterior distributions of the $\eta_r$s are concentrated on the ``true'' values used to simulate each dataset.\\
In the examples 1 and 3 both
$\lambda_{x,r}$ and $ \xi_r$ are always correctly estimated, except in the first regime of the third example where the posterior distribution of $\xi_1$ has non-zero probability on 22$\frac{2 \pi}{36}$, 23$\frac{2 \pi}{36}$ and 24$\frac{2 \pi}{36}$.
Note that when the density of the discrete circular variable is really close to the uniform, the parameters of the IWP becomes weakly identifiable and then we are not surprised that in the third example, the CI of $\lambda_{x,1}$ has length $\approx 133$ and the true value of $\xi_1$ is not inside the associated CI. \\
In the examples 2 and 4, the $\lambda_{x,r}$s are right estimated in the first and third regimes while in the second regime is overestimated. The posterior distributions of the $\xi_r$s are concentrated over the values used to simulate the data in the first and third regime of the example 2 and the first regime of the example 4. In the second regimes of the example 2 and 4, the posteriors of $\xi_k$ are concentrated over $14\frac{2\pi}{36}$ and $13\frac{2\pi}{36}$ respectively, while in the third regime, example 4, is concentrated over 0.\\
The model we propose is able to recover the parameters used to simulate the data in all identifiable situations.
\subsection{Real data application} \label{sec:real}
\begin{table}[t!]
\centering
\begin{tabular}{c|ccc}
& Regime 1 &Regime 2 & Regime 3 \\ \hline \hline
$\hat{\lambda}_{y,r}$ & 3.908 & 8.646 & 13.112 \\
CI & (3.794 4.017)& (8.222 9.061)& (12.633 13.590) \\
$\hat{\lambda}_{x,r}$ & 93.374 & 16.486 & 59.344\\
CI & (92.571 94.145)& (13.304 20.676 )& (52.494 60.536) \\
$\hat{\nu}_r$& 0.156 & 0.002 & 0.002 \\
CI & (0.141 0.172)& (0.000 0.009)& (0.000 0.011) \\
$\hat{\mu}_r$& 5.951 & 0.983 & 4.161 \\
CI & (5.817 6.091 )& (0.885 1.077)& (3.971 4.347) \\
$\hat{c}_r$& 0.242& 0.778 & 0.406 \\
CI & (0.239 0.245)& (0.730 0.817)& (0.399 0.450) \\
\hline
\end{tabular}
\caption{Real data example: posterior estimates $(\hat{})$ and 95\% credible intervals (CI) for $\lambda_{y,r}$, $\lambda_{x,r}$ and $\nu_{r}$.} \label{tab:eqw}
\end{table}
\begin{table}[t!]
\centering
\begin{tabular}{cc|ccc}
&&&Destination \\
&& Regime 1&Regime 2&Regime 3 \\
\hline
& Regime 1& 0.946 &0.037 & 0.017 \\
& & (0.934 0.957)& (0.028 0.047)& (0.011 0.025)\\
Origin & Regime 2& 0.127 & 0.808 & 0.064\\
& & (0.091 0.167)& (0.763 0.850)& (0.039 0.095)\\
& Regime 3 & 0.155 & 0.020 & 0.824\\
& & (0.115 0.200)& (0.005 0.043)& (0.780 0.867)\\
\hline
\hline
\end{tabular}
\caption{Real data example: Posterior mean estimates and 95 \% credible intervals for the transition probability matrix. } \label{tab:traq}
\end{table}
\begin{figure}[t!]
\centering
{\subfigure[Linear densities]{\includegraphics[scale=0.4]{Reallin}}}
{\subfigure[Circular densities]{\includegraphics[scale=0.4]{Realcirc}}}
\caption{Real data example: predictive densities. The solid line is the first regime, the dashed the second and the dotted the third } \label{fig:Ex5}
\end{figure}
After the model fitting we observed that
the posterior distribution of $R$ is concentrated over 3. The three estimated regimes are numbered in increasing order, based on the value of mean wind speeds, $\lambda_{y,r}$, that can be seen in Table \ref{tab:eqw} along with the posterior estimates of $\lambda_{x,r}$, $\nu_r$, and the circular mean and concentration. The predictive posterior densities are depicted in Figure \ref{fig:Ex5}.
With probability 1 the parameter $\eta_r$ is -1 in the first regime and 1 in the second, while in the third is equal to -1 with probability 0.989 and 1 with probability 0.011. The posterior distribution of $\xi_r$ is concentrated over $18 \frac{2 \pi}{36}$ in the first regime, in the second regime it has probability greater than 0 between the values $19 \frac{2 \pi}{36}$ and $31 \frac{2 \pi}{36}$ and its mode, $27 \frac{2 \pi}{36}$,
has probability 0.223. In the third regime the posterior distribution of $\xi_r$ assumes positive values between $26 \frac{2 \pi}{36}$ and $6 \frac{2 \pi}{36}$\footnote{$\xi$ is a discrete circular variable and then the left end of an interval can be greater than the right one} and the modal value, $26 \frac{2 \pi}{36}$, has probability 0.845.
In the first regime we have a mean wind speed of 3.908, while the mean wind direction is 5.951, that corresponds to a direction between North and North-West. The circular concentration is 0.242 and the distribution is close to the discrete uniform, see Figure \ref{fig:Ex5}. The posterior mean value of $\nu_r$, the hurdle probability, is 0.156. According to the empirical Beaufort scale (\cite{scoot2005}), the first regime represents the \emph{light breeze} state, where there are ripples without crests or small wavelets and then there is not cost erosion.
In the second regime, that can be considered as a transition state between the calm (first regime) and the storm (third) regime, the mean wind speed is 8.646 and the mean wind direction is 0.983, roughly North-East, while the circular concentration is 0.778, i.e. the second regime has a directional distribution more concentrated. The hurdle probability is 0.002, really close to zero.
In the third regime the mean wind speed is 13.112 and the mean direction is about South-West, $\hat{\mu}_{3}=4.161$. The circular concentration is 0.406 and again the mean hurdle probability is 0.002.
In this regime, the distribution of the wind speed is fully concentrated between 5 and 23 knots, resulting in an extreme wave height of almost 4 meters in open water.
It is interesting to note that, in this year, the winds with the higher speed are the ones blowing from the sea, more precisely from South West quadrant, resulting in a year with waves with more energy, intensification of erosion and changes in the longshore drift. \\
The posterior distribution of the transition probability matrix, Table \ref{tab:traq}, shows a strong self transition, i.e. the left side interval of the CIs of the self transition are always higher than 0.75.
\section{Discussion} \label{sec:disc}
Motivated by our real data example, we introduced a new HMM for discrete circular-linear variables. Our data have some peculiar features: i) the linear and circular observations are interval-censored, ii) measurements of wind speed equal to 0 and 1 are not reliable, iii) some of the missing observations of the circular variables are informative on the values of the non reliable wind speed measurements. All these features was taken into account when the regime-specific density of the HMM was specified. We introduced a new circular-linear distribution that is suited to model our data. We estimated the model in a non-parametric Bayesian framework and we have shown how specific choices of prior distributions lead to a MCMC algorithm based only on Gibbs steps. We estimated the model on 4 simulated examples and then on the real one.
Future work will find us enriching the model in at least two directions.
First we want to introduce a time-dependent transition matrix. Second, since the Poisson, due to the unit variance-to-mean ratio, can not model over-dispersed data, we are going to change the regime-specific marginal linear density to increase flexibility.
\bibliographystyle{gSCS}
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1,116,691,499,241 | arxiv |
\section{Motivation}
\label{sec:motivation}
A sterile neutrino is a well motivated and widely studied dark matter (DM) candidate.
In the Neutrino Minimal Standard Model ($\nu$MSM) \cite{Asaka:2005an,Asaka:2005pn,Asaka:2006nq}, its relic abundance is produced through its mixing with the active neutrinos via the Dodelson-Widrow mechanism \cite{Dodelson:1993je} for keV scale masses; however, this possibility has now been ruled out by a combination of X-ray and Lyman-alpha measurements \cite{Boyarsky:2006fg,Boyarsky:2006ag, Boyarsky:2005us,Boyarsky:2007ay,Boyarsky:2007ge,Seljak:2006qw, Asaka:2006nq, Boyarsky:2008xj,Horiuchi:2013noa}. The Shi-Fuller mechanism \cite{Shi:1998km} employs resonant production, but requires fine-tuned parameters and faces constraints from structure formation \cite{Horiuchi:2015qri,Schneider:2016uqi}. Thermal freeze-out with additional interactions followed by appropriate entropy dilution can also result in the correct relic abundance \cite{Bezrukov:2009th,Nemevsek:2012cd,Shuve:2014doa,Patwardhan:2015kga}, but is strongly constrained by Big Bang nucleosynthesis \cite{King:2012wg}.
An alternate production mechanism that is compatible with all constraints is the freeze-in mechanism \cite{Chung:1998rq,Hall:2009bx}, where the relic abundance is built up through a feeble coupling to some particle beyond the Standard Model (BSM) present in the early Universe. This possibility has been studied by several groups in several motivated frameworks \cite{Shaposhnikov:2006xi,Petraki:2007gq,Kusenko:2006rh,McDonald:1993ex,Yaguna:2011qn,Merle:2013wta,Adulpravitchai:2014xna,Kang:2014cia,Roland:2014vba,Roland:2015yoa,Frigerio:2014ifa,Kadota:2007mv,Abada:2014zra} (see \cite{Shakya:2015xnx} for a recent summary). While the details differ, all of these frameworks share two common salient features:
(1) a vanishing mixing between the sterile neutrino DM candidate $N_1$ and the active neutrinos, necessary to make $N_1$ stable or very long-lived and to alleviate tension with observations, and
(2) a feeble coupling between $N_1$ and a BSM particle present in the early Universe, which facilitates DM production.
It should be noted that (1) may appear unnatural at face value, but can be rendered technically natural in the limit of a $\mathcal{Z}_2$ symmetry that $N_1$ is charged under, which could be built into the details of the underlying model.
Studies of sterile neutrino DM in the literature are generally performed in non-supersymmetric setups. However, independent of dark matter considerations, there are several compelling reasons to expect the underlying theory of nature to be supersymmetric. The purpose of this paper is to study a supersymmetric extension of the sterile neutrino dark matter framework with properties (1) and (2) above, which are generic, model-independent features of the freeze-in mechanism. In this framework, $N_1$ is part of a supermultiplet that also contains a scalar, the sterile sneutrino $\tilde{N}_1$. The aforementioned $\mathcal{Z}_2$ symmetry necessarily requires $\tilde{N}_1$ to decay into $N_1$; furthermore, as we will see, this decay involves the ``feeble" coupling from (2) above, hence $\tilde{N}_1$ can potentially be long-lived. These features allow for interesting modifications of early Universe cosmology and dark matter properties.
\section{Framework}
The sterile neutrino DM freeze-in framework requires the following Lagrangian terms \cite{Shaposhnikov:2006xi,Petraki:2007gq,Kusenko:2006rh,McDonald:1993ex,Yaguna:2011qn,Merle:2013wta,Adulpravitchai:2014xna,Kang:2014cia,Roland:2014vba,Roland:2015yoa,Frigerio:2014ifa,Kadota:2007mv,Abada:2014zra,Shakya:2015xnx,Merle:2015oja} (we only list terms that will be relevant for our study):
\begin{equation}
\mathcal{L}\supset y_{ij} L_i h N_j+x_i \phi \bar{N}^c_i N_i + \lambda (H^\dagger H) \phi^2.
\label{eq:lagrangian}
\end{equation}
In addition to the three Standard Model (SM)-singlet, sterile neutrinos $N_i$, this setup also features a neutral scalar $\phi$. $x, y$ are dimensionless numbers. The aforementioned requirement of vanishing mixing for $N_1$ translates to $y_{i1}\rightarrow 0$, corresponding to a $\mathcal{Z}_2$ symmetry for $N_1$. The second term leads to freeze-in production of $N_1$ via $\phi\rightarrow N_1N_1$ decays if the coupling $x_1$ is ``feeble", $x_1^2\,\textless\,m_\phi/M_{Pl}$ \cite{Hall:2009bx}, where $M_{Pl}$ is the Planck mass. If $\phi$ obtains a vacuum expectation value, this term also gives rise to Majorana masses for the sterile neutrinos; we do not consider this possibility here. Finally, the third term accounts for the SM interactions of $\phi$ necessary for its presence in the early Universe.
In a supersymmetric theory, each of the above fields is part of a supermultiplet; we denote the supermultiplets as $\Phi$ and ${\mathcal{N}}_i$, with their spin $(0,1/2)$ components being ($\phi,\psi$) and ($\tilde{N}_i,N_i$) respectively. The Lagrangian terms in Eq.\,\ref{eq:lagrangian} can then be generated from the following superpotential:
\begin{equation}
\label{eq:newterms}
W\supset y_{ij} \mathcal{L}_i H_u \mathcal{N}_j+x_i \Phi \mathcal{N}_i\mathcal{N}_i + \sqrt{\lambda} \Phi H_u H_d\,.
\label{eq:superpotential}
\end{equation}
This superpotential further generates the following additional terms (we only list the ones that will be relevant for our study):
\begin{equation}
\mathcal{L}\supset x_i \psi N_i \tilde{N}_i + \sqrt{\lambda}\phi\tilde{H}_u\tilde{H}_d
+\sqrt{\lambda}(\psi h_d\tilde{H_u}+\psi h_u\tilde{H_d})
\label{eq:lagrangian2}
\end{equation}
In addition, the following soft terms are also generated after SUSY breaking:
\begin{equation}
\mathcal{L}_{soft}\supset y_{ij}{A_y}_{ij}\tilde{L}_i h_u \tilde{N}_j+ x_i{A_x}_i \phi\tilde{N}_1\ntilde+\sqrt{\lambda}A_\lambda \phi h_u h_d.
\label{eq:soft}
\end{equation}
Note, in particular, that the second term can give rise to the decay $\phi\rightarrow\tilde{N}_1\ntilde$.
In keeping with previous work on freeze-in of sterile neutrino dark matter \cite{Shaposhnikov:2006xi,Petraki:2007gq,Kusenko:2006rh,McDonald:1993ex,Yaguna:2011qn,Merle:2013wta,Adulpravitchai:2014xna,Kang:2014cia,Roland:2014vba,Roland:2015yoa,Frigerio:2014ifa,Kadota:2007mv,Abada:2014zra, Shakya:2015xnx}, we take $N_1$ to be light (sub-GeV scale). $N_2, N_3$ are taken to be above the GeV scale to ensure they decay before BBN and remain compatible with cosmological constraints. The heavier particles $\phi,\,\psi,$ and $ \tilde{N}_i$ are at some heavy scale $m_{\text{SUSY}}$, the scale of supersymmetry. For concreteness, we also assume R-parity and take the lightest supersymmetric particle (LSP) to be a sub-TeV Higgsino, which therefore makes up a small fraction of dark matter.
In general, several permutations of particle masses and couplings are possible. In this paper, we take $m_\phi\,\textgreater\,m_{\tilde{N}_1}$, so that $\phi$ decays into both the dark matter candidate $N_1$ and its superpartner $\tilde{N}_1$ (this is not strictly necessary for $\tilde{N}_1$ production, as $\tilde{N}_1$ also gets produced through annihilation processes in the early Universe). The $\mathcal{Z}_2$ symmetry forces $\tilde{N}_1$ to necessarily decay into $N_1$, and therefore through the $x_1\psi N_1 \tilde{N}_1$ operator (see Eq.\,\ref{eq:lagrangian2}). If $m_{\tilde{N}_1}\,\textgreater\,m_\psi$, $\tilde{N}_1$ decays as $\tilde{N}_1\rightarrow \psi N_1$. Otherwise, if $m_{\tilde{N}_1}\,\textless\,m_\psi$, the decay occurs either through an off-shell $\psi$ as $\tilde{N}_1\rightarrow N_1\tilde{H}h$, or as $\tilde{N}_1\rightarrow \tilde{H} N_1$ through $\psi-\tilde{H}$ mixing, induced by the final term in Eq.\,\ref{eq:lagrangian2} after electroweak symmetry breaking; the former dominates for $m_{\tilde{N}_1}/\langle H_u^0\rangle\textgreater 10$. We will consider both $m_{\tilde{N}_1}\,\textgreater\,m_\psi$ and $m_{\tilde{N}_1}\,\textless\,m_\psi$ in this paper. In Fig.\,\ref{fig:spectrum}, we show the mass spectrum and the decays relevant for our study in the $m_{\tilde{N}_1}\,\textgreater\,m_\psi$ scenario. Finally, $N_2,\,N_3,\,\tilde{N}_2,$ and $\tilde{N_3}$ decay via the mixings with their active neutrino or sneutrino counterparts.
\begin{figure}[t!]
\includegraphics[width=2.7in]{spectrumdecay}
\caption{\label{fig:spectrum} Particle masses and relevant decays. Supersymmetric particles are shown in green to highlight how the non-supersymmetric sterile neutrino freeze-in framework gets extended. Particles that make up dark matter are denoted by think lines.}
\end{figure}
\section{Formalism}
\label{sec:formalism}
The goal of this paper is to highlight new qualitative features arising in the supersymmetric framework. We focus on scenarios where $\phi$ is in equilibrium at high temperatures $T\,\textgreater\, m_\phi$, and its decays during this period result in the freeze-in production of $N_1$ and $\tilde{N}_1$. No significant production of $N_1$ or $\tilde{N}_1$ occurs after $\phi$ freezes out, as it decays rapidly into lighter SM or SUSY particles. For cases where $m_{\tilde{N}_1}\,\textless\,m_\psi$, we take $m_\psi\gg m_\phi$ so that $\phi$ decays remain the dominant source of $\tilde{N}_1$ and $N_1$ production. We ignore the cases where $\phi$ itself freezes in, which can also produce sterile neutrino dark matter \cite{Merle:2013wta,Adulpravitchai:2014xna,Kang:2014cia,Roland:2014vba,Merle:2015oja}, or where $\psi$ decay is the dominant production mechanism, since they do not demonstrate any qualitatively new features.
The conditions that $\phi$ maintain equilibrium with the thermal bath while $N_1$ and $\tilde{N}_1$ both freeze in from $\phi$ decays enforce the following relations between couplings and masses \cite{Hall:2009bx} (we simplify $x\equiv x_1, A_\phi\equiv {A_x}_1$):
\begin{equation}
\lambda^2\,\textgreater \frac{m_\phi}{M_{\text{Pl}}},~~~x^2\, \textless\, \frac{m_\phi}{M_{\text{Pl}}},~~~x^2\,\frac{A_\phi^2}{m_\phi^2}\, \textless\, \frac{m_\phi}{M_{\text{Pl}}}.
\label{eq:freezeincondition}
\end{equation}
Crucially, note that this feeble coupling $x\ll 1$ results in a long lifetime for $\tilde{N}_1$.
Since each $\tilde{N}_1$ decay produces an $N_1$ particle, both direct $\phi$ decays and $\tilde{N}_1$ decays contribute to the final dark matter population; these contributions are approximately \cite{Hall:2009bx}:
\begin{eqnarray}
\Omega_{N_1} h^2(\phi) &\sim& \frac{10^{24}\, x^2}{2\pi\,S_{N_{2,3}}}\frac{m_{N_1}}{m_\phi}\label{rd1}\\
\Omega_{N_1} h^2(\tilde{N}_1) &\sim& \frac{10^{24}\, x^2}{2\pi\,S_{N_{2,3}}}\frac{m_{N_1}}{m_\phi}\left(\frac{A_\phi}{m_\phi}\right)^2\label{rd2}
\end{eqnarray}
Here, $S_{N_{2,3}}$ ($\sim 1-30$ for GeV scale $N_2,N_3$ \cite{Scherrer:1984fd,Bezrukov:2009th,Asaka:2006ek}) accounts for entropy dilution from the late freeze-out and out-of-equilibrium decay of the other two sterile neutrinos $N_2,N_3$.
Since $\tilde{N}_1$ decays produce Higgsinos, we must ensure that $\tilde{N}_1$ decays before Higgsino freeze-out in order for $N_1$ to form the dominant DM component. Using the radiation-dominated time-temperature relation $H(T) = T^2/M_0$ with $M_0 = \left(\frac{45 M_{Pl}^2}{4\pi^3 g_*}\right)^{1/2}$, where $g_*$ is the number of degrees of freedom in the bath, the temperature of the SM bath when $\tilde{N}_1$ decays is approximately $T_{\text{decay}}\approx (\Gamma_{\tilde{N}_1} M_0)^{1/2}$, where $\Gamma_{\tilde{N}_1}$ is the decay width of $\tilde{N}_1$. In our calculations, we ensure that $T_{\text{decay}}$ is higher than the Higgsino freeze-out temperature $\sim m_{\tilde{H}}/20$.
Sterile neutrino dark matter can be cold, warm, or hot, as characterized by its free-streaming length $\Lambda_{FS}$, defined as the distance travelled by a dark matter particle from its production at time $t_p$ to the present time $t_0$
\begin{equation}
\Lambda_{FS} = \int_{t_p}^{t_0} \frac{\langle v(t) \rangle}{a(t)} \diff t\,.
\end{equation}
Here $v(t)$ and $a(t)$ are the DM velocity and the scale factor respectively at a given time t. As a rough guide, we take $\Lambda_{FS}\lesssim0.01$ Mpc, $0.01\lesssim \Lambda_{FS} \lesssim 0.1$ Mpc, and $0.1 ~\text{Mpc}\lesssim\Lambda_{FS}$ as corresponding to cold, warm, and hot dark matter respectively \cite{Merle:2015oja}.
If $m_{\tilde{N}_1}\gg m_{N_1}$ and $\tilde{N}_1$ decays extremely late, the population of $N_1$ produced from such decays can be relativistic and act as dark radiation. It is well known that a species that forms all of dark matter cannot account for any measurable dark radiation in the Universe \cite{Hasenkamp:2012ii,Merle:2015oja,Reece:2015lch}. However, this constraint can be circumvented in our framework since the hot $N_1$ population produced from $\tilde{N}_1$ decays does not mix with the cold $N_1$ population from $\phi$ decays. The latter population can thus be the dominant dark matter component, while a subdominant, hot population from $\tilde{N}_1$ decays forms dark radiation; we conservatively take this fraction to be $\lesssim 1\%$ (as in \cite{Hooper:2011aj}), which should leave structure formation unaffected. We note that heavy, long-lived $\tilde{N}_1$ can grow to dominate the energy density of the Universe, introducing an intermediate phase of matter domination, subsequently releasing entropy that reheats the thermal bath and dilutes the dark matter abundance. This indeed occurs in parts of our parameter space, and we correct for these effects appropriately.
Such energetic $N_1$ contribute to the effective number of relativistic degrees of freedom $\Delta N_{\rm eff}$ during Big Bang Nucleosynthesis (BBN) (which we take to be at $T_{BBN}=4$ MeV). This contribution can be estimated as
\begin{equation}
\Delta N_{\rm eff}=\left.\frac{\rho_{N_1}}{\rho_{\nu}}\right|_{T=T_{BBN}},
\end{equation}
which compares the sterile neutrino energy density with the energy density of a neutrino species in equilibrium at the same temperature. Current bounds on $\Delta N_{\rm eff}$ at BBN are at the level of $\sim 0.3$ at 1$\sigma$ \cite{Cyburt:2015mya}. With the simplifying assumption that all of the $\tilde{N}_1$ population decays at $T_{decay}$ and $N_1$ is produced with typical energy $m_{\tilde{N}_1}/2$ ($m_{\tilde{N}_1}/3$) in a two (three) body decay process, which gets redshifted by a factor ${S_{N_{2,3}}^{1/3} ({g_*}_{SM}/{g_*}_{BBN})^{1/3}}$ due to subsequent entropy dilution, $\Delta N_{\rm eff}$ can be approximated as (for the three body decay case)
\begin{eqnarray}
&&\Delta N_{\rm eff}\approx\frac{10^{-8}}{S_{N_{2,3}}^{1/3} ({g_*}_{SM}/{g_*}_{BBN})^{1/3}}\,\Omega h^2 \frac{m_{\tilde{N}_1}}{T_{decay}}\frac{\text{GeV}}{m_{N_1}}\nonumber\\
&\approx& 0.2\left(\frac{\Omega h^2}{0.0012}\right) \left(10^{-8}\frac{m_{\tilde{N}_1}}{T_{decay}}\right)\left(\frac{\text{MeV}}{m_{N_1}}\right)\left(\frac{10}{S_{N_{2,3}}}\right)^{1/3}
\label{neffeqn}
\end{eqnarray}
Here, $\Omega h^2$ represents the present relic abundance that originated from $\tilde{N}_1$ decay, as this is the only component that is relativistic at BBN.
While there are stronger constraints on $\Delta N_{\rm eff}$ from the later era of Cosmic Microwave Background (CMB) decoupling, the $N_1$ particles generally redshift and become nonrelativistic by this time \cite{Merle:2015oja}, resulting in weaker constraints, hence we only focus on $\Delta N_{\rm eff}$ during BBN. However, we do note that light (sub-eV) mass sterile neutrinos produced in this manner could contribute to $\Delta N_{\rm eff}$ at CMB decoupling, and might be relevant for alleviating the recent tension between the local and CMB-inferred measurements of the Hubble rate \cite{Bernal:2016gxb}.
\section{Results}
In this section, we investigate modifications to dark matter properties in the supersymmetric framework.
\textit{Abundance and Composition:} The $N_1$ population acts as multi-component dark matter as the fractions produced from $\phi$ and $\tilde{N}_1$ decays do not interact with each other. The two abundances differ by a factor of $(A_\phi/m_\phi)^2$ (see Eqs.\,\ref{rd1},\,\ref{rd2}). Since we expect $A_{\phi}\sim m_\phi\sim m_{\text{SUSY}}$, the two abundances are generally of comparable magnitude. For given values of $m_\phi$ and $m_{N_1}$, the desired relic abundance can be obtained by selecting appropriate values of $x$ and $A_\phi$ as long as Eq.\,\ref{eq:freezeincondition} remains satisfied. Due to the presence of an additional dark matter production mechanism in $\tilde{N}_1$ decays, the supersymmetric framework opens up more parameter space where sterile neutrino dark matter can be realized.
\begin{figure}[t!]
\includegraphics[width=3.2in]{freestreaming1}
\caption{\label{fig:fslength1} Parameter space with cold, warm, and hot dark matter (black, blue, and red regions respectively). For all points in the plot, $\Omega h^2=0.12$, $m_\phi=10^{11}$ GeV, $A_\phi/m_\phi= 10$, $S_{N_{2,3}}=10$.}
\end{figure}
\textit{Free-streaming length:} $\tilde{N}_1$ decays can produce dark matter that is cold, warm, or hot. This is illustrated in Fig.\,\ref{fig:fslength1}, where we delineate combinations of sterile neutrino and sterile sneutrino masses that give rise to cold, warm, or hot dark matter (regions where the full dark matter relic density can be achieved extend beyond the boundaries of this plot). In this plot, $m_{\tilde{N}_1}\,\textgreater\, m_\psi$, so that $\tilde{N}_1$ decays as $\tilde{N}_1\rightarrow\psi N_1$; $m_\phi=10^{11}$ GeV, so that $\phi\rightarrow\tilde{N}_1\ntilde$ is allowed at all points; $A_\phi\,=\,10\,m_\phi$, so that $\tilde{N}_1$ decays account for essentially all of dark matter; and $x$ is chosen to produce the desired relic density $\Omega h^2=0.12$. As expected, heavier $\tilde{N}_1$ or lighter $N_1$ cause dark matter particles to become more energetic, resulting in larger free streaming lengths. Note, however, that the demarcation of cold, warm, and hot regions depends not only on $m_{\tilde{N}_1}$ and $m_{N_1}$ but also on other parameters (in particular, the ones that determine the $\tilde{N}_1$ lifetime); this point is illustrated in Fig.\,\ref{fig:fslength2}, where we show that all three possibilities can be realized for the same choice of $m_{\tilde{N}_1}$ and $m_{N_1}$ (fixed to $10^6$ GeV and $1$ MeV respectively) by varying $m_\phi$ and $A_\phi$.
\begin{figure}[t!]
\includegraphics[width=3.2in]{freestreaming2}
\caption{\label{fig:fslength2} Cold, warm, hot dark matter (black, blue, and red regions respectively) for $m_{N_1}=1$ MeV and $m_{\tilde{N}_1}=10^6$ GeV. We set $S_{N_{2,3}}=10$.}
\end{figure}
\textit{Dark radiation:}
Next, we consider scenarios where extremely energetic $N_1$ from late $\tilde{N}_1$ decays contribute significantly to $\Delta N_{\rm eff}$ during BBN. Here we choose $m_{\tilde{N}_1}\,\textless\, m_\psi$ so that $\tilde{N}_1$ decays through the three-body channel $\tilde{N}_1\rightarrow N_1 \tilde{H} h$ with a long lifetime. As discussed in the previous section, this $N_1$ population can only comprise a subdominant component of dark matter, and we fix its abundance to $1\%$ of the total DM abundance by choosing $A_\phi=0.1\, m_\phi$.
In Fig.\,\ref{fig:neff} we plot $\Delta N_{\rm eff}$ at BBN as a function of $N_1$ and $\tilde{N}_1$ masses from a scan over parameter space, where we scanned over $S_{N_{2,3}}=1-30$. Red, green, blue, and black points represent $\Delta N_{\rm eff}$ in the ranges $\textgreater \,0.5,\, 0.1-0.5,\, 0.01-0.1$, and $\textless\, 0.01$ respectively; we see that large contributions to $\Delta N_{\rm eff}$ comparable to current bounds are possible while satisfying all the enforced constraints. The largest values correspond to $m_{N_1}\sim$ MeV and $m_{\tilde{N}_1}\sim 10^{9}-10^{12}$ GeV: for lighter $\tilde{N}_1$ or heavier $N_1$, the DM particles are not sufficiently relativistic at BBN, whereas heavier $\tilde{N}_1$ (which forces $\phi$ to be heavier) or lighter $N_1$ both require larger $x$ to maintain the correct dark matter abundance (see Eq.\ref{rd1}), which reduces the $\tilde{N}_1$ lifetime.
\begin{figure}[t!]
\includegraphics[width=3.65in]{ssnneff}
\caption{\label{fig:neff} $\Delta N_{\rm eff}$ (BBN) for different $N_1$ and $\tilde{N}_1$ masses. Red, green, blue, and black points denote $\Delta N_{\rm eff}$ in the ranges $\textgreater\, 0.5,\, 0.1-0.5,\, 0.01-0.1$, and $\textless\, 0.01$ respectively. For all points, the $\Delta N_{\rm eff}$ contribution comes from $\tilde{N}_1$ decays, which account for $1\%$ of the dark matter abundance, while $\phi$ decays produce the rest of dark matter.}
\end{figure}
\section{Discussion}
In this paper, we have demonstrated that a supersymmetric extension of the widely studied sterile neutrino dark matter framework with the basic features of dark matter freeze-in, namely an underlying symmetry that stabilizes the dark matter candidate and a feeble coupling that facilitates dark matter production, can
introduce several qualitatively new cosmological features and dark matter properties that are not possible in the non-supersymmetric scenario.
The presence of the superpartner, the sterile sneutrino $\tilde{N}_1$, offers an additional production mechanism for dark matter. In addition to extending the allowed parameter space for sterile neutrino dark matter, this makes possible the scenario of multiple-component dark matter with a single constituent $N_1$, as the fractions produced via different processes ($\phi$ and $\tilde{N}_1$ decays) do not mix, and effectively act as different components. Note that this possibility is unique to freeze-in production, as the two fractions would thermalize in the standard dark matter (freeze-out) histories if such production occurred before freeze-out. $\tilde{N}_1$ decays can be the dominant source of dark matter production, and dark matter produced via its decay can be cold, warm, or hot. The scenario of mixed dark matter (some combination of cold and warm components) might hold interesting implications for structure formation, offering resolution to issues such as the core vs.\ cusp problem and the ``too big to fail" problem \cite{Lovell:2011rd, BoylanKolchin:2011dk}. In scenarios where $\tilde{N}_1$ is long lived and produces $\sim 1\%$ of the dark matter population, $\mathcal{O}(0.1)$ contributions to the effective number of relativistic degrees of freedom $\Delta N_{\rm eff}$ during BBN are possible, which can be probed by near future measurements. Similar results were also discussed in \cite{Roland:2016gli}, but within a constrained framework that only offered limited possibilities. Such mixed dark matter scenarios with a single particle constituent and the associated dark matter phenomenology deserve more attention and careful study.
There are several interesting supersymmetric model building aspects that we have not fully addressed. The $\mathcal{Z}_2$ symmetry that makes the vanishing mixing of $N_1$ technically natural can be embedded into the details of the underlying theory. This $\mathcal{Z}_2$ symmetry also need not be exact, in which case $N_1$ can decay. This prospect is especially appealing given the recent claims of an X-ray line from galaxy clusters at 3.5 keV \cite{Bulbul:2014sua,Boyarsky:2014jta} compatible with decays of a 7 keV sterile neutrino; this direction would warrant further study should the signal persist. Likewise, the most interesting regions of parameter space from a phenomenological point of view require a large hierarchy between $\tilde{N}_1$ and $N_1$ masses; these could emerge naturally from symmetry considerations in the supersymmetric neutrino sector, see $\textit{e.g.}$ \cite{Roland:2014vba,Roland:2015yoa}.
Given the tremendous appeal of supersymmetry as part of the underlying theory of nature, the cosmological aspects discussed in this paper are relevant for any study on sterile neutrino dark matter. Moreover, in the absence of clear observational signals of weak scale supersymmetry or WIMP dark matter, such lines of inquiry might provide hints on the nature and scale of supersymmetry and open up promising avenues of research in the future.
\medskip
\textit{Acknowledgements: } We acknowledge helpful discussions with Samuel Roland. The authors are supported in part by the DoE under grants DE-SC0007859 and DE-SC0011719. BS also acknowledges support from the University of Cincinnati. JW wishes to acknowledge support from the Humboldt Foundation. This work was performed in part at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1066293.
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1,116,691,499,242 | arxiv | \section{Introduction}
Let $G$ be a group.
A subset $P$ of $G$ is {\it a positive cone} of $G$ if it is a subsemigroup which, together with $P^{-1}$ and the identity $\{1\}$, forms a partition of $G$. Elements from $P$ and $P^{-1}$ are called positive and negative respectively, and groups supporting positive cones are called {\em left-orderable} since from every positive cone $P$ a total and left-multiplication-invariant order $\prec$ (a {\em left-order}, for short) can be defined on $G$ by setting $g\prec h$ whenever $g^{-1}h \in P$.
Suppose that $G$ is finitely generated and endow it with a word metric. In this paper, we study the geometry of the positive cones $P$ of $G$, focusing on whether $P$ is (coarsely) connected or not\footnote{Recall that, given a metric space $(X,\dist)$, a subset $Y\subseteq X$ is coarsely connected if there is $N$ such that $\{x\in X\mid \dist(x,Y)\leq N\}$, the $N$-neighborhood of $Y$, is connected}. As we will see below, this geometric information give us algebraic and formal-language complexity information about the positive cone $P$.
Our initial observation is that, although locally the geometry of a positive cone $P$ and the ambient group $G$ might be quite different, when $G$ acts on a (Gromov) hyperbolic space, both the action of $G$ and of $P$ on the boundary look the same.
To be more concrete, let us fix some notation. Suppose that $G$ is acting by isometries on an hyperbolic space $\Gamma$.
For $H$ a subset of $G$ and $x_0\in \Gamma$, the set $Orb_H(x_0)=\{h x_0\mid h\in H\}$ is called the orbit of $x_0$ under $H$ and we denote by $\Lambda(H)$ the accumulation points of $Orb_H(x_0)$ in $\partial \Gamma$ (it is easy to see that this notion is independent of $x_0$).
The action is called {\it non-elementary} if $|\Lambda(G)|\geq 3$.
We will say that the action is {\it of general type} if it is non-elementary and not quasi-parabolic i.e. $G$ does not fix a point of $\Lambda(G)$. (The literature about isometries of Gromov hyperbolic spaces is vast. We recomend \cite{GH} for an introduction.)
\begin{thm}\label{thm: main boundary}
Let $\Gamma$ be a Gromov hyperbolic metric space and suppose that $G\curvearrowright \Gamma$ is an action by isometries of general type.
Let $P$ be a positive cone of $G$.
Then, the $P$-orbits in $\Gamma$ accumulate on every point of $\Lambda(G)$.
\end{thm}
By a theorem of Osin \cite[Theorem 1]{Osin} every non-elementary acylindrical action on a hyperbolic space by isometries is of general type. In particular the theorem applies to hyperbolic groups acting on their Cayley graph, relatively hyperbolic groups acting on their relative Cayley graph \cite[Proposition 5.2]{Osin}, or mapping class groups of surfaces of sufficiently high complexity acting on their curve Complex, to name some classical examples. It is worth noticing that Koberda \cite{koberda} already observed interactions between orderability and the boundary of an hyperbolic group, establishing that for a a residually torsion-free nilpotent hyperbolic group $G$, $\mathrm{Aut}(G)$ acts faithfully on $\partial G$.
Let us see, as an example, how to use Theorem \ref{thm: main boundary} to prove that non-abelian free groups do not have coarsely connected positive cones.
Suppose that $G=F_n$ is a free group of rank $n\geq 2$ and suppose that $\Gamma$ is the Cayley graph of $G$ with respect to a basis.
Thus $\Gamma$ is a tree whose vertices are in correspondence with the elements of $G$ and $G$ acts on $\Gamma$ by left-multiplications.
Fix a positive cone $P$ with corresponding left-order $\prec$.
Then for every $R>0$, there is $g_R\in G$ which is larger (with respect to $\prec$) than every element from $\mathbb{B}(1_G,R)$, the ball of radius $R$ in $G$. By left-invariance, every element of $g_R^{-1}\mathbb{B}(1,R)=\mathbb{B}(g_R^{-1},R)$ is smaller than $1_G$ and thus we conclude that $\mathbb{B}(g_R^{-1},R)$ is contained in $P^{-1}$.
Finally, by Theorem \ref{thm: main boundary}, every component of $\Gamma \setminus\mathbb{B}(g_R^{-1},R)$ contains at least one positive element, and so we can find two elements of $P$ such that any path connecting them must go through $\mathbb{B}(g_R^{-1},R)$. Since $R$ is arbitrary, we conclude that $P$ is not a coarsely connected subset of $\Gamma$.
It is easy to see that (up to adjusting constant) the previous argument is independent of generating sets. So we have proved
\begin{cor}\label{cor: free groups}
Positive cones of non-abelian free groups are not coarsely connected.
\end{cor}
The previous proof exemplifies how we will exploit the interaction of the geometry at infinity of $P$, with the local geometry of $G$.
We use a similar but more involved argument in Theorem \ref{thm: surface group}, where we show that fundamental groups of hyperbolic surfaces do not have coarsely connected positive cones. In this case $G$ acts properly and co-compactly on its Cayley graph, a space which is quasi-isometric to the hyperbolic plane $\mathbb{H}^{2}$.
Finding negative elements separating the boundary (we call such sets {\it negative swamps}) becomes more delicate, and we heavily relies on the ``planarity" of $\mathbb{H}^2$.
The situation for hyperbolic groups in general is much more complex, as there are examples with connected positive cones.
To see this, recall that given a finitely generated group $G$, a non-trivial homomorphism $\phi\colon G\to \mathbb{R}$ belongs to $\Sigma^1(G)$, the Bieri-Neumann-Strebel invariant (BNS invariant for short), if and only if $\phi^{-1}((0,\infty))$ is connected. Moreover, the kernel of $\phi$ is finitely generated if and only if both $\phi$ and $-\phi $ belong to $\Sigma^1(G)$. See \cite{BNS}.
In this light, it is easy to come up with examples of hyperbolic groups with non-trivial BNS invariant, for instance by considering (finitely generated free)-by-$\mathbb Z$ groups that are hyperbolic (see \cite{Brink} for a characterization) or, up to passing to finite index, any fundamental group of a closed hyperbolic 3-manifold (see \cite{Agol}).
Clearly, any (left-orderable)-by-$\mathbb Z$ group is left-orderable (since left-orderability is stable under extensions, see for instance \cite{GOD}), and we can show that all the previous examples support connected positive cones. Indeed, in Section \ref{sec: basics} we provide an easy argument showing
\begin{prop}\label{prop:BNS}
Every finitely generated left-orderable group $G$ with non-empty $\Sigma^1(G)$ admits connected positive cones.
In particular, there are left-orderable hyperbolic groups with connected positive cones.
\end{prop}
This may seem counter-intuitive in view of Theorem \ref{thm: main boundary}, yet, we will show that when positive cones of hyperbolic groups are coarsely connected, then they have to be very distorted.
More precisely, in Theorem \ref{thm combing}, we show that if a positive cone in a non-elementary hyperbolic group is coarsely connected, then for every $\lambda \geq 1$, $c,r\geq 0$ there are pairs of positive elements that {\em can not} be joined by a $(\lambda, c)$-quasi-geodesic supported on the $r$-neighourhood of the positive cone.
\begin{rem} \label{rem: tilde T}It is tempting to believe that the converse of Proposition \ref{prop:BNS} holds in general, namely that having a coarsely connected positive cone implies that the BNS invariant is not trivial. This is, however, not true in general. For instance there are perfect groups (i.e. groups with trivial abelianization) which supports only coarsely connected positive cones. A concrete example is $\widetilde{T}=\langle a,b,c\mid a^2=b^3=c^7=abc\rangle$, which is easily seen to be perfect, left-orderable group such that all its positive cones are connected. See Corollary \ref{cor: Braids are Prieto}.
\end{rem}
Coming back to applications of Theorem \ref{thm: main boundary}, we can show that many HNN extensions and amalgamated free product do not admit coarsely connected positive cones by exploiting the action on its associated Bass-Serre tree.
Curiously, to show that certain groups acting on trees do not admit coarsely connected positive cones, we will need the opposite property for edge stabilizers, namely that for (certain) edge stabilizers all its positive cones are coarsely connected.
Before stating our main result, we need to set some terminology.
\begin{defi}
Let $G$ be a finitely generated left-orderable group endowed with a word metric with respect to a finite generating set and let $H$ be a subgroup of $G$.
\begin{enumerate}
\item We say that $G$ is {\it Prieto} if all its positive cones are coarsely connected
\item We say that $G$ is {\it Hucha with respect to $H$} if for all $r>0$ and all positive cone $P$ of $G$, the $r$-neighbourhood of $P$ is not connected and there are cosets $g_1H$ and $g_2H$ that can not be connected by a path inside the $r$-neighbourhood of $P$.
\end{enumerate}
We will say that a group is Hucha if it is Hucha relative to the trivial subgroup.
\end{defi}
Prototypical examples of Prieto groups are finitely-generated torsion-free abelian groups. It is well known that if one view $\mathbb Z^n$ as the standard lattice in $\mathbb{R}^n$, positive cones essentially correspond to half-spaces defined by hyperplanes in $\mathbb{R}^n$ going through the origin (see Section \ref{sec: Prieto}). Less obvious examples are braid groups and the group $\widetilde{T}$ from Remark \ref{rem: tilde T} among others. See Corollary \ref{cor: Braids are Prieto}. On the other hand, free groups are {\it Hucha groups} with respect to all its infinite index finitely generated subgroups. See Proposition \ref{prop:F2hucha}.
As mentioned before, there is a beautiful interplay between the opposite properties (Prieto/Hucha) to construct new Hucha groups. For instance we can show
\begin{thm}\label{thm: main combination}
Suppose that $A$ has the Hucha property with respect to a subgroup $C$, and suppose that $C$ is Prieto. Then every HNN extension $A*_C t$ and every amalgamated free product of $A*_C B$, where $B$ is any finitely generated group, are Hucha (whenever they are left-orderable).
\end{thm}
Theorem \ref{thm: main combination} is really a corollary of our main technical result, Theorem \ref{thm:induction}, where in addition, we conclude that HNN extensions and amalgameted free product are Hucha with respect to a family of non-trivial subgroups. This allow us, starting from free groups, to inductively build an infinite family of Hucha groups, that we denote by $\mathfrak{H}$.
\begin{defi}\label{def: familia H}
Let $\mathfrak{H}$ be the smallest family of groups containing non-abelian finitely generated free groups, closed under taking free products and taking non-abelian finitely generated subgroups and such that for every $G\in \mathfrak{H}$ and any cyclic centralizer subgroup $C$ of $G$, the group $G*_C A$, where $A$ is finitely generated torsion-free abelian, lies in $\mathfrak{H}$.
\end{defi}
It easily follows from a theorem Howie \cite{Howie} (see Proposition \ref{prop:howie}) that every group in $\mathfrak{H}$ is locally indicable (i.e. every non-trivial finitely generated subgroups maps onto $\mathbb Z$), and hence left-orderable \cite{burnshale}. Further, there are two important families of groups closely related to our family $\mathfrak H$. One is the family of {\em Limit groups} introduced by Sela (see \cite{ChampGuir} for a nice survey). Indeed, it follows from the hierarchical characterization given by Kharlampovich and Mysniakov \cite{KM98}, that {\em non-abelian Limit groups} belong to $\mathfrak H$. The second, is the family of {\em free $\mathbb Q$-groups} introduced by G. Baumslag \cite{GBaumslag}. It turns out that free $\mathbb Q$-groups also admit a hierarchical construction starting from free groups, from which we can deduce that {\em finitely generated subgroups of free $\mathbb Q$-groups} are also contained in $\mathfrak H$. See Section 6 for details.
The main theorem of this paper is
\begin{thm}\label{thm: main}
Any group in $\mathfrak{H}$ is Hucha. In particular, non-abelian limit groups and finitely generated subgroups of free $\mathbb Q$-groups are Hucha.
\end{thm}
In light of Proposition \ref{prop:BNS}, we obtain the following immediate consequence of Theorem \ref{thm: main}. We point out that for the case of non-abelian limit groups this was already known by Kouchloukova \cite{Kouch}.
\begin{cor} The BNS-invariant $\Sigma^1$ of every group in $\mathfrak{H}$ is trivial. In particular, the BNS-invariants $\Sigma^1$ of non-abelian limit groups and finitely generated subgroups of free $\mathbb Q$-groups are trivial.
\end{cor}
Another immediate consequence of the lack of coarse connectivity of a positive cone is that it can not be finitely generated as a semi-group (indeed see Remark \ref{rem: fg pos cone are coarsely connected}). The search for groups supporting or not supporting finitely generated positive cones its being an active line of research (see \cite{GOD} and reference therein for some partial account). For instance, using dynamical techniques it was shown that free-products of groups \cite{rivas free} and fundamental groups of closed hyperbolic surfaces \cite{ABR} do not support finitely generated positive cones (in fact, this method proves something stronger: it rules out the existence of {\em isolated } left-orders). This dynamical approach, roughly, consist of perturbing actions by homeomorphisms of the line of the generators of the group while preserving the group relations, a task that became impractical when dealing with complicated presentations, which is the case for arbitrary limit groups.
In contrast, our method immediately yields
\begin{cor}
No positive cone of a group in $\mathfrak{H}$ is finitely generated as a semi-group. In particular, no positive cone in a non-abelian limit group or in a finitely generated subgroup of free $\mathbb Q$-group is finitely generated as a semi-group.
\end{cor}
In fact, something stronger holds for Hucha groups: they do not admit positive cones that can be described by a regular language over the generators (regular positive cones for short). See Corollary \ref{cor: hucha not regular}. Recall that, roughly, a subset of a finitely generated group is regular, if it can be described by the set of paths in a finite labelled graph (see Section \ref{sec: regular} for a precise definition).
Certainly, finitely generated positive cones are regular, but being regular is a much more stable property: for instance regularity passes to finite index subgroups \cite{Su} while finite generation does not\footnote{An easy example is given by $\mathbb Z^2$, which has no finitely generated positive cones but it is a finite index subgroup of $K=\langle a,b\mid aba^{-1}=b^{-1}\rangle$, which contains a positive cone generated as a semi-group by $a$ and $b$.}.
It is easy to show that finitely presented groups with a regular positive cone have solvable word problem.
A more refined criterium for not admitting regular positive cones was obtained by Hermiller and \u Suni\'c in \cite{HS} where they show that no positive cone in a free product of groups is regular. In fact, finding a geometrical interpretation of their criterium was one of our initial motivations to pursue the present work.
The paper is organized as follows: we start in Section \ref{sec: basics} proving some basic features of the geometry of positive cones. Among other things, we show that a positive cone naturally defines a special geodesic on the group with the property that {\em it goes deep} into the negative cone (see Proposition \ref{prop tool}).
In Section \ref{sec: hyperbolic} we recall the basics of hyperbolic geometry to show Theorem \ref{thm: main boundary}.
In this section we also show that the Hucha property holds for surface groups (Theorem \ref{thm: surface group}) and that there is no quasi-geodesic combings for positive cones of hyperbolic groups (Theorem \ref{thm combing}).
We remark that results of Section \ref{sec: hyperbolic} are not needed elsewhere in the paper so readers interested on Theorem \ref{thm: main} and its applications might want to skip the section.
In Section \ref{sec: prieto and hucha} we introduce the Prieto and Hucha properties, and show basic stability results such as the independence of the generating set that these properties passes to finite index overgroups.
In Section \ref{sec: trees}, we will study left-orderable groups acting on trees and see how the geometry of the tree dominates the geometry of the group to show our combination theorems (Theorem \ref{thm:induction} and Theorem \ref{thm: main combination}).
In Section \ref{sec: limit}, we will review the needed facts of limit groups and $\mathbb{Q}$-groups and show Theorem \ref{thm: main}.
Finally, in Section \ref{sec: regular}, we recall the definition of regular languages, regular positive cones, and observe that Hucha groups do not admit regular positive cones.
\vspace{0.3cm}
\noindent{\textbf{{Acknowledgments}}} We are grateful to R. Potrie, S. Hermiller and Z. \u Suni\'c for their interest in this work, and for many fruitful conversations regarding earlier drafts of this paper. Y.A. acknowledges partial support from the Spanish Government through grants number MTM2014-54896 and MTM2017-82690-P, and through the ''Severo Ochoa Programme for Centres of Excellence in R\&{}D'' (SEV-2015-0554),
and through European Union's Horizon 2020 research and innovation programme under the Marie Sk\l{}odowska-Curie grant agreement No 777822.
C.R. acknowledges partial
support from FONDECYT 1181548.
J.B. acknowledges support from FONDECYT Postdoc 3190719
\section{Notation and basics facts}
\label{sec: basics}
Let $\Gamma=(V\Gamma, E\Gamma)$ be a graph.
A {\it path or $1$-sequence} in $\Gamma$, is a function $\alpha\colon \{0,1,\dots, k\}\to V\Gamma$ where
$\alpha(i)=\alpha(i+1)$ or there is an edge connecting $\alpha(i)$ and $\alpha(i+1)$.
We also use the notation $\{\alpha(i)\}_{i=0}^k$ to denote $\alpha$.
We now can define the combinatorial metric on $V\Gamma$, setting $\dist_\Gamma(u,v)$ to be the minimum $k$ such that there is a path $\{v_i\}_{i=0}^k$ with $v_0=u$ and $v_k=v$.
If no such path exists we set $\dist_\Gamma(u,v)=\infty$.
By $\mathbb{B}_\Gamma(v,r)$ we denote the set $\{u\in V\Gamma \mid \dist_\Gamma(u,v )\leq r\}$, the closed ball of radius $r$.
Let $r\geq 1$.
An {\it $r$-path or $r$-sequence} $\{v_n\}_{n=1}^k$ is a sequence in $\Gamma$ such that $\dist_\Gamma(v_i,v_{i+1})\leq r$.
Let $\lambda \geq 1, c\geq 0$.
An $r$-path is a {\it $(\lambda,c)$-quasi-geodesic} if
$$\dfrac{|i-j|}{\lambda}-c \leq \dist_\Gamma(v_i, v_j) \quad \forall\, 1\leq i,j\leq k.$$
A {\it geodesic} is a $1$-path that is a $(1,0)$-quasi-geodesic.
For infinite $r$-paths we will use the term $r$-rays, $r$-quasi-geodesics or geodesic rays.
A subset $S$ of $V\Gamma$ is {\it coarsely connected} if there is some $r\geq 1$ such that for all $u,v$ in $S$ there is an $r$-sequence supported at $S$ starting at $u$ and ending at $v$.
Equivalently, $S$ is coarsely connected if there is $r\geq 1$ such $\{v\in V\Gamma | \dist(v,S)\leq r\}$ spans a connected subgraph.
Let $G$ be a group and $X$ a generating set of $G$.
Generating sets, unless otherwise stated, will assumed to be symmetric, that is $X=X^{-1}$.
We denote by $\Gamma(G,X)$ the {\it Cayley graph} of $G$ with respect to $X$.
By identifying the group $G$ with the vertices of $\Gamma(G,X)$, $G$ is endowed with a metric $\mathrm{d}_X=\dist_{\Gamma(G,X)}$.
We also write $|g|_X$ for $\mathrm{d}_X(1_G, g)$.
By $\mathbb{B}_X(g,r)$ we denote the ball in $\Gamma(G,X)$ with center $g$ and radius $r$.
We will drop the subscripts $_X$ and $_\Gamma$ when the meaning is clear from the context.
\subsection{General facts about the geometry of positive cones}
Recall that $P\subseteq G$ is a {\it positive cone} of $G$, if $P$ is a sub-semigroup and $G=P\sqcup P^{-1}\sqcup \{1\}$ where $P^{-1}=\{g^{-1}:g\in P\}$.
Moreover, $a\prec b \Leftrightarrow a^{-1}b\in P$ defines a $G$-left-invariant total order on $G$.
Conversely, if $\prec$ is a $G$-left-invariant total order on $G$, then $P=\{g\in G \mid 1_G\prec g\}$ is a positive cone.
Groups with positive cones must be torsion-free.
Given $S\subseteq G$, we denote by $\langle S \rangle^+$ the sub-semigroup generated by $S$.
A positive cone is {\it finitely generated} if there is a finite set $S$ of $G$ such that $P=\langle S \rangle^+$.
\begin{rem}\label{rem: fg pos cone are coarsely connected}
If $P$ is a finitely generated positive cone of a group $G$, then it is coarsely connected subset of the Cayley graph of $G$.
Indeed, let $P=\langle S \rangle^+$ with $S$ finite and $G=\langle X \rangle$.
Let $r=\max_{s\in S} |s|_X$, then there are $r$-paths in $P$ from $1_G$ to $g\in P$ for all $g\in P$.
\end{rem}
We show now that positive cones define preferred ways to go to infinity and that positive cones contain arbitrarily large balls. This will be essential for showing that fundamental groups of hyperbolic surfaces do not have coarsely connected positive cones.
\begin{prop}\label{prop tool}
Let $G$ be generated by a finite symmetric set $X$.
Let $P$ be a positive cone of $G$ with associated order $\prec.$
Let $g_n=\max_{\prec }\mathbb{B}_X(1_G,n)$. The following holds:
\begin{enumerate}
\item The map $\{g_n^{-1}\}_{n=0}^\infty$ is a geodesic ray in $\Gamma(G,X)$. In particular, $|g_n|_X=n$.
\item $\mathbb{B}_X(g_n^{-1},n-1) \subseteq P^{-1}$.
\end{enumerate}
In particular $P^{-1}$ contains $\bigcup_n \mathbb{B}_X(g_n^{-1},n-1)$.
\end{prop}
\begin{proof}
To show 1, for $0< i\leq n$ let $g_i=\max_{\prec} \mathbb{B}(1_G,i)$ and assume, by induction on $n$, that $|g_i|_X=i$ and $g_i=x_i g_{i-1}$ with $x_i\in X$.
The base of induction, $n=1$, $g_1=\max_{\prec } X$ and $g_{0}=\{1_G\}$, clearly holds.
Let $g_{n+1}=\max_{\prec} \mathbb{B}(1_G,n+1)$.
By definition $|g_{n+1}|_X\leq n+1$.
So we need to show that $g_{n+1}\notin \mathbb{B}(1_G,n)$ and that there is some $x\in X$ such that $g_{n+1}=xg_n$.
By left-invariance, $xh\preceq xg_{n}$ for all $x\in X$ and all $h\in \mathbb{B}(1_G, n)$.
Thus
$$\mathbb{B}(1_G, n+1)= \bigcup_{x\in X}x\mathbb{B}(1_G, n)\preceq\max_{\prec}\{xg_n \mid x\in X\}.$$
Thus $g_{n+1}\in \{xg_n\mid x\in X\}$.
It remains to show that $g_{n+1}\notin \mathbb{B}(1_G,n)$.
For that, by totality of the order, for each $x\in X$, either $xg_n\prec g_n$ or $g_n\prec xg_n$,
which, by left-invariance it is equivalent to $g_n\prec x g_n$ or $g_n \prec x^{-1} g_n$.
Thus $\mathbb{B}(1_G, n)\preceq g_n \prec \max_{\prec}\{xg_n \mid x\in X\}$.
This completes the proof of the induction.
It follows that the map $n\mapsto g_n^{-1}$ is a geodesic ray, that is $\dist(g_n^{-1},g_m^{-1})=|m-n|$.
To show 2, we let $b\in \mathbb{B}(1_G, n-1)$.
By definition we have that $g_n\succ b$ and so by left-invariance we get that $1_G\succ g_n^{-1}b$.
\end{proof}
\begin{cor}\label{cor: tool}
Let $X=X^{-1}$ be a finite generating set of $G$ and $P$ a positive cone of $G$.
Every connected component of $P$ in $\Gamma(G,X)$ is infinite and contains a geodesic ray.
\end{cor}
\begin{proof}
Applying Proposition \ref{prop tool} to $P^{-1}$, we get a geodesic ray $\{g_n\}_{i=0}^{\infty}$ starting at $1_G$ and contained (except from $g_0$) in $P$.
Let $g\in P$.
By left-invariance of the action of $G$ on $\Gamma(G,X)$, $\{gg_n\}_{i=0}^\infty$ is a geodesic ray.
Since $P$ is a sub-semigroup $\{gg_n\}_{i=0}^\infty \subseteq P$.
\end{proof}
We finish this section by showing that left-orderable groups with non-trivial BNS invariant enjoys connected positive cones.
\begin{proof}[Proof of Proposition \ref{prop:BNS}]
Let $G$ be a left-orderable finitely generated by $X=X^{-1}$, and assume that $\Sigma^1(G)$ is non-empty.
Then, there is a non-trivial homomorphism $\phi \colon G \to \mathbb{R}$ such that $\phi^{-1}((0,\infty))$ is connected.
Let $x_0\in X$ such that $\phi(x_0)>0$.
It is know that $\{g\in G\mid \phi(x_0)<\phi(g)\}$ is also connected \cite{BNS}.
Now let $K=ker(\phi)$ and $\preceq_K$ be any left-ordering on $K$.
Using $\phi$, we can lexicographically extend $\preceq_K$ to produce a left-ordering of $G$.
Precisely we set $1\prec_G g$ if and only if $\phi(g)>0$ or $\phi(g)=0$ and $1 \prec_K g$ (checking that this is a left-order is left to the reader).
Now, if $g_1$ and $g_2$ are positive in $\preceq_G$ then $g_1x_0$ and $g_2x_0$ both belong to $\{g\in G\mid \phi(x_0)<\phi(g)\}$, which we already pointed out that is connected.
In particular, there is a path made of positive elements connecting $g_1$ and $g_2$.
\end{proof}
\section{Positive cones acting on hyperbolic spaces}
\label{sec: hyperbolic}
In this section we prove Theorem \ref{thm: main boundary}. As stated, the proof works in general for not necesarily proper (i.e. locally compact) metric spaces. Yet, in this section it will only be applied to hyperbolic groups (i.e. proper spaces), thus the reader only interested in these applications can assume that the spaces involved are proper.
We follow \cite{Hamann}.
Let $\Gamma$ be a graph.
The Gromov product of $a$ and $b$ in $V\Gamma$ (with respect to $v$) is defined by
$$(a|b)_v=\dist(a,v)+\dist(b,v)-\dist(a,b).$$
Let $\delta \geq 0$. The graph $\Gamma$ is $\delta$-hyperbolic if for a given base-point $v\in V\Gamma$ we have
$$ (a|b)_v\geq \min \{(a|c)_v, (b|c)_v\}-\delta$$
for all $a,b,c\in V\Gamma$.
With this definition of hyperbolicity, every geodesic triangle in $\Gamma$ is $(4\delta)$-thin.
Moreover, if $\alpha$ is a geodesic from $v$ to $a$, and $\beta$ is a geodesic from $v$ to $b$ then
\begin{equation}\label{eq: gromov product measures cancellation}
\dist(\alpha(t),\beta(t))\leq 2\delta \qquad \forall t\in [0, (a|b)_v].
\end{equation}
Finally, if $\Gamma$ is $\delta$-hyperbolic with respect to some base-point $v$, then it is $\delta(u)$-hyperbolic for any other base-point $u$.
Let $\epsilon > 0$ with $\epsilon' = \exp(\epsilon \cdot \delta) -1 < \sqrt{2} - 1$.
For $a,b\in V\Gamma$, define $\rho_\epsilon (a,b)= \exp (\epsilon \cdot (a|b)_v)$ if $a\neq b$, and $\rho_\epsilon(a,a)=0$. The function $\rho_\epsilon$ is a pseudo-metric. However we can construct a metric on $V\Gamma$ by setting
$$\dist_\epsilon(a,b)= \inf_{n}\{ \sum_{i=1}^n \rho_\epsilon (z_i, z_{i+1}) \mid z_1=a, z_{n+1}=b, z_i\in V\Gamma \} $$
and this metric satisfies that $(1-2\epsilon')\rho_\epsilon (a,b) \leq \dist_\epsilon (a,b) \leq \rho_\epsilon(a,b)$.
Let $\hat{\Gamma}$ be the metric completion of $V\Gamma$ with respect to $\dist_\epsilon$. Then, one can define the Gromov boundary $\partial \Gamma$ of $\Gamma$ as $\hat \Gamma \setminus V\Gamma$.
The topology of the Gromov boundary is independent of the base point $v$ and the parameter $\epsilon$.
Let $G$ be a group acting by isometries on $\Gamma$.
This action naturally extends to a continuous action on $\hat{\Gamma}$ and hence on $\partial \Gamma$.
Given a subset $H$ of $G$ and a vertex $v\in V\Gamma$, we denote by $\Lambda(H)$ the intersection of the closure of the orbit $Hv$ in $\hat{\Gamma}$ (with respect to the metric $\dist_\epsilon$) with $\partial \Gamma$.
The set $\Lambda(H)$ is independent of $v$.
The following theorem classifies the actions of $G$ on $\Gamma$ in terms of the dynamics of $G$ on $\Lambda(G)$.
\begin{thm}\cite[Theorem 2.7]{Hamann}\label{thm: classification actions}
Suppose that $G$ acts by isometries on an hyperbolic geodesic space $\Gamma$.
Then one of the following holds.
\begin{enumerate}
\item[(i)] {\it (elliptic action)} $\Lambda(G)$ is empty.
\item[(ii)] {\it (parabolic action)} $G$ fixes a unique point of $\Lambda(G)$.
\item[(iii)] {\it (dihedral action)} $\Lambda(G)$ has two points.
\item[(iv)] There are two elements $g,h\in G$ such that $|\Lambda( \langle g \rangle )\cup \Lambda(\langle h\rangle) |=4$.
\end{enumerate}
\end{thm}
The action is called {\it non-elementary} if $|\Lambda(G)|> 2$ (in fact this is equivalent to $|\Lambda(G)|=\infty$).
We will say that a non-elementary action is {\it of general type} if it lies on case (iv) of the previous theorem.
By a theorem of Osin \cite[Theorem 1]{Osin} every non-elementary acylindrical action on a hyperbolic space by isometries is of general type.
Recall that the action of $G$ on $\Gamma$ is {\it acylindrical} if for every $r\geq 0$, there are $N$ and $R$ such that for any pair $a,b\in V\Gamma$ with $\dist(a,b)\geq R$, the set $\{g\in G \mid \dist(a,ga)\leq r \text{ and } \dist(b, gb)\leq r\}$ has at most $N$ elements.
This implies, although it is much easier to prove, that
\begin{rem}\label{rem: hyperbolic groups are of general type}
non-elementary proper co-compact actions are of general type. See also \cite[Chapitre 8, Th\'eor\`eme 30]{GH} for a direct proof.
\end{rem}
An element $g\in G$ is called {\it loxodromic} if $|\Lambda(\langle g\rangle)|=2$.
Equivalently, an element $g\in G$ is loxodromic if $n\mapsto g^n v$, $n\in \mathbb Z$, defines a quasi-geodesic. The {\it loxodromic limit set} of $G$, denote $\cL(G)$, is the set $\Lambda( \{g \in G \mid g\;\, \text{loxodromic} \})$.
The loxodromic limit set is {\it bilaterally dense in $\Lambda(G)$} if for every pair of disjoint non-empty open subsets $A$, $B$ of $\Lambda(G)$, there is a loxodromic element $g\in G$ such that $\Lambda(\langle g \rangle )$ has non-trivial intersection with both $A$ and $B$, in other words, the quasi-geodesic $g^nv$ goes from $A$ to $B$.
Theorem \ref{thm: main boundary} follows from the following.
\begin{thm}\cite[Theorem 2.9.]{Hamann}
Suppose that $G$ acts non-elementary by isometries on an hyperbolic graph $\Gamma$. Then
\begin{enumerate}
\item $\Lambda(G)$ is closed without isolated points (i.e is a perfect set).
\item the hyperbolic limit set is bilateraly dense in $\Lambda(G)$ if and only if the action is of general type.
\end{enumerate}
\end{thm}
\begin{proof}[Proof of Theorem \ref{thm: main boundary}]
Let $\eta \in \Lambda(G)$ and $A$ an open neighbourhood of $\eta$ in $\Lambda(G)$.
Since the action is non-elementary, $\Lambda(G)$ is perfect and thus $\eta$ is not isolated and we can assume that there is $\nu\neq \eta$ in $A$.
Since the topology on $\partial \Gamma$ (and $\Lambda(G)$) is Hausdorff, there are disjoint open neighborhoods $U$ and $V$ of $\eta$ and $\nu$ respectively lying in $A$.
Since the action of $G$ is of general type, the hyperbolic limit set is biliaterally dense in $\Lambda(G)$ and thus, there is a loxodromic element $g$ such that $\Lambda(\langle g \rangle)$ meets $U$ and $V$.
Now, if $P$ is positive cone of $G$, we have that either $g$ or $g^{-1}$ lies in $P$.
And thus either $\Lambda(\langle g \rangle^{+})$ or $\Lambda(\langle g^{-1} \rangle^{+})$ lies in $\Lambda(P)$ and thus $\Lambda(P)\cap A \neq \emptyset$.
\end{proof}
\subsection{Positive cones in surface groups}
\label{subsec surface}
As promised, we show how the ideas of the proof of Corollary \ref{cor: free groups} can be adapted to the case of surface groups. The proof uses strongly the topology of $\mathbb{H}^2$. Specifically, that $\mathbb{H}^2$ can be cut into two infinite connected components using a curve.
\begin{thm} \label{thm: surface group}
Suppose that $G$ is the fundamental group of a closed hyperbolic surface. Then, no positive cone in $G$ is coarsely connected.
\end{thm}
\begin{proof}
Assume that $X$ is the standard generating set of $G$, so that the Cayley graph, not only is quasi-isometric to $\mathbb{H}^2$, but also it is embedded as a 1-dimensional topological subspace.
In particular, any path in $G$ connecting two different points of $\partial G$ separates the Cayley graph into two infinite connected components.
We fix a positive cone $P$ and the corresponding left-order $\prec.$
We also fix $N_0>1$.
By Theorem \ref{thm: main boundary}, $P$ accumulates on every point of $\partial G$, so to show that $P$ is not $N_0$-connected, it is enough to build a bi-infinite path $\gamma:\mathbb Z\to G$ such that
\begin{enumerate}
\item[$i)$] the $N_0$-neighbourhood of $\gamma$ is contained in $P^{-1}$ i.e. $\mathbb{B}_X(\gamma(i), N_0)\subseteq P^{-1}\, \forall i\in \mathbb{Z}$,
\item[$ii)$] $\gamma(i)$ converges, as $i$ tends to $\pm\infty$, to two different boundary points of $\partial G$.
\end{enumerate}
We let $g_n=\max_{\prec}\mathbb{B}_X(1_G, n)$.
By Proposition \ref{prop tool}, $n\mapsto g_n^{-1}$ is a geodesic ray such that $\mathbb{B}(g_n^{-1},n)\subseteq P^{-1}$.
We let $\xi=\lim g_n^{-1}\in \Lambda(G)$.
Given any $g\in G$ there exist some generator $x\in X^{\pm 1}$ such that $gx$ does not belong to the stabilizer of $\xi$. Indeed, otherwise $\langle gX^{\pm 1} \rangle\supseteq \langle \{x_1x_2\mid x_1,x_2\in X^{\pm 1}\} \rangle$ is a finite index subgroup of $G$ that stabilizes $\xi$, contradicting that the action of $G$ on itself is of general type (Remark \ref{rem: hyperbolic groups are of general type}).
Take $h\in g_{2N_0+1}^{-1}X^{\pm 1}\subseteq P^{-1}$ not belonging to the stabilizer of $\xi$ and define $\gamma:\mathbb Z\mapsto G$ as:
$$\gamma(n)=g_{2N_0+1+n}^{-1}\text{ if }n\geq 0;\qquad\qquad \gamma(n)=hg_{-(n+1)}^{-1}\text{ if }n<0.$$
We claim that $\gamma$ is the desired path.
Notice that $\lim_{n\to+\infty}\gamma(n)=\xi$, $\lim_{n\to-\infty}\gamma(n)=h\xi$ and $h\xi$ differs from $\xi$ by construction of $h$.
Thus $ii)$ holds.
To see condition $i)$, take $n\in\mathbb Z$.
If $n\geq 0$ the condition follows from the definition of $g_n$ and Proposition \ref{prop tool}.
If $n\in[-N_0,-1]$ we get that $\gamma(n)\in h\mathbb{B}(1_G, N_0)\subseteq \mathbb{B}(g_{2N_0+1}^{-1},N_0+1)$ and therefore $\gamma(n)\mathbb{B}(1_G,N_0)\subseteq \mathbb{B}(g_{2N_0+1}^{-1},2N_0+1)\subseteq P^{-1}$.
Finally, if $n<-N_0$ we get $\gamma(n)=hg_{m}^{-1}$ for some $m\geq N_0$ and therefore $\gamma(n)\mathbb{B}(1_G,N_0)=hg_{m}^{-1}\mathbb{B}(1_G,N_0)\subseteq P^{-1}$ since $h\in P^{-1}$ and $g_{m}^{-1}\mathbb{B}(1_G,N_0)\subseteq g_{m}^{-1}\mathbb{B}(1_G,m)\subseteq P^{-1}$.
\end{proof}
\subsection{Quasi-geodesic combings of positive cones}
\label{subsec combing}
Let $G$ be a non-elementary hyperbolic group and $X$ a finite generating set.
Let $\Gamma=\Gamma(G,X)$ be the Cayley graph.
A {\it $(\lambda, c)$-quasi-geodesic combing} of $P$ is a subset $\cP$ of $(\lambda,c)$-quasi-geodesics paths in $\Gamma$ starting at $1_G$ and ending at vertices of $P$, and such that for each $g\in P$, there is at least a path in $\cP$ from $1_G$ to $g$.
We say that $\cP$ is supported on the $r$-neighbourhood of $P$ if every $p\in \cP$ lies in $\cup_{g\in P} \mathbb{B}_X(g,r)$.
\begin{thm}\label{thm combing}
Let $G$ be a finitely generated and non-elementary hyperbolic group and $P$ a positive cone of $G$. Then, there is no quasi-geodesic combing of $P$ supported on a neighbourhood of $P$.
\end{thm}
\begin{proof}
Let $X$ be a finitely generating set of $G$ and $\Gamma=\Gamma(G,X)$ its associated Cayley graph.
Let $P$ be a positive cone of $G$ and suppose that there exists $\lambda \geq 1$, $c\geq 0$ and $r\geq 0$ and a $(\lambda,c)$-quasi-geodesic combing $\cP$ of $P$ supported on the $r$-neighbourhood of $P$.
Let $\prec$ denote the associated order of $P$.
Let $g_n=\max_\prec\mathbb{B}(1_G,n)$.
By Proposition \ref{prop tool}, $n\mapsto g_n^{-1}$ is a geodesic ray.
We denote by $\xi\in \partial G$ its limit as $n\to \infty$.
Also by Proposition \ref{prop tool}, the horoball $\cup_n \mathbb{B}(g_n^{-1},n-1)$ is contained in $P^{-1}$.
By Theorem \ref{thm: main boundary}, we know that there is a sequence $h_n\in P$ such that $h_n$ converges to $\xi$, {\em i.e.} $(h_n|\xi)_{1_G}\to \infty$.
Thus, for every $N>0$, we can find $n(N),m(N)$ such that $(h_{n(N)}|g^{-1}_{m(N)})_{1}>N$.
Let $\alpha^{N}$ and $\beta^N$ be geodesic paths from $1_G$ to $h_{n(N)}$ and $g_{m(N)}^{-1}$ respectively.
By Proposition \ref{prop tool}, we can assume that $\beta^N(t)=g_t^{-1}$.
By equation \eqref{eq: gromov product measures cancellation} in the beginning of Section \ref{sec: hyperbolic}, $\dist(\alpha^N(t),\beta^N(t))\leq 2\delta$ for all $t\in [0,N]$.
Denote by $p_n$ a path in $\cP$ from $1_G$ to $h_n$.
By the stability of quasi-geodesics, there is constant $D$ (only depending on $\delta$, $\lambda$ and $C$) such that $p_n$ and $\alpha_n$ are at Hausdorff distance at most $D$.
Suppose that $N> D+r+2\delta$.
Since $p_n$ is contained in the $r$-neighbourhood of $P$, there is $z\in P$ such that $\dist(z,\alpha^N(N))\leq D+r$.
Since $\dist(\alpha^N(N), g_N^{-1})\leq 2\delta$, we have that $z\in \mathbb{B}(g_N^{-1}, D+r+2\delta)\subseteq P^{-1}$, which is a contradiction.
\end{proof}
\section{Connectedness properties for positive cones}
\label{sec: prieto and hucha}
In this section we introduce two properties that essentially say that all positive cones have the same connectedness behaviour: either all positive cones are coarsely connected (Prieto property) or all positive cones are not coarsely connected (Hucha property).
\subsection{The Hucha property}
Let $\Gamma$ be a graph.
A subset $S$ of $V\Gamma$ {\it $r$-disconnects} subsets $H_1$ and $H_2$ of $\Gamma$ if any $r$-path from $H_1$ to $H_2$ has non-trivial intersection with $S$. We say that $S$ $r$-disconnects a subset $P$, if $P$ contains a subset $\{u,v\}$ so that $S$ $r$-disconnects $\{u\}$ and $\{v\}$.
\begin{defi}\label{defi:swamp}
Let $P$ be a positive cone of a group $G$, $H\leqslant G$ a subgroup and $\Gamma(G,X)$ a Cayley graph. A subset $S \subseteq P^{-1}$ is a {\it negative swamp of width $r$ for $H$} in $\Gamma(G,X)$ if:
\begin{enumerate}
\item $S$ $r$-disconnects $P$,
\item there are $g_1,g_2\in G$ such that $S$ $r$-disconnects $g_1H$ and $g_2H$.
\end{enumerate}
\end{defi}
\begin{rem}\label{rem: flexibility swamp}
If $S\subseteq P^{-1}$ is a negative swamp of with $r$ for $H$, then the same holds for any $S'$ with $S \subseteq S'\subseteq P^{-1}$. That is, $S'$ is also a negative swamp of width $r$ for $H$.
If $S$ is a negative swamp of with $r$ for $H$, then the same holds for any subgroup $K$ of $H$. That is $S$ is also a negative swamp of width $r$ for $K$.
\end{rem}
\begin{defi}\label{def: Hucha}
Let $G$ be a left-orderable group, $X$ a finite generating set and $\mathcal{H}$ a family of subgroups.
We say that the Cayley graph $\Gamma(G,X)$ is {\it Hucha with respect to $\mathcal{H}$} if for every positive cone $P$ of $G$, for every $H\in \mathcal{H}$ and for every $r>0$, there is is a negative swamp $S$ of width $r$ for $H$.
If $\mathcal{H}$ is the trivial subgroup we just say that $\Gamma(G,X)$ is Hucha.
\end{defi}
\begin{rem}\label{rem: family closed}
By Remark \ref{rem: flexibility swamp}, if $\Gamma(G,X)$ is Hucha with respect to $\cH$, then it is also Hucha with respect to $\{K\leqslant H \mid H\in \cH\}$. Thus one might assume that the family $\mathcal{H}$ is closed under taking subgroups.
\end{rem}
By Remark \ref{rem: family closed}, if $\Gamma(G,X)$ is Hucha with respect to $\mathcal H$ then $\Gamma(G,X)$ is Hucha.
The following is worth noticing.
\begin{prop}\label{prop: Hucha are not coarsely connected}
If $\Gamma(G,X)$ is Hucha, then no positive cone $P$ of $G$ is coarsely connected.
\end{prop}
\begin{proof}
Let $P$ be a positive cone and $r$ a positive integer. Since $\Gamma(G,X)$ is Hucha, there is a negative swamp $S\subseteq P^{-1}$ that $r$-disconnects $P$. In particular, there are $g_1,g_2\in P$ such that every $r$-path connecting them must go through $S$ and thus it can not be supported in $P$. Since $r$ is arbitrary we deduce that $P$ is not coarsely connected.
\end{proof}
The following is easy from the definitions.
\begin{lem}\label{lem: Hucha is geometric}
Being Hucha with respect to $\mathcal H$ is independent of the generating set.
\end{lem}
Therefore, we will say that a group $G$ {\it is Hucha with respect to $\mathcal H$} if some (any) Cayley graph is Hucha with respect to $\cH$.
By Remark \ref{rem: fg pos cone are coarsely connected} we also have the following:
\begin{cor}
Hucha groups do not have finitely generated positive cones.
\end{cor}
Although we will not need this, it is worth recording it.
The proof is easy but will exemplify some of the ideas used later.
\begin{lem}\label{lem: Hucha is preserved by overgroups}
Let $G$ be a finitely generated left-orderable group, $H$ a finite index subgroup such that $H$ is Hucha with respect to a family $\cH$ of subgroups of $H$. Then $G$ is Hucha with respect to the family $\cH$.
\end{lem}
\begin{proof}
Fix a finite generating set $X$ of $G$.
By Lemma \ref{lem: Hucha is geometric}, we can assume that $Y=X\cap H$ generates $H$.
Thus $\Gamma(H,Y)$ is a subgraph of $\Gamma(G,X)$.
Let $R$ such that $H\mathbb{B}(1,R)=G$.
Fix a positive cone $P$ of $G$ with corresponding order $\prec$.
Let $t^{-1}=\max_\prec\mathbb{B}(1,R)$.
Thus $t\mathbb{B}(1,R)\subseteq P^{-1}\cup \{1\}$.
Let $r>0$ and $K\in \cH$. We need to find a negative swamp $S\subseteq P^{-1}$ for $K$ of width $r$.
Notice that $P'=t^{-1}Pt\cap H$ is a positive cone for $H$.
By hypothesis, for every $r'>0$ there exists $S'\subseteq (P')^{-1}$ and $a_1,a_2\in H$ such that $S'$ $r'$-disconnects $a_1K$ and $a_2K$ in $\Gamma(H,Y)$.
We take $r'$ satisfying that $|h|_Y\leq r'$ for every $h\in H$ with $|h|_X\leq 2R+r$.
Note that $S'\subseteq t^{-1} P^{-1} t$. Thus $tS't^{-1}\subseteq P^{-1}$ and therefore $tS'\mathbb{B}(1,R)\subseteq P^{-1}$.
We claim that $tS'\mathbb{B}(1,R)$ is a negative swamp of width $r$ for $K$.
Let $\{g_i\}_{i=0}^m$ be an $r$-sequence in $G$ with $g_0\in ta_1K$ and $g_m\in ta_2K$.
Let $h_0,h_m\in H$ such that $th_0=g_0$, $th_m=g_m$ and for $0<i<m$, let $h_i\in H$ such that $\dist(g_i,th_i)\leq R$.
Then $\{h_i\}_{i=0}^m$ is an $r'$-sequence in $\Gamma(H,Y)$ from $a_1K$ to $a_2K$ and hence there is some $i$ such that $h_i\in S'$.
Thus $th_i\in tS'$ and $g_i\in th_i\mathbb{B}(1,R)$.
It remains to show that $tS'\mathbb{B}(1,R)$ $r$-disconnects $P$.
By hypothesis, there are $a_1,a_2\in P'$ such that $S'$ $r'$-disconnects them.
Note that $ta_it^{-1}\in P$.
Let $\{g_i\}_{i=0}^m$ be an $r$-sequence from $ta_1t^{-1}$ to $ta_2t^{-1}$.
Let $h_0=a_1$, $h_m=a_2$, and for $1<i<m$ let $h_i\in H$ such that $\dist(g_i,th_i)\leq R$.
Arguing as before, since the $r'$-sequence $\{h_i\}_{i=0}^m$ meets $S'$, we conclude that $\{g_i\}_{i=0}^m$ meets $tS'\mathbb{B}(1,R)$.
\end{proof}
We do not know if the Hucha property passes to finite index subgroups since not always a left-order on a finite index subgroup can be extended to a left-order on the ambient group.
We remark that in all the groups that we show that are Hucha, the property is inherited by its finite index subgroups.
However, we know that the Hucha property is not preserved under direct products.
\begin{ex}\label{ex F2xF2}
The free group of rank 2, $F_2$ has the Hucha property with respect to the trivial group. This follows form Corollary \ref{cor: free groups}.
It is known that the BNS-invariant $\Sigma^1(F_2\times F_2)$ is non-empty.
Concretely, the map that sends each standard generator of $F_2\times F_2$ to $1\in\mathbb{Z}$ has finitely generated kernel. Therefore, by Proposition \ref{prop:BNS}, $F_2\times F_2$ has a coarsely connected positive cone.
\end{ex}
\subsection{The Prieto property}\label{sec: Prieto}
We introduce a strong negation of the Hucha property.
\begin{defi}
Let $G$ be a left-orderable group and $X$ a finite generating set of $G$.
We say that $\Gamma(G,X)$ {\it is Prieto} if every positive cone $P$ of $G$ is coarsely connected.
\end{defi}
It is easy to see that Prieto is a geometric property of the group, and thus if a Cayley graph of $G$ is Prieto, then every finitely generated Cayley graph of $G$ is Prieto. We record this fact in the following lemma.
\begin{lem}\label{lem: Prieto is geometric}
Being Prieto is independent of the finite generating set.
\end{lem}
We say that $G$ {\it is Prieto} if there is some finite generating set $Y$ of $G$ such that $\Gamma(G,Y)$ is Prieto. The following is the analogous to Lemma \ref{lem: Hucha is preserved by overgroups} in the Prieto case. We omit the proof, since it is straightforward.
\begin{lem}
Let $G$ be a finitely generated left-orderable group, and $H$ a finite index Prieto subgroup.
Then $G$ is Prieto.
\end{lem}
The easiest example of a Prieto groups are finitely generated torsion free-abelian groups.
\begin{prop}\label{prop: abelian groups are Prieto}
Finitely generated free-abelian groups are Prieto.
\end{prop}
\begin{proof}
We view the Cayley graph $\Gamma$ of $\mathbb Z^n$ with respect to a basis as a lattice in $\mathbb{R}^n$.
It is well known (see \cite[Section 1.2.1]{GOD} and references therein) that for every positive cone $P$ of $\mathbb Z^n$ there exists a hyperplane $\pi$ of $\mathbb{R}^n$ passing through the origin, such that $P$ consists on the lattice points in some connected component of $\mathbb{R}^n \setminus \pi$, and perhaps some points on $\pi$ (if $\pi$ contains lattice points).
Therefore $P$ is a connected subset of $\Gamma$.
\end{proof}
To exhibit more involved examples, recall that given $\prec$ a left-invariant order on a group $G$. A subgroup $H$ of $G$ is {\it cofinal} if for every $g\in G$, there are $h_1,h_2$ such that $h_1\prec g\prec h_2$.
\begin{lem}\label{lem: cofinal center makes P connected}
Let $(G,\prec)$ be a finitely generated left-ordered group.
Suppose that $Z$ is a finitely generated, cofinal and central subgroup. Then $P_\prec=\{g\in G \mid 1\prec g\}$ is coarsely connected.
\end{lem}
\begin{proof}
Fix a finite generating set $X$ of $G$.
Let $g_0,g_m\in P_\prec$ and $\{g_i\}_{i=0}^m$ be a $1$-sequence connecting them.
Since $Z$ is cofinal, there is $z\in Z$ such that $z\prec g_i$ for $0<i<m$.
Withoutloss of generality, assume $z\prec 1$.
Then $\{z^{-1}g_{i}\}_{i=0}^m$ is a $1$-sequence contained in $P$.
Since $Z$ if finitely generated, it is free-abelian and hence Prieto.
Thus there is some $r>0$, such that $P_\prec \cap Z$ (with the induced metric) is $r$-connected.
Thus there is an $r$-path $\{z_i\}_{i=0}^k$ supported in $P_\prec$ from $1$ to $z^{-1}$.
Now the concatenation $\{g_0 z_i\}_{i=0}^{k}$ with $\{z^{-1} g_i\}_{i=0}^m$ followed by $\{ z_ig_m\}_{i=k}^0$ is an $r$-sequence supported in $P_\prec$ from $g_0$ to $g_m$.
\end{proof}
\begin{rem}\label{rem: F_2x Z}
The group $F_2\times \mathbb Z$ is not a Hucha group neither a Prieto group.
This will follow by considering different lexicographic orders. For showing that is not Hucha, we take the order on $F_2$ as the leading order in the lexicographic order. For showing that it is not Prieto, we consider the order on $\mathbb Z$ as the leading one.
Indeed, from Corollary \ref{cor: free groups} we know that positive cones of $F_2$ are not coarsely connected, so, if we fix $P$ a positive cone in $F_2$, it is easy to check that
$P'= (P\times \mathbb Z) \cup (\{1\}\times \mathbb{Z}_{\geq 1})\subseteq F_2\times \mathbb Z$ is positive cone for $F_2\times \mathbb Z$ which not coarsely connected either.
If instead one takes $P'=(F_2\times \mathbb Z_{\geq 1})\cup (P\times \{0\})$ this is a positive cone for $F_2\times \mathbb Z$ on which $\mathbb Z$ is co-final, and by Lemma \ref{lem: cofinal center makes P connected}, $P'$ is coarsely connected.
It is worth noticing that not only $F_2\times \mathbb Z$ has coarsely connected positive cones, but also it has finitely generated ones. This was recently showed by H. L. Su \cite{Su}.
\end{rem}
The braid group $B_n$ can be defined as the mapping class group of a punctured disk with $n$ punctures. E. Artin gave a presentation for $B_n$
$$B_n=\langle \sigma_1,\dots, \sigma_{n-1}\, \mid (\sigma_i \sigma_{i+1} \sigma_{i} = \sigma_{i+1} \sigma_i \sigma_{i+1}\,: \, 1\leq i \leq n-2), (\sigma_i \sigma_j= \sigma_j \sigma_i \,:\, |i-j|>1)\rangle. $$
Iterated Torus knot groups are groups with presentation
$$T_{n_1,\dots,n_k} =\langle a_1,\dots, a_k \mid a_1^{n_1}=a_2^{n_2}=a_3^{n_3}=\dots =a_k^{n_k}\rangle.$$
\begin{cor}\label{cor: Braids are Prieto}
Braid groups, iterated Torus knot groups and $\widetilde{T}=\langle a,b,c\mid a^2=b^3=c^7=abc\rangle $ are Prieto groups.
\end{cor}
\begin{proof}
The left-orderability of these groups is well-known. The orderabilty of $B_n$ was first proved by P. Dehornoy (see \cite[Section 1.2.6]{GOD}). The observation that $\widetilde{T}$ is left-orderable is attributed to W. Thurston and rediscovered by G. Bergman (see \cite[Discussion before Example 3.2.7]{GOD}), while the left-orderability of $T_{n_1,\dots, n_k}$ is discussed in \cite[End of Section 2.2.]{GOD}.
With the given presentation, the center of $B_n$ is generated by $(\sigma_1\sigma_2\dots\sigma_{n-1})^2$.
A. Clay showed that this cyclic group is cofinal in every left-order of $B_n$.
See \cite[Example 3.5.14.]{GOD} for a proof.
By Lemma \ref{lem: cofinal center makes P connected} each positive cone is coarsely connected.
For iterated Torus knot groups the same holds: $z=a_1^{n_1}$ is always cofinal and central.
The centrality is clear from the presentation.
For the cofinality we argue as follows.
Let $\preceq$ be a left-order of $T_{n_1,\ldots,n_k}$ and set $U=\{t\in T_{n_1,\ldots,n_k}\mid t\preceq z^m \text{ for some } m\in \mathbb Z\}$.
We claim that $a_i U\subseteq U $ for all $a_i$.
Indeed, let $t\in U$, $t\prec z^m$. If $a_i t\prec t\prec z^m$ then $a_i t \in U$ by definition.
On the other hand if $t\prec a_i t$ then, inductively, we get that $t\prec a_i^{n_i} t=zt$, so $a_i t\prec z^{m+1}$.
This implies that $U$ is invariant under every generator, so $U=T_{n_1\ldots,n_k}$. In particular $z$ is cofinal.
The preceding argument also shows that $a^2\in \widetilde{T}$ is always (central and) cofinal.
\end{proof}
In contrast with the Hucha property, we know that the Prieto property does not pass to finite index subgroups.
Indeed, the previous corollary shows that $B_3$ is Prieto while the pure braid group $PB_3$ is a finite index subgroup isomorphic to $F_2\times \mathbb Z$, which, by Remark \ref{rem: F_2x Z}, we know that is not Prieto.
We finish this section by showing that, unlike Hucha property (see Example \ref{ex F2xF2}), the Prieto property is stable under direct products.
\begin{prop}
The direct product of Prieto groups is Prieto.
\end{prop}
\begin{proof}
Let $A$ and $B$ be two finitely generated, left-orderable, Prieto groups.
Fix a positive cone $P$ of $G=A\times B$.
Let $X_A$ and $X_B$ be finite generating sets for $A$ and $B$ respectively.
We consider $X= X_A \times \{1\} \cup \{1\} \times X_B$ as generating set for $G$.
Since $A$ and $B$ are Prieto, we assume that $P\cap A$ is an $r$-connected subset of $\Gamma(A, X_A)$.
Similarly, we assume that $P\cap B$ is an $r$-connected subset of $\Gamma(B, X_B)$.
Note that for simplicity we identify $A$ with the subgroup $A\times \{1\}$ and similarly $B$ with $\{1\}\times B$.
Let $g_1=(a_1,b_1)$ and $g_2=(a_2,b_2)$ be two positive elements.
We will connected them with an $r$-path in $\Gamma(G,X)$ supported in $P$.
We can assume without loss of generality that $(a_1,1_B)$ is in $P$.
Indeed, if that was not the case, then $a_1^{-1}\in P \cup \{1\}$ and there is an $r$-path $\{h_i\}_{i=1}^n$ from $1_A$ to $a^{-1}$ in $\Gamma(A, X_A)$ supported on $P\cap A$. Thus $\{(a_1h_i, b_1)\}_{i=1}^n$ is an $r$-path in $P$ from $g_1$ to $(1_A,b_1)$. Let $a'\in P\cap A$, $|a'|_X\leq r$. Then there is an $r$-path from $g_1$ to $(a',b_1)$ supported in $P$ and we can replace $g_1$ by $(a',b_1)$ if necessary.
Repeating the argument, we assume that $a_1,a_2\in P\cap A$ and $b_1,b_2 \in B\cap P$.
Thus there are $r$-paths $\{y_i\}_{i=1}^m$ in $\Gamma(A, X_A)$ from $a_1$ to $a_2$ and $\{z_i\}_{i=1}^l$ in $\Gamma(B,X_B)$ from $b_1$ to $b_2$ both supported on positive elements.
Thus $\{(y_i, b_1)\}_{i=1}^m$ concatenated with $\{(a_2, z_i)\}_{i=1}^l$ is an $r$-path from $g_1$ to $g_2$.
\end{proof}
\section{Groups acting on trees}
\label{sec: trees}
In this section we will show the main technical result of the paper.
We will give a combination theorem that produces Hucha groups.
For that, first we will prove Proposition \ref{prop: P accumulates in ends of T}, which is a self contained proof of Theorem \ref{thm: main boundary} in the special case of groups acting on trees.
There are several reasons of giving this proof. Firstly, it is not lengthy and we produce a statement better adapted for our purposes, sparing some translation effort.
Secondly, to show that certain groups are Hucha with respect to cyclic subgroups, we are using something slightly more general that what was proved on Theorem \ref{thm: main boundary}, namely that for any loxodromic isometry $h$ of the tree, and any neighbourhood $A$ of an end, there is a coset of $\langle h\rangle$ whose limit points lie in $A$.
Throughout this paper, trees are simplicial trees and actions on trees are by graph automorphisms that do not reverse edge orientations.
The key of the section is the strong interplay between the geometry of $T$ and the geometry of $G$ arising from the action of $G$ on $T$. We could say that the geometry of $T$ dominates the geometry of the group.
This allows us to construct negative swamps and to show that certain groups acting on trees are Hucha in Subsection \ref{ssec: creating hucha}.
\subsection{Positive Cones of groups acting on trees}
\label{sec: positive cones accumulate on the bountary of a tree}
Let $G$ be a group acting on a tree $T$.
An element $g\in G$ that stabilizes a vertex of $T$ is called {\it $T$-elliptic}. If $g$ is not $T$-elliptic then it is called {\it $T$-loxodromic}.
If $g$ is a $T$-loxodromic element, then there exists a unique subtree of $T$ homeomorphic to $\mathbb{R}$ on which $g$ acts by translation and we denote it by $\axis_T(g)$.
See for example \cite[Proposition I.4.11]{DicksDunwoody}.
We note that $\axis_T(g)$ is the minimal $\langle g \rangle$-invariant subtree of $T$. We will drop the $T$ from the notation if the action is clear from the context. We say that the action is \emph{minimal} when $T$ is the unique $G$-invariant sub-tree. Denote $T$ mod $G$ the quotient of $T$ under the $G$-action.
When a path is semi-infinite and reduced we will call it a \emph{ray}. We say that two rays $p,q$ of $T$ are equivalent if they have infinite intersection.
An end $\eta$ of $T$ is an equivalence class of rays.
A group $G$ acting on a tree $T$ {\it fixes an end of $T$} if there is a ray $p$ in $T$ such that
$g p \cap p$ is infinite for all $g\in G$.
Recall from Section \ref{sec: hyperbolic} that the action of a group $G$ on a tree $T$ is {\it of general type} if $G$ does not fix a vertex or an end of $T$, and a $G$-orbit of a vertex meets at least 3 different ends of $T$. Also, from Theorem \ref{thm: classification actions}, under the assumption of general type action there are always some element acting loxodromically on $T$ (a direct proof for actions on trees can be found in \cite[Theorem I.4.12]{DicksDunwoody}).
\begin{ex}\label{rem: example}
For the purposes of the paper, the main examples of general type actions on trees are the following: Let $G$ be either an amalgamated free product $A*_C B$, with $C$ a proper subgroup of $A$ and $B$, or an HNN extension $A*_C t$, with $C$ a proper subgroup of $A$.
Then the action of $G$ on its associated Bass-Serre tree is of general type.
If $G=A*_C B$, the Bass-Serre tree has vertex set $G/A \sqcup G/B$ and edge set $G/C$ with $gC$ adjacent to $gA$ and $gB$.
If $G= A*_C t$, the Bass-Serre tree has vertex set $G/A$, edge set $G/C$ with $gC$ adjacent to $gA$ and $gtA$.
The action of $G$ on the Bass-Serre tree is induced by the left permutation action of $G$ on the cosets. See \cite{serre}.
\end{ex}
The following is well-known, we include the proof for completeness.
\begin{lem}\label{lem:h_1h_2}
Let $G\curvearrowright T$ be a general type, co-bounded action of a group $G$ on a tree $T$.
Let $h$ be an element acting loxodromically on $T$.
Then, there are $h_1,h_2\in h^G$, two $G$-conjugates of $h$, such that
$\axis_T(h_1)\cap \axis_T(h_2)$ is empty.
\end{lem}
\begin{proof}
Recall that for $g\in G$, one has that $g\axis_T(h)=\axis_T(ghg^{-1})$.
Since $T$ has more than two ends, there are vertices of $T$ arbitrarily far away from $\axis_T(h)$.
Since $G$ acts co-boundedly, there must be $g\in G$ such that $g\axis_T(h)\neq \axis_T(h)$.
Suppose that for all $g\in G$, $g\axis_T(h)\cap \axis_T(h)$ is infinite.
Since $G$ does not fix an end
there must be $g_1$ and $g_2$ in $G$ such that
$g_1\axis_T(h)\cap g_2 \axis_T(h) \cap \axis_T(h)$ is finite, that is $g_1\axis_T(h)$ and $g_2\axis_T(h)$ could only have infinite intersection with different ends of $\axis_T(h)$.
The action of $\gen{h}$ on $\axis_T(g_1hg_1^{-1})$ slides this axis along $\axis_T(h)$, therefore we can find $n$ such that
$h^n\axis_T(g_1hg_1^{-1})\cap \axis_T(g_2hg_2^{-1})$ is empty.
Now we can take $h_1=h^ng_1hg_1^{-1}h^{-n}$ and $h_2=g_2hg_2^{-1}$.
\end{proof}
We are ready to show that the orbit of a positive cone visits every neighbourhood of every end of $T$.
\begin{prop}\label{prop: P accumulates in ends of T}
Let $G\curvearrowright T$ be a co-bounded, general type action of a group $G$ on a tree $T$.
Let $h\in G$ be an element acting loxodromically on $T$.
Consider $v$ a vertex of $T$, $B$ a bounded subset of $VT$ and $\cC$ an infinite connected component of $T-B$.
Then, there exists $k\in G$ such that $k\langle h \rangle v$ lies in $\cC$.
Moreover, for every positive cone $P$ of $G$ the set $Pv$ has non-trivial intersection with every infinite connected component of $T-B$.
\end{prop}
\begin{proof}
Fix a vertex $v$, a bounded set $B$ of $T$ and an infinite connected component $\mathcal{C}$ of $T-B$.
By Lemma \ref{lem:h_1h_2} there exist loxodromic elements $h_1=g_1hg_1^{-1}$ and $h_2=g_2hg_2^{-1}$ such that
$\axis_T(h_1)\cap \axis_T(h_2)=\emptyset$.
Let $\mathcal{O}_i$ be the orbit of $g_iv$ under $h_i$. That is $\mathcal{O}_i=\{g_ih^n v \mid n\in \mathbb Z\}$.
Let $L$ denote the minimal segment connecting $\mathcal{O}_1$ and $\mathcal{O}_2$.
Since the action of $G$ on $T$ is co-bounded, there is some $g\in G$ such that $g L\subseteq \mathcal{C} $.
Moreover, we can assume that
\begin{equation}\label{eq:sufficiently far away L}
\dist(g L, B)>D.
\end{equation}
where $D$ is as constant to specify later.
We claim that either $g\mathcal{O}_1$ or $g\mathcal{O}_2$ is contained in $\mathcal{C}$.
Suppose that this was not the case.
Thus for $i=1,2$, there are $w_i\in B$ such that $w_i$ lies in the geodesic between two consecutive vertices of $\mathcal{O}_i$ (i.e there is some $n$ such that $w_i$ is in $[g_ih^nv,g_ih^{n+1}v]_T$ the geodesic segment in $T$ from $g_ih^nv$ to $g_ih^{n+1}v$).
Note that $\dist(g_ih^nv,g_ih^{n+1}v)=\dist(hv,v)$.
Therefore $\dist(g\mathcal{O}_1,g\mathcal{O}_2)\leq \dist(w_1,w_2)+2\dist(v,hv)\leq \diam(B)+2\dist(v,hv)$.
We get from \eqref{eq:sufficiently far away L}, that $\dist(w_1,w_2)\geq 2 D+ \diam (L)-2 \dist(v,hv)$.
Putting the two things together we get that $2D +\diam(L)\leq \diam(B)+2\dist(v,hv)$.
Since we can take $D$ arbitrarily big, for an appropriate choice of $g$ we get a contradiction.
Thus either $g\mathcal{O}_1$ or $g\mathcal{O}_2$ in $\cC$ and this completes the proof.
Let $P$ be a positive cone. Since either $gg_1 h g_1^{-1}g^{-1}\in P$ of $gg_1 h^{-1} g_1^{-1}g^{-1}\in P$,
we get that $gg_1\langle h\rangle v$ contains infinitely many elements of $Pv$.
\end{proof}
The following is an strengthening of Corollary \ref{cor: free groups}, and it will be used as the base case for showing that limit groups are Hucha.
\begin{prop}\label{prop:F2hucha}
Finitely generated non-abelian free groups are Hucha with respect to the family of finitely generated, infinite-index subgroups.
\end{prop}
\begin{proof}
Let $F$ be a finitely generated non-abelian free group and $X$ a finite generating set.
Without loss of generality, by Lemma \ref{lem: Hucha is geometric}, we can assume that $X$ is a basis and hence the Cayley graph of $F$ is a tree $T$ with infinitely many ends.
The action of $F$ on $T$ is co-compact, free and of general type.
Fix a positive cone $P$, an infinite-index finitely generated subgroup $H$ and $r\geq 0$.
Denote by $\prec$ the order on $F$ corresponding to $P$.
Let $x_n=\max_{\prec}\mathbb{B}(1_F, n)$ and recall that $x_n^{-1}\mathbb{B}(1_F,n-1)\subseteq P^{-1}$.
Take $S= x_{r+1}^{-1}\mathbb{B}(1_F, r)$.
Since $T$ is a tree, every $r$-path joining two components of $T-S$ must go through $S$.
By Proposition \ref{prop: P accumulates in ends of T} the connected components of $T-S$ contains positive elements.
Thus $P$ is not $r$-connected.
Every finitely generated subgroup of a free group is quasi-convex, and therefore, by \cite[Theorem 4.8]{HruskaWise}, $H$ has {\it bounded packing}, meaning that for every $D$ there is some number $N=N(D)$ such that for any collection of $g_1H,\dots, g_N H$ of $N$ distinct cosets of $H$, we have that there are at least two separated by a distance at least of $D$.
Therefore, there exists cosets $g_1 H$ and $g_2 H$ at distance $>2r$.
Let $\gamma$ be a geodesic path joining $g_1 H$ and $g_2 H$ and assume that $v$ is a vertex in $\gamma$ with $\dist(v, g_1 H)>r$ and $\dist (v,g_2H)>r$.
Since $F$ acts transitively by isometries, we can assume that $v=x_{r+1}^{-1}$ and therefore, $g_1H$ and $g_2H$ are separated by $S=x_{r+1}^{-1}\mathbb{B}(1_F,r)$ and thus, there is not $r$-path in $P$ connecting them.
\end{proof}
\subsection{Cayley graphs of groups acting on trees}
\label{ssec: T dominates}
Consider a group $G$ acting on a a tree $T$. In this subsection we will show how to find generating sets for $G$ which are \emph{adapted} for the action on $T$ (Lemma \ref{lem:adapted}).
This generating sets will be useful to relate the geometries of $G$ and $T$.
Given $v_1,v_2\in VT$ denote $[v_1,v_2]_T$ the geodesic segment between them.
Also, denote $\text{link}_T(v)$ the set of edges in $ET$ which are adjacent to $v$.
We say that a sub-tree $T_0\subseteq T$ is a \emph{fundamental domain} for the $G$-action on $T$ if $T_0$ contains exactly one representative of every edge-orbit under the action.
Also, given $v\in VT$ we denote $G_v$ its stabilizer under the $G$-action, that is $G_v=\{g\in G:gv=v\}$.
Analogously, for $e\in ET$, denote the edge stabilizer by $G_e$.
We need one extra definition.
Given $X$ a generating set for $G$ and $E\subseteq \text{link}_T(v)$, we say that $X$ has $v$-reductions modulo $E$ if: for every $1$-path $\gamma\colon\{0,\ldots,n\}\to\Gamma(G,X)$ with \begin{itemize}
\item $\gamma(i)v\neq v$ for $0<i<n$, and
\item $\gamma(0)v=\gamma(n)v=v$,
\end{itemize} we have that $\gamma(n)\in\gamma(0)G_e$ for some $e\in E$.
It follows that if a generating set $X$ has $v$-reductions modulo $E$, then for every path leaving and then coming back to $G_v$ in $\Gamma(G,X)$, there is and edge $e\in \text{link}_T(v)$ and $g\in G_v$ such that the path leaves from and comes back to (the same) coset $gG_e$. In other words, this is saying that cosets of edge stabilizers are check-points for paths that leave cosets of vertex stabilizers.
The following lemma shows the existence of generators having $v$-reductions as well as some other properties. Its proof is implicit in Serre's \cite{serre} but we include a full proof for completeness.
\begin{lem}\label{lem:adapted}
Let $G$ be a finitely generated group with a co-finite action on a tree $T$ with finitely generated vertex stabilizers.
Given a vertex $v\in VT$, there exists a finite fundamental domain $T_0$ containing $v$ and a symmetric and finite generating set $X$ satisfying:
\begin{enumerate}
\item[(1)] $X\cap G_v$ generates $G_v$,
\item[(2)] $Gv\cap [v,sv]_T=\{v,sv\}$ for every $s\in X$,
\item[(3)] $s T_0\cap T_0$ is non-empty for every $s\in X$,
\item[(4)] There exists a finite subset $E\subseteq \text{link}_T(v)$ so that $X$ has $v$-reduction modulo $E$.
\end{enumerate}
\end{lem}
\begin{proof}
We first construct the set domain $T_0$ and the generating set $X$ and then we verify the properties (1)--(4).
\paragraph{Construction of $T_0$:}
Let $A:=G \backslash T$ be the quotient graph of $T$ by the action of $G$ and let $\pi:T\to A$ its associated quotient map.
Also denote $v_0:=\pi(v)$.
Consider $A_0\subseteq A$ a maximal subtree of $A$ such that $A_0$ contains every edge of $A$ one of its end points (but not both) is $v_0$.
Let $T_0'$ be a lift of $A_0$ containing $v$, that is a sub-tree $T_0'\subseteq T$ so that $\pi$ induces an isomorphism between $T_0'$ and $A_0$.
Finally extend $T_0'$ to a fundamental domain $T_0$ such that every edge in $ET_0\setminus ET_0'$ has one end-point in $T_0'$. Note that $T_0$ is finite because, by hypothesis, the action is co-finite.
\paragraph{Construction of $X$:}
For every $w\in T_0'$ consider a finite and symmetric set $X_w$ so that $\langle X_w \rangle=G_w$.
We can do that because, by hypothesis vertex stabilizers are finitely generated.
Also, for every $e=[w_1,w_2]_T$ in $ET_0\setminus ET_0'$ with $w_1\in T_0'$ consider an element $g_e\in G$ so that $g_e^{-1}w_2\in T_0'$.
We define $$X=(\cup_{w\in T_0'}X_w)\bigcup \{g_{e},g_{e}^{-1}:e\in ET_0\setminus ET_0'\}.$$
\paragraph{Condition (1):} It follows immediately from our definition of $X$.
\paragraph{Conditions (2) and (3):}
For this, take $s\in X$. We will distinguish two cases.
\subparagraph{Case 1:} $s\in G_w$ for some $w\in T_0'$.
In this case, $[v,sv]_T\subseteq [v,w]_T\cup[w,sv]_T$.
In particular, since $T_0'$ meets every vertex orbit under $G$ exactly once, we have that $[v,sv]_T\cap Gv=\{v,sv\}$ which proves condition (2) in this case.
Since $w\in T_0\cap sT_0$ condition (3) follows.
\subparagraph{Case 2:} $s\notin G_w$ for every $w\in T_0'$.
First notice that, since $T_0'$ meets every vertex orbit under $G$ exactly once, if $sT_0'\cap T_0'\neq\emptyset$ then $s\in G_w$ for some $w\in T_0'$. Then, we can assume that $sT_0'\cap T_0'=\emptyset$.
Assume without loss of generality that $s=g_e$ for some $e\in T_0\setminus T_0'$ (the case $s=g_e^{-1}$ is analogous).
Denote $e=[w_1,w_2]_T$ with $w_1\in T_0'$.
By construction of $g_e$ we have that $w_2\in g_eT_0'$.
Then, $w_2\in T_0\cap g_eT_0$ which implies condition (3).
To verify condition (2) note that, since $e$ meets both $T_0'$ and $g_eT_0'$, we have that
$$[v,g_ev]_T\subseteq T_0'\cup e\cup g_eT_0'.$$
If $w_1=v$, by the choice of $A_0$, we have that $w_2=g_ew_1=g_ev$ which implies condition (2) for this sub-case.
On the other hand, if $w_1\neq v$, again by the choice of $A_0$, we have that $w_2\neq v$ and therefore
$Gv\cap T_0'\cup e\cup g_eT_0'=\{v,g_ev\}$ which shows that condition (2) is also verified in this sub-case.
\paragraph{Condition (4):} Define
$$E=\text{link}_{T_0}(v)\cup\{g_e^{-1}e:e\in \text{link}_{T_0}(v)\setminus ET_0'\}.$$
Since $T_0$ is finite, so is $E$. We will show that $X$ has $v$-reduction modulo $E$.
For this consider a $1$-path $\{\gamma_i\}_{i=0}^n$ in $\Gamma(G,X)$ so that $\gamma_0 v=\gamma_n v=v$ and $\gamma_iv\neq v$ for $0<i<n$.
Since condition (2) holds we have that $\{\gamma_i\}_{i=1}^{n-1}$ is supported on a single connected component of $T\setminus \{v\}$ or otherwise we would have $\gamma_iv=v$ for some $0<i<n$ contradicting our assumption.
Denote by $T_{\ast}$ minimal subtree of $T$ containing $v$ and the component of $T\setminus \{v\}$ containing the vertices $\{\gamma_iv\}_{i=1}^{n-1}$.
Denote by $\overline{e}$ the {\it unique } edge in $ET_{\ast}$ adjacent to $v$.
Take $e\in ET_0$ so that $ge=\overline{e}$ for some $g\in G$.
Note that by construction of $T_0$ we have that all the edges of $T_0$ meeting $Ge$ are adjacent to $v$.
Therefore we have that $$\overline{e}=ge\text{ with }e\in \text{link}_{T_0}(v),\text{ for some }g\in G.$$
In order to show that $\gamma$ $v$-reduces modulo $E$, we will distinguish in two cases:
\subparagraph{Case 1:} $\dist_T(v,\gamma_1 v)>1$.
We will first show that $\overline{e}=\gamma_0 e$ for some $e\in \text{link}_{T_0}(v)$.
Since $\dist_T(v,\gamma_1 v)>1$ and $e\in \text{link}_{T_0}(v)$ we get that $\gamma_1 e\neq \overline{e}$.
Therefore $\overline{e}$ does not belong to $\gamma_1 T_0$ because $T_0$ is a fundamental domain.
On the other hand, since condition (3) holds, $\gamma_0 T_0\cup\gamma_1 T_0$ is connected and therefore contains the geodesic segment $[v,\gamma_1 v]_T$ which is contained in $T_{\ast}$.
Then, $\overline{e}\in\gamma_0 T_0$ which implies that $\overline{e}=\gamma_0 e$.
Now, we point out that $\dist_T(\gamma_{n-1}v,v)>1$.
Indeed, if it was not the case we would have that $\gamma_{n-1}v$ would belong to the interior of $[\gamma_0 v,\gamma_1 v]_T$ contradicting condition (2).
Since $\dist_T(\gamma_{n-1}v,v)>1$, we can repeat the previous argument and show that $\overline{e}=\gamma_n e$.
This implies that $\gamma_0^{-1}\gamma_n\in G_{e}$.
Therefore, since $e\in E$, condition (4) holds in this case.
\subparagraph{Case 2:} $\dist_T(v,\gamma_1 v)=1$.
Arguing as in the previous case we can use condition (2) to show that $\dist_T(\gamma_{n-1}v,v)=1$.
Since $\{\gamma_i\}_{i=0}^n$ is a $1$-path in $\Gamma(G,X)$ there are $s_1,s_n\in X$ such that $\gamma_{1}=\gamma_{0}s_1$ and $\gamma_{n}=\gamma_{n-1}s_n$.
We claim that $s_1,s_n\in\{g_e,g_{e}^{-1}\}$ for some $e\in \text{link}_{T_0}(v)\setminus T_0'$.
We proceed to prove the claim for $s_1$, the other case is analogous.
For this, first notice that
\begin{equation}\label{equdist}\dist_T(v,s_1v)=1.
\end{equation}
This excludes the case where $s_1$ is elliptic because whenever $h\in G$ is elliptic we have that $\dist_T(v,hv)$ is even. Therefore $s_1=g_{e_0}^{\pm1}$ for some $e_0\in T_0\setminus T_0'$.
Assume it is the case that $s_1=g_{e_0}$.
As was shown before, it holds that $[v,g_ev]_T\subseteq T_0'\cup e\cup g_eT_0'$.
Then, since $T_0'$ and $g_eT_0'$ do not meet we have that $e\in [v,g_ev]_T$.
Therefore, by equation \eqref{equdist} we have that $e=[v,g_{e_0}v]_T$.
By construction this implies that $e=e_0$ and therefore $s_1=g_e$.
Analogously, assuming that $s_1=g_{e_0}^{-1}$ we deduce that $s_1=g_e^{-1}$.
This finishes the proof of our claim.
Now, in order to finish the proof that condition (4) holds, we will distinguish in two sub-cases according to $s_1=g_e$ or $s_1=g_e^{-1}$.
\subparagraph{Subcase 2A:} $s_1=g_e$.
In this case we have that $\overline{e}=\gamma_0 e$.
Applying condition (2) we deduce that $\dist_T(\gamma_{n-1}v,v)=1$.
On the other hand, if $\gamma_{n-1} e=\overline{e}$ we would have that $\gamma_{0}\gamma_{n-1}^{-1}$ would produce an inversion on $\overline{e}$ (i.e. $\overline{e}$ is fixed but its endpoints swapped by $\gamma_0\gamma_{n-1}^{-1}$) which contradicts that $G$ acts without inversions.
Therefore, applying condition (3) again we get that $\overline{e}$ must belong to $\gamma_{n}T_0$ which implies that $\gamma_{n}e=\overline{e}$.
This shows that $\gamma_0^{-1}\gamma_n\in G_e$.
Since $e\in E$ this proves condition (4) in this sub-case.
\subparagraph{Subscase 2B:} $s_1=g_e^{-1}$.
Since $g_e^{-1}v\notin T_0$ we get that $\gamma_1 v=\gamma_0 g_e^{-1}v\notin\gamma_0 T_0$.
Then, using condition (3) again we deduce that $\overline{e}\in \gamma_1T_0$ and therefore
\begin{equation}\label{equequ}
\overline{e}=\gamma_0g_e^{-1}e.
\end{equation}
Notice that if $\overline{e}\in\gamma_nT_0$ we would have that $\gamma_1\gamma_n^{-1}$ would induce an inversion on $\overline{e}$.
Then, using condition (3) again we deduce that $\overline{e}=\gamma_{n-1}e$ and in particular $v\in \gamma_{n-1}T_0$.
On the other hand $v=\gamma_{n}v=\gamma_{n-1}s_nv$.
Then $s_nv\in T_0$.
Since $s_n\in\{g_e,g_e^{-1}\}$ and $s_nv\in T_0$ we deduce that $s_n=g_e$ and therefore $\overline{e}=\gamma_{n-1}e=\gamma_{n}g_e^{-1}e$.
The last equality together with equation \eqref{equequ} imply that $\gamma_0^{-1}\gamma_n\in G_{g_e^{-1}e}$.
Since $g_e^{-1}e\in E$ this finishes the proof of condition (4).
\end{proof}
\begin{defi}
Let $G$ be a group acting on a tree $T$ with the hypothesis of Lemma \ref{lem:adapted}.
If generating set $X$ of $G$ satisfies the conclusions (1)--(4) of Lemma \ref{lem:adapted} for a pointed sub-tree $(T_0,v)$, we will say that $X$ is a \emph{adapted to} $(T_0,v)$. Lemma \ref{lem:adapted} shows that adapted generators always exist.
\end{defi}
\begin{rem}
If $X$ is a generating set of $G$ adapted to $(T_0,v)$, then $\Gamma(G,X)\setminus G_v$ is not connected.
Indeed, if $g_1v$ and $g_2v$ are in different components of $T\setminus\{v\}$, then for every every path $\{\gamma_i\}_{i=0}^n$ in $\Gamma(G,X)$ joining $g_1$ and $g_2$, by condition (2) in Lemma \ref{lem:adapted}, there exist $i$ such that $\gamma_iv=v$.
\end{rem}
\subsection{Creating Hucha groups}
\label{ssec: creating hucha}
Given a geodesic segment $p=[w_1,w_2]_T$ in $T$ and, denote $T^p_{w_1}$ the minimal subtree of $T$ that contains $w_1$ and the connected components of $T\setminus\{w_1\}$ not containing $w_2$.
Similarly we define $T^p_{w_2}$ as the minimal subtree of $T$ containing $w_2$ and all the connected components of $T\setminus \{w_2\}$ not containing $w_1$.
Given $v\in VT$ denote $\cC^p_{w_i}$ the pre-image of $T_{w_i}$ under the orbit map of $v$.
That is, for $i=1,2$, define $$\cC^p_{w_i}=\{g\in G:gv\in T^p_{w_i}\}.$$
If $p$ is well understood, we might drop it from $T^p_w$ and $\cC^p_w$.
We start this sub-section showing that we can reduce the problem of proving the Hucha property with respect to cyclic subgroups, to showing that negative cones $r$-disconnect subsets of the form $\cC_{w_i}$. More precisely:
\begin{lem}\label{lem:disconecting}
Let $G\curvearrowright T$ be a co-finite, minimal, general type action of a left-orderable group $G$ on a tree $T$.
Let $v\in VT$ and $X$ be a generating set of $G$.
Assume that for every positive cone $P$ and any $r>0$ there exists a geodesic segment $[v_1,v_2]$ in $T$ and $S$ contained in $P^{-1}$ so that $S$ $r$-disconnects $\cC_{v_1}$ from $\cC_{v_2}$ inside $\Gamma(G,X)$.
Then $G$ has the Hucha respect to subgroups acting elliptically on $T$ and cyclic subgroups acting loxodromically.
In particular, $G$ has the Hucha respect to all cyclic subgroups.
\end{lem}
\begin{proof}
Take $\prec$ a left-order on $G$ with positive cone $P$.
We first show that $G$ has the Hucha with respect to cyclic subgroups.
Let $H$ be either an elliptic subgroup or cyclic subgroup acting loxodromically.
Take $r>0$.
In order to prove that $G$ is Hucha respect to $H$, we need to find a negative swamp in $P^{-1}$ of width $r$ for $H$.
For this consider the geodesic segment $[v_1,v_2]$ and a subset $S\subseteq P^{-1}$ (provided by the hypothesis of the lemma) so that $S$ $r$-disconnects $\cC_{v_1}$ from $\cC_{v_2}$ in $\Gamma(G,X)$.
To show that $S$ is our negative swamp, it is enough to show that:
\begin{enumerate}
\item $P\cap \cC_{v_i}$ is non-empty for $i=1,2$,
\item there exist co-clases $h_iH$ such that $h_iH$ is contained in $\cC_{v_i}$ for $i=1,2$.
\end{enumerate}
Since $G$ acts minimally on $T$ we get that $T_{v_i}$ is infinite for $i=1,2$.
On the other hand, the action is of general type which implies that there exists some $h\in G$ acting loxodromically on $T$.
Then, applying Proposition \ref{prop: P accumulates in ends of T} we conclude that $Pv$ intersect both $T_{v_1}$ and $T_{v_2}$
which implies that $P\cap \cC_{v_i}$ is non-empty for $i=1,2$ as desired.
In order to show point 2 will discuss according to the action of $H$ on $T$.
If $H$ is elliptic, then there exists $v\in VT$ so that $H\subseteq G_v$.
On the other hand, since $T_{v_i}$ is infinite for $i=1,2$, by the co-finiteness of the action, we deduce that the $G$-orbit of $v$ intersects both $T_{v_1}$ and $T_{v_2}$, namely, there exists $h_1,h_2\in G$ so that $h_1v\in T_{v_1}$ and $h_2v\in T_{v_2}$. This implies that $h_iHv\subseteq \cC_{v_i}$ for $i=1,2$ and therefore $h_iH\subseteq \cC_{v_i}$ for $i=1,2$ as desired.
If $H$ is infinite cyclic acting loxodromically, we are in condition to apply Proposition \ref{prop: P accumulates in ends of T} to show that there exists $h_1,h_2\in G$ so that $h_i H v\subseteq T_{v_i}$ for $i=1,2$.
This implies that $h_iH\subseteq \cC_{v_i}$ for $i=1,2$ as desired.
\end{proof}
We proceed to prove our combination theorems.
The fact that when $X$ is an adapted generating set of $G$, the stabilizers of vertices (or edges) of $T$ disconnect $\Gamma(G,X)$ will be crucial.
We start with the easy case where a group $G$ acts on a tree $T$ having an edge with trivial stabilizer.
First, we need to introduce a definition. Given an $r$-path $\gamma\colon\{0,\ldots,n\}\to\Gamma(G,X)$ and a $1$-path $\overline{\gamma}\colon\{0,\ldots,m\}\to\Gamma(G,X)$, we say that $\overline{\gamma}$ is a \emph{geodesic interpolation} of $\gamma$ if there exists a monotone map $\sigma\colon\{0,\ldots,n\}\to\{0,\ldots,m\}$ satisfying:\begin{itemize}
\item $\gamma=\overline{\gamma}\circ\sigma$
\item $\{\overline{\gamma}_j\}_{j=\sigma(i)}^{\sigma(i+1)}$ is a geodesic for $i=0,\ldots,n-1$.
\end{itemize}
\begin{prop}\label{prop: free product Hucha}
Suppose that $G\curvearrowright T$ is a co-finite, minimal, general type action of finitely generated left-orderable group $G$ on a tree $T$.
Assume there exists $e\in ET$ with trivial stabilizer and that vertex stabilizers are finitely generated. Then $G$ has the Hucha with respect cyclic subgroups and subgroups acting elliptically on $T$.
\end{prop}
\begin{proof}
By Lemma \ref{lem:adapted}, we can consider a finite generating set $X$ and a finite pointed sub-tree $(T_0,v)$ so that $X$ is adapted to $(T_0,v)$.
Take $\prec$ a left-order on $G$ with positive cone $P$ and $r>0$.
We will find a subset $S\subseteq P^{-1}$ and a geodesic segment $[w_1,w_2]$ so that $S$ $r$-disconnects $\cC_{w_1}$ from $\cC_{w_2}$ in $\Gamma(G,X)$.
Then, the proposition will follow from Lemma \ref{lem:disconecting}.
For this, take $g\in G$ so that $\max_\prec\mathbb{B}_X(1,r)\prec g^{-1}$.
So we have that $g\mathbb{B}_X(1,r)\subseteq P^{-1}$.
Take $e_0\in ET_0$ with trivial stabilizer and consider the geodesic segment $[w_1,w_2]_T$ defined as $[w_1,w_2]_T=ge_0$.
We claim that $S\coloneqq g\mathbb{B}_X(1,r)$ $r$-disconnects $\cC_{w_1}$ from $\cC_{w_2}$.
To see this, take $\gamma=\{\gamma_i\}_{i=0}^n$ an $r$-path between $h_1\in\mathcal{C}_{w_1}$ and $h_2\in \cC_{w_2}$, and $\overline{\gamma}=\{\overline{\gamma}_i\}_{i=0}^m$ a geodesic interpolation of $\gamma$.
By condition (3) in Lemma \ref{lem:adapted} we have that $\overline{\gamma}_{i+1}T_0\cap\overline{\gamma}_iT_0\neq\emptyset$ for every $i$.
Then, $ge_0\in \overline{\gamma}_iT_0$ for some $i$.
Since $e_0$ has trivial stabilizer, this implies that $\overline{\gamma}_i=g$ and therefore $\gamma$ intersects $g\mathbb{B}_X(1,r)$ as desired.
\end{proof}
\begin{cor}\label{cor:productolibre}
Let $A,B$ be two non-trivial, finitely generated left-orderable groups, then $G=A*B$ has the Hucha with respect to $A$, $B$ and all cyclic subgroups.
\end{cor}
\begin{proof} Consider $G\curvearrowright T$ the Bass-Serre action of $G$ on the Bass-Serre Tree associated to the free product. Note that the edge stabilizers of this action are trivial.
Then, the fact that $G$ has the Hucha with respect to cyclic subgroups follows by applying Proposition \ref{prop: free product Hucha} to this action. To see that $G$ has the Hucha with respect to $A$ and $B$ notice that these subgroups act elliptically.
\end{proof}
We now deal with the case where the edge stabilizers are Prieto.
The next important lemma is the key for proving Theorem \ref{thm:induction}.
Recall that if $X$ is generating set adapted to $(T_0,v)$, then $X_v=X\cap G_v$ generates $G_v$.
\begin{lem}\label{lem:projection}
Suppose that $G\curvearrowright T$ is a co-finite, minimal, general type action of finitely generated left-orderable group $G$ on a tree $T$.
Let $v$ be a vertex on a subtree $T_0$ and $X$ a finite generating set of $G$ adapted to $(T_0,v)$ with $v$-reductions modulo the finite set $E\subseteq \text{link}_T(v)$.
Let $P$ be a positive cone of $G$ and assume that there exists $r_0>0$ so that $P\cap G_e$ is $r_0$-connected in $\Gamma(G_v,X_v)$ for every $e\in E$.
Then, given an $r$-path $\gamma=\{\gamma_i\}_{i=0}^n$ between $g_1$ and $g_2$ where $g_1,g_2\in G_v$ we have that:
\begin{enumerate}
\item either $\gamma$ meets $P^{-1}\mathbb{B}(1,r)$,
\item or there exists $\delta=\{\delta_i\}_{i=0}^m$ a $r_0$-path in $\Gamma(G_v,X_v)$ supported on $P$ and joining $g_1$ with $g_2$.
\end{enumerate}
\end{lem}
\begin{proof}
Take $\overline{\gamma}$ a geodesic interpolation of $\gamma$ in $\Gamma(G,X)$.
Since $\gamma$ is an $r$-path, if $\overline{\gamma}$ meets $P^{-1}$ then $\gamma$ meets $P^{-1}\mathbb{B}(1,r)$ and we are done. Suppose it is not the case, that is, $\overline{\gamma}$ is supported on $P$.
In this case we will find a positive $r_0$-path in $\Gamma(G_v,X_v)$ with the same endpoints than $\gamma$.
We say that an $r_0$-sequence $\delta=\{\delta_i\}_{i=0}^n$ in $\Gamma(G,X)$ is admissible if it is a path supported on $P$ with $\delta_0v=\delta_n v=v$ and satisfying that if $\dist_X(\delta_i,\delta_{i+1})>1$ then $\delta_i$ and $\delta_{i+1}$ are in $G_v$ and the $\dist_{X_v}(\delta_i,\delta_{i+1})\leq r_0$.
Informally speaking, $\delta$ only can jump inside $G_v$ and when it does at most at distance $r_0$ with respect to $X_v$.
We define the defect of $\delta$ as $D(\delta)=\sharp\{i:\delta_i v\neq v\}$.
Notice that $\delta$ is supported in $G_v$ if and only if $D(\delta)=0$.
Also notice that point (2) in the statement can be rephrased as: there exists an admissible path $\delta$ joining $g_1$ with $g_2$ with $D(\delta)=0$.
We will show a procedure to reduce the defect of admissible $r_0$-paths with positive defect, while preserving the endpoints.
Suppose that $\delta$ is admissible with positive defect.
Since $D(\delta)>0$ there must exist $0\leq i<j\leq n$ such that $\delta_i v=\delta_j v=v$ and $\{\delta_k\}_{k=i+1}^{j-1}$ is a $1$-path not meeting $G_v$.
Then, since $X$ has $v$-reductions modulo $E$, we deduce that $\delta_i^{-1}\delta_j$ or $\delta_j^{-1}\delta_i$ belong to $G_e\cap P$ for some $e\in E$.
Suppose without loss of generality that it is the case for $\delta_i^{-1}\delta_j$.
Therefore, by hypothesis we have that there exists a $r_0$-path $\alpha=\{\alpha_k\}_{k=0}^m$ in $\Gamma(G_v,X_v)$ supported on $P\cup\{1_G\}$ and joining $1_G$ with $\delta_i^{-1}\delta_j$.
Define $\beta$ as the concatenation of the admissible $r_0$-paths $\{\delta_k\}_{k=0}^i$ followed by $\delta_i \{\alpha_k\}_{k=0}^m$, and finally followed by $\{\delta_k\}_{k=j}^n$.
It is straightforward to check that $\beta$ is an admissible path with strictly smaller defect.
Repeating this procedure finitely many times we obtain an admissible path with zero defect. This finishes the proof.
\end{proof}
We are ready to state and prove our main Theorem.
\begin{thm}\label{thm:induction}
Suppose that $G\curvearrowright T$ is a co-finite, minimal, general type action of a finitely generated left-orderable group $G$ on a tree $T$. Suppose further that all vertex stabilizers are finitely generated.
Consider $v\in VT$ and $H\in\cP=\{G_e: e\in \text{link}_T(v)\}$, and assume that $G_v$ is Hucha with respect to $H$ and all the groups of $\cP$ are Prieto. Then $G$ is Hucha with respect to the family of cyclic subgroups of $G$ and subgroups of $G$ acting elliptically on $T$.
\end{thm}
\begin{proof}
By Lemma \ref{lem:adapted} we can consider a generating set $X$ and a finite pointed tree $(T_0,v)$ so that $X$ is adapted to $(T_0,v)$.
Consider a positive cone $P$ with corresponding left-order $\prec$ and $r>0$, we will find a geodesic segment $[w_1,w_2]_T$ in $T$ and a subset $S$ included in $P^{-1}$ so that $S$ $r$-disconnects $\cC_{w_1}$ from $\cC_{w_2}$ in $\Gamma(G,X)$.
Then, since $P$ and $r$ are arbitrary we can apply Lemma \ref{lem:disconecting} to conclude that $G$ has the Hucha with respect to cyclic subgroups.
Consider $h\in G$ so that $g\prec h$ for every $g\in\mathbb{B}(1,r)$.
Then, since $h^{-1}(hP^{-1}h^{-1})h=P^{-1}$ we get that $h^{-1}(hP^{-1}h^{-1})\mathbb{B}(1,r)\subseteq P^{-1}$.
Denote $\prec_{\ast}$ the left-order on $G$ with positive cone $P_{\ast}:=hPh^{-1}$.
Then, re-writing we have $$h^{-1}P_{\ast}\mathbb{B}(1,r)\subseteq P^{-1}$$
By definition of an adapted generating set, there exists a finite subset $E\subseteq \text{link}_T(v)$ so that $X$ has $v$-reduction modulo $E$.
Since $E\subseteq\cP$ is finite and the subgroups in $\cP$ are Prieto, there exists $k_0>0$ so that $P_{\ast}\cap G_e$ is $k_0$-connected in $\Gamma(G_v,X_v)$ for every $e\in E$.
On the other hand, since $G_v$ is Hucha respect to $H$, we can find $a_1,a_2\in G_v$ so that $P_{\ast}^{-1}\cap G_v$ $k_0$-disconnects $a_1H$ from $a_2H$ inside $\Gamma(G_v,X_v)$.
Take an edge $e=[v,w]_T$ with $G_e=H$ and denote $a_1e=[v,w_1]_T$ and $a_2e=[v,w_2]_T$.
Consider the geodesic segment $[w_1,w_2]_T$.
We claim that $P_{\ast}\mathbb{B}(1,r)$ $r$-disconnects $\cC_{w_1}$ from $\cC_{w_2}$.
To see this, take $h_i\in \cC_{w_i}$ for $i=1,2$ and an $r$-path ${\gamma}=\{\gamma_i\}_{i=0}^m$ in $\Gamma(G,X)$ joining them.
Consider $\overline{\gamma}=\{\overline{\gamma}_{i}\}_{i=0}^n$ a geodesic interpolation of $\gamma$.
Let $\sigma\colon \{0,\ldots, m\}\to \{0,\ldots,n\}$ such that $\gamma_i=\overline{\gamma}_{\sigma(i)}$.
Since $X$ is adapted to $(T_0,v)$ we have that $\overline{\gamma}_{i+1}T_0\cap\overline{\gamma}_{i}T_0\neq\emptyset$ for $i=0,\ldots,n-1$.
Then, the definition of the $\cC_{w_i}$ imply that there exists $0<i<j<n$ so that $a_1e\in\overline{\gamma}_iT_0$ and $a_2e\in\overline{\gamma}_jT_0$.
Note that this implies that $\overline{\gamma}_i\in a_1H$ and $\overline{\gamma}_j\in a_2H$.
If either $\overline{\gamma}_i$ or $\overline{\gamma}_j$ are in $P_{\ast}$ then $\gamma$ meets $P_{\ast}\mathbb{B}(1,r)$ and we are done.
Let $p$ (resp. $q$) be the smallest (resp. greatest) integer such that $\sigma(p)\geq i$ (resp. $\sigma(q)\leq j$).
Consider the $r$-path $\beta=\{\beta_k\}_{p-1}^{q+1}$ with $\beta_{p-1}=\overline{\gamma}_i$, $\beta_{q+1}=\overline{\gamma}_j$ and $\beta_k=\gamma_k$ for $p\leq k\leq q$.
Note that $\beta$ is an $r$-path from $\overline{\gamma}_i\in a_1H$ to $\overline{\gamma}_j\in a_2H$.
Because all groups of $\cP$ are Prieto, we can apply Lemma \ref{lem:projection} to $P_{\ast}$, $k_0$, the $r$-path $\beta$, we would get that there exist a $k_0$-path in $\Gamma(G_v,X_v)$ supported on $P_{\ast}$ joining joining $a_1H$ with $a_2H$.
But this is absurd by the choice of $a_1$ and $a_2$.
Finally, from the claim we have that, $h^{-1}P_{\ast}\mathbb{B}(1,r)$ $r$-disconnects the sets $\cC_{h^{-1}w_1}$ and $\cC_{h^{-1}w_2}$ associated to the segment $[h^{-1}w_1,h^{-1}w_2]_T$.
Since $h^{-1}P_{\ast}\mathbb{B}(1,r)\subseteq P^{-1}$ the theorem follows.
\end{proof}
\begin{rem}
Note that $F_2\times \mathbb Z$ is the amalgamated free product of two groups isomorphic to $\mathbb Z^2$ along an infinite cyclic subgroup. We know that this group is not Hucha nor Prieto (Remark \ref{rem: F_2x Z}), thus the fact that at least one vertex stabilizer is Hucha is essential in Theorem \ref{thm:induction}.
\end{rem}
\begin{cor}
Let $A$ be Hucha with respect to $C\leqslant A$, where $C$ is a proper Prieto subgroup.
For any group $B$ such that $G=A*_CB$ is left-orderable, $G$ is Hucha with respect to $\{A, B\}$ and the collection of the cyclic subgroups.
\end{cor}
\begin{proof} The proof of this Corollary is analogous to that of Corollary \ref{cor:productolibre}.
\end{proof}
By a result of Bludov and Glass \cite{BludovGlass}, the amalgamation of left-orderable groups over a cyclic subgroup is again left-orderable.
\begin{cor}
Let $A$ be Hucha with respect to $C\leqslant A$ with $C$ cyclic and let $B$ be a left-orderable group.
Then $A*_CB$ is Hucha with respect to $A$ and $B$ and the collection of infinite cyclic subgroups.
\end{cor}
A one-relator group $G$ is called {\it cyclically pinched} if $G=F_1*_{\langle a =b \rangle} F_2$ with $F_1,F_2$ finitely generated free groups, $a\in F_1-\{1\}$ and $b\in F_2-\{1\}$.
\begin{cor}
Cyclically pinched $(\geq 3)$-generated one-relator groups are Hucha with respect to the collection of infinite cyclic subgroups.
\end{cor}
\section{A family of Hucha groups}
\label{sec: limit}
In this section we prove Theorem \ref{thm: main}.
Let $\mathfrak{H}_0$ be the family of non-abelian finitely generated free groups.
For $i>0$, let $\mathfrak{H}_i$ be the closure under free products of the family consisting of finitely generated non-abelian subgroups, of groups of the form $G*_C A$ where $G\in \mathfrak{H}_{i-1}$, $C$ is a cyclic centralizer subgroup of $G$ and $A$ is finitely generated abelian.
Thus, the family $\mathfrak{H}$ of Definition \ref{def: familia H}, is equal to $\cup_{i\geq 0}\mathfrak{H}_i$.
\begin{prop}\label{prop:howie}
If $G\in \mathfrak{H}$ then $G$ is locally indicable (and hence left-orderable).
\end{prop}
\begin{proof}
Since every finitely generated subgroup of a free group is free, the groups in $\mathfrak{H}_0$ are locally indicable.
It is easy to see that local indicability is preserved under free products and taking subgroups.
So it remains to show that if $G\in \mathfrak{H}_n$, $C=\langle c\rangle$ is the centralizer of some $c\in G$ and $A$ is a finitely generated abelian group, then $G*_CA$ is locally indicable.
Let $H,K$ be locally indicable groups and $r\in H*K$ an element acting loxodromically on the Bass-Serre tree of the free product.
Howie \cite{Howie} showed that a one-relator quotient $H*K/\langle \langle r \rangle \rangle$ is again locally indicable if and only if $r\in H*K$ is not a proper power (see \cite[Appendix]{ADL} for an alternative proof).
Since $G*_C A=G*A/\langle\langle ca^{-1}\rangle \rangle$ for some $a\in A-\{1\}$, and the element $ca^{-1}$ acts loxodromically (on the Bass-Serre tree of the free product) and it is not a proper power, we deduce that $G*_CA$ is locally indicable.
\end{proof}
A group $G$ is called {\it CSA} if all its maximal abelian
subgroups are malnormal, that means that if $A$ is a maximal abelian subgroup of $G$ and $g\in G$, then $gHg^{-1}\cap H\neq \{1\}$ implies $g\in H$.
In \cite{MiRe} define the class CSA$^*$ as the CSA groups with no elements of order 2. It is proved that CSA$^*$ is closed under free products \cite[Theorem 4]{MiRe}, and the following construction: if $L$ is CSA$^*$, $C$ is a centralizer of an element of $L$ and $A$ is torsion-free finitely generated abelian, then $L*_CA$ is CSA$^*$ \cite[Theorem 5]{MiRe}.
Since CSA clearly passes to subgroups and free groups are CSA we have.
\begin{lem}\label{lem: CSA}
If $G\in \mathfrak{H}$ then $G$ is CSA.
\end{lem}
Recall that a simplicial $G$-tree is $k$-acylindrical if the fixed point set of any $g\in G$ has diameter at most $k$ (i.e. any set of diameter $>k$ has trivial stabilizer).
\begin{lem}\label{lem: acylindrical}
If $T$ is the Bass-Serre tree of $G=A*_CB$ with $C$ malnormal in $A$ then the action is 2-acylindrical.
\end{lem}
\begin{proof}
Indeed, the stabilizer of an edge $e \in ET$ is a conjugate of $C$ (we can assume by $G$-equivariance that it is $C$).
Let $v$ be the vertex adjacent to $e$ with stabilizer $L$.
All edges adjacent to $v$ have stabilizer $lCl^{-1}$ with $l\in L\setminus C$.
By malnormality of $C$ in $L$, $C\cap lCl^{-1}$ is trivial.
Thus $C$ does not fix any other edge adjacent to $v$ different from $e$.
Now any subset of diameter $\geq 3$, up to $G$-equivariance, contains a path of length $\geq 2$ that has $v$ as an internal vertex and hence it has trivial stabilizer.
\end{proof}
We need the following fact.
\begin{lem}\cite[Lemma 2]{Cohen}\label{lem:fg-vertex groups}
Let $G$ be a finitely generated group acting co-finitely on a tree $T$.
If the stabilizers of edges adjacent to $v\in VT$ are finitely generated, then the stabilizer of $v$ is finitely generated.
\end{lem}
We can prove now our main theorem.
\begin{proof}[Proof of Theorem \ref{thm: main}]
We will prove by induction that groups in $\cup_{i=0}^n\mathfrak{H}_i$ are Hucha with respect to the family of cyclic centralizers. The base case of induction ($n=0$) is a consequence of Proposition \ref{prop:F2hucha}.
Thus assume that the induction hypothesis hold for $n>0$. Have to prove the case $n+1$.
Let $L\in \mathfrak{H}_{n}$, $C\leqslant L$ a subgroup of a cyclic centralizer and $A$ a finitely generated abelian group.
Without loss of generality we assume that $C$ is a proper subgroup of $A$ (since if not $L*_CA=L$).
Let $T$ be the associated Bass-Serre tree to $L*_CA$.
By Lemma \ref{lem: CSA}, $C$ is malnormal in $L$, which implies (Lemma \ref{lem: acylindrical}) that the action on the $T$ is $2$-acylindrical.
In particular, if $H$ is a subgroup $L*_CA$ with elements acting loxodromically on $T$ and such that the minimal $H$-tree is homeomorphic to $\mathbb{R}$, then it must be virtually cyclic (see for example \cite[Theorem 1.1.]{Osin}), and since $L*_C A$ is torsion-free (Proposition \ref{prop:howie}), $H$ is must be cyclic.
Let $G\leqslant L*_C A$ be a non-abelian finitely generated subgroup.
We have to show that $G$ is Hucha with respect to its cyclic centralizers.
If $G$ is, up to conjugation, a subgroup of $L$.
Then, by induction hypothesis $G$ is Hucha with respect to its cyclic centralizers.
So we restrict to the case where $G$ is not contained, up to conjugation, in $L$.
Note that, since $G$ is non-abelian, it is not contained, up conjugation, in $A$.
Thus the action of $G$ on the Bass-Serre tree of $L*_CA$ has no-global fixed point.
By \cite[Proposition I.4.11]{DicksDunwoody} $G$ contains elements acting loxodromically on $T$.
Consider $T'\subseteq T$ the minimal $G$-invariant subtree of $T$.
This is the subtree consisting on the union of the axis of the elements of $G$ acting loxodromically.
Since $G$ is finitely generated, $G\backslash T'$ is finite
(see \cite[Proposition I.4.13]{DicksDunwoody}).
By the previous discussion, if $T'$ has two ends, $G$ must be cyclic, contradicting that $G$ is non-abelian.
In fact, since the action of $G$ of $T'$ is acylindicral, any element acting loxodromically on $T$ has cyclic stabilizer.
Finally, since $L*_C A$ does not fix any end of $T$, $G$ does not fix any end of $T'$.
Thus, the action of $G$ on $T'$ is of general type and co-compact, with edge stabilizers that are trivial or cyclic, and vertex stabilizers that are isomorphic to subgroups of $A$ or subgroups of $L$.
Moreover, by Lemma \ref{lem:fg-vertex groups} the vertex groups are finitely generated.
If $G$ acts on $T'$ with trivial vertex stabilizers, then $G$ is free, and thus $G\in \mathfrak{H}_0$ and there is nothing to prove.
Suppose that $G$-stabilizer of all vertices of $T'$ are non Hucha.
Thus, all $G$-stabilizers of all vertices of $T'$ must be abelian, since by induction, all non Hucha finitely generated subgroups of $L$ are the abelian ones.
If some edge has trivial stabilizer, by Proposition \ref{prop: free product Hucha}, $G$ is Hucha with respect to the subgroups acting loxodromically on $T'$ and the subgroups of the vertex stabilizers. In particular it is Hucha with respect to the cyclic centralizers.
Thus we can assume that all edge stabilizer are non-trivial and cyclic.
Let $e$ be an edge of $T'$ adjacent to $u$, and suppose that the stabilizer of $u$ is a conjugate of a subgroup of $L$.
Since $G_u$ is abelian and $G_e\leqslant G_u$ is a centralizer of some element (restricted to $G_u$) we get that $G_u=G_e$.
We can $G$-equivariantly collapse the edge $e$ and we will still have an action on a tree with the previous properties, and we could repeat this argument and continue $G$-equivariantly collapsing edges, obtaining that $G$ is abelian, a contradiction.
Thus the only case remaining is when there is some $v\in T'$ with $G_v$ an Hucha group.
Thus $G_v$ must be (up to conjugation) a finitely generated subgroup of $L$ and thus, by induction, it is Hucha with respect to all its cyclic centralizers.
Since the stabilizers of edges adjacent to $v$ are either trivial or Prieto, we can use Theorem \ref{thm:induction} to conclude that $G$ is Hucha with respect to all its cyclic centralizers.
\end{proof}
\paragraph{Application to Limit groups.}
There are several equivalent definitions in the literature for {\it limit groups.}
See for example \cite{ChampGuir} for a nice survey. The quickest definition is that $G$ is a limit group if it is a finitely generated fully residually free group, meaning that for any finite set $S\subseteq G$ there is a homomorphism $\phi \colon G\to F$ where $F$ is a free group and $\phi$ is injective restricted to $S$.
However, to show that limit groups belong to $\mathfrak{H}$ we need a different characterization.
Let $H$ be a group and $Z$ the centralizer of some non-trivial element of $H$.
A {\it free extension of $H$ by the centralizer $Z$} is a group of the form $H*_Z(Z\times A)$ for some finitely generated free abelian group $A$.
Let $\textrm{ICE}_0$ denote the class of finitely generated free groups. For $i>0$, let $\textrm{ICE}_i$ denote the groups that are free extensions by cyclic centralizers of groups in $\textrm{ICE}_{i-1}$. Finally let $\textrm{ICE}$ denote the union of $\textrm{ICE}_i$ $i\in \mathbb{Z}_\geq 0$.
An important and very useful result for us is the following.
\begin{thm}\cite[Thm 4.]{KM98}.\label{thm:KM} The following are equivalent
\begin{enumerate}
\item $G$ is a limit group
\item $G$ is a finitely generated subgroup of a group in $\textrm{ICE}$
\end{enumerate}
\end{thm}
Clearly, in view of Theorem \ref{thm:KM} we have that non-abelian limit groups are in $\mathfrak{H}$.
\paragraph{Application to free $\mathbb Q$-groups.}
Recall that $H$ is a $\mathbb Q$-group if for every $h\in H$ and all $n\in \mathbb N$ there is a unique $g\in H$ such that $g^n=h$.
This allows to define an action of $\mathbb Q$ on $H$, and we denote the image of the action of $\alpha\in \mathbb Q$ on $h\in H$ by $h^\alpha$.
If $\alpha=p/q$ then $h^\alpha$ is $g^p$ where $g$ is the unique element such that $g^q=h$.
It is easy to check that indeed this in an action.
A $\mathbb Q$-group $G^\mathbb Q$ together with an homomorphism $\phi \colon G \to G^\mathbb Q$ is called the {\it tensor $\mathbb Q$-completion of the group $G$} if it satisfy the following universal property:
for any $\mathbb Q$-group $H$ and a homomorphism $f \colon G\to H$
there exists a unique $\mathbb Q$-homomorphism $\psi \colon G^\mathbb Q \to H$ (a homomorphism that commutes with the $\mathbb Q$-action) such that $f=\psi \circ \phi$.
$\mathbb Q$-groups were introduce by G. Baumslag in \cite{GBaumslag} where he showed that $\mathbb Q$-tensor completions exists and are unique.
A free $\mathbb{Q}$-groups is the tensor $\mathbb Q$-completion of a free group $F$.
Crucially, Baumslag also showed that $F^\mathbb Q$ can be obtained from a free group $F$ by interatively adding roots to $F$ i.e.
there is chain
$$F=F_0<F_1<F_2< \dots $$
such that $F_{n+1}$ is obtained from $F_{n}$ by adding a some $q$-th root of an element generating its own centralizer i.e. if $c\in F_n$ and $\langle c \rangle =\{g\in F_n \colon gc=cg\}$ then $F_{n+1}=F_n*_{c=t^q}\langle t \mid \;\rangle$.
There are some subtleties about how to construct this ascending chain (see \cite[Section 8]{MiRe}) however, clearly any finitely generated subgroup of a $\mathbb Q$-free group is a subgroup of a finite iterative addition of roots to a free group $F$ and hence it lies in $\mathfrak{H}$ \footnote{Since finitely generated subgroups of free $\mathbb Q$-groups are in $\mathfrak{H}$, Proposition \ref{prop:howie} then implies that free $\mathbb Q$-groups are left-orderable. In fact, G Baumslag ask whether these groups are residually torsion free nilpotent a fact recently proved by A. Jaikin \cite{Jaikin}. In particular, free $\mathbb Q$-groups are bi-orderable, that is they admit left-orders which are also right-invariant. }.
\begin{ex}
An example of a group in $\mathfrak{H}$ is $G=\langle a,b,c\mid a^2b^2c^2\rangle$, the fundamental group of the connected sum of 3-projective planes.
It follows from a theorem of Lyndon \cite{Lyndon} that the equation $a^2b^2c^2=1$ in a free group implies that $a,b,c$ commute.
Hence $G$ is not a limit group.
On the other hand, $G$ is obtained from the free group $\langle a,b\mid \;\rangle$, by adding a square root to $a^2b^2$ and hence it is a finitely generated subgroup of a free $\mathbb Q$-group.
\end{ex}
\section{Regular sets are coarsely connected}
\label{sec: regular}
Let $X$ be a set.
Recall that $X^*$ denotes the {\it free monoid generated by $X$} and consists on the set of finite words on $X$ together with concatenation. If $X\subseteq G$ generates the group $G$, then there is a natural monoid epimorphism $\mathrm{ev}\colon X^*\to G$, called {\it evaluation map}, that is induced by viewing each element of $x\in X$ as element of $G$.
\begin{defi}\label{def:regular language}
A {\it finite state automaton} is a 5-tuple $(\cS,\cA,s_0,X,\tau)$, where $\cS$ is a set whose elements are called {\it states}, $\cA$ is a subset of $\cS$ of whose states are called {\it accepting states}, a distinguished element $s_0\in \cS$ called {\it initial state}, a finite set $X$ called the {\it input alphabet} and a function $\tau\colon \cS\times X\to \cS$ called {\it the transition function}. We extend $\tau$ to a function $\tau\colon \cS\times X^*\to \cS$ recursively, by setting $\tau(s,wx)=\tau(\tau(s,w),x)$ where $w\in X^*$, $x\in X$ and $s\in \cS$.
A {\it language $\cL$ over $X$} is a subset of $X^*$, and $\cL$ is {\it regular} if there is a finite state automaton $(\cS,\cA,s_0,X,\tau)$
such that $$\cL=\{w\in X^* \mid \tau(s_0,w)\in \cA\}.$$
\end{defi}
\begin{prop}\label{prop:r-connected}
Let $G$ be a group generated by a finite set $X$.
If $\mathcal{L}\subseteq X^*$ is a regular language, then $\mathrm{ev}(\mathcal{L})$ is an $r$-connected subset of $\Gamma(G,X)$ for some $r$.
\end{prop}
\begin{proof}
Assume that $\cL$ is a regular language over $X$ and $(\cS,\cA,s_0,X,\tau)$ is a finite state automaton accepting $\cL$.
We can view the automaton as a directed graph with vertices $\cS$ and edges $\cS\times X$ where $(s,x)$ is an edge from $s$ to $\tau(s,x)$.
We call this graph, the {\it directed automaton graph}.
Let $r=|\cS|$.
Let $w\equiv x_1x_2\dots x_n\in \cL$.
We will show that we can go from $1$ to $\mathrm{ev}(w)$ with an $(2r+1)$-path $\{v_i\}_{i=0}^n$ where $v_i\in \mathrm{ev}(\cL)$.
Indeed, let $w_i$ be the prefix of length $i$ of $w$, i.e. $w_i=x_1x_2\dots x_i$.
It is enough to find $v_i\in \mathrm{ev}(\cL)$ such that $\dist(\mathrm{ev}(w_i), v_i)\leq r$.
Note that $w_i$ is a prefix of some word in $\cL$, so we take a word of minimal length $w'_i$ with the property that $w_i w_i'\in \cL$.
Now, $w_i'$ gives a path in directed automaton graph from $\tau(s_0, w_i)$ to some vertex in $\cA$.
By minimality, this path can not repeat a vertex and thus $\ell(w_i')\leq r$.
Note that $w_i'$ will be the label of some path in the Caley graph from $\mathrm{ev}(w_i)$ to $\mathrm{ev}(w_iw_i')$ of length at most $r$.
Thus we can take $v_i$ to be $\mathrm{ev}(w_iw_i')$.
\end{proof}
\begin{defi}
A positive cone $P$ of a finitely generated group $G$ is {\it regular} if there is a finite generating set $X$ of $G$ and a regular language $\cL$ over $X$ such that $P=\mathrm{ev}(\cL)$.
\end{defi}
\begin{rem}
Finitely generated (as subsemigroups) positive cones are regular.
\end{rem}
\begin{cor}\label{cor: hucha not regular}
If $G$ is Hucha, then $G$ has no regular positive cone.
\end{cor}
\begin{proof}
Let $P$ be a positive cone for $G$ and $X$ a finite generating set of $G$.
Suppose that there is a regular language $\mathcal{L}\subseteq X^*$ such that the evaluation of $\mathcal{L}$ in $G$ is $P$.
By Proposition \ref{prop:r-connected}, there is some $r>0$ such that $P$ is an $r$-connected subset of $\Gamma(G,X)$.
This contradicts Proposition \ref{prop: Hucha are not coarsely connected}.
\end{proof}
\begin{rem}\label{rem: combing}
Let $G$ be a group and $X$ a generating set.
If $\cL$ is a regular language over $X$ such that $\mathrm{ev}(\cL)$ is a positive cone $P$,
then the set of paths in $\Gamma(G,X)$ starting at $1_G$ and with label $w\in \cL$ is a combing of $P$ and by Proposition \ref{prop:r-connected} this combing is supported in the $r$-neighbourhood of $P$ for some $r$.
\end{rem}
Suppose that $X$ is a finite generating set of a group $G$.
A language $\cL\subseteq X^*$ is {\it quasi-geodesic} if there are $\lambda\geq 1$ and $c\geq 0$ such that each word $w\in \cL$ labels a $(\lambda,c)$-quasi-geodesic path.
In \cite[Question 8.7]{calegari} Calegari depicted an argument showing that on the fundamental group of an hyperbolic 3-manifold, no positive cone language is both regular and quasigeodesic. Recently H. L. Su \cite{Su} showed, using Calegari's ideas, that in fact no positive in the (much larger) class of acylindrically hyperbolic group can be described by a regular and quasigeodesic language. With our tools we can easily recover Su's theorem in restriction to the class of $\delta$-hyperbolic groups.
\begin{cor}
Non-elementary hyperbolic groups do not admit regular quasi-geodesic positive cones.
\end{cor}
\begin{proof}
By Remark \ref{rem: combing}, if $P$ is a regular quasi-geodesic positive cone, then it admits a quasi-geodesic combing supported on an $r$-neighbourhood of $P$ for some $r$. This contradicts Theorem \ref{thm combing}
\end{proof}
\begin{small}
|
1,116,691,499,243 | arxiv | \section{INTRODUCTION}
In conventional chemical evolution models, stars are assumed to form
from well-mixed gas clouds at a rate proportional to some power of the
gas density (Schmidt 1959), thus inheriting the abundance pattern of
the gas at that time (e.g., Tinsley 1980). This simplified treatment
of the complex star formation process has been remarkably successful
in accounting for the general features of the chemical compositions of
nearby stars and H$\;${\scriptsize II} regions in a consistent manner
(e.g., Pagel \& Patchett 1975; Matteucci \& Greggio 1986; Yoshii,
Tsujimoto, \& Nomoto 1996). However, Shigeyama \& Tsujimoto (1998,
hereafter ST98) and Tsujimoto \& Shigeyama (1998, hereafter TS98)
argued that these conventional models cannot be applied to the early
stage of the Galactic halo, because observed abundance patterns of
extremely metal-deficient stars in the range of $-4<$[Fe/H]$<-2.5$
(McWilliam et al.~1995; Ryan, Norris, \& Beers 1996) are incompatible
with those predicted by these simple formulations. (see also Audouze
\& Silk 1995).
As an alternative, they proposed a scenario in which stars are born
from each supernova remnant (SNR), following a sweeping up of the
interstellar gas and formation of a dense shell immediately behind the
radiative shock front. The dense shell is dynamically overstable
(Vishniac 1983; Ryu \& Vishniac 1987), so that it is broken into a few
thousand fragments, in each of which the self-gravity is unimportant.
According to Nakano (1998) it is likely that, for at least some of
these fragments, the ambient pressure outside the fragments exceeds
the critical value above which no equilibrium state exists. Such
fragments will dynamically contract and eventually form stars.
This hypothesis sheds light on the entire process of early-epoch star
formation. Galactic chemical evolution is envisioned as a successive
sequence of supernova (SN) explosions, shell formation, and resulting
star formation. In this picture, the abundance pattern of each star
is set not only by the heavy elements ejected from the SN explosion
which directly preceeds local star formation, but also incorporates
those elements already present in the interstellar gas which are swept
up by the SNR. The predicted stellar abundance patterns are thus
different from those of the gas at the time when stars form. This
difference can be quite large in the early stage of Galactic
evolution, when the metallicity in the gas is very low. The abundance
ratios of low-metallicity stars are predicted to exhibit a large
star-to-star scatter, depending in detail on the abundance patterns of
SN ejecta with different progenitor
masses.
By incorporation of a sequence of SN-induced star formation into the
enrichment process of heavy elements, we describe the evolution of
stellar abundances in the Galactic halo characterized by the
inhomogeneous mixing of chemical compositions in dense shells and the
ambient medium.
\section{CHEMICAL EVOLUTION IN A CLOUD}
We assume that the star-forming process is confined to separate
clouds, of mass $M_c$, which make up the entire halo. In this section
we present a formulation to describe the chemical evolution in a cloud
that is initially composed of metal-free Population III stars (Pop
III) and gas which has yet to form stars. The mass fraction of such
stars in our model is a parameter hereafter denoted by $x_{\rm III}$,
the massive ones of which eventually explode as Pop III SNe to
initiate chemical evolution.
All stars of subsequent generations are assumed to form in SNR shells.
The mass fraction of each shell which turns into stars is assumed to
be constant, and is denoted by $\epsilon$. Heavy elements ejected
from an SN are assumed to be trapped and well-mixed within the SNR
shell. Some of these elements go into stars of the next generation,
and the rest is left in the gas that will be mixed with the ambient
medium. The above process will repeat with increasing metallicity
until SNRs can no longer sweep up enough gas to form shells. No stars
will form from SNRs after this happens, and the process terminates.
When stars form as above, the star formation rate (SFR)
${\dot M}_\ast(t)$ induced by Pop III SNe at time $t$ measured
from the birth of their progenitor stars is given by
\begin{equation}
{\dot M}_\ast (t)=\epsilon M_{\rm sh}({m_t {\rm ,}\, t}) {\phi
({m_t})\over m_t}
\left| {dm_t\over dt} \right| x_{\rm III}M_c {\rm ,}
\end{equation}
and the SFR for later generations can be expressed as
\begin{equation}
{\dot M}_\ast (t)=\int_{\max(m_t,\, m_{{\rm SN}, l})}^{m_u}
\hspace{-1.5cm}dm \epsilon M_{\rm sh}({m {\rm ,}\, t})
{\phi(m)\over m }{\dot M}_\ast ({t-\tau (m)}){\rm ,}
\end{equation}
where $\tau(m)$ denotes the lifetime of a star of mass $m$, and $m_t$
is the stellar mass for which $\tau(m)=t$. The initial mass function
(IMF) $\phi (m)$ used here is a Salpeter one, with a slope index of
$-1.35$. An upper mass limit of stars is assumed to be
$m_{u}=50\,M_\odot$; a lower mass limit of $m_{l}=0.05\,M_\odot$
(Tsujimoto et al.~1997) is also assumed. The lower mass limit for
stars that explode as SNe is taken to be $m_{{\rm SN}, l}=10\,
M_\odot$. The mass of the shell $M_{\rm sh}(m,t)$ formed at time $t$
from an SN with progenitor mass $m$ is a sum of the mass of the SN
ejecta $M_{\rm ej}(m)$ and the mass of the swept-up gas $M_{\rm
sw}(m,t)$;
\begin{equation}
\label{eqn:masssh}
M_{\rm sh}({m{\rm ,}\, t})=M_{\rm ej}(m)+M_{\rm sw}({m{\rm ,}\,
t}){\rm ,}
\end{equation}
\noindent with
\begin{equation}
\label{eqn:masssw}
M_{\rm sw}({m{\rm ,}\, t})=\rho _{\rm g}(t){R^3 }
_{\rm SN}\left(E\left(m\right) ,\rho _{\rm g}(t)\right),
\end{equation}
where $\rho_{\rm g}(t)$ is the density of the interstellar gas and
$R_{\rm SN}\left(E(m),\rho_{\rm g}(t)\right)$ is the maximum radius of
the SNR shell. $M_{\rm sw}(m,t)$ is insensitive to time $t$ because
$M_{\rm sw}$ is proportional to $\rho_{\rm g}(t)^{-0.06}$ (see ST98).
The SN explosion energy is assumed to be $E(m)=10^{51}$ ergs,
irrespective of stellar mass $m$. Thus a constant value of $M_{\rm
sw}(m,t) = 6.5\times10^4\,M_\odot$ will be
used below.
Utilizing the SFR introduced in equations (1) and (2), the mass
$M_{\rm g}(t)$ and the metallicity $Z_{\rm g}(t)$ of the gas will change
with time according to the formulae:
\begin{equation}
\label{eqn:chevg}
\hspace{-12pt}\displaystyle{dM_{\rm g} \over dt}=-{\dot M}_\ast
(t)+\int_{\max ({m_t{\rm ,}\, m_l})}^{m_{u}}\hspace{-1.2cm}
dmM_{\rm ej}(m){\phi
(m) \over m}{\dot M}_\ast ({t-\tau (m)}) {\rm ,}
\end{equation}
\noindent and
\begin{displaymath}
\displaystyle{d({z_{\rm g}M}_{\rm g}) \over dt}
=-\int_{\max ({m_t{\rm ,}\, m_{{\rm SN}, \,l}})}^{m_{u}}
\hspace{-1.8cm}dm
Z_\ast ({m {\rm ,}\, t})\epsilon M_{\rm sh}({m {\rm ,}\, t})
{\phi (m)\over m }
\end{displaymath}
\begin{displaymath}
\times{\dot M}_\ast ({t-\tau (m)})+\int_{\max
({m_t{\rm ,}\, m_l})}^{m_{u}}\hspace{-1.2cm}
dm\left(M_{\rm ej}(m)-M_{\rm z}(m) \right){\phi (m)
\over m}
\end{displaymath}
\begin{displaymath}
\times\int_{\max ({m_{t-\tau (m)}{\rm ,}\, m_{{\rm SN},
\,l}})}^{m_u}\hspace{-2.2cm}
dm^\prime Z_\ast ({m^\prime
{\rm ,}\, t-\tau (m)})\epsilon M_{\rm sh}({m^\prime {\rm ,}\, t-\tau
(m)}){\phi ({m^\prime }) \over m^\prime }
\end{displaymath}
\begin{displaymath}
\times{\dot M}_\ast ({t-\tau(m)-\tau ({m^\prime })})
+\int_{\max ({m_t{\rm ,}\, m_l})}^{m_{u}}\hspace{-1.2cm}
dmM_{\rm z}(m){\phi (m)\over m}
\end{displaymath}
\begin{equation}
\label{eqn:chevm}
\hspace{-12pt}\times\int_{\max ({m_{t-\tau (m)}{\rm ,}\,
m_{{\rm SN}, l}})}^{m_{u}}\hspace{-2.7cm}
dm^\prime \epsilon M_{\rm sh}({m^\prime
{\rm ,}\, t-\tau(m)}){\phi ({m^\prime }) \over m^\prime }{\dot
M}_\ast({t-\tau(m)-\tau({m^\prime })}),
\end{equation}
where $M_{\rm z}(m)$ is the mass of all synthesized heavy elements
ejected from a star with mass $m$, and $Z_\ast(m,\, t)$ is the metallicity
of stars born at time $t$ from an SNR shell with progenitor mass $m$, which
is defined as
\begin{equation}
\label{eqn:massms}
Z_\ast ({m{\rm ,}\, t})={M_{\rm
z}(m)+Z_{\rm g}(t)M_{\rm sw}({m{\rm ,}\, t}) \over M_{\rm
ej}(m)+M_{\rm sw}({m{\rm ,}\, t})}.
\end{equation}
Since $M_\ast(0)=x_{\rm III}M_c$ and $M_{\rm g}(0)=(1-x_{\rm III})M_c$
at $t=0$, after specifying $x_{\rm III}$ and $M_c$ and setting $Z_{\rm
g}(0)=0$, equations (5) and (6) are then
integrated with (1)-(4), and (7).
Star formation stops when the total mass involved in SNR shells
reaches the total mass of gas, i.e.,
\begin{equation}\label{eqn:term}
\hspace{-12pt}\int_{t-\Delta t}^t\hspace{-15pt}dt'
\int_{\max({m_{t'}{\rm ,}\, m_{{\rm SN{\rm ,}}\,
l}})}^{m_{u}}\hspace{-60pt}dmM_{\rm sh}({m{\rm ,}\, t'})
{\phi (m) \over m}{\dot M}_\ast({t'-\tau ({m})})
= M_{\rm g}(t),
\end{equation}
where $\Delta t$ is the time for the turbulent motion in the ambient
medium to completely destroy the shape of an SNR shell. When this
condition is satisfied, the cloud is composed of stars, hot tenuous
gas with negligible mass, and dense blobs of gas (an SNR shell would
be broken into blobs) which fill in a negligible volume of the cloud.
In the following calculation $\Delta t = 3\times 10^6$ yr is adopted
which corresponds to the crossing time of the turbulent flow over an
SNR.
The heavy-element yields from SNe are estimated directly from the
observed abundances in metal-poor stars, as opposed to the presently
uncertain predictions of theoretical supernovae models, using the
procedure proposed by ST98 and TS98. We note that the europium yield
is estimated from the data of Ryan et al.~(1996).
\section{RESULTS}
\begin{figure}[ht]
\begin{center}
\leavevmode
\epsfxsize=\columnwidth\epsfbox{fig1.ps}\hfil
\end{center}
\caption{The relation between the duration $\Delta T$ of star
formation in a cloud and a probability $p_{\rm III}$ of observing Pop
III stars as a fraction of their number among all the long-lived stars
that have ever formed with $m<1 M_\odot$. The dashed lines denote the
strict lower bound of $p_{\rm III}$ for each mass $M_c$ of a cloud.}
\end{figure}
\begin{figure}[ht]
\begin{center}
\leavevmode
\epsfxsize=\columnwidth\epsfbox{fig2.ps}\hfil
\end{center}
\caption{The frequency distribution of the long-lived halo field stars
for [Fe/H] $<-1$ against the iron abundance, compared with the
observations of Ryan \& Norris (1991). The observational data for
[Fe/H]$>-1$ are not shown because these data do not correspond to the
number pure halo stars due to an enormous contamination by disk stars
(see Ryan \& Norris 1991). The model curves have been convolved with
Gaussians having $\sigma$=0.3 dex (solid line) and $\sigma$=0.15 dex
(dotted line), respectively.}
\end{figure}
\begin{figure}[ht]
\begin{center}
\leavevmode
\epsfxsize=\columnwidth\epsfbox{fig3.ps}\hfil
\end{center}
\caption{The predicted [Fe/H] distribution functions of the
long-lived stars at different ages of $4\times10^8, 8\times10^8,
1.2\times10^9$ yrs, in the age-[Fe/H] plane. The total number of stars
in each case is normalized to $100$. The dashed line denotes the
age-metallicity relation of the gas.}
\end{figure}
\begin{figure}[ht]
\begin{center}
\leavevmode
\epsfxsize=\columnwidth\epsfbox{eufe.eps}\hfil
\end{center}
\caption{The color-coded frequency distribution of the long-lived
stars in the [Eu/Fe]$-$[Fe/H] plane convolved with a Gaussian having
$\sigma$=0.2 dex for [Eu/Fe] and $\sigma$=0.15 dex for [Fe/H], while
there is no convolution for the inset. The dashed line in the inset
denotes the [Eu/Fe]$-$[Fe/H] relation for the gas. The symbols
represent the data taken from various references (open squares:
McWilliam et al.~1995; filled squares: Ryan et al.~1996; plus signs:
Luck \& Bond 1985; open triangles: Gilroy et al.~1988; crosses: Magain
1989).}
\end{figure}
From the formulation in the preceding section, we calculate the
chemical evolution for Galactic halo stars. The free parameters in
our model are the mass fraction $x_{\rm III}$ of metal-free Pop III
stars initially formed in each cloud and the mass fraction $\epsilon$
of stars formed in a dense shell of each SNR. The values of these
parameters are chosen to reproduce the observed [Fe/H] distribution
function of halo field stars for [Fe/H]$<-1$. If $x_{\rm III}$ is
larger than about $1\times10^{-2}$, the total gas swept up by the
first SNRs exceeds the entire amount of available gas, and thus star
formation stops at the first or second generation. In order for the
enrichment process to continue as a sequence of SN-induced star
formation, $x_{\rm III}$ must be less than about $1\times10^{-2}$. In
addition, $\epsilon$ must be strictly confined within a narrow range
so that nearly one (slightly larger than one) massive star is born
from each SNR. This might imply that star formation is suppressed and
self-regulated by the photoionization from an OB star. A value of
$\epsilon=4.3\times10^{-3}$ is found to give the best fit to the
observed [Fe/H] distribution function for various values of $x_{\rm
III} (<1\times10^{-2})$. If $\epsilon$ slightly exceeds this critical
value, star formation will soon stop with little enrichment by heavy
elements. On the other hand, if $\epsilon$ is taken slightly below
its critical value, star formation will proceed until most of the gas
is used up. In such a scenario, the majority of halo stars become too
metal-rich to be consistent with observations.
The duration $\Delta T$ of star formation in a cloud is related to
$x_{\rm III}$ in the sense that a smaller $x_{\rm III}$ gives a longer
$\Delta T$. Here we define a probability $p_{\rm III}$ of observing
Pop III stars as a fraction of their number among all of the
long-lived stars that have ever formed with $m<1 M_\odot$. The
probability $p_{\rm III}$ decreases with increasing $\Delta T$ -- this
relation is shown by the thick line in Figure 1. Each value of
$p_{\rm III}$ corresponds to about ten times $x_{\rm III}$, because
about 10 \% of $M_{\rm g}(0)$ has been converted to stars in the end.
We note that the most metal-poor halo stars exhibit abundance patterns
of genuine SN II origin, showing the overabundance of
$\alpha$-elements relative to iron Fe (e.g., Wheeler, Sneden, \&
Truran 1989), which means that they were born before SNe Ia start to
contaminate the gas with their products. If the lower bound for the
lifetime of SN Ia is set at 1 Gyr (Yoshii, Tsujimoto, \& Kawara 1998),
the duration $\Delta T$ has an upper bound, and the probability
$p_{\rm III}$ should be larger than $10^{-4}$. However, such an
argument does not hold if relatively few SNe Ia occur at [Fe/H]$<-1$,
as argued by Kobayashi et al.~(1998).
One absolute condition under which the SN-induced star formation
proceeds in a cloud is that at least one massive star of Pop III must
initially explode as an SN. Therefore, for each mass $M_c$ of a cloud
a lower bound of $p_{\rm III}$ exists, and is indicated in Figure 1 by
the vertical dashed line. If $M_c$ is in a range of several $10^{6-7}
M_\odot$, which is inferred from the mass scale of the Galactic
globular clusters, one Pop III star might be observed in a complete
sample of $10^{3-4}$ halo stars. This estimate depends critically on
the assumed IMF for Pop III stars. For instance, if we adopt the
theoretical IMF for metal-free stars proposed by Yoshii \& Saio
(1986), the expected $p_{\rm III}$ is reduced by a factor of $15-45$.
Figure 2 shows the predicted stellar [Fe/H] distribution function for
$x_{\rm III}=10^{-6}$, as compared with the data obtained by Ryan \&
Norris (1991). The result has been convolved with a Gaussian of
$\sigma=0.15$ dex (dotted line) and $\sigma=0.3$ dex (solid line)
because measurement errors in [Fe/H] lie between these values (Ryan \&
Norris 1991). Rather good agreement with the data is obtained for
[Fe/H]$<-1$ using the Salpeter IMF, without decreasing the
heavy-element yield by some manipulation such as mass loss from the
halo (Hartwick 1976; Searle
\& Zinn 1978; Bond 1981; Laird et al.~1988; Ryan \& Norris 1991).
The termination of star formation at the gas metallicity of
[Fe/H]$\sim -1.5$ produces the peak of stellar frequency at
[Fe/H]$\sim -1.6$ as observed. Only about 10\% of the initial gas has
been converted to halo stars, and the rest is left as the remaining
gas which may fall onto the disk to be consumed in formation of disk
stars.
Figure 3 shows the predicted [Fe/H] distribution function of stars
formed at a given age in the age$-$[Fe/H] plane. Three distribution
functions shown at different ages of $4\times10^8$, $8\times10^8$,
$1.2\times10^9$ yrs, normalized in such a way that the total number of
stars is 100. The dashed line denotes the age-metallicity relation of
the gas. It is important to note from this figure that the stellar
metallicity {\it does not} correspond to a unique age, but that the
scatter in stellar [Fe/H] among stars formed at a given age
progressively diminishes with time. As can be appreciated from
inspection of the figure, some stars with abundance [Fe/H]$ > -3.0$
will form at essentially the same time as stars with abundances much
lower than this value.
Our primary concern is whether the observed stellar elemental
abundance patterns in the most metal-deficient halo stars can be
predicted by our model. The abundance ratio [Eu/Fe] is a suitable
probe for this purpose, because this ratio in extremely
metal-deficient stars (McWilliam et al.~1995; Ryan et al.~1996) spans
over 2 dex, far exceeding the measurement errors. Figure 4 shows the
color-coded frequency distribution of stars in the [Eu/Fe]$-$[Fe/H]
plane, normalized to unity when integrated over the entire area (see
the color bar for the scale). In order to compare with the data, the
frequency distribution has been convolved with a Gaussian with
$\sigma=0.2$ dex for [Eu/Fe] and $\sigma=0.15$ dex for [Fe/H]. For
illustrative purposes the frequency distribution without convolution
is also shown in the inset. The predicted [Eu/Fe]$-$[Fe/H] relation
for the gas is shown by the dashed line. The predicted scatter in
[Eu/Fe] becomes smaller toward larger [Fe/H] and converges to a
plateau at $-2<$[Fe/H]$<-1$. This tendency, which is also seen in
other elements, is understood by considering that the major
contribution to stellar metallicity has switched from the ejecta of
SNe to the metallicity in the interstellar gas swept up by SNRs (see
eq.[7]) in this abundance interval.
\section{CONCLUSION}
We have presented a new model for Galactic chemical evolution based on
the SN-induced star formation hypothesis.
An overview of our model
applied to the Galactic halo is as follows -- (1) The metal-free Pop
III stars form by some (as yet unspecified) mechanism in primordial
gas clouds of the Galactic halo. (2) The most massive stars among
them explode as Type II SNe, which trigger a series of star formation
events. (3) Star formation terminates when dense shells cannot be
formed from SNRs. (4) About 90 \% of the cloud mass remains unused in
star formation and may fall onto the still-forming Galactic disk.
Our model predicts that stellar abundance patterns are different from
those of the interstellar gas at the time when stars form.
Calculations presented here have shown that the observed features of a
large scatter in the europium to iron abundance ratios and the
metallicity distribution function with $\langle {\rm [Fe/H]}
\rangle\sim -1.6$ for halo field stars are naturally reproduced. A
particularly attractive feature of our model is that agreement with
this mean stellar metallicity is achieved without decreasing the
value of heavy-element yields from SNe.
Our model strongly suggests the existence of metal-free Pop III stars,
because it is the first SN explosions among them that trigger a series
of SN-induced star formation. Provided that star formation is
confined to separate clouds with mass of $10^{6-7} M_\odot$, and that
metal-free stars have a Salpeter IMF, it is estimated from the model
that one Pop III star would be found in a sample of $10^{3-4}$ halo
stars. Ongoing surveys which seek to increase the numbers of stars
known at the lowest metal abundances (e.g., Beers, Preston, \&
Shectman 1992; Christlieb et al.~1999)
provide the best hope for the eventual detection of this very {\it
first} generation of stars. However, the probability of detecting
them would be smaller if the primordial IMF is enhanced on the massive
part compared to the Salpeter IMF (Yoshii \& Saio 1986) and/or if the
accretion of interstellar gas contaminates the surface of metal-free
stars (Yoshii 1981). Both of these possible complications are
suitable for direct tests via high-resolution observations of a
sufficiently large sample of dwarf and giant stars with [Fe/H]$ <
-2.0$, which already exists (see Beers 1999 for a review). For
an insight into the star formation process, it may be interesting
to consider that stars can be born from interstellar gas disturbed by
shock waves, implying that the first generation stars (Pop III stars)
could have been born in a dense shell formed by shock waves triggered
by a cloud-cloud collision in a protogalaxy.
\bigskip
This work has been partially supported by COE research (07CE2002) of
the Ministry of Education, Science, Culture, and Sports in Japan.
We would like to thank T. C. Beers for many fruitful discussions.
|
1,116,691,499,244 | arxiv | \section{Introduction}
In order to facilitate the planning of early intervention and treatment in Alzheimer’s disease (AD), one of the leading cause of death for older people in the United States, over the past few years, several machine learning-based approaches have been proposed to assist in the early computer aided diagnosis (CAD) of AD [1]. Using multiple modalities of biomarkers that have been proved to be sensitive in assessing the progression of AD, those proposed approaches aimed at the effective and accurate classification performance, especially for discriminating the subtler prodromal stage of AD, mild cognitive impairment (MCI) group, from the cognitively normal control (CN) group. Among those modalities, structural magnetic resonance imaging (MRI) is currently widely used for analyzing the gradual progression of atrophy patterns in key brain regions [1]. And utilizing structural MRI images, the recent proposed Gaussian discriminant analysis (GDA)-based approaches have been demonstrated to be effective and accurate in the classification of AD and MCI [2]. Moreover, since most of the current studies assumed only binary classification, where classification experiments were based on two-group comparisons, i.e., CN vs. MCI, CN vs. AD, and MCI vs. AD, such studies commonly applied local feature extraction methods. For different comparisons, the feature sets in the classification are very various from each other, which also limit the clinical diagnosis for a given patient, which could belong in any of the three groups [2].
This study aims to introduce a novel global feature extraction model based on the recent proposed GDA-based dual high-dimensional decision spaces, so that can significantly improve the early CAD performance. Instead of extracting three different sets of features for those three comparisons (CN vs. MCI, CN vs. AD, and MCI vs. AD), respectively, in this study, only one optimal set of features will be generated. The optimal set of features in this study indicates that the entorhinal cortex is the most significant cortical region associated with the progression of AD, which is consistent with recent studies concluding that the entorhinal cortex is the signature region to be implicated in AD [3]–[5]. And the top 5 regions of interest, entorhinal, middle temporal, inferior temporal, fusiform, and parahippocampal are all AD signature regions investigated previously [6]. Furthermore, for the dual high-dimensional decision spaces, a global GDA-based classifier used for combined training will be proposed as well and achieve a significant improvement of the early CAD performance.
\section{Material}
\subsection{Subjects}
The data used in the preparation of this study were obtained from the Alzheimer’s disease Neuroimaging Initiative (ADNI) database (adni.loni.usc.edu), as part of the ADNI1: Complete 1Yr 1.5T collection and their assessments at baseline, which includes 628 individuals (190 CN, 305 MCI, and 133 AD) [7]. The ADNI was launched in 2003 as a public-private partnership, led by Principal Investigator Michael W. Weiner, MD. The primary goal of ADNI has been to test whether serial magnetic resonance imaging (MRI), positron emission tomography (PET), other biological markers, and clinical and neuropsychological assessment can be combined to measure the progression of MCI and early AD. The primary phenotype is diagnostic group and MMSE. All source imaging data consist of 1.5 Tesla T1-weighted MRI volumes in the NIfTI (.nii.gz) format from the ADNI1: Complete 1Yr 1.5T Data Collection.
\subsection{MRI data pre-processing}
Using three neuroimaging software pipelines: FreeSurfer [8], Advanced Normalization Tools (ANTs) [9], and Mindboggle [10], the original MRI data were pre-processed following the instruction provided by Alzheimer's Disease Big Data DREAM Challenge \#1 [11]. Tables of morphometric data were derived from the images using the following seven shape measures for all 25 FreeSurfer labeled cortical regions for both left and right hemispheres of the brain: 1) surface area; 2) travel depth; 3) geodesic depth; 4) mean curvature; 5) convexity; 6) thickness; 7) volume. FreeSurfer pipeline (version 5.3) was applied to all T1-weighted images to generate labeled cortical surfaces, and labeled cortical and noncortical volumes. Templates and atlases used by ANTs and Mindboggle could be found on the Mindboggle website [12]. The aforementioned pre-processed MRI data of the 25 labeled cortical regions were used to generate two 175-variable ($7\times25$) vector discriminator, for each subject (one 175-variable vector per hemisphere). Any subjects that involved abnormal variables were eliminated from the dataset for further investigation, for example, some of their cortical regions having measurements of some features to be zero.
\section{Methods}
\subsection{ANOVA-based feature ranking}
The analysis of variance (ANOVA) was carried out on each of the 175 features of the two brain hemispheres between all three groups to determine the significance of each feature in terms of classification outcome and all features were thereafter ranked according to their p-values. For the purpose of comparing the performance between the local and global feature extraction models, ANOVA was also performed on each of the 175 features of the two brain hemispheres between any two groups (i.e., CN vs. MCI, CN vs. AD, and MCI vs. AD), which would show that the significant features of each comparison are much different to others in Section \ref{res} .
\subsection{Incremental error analysis}
In order to maintain only few but key features for achieving best classification performance, an incremental error analysis was employed to determine how many of the top-ranked features ought to be involved in the final GDA-based classifier. In the initial phase, the proposed GDA-based classifier only used the first-ranked feature. The error analysis was performed whereby introducing the next top-ranked feature in the classifier at each subsequent phase, and recording the corresponding classification statistics, which then would be compared with the previous phase. In this study, as a machine learning classification problem, the performance was estimated by the F1 score, and until the performance can be no longer improved, the optimal sets of features should have been obtained.
\subsection{Global GDA-based classifier}
Taking advantage of GDA, an important generative learning algorithm for high dimensional classification problems, some proposed studies were capable of recognizing the different patterns between any given groups (i.e., CN vs. MCI, CN vs. AD, and MCI vs. AD) [2]. The fundamental part of GDA is the multivariate Gaussian distribution or multivariate normal distribution, which could take the correlations among all features into account of the proposed global classifier. The performance of the proposed global GDA-based model was measured by some classification related statistics based on tenfold cross validation using $80\%$ of the noise-free detected data points, including the F1 score, accuracy (ACC), sensitivity (SEN), specificity (SPE) and so on. After the optimal sets of features were obtained, the remaining $20\%$ of the noise-free detected data were utilized as held-out test data to estimate the classification performance.
As the same as aforementioned ANOVA-based feature ranking process, both local and global classifiers were validated. For the local model, the GDA-based classifier was applied to each hemisphere of the brain separately, and corresponding to each hemisphere, an optimal set of features was generated, and the two sets were then used as the final optimal sets of features. In contrast with the local model, the proposed global model implemented a combined incremental error analysis on the 175 features of the left and right hemispheres of the brain, so that all combinations of the two sets of 175 features were taken into consideration for obtaining the final optimal sets of features. In terms of the dual decision space CAD, if either one of the two decision spaces produced a positive result, the corresponding subject should be classified as such.
\section{Results and conclusion}
\label{res}
\subsection{Top-ranked features}
\begin{table}[t]
\caption{Number of significant features selected for each model}
\label{table1}
\centering
\begin{tabular}{lllll}
\toprule
Model
& \multicolumn{3}{c}{Local} & Global \\
\cmidrule{2-4}
Comparison & CN vs. MCI & CN vs. AD & MCI vs. AD & All \\
\cmidrule{2-5}
Side of brain
& \multicolumn{4}{c}{Number of significant variables (\textit{p-value} < 0.01)} \\
\midrule
Left & 50 & 79 & 51 & 71 \\
Right & 44 & 68 & 41 & 66 \\
\bottomrule
\end{tabular}
\end{table}
After pre-processing the MRI images, 9 subjects were eliminated because of the noisy data which included measurements with zero values, so the final data used in the classification experiment included 619 individuals, among them 187 CN, 301 MCI, and 131 AD. As mentioned earlier, in the local feature ranking, ANOVA was performed for CN vs. MCI, CN vs. AD, and MCI vs. AD using two 175-feature vectors corresponding to the left and right hemispheres of the brain. For each comparison, all features found at 0.01 level of significance (LOS) out of all 175 features for each side of the brain were used for the tenfold cross validation. In the global case, ANOVA was carried out on the two sets of 175-feature vectors between all three groups. For each model, the number of selected features is shown in Table \ref{table1}.
In the local feature ranking, for each comparison, the top ranked feature sets are much different to others. Nevertheless, in both local and global feature ranking results, it could be observed that the most significant cortical region associated with the progression of AD is the entorhinal cortex, that is consistent with recent studies indicating that indeed the entorhinal cortex is the first area to be implicated in AD [3]–[5]. Furthermore, the entorhinal cortex has been also proven as a major source of projections to the hippocampus [13]. The other four significant regions of interest identified from the global feature ranking, middle temporal, inferior temporal, fusiform, and parahippocampal are all AD signature regions investigated previously [6], which provide credence to the validity of our ANOVA-based feature ranking method.
\subsection{Optimal feature sets}
The summary of tenfold cross validation performance is demonstrated in Table \ref{table2}. For both models, it can be observed that the final optimal sets obtained by the global GDA-based classifiers are different from the ones obtained for each hemisphere without combining the two decision spaces together. For all comparisons, the performance was improved significantly after applying the global classifiers. Moreover, as shown in Table \ref{table3}, for the most difficult two groups to delineate, CN vs. MCI, the global model yielded significant improvement of all classification statistics comparing to those recently reported cross validation results [14]–[16], that the proposed study achieves the best performance.
\subsection{Classification performance on held-out test data}
In order to obtain a reliable measure of the proposed CAD system performance, the remaining $20\%$ of the noise-free detected data points were used as the held-out test data applying the acquired optimal sets of features that results are presented in Table \ref{table4}, which achieved significant performance improvement even in contrast to those recent reported cross validation results in Table \ref{table3}, especially, for discriminating the most challenging MCI group from CN.
\begin{table}[t]
\begin{adjustwidth}{-1in}{-1in}
\caption{Summary of tenfold cross validation performance for each model}
\label{table2}
\centering
\begin{tabular}{lllllllllllll}
\toprule
Comparison
& \multicolumn{4}{c}{CN vs. MCI}
& \multicolumn{4}{c}{CN vs. AD}
& \multicolumn{4}{c}{MCI vs. AD} \\
\cmidrule{2-13}
Feature ranking
& \multicolumn{2}{c}{Local} & \multicolumn{2}{c}{Global}
& \multicolumn{2}{c}{Local} & \multicolumn{2}{c}{Global}
& \multicolumn{2}{c}{Local} & \multicolumn{2}{c}{Global} \\
\cmidrule{2-13}
Classifier
& Loc. & Glo. & Loc. & Glo.
& Loc. & Glo. & Loc. & Glo.
& Loc. & Glo. & Loc. & Glo. \\
\midrule
F1 score$\%$
& 88.11 & 92.08 & 83.62 & 95.30 & 92.66 & 95.89 & 91.67 & 96.23 & 74.69 & 81.41 & 69.14 & 78.49 \\
Accuracy$\%$
& 83.36 & 90.26 & 75.90 & 94.10 & 93.85 & 96.54 & 93.08 & 96.92 & 82.57 & 89.43 & 78.57 & 88.57 \\
Sensitivity$\%$
& 94.17 & 92.08 & 100.0 & 97.08 & 91.82 & 95.45 & 90.00 & 92.73 & 81.82 & 73.64 & 76.36 & 66.36 \\
Specificity$\%$
& 68.67 & 87.33 & 37.33 & 89.33 & 95.33 & 97.33 & 95.33 & 100.0 & 82.92 & 96.67 & 79.58 & 98.75 \\
\midrule
Side of the brain
& \multicolumn{12}{c}{Optimal Feature Sets} \\
\cmidrule{2-13}
Left & 36 & 5 & 64 & 69 & 6 & 10 & 6 & 4 & 2 & 48 & 1 & 1 \\
Right & 34 & 44 & 65 & 1 & 5 & 44 & 5 & 42 & 2 & 4 & 2 & 61 \\
\bottomrule
\end{tabular}
\end{adjustwidth}
\end{table}
\begin{table}[t]
\begin{adjustwidth}{-1in}{-1in}
\caption{Summary of tenfold cross validation performance for each model}
\label{table3}
\centering
\begin{tabular}{llllllll}
\toprule
\multicolumn{2}{c}{Comparison}
& \multicolumn{3}{c}{CN vs. MCI}
& \multicolumn{3}{c}{CN vs. AD} \\
\cmidrule{3-8}
Reference & Modalities
& ACC $\%$ & SEN $\%$ & SPE $\%$
& ACC $\%$ & SEN $\%$ & SPE $\%$ \\
\midrule
Khedher L., et al. (2015) [14] & MRI
& 81.89 & 82.61 & 81.62
& 88.49 & 91.27 & 85.11 \\
Tong T., et al. (2017) [15] & MRI+PET+CSF+Genetic
& 79.50 & 85.10 & 67.10
& 91.80 & 88.90 & 94.70 \\
Khedher L., et al. (2017) [16] & MRI
& 79.00 & 82.00 & 76.00
& 89.00 & 92.00 & 86.00 \\
Proposed Study & MRI
& \textbf{94.10} & \textbf{97.08} & \textbf{89.33}
& \textbf{96.92} & \textbf{92.73} & \textbf{100.0} \\
\bottomrule
\end{tabular}
\end{adjustwidth}
\end{table}
\begin{table}[b]
\caption{Summary of the classification performance using held-out test data}
\label{table4}
\centering
\begin{tabular}{lllll}
\toprule
Comparison
& F1 score $\%$ & Accuracy$\%$ & Sensitivity$\%$ & Specificity$\%$ \\
\cmidrule{2-5}
CN vs. MCI
& \textbf{91.06} & \textbf{88.78} & \textbf{91.80} & \textbf{83.78} \\
CN vs. AD
& \textbf{90.48} & \textbf{93.10} & \textbf{90.48} & \textbf{94.59} \\
\bottomrule
\end{tabular}
\end{table}
\subsubsection*{Acknowledgments}
We acknowledge the critical support provided by the National Science Foundation under grants: CNS-1532061, CNS-1551221, CNS-1642193, IIP 1338922, and CNS-1429345. The generous support of the Ware Foundation is greatly appreciated. This research is also supported through the Florida Department of Health, Ed and Ethel Moore Alzheimer's Disease Research Program, and Florida ADRC (Alzheimer’s Disease Research Center) (1P50AG047266-01A1).
Data used in preparation of this article were obtained from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database (adni.loni.usc.edu). As such, the investigators within the ADNI contributed to the design and implementation of ADNI and/or provided data but did not participate in analysis or writing of this report. A complete listing of ADNI investigators can be found at: \url{http://adni.loni.usc.edu/wp-content/uploads/how\_to\_apply/ADNI\_Acknowledgement\_List.pdf}. Data collection and sharing for this project was funded by the Alzheimer's Disease Neuroimaging Initiative (ADNI) (National Institutes of Health Grant U01 AG024904) and DOD ADNI (Department of Defense award number W81XWH-12-2-0012).
\section*{References}
\small
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[2] Fang, C., Li, C., Cabrerizo, M., Barreto, A.\ et al. (2017) A Novel Gaussian Discriminant Analysis-based Computer Aided Diagnosis System for Screening Different Stages of Alzheimer’s Disease. {\it Proceedings of 17th IEEE BIBE}:279–284.
[3] Braak, H.\ \& Tredici, K.D.\ (2012) Alzheimer’s disease: pathogenesis and prevention. {\it Alzheimers Dement.}, {\bf 8}(3):227–233.
[4] Gómez-Isla, T., Price, J.L., McKeel, D.W., Morris, J.C., Growdon, J.H.\ \& Hyman, B.T.\ (1996) Profound loss of layer II entorhinal cortex neurons occurs in very mild Alzheimer's disease. {\it J. Neurosci.}, {\bf 16}(14):4491–4500.
[5] Khan, U.A., Liu, L., Provenzano, F.A., Berman, D.E., Profaci, C.P., Sloan, R., Mayeux, R., Duff, K.E.\ \& Small, S.A.\ (2013) Molecular drivers and cortical spread of lateral entorhinal cortex dysfunction in preclinical Alzheimers disease. {\it Nat. Neurosci.}, {\bf 17}(2):304–311.
[6] Bakkour, A., Morris, J.C.\ \& Dickerson B.C.\ (2009) The cortical signature of prodromal AD: regional thinning predicts mild AD dementia. {\it Neurology}, {\bf 72}:1048-1055.
[7] University of South California,\ ADNI | Standardized MRI Data Sets. ADNI. [Online]. Available: \url{http://adni.loni.usc.edu/methods/mri-analysis/adni-standardized-data/} [Accessed: 02-Aug-2017].
[8] Reuter, M., Schmansky, N.J., Rosas, H.D.\ \& Fischl, B.\ (2012) Within-subject template estimation for unbiased longitudinal image analysis. {\it Neuroimage}, {\bf 61}(4):1402–1418.
[9] Tustison, N.J., Cook, P.A., Klein, A., Song, G., Das, S.R., Duda, J.T., Kandel, B.M., Strien, N.V., Stone, J.R., Gee, J.C.\ \& Avants, B.B.\ (2014) Large-scale evaluation of ANTs and FreeSurfer cortical thickness measurements. {\it Neuroimage}, {\bf 99}:166–179.
[10] Klein, A., Ghosh, S.S., Bao, F.S., Giard, J., Hame, Y., Stavsky, E., Lee, N., Rossa, B., Reuter, M., Neto, E.C.\ \& Keshavan, A.\ (2017) Mindboggling morphometry of human brains. {\it PLoS Comput. Biol.}, {\bf 13}(2):e1005350.
[11] Alzheimer's Disease Big Data DREAM Challenge 1. [Online]. Available: \url{https://www.synapse.org/#!Synapse:syn2290704/wiki/60828} [Accessed: 02-Aug-2017].
[12] Mindboggle Data. Mindboggle-101. [Online]. Available:\url{http://www.mindboggle.info/data.html} [Accessed: 02-Aug-2017].
[13] Squire, L.R.\ \& Zola-Morgan, S.\ (1991) The medial temporal lobe memory system. {\it Science}, {\bf 253}(5026):1380–1386.
[14] Khedher, L., Ramírez, J., Górriz, J.M., Brahim, A., Segovia, F.\ et al. (2015) Early diagnosis of Alzheimer’s disease based on partial least squares, principal component analysis and support vector machine using segmented MRI images. {\it Neurocomputing}, {\bf 151}(1):139–150.
[15] Tong, T., Gray, K., Gao, Q., Chen, L.\ \& Rueckert, D.\ (2017) Multi-modal classification of Alzheimer's disease using nonlinear graph fusion. {\it Pattern Recognition}, {\bf 63}:171–181.
[16] Khedher, L., Illán, I., Górriz, J.M., Ramírez, J., Brahim, A.\ \& Meyer-Baese, A.\ (2017) Independent Component Analysis-Support Vector Machine-Based Computer-Aided Diagnosis System for Alzheimer's with Visual Support. {\it Int. J. Neural Syst.}, {\bf 27}(3):1650050.
\end{document}
|
1,116,691,499,245 | arxiv | \section{Introduction}
\label{sec:intro}
Topological defects such as monopoles, strings and domain walls~\cite{Zeldovich:1974uw, Vilenkin:1982ks} can arise in many gauge theories including grand unification. In addition, there can appear (hybrid) configurations such as monopoles connected via strings or walls bounded by strings. Two well known examples of the latter arise in $SO(10)$~\cite{Kibble:1982ae} and axion models~\cite{Kibble:1982dd}. Stable or sufficiently long lived domain walls, associated with symmetry breaking scales comparable to or larger than in the Standard Model (SM) will sooner or later become the dominant energy component of the early universe. As a consequence such domain walls pose a serious challenge in cosmology and should therefore be avoided in realistic model building (some possibilities were recently discussed in~\cite{Sato:2018nqy}).
Domain walls are well known to appear associated with the spontaneous breaking of the Peccei-Quinn symmetry \cite{Sikivie:1982qv}. Here we note that also the ``weak'' $SU(2)_L$ may be associated with the presence of domain walls. This may happen in the context of spontaneous violation of lepton number symmetry. Indeed, such models in which lepton number is violated by a gauge singlet Higgs vacuum expectation value (vev) \cite{chikashige:1981ui,Schechter:1981cv} provides an attractive way to generate Majorana masses for neutrinos~\cite{Valle:2015pba}, as needed to account for current neutrino data~\cite{deSalas:2017kay}.
In addition, it implies the existence of a physical Nambu-Goldstone boson, called Majoron. The latter may pick up a mass from explicit symmetry breaking by gravity effects~\cite{Rothstein:1992rh,Akhmedov:1992et,Akhmedov:1992hi,King:2017nbl}. Under such circumstances the Majoron may provide a good dark matter candidate~\cite{berezinsky:1993fm,Lattanzi:2007ux,Bazzocchi:2008fh,Lattanzi:2013uza,Lattanzi:2014mia,Kuo:2018fgw}~\footnote{Since gravitational effects are not calculable in a reliable way, here we prefer not to invoke their existence.}.
The origin of the domain wall problem in this case stems from the existence of an unbroken residual subgroup $Z_2$ arising from the spontaneous lepton number violation, which clashes with the unbroken $Z_3$ from the non-perturbative instanton effects associated with the weak $SU(2)_L$.
This implies that the domain wall problem associated with the weak $SU(2)_L$ exists in a broad class of Majoron models of neutrino mass generation. A standard mechanism for evading the domain wall problem is to invoke a suitable inflationary phase during their formation such that the walls are inflated away. In this letter we propose a more direct resolution of the domain wall problem which does not rely on inflation. We present various possible mechanisms for having realistic Majoron models, with and without supersymmetry, which allow spontaneous lepton number violation to occur without encountering a domain wall problem.
\section{Global Lepton Number and Domain Wall problem}
\label{sec:ssb-of-lep-no}
Apart from the gauge symmetries it is well know that in the {Standard Model } there are two ``accidental'' global $U(1)$ symmetries namely the Baryon number $U(1)_B$ and the Lepton number $U(1)_L$ symmetries.
Although, accidental within {Standard Model }, these symmetries nonetheless play a very important role.
The baryon number symmetry $U(1)_B$ is responsible for the stability of the proton and the lepton number symmetry plays a key role in neutrino mass generation and in determining the Dirac or Majorana nature of neutrinos.
Lepton number in the {Standard Model } is conserved at the Lagrangian level to all orders in perturbation theory.
However, lepton number is an anomalous symmetry, hence it is explicitly broken by non-perturbative effects~\cite{tHooft:1976rip}
\footnote{Note that both baryon and lepton numbers are anomalous symmetries however a particular combination $U(1)_{B-L}$ is anomaly free. The other orthogonal combination $U(1)_{B+L}$ remains anomalous and hence it is explicitly broken by non-perturbative effects~\cite{tHooft:1976rip}.}.
In particular, owing to the $[SU(2)_L]^2\times U(1)_L$ anomaly, the non-perturbative instantons will explicitly break the initial lepton number symmetry $U(1)_L$ down to the discrete $Z_N$ subgroup, with
\begin{equation}
N = \sum_R N(R) \times L(R) \times T(R) = 3 \times 1 \times 1 = 3\,,
\label{eq:inst-z3}
\end{equation}
where $N(R)$ is the number of copies of a given fermion in representation R, $L(R)$ is the lepton number of the fermion and $T(R)$ is the $SU(2)_L$ Dynkin multiplicity index.
For the $SU(2)_L$ group, the index $T(R)$ for the lowest representations, singlets, doublets and triplets are respectively, $T(1) = 0$, $T(2) = 1$ and $T(3) = 4$.
It is clear, from \eqref{eq:inst-z3}, that the non-perturbative instantons associated with the weak $SU(2)_L$ break $U(1)_L \to Z_3$.
Notice also that the threefold family replication in the {Standard Model } plays a crucial role in dictating the breaking $U(1)_L \to Z_3$.
The residual $Z_3$ symmetry is exact at the classical and quantum level, implying the existence of degenerate vacua in our theory.
Notice also that, in contrast to the case of axions, where the anomaly is related to the $U(1)$ Peccei-Quinn charge assignments, in the case of the lepton number there is an anomaly intrinsically associated with the chiral nature of weak $SU(2)_L$.
Although the tunneling rate from one vacuum to another due to instantons is extremely small at zero temperature\footnote{This rate is proportional to $exp(-2\pi/\alpha_W)$ and thus unimportant for our discussion \cite{Gross:1980br}}, sphaleron induced transitions between the vacua become relevant at higher temperatures \cite{Kuzmin:1985mm}. Moreover, it is argued that frequent transitions between the vacua occur even above the critical temperature $T_c$ for the electroweak transition\footnote{Our argument below involves only temperatures between around 200 GeV and $T_c$, where the sphalerons operate, and the transitions from one vacuum to another are very frequent \cite{Arnold:1987zg}.}. Thus, the B and L violating reactions at high temperatures are fast, so that the $U(1)_L$, $U(1)_B$ are explicitly broken by non-perturbative effects
down to discrete $Z_3$ symmetries.
If the {Standard Model } is the final gauge theory, the non-perturbative breaking of lepton number won't be a serious issue. However, a dynamical understanding of the smallness of neutrinos mass often requires that lepton number is further broken down either explicitly or spontaneously by the new physics associated to neutrino mass generation.
A popular and well studied scenario is the case of spontaneous breaking of the lepton number~\cite{chikashige:1981ui,Schechter:1981cv}.
This is a specially attractive scenario that not only leads to Majorana masses, but also implies the existence of a Nambu-Goldstone boson, called Majoron. It breaks the global $U(1)_L$ lepton number symmetry down to a $Z_2$ subgroup through the vev of a $\mathrm{SU(3)_c \otimes SU(2)_L \otimes U(1)_Y}$ singlet scalar carrying two units of lepton number.
However, we notice the mismatch between the unbroken residual subgroup $Z_2$ arising from the spontaneous lepton number violation and the subgroup $Z_3$ that is left unbroken by the nonperturbative effects.
Owing to this mismatch the domain walls will appear.
For temperatures between 200 GeV and the electroweak critical temperature
$T_c$ the tunneling rate between the vacua connected by the $Z_3$ subgroup which remains explicitly unbroken by instantons is very frequent.
The barrier separating different vacua, related by the $Z_3$, has static energy $E_{sph} (T)$, the sphaleron mass, and the width is of order $m_W^{-1}$. This is the size of the ``restricted instanton'' which minimizes the height of the barrier and corresponds to the sphaleron (see \cite{Lazarides:2001zd} and references therein). Since the wall thickness is much smaller than the horizon size at these temperatures, the walls are expected to be present.
The mass per unit surface is $\simeq v^2m_W$, where $v$ is the order parameter, i.e. the vev that breaks $U(1)_L$. Even one such wall per horizon would provide an energy density $\simeq3v^2m_W/4t$ ($t$ is the cosmic time). This exceeds the radiation energy density $\rho_r=(\pi^2/30)g_*^{1/2}T^4$ ($g_*=106.75$ is the effective number of degrees of freedom) at a cosmic temperature $T>200$ GeV if the vev ($v$) is larger than about $(8\pi/3)^{1/2}(g_*/10)^{1/4}(m_W m_{\rm P})^{1/2} \simeq 8.2\times 10^{10}~{\rm GeV}$ ($m_{\rm P}=2.44\times 10^{18}~{\rm GeV}$ is the reduced Planck mass).
Such values for $v$ are very reasonable if they are to generate, say, the right handed neutrino masses within a type-I seesaw mechanism~\cite{Minkowski:1977sc,GellMann:1980vs,Yanagida:1979as,Schechter:1980gr,mohapatra:1980ia}.
Right after the wall domination, the Schwarzschild radius corresponding to the mass within each horizon becomes bigger than the horizon itself and the system becomes unstable and collapses into black holes leading to a cosmological catastrophe \cite{Vilenkin:1981zs}.
Therefore, unless a suitable remedy is provided, we expect the standard high-scale type-I seesaw Majoron model of neutrino mass generation to be cosmologically inconsistent due to the existence of such domain walls. Note however that low-scale scenarios, such as the inverse seesaw Majoron schemes~\cite{Mohapatra:1986bd,GonzalezGarcia:1988rw,Joshipura:1992hp,Berezhiani:1992cd} constitute a potential way out. This is because in that case the lepton number violating order parameter can lie much below the electroweak
scale, where sphaleron effects are negligible and $U(1)_L$ can be regarded as an exact continuous symmetry.\\[-.2cm]
The above spontaneous breaking of $U(1)_L \to Z_2$ by the vev of a field carrying two units of lepton number can be connected to neutrino masses in full generality at the operator level.
Consider the $U(1)_L$ invariant effective operator
\begin{equation}
\frac{1}{\Lambda^2} \bar{L}^c H H \sigma L\,.
\label{eq:maj-weinberg}
\end{equation}
In \eqref{eq:maj-weinberg} the field $L$ is the $SU(2)_L$ lepton doublet, $H$ is the Higgs doublet and $\sigma$ is a {Standard Model } gauge singlet
scalar field charged under the $U(1)_L$ symmetry.
Also, $\Lambda$ is the cutoff scale for the effective operator above which the full Ultra-Violet complete theory should be specified.
This operator is $U(1)_L$ invariant if $\sigma$ has charge $-2$ under the $U(1)_L$ symmetry.
After $\sigma$ develops a non-zero vev, $\vev{\sigma}$, $U(1)_L$ is broken down to $Z_2$ and the expression in Eq. \eqref{eq:maj-weinberg} reduces to the famous Weinberg operator~\cite{weinberg:1980bf}. Again the CP odd part of $\sigma$ will be a Nambu-Goldstone boson, the Majoron. Here CP denotes the combined action of charge conjugation (C) and parity (P).
\section{Solutions to the Domain Wall problem}
\label{sec:solutions}
In this section we consider alternative solutions to the domain wall problem which arises from the spontaneous breaking of $U(1)_L$ by the vev of a lepton-number-carrying scalar field. We focus attention on Majoron-type models characterized by the spontaneous breaking of lepton number at high-scale.
The examples in Sec.\ref{sec:T}, \ref{sec:susyT} and \ref{sec:Diracon} involve only the Standard Model gauge structure. On the other hand the model considered in Sec.\ref{sec:family} requires an extension of the {Standard Model } with a gauge family symmetry.
\subsection{Majoron with Singlet-Triplet Seesaw}
\label{sec:T}
The simplest solution of the domain wall problem in the Majoron model uses only the usual {Standard Model } gauge framework. It requires, in addition to the {Standard Model } fields, the following new ones with their $\mathrm{SU(3)_c \otimes SU(2)_L \otimes U(1)_Y}$ quantum numbers indicated in parenthesis and subscripts denoting their charges under $U(1)_L$:
%
\begin{equation}
\nu_R = (1,1,0)_{+1}\,, \,\, \Sigma_R = (1,3,0)_{+1}\,, \,\, \sigma = (1,1,0)_{-2}\,,
\end{equation}
where the first field ($\nu_R$) is a gauge singlet right-handed neutrino present in seesaw schemes~\cite{Minkowski:1977sc,GellMann:1980vs,Yanagida:1979as,Schechter:1980gr,mohapatra:1980ia} (with arbitrary multiplicity, which we take equal to one for simplicity, given that this is sufficient to account for the current neutrino oscillation data).
%
The second field ($\Sigma_R$) is a $SU(2)_L$ triplet right-handed fermion
and, the last field is the complex scalar whose vev $\vev{\sigma}$ is responsible for the spontaneous lepton number breaking.
%
The Lagrangian will now contain the following new couplings:
%
%
\begin{equation}
\mathcal{L}_{new}=y^{D i}_{\nu_R}\bar{L}^i \tilde{H} \nu_R \, + \, y^{D i}_\Sigma \bar{L}^i \tilde{H} \Sigma_R
\, + \, y^M_\Sigma \sigma \bar{\Sigma}^c_R \Sigma_R \, + \, y^M_{\nu_R} \sigma \bar{\nu}^c_R \nu_R \,.
\end{equation}
%
where $\tilde{H} = i \tau_2 H^*$ with $\tau_2$ denoting the second Pauli matrix.
%
After electroweak symmetry breaking the Higgs field will get a vev $\vev{ H} = v$ and we will have a seesaw-like mechanism for light neutrinos
with mass matrix $m_\nu = M_D^T M_R^{-1} M_D$ where
%
\begin{equation}
\begin{split}
&M_D=\left(\begin{array}{ccc}
vy^{D1}_\nu&vy^{D2}_\nu&vy^{D3}_\nu\\
vy^{D1}_\Sigma&vy^{D2}_\Sigma&vy^{D3}_\Sigma\\
\end{array}\right)\,,\\&
M_R=\left(\begin{array}{cc}
y^M_{\nu_R}\vev{\sigma}&0\\
0&y^M_\Sigma\vev{\sigma}\\
\end{array}\right)\,.
\end{split}
\end{equation}
The resulting matrix, $m_\nu$, has rank 2 leaving one light neutrino massless. Note that, since $\Sigma_R$ has non-trivial $SU(2)_L$ quantum numbers it produces a significant change in the $[SU(2)_L]^2\times U(1)_L$ anomaly, which is now given by
%
\begin{equation}\label{anomaly}
N = \sum_R N(R) \times L(R) \times T(R) = 3\times 1\times 1-1\times 1\times 4 = -1~,
\end{equation}
%
By computing the anomaly factor one sees that the domain wall problem is absent in this extension. Therefore, the heavy triplet $\Sigma_R$ acts as an auxiliary Majorana field to address the domain wall issue. Moreover, it also acts as heavy messenger for small neutrino mass generation through the seesaw mechanism.
\subsection{ Majoron seesaw within supersymmetry}
\label{sec:susyT}
The simple solution illustrated in the previous section can be generalized within a supersymmetric (SUSY) context. We present here a simple supersymmetric model which also addresses the domain wall problem. The particle content and charges of the superfields are as shown in Table \ref{Tab:sec2}.
\begin{table}[ht]
\begin{center}
\begin{tabular}{| c c c c c c c | }
\hline
Superfields \hspace{0.5cm} & $SU(3)_C$ \hspace{0.5cm} & $SU(2)_L$ \hspace{0.5cm} & $U(1)_Y$ \hspace{0.5cm} & $U(1)_B$ \hspace{0.5cm} & $U(1)_L$ \hspace{0.5cm} & $U(1)_R$ \hspace{0.5cm} \\
\hline \hline
$Q_i$ \hspace{0.5cm} & 3 \hspace{0.5cm} & 2 \hspace{0.5cm} & 1/6 \hspace{0.5cm} & 1/3 \hspace{0.5cm} &
0 \hspace{0.5cm} & 1 \hspace{0.5cm} \\
$u^c_i$ \hspace{0.5cm} & $\bar{3}$ \hspace{0.5cm} & 1 \hspace{0.5cm} & -2/3 \hspace{0.5cm} & -1/3 \hspace{0.5cm} &
0 \hspace{0.5cm} & 1 \hspace{0.5cm} \\
$d^c_i$ \hspace{0.5cm} & $\bar{3}$ \hspace{0.5cm} & 1 \hspace{0.5cm} & 1/3 \hspace{0.5cm} & -1/3 \hspace{0.5cm} &
0 \hspace{0.5cm} & 1 \hspace{0.5cm} \\
\hline
$L_i$ \hspace{0.5cm} & 1 \hspace{0.5cm} & 2 \hspace{0.5cm} & -1/2 \hspace{0.5cm} & 0 \hspace{0.5cm} &
1 \hspace{0.5cm} & 1 \hspace{0.5cm} \\
$e^c_i$ \hspace{0.5cm} & 1 \hspace{0.5cm} & 1 \hspace{0.5cm} & 1 \hspace{0.5cm} & 0 \hspace{0.5cm} &
-1 \hspace{0.5cm} & 1 \hspace{0.5cm} \\
$\nu^c_i$ \hspace{0.5cm} & 1 \hspace{0.5cm} & 1 \hspace{0.5cm} & 0 \hspace{0.5cm} & 0 \hspace{0.5cm} &
-1 \hspace{0.5cm} & 1 \hspace{0.5cm} \\
\hline
$T$ \hspace{0.5cm} & 1 \hspace{0.5cm} & 3 \hspace{0.5cm} & 1 \hspace{0.5cm} & 0 \hspace{0.5cm} &
-1 \hspace{0.5cm} & 1 \hspace{0.5cm} \\
$\bar{T}$ \hspace{0.5cm} & 1 \hspace{0.5cm} & 3 \hspace{0.5cm} & -1 \hspace{0.5cm} & 0 \hspace{0.5cm} &
0 \hspace{0.5cm} & 1 \hspace{0.5cm} \\
\hline \hline
$H_u$ \hspace{0.5cm} & 1 \hspace{0.5cm} & 2 \hspace{0.5cm} & 1/2 \hspace{0.5cm} & 0 \hspace{0.5cm} &
0 \hspace{0.5cm} & 0 \hspace{0.5cm} \\
$H_d$ \hspace{0.5cm} & 1 \hspace{0.5cm} & 2 \hspace{0.5cm} & -1/2 \hspace{0.5cm} & 0 \hspace{0.5cm} &
0 \hspace{0.5cm} & 0 \hspace{0.5cm} \\
\hline
$S$ \hspace{0.5cm} & 1 \hspace{0.5cm} & 1 \hspace{0.5cm} & 0 \hspace{0.5cm} & 0 \hspace{0.5cm} &
0 \hspace{0.5cm} & 2 \hspace{0.5cm} \\
$\phi$ \hspace{0.5cm} & 1 \hspace{0.5cm} & 1 \hspace{0.5cm} & 0 \hspace{0.5cm} & 0 \hspace{0.5cm} &
-1 \hspace{0.5cm} & 0 \hspace{0.5cm} \\
$\bar{\phi}$\hspace{0.5cm} & 1 \hspace{0.5cm} & 1 \hspace{0.5cm} & 0 \hspace{0.5cm} & 0 \hspace{0.5cm} &
1 \hspace{0.5cm} & 0 \hspace{0.5cm}
\\
\hline
\end{tabular}
\end{center}
\caption{ Particle content and charges. $\rm U(1)_R$ is an R-Symmetry under which the superpotential $\mathcal{W}$ has R-charge of $2$ units. }
\label{Tab:sec2}
\end{table}
In addition to the usual minimal supersymmetric standard model (MSSM) superfields and the right-handed neutrinos ($\nu^c$), one adds the $SU(2)_L$ triplet superfields $T, \bar{T}$ and the gauge singlet superfields $S, \phi, \bar{\phi}$ with charges as listed in Table \ref{Tab:sec2}.
The superpotential of our model is given by
\begin{eqnarray}
\mathcal{W} & = & \kappa \, S(\bar{\phi} \phi - M^2) \, + \, y^u_{ij} \, H_u Q_i u^c_j \, + \, y^d_{ij} \, H_d Q_i d^c_j
\, + \, y^\nu_{ij} \, H_u L_i \nu^c_j \, + \, y^e_{ij} \, H_d L_i e^c_j \nonumber \\
& + & \lambda \, S H_u H_d \, + \, y^T_i \, T L_i H_d \, + \, y^{'T} \, \bar{\phi} T \bar{T} \, + \, y^\phi_{ij} \, \frac{\bar{\phi}^2 \nu^c_i \nu^c_j}{m_P} \, ,
\label{spot}
\end{eqnarray}
where $i,j = 1,2,3$ are generation indices.
Owing to the presence of the triplet superfield $T$, the $[SU(2)_L]^2\times U(1)_L$ anomaly is again found to be
\begin{equation}\label{susy-anomaly}
N = \sum_R N(R) \times L(R) \times T(R) = 3\times 1\times 1-1\times 1\times 4 = -1\,.
\end{equation}
Thus, unlike the usual Majoron models, here the instanton effects will break $U(1)_L \to Z_1$, avoiding the domain wall problem. As in the previous case, this holds irrespective of the number of right-handed neutrino superfields, the minimal realistic model has just one.
Notice that this solution differs from the standard seesaw mechanism in that the Majoron coming from the imaginary parts of the $\phi,\bar\phi$ scalars carry one unit of lepton number, instead of two.
Moreover, our model has other attractive features which make it quite appealing.
Apart from solving the domain wall problem, it automatically addresses the so-called ``$\mu$-problem'' of the MSSM~\cite{Dvali:1997uq}.
In addition we also have a R-symmetry which contains the usual R-parity of the MSSM, forbidding all the potentially dangerous terms in the superpotential, \eqref{spot}.
Finally, right-handed neutrino masses arise through the non-renormalizable term $\bar{\phi}^2 \nu^c \nu^c / m_P$, where we take the high scale as $m_P$.
\subsection{$SU(3)_{\rm lep}$ family symmetry for leptons}
\label{sec:family}
Consider now a $SU(3)_{\rm lep}$ gauge extension of the {Standard Model } scenario. Let quarks be singlets under this group, while leptons transform under it in a vector-like way\footnote{Note that this differs from the usual $SU(3)_{\rm lep}$ family symmetry used to address the observed fermion mass hierarchy \cite{Berezhiani:1983hm,Berezhiani:1990wn}.},
\begin{equation}
\begin{split}
& L = (1,2,-1/2,3)\,,\\&
e_R = (1,1,-1,3)\,,\\&
\nu_R = (1,1,0,3)\,,
\end{split}
\end{equation}
with the first three entries in parenthesis indicating the standard model charges and the last entry the $SU(3)_{\rm lep}$ representation. This extension has several consequences. First of all, right-handed neutrinos cannot have a bare mass term. Their masses must be generated through the spontaneous violation of U(1)$_{\rm L}$. This is related with the breaking of $SU(3)_{\rm lep}$ and is achieved by the vev of a flavour sextet scalar field $\sigma$ with lepton number $-2$ via the coupling
\begin{equation}
\sigma \bar{\nu}^c_R \nu_R\,.
\end{equation}
The second and more important implication is that this extension automatically solves the domain wall problem. The reason is that the center
of $SU(3)_{\rm lep}$ which is $Z_3$, exactly coincides with the discrete $Z_3$ subgroup of U(1)$_{\rm L}$ left unbroken by the anomaly. Since this accidental subgroup can be embedded in the continuous gauge group $SU(3)_{\rm lep}$, the degenerate minima are now connected by a gauge transformation, so that any difference among them becomes unphysical. In this way, the domain wall problem is solved. This is a Majoron variant of the domain wall axion solution given in the context of Grand Unified Theory (GUT) in Ref.~\cite{Lazarides:1982tw, Barr:1982bb}.
\subsection{Diracon Solution}
\label{sec:Diracon}
Another possible solution to the domain wall problem is obtained by enforcing that the spontaneous lepton number breaking is such that $U(1)_L \to Z_3$ instead of $ Z_2$. In this case there is no mismatch between the residual subgroup preserved by the anomaly and that preserved by the spontaneous lepton number violation due to $\vev{\sigma}$, so the domain wall problem will be automatically solved.
Clearly, the $U(1)_L \to Z_3$ spontaneous breaking cannot be accomplished within the framework of the canonical Majoron model.
In fact, if $Z_3$ is the residual unbroken symmetry then neutrinos cannot be Majorana particles.
However, we note that for Dirac neutrinos the $U(1)_L \to Z_3$ breaking is viable, and will lead to a solution of the domain wall problem within a variant of the ``Diracon models''~\cite{Bonilla:2016zef,Bonilla:2016diq}.
To see this Diracon solution, the first thing is to realize that the lepton number of right handed neutrinos $\nu_R$ need not be the same as that of the left handed neutrinos~\cite{Bento:1991bc,Peltoniemi:1992ss}.
In fact, a non-conventional lepton number assignment of $(4, 4, -5)$ for the three generations of $\nu_{i,R}$; $i = 1,2,3$, proposed in~\cite{Ma:2014qra,Ma:2015mjd} is equally acceptable.
If the $\nu_{i,R}$ transform with such non-conventional charges under $U(1)_L$ then one cannot write down the tree level Dirac term
$\bar{L} \tilde{H} \nu_{i,R}$ nor the Majoron Weinberg operator of \eqref{eq:maj-weinberg}.
However, one can still write down the following $U(1)_L$ invariant operators
\begin{equation}
\frac{1}{\Lambda} \bar{L} \tilde{H} \chi \nu_{i,R}\,, \quad \frac{1}{\Lambda^2} \bar{L} \tilde{H} \chi^* \chi^* \nu_{3,R} \, ,
\label{eq:dir-op}
\end{equation}
where $\nu_{i,R}$; $i = 1,2$ are the two right handed neutrinos carrying charge $4$ units under $U(1)_L$, and $\nu_{3,R}$ has $U(1)_L$ charge of $-5$. Also, the field $\chi$ has charge of $-3$ under $U(1)_L$.
It can be easily seen that the vev of the $\chi$ field will spontaneously break $U(1)_L \to Z_3$ with the resulting neutrinos being Dirac in nature.
Furthermore, the CP odd part of $\chi$ will be a Nambu-Goldstone boson which we call Diracon and is associated with the Dirac mass generation of the neutrinos.
Now, since the $U(1)_L$ in this case is spontaneously broken to the same residual subgroup $Z_3$ as that preserved by the non-perturbative $SU(2)_L$ instantons, there is no mismatch and hence the problem of domain walls is automatically avoided.
\section{Conclusions}
\label{sec:con}
We have shown that if the global lepton number symmetry is broken spontaneously in a post-inflationary epoch, then it can lead to the formation of cosmological domain walls. Since the presence of these domain walls may spoil the standard picture of cosmological evolution, we have studied the conditions to prevent their formation as a result of spontaneous symmetry breaking. We have shown that the simplest seesaw Majoron models of neutrino masses have, in principle, a domain wall problem associated with the chiral $SU(2)_L$ gauge group describing the weak interaction. We have also provided some explicit and realistic solutions which allow a safe spontaneous breaking of lepton number, free of domain walls. Some of these models involve new particles that could potentially lead to phenomenological implications.
\begin{acknowledgments}
The work of M.R., R.S. and J.W.F.V is supported by the Spanish grants FPA2017-85216-P (AEI/FEDER, UE), SEV-2014-0398 and PROMETEO/2018/165 (Generalitat Valenciana) and the Spanish Red Consolider MultiDark FPA2017-90566-REDC. M.R. is also supported by Formaci\'on de Profesorado Universitario (FPU) grant FPU16/01907.
Q.S. is supported in part by the DOE grant, No. DE-SC0013880. He also thanks the Alexander von Humboldt Foundation for providing support
through their research prize. Q.S. also acknowledges hospitality of the Instit\"{u}t f\"{u}r Theoretische Physik during his stay in Heidelberg.
\end{acknowledgments}
|
1,116,691,499,246 | arxiv | \section{Introduction}
Multi-agent systems are widely studied in the past decades because of their applications in various areas such as formation control, distributed sensor network, and so on \cite{fax2004information, cortes2005coordination, yu2009distributed}. Generally, the target of distributed consensus control is to achieve an agreement for the states of all the systems connected in a network by designing a suitable controller for each agent depending on locally available information from itself and its neighbors.
Recently, consensus control problems for integer-order multi-agent systems have been broadly investigated, see for examples \cite{olfati2004consensus, wang2014distributed, wang2017distributed, cheng2011necessary, cheng2016convergence, huang2015adaptive, huang2017smooth, chen2009reaching, zheng2018average}, and references therein.
As for the consensus control problem of fractional-order multi-agent systems, it is first studied in \cite{cao2009distributed}, where the conditions for achieving consensus in term of network structure and the number of agents are provided. In \cite{sun2011convergence}, where an undirected graph is considered, a consensus protocol with switching order is raised to increase convergence speed. Consensus involving communication delays is addressed in \cite{shen2012necessary}. Observer-based leader-following consensus problem is investigated in \cite{yu2016observer}, where the leader is described as a second-order integer model while the followers are fractional-order systems with order $\alpha \in(0,2)$. Consensus for incommensurate nonlinear fractional-order multi-agent systems with system uncertainties and external disturbances is addressed in \cite{shahvali2019distributed}.
All the works above are studied with continuous control, which requires control signals to be updated and transmitted continuously. In comparison with continuous control, sampled-data control for continuous-time systems, which can be found in for example \cite{hu2007sampled} and \cite{shen2016stabilization}, possesses various of benefits such as low cost and be more practical in implementation.
Many research works for integer-order multi-agent systems with sampled-data control have also been done and a survey on this topic is provided in \cite{ge2018survey}. However, results on sampled-data control of fractional-order multi-agent systems are still limited. The consensus problem of such systems with directed graph via sampled-data control is investigated in \cite{yu2017necessary, su2019necessary, liu2019consensus}. In \cite{chen2019synchronization}, consensus of linear fractional-order multi-agent systems over a communication topology, whose coupling structure is not necessary to be Laplacian, is studied. Event-triggering sampled-data control of fractional-order multi-agent systems is proposed in \cite{chen2019consensus}, in which the networked graph for the agents is assumed to be undirected.
From the definitions of fractional integral and derivative reviewed in Section \ref{sec:Preliminaries}, it can be observed that they can both be treated as weighted integral, which reveals the properties of fractional-order calculus: hereditary and infinite memory. Due to these unique properties, the initial values and the ``history" of the variables in the entire interval of integration play extremely important roles in solving fractional-order equations. It is noted that such properties are not considered in the existing literature on sampled-data control of fractional-order multi-agent systems mentioned above. Instead, by dividing the entire time history to sampling intervals, the solutions of fractional-order equations at the beginning of each sampling interval are used as the new initial values for this sampling interval. As a result, the closed-loop fractional-order
systems obtained are transformed to discrete-time systems, where the evolutions of step-forward states only rely on the current states and control inputs. However, when taking these properties into account, the evolutions should also depend on the previous states at all the discrete-time sampling instants starting from the initial time $t_0$.
Motivated by the discussions above, we address the consensus issue with the consideration of such unique characteristics of fractional-order calculus. With our proposed control design scheme, it is shown that the closed-loop system is globally stable and asymptotic consensus for fractional-order multi-agent systems is achieved. Different from existing research works,
the resulting fractional-order system is rigorously analyzed by considering time evolution of the system states which depends on their initial values at $t_0$ and also all the previous control signals. A new challenge that is to establish the global boundedness of the proposed distributed controllers which contain all the previous control inputs is overcome in this paper. Simulation studies illustrate the effectiveness of the proposed control scheme and also reveal its accuracy on achieving asymptotic consensus compared to an existing approach.
The rest of the paper is organized as follows. Preliminaries are provided and the class of fractional-order multi-agent systems considered is described in Section \ref{sec:Preliminaries}. In Section \ref{sec:controller}, the design of distributed controllers is presented in detail with analysis. In Section \ref{sec:simulation}, the scheme is illustrated by simulation studies with comparison to that in \cite{yu2017necessary}. Finally the paper is concluded in Section \ref{sec:conclusion}.
\section{Preliminaries and Problem Formulation}
\label{sec:Preliminaries}
\subsection{Preliminaries}
\emph{Definition 1 \cite{podlubny1998fractional}:} The fractional integral of an integrable function $f(t)$ with $\alpha\in\mathbb{R}^{+}$ and initial time $t_0$ is
\begin{equation}
{}_{t_0}\!{\mathit{I}}_t^{\alpha}f(t) = \frac{1}{\Gamma(\alpha)}\int_{t_0}^{t}\frac{f(\tau)}{(t-\tau)^{1-\alpha}}\,d\tau
\label{eq:eqI}
\end{equation}
where $\Gamma(\cdot)$ denotes the well-known Gamma function, which is defined as $\Gamma(z)=\int_{0}^{\infty}e^{-t}t^{z-1}\,dt$, where $z \in \mathbb{C}$. One of the significant properties of Gamma function is \cite{xue2017fractional}: $\Gamma(z+1) = z\Gamma(z),\, \Gamma(n+1) = n\Gamma(n)=n(n-1)\Gamma(n-1)=\cdots=n!,\, \Gamma(-n)= \infty$, where $n \in \mathbb{N}_0=\{n|n \ge 0,n \in \mathbb{N}\}$.
\emph{Definition 2 \cite{podlubny1998fractional}:} The Caputo fractional derivative of a function is defined as
\begin{equation}
\begin{aligned}
{}_{t_0}^{C}\!{\mathcal{D}}_t^{\alpha}f(t) &= {}_{t_0}{\mathit{I}}_t^{(m-\alpha)}\frac{\mathrm{d}^m}{\mathrm{d}t^m}f(t)\\
&=\frac{1}{\Gamma(m-\alpha)}\int_{t_0}^{t} \frac{f^{(m)}(\tau)}{(t-\tau)^{\alpha-m+1}}\,d\tau\\
\end{aligned}
\label{eq:eqCd}
\end{equation}
where $m - 1 < \alpha < m \in\mathbb{Z}^+$. From (\ref{eq:eqCd}) we can observe that the Caputo derivative of a constant is $0$. Another commonly used fractional derivative is named Riemann-Liouville (RL) and the RL fractional derivative of a function $f(t)$ is denoted as ${}_{t_0}^{RL}\!{\mathcal{D}}_t^{\alpha}f(t)$. Different from the Caputo derivative, RL derivative of a constant is not equal to $0$ \cite{lakshmikantham2009theory,podlubny1998fractional}.
\begin{comment}
The Riemann-Liouville (RL) fractional derivative of a function is given by
\begin{equation}
\begin{aligned}
{}_{t_0}^{RL}\!{\mathcal{D}}_t^{\alpha}f(t) = \frac{1}{\Gamma(m-\alpha)}\frac{\mathrm{d}^m}{\mathrm{d}t^m}\int_{t_0}^{t} \frac{f(\tau)}{(t-\tau)^{\alpha-m+1}}\,d\tau\\
\end{aligned}
\label{eq:eqRLd}
\end{equation}
where $m - 1 < \alpha < m \in\mathbb{Z}^+$. Different from the Caputo derivative, RL derivative of a constant is not equal to $0$ \cite{Lakshmikantham2009TheoryOF,podlubny1998fractional}. Instead, its derivative is provided as
\begin{equation}
{}_{t_0}^{RL}\!{\mathcal{D}}_t^{\alpha}C=\frac{(t-t_0)^{-\alpha}}{\Gamma(1-\alpha)}C,\quad 0 < \alpha < 1.
\label{eq:eqRLd_c}
\end{equation}
According to \cite{kilbas2006theory}, for $\alpha \notin \mathbb{N}_0$ and $Re(\alpha) \ge 0$, if $f(t)$ has Caputo and RL fractional derivatives, then the relationship between Caputo and RL fractional derivatives is described as follow:
\begin{equation}
{}_{t_0}^{C}\!{\mathcal{D}}_t^{\alpha}f(t) = {}_{t_0}^{RL}\!{\mathcal{D}}_t^{\alpha}f(t) - \sum_{k=0}^{m-1}\frac{f^{(k)}(t_0)}{\Gamma(k-\alpha+1)}(t-t_0)^{k-\alpha}
\label{eq:eqR1}
\end{equation}
where $m - 1 < \alpha < m \in\mathbb{Z}^+$. The Caputo and RL fractional derivatives are the same under the conditions zero initial conditions, namely $f(t_0)=f'(t_0)=\cdots=f^{(m-1)}(t_0)=0$.
\end{comment}
The initial values are needed in order to obtain the unique solution for fractional differential equation ${}_{t_0}\!{\mathcal{D}}_t^{\alpha}x(t)=f(x,t)$, ($m - 1 < \alpha < m \in\mathbb{Z}^+$ and $t\ge t_0$). According to \cite{li2007remarks,podlubny1998fractional} and \cite{bandyopadhyay2015stabilization}, fractional differential equations with Caputo-type derivative have initial values that are in-line with integer-order differential equations, i.e.\ $x(t_0),x'(t_0),\dots,x^{(m-1)}(t_0)$, which contain specific physical interpretations.
Therefore, Caputo-type fractional systems are frequently employed in practical analysis.
\emph{Definition 3 \cite{li2009mittag,li2010stability, zhang2011asymptotical}:} For fractional nonautonomous system ${}_{t_0}\!{\mathcal{D}}_t^{\alpha}x_i(t)=f_i(x,t)$ ($i = 1,2,\dots,n$), where $0 < \alpha < 1$, initial condition $x(t_0) = [x_1(t_0),x_2(t_0),\dots,x_n(t_0)]^\mathrm{T} \in \mathbb{R}^n$, $f_i(x,t): [t_0, \infty) \times \Omega \rightarrow \mathbb{R}^n$ is locally Lipschitz in $x$ and piecewise continuous in $t$ (which insinuates the existence and uniqueness of the solution to the fractional systems \cite{podlubny1998fractional}), ${}_{t_0}\!{\mathcal{D}}_t^{\alpha}$ denotes Caputo or RL fractional derivative and $\Omega \in \mathbb{R}^n$ stands for a region that contains the origin $x = [0,0,\dots,0]^\mathrm{T}$, the equilibrium $x^* =[x_1^*,x_2^*,\dots,x_n^*]^\mathrm{T}$ of this system is defined as ${}_{t_0}\!{\mathcal{D}}_t^{\alpha}x^*=f_i(x^*,t)$ for $t \ge t_0$.
\emph{Lemma 1 \cite{lakshmikantham2009theory}:} If $x(t)\in \mathbb{R}^n$ satisfies
\begin{equation}
\begin{aligned}
{}_{t_0}^{C}\!{\mathcal{D}}_t^{\alpha}x(t) &= f(x,t), \; x(t_0)=x_0,
\end{aligned}
\label{eq:eqL1_1}
\end{equation}
where $0<\alpha<1$ and $f(x,t) \in L_1[t_0,T]$, then it also satisfies the Volterra fractional integral
\begin{equation}
\begin{aligned}
x(t)=x_0+\frac{1}{\Gamma(\alpha)}\int_{t_0}^t (t-\tau)^{\alpha-1}f\big(x(\tau),\tau\big)d\tau,
\end{aligned}
\label{eq:eqL1_2}
\end{equation}
with $t_0\leq t \leq T$ and vice versa.
\emph{Lemma 2:} For $\forall j \in\mathbb{Z}^+$ and $0<\alpha<1$, the following results hold\\
\indent1) $0<\lvert (j+1)^\alpha-2j^\alpha+(j-1)^\alpha\rvert <1$,\\
\indent2) $\lim_{j \rightarrow \infty} \lvert (j+1)^\alpha -2j^\alpha+(j-1)^\alpha\rvert = 0$.
\emph{Proof:} Define
\begin{equation}
\begin{aligned}
f(j)=(j+1)^\alpha -2j^\alpha+(j-1)^\alpha
\end{aligned}
\label{eq:eqL2_1}
\end{equation}
and
\begin{equation}
\begin{aligned}
g(s)&=s^\alpha-(s-1)^\alpha.
\end{aligned}
\label{eq:eqL2_g}
\end{equation}
It can be shown from (\ref{eq:eqL2_g}) that, for $\forall s \in\mathbb{R}^+$,
\begin{equation}
\begin{aligned}
g(s)&> 0,\\
\frac{dg(s)}{ds}=\alpha s^{\alpha-1}&-\alpha (s-1)^{\alpha-1} < 0.
\end{aligned}
\label{eq:eqL2_2}
\end{equation}
From (\ref{eq:eqL2_2}), we can obtain that $g(s)$ is monotonically decreasing and $\lim_{s\rightarrow \infty}g(s)=0$, which further implies that $f(j)<0$ and will monotonically tends to 0 as $j$ tends to $\infty$. Additionally, it can be easily checked that $\lvert f(1)\rvert < 1$ for $0<\alpha<1$. This complete the proof.
\subsection{Problem Formulation}
In this paper, Caputo-type definition of the fractional derivatives is utilized. A group of $\mathit{N}$ fractional-order agents are governed by
\begin{equation}
\begin{aligned}
{}_{t_0}^{C}\!{\mathcal{D}}_t^{\alpha}&x_{i}(t) = u_i(t), \text{ for } i=1,2,\cdots,N
\end{aligned}
\label{eq:eqPro}
\end{equation}
where the fractional-orders of all the states are equal to $\alpha \in (0, 1)$,
$x_i\in \mathbb{R}$ and $u_i \in \mathbb{R}$ represent the measurable state and control input of $i$-th agent, respectively.
\emph{Remark 1:} All the agents in this paper are in one-dimensional space for convenience. The results established can be easily extended to $n$-dimensional space by applying the Kronecker product.
In this paper, the control problem is to design distributed controller $u_i$ for each agent described in (\ref{eq:eqPro}) to achieve the following objectives: 1) all the signals in the closed-loop systems are globally bounded; 2) asymptotic consensus for fractional-order systems (\ref{eq:eqPro}) is ensured, i.e.\ $\lim_{t\rightarrow \infty}\lVert x_i(t)-x_j(t)\rVert=0,\, \forall i, j = 1,2,\cdots,\mathit{N}$, and additionally, $\lim_{{t \rightarrow \infty}}x_i(t)=\frac{\sum_{i=1}^N x_i(t_0)}{N}$.
Suppose that the communications among the $\mathit{N}$ agents can be represented by a directed graph $\mathcal{G}\triangleq (\mathcal{V},\mathcal{E})$ where $\mathcal{V}=\{1,2,\cdots,\mathit{N}\}$ means the set of indexes (or vertices) corresponding to each agent, $\mathcal{E} \subseteq \mathcal{V} \times \mathcal{V}$ is the set of edges between two distinct agents. An edge $(i,j) \in \mathcal{E}$ denotes that agent $j$ can obtain information from agent $i$, but not necessarily vice versa. In this case, agent $i$ is called a neighbor of agent $j$ and we indicate the set of neighbors for agent $i$ as $\mathcal{N}_i$. In this paper, $(i,i) \notin \mathcal{E}$ and $i\notin \mathcal{N}_i$ since self edges $(i,i)$ are not allowed. $\mathit{A}=[a_{ij}]\in \mathbb{R}^{N\times N}$ is the connectivity matrix with $a_{ij}=1$ if $(j,i)\in \mathcal{E}$ and $a_{ij}=0$ if $(j,i)\notin \mathcal{E}$ defined. Throughout this paper, the diagonal elements $a_{ii}=0$. An in-degree matrix $\triangle$ is introduced as $\triangle = \text{diag}(\triangle_i)\in \mathbb{R}^{N \times N}$ with $\triangle_i=\sum_{j\in \mathcal{N}_i}a_{ij}$ being the $i$-th row sum of $\mathit{A}$. Therefore, the Laplacian matrix of $\mathcal{G}$ is defined as $\mathcal{L}=\triangle-\mathit{A}$. A digraph is strongly connected if there is a directed path that connects any two arbitrary nodes of the graph and is balanced if for all $i \in \mathcal{V}$, $\sum_{j \ne i}a_{ij}=\sum_{j \ne i}a_{ji}$.
\emph{Notations:} $\lVert \cdot \rVert$ is the Euclidean norm of a vector. $I_N$ denotes an identity matrix with dimension equals to $N$. $\mathbf{1}_N=[1,1,\cdots,1]^{\mathrm{T}}\in \mathbb{R}^N$.
\emph{Assumption 1:} The digraph $\mathcal{G}$ is strongly connected and balanced.
\section{Distributed Controller Design and Stability Analysis}
\label{sec:controller}
\subsection{Distributed Controller Design}
To achieve the above objectives, distributed controller is designed for each local agent based on periodic sampled-data control technology. The sampling instants are described by a discrete-time sequence $\{t_k\}$ with $t_0 < t_1 < \cdots < t_k < \cdots$ and $t_{k+1}-t_k = h$, where $h>0$ is the sampling period.
For $t\in [t_0, t_1)$, let $x(t)=[x_1(t),x_2(t),\cdots,x_N(t)]^{\mathrm{T}}$ and $\underline{u}_0=[u_{1,0},u_{2,0},\cdots,u_{N,0}]^{\mathrm{T}}$ where $u_{i,0}$ denotes the control signal for the $i$-th agent within this time interval. According to Lemma 1, the value of $x(t_1)$ can be computed as follows
\begin{equation}
\begin{aligned}
x(t_1)&=x(t_0)+\frac{1}{\Gamma(\alpha)}\int_{t_0}^{t_1}(t_1-\tau)^{\alpha-1}\underline{u}_0d\tau\\
&=x(t_0)+\underline{u}_0\frac{h^\alpha}{\Gamma(\alpha+1)}.
\end{aligned}
\label{eq:eqxi1_1}
\end{equation}
By designing the controller for the $i$-th agent as
\begin{equation}
u_{i,0}=\gamma \sum_{j\in \mathcal{N}_i}a_{ij}\big(x_j(t_0)-x_i(t_0)\big),
\label{eq:equ0i}
\end{equation}
where $\gamma > 0$, the closed-loop systems in this sampling period can be expressed as
\begin{equation}
\begin{aligned}
x(t_1)&=x(t_0)-\gamma\mathcal{L}x(t_0)\frac{h^\alpha}{\Gamma(\alpha+1)}.
\end{aligned}
\label{eq:eqxi1_2}
\end{equation}
\begin{comment}
\emph{For $t\in [t_1, t_2)$:} Let the controller for $i$-th agent be
\begin{equation}
\begin{aligned}
u_{i,1}=\gamma \sum_{j\in\mathcal{N}_i}a_{ij}\big(x_j(t_1)-x_i(t_1)\big)-(2^\alpha-2)u_{i,0},
\end{aligned}
\label{eq:equ1i}
\end{equation}
then we can calculate $x(t_2)$ as
\begin{equation}
\begin{aligned}
x(t_2)=&x(t_0)+\frac{1}{\Gamma(\alpha)}\int_{t_0}^{t_1}(t_2-\tau)^{\alpha-1}\underline{u}_0d\tau\\&+\frac{1}{\Gamma(\alpha)}\int_{t_1}^{t_2}(t_2-\tau)^{\alpha-1}\underline{u}_1d\tau\\
=&x(t_0)+\frac{\underline{u}_0}{\Gamma(\alpha+1)}[(2h)^\alpha-h^\alpha]+\frac{\underline{u}_1}{\Gamma(\alpha+1)}h^\alpha\\
=&x(t_1)+\frac{\underline{u}_0}{\Gamma(\alpha+1)}[(2h)^\alpha-2h^\alpha]+\frac{\underline{u}_1}{\Gamma(\alpha+1)}h^\alpha\\
=&x(t_1)-\gamma\mathcal{L}x(t_1)\frac{h^\alpha}{\Gamma(\alpha+1)},
\end{aligned}
\label{eq:eqxi2}
\end{equation}
where $\underline{u}_1=[u_{1,1},u_{2,1},\cdots,u_{N,1}]^{\mathrm{T}}$.
\end{comment}
Now for $t\in [t_k, t_{k+1})\,(k=1,2,\cdots)$, define $\underline{u}_k=[u_{1,k},u_{2,k},\cdots,u_{N,k}]^{\mathrm{T}}$. Since the fractional-order derivative of $x$ depends on all the historical values of $x$, thus not only the values of $x$ at present instant but also all of its previous values are needed to determine the future behavior of fractional-order systems. Therefore, $x(t_{k+1})$ should be expressed in terms of $x(t_0)$ and $\underline{u}_j (j=0,1,\cdots,k)$ as follows
\begin{equation}
\begin{aligned}
x(t_{k+1})=&x(t_0)+\frac{1}{\Gamma(\alpha)}\int_{t_0}^{t_1}(t_{k+1}-\tau)^{\alpha-1}\underline{u}_0d\tau+\cdots+\frac{1}{\Gamma(\alpha)}\int_{t_k}^{t_{k+1}}(t_{k+1}-\tau)^{\alpha-1}\underline{u}_k d\tau\\
=&x(t_k)+\sum_{l=1}^k\frac{[(l+1)h]^\alpha-2(lh)^\alpha+[(l-1)h]^\alpha}{\Gamma(\alpha+1)}\underline{u}_{k-l}+\frac{\underline{u}_k}{\Gamma(\alpha+1)}h^\alpha.
\end{aligned}
\label{eq:eqxik_1}
\end{equation}
Then based on (\ref{eq:eqxik_1}), we design the distributed controller for the $i$-th agent as
\begin{equation}
\begin{aligned}
u_{i,k}=&\gamma \sum_{j\in\mathcal{N}_i}a_{ij}\big(x_j(t_k)-x_i(t_k)\big)-\sum_{l=1}^{k}[(l+1)^\alpha-2l^\alpha+(l-1)^\alpha]u_{i,(k-l)},
\end{aligned}
\label{eq:equki}
\end{equation}
which results in the following closed-loop systems
\begin{equation}
\begin{aligned}
x(t_{k+1})=&x(t_k)-\gamma\mathcal{L}x(t_k)\frac{h^\alpha}{\Gamma(\alpha+1)}.
\end{aligned}
\label{eq:eqxik_2}
\end{equation}
\emph{Remark 2:} Existing studies on sampled-data control of fractional-order multi-agent systems divide the whole time interval $[t_0,t]$ into sampling intervals $[t_k,t_{k+1}],\,k=0,1,\cdots$ and solve the fractional-order equations on each sampling interval by treating $x(t_k)$ as the initial value for this corresponding time period. Through this way, the closed-loop fractional-order systems are simply modeled by simplified discrete-time systems in such a way that $x(t_{k+1})$ is expressed as a function only of $x(t_k)$ and $\underline{u}_k$, as is done in the integer-order derivative systems discretization. However, due to the hereditary and infinite memory properties of fractional-order derivative, when converting the actual initial value $x(t_0)$ into $x(t_k)$ for solving fractional-order equations, the second term $\sum_{l=1}^k\frac{[(l+1)h]^\alpha-2(lh)^\alpha+[(l-1)h]^\alpha}{\Gamma(\alpha+1)}\underline{u}_{k-l}$ on the right-hand side of (\ref{eq:eqxik_1}) cannot be neglected. On the contrary, the control scheme design and system analysis in this paper are carried out strictly by bearing the unique properties of fractional-order calculus in mind, which specifically can be seen from (\ref{eq:eqxik_1}), (\ref{eq:equki}) and the boundedness analysis of control signals given later.
\emph{Remark 3:} Note that the second term on the right-hand side of (\ref{eq:eqxik_1}) depends on all the previous control signals $\underline{u}_j,\,j=0,1,\cdots,k-1,$ and thus it cannot be assumed bounded before establishing system stability, giving arise to a challenge in controller design and analysis. Such a challenge is overcome in our proposed controller in (\ref{eq:equki}) which consists of two parts. The first part $u_{i1,k}=\gamma \sum_{j\in\mathcal{N}_i}a_{ij}\big(x_j(t_k)-x_i(t_k)\big) (k=0,1,2,\cdots)$ is designed for achieving consensus of fractional-order multi-agent systems and the second part $u_{i2,k}=-\sum_{l=1}^{k}[(l+1)^\alpha-2l^\alpha+(l-1)^\alpha]u_{i,(k-l)}$ which exists for $t \geq t_1$ aims at compensating for the effect caused by the hereditary and infinite memory properties of fractional-order calculus. It can be observed from (\ref{eq:eqxi1_2}) and (\ref{eq:eqxik_2}) that our designed distributed controllers allow us to analyze the stability of the closed-loop systems in the same way as that of the integer-order discrete-time closed-loop systems, which will be demonstrated in the next subsection.
\subsection{Stability Analysis}
Our main result is presented in the following theorem, where a stability criterion is given.
\emph{Theorem 1:} Consider the closed-loop systems consisting of fractional systems (\ref{eq:eqPro}) and sampled-data based distributed controllers (\ref{eq:equ0i}) and (\ref{eq:equki}). All the signals in the closed-loop systems are globally bounded and asymptotic consensus is achieved, i.e.\ $\lim_{t\rightarrow \infty}\lVert x_i(t)-x_j(t)\rVert=0,\, \forall i, j = 1,2,\cdots,\mathit{N}$, if the design parameters $h$ and $\gamma$ satisfy
\begin{equation}
0 < \gamma\frac{h^\alpha}{\Gamma(\alpha+1)} <\frac{1}{\triangle_{\text{max}}}
\label{eq:eqh}
\end{equation}
and
\begin{equation}
\bigg[\beta^{k+1}+\sum_{l=1}^{k+1}\lvert f(l)\rvert \bigg] \leq 1,
\label{eq:eqT2}
\end{equation}
in which $\triangle_{\text{max}}$ is the maximum degree of the graph $\mathcal{G}$, $f(l)$ is defined in (\ref{eq:eqL2_1}) and
\begin{equation}
\beta = 1-\gamma \lambda_2(\mathcal{L}_s)\frac{h^\alpha}{\Gamma(\alpha+1)},
\label{eq:eqbeta}
\end{equation}
where $\mathcal{L}_s=\frac{\mathcal{L}+\mathcal{L}^\mathrm{T}}{2}$ and $\lambda_2(\mathcal{L}_s)$ represents the second smallest eigenvalue of $\mathcal{L}_s$. Moreover, since the digraph is balanced, asymptotic average-consensus can be achieved, i.e. $\lim_{{t \rightarrow \infty}}x_i(t)=\frac{\sum_{i=1}^N x_i(t_0)}{N}$.
\emph{Proof:} As mentioned above, the main challenge is how to achieve global stability of the resulting systems in the presence of the second term on the right-hand side of (\ref{eq:eqxik_1}). For this purpose, an additional control action is proposed in (\ref{eq:equki}), in order to compensate for the effects of this term. However, with this new control term which is the weighted sum of previous control signals, it is difficult to show the boundedness of control inputs. To overcome this difficulty, we first establish the following relationship
\begin{equation}
\lVert \underline{u}_k\rVert \leq \lVert \underline{u}_0\rVert ,\,\text{for all}\, k \geq 0.
\label{eq:eqL3}
\end{equation}
Now we define error vectors as
\begin{equation}
e(t_k)=\mathcal{L}x(t_k),\, k=0,1,\cdots.
\label{eq:eqek}
\end{equation}
Then the proof of (\ref{eq:eqL3}) is completed through mathematical induction as detailed below.
\emph{Step 1:} According to (\ref{eq:equ0i}), (\ref{eq:equki}) and (\ref{eq:eqek}), $\underline{u}_0$ and $\underline{u}_1$ can be respectively expressed as follows
\begin{equation}
\underline{u}_0 = -\gamma e(t_0),
\label{eq:equ0}
\end{equation}
\begin{equation}
\underline{u}_1 = -\gamma e(t_1) + \lvert f(1) \rvert \underline{u}_0.
\label{eq:equ1}
\end{equation}
Therefore, we can have
\begin{equation}
\begin{aligned}
\lVert \underline{u}_1\rVert =& \big\lVert -\gamma e(t_1) + \lvert f(1) \rvert \underline{u}_0 \big\rVert \\
\leq & \gamma \lVert e(t_1) \rVert +\lvert f(1) \rvert \cdot\lVert \underline{u}_0 \rVert\\
\leq& \gamma \beta\lVert e(t_0) \rVert +\lvert f(1) \rvert \cdot\lVert \underline{u}_0 \rVert \\
= & \big[ \beta +\lvert f(1) \rvert \big]\cdot \lVert \underline{u}_0 \rVert.
\end{aligned}
\label{eq:equ1u0}
\end{equation}
\begin{comment}
Let
\begin{equation}
\beta = 1-\gamma \lambda_2(\mathcal{L}_s)\frac{h^\alpha}{\Gamma(\alpha+1)},
\label{eq:eqbeta}
\end{equation}
where $\mathcal{L}_s=\frac{\mathcal{L}+\mathcal{L}^\mathrm{T}}{2}$ and $\lambda_2(\mathcal{L}_s)$ represents the second smallest eigenvalue of $\mathcal{L}_s$. Thus for (\ref{eq:equ1u0_1}), we get
\begin{equation}
\begin{aligned}
\lvert \underline{u}_1^{\mathrm{T}}\underline{u}_0 \rvert \leq& \Big\lvert \gamma^2\beta e(t_0)^\mathrm{T}e(t_0)+\lvert f(1)\rvert\underline{u}_0^\mathrm{T}\underline{u}_0 \Big\rvert \\
\leq & \Big\lvert \beta + \lvert f(1)\rvert \Big\rvert \cdot \lvert \underline{u}_0^{\mathrm{T}}\underline{u}_0 \rvert.
\end{aligned}
\label{eq:equ1u0_2}
\end{equation}
\end{comment}
By designing $\gamma$ and $h$ in such a way that $\big[\beta + \lvert f(1)\rvert\big] \leq 1$, then we can have $ \lVert \underline{u}_1\rVert \leq \lVert \underline{u}_0\rVert $.
\emph{Step 2:} Assuming that $ \lVert \underline{u}_k\rVert \leq \lVert \underline{u}_0\rVert $ holds for $k>1$. According to (\ref{eq:equki}) and (\ref{eq:eqek}), $ \lVert \underline{u}_{k+1}\rVert $ can be expressed as
\begin{equation}
\begin{aligned}
\lVert \underline{u}_{k+1}\rVert = & \bigg\lVert-\gamma e(t_{k+1})+\sum_{l=1}^{k+1}\lvert f(l) \rvert \underline{u}_{k+1-l} \bigg\rVert \\
\leq & \gamma \lVert e(t_{k+1}) \rVert+\sum_{l=1}^{k+1}\lvert f(l) \rvert \cdot \lVert \underline{u}_{k+1-l}\rVert \\
\leq & \gamma \beta^{k+1}\lVert e(t_0)\rVert +\sum_{l=1}^{k+1}\lvert f(l)\rvert \cdot\lVert \underline{u}_{k+1-l}\rVert .
\end{aligned}
\label{eq:equk+1u0_1}
\end{equation}
Since for $1 \leq l \leq k+1$, $ \lVert \underline{u}_{k+1-l}\rVert \leq \lVert \underline{u}_{0}\rVert $ holds, hence
\begin{equation}
\begin{aligned}
\lVert \underline{u}_{k+1}\rVert \leq & \beta^{k+1}\lVert \underline{u}_0\rVert +\sum_{l=1}^{k+1}\lvert f(l)\rvert \cdot\lVert \underline{u}_{0}\rVert \\
=& \bigg[ \beta^{k+1}+\sum_{l=1}^{k+1} \lvert f(l) \rvert \bigg]\cdot\lVert \underline{u}_{0}\rVert.
\end{aligned}
\label{eq:equk+1u0_2}
\end{equation}
If $\gamma$ and $h$ are chosen to satisfy (\ref{eq:eqT2}), then we have (\ref{eq:eqL3}).
Since $\lVert\underline{u}_0 \rVert= \gamma \lVert e(t_0)\rVert$ is bounded, therefore the global boundedness of all control signals are guaranteed.
Furthermore, (\ref{eq:eqxik_2}) can be rewritten as
\begin{equation}
\begin{aligned}
x(t_{k+1})=Px(t_k)
\end{aligned}
\label{eq:eqxik_3}
\end{equation}
where $P=I_N-\gamma\frac{h^\alpha}{\Gamma(\alpha+1)}\mathcal{L}$ is Perron matrix of digraph $\mathcal{G}$ with $\gamma\frac{h^{\alpha}}{\Gamma(\alpha+1)}$ treated as an integrated gain. Under condition (\ref{eq:eqh}), all the eigenvalues of $P$ are within the unit circle, which further implies the global boundedness of $x(t)$.
Also, from (\ref{eq:eqxik_2}) it can be noticed that the $i$-th agent in the closed-loop systems is described as
\begin{equation}
x_i(t_{k+1})=x_i(t_k)+\gamma\frac{h^{\alpha}}{\Gamma(\alpha+1)}\sum_{j\in\mathcal{N}_i}a_{ij}\big(x_j(t_k)-x_i(t_k)\big).
\label{eq:eqTh1}
\end{equation}
Hence, under Assumption 1, the proof of the condition (\ref{eq:eqh}) for reaching asymptotic consensus follows from that of Theorem 2 in \cite{olfati2007consensus} and thus the theorem is proved.
\section{Illustrative Example}
\label{sec:simulation}
In this section, an example is presented to demonstrate the proposed design control scheme and verify the established theoretical results. By comparing to an existing scheme in \cite{yu2017necessary}, it is revealed that the proposed controller can achieve asymptotic consensus in a more precise way.
Consider a group of five fractional-order agents with the following dynamics
\begin{equation}
\begin{aligned}
{}_{t_0}^{C}\!{\mathcal{D}}_t^{\alpha}&x_{i}(t) = u_i(t), \, i=1,2,\cdots,5,
\end{aligned}
\label{eq:eqsim}
\end{equation}
where $\alpha=0.9$ and initial values of states are $x(t_0)=[4.5,5,6,1.5,-1]^{\mathrm{T}}$.
The connection weights of the graph are $a_{14}=a_{15}=a_{21}=a_{23}=a_{31}=a_{32}=a_{42}=a_{43}=a_{54}=1$ and other entries of $\mathit{A}$ are equal to zero.
Based on (\ref{eq:eqh}) to (\ref{eq:eqbeta}), the designed control parameter and sampling period are respectively selected as $\gamma =0.15$ and sampling period $h=0.85s$. The simulation results are shown in Fig.~\ref{fig:x} to Fig.~\ref{fig:betaf}. From Fig.~\ref{fig:x}, asymptotic consensus and average-consensus is realized with $\lim_{{t \rightarrow \infty}}x_i(t)=x_{\text{final}}=\frac{\sum_{i=1}^5 x_i(t_0)}{5}=3.2$. The value of $\underline{u}^{\mathrm{T}}\underline{u}$ can be observed from Fig.~\ref{fig:u2}, which shows the boundednesses of all control signals. Furthermore, for the purpose of verifying the condition for ensuring all the control signals are globally bounded, the value of $\big[\beta^{k+1}+\sum_{l=1}^{k+1}\lvert f(l)\rvert \big]$ is given in Fig.~\ref{fig:betaf}, from which it can be noticed that inequality (\ref{eq:eqT2}) holds for $\forall k =0,1,\cdots$ with the selected design parameters.
To better illustrate the accuracy and effectiveness of our proposed control algorithm, a comparative simulation study between the control scheme in \cite{yu2017necessary} and in this paper is implemented under the same control parameters.
The mean absolute error $r(t)=\frac{1}{5}\sum_{i=1}^5\lvert x_i(t) - x_{\text{final}}\rvert$ with the proposed scheme and control method in \cite{yu2017necessary} are displayed in Fig.~\ref{fig:r}. Although the control signals under the control scheme in \cite{yu2017necessary} share similar magnitude with our proposed control inputs, which can be observed in Fig.~\ref{fig:u22}, it can be seen from Fig.~\ref{fig:xxxx} and Fig.~\ref{fig:r} that the mean absolute error with control scheme in \cite{yu2017necessary} is larger compared to that with the control scheme in this paper.
\begin{figure}
\centering
\includegraphics[width=10.4cm,height=6.4cm]{x_new.eps}
\caption{The states of (\ref{eq:eqsim}) with the proposed control scheme.}
\label{fig:x}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=10.4cm,height=6.4cm]{u2_new.eps}
\caption{The value of $\underline{u}^{\mathrm{T}}\underline{u}$ of the proposed controller.}
\label{fig:u2}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=10.4cm,height=6.4cm]{betaf_new.eps}
\caption{The value of $\big[\beta^{k+1}+\sum_{l=1}^{k+1}\lvert f(l)\rvert \big]$ under the proposed control scheme.}
\label{fig:betaf}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=10.4cm,height=6.4cm]{xx_new.eps}
\caption{The states of (\ref{eq:eqsim}) with control scheme in \cite{yu2017necessary}.}
\label{fig:xxxx}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=10.4cm,height=6.4cm]{rr_new21.eps}
\caption{The mean absolute error $r$ with proposed scheme and scheme in \cite{yu2017necessary}.}
\label{fig:r}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=10.4cm,height=6.4cm]{u22_new.eps}
\caption{The value of $\underline{u}^{\mathrm{T}}\underline{u}$ of the controller in \cite{yu2017necessary}.}
\label{fig:u22}
\end{figure}
\section{Conclusion}
\label{sec:conclusion}
In this paper, a distributed consensus sampled-data based control scheme for multi-agent systems with fractional-order $\alpha \in (0,1)$ is proposed. By taking the hereditary and infinite memory properties of fractional-order calculus into account, a new control scheme is designed to not only ensure system stability, but also achieve asymptotic consensus. Simulation results also illustrate the accuracy and effectiveness of the proposed control algorithm.
\bibliographystyle{unsrt}
\begin{comment}
|
1,116,691,499,247 | arxiv | \section{Introduction}\label{sec:intro}
Unconditional image generation has been a hot research topic in the last
decade. In image generation, a generative model is trained to learn the
image data distribution from a finite set of training images. Once
trained, the generative model can synthesize images by sampling from the
underlying distribution.
GANs have been widely used for unconditional image generation with
impressive visual quality in recent years
\cite{goodfellow2014generative}. Despite the evident advantages of GANs,
their training is a non-trivial task: GANs are sensitive to training
hyperparameters and generally suffer from convergence issues
\cite{hoshen2019non}. This concern has led to the development of
improved GAN training methods \cite{gulrajani2017improved} or
non-adversarial approaches \cite{hoshen2019non}. Yet, the great majority
of existing generation techniques utilize deep learning (DL), a method
for learning deep neural networks, as the modeling backbone.
A neural network is typically trained using a large corpus of data over
long episodes of iterative updates. Therefore, training a neural network
is often a time-consuming and data-hungry process. To ensure the
convergence of deep neural networks (DNNs), one has to carefully select
(or design) the neural network architecture, the optimization objective
(or the loss function), and the training hyper-parameters. DL-based
generative models are often specifically engineered to perform a single
task. They cannot be easily generalized to different related generative
applications. The architectures of neural networks for unconditional
image generation have to be re-designed for image super-resolution or
attribute-guided image generation. Last but not the least, due to
the non-linearity of neural networks, understanding and explaining their
performance is a standing challenge.
To address the above-mentioned concerns, this paper presents an
alternative approach for unconditional image generation based on
successive subspace learning (SSL) \cite{kuo2016understanding,
kuo2017cnn, kuo2018data, kuo2019interpretable}. The resulting method,
called progressive attribute-guided extendable robust image generative
(PAGER{}) model, has several advantages, including mathematical
transparency, progressive content generation, lower training time,
robust performance with fewer training samples, and extendibility to
conditional image generation.
PAGER{} consists of three modules: 1) core generator, 2) resolution
enhancer, and 3) quality booster. The core generator learns the
distribution of low-resolution images and performs unconditional image
generation. The resolution enhancer increases image resolution via
conditional generation. Finally, the quality booster adds finer details
to generated images.
To demonstrate the generative performance of PAGER{}, we conduct
extensive experiments on MNIST, Fashion-MNIST, and CelebA datasets. We
show that PAGER{} can be trained in a fraction of the time required
for training DL based models and still achieve a similar generation
quality. We then demonstrate the robustness of PAGER{} to the training
size by reducing the number of training samples. Next, we show that
PAGER{} can be used in image super resolution, high-resolution image
generation, and attribute-guided face image generation. In particular,
the modular design of PAGER{} allows us to use the conditional
generation modules for image super resolution and high-resolution image
generation. The robustness of PAGER{} to the number of training
samples enables us to train multiple sub-models with smaller subsets of
data. As a result, PAGER{} can be easily used for attribute-guided
image generation.
The rest of this paper is organized as follows. Related work is reviewed
in Sec.~\ref{sec:related}. The PAGER{} method is proposed in
Sec.~\ref{sec:method}. Experimental results are reported in
Sec.~\ref{sec:experiments}. Extendability and applications of PAGER{}
are presented in Sec.~\ref{sec:applications}. Finally, concluding
remarks and possible future extensions are given in
Sec.~\ref{sec:conclusion}.
\section{Related Work}\label{sec:related}
\subsection{DL-based Image Generative Models}\label{subsec:DL}
DL-based image generative models generally consist of two components:
feature extraction and generation. The feature extraction component
converts images into a feature domain where probability density
functions can be learned in the form of a distribution. The generation
component samples a feature vector from the underlying distribution and
converts it to an image.
DL-based image generative models can be categorized into two main
classes: adversarial-based and non-adversarial-based models. GANs
\cite{goodfellow2014generative} are adversarial-based generative models
that consist of a generator and a discriminator. The training procedure
of a GAN is a min-max optimization where the generator learns to
generate realistic samples that are not distinguishable from those in
the original dataset and the discriminator learns to distinguish between
real and fake samples. Once the GAN model is trained, the generator
model can be used to draw samples from the learned distribution.
Examples of non-adversarial DL-based generative models include
variational auto-encoders (VAEs) \cite{kingma2013auto}, flow-based
models \cite{dinh2014nice, dinh2016density}, GLANN \cite{hoshen2019non},
and diffusion-based models \cite{dhariwal2021diffusion, ho2022cascaded}.
VAEs have an encoder/decoder structure that learns variational
approximation to the density function. Then, they generate images from
samples of the Gaussian distribution learnt through the variational
approximation. Flow-based methods apply a series of invertible
transformations on data to transform the Gaussian distribution into a
complex distribution. Following the invertible transformations, one can
generate images from the Gaussian distribution. GLANN
\cite{hoshen2019non} employs GLO \cite{bojanowski2017optimizing} and
IMLE \cite{li2018implicit} to map images to the feature and the noise
spaces, respectively. The noise space is then used for sampling and
image generation. Recently, diffusion-based models are developed for
image generation. During the training process, they add noise to images
in multiple iterations to ensure that the data follows the Gaussian
distribution ultimately. For image generation, they draw samples from
the Gaussian distribution and denoise the data in multiple gradual steps
until clean images show up.
Despite impressive results of DL-based generative models, they are
mathematically not transparent due to their highly non-linear
functionality. Furthermore, they are often susceptible to unexpected
convergence problems \cite{hoshen2019non}, long training time, and
dependency on large training dataset size. As we show in our
experiments, PAGER{} addresses the aforementioned concerns while
maintaining the quality of the images generated by DL-based techniques.
\subsection{Unconditional and Conditional Image Generation}
In unconditional image generation, sample images are drawn from an
underlying distribution without any prior assumption on the images to be
generated. In conditional image generation, samples are generated under
a certain assumption. One example of the latter is the generation of a
high-resolution image given a low-resolution image. The proposed
PAGER{} method contains both unconditional and conditional image
generation techniques. Its core generator module employs the
unconditional image generation technique. Its resolution enhancer and
quality booster modules perform conditional image generation. Although
PAGER{} is an unconditional image generator by itself, it can be
easily extended to conditional image generation with rich applications.
We will elaborate this point with three examples, namely,
attribute-guided face image generation, image super resolution, and
high-resolution image generation. Each task is elaborated below.
\textbf{Attribute-guided face image generation:} For a set of required
facial attributes, the goal is to generate face images that meet the
requirements. \cite{lu2018attribute} performs attribute-guided face
image generation using a low-resolution input image. It modifies the
original CycleGAN \cite{zhu2017unpaired} architecture and its loss
functions to take conditional constraints during training and inference.
In \cite{kowalski2020config}, synthetic labeled data are used to
factorize the latent space into sections which associate with separate
aspects of face images. It designs a VAE with an additional attribute
vector to specify the target part in the factorized latent space.
\cite{qian2019make} proposes to learn a geometry-guided disentangled
latent space using facial landmarks to preserve generation fidelity. It
utilizes a conditional VAE to sample from a combination of
distributions. Each of them corresponds to a certain attribute.
\textbf{Image super-resolution:} The problem aims at generating a
high-resolution image that is consistent with a low-resolution image
input. One solution is the example-based method
\cite{freeman2002example}. Others include auto-regressive models and
normalized flows \cite{van2016conditional,parmar2018image,
yu2020wavelet}. Quite a few recent papers adopt the DL methodology
\cite{dong2014learning}. Another line of work treats super-resolution as
a conditional generation problem, and utilize GANs or diffusion-based
models as conditional generative tools which use low-resolution images
as the generation condition \cite{ledig2017photo,chen2018fsrnet,
saharia2021image}.
\textbf{Progressive generation of very-high-resolution Images:}
Generation of a very-high-resolution image of high quality is
challenging and treated as a separate research track. It demands a
progressive approach in training and generation to maintain the model
stability and generation quality. There exist both GAN-based and
diffusion-based very-high-resolution image generation solutions
\cite{karras2017progressive, ho2022cascaded}.
Our PAGER{} method can be trained for unconditional image generation
as well as for conditional image generation such as attribute-guided
face image generation and image super-resolution. In principle, it can
also be used for progressive generation of very-high-resolution images.
Our PAGER{} serves as a general framework that can bridge different
generation models and applications.
\subsection{Successive Subspace Learning (SSL)}\label{subsec:SSL}
In order to extract abstract information from visual data, spectral or
spatial transforms can be applied to images. For example, the
Fourier transform is used to capture the global spectral information of
an image while the wavelet transform can be exploited to extract the
joint spatial/spectral information. Two new transforms, namely, the
Saak transform \cite{kuo2018data} and the Saab transform
\cite{kuo2019interpretable}, were recently introduced by Kuo {\em et al.}
\cite{kuo2016understanding, kuo2017cnn, kuo2018data,
kuo2019interpretable} to capture joint spatial/spectral features. These
transforms are derived based on the statistics of the input without
supervision. Furthermore, they can be cascaded to find a sequence of
joint spatial-spectral representations in multiple scales, leading to
Successive Subspace Learning (SSL). The first implementation of SSL is
the PixelHop system \cite{chen2020pixelhop}, where multiple stages of
Saab transforms are cascaded to extract features from images. Its second
implementation is PixelHop++, where channel-wise Saab transforms are
utilized to achieve a reduced model size while maintaining an effective
representation \cite{chen2020pixelhop++}.
SSL has been successfully applied to many image processing and computer
vision applications. Several examples include unconditional image
generation \cite{lei2022genhop, lei2020nites, lei2021tghop}, point cloud
analysis \cite{zhang2020pointhop, zhang2020pointhop++,
zhang2020unsupervised, kadam2020unsupervised, zhang2021gsip,
kadam2021gpco, kadam2021r}, fake image detection
\cite{chen2021defakehop, chen2021geo, chen2022defakehop++,
zhu2021pixelhop}, face recognition \cite{rouhsedaghat2021facehop,
rouhsedaghat2021low}, medical diagnosis \cite{liu2021voxelhop,
monajatipoor2021berthop}, low light enhancement \cite{azizi2020noise},
anomaly detection \cite{zhang2021anomalyhop}, to name a few. Inspired
by the success of SSL, we adopt this methodology in the design of a new
image generative model as elaborated in the next section.
\subsection{SSL-based Image Generative Models}\label{subsec:GenHop}
GenHop \cite{lei2022genhop} is the contemporary SSL-based image
generative model in literature. GenHop utilizes SSL for feature
extraction. It applies independent component analysis (ICA) and
clustering to obtain clusters of independent feature components at the
last stage of SSL. Then, it finds a mapping between the distribution of
ICA features and Guassian distributions. In this work, we do not
perform ICA but model the distribution of SSL features via GMMs
directly. As compared to GenHop, our approach offers several attractive
features. First, it has lower computational complexity and demands less
memory. Second, our method offers a progressive and modular image
generation solution. It is capable of conditional and attribute-guided
image generation. It can also be easily extended to other generative
applications such as super-resolution or high-resolution image
generation.
\section{Proposed PAGER{} Method}\label{sec:method}
The PAGER{} method is presented in this section. First, our research
motivation is given in Sec. \ref{subsec:motivation}. Then, an overview
on PAGER{} and its three modules are described in Sec.
\ref{subsec:overview}. Finally, our attribute-guided face image
generation is elaborated in Sec. \ref{subsec:attributes}.
\subsection{Motivation}\label{subsec:motivation}
A generative model learns the distribution of the training data in the
training phase. During the generation phase, samples are drawn from the
distribution as new data. To improve the accuracy of generative image
modeling, gray-scale or color images should be first converted into
dimension-reduced latent representations. After converting all training
images into their (low-dimensional) latent representation, the
distribution of the latent space can be approximated by a multivariate
Gaussian distribution. For learning the latent representation, most
prior work adopts GAN-, VAE-, and diffusion-based generative models;
they train neural networks that can extract latent representations from
an image source through a series of nonlinear transformations.
Similarly, we need to learn such a transformation from the image space
to the latent representation space.
\begin{figure*}[t]
\includegraphics[width=\linewidth]{figures/gauss.png}
\caption{Example distributions from RGB pixels (left block) and Saab
transforms (right block). The top figures correspond to single vector
dimensions ($I_0\dots I_2$ in RGB and $X_0\dots X_2$ in Saab domains). The
bottom figures correspond to joint distributions. Distributions are
extracted from the first three components of CelebA
images.}\label{fig:gmm}
\end{figure*}
In this work, we utilize SSL pipleline, rather than neural networks, to achieve the transformation to the latent representation space. An SSL pipeline consists of consecutive Saab transforms. In essence, the SSL pipeline receives an image $I\in \mathbb{R}^{w\times h\times c}$ and converts it into a latent feature vector $X\in \mathbb{R}^n$, where $w$, $h$ and $c$ are the pixel numbers of the width, height and the color number of an image while $n$ is the dimension of the latent vector. In the remainder of this paper, we refer to the latent space obtained by SSL as the \textit{core space}. The Saab transform utilizes mean calculation and PCA computation to extract features from its input. Due to the properties of PCA, the $i$-th and $j$-th components in the core space are uncorrelated for $i\neq j$. This property facilitates the use of Gaussian priors for generative model learning over the core space.
Fig.~\ref{fig:gmm} illustrates the distributions of input image pixels ($I$) and Saab outputs ($X$). In this example, we plot the distributions of the first, second and third components of $I$ (i.e., the RGB values of the upper-left pixel of all source images) and $X$ (i.e., the Saab transform coefficients).
The RGB components are almost uniformly distributed in the marginal probability. They are highly correlated as shown in the plot of joint distributions. In contrast, Saab coefficients are close to the Gaussian distribution and they are nearly uncorrelated. While the distributions of one- and two-dimensional components of $X$ are very close to Gaussians, the distribution of higher-dimensional vectors might not be well modeled by one multivariate Gaussian distribution. For this reason, we employ a mixture of Gaussians to represent the distribution of the core space.
\begin{figure*}[t]
\includegraphics[width=\linewidth]{figures/overview.png}
\caption{Overview of PAGER{} generation method.}\label{fig:overview}
\end{figure*}
\subsection{System Overview}\label{subsec:overview}
An Overview of the PAGER{} generation method is shown in
Fig.~\ref{fig:overview}. PAGER{} is an unconditional generative model with a progressive approach in image generation. It starts with unconditional generation in a low-resolution regime, which is performed by the core generator. Then, it sequentially increases the image resolution and quality through a cascade of two conditional generation modules: the resolution enhancer and the quality booster.
\subsubsection{Module 1: Core Generator}\label{sec:generation}
The core generator is the unconditional generative module in PAGER{}. Its goal is to generate low-resolution (e.g., $4\times4\times 3$) color images. This module is trained with images of shape $2^d\times 2^d\times 3$ (e.g., $d=2$). It applies consecutive Saab transforms on input images $\{I_i\}_{i=1}^{M}$ using PixelHop++ structure \cite{chen2020pixelhop++}, ultimately converting images into $n$-dimensional vectors $X\in \mathbb{R}^n$ ($n=2^d\times2^d\times3$) in core space. The goal of the core generator is to learn the distribution of $\{X_i\}_{i=1}^{M}$. We use $\mathcal{X}$ to denote a random variable within $\{X_i\}_{i=1}^{M}$, representing observed samples in core space. Let $P(\mathcal{X})$ be the underlying distribution of $\mathcal{X}\in \mathbb{R}^n$. The generation core $G$ attempts to approximate the distribution $P(\mathcal{X})$ with a distribution $G(\mathcal{X})$.
DL-based methods utilize iterative end-to-end optimization of neural networks to achieve this objective. In PAGER{}, we model the underlying
distribution of the core space using the Gaussian Mixture Model (GMM), which is highly efficient in terms of training time. This is feasible since we use SSL to decouple random variables, which we illustrated in Sec. \ref{subsec:motivation}. The conjunction of multi-stage Saab (SSL) features and GMMs can yield a highly accurate density modeling. Formally, the GMM approximation of $G(\mathcal{X})$ is defined as follows:
\begin{equation}\label{eq:GMM}
G(\mathcal{X})=\sum_{k=1}^{K} p_k \mathcal{N}(\mathcal{X},\mu_k,\Sigma_k),
\end{equation}
where $\mathcal{N}(\mathcal{X},\mu_k,\Sigma_k)$ is a multi-variate
normal distribution with mean $\mu_k$ and diagonal covariance matrix
$\Sigma_k$, and $p_k$ is a binary random variable. We have $p_k=1$ with
probability $P_k$, $p_k=0$ with probability $(1-P_k)$ and
$\sum_{k=1}^{K} P_k=1$. In other words, only one of the $K$ Gaussian
models will be selected at a time, and the probability of selecting
the $k$-th Gaussian model is $P_k$ in such a GMM. The parameters of the
GMM can be determined using the Expectation Maximization (EM) algorithm
~\cite{reynolds2009gaussian}. Once such a GMM model is obtained, one
can draw a sample, $X$, randomly and proceed to Modules 2 and 3.
The need for Modules 2 and 3 is explained below. $G(\mathcal{X})$ is
learned from observations $X_i$, $i=1 \cdots M$. When the dimension,
$n$, of the core space is large, estimating $G(\mathcal{X})$ becomes
intractable and the approximation accuracy of GMM would drop. For this
reason, the unconditional generation process is constrained to a
low-dimensional space. Then, we employ conditional generative models (modules 2 and 3) to
further increase image resolution and quality.
\subsubsection{Module 2: Resolution Enhancer}\label{subsec:module_2}
We represent image $I_d$ as the summation of its DC and AC components:
\begin{eqnarray}\label{eq:DC_AC_decomposition}
I_d &=& DC_d + AC_d, \\
DC_d &=& U(I_{d-1}),
\end{eqnarray}
where ${I_{d}}$ is an image of size ${2^{d}\times 2^{d}}$, $U$ is the
Lanczos image interpolation operator, $DC_d$ is the interpolated image
of size ${2^{d}\times 2^{d}}$ and $AC_d$ is the residual image of size
$2^d \times 2^d$. The above decoupling of DC and AC components of an
image allows to define the objective of the resolution enhancer. It
aims to generate the residual image $AC_d$ conditioned on $DC_d$. In
Fig.~\ref{fig:overview}, a multi-stage cascade of resolution enhancers
is shown. The detail of a representative resolution enhancer is
highlighted in the lower subfigure.
To train the resolution enhancer, we first decouple the DC and AC of
training samples. Then, we extract SSL features from the DC and build a
GMM model with $K$ components, denoted by $G_{DC}$. By this method, we
learn a distribution of the DC at a certain image resolution. Note that
each DC from a training image belongs to one of the Gaussian models in
$G_{DC}$. Therefore, DCs (and their associated AC) are clustered into
$K$ classes using $G_{DC}$. We gather the AC of each class and build a
corresponding GMM, denoted by $G_{AC,k}$ where $k\in\{1, \cdots, K\}$.
In total, we learn $K+1$ GMMs: $\{G_{DC}, \, G_{AC,1} \, \dots \,
G_{AC,K}\}$.
At the test time, the resolution enhancer receives the low resolution
image $I_{d-1}$, and upsamples it to obtain the interpolated DC, i.e.,
$DC_d = U(I_{d-1})$. Then, the resolution enhancer converts the DC to
its SSL features and classifies it into one of $K$ clusters using
$G_{DC}$. Mathematically, we have
\begin{eqnarray}
X_{DC} &=& \mbox{SSL} (DC_d), \\
y &=& \mbox{arg}_k \max {\{\mathcal{N}(X_{DC}, \mu_k, \Sigma_k)\}_{k=1}^{K}},
\end{eqnarray}
where $\mathcal{N}(X_{DC}, \mu_k, \Sigma_k)$ is the probability score of
$X_{DC}$ according to the $k$-th component of $G_{DC}$, and the
classification label $y$ is the maximizer index. In other words, the
resolution enhancer identifies a cluster of samples that are most
similar to $DC_{d}$. Next, the resolution enhancer draws a sample from
the $AC$ distribution corresponding to class $y$:
\begin{equation}
X_{AC} \sim G_{AC,y}(\mathcal{X}_{AC}).
\end{equation}
With the above two-step generation, the resolution enhancer generates
$X_{AC}$ conditioned on $X_{DC}$. Afterwards, $X_{AC}$ is converted to
the RGB domain using the inverse SSL transform:
\begin{equation}\label{eq:ac_inverse}
AC_d = \mbox{SSL}^{-1}(X_{AC}).
\end{equation}
The computed AC component is masked and added to the DC to yield
the higher resolution image via
\begin{eqnarray}\label{eq:mask}
I_d &=& DC_d+\widehat{AC}_d, \\
\widehat{AC}_d &=& M(DC_d)\odot AC_d,
\end{eqnarray}
where $M(DC_d)$ is a mask and $\odot$ denotes elementwise
multiplication. The mask is derived from the edge information obtained
by the Canny edge detector~\cite{canny1986computational}. The masking
operation serves two objectives. First, it prevents details from being
added to smooth regions of the DC component. Second, it suppresses
unwanted noise. Once $I_d$ is generated, it is cropped into four
non-overlapping regions, and each region goes through another resolution
enhancement process. The process is recursively applied to each
sub-region to further enhance image quality. In our experiments, we
continue the recursion until a cropped windows size of $2\times 2$ is
reached.
\subsubsection{Module 3: Quality Booster}\label{subsec:module_3}
The right subfigure of Fig.~\ref{fig:overview} presents the quality
booster module. It follows the resolution enhancer by adding detail and
texture to the output of the resolution enhancer. It exploits the
locally linear embedding (LLE)~\cite{roweis2000nonlinear} scheme and
adds extra residue values that are missed by the resolution enhancer.
LLE is a well known method in building correspondence between two
components in image super resolution~\cite{chang2004super,
johnson2019billion} or image restoration~\cite{huang2016visible}. To
design the quality booster, we decompose the training dataset, enhance
the DC component, and compute the residuals as follows:
\begin{eqnarray}
I_d & = & DC_d+AC_d, \\
E_d & = & \mbox{Enhancer}(DC_d), \\
R_d & = & I_d- E_d,
\end{eqnarray}
where ${I}_d$ represents a ${2^d\times 2^d}$ training image, ${E_d}$ is
the result of applying the enhancer module to the DC component of the
image, and $R_d$ is the residual image. During training, the quality
booster stores $E_d^i$ and $R_d^i$, $i=1, \cdots, M$ from $M$ training
samples. In generation, the quality booster receives image $E_d$ and
uses the LLE algorithm to estimate the residual image for image $E_d$
based on $E_d^i$ and $R_d^i$ from the training dataset. It approximates
the residual image with a summation of several elements within $R_d^i$.
Readers are referred to \cite{roweis2000nonlinear} for details of LLE
computation. Similar to the enhancer module, the computed $R_d^i$ is
masked and added to $E_d$ to boost its quality.
\subsection{Attribute-Guided Face Image Generation}\label{subsec:attributes}
In attribute-guided face image generation, the goal is to synthesize
face images that have certain properties. Let $A\in\{-1, +1\}^T$ denote
a set of $T$ binary attributes. The goal is to synthesize an image that
satisfies a query ${\bf q}\in\{-1, 0, +1\}^T$, where -1, 0, +1 denote
negative, don't care, and positive attributes. For instance, if the
attribute set is \{\textit{male}, \textit{smiling}\}, the query ${\bf
q}=[-1, +1]$ requests an image of a female smiling person, and the query
${\bf q}=[0,-1]$ request an image (of any gender) that is not smiling.
Without loss of generality, we explain the attribute-guided generation
process with $T=7$. The attributes selected from attribute labels in
CelebA dataset include `gender', `smiling', `blond hair', `black hair',
`wearing lipstick', `bangs' and `young'. Given these seven binary
attributes, there are $2^7=128$ subsets of data that correspond to each
unique set of selected attributes. However, some of the attribute
combinations might not be abundant in the training data due to the
existing correlation between the attributes. For instance, `wearing
lipstick', `bangs', and `gender' are highly correlated. Thus, instead of
considering all 128 combinations, we partition the attributes of
training data into $K$ subsets using k-means clustering (we set $K=10$
in our experiments). Based on the attribute clusters, we create $K$
data subsets and train a separate PAGER{} model for each subset.
At generation time, the goal is to synthesize a sample with a given
attribute set, ${\bf q}\in\{-1, 0, +1\}^7$. To determine which of the 10
models best represents the requested attribute set, we compute the
Cosine distance of ${\bf q}$ to each of the cluster centers and select
the model that gives the minimum distance. Then, we draw samples from
the corresponding model. Fig.~\ref{fig:attributes} shows generated
images corresponding to 15 different attribute vectors. We see that the
attribute-based generation technique can successfully synthesize images
with target attributes while preserving diversity and fidelity.
\begin{figure*}[t]
\includegraphics[width=\linewidth]{figures/attributes_1.png}
\caption{Examples of attribute-guided generated images for CelebA with various attribute
combinations.}\label{fig:attributes}
\end{figure*}
\section{Experiments}\label{sec:experiments}
\subsection{Experimental Setup}
We perform experiments on three datasets: MNIST, Fashion-MNIST, and
CelebA. They are commonly used for learning unconditional image
generative models. We briefly explain the experimental settings of
PAGER{} for each dataset below.
\textbf{CelebA.} The dataset is a set of colored human face images.
Suppose that there are $2^d\times 2^d$ pixels per image. To derive Saab
features and their distributions, we apply $d$-stage cascaded Saab
transforms. At each stage, the Saab filter has a spatial dimension of
$2\times2$ with stride $2$. The number of GMM components in the core
generator is $500$. The core generator synthesizes color images of size
$4\times 4$. Higher resolution images are generated conditioned on the
previous resolution with the resolution enhancer and the quality booster
modules in cascade ($4\times 4 \rightarrow 8\times 8 \rightarrow
16\times 16 \rightarrow 32\times 32$). The resolution enhancer has $100$
GMM components for the DC part and $3$ GMM components for the AC part at
each stage. LLE in the quality booster module is performed using $2$
nearest neighbors.
\textbf{MNIST and Fashion-MNIST.} The two datasets contain gray-scale
images of digits and clothing items, respectively. The generation
pipeline for these datasets is similar to CelebA except that the core
generator synthesizes $16\times 16$ padded gray-scale images for each of the
$10$ classes. The $16\times 16$ images are converted to $32\times 32$
with a single stage of resolution enhancer and quality booster. Finally, they are cropped to $28\times 28$.
\begin{figure*}[t]
\centering
\includegraphics[width=\linewidth]{figures/ThreeDatasets_2.png}
\caption{Examples of PAGER{} generated images for MNIST (top), Fashion-MNIST (middle), and CelebA (bottom) datasets.} \label{fig:mnist}
\end{figure*}
\subsection{Evaluation of Generated Image Quality}
{\bf Subjective Evaluation.} We show image samples of resolution $32
\times 32$ generated by PAGER{} for MNIST, Fashion-MNIST and CelebA in
Fig.~\ref{fig:mnist}. Generated images learned from MNIST represent the structure of digits accurately and with rich diversity. Images
generated from Fashion-MNIST show diverse examples for all classes with fine details and textures. Generated images for CelebA are semantically meaningful and with fine and diverse detail in skin tone, eyes, hair and lip color, gender, hairstyle, smiling, lighting, and angle of view.
{\bf Objective Evalution.} We use the Frechet Inception Distance
(FID) \cite{heusel2017gans} score to perform quantitative comparison of our method with prior art. FID is a commonly used metric to evaluate the performance of generative models. It considers both diversity and fidelity of generated images. We follow the procedure described in \cite{lucic2018gans} to obtain the FID scores; an Inception neural network extracts features from a set of 10K generated images as well as another set of 10K real (test) images. Two multivariate Gaussians are fit to the extracted features from two sets separately. Then, the Frechet distance between their mean vectors and covariance matrices is calculated. A smaller FID score is more desirable as it indicates a better match between the synthesized and real test samples.
\begin{table}[t]
\caption{Comparison of FID scores for MNIST, Fashion-MNIST and CelabA datasets.}\label{tab:fid_score}
\centering
\begin{tabular}{l|lll}
Method & MNIST & Fashion & CelabA \\ \hline
MM GAN \cite{goodfellow2014generative} & 9.8 & 29.6 & 65.6 \\
NS GAN \cite{goodfellow2014generative} & 6.8 & 26.5 & 55.0 \\
LSGAN \cite{mao2017least} & 7.8 & 30.7 & 53.9 \\
WGAN \cite{arjovsky2017wasserstein} & 6.7 & 21.5 & 41.3 \\
WGAN-GP \cite{gulrajani2017improved} & 20.3 & 24.5 & 30.0 \\
DRAGAN \cite{kodali2017convergence} & 7.6 & 27.7 & 42.3 \\
BEGAN \cite{berthelot2017began} & 13.1 & 22.9 & 38.9 \\ \hline
VAE \cite{kingma2013auto} & 23.8 & 58.7 & 85.7 \\
GLO \cite{bojanowski2017optimizing} & 49.6 & 57.7 & 52.4 \\
GLANN \cite{hoshen2019non} & 8.6 & 13.0 & 46.3 \\ \hline
GenHop \cite{lei2022genhop} & 5.1 & 18.1 & 40.3 \\ \hline
PAGER{} (Ours) & 9.5 & 19.3 & 43.8 \\
\end{tabular}
\end{table}
The FID scores of various methods for MNIST, Fashion-MNIST and CelabA
datasets are compared in Table~\ref{tab:fid_score}. Methods in the
first and second sections are both based on DL. Methods in the first
section are adversarial generative models while those in the second
section are non-adversarial. The results of the first and second
sections are taken from \cite{lucic2018gans} and \cite{hoshen2019non},
respectively. GenHop in Section 3 does not use a neural network
backbone. Its results are taken from \cite{lei2022genhop}. We see from
Table~\ref{tab:fid_score} that the FID scores of PAGER{} are
comparable with prior generative models. In training
PAGER{} models for Table~\ref{tab:fid_score}, we used 100K training
images from CelebA dataset and 60K training images from MNIST and
Fashion-MNIST datasets, with no augmentation.
\begin{figure*}[t]
\centering
\includegraphics[width=\linewidth]{figures/training_size_comparison_2.png}
\caption{Comparison of FID scores of six benchmarking methods with six
training sizes (1K, 2K, 5K, 10K, 20K and 60K) for the MNIST dataset.
The FID scores of PAGER{} are significantly less sensitive with
respect to smaller training sizes.} \label{fig:training_comparison}
\end{figure*}
\subsection{Other Performance Metrics}
In this section we study additional performance metrics: robustness to
the number of training samples and training time.
{\bf Robustness to training dataset sizes.}
Fig.~\ref{fig:training_comparison} presents the FID score of PAGER{}
and five DL-based generative models (MM GAN, LSGAN, WGAN, WGAN-GP and
GLANN) when the number of training samples is set to 1K, 2K, 5K, 10K,
20K and 60K. To produce the FID scores of the GAN-based related
work, we use the open-source implementation by PyTorch-GAN
\footnote{\url{https://github.com/eriklindernoren/PyTorch-GAN}}.
For GLANN, we use the implementation provided by the authors. Since
GLANN is not trained with less than 10K samples, its FID scores for 1K,
2K and 5K samples are not available. It is worth noting that the FID
scores for 60K training samples of some prior work in
Fig.~\ref{fig:training_comparison} are different than those in
Table~\ref{tab:fid_score}. This happens because some of prior generative
models (e.g., MM GAN, LSGAN, and WGAN) are too sensitive to training
hyper-parameters and/or data augmentation \cite{lucic2018gans}. The scores reported in Fig.~\ref{fig:training_comparison} are the best
FID scores obtained using the default hyper-parameters in the
open-source library. We see from Fig.~\ref{fig:training_comparison}
that PAGER{} is least affected by the number of training samples. Even
with the number of training samples as small as 1K, PAGER{} has an FID
score of 16.2 which is still better than some prior works' original FID
scores presented in Table \ref{tab:fid_score}, such as WGAN-GP, VAE and
GLO. Among prior works, GLANN is less sensitive to training size but
cannot be trained with less than 10K samples.
{\bf Comparison of Training Time.} The training time of PAGER{} is
compared with prior work in Table~\ref{tab:runtime} on two platforms.
\begin{itemize}
\item CPU (Intel Xeon 6130): The CPU training time of PAGER{} is
slightly more than 4 minutes, which is significantly less than all other
methods as shown in Table~\ref{tab:runtime}. The normalized CPU
training times of various methods against PAGER{} are visualized in
the left subfigure of Fig.~\ref{fig:time_ratio}. PAGER{} is $11\times$
faster than WGAN and $325\times$ faster than LSGAN.
\item GPU (NVIDIA Tesla V100): The GPU training time of PAGER{} is
around 3 minutes, which is again less than all other methods as shown in
Table~\ref{tab:runtime}. The normalized GPU run times of various
methods are also visualized in the right subfigure of
Fig.~\ref{fig:time_ratio}. PAGER{} is $9\times$ faster than WGAN and
$48\times$ faster than GLANN.
\end{itemize}
\begin{table}[t]
\caption{Training time comparison.}\label{tab:runtime}
\centering
\begin{tabular}{l|ll}
Method & CPU & GPU \\ \hline
MM GAN \cite{goodfellow2014generative} & 93m14s & 33m17s \\
LSGAN \cite{mao2017least} & 1426m23s & 45m52s \\
WGAN \cite{arjovsky2017wasserstein} & 48m11s & 25m55s \\
WGAN-GP \cite{gulrajani2017improved} & 97m9s & 34m7s \\ \hline
GLO \cite{bojanowski2017optimizing} & 1090m7s & 139m18s \\
GLANN \cite{hoshen2019non} & 1096m24s & 142m19s \\ \hline
PAGER{} (Ours) & 4m23s & 2m59s \\
\end{tabular}
\end{table}
\begin{figure}[t]
\centering
\includegraphics[width=0.8\columnwidth]{figures/TimeRatio.png}
\caption{Comparison of normalized training time, where each bar represents
the training time corresponding to those shown in Table~\ref{tab:runtime}
and normalized by training time of PAGER{}.} \label{fig:time_ratio}
\end{figure}
{\bf Joint Consideration of FID Scores and Training Time.} To provide a
better picture of the tradeoff between training time and FID score, we
present both of these metrics in Fig.~\ref{fig:FID_time}. On this
figure, points that are closer to the bottom left are more desirable. As
seen, PAGER{} significantly outperforms prior art when considering FID
scores and training time jointly.
\begin{figure}[t]
\centering
\includegraphics[width=0.75\columnwidth]{figures/FID_time.png}
\caption{Comparison of joint FID scores and GPU training time of PAGER{} with related
work in the generation of MNIST-like images. PAGER{} provides the best
overall performance since it is closest to the left-bottom corner.}
\label{fig:FID_time}
\end{figure}
\subsection{Discussion}
Based on the above experimental results, we can draw the following
conclusions.
\begin{itemize}
\item \textbf{Quality image generation.} The FID scores of PAGER{} are
comparable with state-of-the-art DL-based image generation techniques on
common datasets. This indicates that PAGER{} can generate images of
similar quality to prior art. \item \textbf{Efficient training.}
PAGER{} can be trained in a fraction of the time required by DL-based
techniques. For example, our MNIST generative model is trained in 4
minutes on a personal computer's CPU while the fastest prior work
demands 25-minute training time on an industrial GPU. The efficiency of
PAGER{} is achieved by the development of a non-iterative training
scheme. CPU-based efficient training implies smaller energy consumption
and carbon footprint than GPU-based DL methods. This is a major
advantage of PAGER{}.
\item \textbf{Robustness to training sample size.} PAGER{} can still
yield images of reasonable quality even when the number of training
samples is drastically reduced. For example, in Fig.~\ref{fig:training}
we show that the number of training samples can be reduced from 100K to
5K with only a negligible drop in the generated image quality for the
CelebA dataset.
\end{itemize}
\begin{figure}[t]
\centering
\includegraphics[width=\columnwidth]{figures/training_size_2.png}
\caption{Comparison of PAGER{}'s FID scores with six training
sample sizes for CelebA, Fashion-MNIST and MNIST datasets. We see that
the FID scores do not increase significantly as the training samples
number is as low as 5K for CelebA and 1K for MNIST and Fashion-MNIST.}
\label{fig:training}
\end{figure}
\begin{figure*}[t]
\centering
\includegraphics[width=\linewidth]{figures/SR_1.png}
\caption{Illustration of PAGER{}'s application in image
super-resolution for CelebA images: Two top rows starting from
resolution $4\times 4$ (left block) and $8\times 8$ (right block) and
ending at resolution $32\times 32$. Two middle rows starting from
resolution $8\times 8$ (left block) and $16\times 16$ (right block) and
ending at resolution $64\times 64$. Two bottom rows starting from
resolution $16\times 16$ (left block) and $32\times 32$ (right block)
and ending at resolution $128\times 128$.}\label{fig:superresolution}
\end{figure*}
\section{Comments on Extendability}\label{sec:applications}
In this section, we comment on another advantage of PAGER{}. That is,
PAGER{} can be easily tailored to other contexts without re-training.
We elaborate on three applications at the conceptual level.
\begin{itemize}
\item \textbf{Super Resolution.} PAGER{}'s two conditional image
generation modules (i.e., the resolution enhancer and the quality
booster) can be directly used for image super resolution with no
additional training. These modules enhance the image resolution from an
arbitrary dimension $2^d\times 2^d$ to $2^{d+k}\times 2^{d+k}$, where
$k$ is the number of consecutive resolution enhancer and quality booster
modules needed to achieve this task. Fig.~\ref{fig:superresolution}
shows several examples starting from different resolutions and ending at
resolutions $32\times 32$, $64\times 64$ and $128\times 128$.
\item \textbf{Attribute-guided Face Image Generation.} To generate human
face images with certain characteristics (e.g., a certain gender, hair
color, etc.) we partition the training data based on the underlying
attributes and construct subsets of data (Sec.~\ref{subsec:attributes}).
Each subset is used to train a different core generator that represents
the underlying attributes. Examples of such attribute-guided face
generation are presented in Figure~\ref{fig:attributes}. The
feasibility of training PAGER{} using a subset of training data is a
direct result of its robustness to the training dataset size. It was
empirically evaluated in Fig.~\ref{fig:training}. The mean FID score of
CelebA-like image generation changes only 6\% when the number of
training samples is reduced from 100K to as low as 5K.
\item \textbf{High-Resolution Image Generation.} PAGER{} can be easily
extended to generate images of higher resolution. To achieve this
objective, we can have more resolution enhancer and quality booster
units in cascade to reach the desired resolution. We present several
generated CelebA-like samples of $128\times 128$ pixels in
Fig.~\ref{fig:highres}.
\end{itemize}
\begin{figure*}[t]
\includegraphics[width=\linewidth]{figures/highres3.png}
\caption{Examples of generated CelebA-like images of resolution
$128\times 128$.}\label{fig:highres}
\end{figure*}
\section{Conclusion and Future Work}\label{sec:conclusion}
A non-DL-based generative model for visual data generation called
PAGER{} was proposed in this work. PAGER{} adopts the successive
subspace learning framework to extract multi-scale features and learns
unconditional and conditional probability density functions of extracted
features for image generation. The unconditional probability model is
used in the core generator module to generate low-resolution images to
control the model complexity. Two conditional image generation modules,
the resolution enhancer and the quality booster, are used to enhance the
resolution and quality of generated images progressively. PAGER{} is
mathematically transparent due to its modular design. We showed that
PAGER{} can be trained in a fraction of the time required by DL-based
models. We also demonstrated PAGER{}'s generation quality as the
number of training samples decreases. We then showed the extendibility
of PAGER{} to image super resolution, attribute-guided face image
generation, and high resolution image generation.
The model size of PAGER{} is primarily determined by the sizes
of the quality booster. The number of parameters is about 46 millions.
The large quality booster size is due to the use of LLE in predicting
residual details. We do not optimize the LLE component in the current
implementation. As a future topic, we would like to replace it with a
lightweight counterpart for model size reduction. For example, We might
replace LLE with GMMs to learn the distribution of residual textures, to
reduce the model size significantly. With these techniques, we aim to
reduce to the model size to less than 10 million parameters.
\section{Acknowledgments}
The authors acknowledge the Center for Advanced Research Computing (CARC) at the University of Southern California for providing computing resources that have contributed to the research results reported within this publication. URL: \url{https://carc.usc.edu}.
\bibliographystyle{unsrt}
|
1,116,691,499,248 | arxiv | \section{Introduction}
\label{sec:intro}
Asymptotic symmetries play a central role in the understanding of gauge theories. Gauge transformations can be divided into two important categories. On the one hand, we have the small gauge transformations, that are just redundancies in the description of the theory. On the other, there are the large gauge transformations, that transform the field to a physically inequivalent state. In this sense they are true symmetries of the theory and one can define their corresponding charges via Noether's theorem. These charges like the ADM mass in gravity, or the electric charge in the theory of electromagnetism, are defined as an integral over the boundary of some Cauchy slice.
The first studied examples in asymptotically flat space-times were on space like slices, whose boundary is spatial infinity. However, it turned out that doing the analysis on a null surface leads to a more interesting and rich symplectic structure. In the case of four dimensional gravity, this work was pioneered by Bondi, Metzner and Sachs \cite{1}, \cite{7}. They found that the asymptotic symmetry group of asymptotically flat space-times is much larger than the naively expected Poincar\'{e} group. This symmetry group also gave rise to an infinite number of conservation laws, which can be roughly interpreted as the conservation of different modes of the mass and angular momentum flux. Later, it was proven that the BMS group is also a symmetry of the gravitational scattering matrix \cite{7} - the Ward identities of the corresponding conserved charges are equivalent to soft gravitational theorems \cite{8} (for a full review see \cite{20}). Therefore, asymptotic symmetries are very important for the quantization of the theory. Similar results have been found in other gauge theories \cite{9}, \cite{10}, \cite{11}, \cite{19}.
All this work has been insightful and fruitful. However, although the usual charges are very well understood, the dynamics of the elusive dual charges is still smewhat a mystery. In the case of electromagnetism, it was long believed that magnetic monopoles do not exist. That was until Dirac managed to construct a magnetic monopole from a carefully selected couple of gauge fields, that are each singular along a line starting from the origin. Using the ambiguity of the gauge connection, these singularities were avoided. More recently a quantization of the Maxwell theory was achieved with explicit electromagnetic duality and the corresponding duality charges were constructed \cite{23}. For the Einstein theory of gravity, the equivalent of this exotic solution is the Taub-NUT metric \cite{12}, \cite{13}. Indeed in \cite{14} it was shown that, the free data for the Einstein equations, written in Bondi gauge, contains fields that resemble Maxwell gauge fields. Choosing these gauge fields to be the Dirac monopole, one obtains the Taub-NUT metric. What is even more interesting is that, just as the usual NUT charge is the dual of the Bondi mass, an infinite number of charges, dual to the BMS charges, have been found \cite{15}. More recently their existence was proven rigorously \cite{16} in the Hamiltonian formalism of General Relativity. The correct way to achieve this is to add a topological term to the usual Einstein-Palatini action. While this term does not change the equations of motion, it does lead to a different symplectic structure of the theory and therefore to additional Hamiltonian charges. It is like the $\theta $ term in Yang-Mills theory.
The goal of this paper is to make the first step towards extending the study of dual charges to supergravity. Here we study the magnetic charges of the massless Rarita-Schwinger field in four-dimensional space-time on a fixed background. The asymptotic symmetries of the field in four dimensions have already been studied to some extend at null-infinity \cite{5} and spatial infinity \cite{2}. Here we do a similar analysis. We employ the covariant phase formalism (for a good review see \cite{17}, \cite{18}), to study the usual Rarita-Schwinger action plus a new boundary term, that will give rise to the new magnetic charges. Our study will differ from the previous ones in the choice of boundary conditions. We carefully selected them to allow for a finite and non-degenerate symplectic form and an infinite number of finite conserved Noether charges.
\section{Asymptotic behaviour}
Before introducing the action it is wise to first describe the setting. The Rarita-Schwinger field will be put on a fixed asymptotically flat four-dimensional space-time with vanishing Ricci tensor. Written in the Bondi gauge, this metric is \cite{1},
\begin{flalign}
ds ^2 = - e ^{2\beta}f^2 du ^2 - 2 e^{2\beta} dudr +g_{AB} (dx ^{A} - U^A du )(dx ^{A} - U^A du)
\end{flalign}
where the transverse part of the metric is written in the form $g_{AB} = r^2 h_{AB}$. The boundary conditions on the components of the metric are,
\begin{flalign}
& h _{A B}= \gamma _{AB} +\frac{C _{AB}}{r} + O(r^{-2})\\
&f ^2 (u, r, x^A)= 1 - \frac{2 M (u, x^A )}{r} +O(r^{-1}) \quad e ^{2\beta } =1+O (r^{-2} ) \quad U^A =- \frac{1}{2r^2} D_B C^{AB} + O(r^{-3})
\end{flalign}
where only the relevant components for the subsequent calculations are shown. The symbol $D_B$ is the covariant derivative with respect to the metric on the unit two-sphere $\gamma _{AB}$. The term $M(u, x^A)$ appearing is the Bondi mass. Unlike the ADM mass, it can depend on time and the angular coordinates. Furthermore, the subleading term $C_{AB}$ describes gravitational waves. There is an additional gauge freedom, that is used to set $\partial _r \det \left( \frac{g_{AB}}{r^2} \right) = 0 $. This implies that $\gamma ^{AB} C_{AB} = 0$, leaving out two degrees of freedom of this tensor, corresponding to the two polarizations of the gravitational wave.
In order to be able to talk about spinor vectors on this space-time, we need to introduce frame fields $e^a _{\mu }$,
\begin{flalign}
g ^{\mu \nu} e^a _{\mu } e^b _{\nu} = \eta ^{ab}
\end{flalign}
where $\eta ^{ab}$ is the standard Minkowski metric with signature $(-, + , +, +)$. An equivalent choice for a vierbein basis, would be a Lorentz transformation of the one, written above.The space-time is without torsion so the first structure equation is
\begin{flalign}
de^a + \omega ^a \,_{b} \wedge e^b = 0
\end{flalign}
Explicitly the frame fields for the asymptotically flat space-time can be chosen to be:
\begin{flalign}
&e^0 = \frac{e^{\beta }}{f}dr + e^{\beta } f du\quad
e^1= \frac{e^{\beta }}{f}dr \\
&e ^i = r E ^i _A \left ( dx^A - U^A du \right) \quad E ^i _A E^j _B \delta _{ij} =h _{AB} \quad i, j \in \{2, 3 \}
\end{flalign}
The expressions for the components of the spin connection $\omega $ are tedious and therefore, are put in the appendix.
Just as we did for the metric, we assume that the spinor field is analytic in \protect $1/r$, and its asymptotic behaviour at large \protect $r$ is,
\begin{flalign}
\psi _A = \psi _A ^{(0)} (x^A, u ) + O(r^{-1} ) \quad \psi _{u } = \frac{\psi _u ^{(-1)} (x^A, u )}{r} + O(r^{-2} ) \quad \psi _{r } = \frac{\psi _r ^{(-1)} (x^A, u)}{r} + O(r^{-2})
\end{flalign}
This describes the behaviour of the fields at $\mathcal{I}^+$ and it will be important when studying the Noether charges of the theory. It will ensure that these charges along with the energy and momentum are finite. The latter statement is tangential to the current discussion and is therefore shown in the appendix \ref{T}. The full set of diffeomorphism charges for the $\mathcal{N} = 1$ supergravity theory is studied in an upcoming paper. Notice that the above conditions are different than the boundary conditions, proposed by \cite{5}, by order of $r$ in each component. Furthermore, they differ from \cite{2}, where the leading order of $\psi _A $ was pure gauge.
We also need to fix the behaviour of the field at large $u$. We demand that,
\begin{flalign}
\lim _{u \to + \infty} \psi _{\mu } = \lim _{u \to - \infty} \psi _{\mu } < \infty
\end{flalign}
where $< \infty $ indicates that the limit is finite. The above statement simply means that whatever $u$ dependence the field has, it vanishes at large $|u|$.
We now move to think about what variations are allowed. We divide them into two categories - gauge and not gauge transformations. Our condition on the allowed variations is that the variational principle is well-defined (which is shown in the next section) and that the symplectic form is finite. For generic non-gauge variations, we demand that they do not change the leading behaviour of the field.
\begin{flalign}\label{10}
\delta \psi _A = O(r^{-1}) \quad \delta \psi _u = O(r^{-2}) \quad \delta \psi _A = O(r^{-2})
\end{flalign}
On the other hand, gauge transformations act on a specific way on the field, that allows for more flexibility. In particular the gauge transformation of the Rarita-Schwinger field is,
\begin{flalign}
\psi _{\mu} \rightarrow \psi _{\mu} + \nabla _{\mu } \epsilon
\end{flalign}
where \protect $ \epsilon $ is an anti-commuting spinor. One can think of this as the spinor analog of the \protect $U(1)$ gauge symmetry of electrodynamics. It turns out that the asymptotic behaviour for \protect $\epsilon$ that gives finite, but non-vanishing charges is,
\begin{flalign}
\epsilon = \frac{\epsilon ^{(-1)} (x^A, u)}{r} + O(r^{-2}) \quad \lim _{u \to \pm \infty } \epsilon < \infty
\end{flalign}
We see that this is not compatible with \protect \eqref{10}, because \protect $\delta \psi _{u} = \nabla _{u } \epsilon = O(r^{-1})$. However, as we will see in the next sections, the variational principle and the symplectic form will still be well-defined, in the case, where the transformation is gauge.
These are all the assumptions we make about the asymptotic behaviour of the dynamical field \protect $\psi _{\mu }$ and the background field \protect $g_{\mu \nu }$. As already explained, the calculations done in this paper are well defined because of the above assumptions.
\section{The action }
The goal of this section is to introduce a new term to the action of the massless Majorana spinor vector field. The usual action is the Rarita-Schwinger, denoted in this paper by $S_{RS}$, while the new contribution is a boundary term, similar to the $\theta $ term in Yang-Mills theory and the Nieh-Yan and Pontryagin terms in the bosonic theory of gravity.
Without further ado, we introduce the action for the Rarita-Schwinger field with a magnetic term,
\begin{flalign}
\label{action}
S=S_{RS} + S_{\theta} = \int _{\mathcal{M}} d^4x \, e ^{\mu \nu \sigma \rho } \bar{\psi } _{\mu } \gamma _5 \gamma _{\sigma }\nabla _{\nu } \psi _{\rho } + \mathbb{\theta _{\ast} } \int _{\mathcal{M}} d^4 x \, e ^{\mu \nu \sigma \rho } \bar{\psi } _{\mu } \gamma _{\sigma } \nabla _{\nu } \psi _{\rho }
\end{flalign}
The term $e^{\mu \nu \rho \sigma}$ is the alternating symbol and it obeys $e^{ur\theta \phi} =1$ and $\gamma _{\mu} = e^a _{\mu} \gamma _a$, where $a$ is a spinor index. At first glance it doesn't seem like the added term is topological. However, we recall that a Majorana spinor obeys $\bar{\psi } = \psi ^{\dagger } \gamma ^0=\psi ^T C$, where $C$ is the charge conjugation matrix. This matrix is antisymmetric and has the following conjugation property with the gamma matrices, $\gamma _{\mu } ^T = -C \gamma _{\mu } C^{-1}$. Thanks to this, one can show that $\bar{\psi } _{\mu } \gamma _{\nu }\psi _{\rho} = - \bar{\psi } _{\rho } \gamma _{\nu } \psi _{\mu}$ and more importantly that $ \bar{\psi } _{\mu } \gamma _{\nu } \nabla _{\sigma} \psi _{\rho} = - \left(\nabla _{\sigma }\bar{\psi } _{\rho } \right) \gamma _{\nu } \psi _{\mu}$. This allows to re-write the action as,
\begin{flalign}
S = \int d^4 x e ^{\mu \nu \sigma \rho } \, \bar{\psi } _{\mu } \gamma _5 \gamma _{\sigma }\nabla _{\nu } \psi _{\rho } + \frac{\mathbf{\theta _{\ast}}}{2} \int d^4 x \, \nabla _{\nu } (e ^{\mu \nu \sigma \rho } \bar{\psi } _{\mu } \gamma _{\sigma } \psi _{\rho } )
\end{flalign}
Now it is clear that the additional action term is a boundary term and therefore does not contribute to the equations of motion.
We now proceed to set the boundary conditions for the spinor field. The first thing to require is to have a well-posed variational principle. This means that on-shell $\delta S = 0$.
The variation of the action is given by,
\begin{flalign}
\delta S =& \int d^4 x \, \left[ \frac{\partial L}{\partial \psi ^a } -\partial _{\mu } \left( \frac{\partial L}{\partial (\partial _{\mu } \psi ^a )} \right) \right] \delta \psi ^a +\partial _{\mu } \left( \frac{\partial L}{\partial (\partial _{\mu } \psi ^a )} \delta \psi ^a \right)
\end{flalign}
The boundary term is the pre-symplectic potential, while the bulk one gives rise to the equations of motion.
\begin{flalign}
&e ^{\lambda \nu \sigma \rho } \nabla _{\nu} \bar{\psi } _{\mu } \gamma _5 \gamma _{\sigma } =0
\end{flalign}
The equations of motion for the $\theta $ term is derived in the exact same way, only without the $\gamma _5$ matrix, namely $e ^{\lambda \nu \sigma \rho } \nabla _{\nu} \bar{\psi } _{\mu } \gamma _{\sigma } =0$. Since $\gamma ^5$ is invertible we have the following condition,
\begin{flalign}
e ^{\lambda \nu \sigma \rho } \nabla _{\nu} \bar{\psi } _{\mu } \gamma _5 \gamma _{\sigma } =0 \ \Leftrightarrow \ e ^{\lambda \nu \sigma \rho } \nabla _{\nu} \bar{\psi } _{\mu } \gamma _{\sigma } =0
\end{flalign}
Therefore, the additional term, has no contribution to the equation of motion. We now turn to study the pre-symplectic potential in order to understand why it vanishes on the boundary. Because the $\theta ^{\ast }$ term in the action can be re-written as a boundary term, its contribution to the pre-symplectic potential is straightforward,
\begin{flalign}
\delta S_b = \frac{\mathbf{\theta _{\ast}}}{2} \int _{\partial \mathcal{M}} d \Sigma _{\nu } \, e ^{\mu \nu \sigma \rho } \bar{\psi } _{\mu } \gamma _{\sigma } \delta \psi _{\rho }
\end{flalign}
The contribution from the usual Rarita-Schiwnger action is,
\begin{flalign}
\delta S_{RS} = \int _{\partial \mathcal{M}} d \Sigma _{\nu }\, e ^{\mu \nu \sigma \rho } \bar{\psi } _{\mu } \gamma _{\sigma } \gamma _5 \delta \psi _{\rho }
\end{flalign}
Therefore on-shell the variation of the action is,
\begin{flalign}\label{theta}
\delta S _{\text{on-shell }}= \int _{\partial M } \, d\Sigma _{\nu } \, \Theta ^{\nu } = \int _{\partial M } \, d\Sigma _{\nu } \, e ^{\mu \nu \sigma \rho } \, \left( \bar{\psi } _{[\mu } \gamma _{\sigma } \gamma _5 \delta \psi _{\rho ] } + \frac{\mathbf{\theta _{\ast}}}{2} \bar{\psi } _{[\mu } \gamma _{\sigma } \delta \psi _{\rho ] } \right)
\end{flalign}
The boundary of the manifold is the union of the limit of three different sequences of hypersurfaces \cite{24}. We evaluate the variation of the action at the surfaces defined by $u = \pm const $ and $r = const$ and then take the limit of the result as $u \to \pm \infty $ and $r \to \pm \infty $. Using the boundary conditions we have defined in the previous section is is not difficult to see that,
\begin{flalign}
\delta S _{\text{on-shell }} = \lim _{r \to \infty } \int _{\Sigma _1} d \Sigma _r \Theta ^r + \lim _{ u \to + \infty } \int _{\Sigma _1} d \Sigma _u \Theta ^u + \lim _{u \to - \infty } \int _{\Sigma _1} d \Sigma _u \Theta ^u =0
\end{flalign}
We see that on-shell the linear variation of the action, with respect to any variation of the field vanishes. For gauge transformations, there is even an easier way to see why they leave the action invariant on-shell.
\begin{equation}
\begin{split}
S \rightarrow \, S +& \int d^4 x\, e ^{\mu \nu \sigma \rho } \bar{\psi}_{\mu}\gamma _5 \gamma _{\sigma }\nabla _{\nu } \nabla _{\rho } \epsilon + \theta _{\ast} \int d^4 x\, e ^{\mu \nu \sigma \rho } \bar{\psi} _{\mu} \gamma _{\sigma } \nabla _{\nu} \nabla _{\rho } \epsilon \\
+&\int d^4 x \,e ^{\mu \nu \sigma \rho } \nabla _{\mu} \bar{\epsilon}\gamma _5 \gamma _{\sigma } \nabla _{\nu } \psi _{\rho} + \theta _{\ast} \int d^4 x \, e ^{\mu \nu \sigma \rho } \nabla _{\mu }\bar{\epsilon} \gamma _{\sigma } \nabla _{\nu} \psi _{\rho} = S
\end{split}
\end{equation}
The second line vanishes because of the equation of motion. In the first line we have we have,
\begin{flalign}
&e ^{\mu \nu \sigma \rho } \gamma _{\sigma} [ \nabla _{\nu}, \nabla _{ \rho } ] \epsilon = \frac{1}{4}
e ^{\mu \nu \sigma \rho } \gamma _{\sigma}R_{\nu \rho a b} \gamma ^{ab} \epsilon = \frac{\sqrt{g}}{4} R_{\nu \rho a b} \gamma ^{\mu \nu \rho } \gamma ^{ab} \epsilon
\end{flalign}
This vanishes, because of the Bianchi identity of Riemann tensor and because the Ricci tensor vanishes. For the details of this calculations see \cite{21}. It is interesting that the invariance of the action under gauge transformations is independent of the boundary conditions for the field $\psi $. This shows yet again that gauge transformations are special.
\section{Symplectic structure}
In this section we study the symplectic structure of the theory. We start with a brief review of the covariant phase space formalism for gauge theories.
The covariant phase space of a gauge theory is the space of solutions to the field equations with particular boundary conditions. The symmetries of the theory are gauge transformations, that preserve the gauge condition on the fields, and leave the action invariant. If the transformations are "large", they will have a non-vanishing conjugate charges. Without going into too much details, the recipe for calculating these charges is the following~\cite{17}. Firstly, the variation of the Lagrangian on shell is given by by a total derivative,
\begin{gather}\label{3.1}
\delta L (\phi, \delta \phi ) \approx d \Theta (\phi, \delta \phi )
\end{gather}
The boundary term $\Theta $ is called the pre-symplectic potential. The pre-symplectic current and pre-sympelctic form on phase space are defined respectively as,
\begin{flalign}\label{3.2}
& w (\phi, \delta _1 \phi, \delta _2 \phi ) = \delta _1 \Theta (\phi, \delta _2 \phi ) - \delta _2 \Theta (\phi, \delta _1 \phi )\\
& \tilde{ \Omega } (\phi, \delta _1 \phi, \delta _2 \phi ) = \int _{\Sigma } w (\phi, \delta _1 \phi, \delta _2 \phi )
\end{flalign}
We have used the $w$, to denote the pre-symplectic current, instead of the usual $\omega$, because $\omega $ is already reserved for the spin connection. In order to construct the symplectic form $\Omega$ from the pre-symplectic form, one should quotient out the degenerate directions $Y^b$ of the pre-symplectic form \cite{21}. These satisfy the property that $\Omega _{ab} Y^{a} X^b = 0$ for any $X^b$. Note that in this equation the latin letters are indices on the infinite dimensional phase space, not the spinor indices.
The Hamiltonian conjugate to a gauge transformation is defined as,
\begin{flalign} \label{dH}
& \cancel{\delta} H [\epsilon ] = \Omega _{ab} X^b [\epsilon ]= \int _{\Sigma } w (\phi, \delta _{ \epsilon} \phi, \delta \phi )
\end{flalign}
where $X^b [\epsilon ]$ is a vector field in phase space, which connects field configurations, related by the gauge transformation generated by $ \epsilon$. The integral is taken over a Cauchy surface. Later we will pick it to be future null-infinity (plus future time-infinity). Equation \eqref{dH} can be rewritten as in \cite{3},
\begin{flalign}\label{H}
\cancel{\delta } H _{ \epsilon } = \int _{ \partial \Sigma } \delta \mathcal{Q} - I_{\epsilon} \cdot \Theta
\end{flalign}
where $I_{\epsilon} = \epsilon \cdot \frac{\delta }{\delta \phi }$ and $\mathcal{Q}$ is the Noether charge. The corresponding Noether current is \cite{3}:
\begin{gather}
j=d\mathcal{Q} = \Theta - I _{ \epsilon} L
\end{gather}
In equation \eqref{dH}, I have deliberately used the symbol $\cancel{\delta } $ instead of $\delta $. This is because $\cancel{\delta } H _{\xi }$ need not be an exact one form on phase space. This can be due to two reasons - the transformation in question is not canonical, or because of the presence of flux of the charge $\mathbf{F}$ through null-infinity \cite{3}. In general, one makes the flux vanishing by fixing the right boundary conditions. Equipped with the tools of covariant phase space formalism, we now proceed to study the symplectic structure of our theory.
By using \eqref{3.2} and \eqref{theta} we can derive the pre-symplectic current,
\begin{flalign}
&w(\psi, \delta _1 \psi, \delta _2 \psi ) =\delta _1 \Theta (\psi, \delta _2 \psi) - \delta _2 \Theta (\psi, \delta _1 \psi)=\\ \label{3.12}
&\delta _1 \bar{\psi } _{[\mu } \gamma _{\sigma } \gamma _5 \delta _2 \psi _{\rho ] } - \delta _2 \bar{\psi } _{[\mu } \gamma _{\sigma } \gamma _5 \delta _1 \psi _{\rho ] } + \underbrace{\frac{\mathbf{\theta _{\ast}}}{2} \delta _1 \bar{\psi } _{[\mu } \gamma _{\sigma } \delta _2 \psi _{\rho ] } - \frac{\mathbf{\theta _{\ast}} }{2} \delta _2 \bar{\psi } _{[\mu } \gamma _{\sigma } \delta _1 \psi _{\rho ] } }_{=0}
\end{flalign}
Let's look at equation \eqref{3.12}. On one hand, from the properties of the Majorana spinors, discussed in the first section, we have $- \delta _2 \bar{\psi } _{[\mu } \gamma _{\sigma } \delta _1 \psi _{\rho ] } = + \delta _1 \bar{\psi } _{[\rho } \gamma _{\sigma } \delta _2 \psi _{\mu ] } $. On the other, the pre-symplectic current is antysiymmetrized over its indices. Therefore, the $\theta _{\ast}$ contribution to the pre-symplectic current vanishes.The symplectic structure of the theory is unaffected by the introduction of a boundary term in the action.
The pre-symplectic form on null-infinity is thus,
\begin{flalign}
\tilde{\Omega }= \int_{\mathcal{I} ^+} \, du d\theta d\phi \, \delta _1 \bar{\psi } _{[u } \gamma _{\theta} \gamma _5 \delta _2 \psi _{\phi] }
\end{flalign}
The corresponding Hamiltonian to a gauge transformation is,
\begin{flalign}
&\cancel{\delta} H _{ \epsilon} = \int _{\Sigma } \delta \Theta (\psi, \nabla _{\mu } \epsilon ) - \delta _{ \epsilon} \Theta (\psi, \delta \psi)
\end{flalign}
\begin{equation}
\begin{split}
&\delta _{ \epsilon } \Theta (\psi, \delta \psi) _{\mu \sigma \rho } = \nabla _{[\mu } \bar{ \epsilon } \gamma _{\sigma } \gamma _5 \delta \psi _{\rho ] } +\frac{\theta _{\ast}}{2} \nabla _{[\mu } \bar{ \epsilon} \gamma _{\sigma } \delta \psi _{\rho ] } = \\
&\nabla _{[\mu } \left(\bar{ \epsilon } \gamma _{\sigma } \gamma _5 \delta \psi _{\rho ] } +\frac{\theta _{\ast}}{2} \bar{ \epsilon } \gamma _{\sigma } \delta \psi _{\rho ] } \right) + \bar{ \epsilon } \gamma _{\sigma } \gamma _5 \delta \nabla_{\mu} \psi _{\rho ] } + \frac{ \theta _{\ast}}{2} \bar { \epsilon } \gamma _{\sigma } \nabla _{\mu } \delta \psi _{\rho ] } =d\vartheta
\end{split}
\end{equation}
In the above expression we have written the variation of the symplectic potential along a gauge transformation as a total derivative, using the linearized equations of motion. This is a special property of gauge theories - the charges live on the boundary.
The above equations can be re-written more formally thanks to the identity,
\begin{flalign}
\iota _{\nabla _{\mu } \epsilon ^a } \frac{\delta }{\delta \psi ^a _{\mu }} (...) = d \left( \epsilon ^a \frac{\delta }{\delta \psi ^a _{\mu }} (...) \right) + \epsilon ^a \frac{\delta }{\delta \psi ^a _{\mu }} (d(...))
\end{flalign}
where "d" denotes the exterior derivative on the space-time manifold and $a$ is a spinor index. This identity is the similar of the Cartan's magic formula, but for fermions. Recalling that we defined $I_{\epsilon } = \epsilon ^a \frac{\delta }{\delta \phi ^a}$ this can be rewritten more compactly,
\begin{flalign}
\delta _\epsilon \Theta (\psi, \delta \psi) & = \iota _{\nabla _{\mu } \epsilon ^a } \Theta (\psi, \delta \psi) = d I_{\epsilon } \Theta (\psi, \delta \psi) + I_{\epsilon } d \Theta (\psi, \delta \psi)\\
d\vartheta & = d I_{\epsilon } \Theta (\psi, \delta \psi)
\end{flalign}
The first equality holds because of the way $\Theta $ is constructed from the field and its derivatives. More precisely it a n-2 form on space-time, constructed covariantly. Going back to $\cancel{\delta} H_e$ we have now,
\begin{flalign}
\begin{split}\label{noethercurrent}
\cancel{ \delta } H_{\epsilon } = &\int _{\Sigma } \delta \underbrace{( \Theta (\psi , \nabla _{\mu } \psi ) - \bar{\epsilon } \gamma _{\sigma } \gamma _5 \nabla _{\mu} \psi _{\rho ] } - \frac{\theta _{\ast}}{2} \bar{e} \gamma _{\sigma } \nabla _{\mu } \psi _{\rho ] } ) } _{ \text{Noether current } = \ast j} + d\vartheta \\
\approx & \int _{\Sigma } \delta (\nabla _{[ \mu }(\bar {\psi } _{\mu } \gamma _5 \gamma _{\sigma ] } \epsilon + \frac{\theta _{\ast}}{2} (\bar {\psi } _{\mu } \gamma _{\sigma ] } \epsilon )) + d\vartheta \\
= & \int _{\partial \Sigma } \bar{\epsilon } \gamma _{[\sigma } \gamma _5 \delta \psi _{\rho ] }
\end{split}
\end{flalign}
The $\theta$ contribution vanishes once again. At this point, one may think that the new action term is completely inconsequential. However, because of the peculiar properties of spinors, we will see that the Noether charges will be affected by the magnetic term, even though the Hamiltonian charges are not. This is the subject of the next section.
\section{Noether charge}
If a gauge theory, described by a Lagrangian, admits global symmetries we can apply the generalized Noether theorem \cite{17}. This theorem states that there exists a bijection between the gauge parameters, and the equivalence class of $d-2$ forms $\mathcal{Q}$ that are closed on shell. Two such forms are equivalent if on-shell they differ by a $d-3$ form of the type $dk$. The integral over the boundary of a Cauchy slice of these $d-2$ forms is the Noether charge. Looking back at \eqref{H}, we see that the Noether charge is part of the integrable part of the Hamiltonian charge.
In this section we will calculate the Noether charge for the gravitino field with action \eqref{action}. The exterior derivative of the Noether charge density is given in \eqref{noethercurrent},
\begin{flalign}
\ast j=d \ast \mathcal{Q}=\Theta -I_{\epsilon } \ast L = \theta (\psi, \nabla _{\mu } \epsilon ) - \bar{\epsilon } \gamma _{[\sigma } \gamma _5 \nabla _{\mu} \psi _{\rho ] } - \frac{\theta _{\ast} }{2} \bar{\epsilon } \gamma _{[\sigma } \nabla _{\mu } \psi _{\rho ] }
\end{flalign}
where $\ast$ denotes the Hodge dual. The $I_{\epsilon } \ast L$ part of the equation vanishes, because of the equations of motion. This leads to,
\begin{flalign}
d \ast \mathcal{Q} [\epsilon ] _{\rho \mu \sigma } = &\bar{\psi } _{[\mu } \gamma _{\sigma } \gamma _5 \nabla _{\rho ] } \epsilon + \frac{\theta _{\ast}}{2} \bar{\psi } _{[\mu } \gamma _{\sigma }\nabla _{\rho ] } \epsilon \approx \nabla _{[ \rho } \bar{\psi } _{\mu } \gamma _{\sigma ] } \gamma _5 \epsilon + \frac{\theta _{\ast} }{2} \nabla _{[ \rho } \bar{\psi } _{\mu } \gamma _{\sigma ] } \epsilon \\
\ast \mathcal{Q} [\epsilon ] _{\mu \sigma } = & \bar{\psi } _{[\mu } \gamma _{\sigma ] } \gamma _5 \epsilon + \frac{\theta _{\ast} }{2} \bar{ \psi } _{[ \mu } \gamma _{\sigma ] } e\\
\mathbf{Q} [ \epsilon ] = & \int _{\partial \Sigma } \ast \mathcal{Q} [ \epsilon ]
\end{flalign}
As promised, Noether charge, has a a non-vanishing magnetic contribution, despite the fact that the Hamiltonian charge does not. One can think of this peculiarity in the following way. The non-integrable part of $\cancel{\delta } H$ is $I_{\epsilon } \Theta $. It characterizes the flux of this charge through null-infinity. For the usual charge we have that $\delta \mathcal{Q} =- I_{\epsilon } \Theta $, which means that all of the charge is contained in the bulk. In contrast for the magnetic part, the relationship is $ \delta \mathcal{Q} _{\theta _{\ast}}= I_{\epsilon } \Theta _{\theta _{\ast}}$ - all of the magnetic charge leaks through infinity.
Now let's look at the algebra of the charges:,
\begin{flalign}
\begin{split}
[ \mathbf{Q} [ \epsilon _1], \mathbf{Q} [ \epsilon _2] ] \equiv & \frac{1}{2} (\delta _{ \epsilon _1 }\mathbf{Q} [ \epsilon _2] - \delta _{\epsilon _2}\mathbf{Q} [\epsilon _1] ) \\
= & \int _{\partial \Sigma } \bar{\psi } _{[\mu } \gamma _{\sigma ] } \gamma _5 [ \epsilon _1, \epsilon _2 ]+ \frac{\theta _{\ast} }{2} \bar{ \psi } _{[ \mu } \gamma _{\sigma ] } [ \epsilon _1 , \epsilon _2 ] \\
& + \int _{\partial \Sigma } \nabla _{[\mu } \bar{\epsilon }_1 \gamma _{\sigma ]} \gamma _5 \epsilon _2 +\frac{\theta _{\ast} }{2}\int _{\partial \Sigma } \nabla _{[\mu } \bar{\epsilon }_1 \gamma _{\sigma ]} \epsilon _2 - (\epsilon _1 \leftrightarrow \epsilon _2)\\
= & \underbrace{ \mathbf{Q} \left[ [\epsilon _1, \epsilon _2 ] \right] }_{=0}+ \text{central charge}
\end{split}
\end{flalign}
The underlying algebra is abelian, so the result of the commutator is just the central charge. The central charge has two components- the usual one from the Rarita-Schwinger field and the magnetic one. In \cite{5}, it was shown that, if one defines a vector field as $\xi^{\mu} = \bar{ \epsilon } _1 \gamma ^{\mu } \epsilon _2$ and uses the linearized spin connection $\delta \omega$ in the calculation for the usual central charge, one obtains the super-translation charge, generated by $\xi ^{\mu}$. The $\theta $ contribution to the central charge vanishes because it is the integral of a total derivative on the sphere.
\section{Explicit form of the Noether charge}
In this section we compute the explicit form of the Noether charge. We start by fixing the gauge,
\begin{flalign}\label{gauge}
\gamma ^{\mu } \psi _{\mu } = 0
\end{flalign}
This leaves the the equations of motion in the form,
\begin{flalign}
\gamma ^{\mu } \nabla_{\mu } \psi _{\nu } = 0
\end{flalign}
Explicitly the charge is,
\begin{flalign}
\begin{split}\label{Q}
\mathbf{Q} [\epsilon ] = &\int _{S^2 } d\theta d\phi \, \bar{\psi } _{\phi}\gamma _5 \gamma _{\theta } \epsilon + \frac{\theta _{\ast} }{2} \bar{\psi } _{\phi} \gamma _{\theta} \epsilon - (\theta \leftrightarrow \phi ) \\
=&- \int _{S^2 } d\theta d\phi \bar{\psi } _{A} \gamma ^A \gamma _5 \gamma _{\phi }\gamma _{\theta } \epsilon + \frac{\theta _{\ast} }{2} \bar{\psi } _{A} \gamma ^A \gamma _{\phi} \gamma _{\theta} \epsilon
\end{split}
\end{flalign}
where $S^2$ is the two sphere at null-infinity. In order to evaluate the charge we will need the expression for both the gauge spinor and the angular components of the Rarita-Schwinger field. Therefore we need to solve their equations of motion. We start by looking at the gauge spinor, which is simpler and its solution will help us with the subsequent calculations.
The gauge spinor satisfies the Dirac equation as imposed by the gauge condition \eqref{gauge},
\begin{flalign}
\begin{split}
& \gamma ^{\mu } \nabla_{\mu } \epsilon =0 \quad \epsilon = \sum _{n \geq 1 } \frac{ \epsilon ^{-n}}{r^n} \quad \gamma ^{\mu} \nabla_{\mu } - \gamma ^r \partial _r= \sum _{n \geq 1 } \frac{v ^{-n}}{r^n}\\
& \gamma ^{\mu } \nabla_{\mu } \epsilon = \sum_{n \geq 0 } \sum _{m=0} ^{m=n} \frac{\left( \nabla^{-m} \epsilon ^{m-n} +(m-n+1) \gamma ^{r-m} \epsilon ^{m-n+1}\right)}{r^n} =0
\end{split}
\end{flalign}
At first and second order in $r$ the equations of motion are the following,
\begin{flalign}
& n=1 : \gamma ^{u 0} \partial _u \epsilon ^{-1}=0 \\\label{n1}
& n=2 : \gamma ^{A(-1)} \partial _A \epsilon ^{-1} + \frac{1}{2} \cot \theta \gamma ^2 \epsilon ^{-1} +\gamma ^{u 0}\partial _{u} M \epsilon ^{-1} +\gamma ^{u 0} \partial _u \epsilon ^{-2} =0
\end{flalign}
The first equation leaves us with,
\begin{flalign}
&\epsilon ^{-1}=\rho(x^A) + \gamma ^{u(0)} \beta (u, x^A)
\end{flalign}
Notice that we can have a $u$ dependency, because $\gamma ^{u(0)} \gamma ^{u(0)} = 0 $. Similarly to the gravitational case, working at null-infinity allows us to have more interesting dynamics. Using the representation of the gamma matrices, given in the appendix, we can write $\left( \gamma ^{u(0)} \beta ^{-1} \right) ^T= \begin{pmatrix} \eta ,& \xi \end{pmatrix}$, where $\eta ^T=\begin{pmatrix} \eta ^1 , & \eta ^1 \end{pmatrix}$ and $\xi ^T= \begin{pmatrix} \xi _1 , & -\xi _1\end{pmatrix}$.We also have $ \left( \rho ^{-1} \right)^T = \begin{pmatrix} \iota ,& \lambda \end{pmatrix} $. The equations for the chiral spinors decouple and will be solved separately . We will proceed to treat the equations of $\rho$ and $\beta $ separately and in a different fashion. We assume separation of variables for the components of both spinors. Let's first look at $\rho$. It satisfies the time independent part of equation \eqref{n1}.
\begin{flalign}
&\gamma ^{A(-1)} \partial _A \rho + \frac{1}{2} \cot \theta \gamma ^2 \rho = 0
\end{flalign}
From here, we can extract the equations for the two components of the chiral spinor $\lambda$.
\begin{flalign}
& \partial _{\theta } (\lambda _1 + \lambda _2)- \frac{i}{\sin \theta } \partial _{\phi } (\lambda _1 + \lambda _2) + \frac{1}{2} \cot \theta (\lambda _1 + \lambda _2)= 0\\
& \partial _{\theta } (\lambda _1 - \lambda _2)+ \frac{i}{\sin \theta } \partial _{\phi } (\lambda _1 - \lambda _2) + \frac{1}{2} \cot \theta (\lambda _1 - \lambda _2)= 0
\end{flalign}
This is solved by $\lambda _1 - \lambda _2 =\frac{A}{\sqrt{\sin \theta }} e^{-im \phi } \left( \cot \theta /2 \right) ^{m}$ and $\lambda _1 + \lambda _2 = \frac{B}{\sqrt{\sin \theta }} e^{im \phi } \left( \cot \theta /2 \right) ^{m}$. The same set of equations and solutions hold for $\iota$. This sets everything we need to know about $\rho$.
Now we move on to studying the equation for components of $\eta$ and $\xi$.
\begin{flalign}
\begin{split}\label{5.20}
& \partial _{\theta }\xi _1 - \frac{i}{\sin \theta } \partial _{\phi } \xi _1 + \frac{1}{2} \cot \theta \xi _1 =-i \partial _u M(\lambda _1 - \lambda _2) - i \partial _u (\epsilon ^{-2} _3 - \epsilon ^{-2} _4) \\
&- \bar{\eth } _{1/2} \xi _1 = -i \partial _u M(\lambda _1 - \lambda _2) - i \partial _u (\epsilon ^{-2} _3 - \epsilon ^{-2} _4)
\end{split}
\end{flalign}
\begin{flalign}
\begin{split}\label{5.21}
& \partial _{\theta }\eta _1 + \frac{i}{\sin \theta } \partial _{\phi } \eta _1 + \frac{1}{2} \cot \theta \eta _1 = i \partial _u M(\lambda _1 - \lambda _2) + 2 \partial _u (\epsilon ^{-2} _1 + \epsilon ^{-2} _2) \\
&-\eth _{-1/2} \eta _1 = i \partial _u M(\lambda _1 - \lambda _2) + 2 \partial _u (\epsilon ^{-2} _1 + \epsilon ^{-2} _2)
\end{split}
\end{flalign}
The operators $\eth$ and $\bar{\eth }$ are defined in \cite{4}. The first thing we notice is that $\xi _1$ and $\eta _1$ should be expressed in terms of spherical harmonics of spin weight $1/2$ and $-1/2$ respectively. Furthermore, we observe that we have two equations and four unknown functions. Therefore we claim that $\xi _1$ and $\eta _1$ can be expressed as products of arbitrary functions of $u$ and a spin $\pm 1/2$ spherical harmonics. Then the above equations can be viewed as equations for $\epsilon ^{-2}$. Of course, $\epsilon ^{-2}$, will have another equation of motion, but in that $\epsilon ^{-3}$ will appear etc. We see that whatever choice we make for leading order component of the gauge spinor, all the subsequent components will adjust to accommodate it. Thus. we have a complete freedom for $\epsilon ^{-1}$. What is left now is to impose the Majorana condition $\left(\epsilon ^{-1} \right)^{ \dagger} \gamma ^0 = \left( \epsilon ^{-1} \right) ^T C$.
The final piece of the puzzle we need is the behavior of the field itself. Its equations of motion at first and second order in $r$ are,
\begin{itemize}
\item{$\psi _A$}:
\begin{flalign}\label{uA}
O(1) : & \gamma ^{u(0)} \partial _u \psi _A =0 \\\label{A2}
O(r^{-1}) : &\gamma ^B \partial _B \psi _A + \gamma ^1 \frac{1}{r}\psi _A + \frac{1}{2r} \cot \theta \gamma ^2 \psi _A -\frac{1}{r} \gamma ^r \psi _A - \frac{r}{2} \gamma ^u g^{BE} \partial _u C_{EA} \psi _B\\
& - \gamma ^C \Gamma ^B _{CA} \psi _B+ h _{AB} \gamma ^B \psi _{u} +\frac{1}{2} \partial _u C_{AB} \gamma ^B \psi _r + \gamma ^{u(0)} \partial _u \psi _A ^{-1} =0
\end{flalign}
\item{$\psi _u$:}
\begin{flalign}
O(r^{-1}) : & \gamma ^{u } \partial _{u } \psi _u =0 \\ \label{u2}
O(r^{-2}): & r \gamma ^A \partial _A \psi _u + \frac{1}{2} \cot \theta \gamma ^2 \psi _u + \gamma ^u \partial _u M ( \psi _u - \psi _r ) + \gamma ^{u } \partial _{u } \psi _u ^{-2} - r \gamma ^A \partial _u C_{A} ^B \psi _B =0
\end{flalign}
\item{$\psi _r$:}
\begin{flalign}
O(r^{-1}) : &\gamma ^{u } \partial _{u } \psi _r =0\\ \label{r2}
O(r^{-2}) : & r \gamma ^A \partial _A \psi _r + \frac{1}{2} \cot \theta \gamma ^2 \psi _r +\gamma ^u \partial _u M \psi _r + \gamma ^{u } \partial _{u } \psi _r ^{-2} - \gamma ^A \psi _A =0
\end{flalign}
\end{itemize}
We see that the leading order components of the field satisfy $ \phi _{\mu} = \phi _{\mu} (x^A) + \gamma ^{u(0)}\varphi _{\mu} (u, x^A)$. We also notice that the equations for $\psi _A$ are quite complicated to solve. However, we don't need to solve for $\psi _{\theta}$ and $\psi _{\phi }$ individually. From the expression for the charge \eqref{Q} it is clear that what we need is $\gamma ^A \psi _A$. The gauge condition $\gamma ^{\mu } \psi _{\mu}$ can help us here. We demand that $u$ dependent and independent part vanish separately:
\begin{flalign} \label{gauge2}
\gamma ^{u(0)} \varphi _{u} + \gamma ^{r (0)} \varphi _{r} + \gamma ^{A(1)} \varphi _{A }= 0\\
\gamma ^{r (0)} \gamma ^{u(0)} \phi _{r} + \gamma ^{A {1}} \gamma ^{u(0)} \phi _{A} = 0
\end{flalign}
We see that instead of looking at $\gamma ^A \psi _A$, we can look at the equations for $\varphi _r$, $\varphi_u$ and $\phi _u$, which are much simpler. We will do this in the following way - we will extract the $u$ dependent and $u$ independent part of equations \eqref{r2} and \eqref{u2} set them separately to be equal to zero.
We begin by studying $\varphi _u (x^A)$. It's equation of motion is,
\begin{flalign}
\gamma ^{A(-1)} \partial _A \varphi _u + \frac{1}{2} \cot \theta \gamma ^2 \varphi _u = 0
\end{flalign}
This is the exact same equation as for $\rho(x^A)$ and therefore it has the same solutions.
We move on to investigate $\varphi _r$. Multiplying \eqref{r2} by $\gamma ^{u(0)}$ and replacing $ \gamma ^{u(0)} \gamma ^{A {1}}\varphi _{A} $, by $- \gamma ^{u(0)} \gamma ^{r (0)} \varphi _{r} $ we get:
\begin{flalign}
\gamma ^{u (0)} \left(\gamma ^A \partial _A \varphi _r + \gamma ^1 \varphi _r + \frac{1}{2} \cot \theta \gamma ^2 \varphi _r \right) =0
\end{flalign}
The above equation means one of two things. Either $ \varphi _r $ is a covariantly constant spinor or the thing in brackets is in the image of $\gamma ^{u(0)}$. When this happens every component of $\varphi _r$ is a linear sum of spin $\pm 1/2$ spherical harmonics, with some conditions on the coefficients. The details of this calculations are quite long and are therefore, given in the appendix.
We finally look at $ \gamma ^{u(0)}\phi _r(u, x^A)$. This equation for it is,
\begin{flalign}
&\gamma ^{u(0)} \left( \gamma ^{A(-1)} \partial _A\phi _r(u, x^A)+ \frac{1}{2} \cot \theta \gamma ^2 \phi _r(u, x^A) + \partial _u M \varphi _r +\partial _{u } \psi _r ^{-2} - \gamma ^{1} \phi _{r} \right) = 2 \gamma ^0 \gamma ^1 \phi _{r}\\
& \gamma ^{u(0)} \gamma ^0 \gamma ^1 \phi _r = 0 \, \Leftrightarrow \, \gamma ^{u(0)} \phi _r = 0
\end{flalign}
Thus we see that $u$ dependent part of $\psi _r ^{-1}$ and consequently of $\psi _A ^0$ vanish.
\subsection{Result}
To sum up we have,
\begin{flalign}
&\gamma ^{A(-1)} \psi _A ^{0} = - \gamma ^{u(0)} \varphi _{u} - \gamma ^{r (0)} \varphi _r \quad \epsilon ^{-1} = \rho + \gamma ^{u(0)} \beta \\
& \gamma ^{u(0)} \varphi _{u} = \frac{\left( \cot \theta /2 \right) ^{m}}{\sqrt{\sin \theta }} \begin{pmatrix} i A \sin m \phi \\ i A \sin m\phi \\-B \cos m \phi \\ B \cos m \phi \end{pmatrix} \quad \varphi _r = \begin{pmatrix} \zeta \\ \chi \end{pmatrix} \\
& \gamma ^{u(0)} \beta= \sum _{lm } \begin{pmatrix} \Im \{ a(u) _{1/2} ^{lm} \, _{1/2} Y_{lm } \} \\ \Im \{ a(u) _{1/2} ^{lm} \, _{1/2} Y_{lm } \} \\ -\Re \{ b(u) _{-1/2} ^{lm} \, _{-1/2} Y_{lm } \} \\ \Re \{ b(u) _{-1/2} ^{lm} \, _{-1/2} Y_{lm } \} \end{pmatrix} \quad \rho = \frac{\left( \cot \theta /2 \right) ^{m}}{2 \sqrt{\sin \theta }} \begin{pmatrix}i C \sin m\phi \\ i D \sin m \phi \\ F \cos m \phi \\
G \cos m \phi \end{pmatrix}
\end{flalign}
where the expression for $\zeta$ and $\chi$ are quite long and are in the appendix.
The charge is,
\begin{flalign}
\mathbf{Q} [\epsilon ] = & \int _{S^2 } d\theta d\phi \, \sin
\theta \left[\bar{\varphi } _r \gamma ^0 \epsilon ^0 +
\bar{\varphi } _u \gamma ^{u(0)}\gamma _5 \gamma _2 \gamma _3
\rho - \frac{\theta _{\ast} }{2} \bar{\varphi } _r \gamma ^0
\gamma _5 \epsilon ^0 + \frac{\theta _{\ast} }{2} \bar{\varphi }
_u \gamma ^{u(0)} \gamma _2 \gamma _3 \rho \right]
\end{flalign}
\section{Conclusion}
We just discovered and infinite number of electric and magnetic charges at null-infinity for thefree massless Rarita-Schwinger field. We were able to compute those charges explicitly. They exhibit some peculiar properties. Firstly, we notice that even though we have non-vanishing Noether magnetic charges, the symplectic structure of the theory is unaltered by the presence of the new boundary term. Furthermore, the charges can have an arbitrary time dependence. As mentioned in the introduction, the next step is to extend this work in supergravity. Hopefully, the interplay of the spinor magnetic charges and the gravitational dual charges will shed new light on the present work.
|
1,116,691,499,249 | arxiv | \section{Introduction}
One of the most robust predictions of inflationary cosmology is that the
universe after inflation becomes extremely flat, which corresponds to $\Omega =
1$. Here $\Omega = {\rho\over \rho_c}$,\, $\rho_c$ being the energy density of
a
flat universe. There were many good reasons to believe that this prediction
was quite generic. The only way to avoid this conclusion is to assume that the
universe inflated only by about $e^{60}$ times. Exact value of the number of
e-foldings $N$ depends on details of the theory and may somewhat differ from
60. It is important, however, that in any particular theory inflation by an
extra factor $e^2$ would make the universe with $\Omega = 0.5$ or with
$\Omega = 1.5$ almost exactly flat. Meanwhile, the typical number of
e-foldings, say, in chaotic inflation scenario in the theory ${m^2\over 2}
\phi^2$ is not 60 but rather $10^{12}$ \cite{MyBook}.
One can construct models where
inflation leads to expansion of the universe by the factor $e^{60}$. However,
in most of such models small number of e-foldings simultaneously implies that
density perturbations
are extremely large. Indeed, in most inflationary models the process of
inflation begins at the point where density perturbations
${\delta\rho\over\rho}
\sim {H^2\over \dot\phi}$ are very large. The simplest example is the original
new inflation scenario \cite{New}, where inflation begins at the top of the
effective potential
with $\dot\phi = 0$. If there are less than $60$ e-foldings from this moment to
the end of inflation, then we would see extremely large density perturbations
on the scale of horizon.
It may be possible to overcome this obstacle by a specific choice of the
effective potential. However, this would be only a partial solution. Indeed, if
the universe
does not inflate long enough to become flat, then by the same token it
does not inflate long enough to become homogeneous and isotropic.
Thus, the main reason why it is difficult to construct inflationary models
with $\Omega \not = 1$ is not the issue of fine tuning of the parameters of the
models, which is necessary to obtain the universe inflating exactly $e^{60}$
times, but the problem of obtaining a homogeneous universe after inflation.
Fortunately, it is possible to solve this problem, both for a closed universe
\cite{Lab,Omega} and for an open one \cite{Omega}--\cite{BGT}. The
main idea is to use the well known fact that the region of space created in the
process of a quantum tunneling tends to have a spherically symmetric shape,
and homogeneous interior, if the tunneling process is suppressed strongly
enough. Then such bubbles of a new phase tend to evolve (expand) in a
spherically symmetric
fashion. Thus, if one
could associate the whole visible part of the universe with an interior of one
such region, one would solve the homogeneity problem, and then all other
problems
will be solved by the subsequent relatively short stage of inflation.
For a closed universe the realization of this program is relatively
straightforward
\cite{Lab,Omega}. One should consider the process of quantum creation of a
closed inflationary universe from ``nothing.'' If the probability of such a
process is exponentially suppressed (and this is indeed the case if inflation
is possible only at the energy density much smaller than the Planck density
\cite{Creation}), then the universe created that way will be rather
homogeneous from the very beginning.
The situation with an open universe is much more complicated. Indeed, an open
universe is infinite, and it may seem impossible to create an infinite universe
by a tunneling process. Fortunately, this is not the case: any bubble formed in
the process of the false vacuum decay looks from inside like an infinite open
universe \cite{CL}--\cite{STYY}.
If this universe continues inflating
inside the bubble \cite{Gott,BGT,Omega}, then we obtain an open inflationary
universe.
These possibilities became a subject of an active investigation only very
recently, and there are still many questions to be addressed. First of all, the
bubbles created by tunneling are not {\it absolutely} uniform even if the
probability of tunneling is very small. This may easily spoil the whole
scenario since in the
end of the day we need to explain why the microwave background radiation is
isotropic with an accuracy of about $10^{-5}$. Previously we did not care much
about initial homogeneities, but if the stage of inflation is short, we will
the see original
inhomogeneities imprinted in the perturbations of the microwave background
radiation.
The second problem is to construct realistic inflationary models where all
these ideas could be realized in a natural way. Whereas for the closed universe
this problem can be easily solved \cite{Lab,Omega}, for an open universe we
again meet complications. It would be very nice to to obtain an open universe
in a theory of just one scalar field \cite{BGT}. However, in practice it is
not very easy to obtain a satisfactory model of this type. Typically one is
forced either to introduce very complicated effective potentials, or consider
theories with nonminimal kinetic terms for the inflaton field \cite{Bucher}.
This makes the models not only
fine-tuned, but also rather complicated. It is very good to know that the
models of such type in principle can be constructed, but it is also very
tempting to find a
more natural realization of the inflationary universe scenario which would
give
inflation with $\Omega < 1$.
This goal
can be achieved if one considers models of two
scalar fields \cite{Omega}. One of them may be the standard inflaton
field
$\phi$ with a relatively small mass, another may be, e.g., the scalar field
responsible for the symmetry breaking in GUTs. The presence of two scalar
fields allows one to obtain the required bending of the inflaton potential by
simply changing the definition of the inflaton field in the process of
inflation. On the first stage the role of the inflaton is played by a heavy
field with a steep barrier in its potential, while on the second stage the
role of the inflaton is played by a light field, rolling in a flat direction
``orthogonal'' to the direction of quantum tunneling. This change of the
direction of evolution in the space of scalar fields removes the naturalness
constraints for the form of the potential, which are present in the case of one
field.
Inflationary models of this type
are quite simple, yet they have many interesting features. In these models
the universe consists of infinitely many expanding bubbles immersed into
exponentially expanding false vacuum state. Each of these bubbles inside looks
like an open universe, but the values of $\Omega$ in these universes may take
any value from $1$ to $0$.
In some of these models the situation is even more complicated: Interior of
each bubble looks like an infinite universe with an effective value of
$\Omega$
slowly decreasing to $\Omega = 0$ at an exponentially large distance from the
center of the bubble. We will call such universes quasiopen. Thus, rather
unexpectedly, we are obtaining a large variety of interesting and previously
unexplored possibilities.
In this paper we will continue our discussion of inflationary models with
$\Omega \not = 1$. In Section 2 we will describe a model of a closed
inflationary universe. In Section 3 we will consider the possibility to
implement an open inflation scenario in the theory of one
scalar field. In Section 4 we discuss the issue of a
spherical symmetry of the bubbles produced by a tunneling process. In
Sections \ref{Simplest} --
\ref{Natural} we will describe several different models of an open inflationary
universe. Finally, in the last Section of the paper we will summarize our
results and discuss the most
important question: What does inflationary cosmology say now to those
trying to determine $\Omega$ from observational data?
\section{\label{Closed} Closed inflationary universe}
For a long time it was not quite clear how can one obtain a homogeneous closed
inflationary universe. In \cite{BGT} it was even argued that this is
impossible. Fortunately, it turns to be a relatively easy task
\cite{Lab, Omega}. For
example, one can consider a particular version of the chaotic inflation
scenario
\cite{Chaotic} with the effective potential
\begin{equation}\label{1}
V(\phi) = {m^2 \phi^2\over 2}\, \exp{\Bigl({\phi\over CM_{\rm P}}\Bigr)^2} \ .
\end{equation}
Potentials of a similar type often appear in supergravity. In this theory
inflation occurs only in the interval ${M_{\rm P}\over 2} {\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\ } \phi {\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\ }
CM_{\rm P}$. The most natural way to realize inflationary scenario in this
theory is
to assume that the universe was created ``from nothing'' with the field $\phi$
in the interval ${M_{\rm P}\over 2} {\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\ } \phi {\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\ }
CM_{\rm P}$. The universe at the moment of its creation has a size $H^{-1}$,
and then it begins inflating as $H^{-1} \cosh{Ht}$. According to
\cite{Creation}--\cite{Vilenkin}, the probability of creation of an
inflationary
universe is
suppressed by
\begin{equation}\label{2}
P \sim \exp\Bigl(-{3M^4_{\rm P}\over 8V(\phi)}\Bigr) \ .
\end{equation}
Therefore the maximum of the probability appears near the upper range of values
of the field $\phi$ for which inflation is possible, i.e. at $\phi_0 \sim C
M_{\rm P}$ (see more discussion about this below).
The probability of such an event will be so strongly suppressed that the
universe will be formed almost ideally homogeneous and spherically symmetric.
As pointed out in \cite{Lab}, this solves the homogeneity, isotropy and
horizon problems even before inflation really takes over. Then the size of the
newly born universe in this model expands by the factor $\exp({2\pi
\phi_0^2M_{\rm P}^{-2}})\sim \exp({2\pi C^2})$ during the
stage of inflation \cite{MyBook}. If $C {\
\lower-1.2pt\vbox{\hbox{\rlap{$>$}\lower5pt\vbox{\hbox{$\sim$}}}}\ } 3$, i.e.
if $\phi_0 {\
\lower-1.2pt\vbox{\hbox{\rlap{$>$}\lower5pt\vbox{\hbox{$\sim$}}}}\ } 3M_{\rm P}
\sim 3.6\times 10^{19}$ GeV, the
universe expands more than $e^{60}$ times, and it becomes very flat.
Meanwhile, for $C \ll 3$ the universe always remains ``underinflated'' and very
curved, with $\Omega \gg 1$. We emphasize again that in this particular model
``underinflation" does not lead to any problems with homogeneity and isotropy.
The only problem with this model is that in order
to obtain $\Omega$ in the interval between $1$ and $2$ at the present time one
should have the constant $C$ to be fixed somewhere near $C = 3$ with an
accuracy of few percent. This is a fine-tuning, which does not sound very
attractive. However, it is important to realize that we are not talking about
an exponentially good precision; accuracy of few percent is good enough.
A similar result can be obtained even without changing the shape of the
effective potential. It is enough to assume that the field $\phi$ has a
nonminimal interaction with gravity of the form $-{\xi\over 2} R\phi^2$. In
this case inflation becomes impossible for $\phi > {M_{\rm P}\over
\sqrt{8\pi\xi}}$
\cite{Maeda,BL95}. Thus in order to ensure that only closed inflationary
universes can be produced during the process of quantum creation of the
universe in the theory ${m^2\over 2} \phi^2$ it is enough to assume that
${M_{\rm P}\over \sqrt{8\pi\xi}} < 3M_{\rm P}$. This gives a condition $\xi >
{1\over 72\pi} \sim
4\times10^{-4}$.
To make sure that this mechanism of a closed universe creation is viable one
should check that the universe produced that way is sufficiently homogeneous.
Even though one may expect that this is guaranteed by the large absolute value
of
the gravitational action, one should check that the asymmetry of the
universe at the moment of its creation does not exceed an extremely small value
$\sim 10^{-5}$, since otherwise our mechanism will produce anisotropy
of the microwave background radiation exceeding its experimentally established
value ${\Delta T\over T} \sim 10^{-5}$.
Calculation of probability of creation of a closed universe is a very
controversial subject, and determination of its quantum state and of its
possible asymmetry is even more complicated. However, one can make a simple
estimate and show that the absolute value of the action $|S| = {3M^4_{\rm
P}\over 16V(\phi)} $ on the tunneling trajectory describing the universe
formation will change by $\Delta |S| \sim O(1)$ if one adds perturbation of the
standard amplitude $\delta \phi \sim {H\over 2\pi}$ to the homogeneous
background of the scalar field $\phi$. Trajectories with $\Delta |S| < 1$ are
not strongly suppressed as compared with the original tunneling trajectory.
Therefore tunneling into configurations with
$\delta \phi \sim {H\over 2\pi}$ can be possible, whereas we expect that
tunneling with creation of the universes with $\delta \phi \gg {H\over 2\pi}$
should be exponentially suppressed as compared with the tunneling with creation
of the universes with $\delta \phi \sim {H\over 2\pi}$.
This result suggests
that expected deviations of the scalar field from homogeneity at the moment of
the universe
creation have the usual quantum mechanical amplitude ${H\over 2\pi}$ which is
responsible for
galaxy formation and anisotropy of the microwave background radiation in the
standard versions of inflation in a flat universe. In addition to this, there
will be certain irregularities of the shape of the original bubble, since its
size $\sim H^{-1}$ at the tunneling
is defined with an accuracy $\sim M^{-1}_{\rm P}$. The resulting anisotropy
$\sim
{H\over
M_{\rm P}}$ is similar to the amplitude of gravitational waves produced during
inflation. In other words, both scalar and gravitational perturbations induced
at the moment of the universe creation are expected to be
{\it of the same magnitude} as if the universe were inflating for an
indefinitely long time. Therefore tunneling may play the same role as inflation
from the point of view of the homogeneity and isotropy problems
\cite{Lab,Omega}.
This possibility is very intriguing. Still, we do not want to insist that
our conclusions are unambiguous. For example, one could
argue that it is much
more natural for the universe to be created with the density very close to the
Planck density. However, the effective potential (\ref{1}) at the Planck
density is extremely steep. Therefore such a universe will not typically
enter the
inflationary regime, and will recollapse within a very short time. It could
survive long enough for
inflation to occur only if it was extremely large and relatively homogeneous
from the very beginning. If the probability of creation of such a large
universe is
smaller than the probability of a direct creation of a homogeneous closed
inflationary universe which we studied above, all our conclusions will remain
intact. Some preliminary estimates of the probability of creation of a large
universe which subsequently enters the stage of inflation can be found in
\cite{DA}; however, this issue requires a more detailed investigation.
Leaving apart this cautious note, our main conclusion is that it may be
possible to make inflation short and the universe
closed and homogeneous. The
remaining problem is to understand why our universe does not have $\Omega =
100$?
But in fact it is very easy to answer this question: Value of $\Omega$
changes in a closed universe while it expands. It spends only a small fraction
of its lifetime in a state with $\Omega \gg 1$. About a half of its lifetime
before the closed universe
becomes collapsing it has $\Omega$ only slightly greater than 1. Therefore a
considerable fraction of all observers who may populate a closed universe
should live there at the time when $\Omega$ is not much greater than $1$. It is
as
simple as that. The situation with an open universe is much more complicated,
since an open universe spends only a finite amount of time at the beginning of
its evolution in a state with $\Omega \sim 1$, and then $\Omega$ decreases and
stays for indefinitely long time in a state with $\Omega \ll 1$ ($\Omega \to
0$ for $t \to \infty$).
\section{\label{Onefield} Open universe in the models with one scalar
field}
As we have already mentioned in the Introduction, the way to obtain an open
homogeneous inflationary universe is to use the mechanism outlined in
\cite{CL}--\cite{BGT}. An open universe corresponds to an interior of a
single bubble
appearing in the decaying false vacuum. This picture can be consistent with
observations only if the probability of tunneling with the bubble formation is
sufficiently small, so that the bubbles do not collide until the typical size
of the bubble becomes greater than the size of the observable part of our
universe. The corresponding constraints are very easy to satisfy in the
theories with small coupling constants, where the tunneling rate is very small
\cite{Gott}. However, if one modifies the theory in such a way that the
probability of the bubble formation at some moment becomes so large that the
phase transition completes in the whole universe (see e.g. \cite{Occh}), then
there will be a danger that an observer inside the bubble will see
inhomogeneities created by other bubbles. Therefore will not study here
theories of such type.
It is not very easy to find a reasonable model which will lead to
tunneling and inflation by about $e^{60}$ times after the bubble formation. The
simplest idea \cite{Bucher} is to realize this scenario in the chaotic
inflation model with the effective potential
\begin{equation}\label{o1}
V(\phi) = {m^2\over 2} \phi^2 - {\delta\over 3} \phi^3 + {\lambda\over 4}\phi^4
\ .
\end{equation}
In order to obtain an open inflationary
universe in this model it is necessary to adjust parameters in such a way as
to ensure that the
tunneling creates bubbles containing the field $\phi \sim 3 M_{\rm P}$. In
such a case the
interior of the bubble after its formation inflates by about $e^{60}$ times,
and $\Omega$ at the present time may become equal to, say, $0.3$. This
requires a fine tuning of the effective potential. If, for example, tunneling
occurs not to $\phi \sim 3 M_{\rm P}$ but to $\phi \sim 3.1 M_{\rm P}$ then the
universe will become practically flat. It is worth noting, however, that
the required fine tuning is about the same order as for the closed universe
model described in Section \ref{Closed}, i.e.\ few percent.
Fine tuning is not the main difficulty of this model. The tunneling
should occur to the part of the potential which almost does not change its
slope during inflation at smaller $\phi$, since otherwise one does not obtain
scale-invariant density
perturbations. One of the necessary conditions is that the barrier should be
very narrow. Indeed, if $V'' \ll H^2$ at the barrier, then the tunneling occurs
to its top, as in the Hawking-Moss case \cite{HM}. Original interpretation of
this possibility was rather obscure; understanding came after the development
of stochastic approach to inflation. What happens is that the inflaton field
due to long-wave quantum fluctuations experiences Brownian motion. Occasionally
this field may jump from the local minimum of the effective potential to its
local maximum, and then slowly roll down from the maximum to the global minimum
\cite{Star,Gonch,Lab}. If this
happens, one obtains unacceptably large density perturbations
${\delta\rho\over \rho} \sim {H^2\over \dot\phi} > 1$ on the large scale, since
$\dot\phi = 0$ at
the local maximum of $V(\phi)$.
Unfortunately, this is exactly the case for the model (\ref{o1}) \cite{Bucher}.
Indeed, the local minimum of the effective potential in this model appears at
\begin{equation}\label{o2}
\phi = {\delta \over 2 \lambda} + \gamma \ ,
\end{equation}
where
\begin{equation}\label{o3}
\gamma = \sqrt{ {\delta^2 \over 4 \lambda^2} - {m^2\over \lambda} } \ .
\end{equation}
The local minimum of the effective potential appears for ${\delta > 2
\sqrt\lambda\, m }$, and it becomes unacceptably deep (deeper than the minimum
at $\phi = 0$) for ${\delta > {3 \sqrt\lambda\over \sqrt 2}\, m }$. Thus in
the whole region of interest one can use a simple estimate ${\delta \sim 2
\sqrt\lambda\, m }$ and represent $\gamma$ in the form $\beta{m\over 2
\sqrt{2\lambda}}$, with $\beta < {1}$. The local maximum of the potential
appears at $\phi = {\delta \over 2 \lambda} - \gamma$. Tunneling should occur
to some point with $3 M_{\rm P} < \phi < {\delta \over 2 \lambda} - \gamma$,
which
implies that ${\delta \over \lambda} > 6 M_{\rm P}$.
The best way to study tunneling in this theory is to introduce the field $\chi$
in such a way that $\chi = 0$ at the local minimum of $V(\phi)$:
\begin{equation}\label{o4}
\chi = -\phi +{\delta \over 2 \lambda} + \gamma \ .
\end{equation}
After simple algebra one can show that if the local minimum is not very deep
($\beta \ll 1$), the effective potential (\ref{o1}) can be represented as
\begin{equation}\label{o5}
V(\chi) \sim {m^2\delta^2 \over 48\lambda^2} + { \sqrt 2\beta m^2 } \chi^2 -
{\delta\over 3} \chi^3 + {\lambda\over 4}\chi^4 \ .
\end{equation}
The Hubble constant in the local minimum is given by
$H^2 \sim { \pi \delta^2 m^2\over 18\lambda^2 M_{\rm P}^2} > {2\pi}\, m^2$,
which
is much greater than the effective mass squared of the field $\chi$ for
$\beta \ll 1$, $m^2_\chi = {2\sqrt 2\beta m^2}$. In this regime tunneling
should occur to the local maximum of the effective potential, which should lead
to disastrous consequences for the spectrum of density perturbations.
A possible way to overcome this problem is to consider the case ${\delta
\approx {3 \sqrt\lambda\over \sqrt 2}\, m }$\, ($\beta \approx 1$). Then the
two minima of the effective potential become nearly degenerate in energy, and
the Hubble constant becomes much smaller than $m_\chi$. (In this regime our
estimate for $H$, which was valid for $\beta \ll 1$, should be improved.)
However, in such a situation we will have other problems. In this regime
tunneling occurs almost exactly to the minimum of the effective potential at
$\phi = 0$. Therefore it becomes difficult to have any inflation at all after
the tunneling, and the problem of fine tuning becomes especially severe.
Note that this problem is rather general. Its origin is in the condition that
in the inflationary universe scenario the curvature of the effective
potential (the mass squared of the inflaton field) $60$
e-foldings prior to the end of inflation always is much smaller than $H^2$.
Therefore one should bend the effective potential in a quite dramatic way in
order to create a local minimum at large $\phi$ and to make the curvature of
the effective potential in this minimum much greater than $H^2$. One can
avoid this problem by introducing non-minimal kinetic terms
in the Lagrangian of the inflaton field \cite{Bucher}, but this is just
another representation of the artificial bending of the effective potential.
Of course, it may happen that the bending of the effective potential can
appear in a natural way in the theories based on supergravity and superstrings.
The simplest idea is to multiply the effective potential (\ref{o1}) by the
factor of the type $\exp{\Bigl({\phi\over CM_{\rm P}}\Bigr) } $ or
$\exp{\Bigl({\phi\over CM_{\rm P}}\Bigr)^2} $, like in Eq.\ (\ref{1}). At small
$\phi$ these factors will not influence the shape of the effective potential,
but at large $\phi$ they will make it very curved. This is
exactly what we need to avoid the Hawking-Moss tunneling.
Still the necessity to make all these tricks with bending the potential and
making it very curved at some fine-tuned value of the scalar field $\phi$ do
not make the models of this type particularly attractive.
Therefore in the next sections we will make an attempt
to find some simple models where an open inflationary universe can appear in a
more
natural way. However, before doing so we will consider one more problem which
should be addressed in all versions of the open inflationary universe scenario.
\section{\label{Bubbles} Tunneling probability and spherical symmetry}
In the previous section we have assumed that the bubbles are exactly
spherically symmetric (or, to be more accurate, $O(3,1)$-symmetric \cite{CL}).
Meanwhile in realistic situations this condition may be violated for several
reasons. First of all, the bubble may be formed not quite symmetric. Then its
shape may change even further due to growth of its initial inhomogeneities and
due to quantum fluctuations which appear during the bubble wall expansion. As
we will see, this may cause a lot of problems if one wishes to maintain the
degree of anisotropy of the microwave background radiation inside the bubble at
the level of $10^{-5}$.
First of all, let us consider the issue of symmetry of a bubble at the moment
of its formation. For simplicity we will investigate the models where tunneling
can be described in the thin wall approximation. We will neglect gravitational
effects, which is possible as far as the initial radius $r$ of the bubble is
much smaller than $H^{-1}$. In this approximation (which works rather well for
the models to be discussed)
euclidean action of the $O(4)$-symmetric instanton describing bubble formation
is given by
\begin{equation}\label{o6}
S = - {\epsilon\over 2} \pi^2 r^4 + 2\pi^2 r^3 s \ .
\end{equation}
Here $r$ is the radius of the bubble at the moment of its formation, $\epsilon$
is the difference of $V(\phi)$ between the false vacuum $\phi_{\rm initial}$
and the true vacuum $\phi_{\rm final}$, and $s$ is the surface tension,
\begin{equation}\label{o7}
s = \, \int_{\phi_{\rm initial}}^{\phi_{\rm final}} \sqrt{ 2(V(\phi) -
V(\phi_{\rm
final}))}\, d\phi \ .
\end{equation}
The radius of the bubble can be obtained from the extremum of (\ref{o6})
with respect to $r$:
\begin{equation}\label{o8}
r = {3s\over \epsilon } \ .
\end{equation}
Let us check how the action $S $ will change if one consider a bubble of a
radius $r + \Delta r$. Since the first derivative of $S $ at its extremum
vanishes, the change will be determined by its second derivative,
\begin{equation}\label{09}
\Delta S = {1\over 2} S'' (\Delta r)^2 = 9\pi^2\, {s^2\over \epsilon}\, (\Delta
r)^2 \ .
\end{equation}
Now we should remember that all trajectories which have an action different
from the action at extremum by no more than $1$ are quite legitimate. Thus the
typical deviation of the radius of the bubble from its classical value
(\ref{o8}) can be estimated from the condition $\Delta S \sim 1$, which gives
\begin{equation}\label{o10}
|\Delta r| \sim {\sqrt\epsilon\over 3\pi \,s} \ .
\end{equation}
Note, that even though we considered spherically symmetric perturbations, our
estimate is based on corrections proportional to $(\delta r)^2$, and
therefore it should remain valid for perturbations which have an amplitude
$\Delta r$, but change their sign in different parts of the bubble surface.
Thus, eq. (\ref{o10}) gives an estimate of a typical degree of asymmetry of the
bubble at the moment of its creation:
\begin{equation}\label{o11}
A(r) \equiv {|\Delta r| \over r} \sim {\epsilon\sqrt\epsilon\over 3\pi \,s^2}
\ .
\end{equation}
This simple estimate exactly coincides with the corresponding result obtained
by Garriga and Vilenkin \cite{VilGarr} in their study of quantum fluctuations
of bubble walls. It was shown in \cite{VilGarr} that when an empty bubble
begins
expanding, the typical deviation $\Delta r$ remains constant. Therefore the
asymmetry given by the ratio ${|\Delta r| \over r}$ gradually vanishes. This is
a pretty general result: Waves produced by a brick falling to a pond do not
have the shape of a brick, but gradually become circles.
However, in our case the situation is somewhat more complicated. The wavefront
produced by a brick in inflationary background preserves the shape of the brick
if its size is much greater than $H^{-1}$. Indeed, the wavefront moves with
the speed approaching the speed of light, whereas the distance between
different parts of a region with initial size greater than $H^{-1}$ grows with
a much greater (and ever increasing) speed. This means that inflation
stretches the wavefront without changing its shape on scale much greater than
$H^{-1}$. Therefore during inflation which
continues inside the bubble the symmetrization of its shape occurs only in
the very beginning, until the radius of the bubble approaches $H^{-1}$. At this
first stage expansion of the bubble occurs mainly due to the motion of the
walls rather than due to inflationary stretching of the universe, and our
estimate of the bubble wall asymmetry as well as the results obtained by
Garriga and Vilenkin for the empty bubble remain valid. At the moment when the
radius of the bubble becomes equal to $H^{-1}$ its asymmetry
becomes
\begin{equation}\label{o12}
A(H^{-1}) \sim {|\Delta r| H} \sim {\sqrt\epsilon H\over 3\pi \,s} \ ,
\end{equation}
and the subsequent expansion of the bubble does not change this value very
much. Note that the Hubble constant here is determined by the vacuum energy
{\it
after} the tunneling, which may differ from the initial energy density
$\epsilon$.
The deviation of the shape of the
bubble from spherical symmetry implies that the beginning of the second stage
of inflation inside the bubble will be not exactly synchronous, with the delay
time $\Delta t \sim \Delta r$. This, as usual, may lead to adiabatic density
perturbations on the
horizon scale of the order of $H\Delta t$, which coincides with the bubble
asymmetry $A$ after its size becomes greater than $H^{-1}$, see Eq.\
(\ref{o12}).
To estimate this contribution to density perturbations, let us consider again
the
simplest model with the effective potential (\ref{o1}). Now we will consider it
in the limit $\beta - 1 \ll 1$ which implies that the two minima have almost
the same depth, which is necessary for validity of the thin wall
approximation. In this case $2\delta^2 = 9 M^2\lambda$, and the effective
potential (\ref{o1}) looks approximately like ${\lambda \over 4} \phi^2 (\phi
- \phi_0)^2$, where $\phi_0 = {2\delta\over 3\lambda} = \sqrt {2\over \lambda}
{M}$ is the position of the local minimum of the effective
potential. The surface tension in this model is given by $s =
\sqrt{\lambda\over 2} {\phi_0^3\over 6} = { M^3\over 3\lambda}$ \cite{Tunn}.
We will also introduce a phenomenological parameter $\mu$, such that $\mu
{M^4\over 16\lambda} = \epsilon$. The smallness of this parameter controls
applicability of the thin-wall approximation, since the value of the effective
potential near the top of the potential barrier at $\phi = \phi_0/2$ is given
by $M^4\over 16\lambda$. Then our estimate of density
perturbations associated with the bubble wall (\ref{o12}) gives
\begin{equation}\label{o14}
\left. {\delta\rho\over \rho}\right|_{\rm bubble} \sim A(H^{-1}) \sim
{\sqrt{\mu\lambda}
H\over 4 \pi
M} \ .
\end{equation}
Here $H$ is the value of the Hubble constant at the beginning of inflation
inside the bubble.
In order to have $\left. {\delta\rho\over \rho}\right|_{\rm bubble} {\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\ } 5 \times
10^{-5}$ (the number $5 \times 10^{-5}$ corresponds to the amplitude of density
perturbations in the COBE normalization) one
should have
\begin{equation}\label{o17}
\left. {\delta\rho\over \rho}\right|_{\rm bubble} \sim \, {\sqrt{\mu\lambda}
H\over 4 \pi
M}\, {\ \lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\ }\,
5 \times 10^{-5} \ .
\end{equation}
For $H\ll M$ perturbations produced by the bubble walls may be sufficiently
small even if the coupling constants are relatively large and the bubbles at
the moment of their formation are very inhomogeneous.
There is a long way from our simple estimates to the full theory of
anisotropies of
cosmic microwave background induced by fluctuations of the domain wall. In
particular, the significance of this effect will clearly depend on the value of
$\Omega$.
The constraint (\ref{o17}) may appear only if one can ``see'' the
scale at which the bubble walls have imprinted their fluctuations. If inflation
is long enough,
this scale becomes
exponentially large, we do not see the fluctuations due to bubble walls, but
then we return to the standard
inflationary scenario of a flat inflationary universe. However, for $\Omega \ll
1$ inflation is short, and it does not preclude us from seeing perturbations in
a vicinity of the bubble walls \cite{Open}.\footnote{One should distinguish
between
the
infinite size of an open universe and the finite distance from us to the
bubble walls along the light cone.}
In such a case one should take
the constraint (\ref{o17}) very seriously.
In the open universe the form of the spectrum of CMBR temperature
fluctuations can be substantially different from
the form of the spectrum of density fluctuations because of the integral
Sachs-Wolfe effect \cite{Open} (see, in particular, the paper by Lyth and
Woszczyna and references therein). In addition to this, the perturbations
discussed above occur on a super-curvature scale. Therefore, they provide a
natural source for the Grishchuk-Zeldovich effect (according to Lyth and
Woszczyna
\cite{Open} only the modes which are not in conformal vacuum can be
responsible for this), while the density fluctuations produced during the
secondary inflation are not likely to contain super-curvature modes.
One can show that in the theories of one
scalar field similar to the model discussed in the previous section (if this
model would work) the condition (\ref{o17}) is almost automatically satisfied.
Meanwhile in the
models of two different scalar fields, which we are going to discuss now, this
condition may lead
to additional restrictions on the parameters of the models.
\section{\label{Simplest} The simplest model of a (quasi)open inflationary
universe}
As we have seen in Sect. \ref{Onefield}, it is rather difficult to obtain an
open universe in the models of one scalar field with simple potentials, such
as, e.g., ${m^2\over 2} \phi^2 - {\delta\over 3} \phi^3 + {\lambda\over
4}\phi^4$. In this section we will explore an extremely simple model of two
scalar fields, where the universe after inflation becomes open (or quasiopen,
see below) in a very natural way \cite{Omega}.
Consider a model of
two noninteracting scalar fields, $\phi$ and $\sigma$, with the effective
potential
\begin{equation}\label{3}
V(\phi, \sigma) = {m^2\over 2}\phi^2 + V(\sigma) \ .
\end{equation}
Here $\phi$ is a weakly interacting inflaton field, and $\sigma$, for example,
can be the field responsible for the symmetry breaking in GUTs. We will assume
that $V(\sigma)$ has a local minimum at $\sigma = 0$, and a global minimum at
$\sigma_0 \not = 0$, just as in the old inflationary
theory. For definiteness, we will assume that this potential is given by
${M^2\over 2} \sigma^2 -
{\alpha M } \sigma^3 + {\lambda\over 4}\sigma^4 + V(0)$, with $V(0) \sim
{M^4\over 4 \lambda}$, but it is not essential;
no fine tuning of the shape of this potential will be required.
Note that so far we did not make any unreasonable complications to the standard
chaotic inflation scenario; at large $\phi$ inflation is driven
by the field $\phi$, and the GUT potential is necessary in the theory anyway.
In order to obtain density perturbations of the necessary amplitude the mass
$m$ of the scalar field $\phi$ should be of the order of $10^{-6} M_{\rm P}
\sim
10^{13}$ GeV \cite{MyBook}.
Inflation begins at $V(\phi, \sigma) \sim M_{\rm P}^4$. At this stage
fluctuations of
both fields are very strong, and the universe enters the stage of
self-reproduction, which finishes for the field $\phi$ only when it becomes
smaller than $M_{\rm P} \sqrt{M_{\rm P}\over m}$ and the energy density drops
down to $m
M_{\rm P}^3 \sim 10^{-6} M_{\rm P}^4$ \cite{MyBook}. Quantum fluctuations of
the field
$\sigma$ in some parts of the universe put it directly to the absolute minimum
of
$V(\sigma)$, but in some other parts the scalar field $\sigma$ appears in the
local minimum of $V(\sigma)$ at $\sigma = 0$. We will follow evolution of such
domains. Since the energy density in such
domains will be greater, their volume will grow with a greater speed, and
therefore they will be especially important for us.
One may worry that all
domains with $\sigma = 0$
will tunnel to the minimum of $V(\sigma)$ at the stage when the field $\phi$
was very large and quantum fluctuations of the both fields were large too.
This may happen if the Hubble constant induced by the scalar field $\phi$ is
much greater than the curvature of the potential $V(\sigma)$:
\begin{equation}\label{s1}
{m\phi\over M_{\rm P}} {\
\lower-1.2pt\vbox{\hbox{\rlap{$>$}\lower5pt\vbox{\hbox{$\sim$}}}}\ } M \ .
\end{equation}
This decay can be easily suppressed if one introduces a
small interaction $g^2\phi^2\sigma^2$ between these two fields, which
stabilizes the state with $\sigma = 0$ at large $\phi$. Another possibility,
which we have already mentioned in Sect. \ref{Closed}, is to add a
nonminimal interaction with gravity of the form $-{\xi\over 2} R\phi^2$, which
makes inflation impossible for $\phi > {M_{\rm P}\over 8\phi\xi}$. In this case
the
condition (\ref{s1}) will never be satisfied. However, there is a much simpler
answer to this worry. If the effective potential of the field $\phi$ is so
large that the field $\phi$ can easily jump to the true minimum of $V(\sigma)$,
then the universe becomes divided into infinitely many domains with all
possible values of $\sigma$ distributed in the following way
\cite{Star,MyBook}:
\begin{equation}\label{s2}
{P(\sigma= 0)\over P(\sigma = \sigma_0)} \sim \exp\left({3M^4_{\rm P}\over 8
V(\phi,0)} - {3M^4_{\rm P}\over 8V(\phi,\sigma)}\right) = \exp\left({3M^4_{\rm
P}\over 4(m^2\phi^2 + 2V(0))} - {3M^4_{\rm P}\over 4 m^2\phi^2}\right)\ .
\end{equation}
One can easily check that at the moment when the field $\phi$ decreases to ${M
M_{\rm P}\over m}$ and the condition (\ref{s1}) becomes violated, we will
have
\begin{equation}\label{s3}
{P(0)\over P(\sigma_0)} \sim \exp\left(-{C\over \lambda}\right) \ ,
\end{equation}
where $C$ is some constant, $C = O(1)$. After this moment the probability of
the false vacuum decay typically becomes much smaller. Thus the fraction of
space which survives in the false vacuum state $\sigma = 0$ until this time
typically is very small, but finite (and calculable). It is important, that
these rare domains with $\sigma = 0$ eventually will dominate the volume of the
universe since if the probability of the false vacuum decay is small enough,
the volume of the domains in the false vacuum will continue growing
exponentially without end.
The main idea of our scenario can be explained as follows. Because the fields
$\sigma$ and
$\phi$ do not interact with each other, and the dependence of the probability
of tunneling on the vacuum energy at the GUT scale is negligibly small
\cite{CL}, tunneling to the minimum of $V(\sigma)$ may occur with approximately
equal
probability at all sufficiently small values of the field $\phi$ (see, however,
below). The
parameters of the bubbles of the field $\sigma$ are determined by the mass
scale $M$ corresponding to the effective potential $V(\sigma)$. This mass scale
in
our model is much greater than $m$. Thus the duration of tunneling in the
Euclidean ``time'' is much smaller than $m^{-1}$. Therefore the field $\phi$
practically does not change its value during the tunneling. If
the probability of decay at a given $\phi$ is small enough, then it does not
destroy the whole vacuum state $\sigma = 0$ \cite{GW}; the bubbles of the new
phase are produced all the way when the field $\phi$ rolls down to $\phi =
0$. In this process the universe becomes filled with
(nonoverlapping) bubbles immersed in the false vacuum state with $\sigma = 0$.
Interior of each of these bubbles represents an open universe. However, these
bubbles contain {\it different} values of the field $\phi$, depending on the
value of this field at the moment when the bubble formation occurred. If the
field $\phi$ inside a bubble is smaller than $3 M_{\rm P}$, then the universe
inside
this bubble will have a vanishingly small $\Omega$, at the age $10^{10}$ years
after the end of inflation it will be practically empty, and life of our type
would not exist there. If the field $\phi$ is much greater than $3 M_{\rm P}$,
the
universe inside the bubble will be almost exactly flat, $\Omega = 1$, as in the
simplest version of the chaotic inflation scenario. It is important, however,
that {\it in an eternally existing self-reproducing universe there will be
infinitely many universes containing any particular value of $\Omega$, from
$\Omega = 0$ to $\Omega = 1$}, and one does not need any fine tuning of the
effective potential to obtain a universe with, say, $0.2 <\Omega < 0.3$
Of course, one can argue that we did not solve the problem of fine tuning, we
just transformed it into the fact that only a very small percentage of all
universes will have $0.2 <\Omega < 0.3$. However, first of all, we
achieved our goal in a very simple theory, which does not require any
artificial potential bending and nonminimal kinetic terms. Then, there may be
some reasons why it is preferable for us to live in a universe with a small
(but not vanishingly small) $\Omega$.
The simplest way to approach this problem is to find how the probability
for the bubble productiondepends on $\phi$. As we already pointed out, for
small $\phi$ this dependence is not very strong. On the other hand, at large
$\phi$ the probability rapidly grows and becomes quite large at $\phi > {M
M_{\rm P}\over m}$. This may suggest that the bubble production typically
occurs at
$\phi > {M M_{\rm P}\over m}$, and then for ${M\over m} \gg 3$ we typically
obtain
flat universes, $\Omega = 1$. This is another manifestation of the problem of
premature decay of the state $\sigma = 0$ which we discussed above. Moreover,
even if the probability to produce the universes with different $\phi$ were
entirely $\phi$-independent, one could argue that the main volume of the
habitable parts of the universe is contained in the bubbles with $\Omega = 1$,
since the interior of each such bubble inflated longer. Again, there exist
several ways of resolving this problem: involving coupling $g^2\phi^2\sigma^2$,
which stabilizes the state $\sigma = 0$ at large $\phi$, or adding nonminimal
interaction with gravity of the form $-{\xi\over 2} R\phi^2$, which makes
inflation impossible for $\phi > {M_{\rm P}\over \sqrt{8\pi\xi}}$. In either
way one can
easily suppress production of the universes with $\Omega = 1$. Then the
maximum of probability will correspond to some value $\Omega < 1$, which can be
made equal to any given number from $1$ to $0$ by changing the parameters $g^2$
and $\xi$. \footnote{Thus we disagree with the statement made in \cite{Occh}
that this model
typically predicts empty universes.}
However, calculation of probabilities in the context of the theory of a
self-reproducing universe is a very ambiguous process. For example, we may
formulate the problem in a different way. Consider a domain of the false vacuum
with $\sigma = 0$ and $\phi = \phi_1$. After some evolution it
produces one or many bubbles with $\sigma = \sigma_0$ and the field $\phi$
which after some time becomes equal to $\phi_2$. One may argue that the most
efficient way this process may go is the way which in the end produces the
greater volume. Indeed, for the inhabitants of a bubble it does not matter how
much time did it take for this process to occur. The total number of
observers produced by this process will depend on the total volume of the
universe at the hypersurface of a given density, i.e. on the hypersurface of a
given $\phi$. If the domain instantaneously tunnels to the state $\sigma_0$
and $\phi_1$, and then the field $\phi$ in this domain slowly rolls from
$\phi_1$ to $\phi_2$, then the volume of this domain grows $\exp
\Bigl({2\pi\over M_{\rm P}^2} (\phi_1^2 -\phi_2^2)\Bigr)$ times \cite{MyBook}.
Meanwhile, if the tunneling takes a long time, then the field $\phi$ rolls down
extremely slowly being in the false vacuum state with $\sigma = 0$. In this
state the universe expands much faster than in the state with $\sigma =
\sigma_0$. Since it expands much faster, and it takes the field much longer to
roll from $\phi_1$ to $\phi_2$, the trajectories of this kind bring us much
greater volume. This may serve as an argument that most of the volume is
produced by the bubbles created at a very small $\phi$, which leads to the
universes with very small $\Omega$.
One may use another set of considerations, studying all trajectories beginning
at $\phi_1, t_1$ and ending at $\phi_2, t_2$. This will bring us another
answer, or, to be more precise, another set of answers, which will depend on
the choice of the time parametrization \cite{LLM}. A very interesting approach
was recently proposed by Vilenkin, who suggested to introduce a particular
cutoff procedure which (almost) completely eliminates dependence of the final
answer on the time parametrization \cite{VilNew}. A more radical possibility
would be to integrate over all time parametrizations. This task is very
complicated, but it would completely eliminate dependence of the final answer
on the time parametrization \cite{OPEN}.
There is a very deep reason why the calculation of the probability to obtain a
universe with a given $\Omega$ is so ambiguous. For those who will live inside
a bubble there will be no way to say at which stage (at which
time from the point of view of an external observer) this bubble was produced.
Therefore one should compare {\it all} of these bubbles produced at all
possible times. The self-reproducing universe should exist for indefinitely
long time, and therefore it should contain infinitely many bubbles with all
possible values of $\Omega$. Comparing infinities is a very ambiguous task,
which gives results depending on the procedure of comparison. For example, one
can consider an infinitely large box of apples and an infinitely large box of
oranges. One may pick up one apple and one orange, then one apple and one
orange, over and over again, and conclude that there is an equal number of
apples and oranges. However, one may also pick up one apple and
two oranges, and then one apple and two oranges again, and conclude that there
is twice as many oranges as apples. The same situation happens when one tries
to compare the number of bubbles with different values of $\Omega$. If we
would know how to
solve the problem of measure in quantum cosmology, perhaps we would be able to
obtain
something similar to an open universe in the trivial $\lambda\phi^4$ theory
without any first order phase transitions
\cite{OPEN}. In the meantime, it is already encouraging that in our scenario
there are infinitely many inflationary universes with all possible value of
$\Omega < 1$. We can hardly live in the empty bubbles with $\Omega = 0$. As for
the choice between the bubbles with different nonvanishing values of $\Omega <
1$, it is quite possible that eventually we will find out an unambiguous way
of predicting the most probable value of $\Omega$, and we are going to continue
our work in this direction. However, it might also happen that this question
is as meaningless as the question whether it is more probable to be born as a
chinese rather than as an italian. It is quite conceivable that the only way to
find out in which of the bubbles do we live is to make observations.
Some words of caution are in order here. The bubbles produced in our simple
model
are not {\it exactly} open universes. Indeed, in the models discussed in
\cite{CL}--\cite{BGT} the time of reheating (and the temperature of the
universe after the reheating) was synchronized with the value of the scalar
field inside the bubble. In our case the situation is very similar, but not
exactly. Suppose that the Hubble constant induced by $V(0)$ is much
greater than the Hubble constant related to the energy density of the scalar
field $\phi$. Then the speed of rolling of the scalar field $\phi$ sharply
increases inside the bubble. Thus, in our case the field $\sigma$ synchronizes
the motion of the field $\phi$, and then the hypersurface of a constant field
$\phi$ determines the hypersurface of a constant temperature. In the models
where the rolling of the field $\phi$ can occur only inside the bubble (we will
discuss such a model shortly) the synchronization is precise, and everything
goes as in the models of refs. \cite{CL}--\cite{BGT}. However, in our simple
model the scalar field $\phi$ moves down outside the bubble as well, even
though it does it very slowly. Thus, synchronization of motion of the
fields $\sigma$ and $\phi$ is not precise; hypersurface of a constant $\sigma$
ceases to be a hypersurface of a constant density. For example, suppose that
the field $\phi$ has taken some value $\phi_0$ near the bubble wall when the
bubble was just formed. Then the bubble expands, and during this time the field
$\phi$ outside the wall decreases, as $\exp \Bigl(-{m^2t\over 3 H_1}\Bigr)$,
where $H_1 \approx H(\phi = \sigma = 0)$ is the Hubble constant at the first
stage of inflation, $H_1 \approx \sqrt{8\pi V(0)\over 3 M_{\rm P}^2}$
\cite{MyBook}. At the moment
when the bubble expands $e^{60}$ times, the field $\phi$ in the region just
reached by the bubble wall decreases to $\phi_o\exp \Bigl(-{20 m^2\over
H^2_1}\Bigr)$ from its original value $\phi_0$. The universe inside the bubble
is a homogeneous open universe only if this change is negligibly small. This
may not be a real problem. Indeed, let us assume that $V(0) ={\tilde M}^4$,
where ${\tilde M} =
10^{17}$ GeV. (Typically the energy density scale $\tilde M$ is related to the
particle mass as follows: ${\tilde M} \sim \lambda^{-1/4} M$.) In this case
$H_1 = 1.7 \times 10^{15}$ GeV, and for $m =
10^{13}$ GeV one obtains ${20 m^2\over H_1^2} \sim 10^{-4}$. In such a case
a typical degree of distortion of the picture of a homogeneous open universe is
very small.
Still this issue requires careful investigation. When the bubble wall continues
expanding even further, the scalar field outside of it eventually drops down to
zero. Then there will be no new matter created near the wall. Instead of
infinitely large homogeneous open universes we are obtaining spherically
symmetric islands of a size much greater than the size of the observable part
of our universe. We do not know whether this unusual picture is an advantage or
a
disadvantage of our model. Is it possible to consider different parts of the
same
exponentially large island as domains of different ``effective'' $\Omega$? Can
we attribute some part of the dipole anisotropy of the microwave background
radiation to the possibility that we live somewhere outside of the center of
such island? In any
case, as we already mentioned, in the limit $m^2 \ll H_1^2$
we do not expect that the small deviations of the geometry of space inside the
bubble from the geometry of an open universe can do much harm to our model.
Our model admits many generalizations, and details of the scenario which we
just discussed depend on the values of parameters. Let us forget for a moment
about all complicated processes which occur when the field $\phi$ is rolling
down to $\phi = 0$, since this part of the picture depends on the validity of
our ideas about initial conditions. For example, there may be no
self-reproduction of inflationary domains with large $\phi$ if one considers an
effective
potential of the field $\phi$ which is very curved at large $\phi$, as in eq.
(\ref{1}). However, there will be self-reproduction of the universe in a state
$\phi = \sigma = 0$, as in the old inflation scenario. Then the main portion of
the volume of the universe will be determined by the processes which occur when
the fields $\phi$ and $\sigma$ stay at the local minimum of the
effective potential, $\phi = \sigma = 0$. For definiteness we will assume
here that $V(0) = {\tilde M}^4$, where ${\tilde M}$ is the stringy scale,
${\tilde M} \sim 10^{17} -
10^{18}$ GeV. Then the Hubble constant $H_1 = \sqrt{8\pi V(0)\over 3M^2_{\rm
P}} \sim
\sqrt{8\pi \over 3} {{\tilde M}^2\over M_{\rm P}}$ created by the energy
density
$V(0)$ is
much greater than $m \sim 10^{13}$ GeV. In such a case the scalar field $\phi$
will not stay exactly at $\phi = 0$. It will be relatively homogeneous on the
horizon scale $H_1^{-1}$, but otherwise it will be chaotically distributed
with
the dispersion $\langle\phi^2\rangle = {3H^4\over 8\pi^2m^2}$ \cite{MyBook}.
This means that the field $\phi$ inside each of the bubbles produced by the
decay of the false vacuum can take any value $\phi$ with the probability
\begin{equation}\label{4}
P \sim \exp\left(-{\phi^2\over 2 \langle\phi^2\rangle}\right) \sim
\exp\left(-{3m^2 \phi^2M_{\rm P}^4\over 16 {\tilde M}^8}\right) \ .
\end{equation}
One can check that for ${\tilde M} \sim 4.3\times10^{17}$ GeV the typical value
of the
field $\phi$ inside the bubbles will be $\sim 3\times 10^{19}$ GeV. Thus, for
${\tilde M} > 4.3\times10^{17}$ GeV most of the universes produced during the
vacuum
decay will be flat, for ${\tilde M} < 4.3\times10^{17}$ GeV most of them will
be open.
It is interesting that in this version of our model the percentage of open
universes is determined by the stringy scale (or by the GUT scale). However,
since the process of bubble production in this scenario goes without end,
the total number of universes with any particular value of $\Omega < 1$ will
be infinitely large for any value of ${\tilde M}$. Thus this model shows
us is the
simplest way to resurrect some of the ideas of the old inflationary theory with
the help of chaotic inflation, and simultaneously to obtain inflationary
universe with $\Omega < 1$.
Note that this version of our model will not suffer for the problem of
incomplete synchronization. Indeed, the average value of the field $\phi$ in
the false vacuum outside the bubble will remain constant until the bubble
triggers its decrease.
However, this model, just as its previous version, may suffer from another
problem. The Hubble constant $H_1$ before the tunneling in this model was much
greater
than the Hubble constant $H_2$ at the beginning of the second stage of
inflation. Therefore the fluctuations of the
scalar field before the tunneling were very large, $\delta \phi \sim {H_1\over
2 \pi}$, much greater than the
fluctuations generated after the tunneling, $\delta \phi \sim {H_2\over 2
\pi}$. This may lead to very large
density perturbations on the scale comparable to the size of the bubble. For
the models with $\Omega = 1$ this effect would not cause any problems since
such
perturbations would be far away over the present particle horizon, but for
small $\Omega$ this
may lead to unacceptable anisotropy of the microwave background radiation.
Fortunately, this may not be a real difficulty. A possible solution is very
similar to the bubble symmetrization described in the previous section.
Indeed, let us consider more carefully how the long wave perturbations produced
outside the bubble may penetrate into it. At the moment when the bubble is
formed, it has a size (\ref{o3}), which is smaller than $H_1^{-1}$
\cite{CL}. Then the bubble walls begin moving with the speed gradually
approaching the speed of light. At this stage the comoving size of the bubble
(from the point of view of the original coordinate system in the false vacuum)
grows like
\begin{equation}\label{n1}
r(t) = \int_{0}^{t}{dt e^{-H_1 t}} = H_1^{-1} (1 - e^{-H_1 t}) \ .
\end{equation}
During this time the fluctuations of the scalar field $\phi$ of the amplitude
${H_1\over 2\pi}$ and of the wavelength $H_1^{-1}$, which previously were
outside the bubble, gradually become covered by it. When these perturbations
are outside the bubble, inflation with the Hubble constant $H_1$ prevents them
from oscillating and moving. However, once these perturbations penetrate inside
the bubble, their amplitude becomes decreasing \cite{MZ,SP}. Indeed, since the
wavelength of the perturbations is $\sim H_1^{-1} \ll H_2^{-1} \ll m^{-1}$,
these
perturbations move inside the bubbles as relativistic particles, their
wavelength grow as $a(t)$, and their amplitude decreases just like an
amplitude of electromagnetic field, $\delta\phi \sim a^{-1}(t)$, where $a$ is
the scale factor of the universe inside a bubble \cite{MZ}. This process
continues until the wavelength of each perturbation reaches $H_2^{-1}$ (already
at the second stage of inflation). During this time the wavelength grows
${H_1\over H_2}$ times, and the amplitude decreases ${H_2\over H_1}$ times, to
become the standard amplitude of perturbations produced at the second stage of
inflation: $ {H_2\over H_1}\, {H_1\over 2\pi} = {H_2\over 2\pi}$.
In fact, one may argue that this computation was too naive, and that these
perturbations should be neglected altogether. Typically we treat long wave
perturbations in inflationary universe like classical wave for the reason that
the waves with the wavelength much greater than the horizon can be interpreted
as states with extremely large occupation numbers \cite{MyBook}. However, when
the new born perturbations (i.e. fluctuations which did not acquire an
exponentially large wavelength yet) enter the bubble (i.e. under the horizon),
they effectively return to the realm of quantum fluctuations again. Then one
may argue that one should simply forget about the waves with the wavelengths
small enough to fit into the bubble, and consider perturbations created at the
second stage of inflation not as a result of stretching of these waves, but as
a new process of creation of perturbations of an amplitude ${H_2\over 2\pi}$.
One may worry that perturbations which had wavelengths somewhat greater than
$H_1^{-1}$ at the moment of the bubble formation cannot completely penetrate
into the bubble. If, for example, the field $\phi$ differs from some constant
by $+{H_1\over 2\pi}$ at the distance $H_1^{-1}$ to the left of the bubble at
the moment of its formation, and by $-{H_1\over 2\pi}$ at the distance
$H_1^{-1}$ to the right of the bubble, then this difference remains frozen
independently of all processes inside the bubble. This may suggest that there
is some unavoidable asymmetry of the distribution of the field inside the
bubble. However, the field inside the bubble will not be distributed like a
straight line slowly rising from $-{H_1\over 2\pi}$ to $+{H_1\over 2\pi}$.
Inside
the bubble the field will be almost homogeneous; the inhomogeneity $\delta \phi
\sim -{H_1\over 2\pi}$ will be concentrated only in a small vicinity near the
bubble wall.
Finally we should verify that this scenario leads to bubbles which are
symmetric enough, see eq. (\ref{o17}). Fortunately, here we do not have any
problems. One can easily check that for our model with $m \sim 10^{13}$ GeV and
$\tilde M \sim \lambda^{-1/4} M > 10^{17} GeV$ the condition (\ref{o17}) can be
satisfied even for not very small values of the coupling constant $\lambda$.
The arguments presented above should be confirmed by a more detailed
investigation of the vacuum structure inside the expanding bubble in our
scenario. If, as we hope, the result of the investigation will be positive, we
will have an
extremely simple model of an open inflationary universe. In the meantime, it
would be nice to have a model where we do not have any problems at all with
synchronization and
with large fluctuations on the scalar field in the false vacuum. We will
consider such a model in the next section.
\section{\label{Hybrid} Hybrid inflation with $\Omega < 1$}
The model to be discussed below \cite{Omega} is a version of the hybrid
inflation scenario
\cite{Hybrid}, which is a slight generalization (and a simplification) of our
previous model
(\ref{3}):
\begin{equation}\label{4a}
V(\phi,\sigma) = {g^2\over 2}\phi^2\sigma^2 + V(\sigma) \ .
\end{equation}
We eliminated the massive term of the field $\phi$ and added explicitly the
interaction ${g^2\over 2}\phi^2\sigma^2$, which, as we have mentioned already,
can be useful (though not necessary) for stabilization of the state $\sigma =
0$ at large $\phi$. Note
that in this model the line $\sigma = 0$ is a flat direction in the
($\phi,\sigma$) plane. At large $\phi$ the only minimum of the effective
potential with respect to $\sigma$ is at the line $\sigma = 0$. To give a
particular example, one can take $V(\sigma) = {M^2\over 2} \sigma^2 -{\alpha M
} \sigma^3 + {\lambda\over 4}\sigma^4 +V_0$. Here $V_0$ is a constant which is
added to ensure that $V(\phi,\sigma) = 0$ at the absolute minimum of
$V(\phi,\sigma)$. In this case the minimum of the potential $V(\phi,\sigma)$
at $\sigma \not = 0$ is deeper than the minimum at $\sigma = 0$ only for $\phi
< \phi_c$, where $\phi_c = {M\over g}\sqrt{{2\alpha^2\over \lambda} -1}$. This
minimum for $\phi = \phi_c$ appears at $\sigma = \sigma_c = {2\alpha M\over
\lambda}$.
The bubble formation becomes possible only for $\phi < \phi_c$. After the
tunneling the field $\phi$ acquires an effective mass $m = g\sigma$ and begins
to move towards $\phi = 0$, which provides the mechanism for the second stage
of inflation inside the bubble. In this scenario evolution of the scalar field
$\phi$ is exactly synchronized with the evolution of the field $\sigma$, and
the universe inside the bubble appears to be open.
Effective mass of the field $\phi$ at the minimum of $V(\phi,\sigma)$ with
$\phi = \phi_c$, $\sigma = \sigma_c = {2\alpha M\over \lambda}$ is $m =
g\sigma_c = {2g\alpha M\over \lambda}$. With a decrease of the field $\phi$
its effective mass at the minimum of $V(\phi,\sigma)$ will grow, but not
significantly. For simplicity, we will consider the case $\lambda = \alpha^2$.
In this case it can be shown that $V(0) = 2.77\, {M^4\over \lambda}$, and the
Hubble constant before the phase transition is given by $4.8\, {M^2\over \sqrt
\lambda M_p}$. One should check what is necessary to avoid too large density
perturbations (\ref{o17}). However, one should take into account that the mass
$M$ in (\ref{o17}) corresponds to the curvature of the effective potential
near $\phi = \phi_c$ rather than at $\phi = 0$. In our case this implies that
one should use $\sqrt 2 M$ instead of $M$ in this equation. Then one obtains
the following constraint on the mass $M$: \ $M\sqrt \mu {\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\ }
2\times10^{15}$ GeV. Note that the thin wall approximation (requiring $\mu
\ll 1$) breaks down far away from $\phi = \phi_c$. Therefore in general eq.
(\ref{o17}) should be somewhat improved. However for $\phi \approx \phi_c$ it
works quite well. To be on a safe side, we will take $M = 5\times 10^{14}$
GeV. Other parameters may vary; one may consider, e.g., the theory with $g
\sim 10^{-5}$, which gives
$\phi_c = {M\over g} \sim 5\times 10^{19}\ \mbox{GeV} \sim 4M_{\rm P}$.
The effective mass $m$ after the phase transition is equal to ${2gM\over \sqrt
\lambda}$ at $\phi = \phi_c$, and then it grows by only $25\%$ when the field
$\phi$ changes all the way down from $\phi_c$ to $\phi = 0$. As we already
mentioned, in order to obtain the proper amplitude of density perturbations
produced by inflation
inside the bubble one should have $m \sim 10^{13}$ GeV. This corresponds to
$\lambda = \alpha^2 = 10^{-6}$.
The bubble
formation becomes possible only for $\phi < \phi_c$. If it happens in the
interval $4M_{\rm P} > \phi > 3 M_{\rm P}$, we obtain a flat universe. If it
happens at $\phi < 3M_{\rm P}$, we obtain an open universe. Depending on the
initial value of the field $\phi$, we can obtain all possible values of
$\Omega$, from $\Omega = 1$ to $\Omega = 0$. The value of the Hubble constant
at the minimum with $\sigma \not = 0$ at $\phi = 3M_{\rm P}$ in our model does
not differ much from the value of the Hubble constant before the bubble
formation. Therefore we do not expect any specific problems with the large
scale density perturbations in this model.
Note also that the probability of tunneling at large $\phi$ is very small
since the depth of the minimum at $\phi \sim \phi_c$, $\sigma \sim \sigma_c$
does not differ much from the depth of the minimum at $\sigma = 0$, and there
is no tunneling at all for $\phi > \phi_c$. Therefore
the number of flat universes produced by this mechanism will be strongly
suppressed as compared with the number of open universes, the degree of this
suppression being very sensitive to the value of $\phi_c$. Meanwhile, life of
our type is impossible in empty universes with $\Omega \ll 1$. This may provide
us with a tentative explanation of the small value of $\Omega$ in the context
of our model (see, however, discussion of uncertainties related to this issue
in Sect. \ref{Simplest}).
\section{\label{Natural} ``Supernatural'' inflation with $\Omega < 1$ }
Natural inflation has been proposed some time ago \cite{BG,Natural} as a
model in which the inflaton field has a self-coupling constant whose
smallness, required by the amplitude of cosmological inhomogeneities, is
protected by the approximate global symmetry of the underlying particle
physics. The pseudo-Nambu-Goldstone boson (PNGB) field $\phi$, which serves
as an inflaton in these models, would have been exactly massless if not for
explicit
$U(1)$ symmetry breaking induced by non-perturbative effects. The hierarchy
between the scale of spontaneous symmetry breaking with generation of
Nambu-Goldstone mode and the scale of explicit symmetry breaking which gives a
mass to
this mode is exploited to explain the smallness of the effective mass.
We will consider the case when the PNGB mode is described by the pseudo-scalar
field $\phi$, appearing as a phase of a complex scalar $\Phi$. The scalar
sector
of the effective theory of the pseudo-Nambu-Goldstone mechanism is in general
described by the action:
\begin{equation} \label{axionaction}
S(\Phi) = \int{ d^4x \, \sqrt{-g} \left( g^{\mu \nu} \partial_{\mu} \Phi^*
\partial_{\nu} \Phi
- V_0(|\Phi|) - V_1(\Phi) \right)}.
\end{equation}
In notation $\Phi(x) = {f(x)\over \sqrt 2} \, e^{i{\phi(x) \over f_0}}$ the
field $f(x)$ is the radial component, the PNGB field $\phi(x)$ is the phase
component, and $f_0$ is a dimensional parameter which is equal to the value of
the scalar field $f(x)$ after symmetry breaking. The function
$V_0(|\Phi|)=V_0(f)$ is the spontaneous symmetry breaking part of the
potential for the complex scalar which remains globally $U(1)$ symmetric under
the transformation $\phi(x) \rightarrow \phi(x) + c$. In the version
of this theory considered in \cite{Natural} this potential was taken in the
simplest form
\begin{equation}\label{mexicanhat}
V_0(f) = \frac{\lambda}{4} \left( f^2 - f_0^2 \right)^2 \ .
\end{equation}
The term
$V_1(\Phi)=V_1(f, \phi)$ is the explicit
$U(1)$ symmetry breaking potential which in many models takes the following
form in the limit $f \rightarrow f_0$:
\begin{equation}\label{explicit}
V_1(f, \phi) = \Lambda^4(f) \, \left(1- \cos\Bigl({\phi \over f_0 } -
\bar{\theta}\Bigr) \right).
\end{equation}
Here $\Lambda^4(f)$ is some relatively slowly varying (in comparison with
$V_0(f)$) function of the radial field, which vanishes at $f=0$. This term
may appear due to instantons in a theory with a gauge group with a
``confinement'' scale $\Lambda$, just like the term which is responsible
for the axion mass. Another reason for appearance of such terms is the
possibility that quantum gravity violates global symmetries
\cite{AW}--\cite{LLKS}. This violation can be described by adding vertex
operators of the type
${g_{nm}\over M_{\rm P}^{m+n-4}}\, (\Phi^n\Phi^{*m}e^{-i\theta(n-m)} + h.c.)$.
After
spontaneous
symmetry breaking the terms with the minimal degree of global symmetry
violation ($|n-m|= 1$) lead to appearance of the terms of the type of
(\ref{explicit}).
The standard assumption of this scenario is that the effective potential
$V_1(f, \phi)$ is much smaller than $V_0(f)$ everywhere except for $f \approx
f_0$. Therefore the functional form of
$\Lambda^4(f)$ should not be very important as far as (\ref{explicit}) is the
leading term in the PNGB effective potential at low energies, but does not give
a
significant contribution into the potential of the radial field away from the
value $f_0$. Thus in what follows we will simply write $\Lambda \equiv
\Lambda(f_0)$ instead of $\Lambda(f)$. Without loss of generality we can assume
$\bar{\theta}=0$ as long
as there are no other terms in the low energy potential depending on it.
The potential in this theory resembles a slightly tilted mexican hat. In
this scenario inflation occurs when the field $\phi$ falls from the
maximum of the potential (\ref{explicit}) at $\phi \sim \pi f_0$ (for
$\bar{\theta}=0$) to $\phi = 0$. Just
like in the ordinary chaotic inflation scenario, the necessary condition for
inflation to occur is $\pi f_0 {\
\lower-1.2pt\vbox{\hbox{\rlap{$>$}\lower5pt\vbox{\hbox{$\sim$}}}}\ } M_{\rm
P}$.
Inflation will be long enough
and density perturbations produced during this process will have sufficiently
flat spectrum for $f_0 {\
\lower-1.2pt\vbox{\hbox{\rlap{$>$}\lower5pt\vbox{\hbox{$\sim$}}}}\ } M_{\rm
P}$. For
definiteness, and in agreement with
\cite{Natural}, we will assume here that $f_0 \sim M_{\rm P}$.
The parameter $\Lambda$ is determined by normalization of density
perturbations produced during inflation.
For $f_0 \sim
M_{pl} \sim 10^{19}$ GeV one must have $\Lambda(f_0)
\sim M_{GUT} \sim 10^{16}$ GeV (see \cite{Natural} for detailed references).
In order for the radial part of the field to remain frozen and to not
participate in natural inflation there has to be the case that $\lambda >{32
\pi \over 3} \, {\Lambda^4 \over M_{\rm P}^2 f_0^2} \sim 10^{-12}$, which also
ensures that the top of the ``mexican hat'' is higher than the highest energy
of the axion $\Lambda^4$. The coupling constant $\lambda$ could be as large as
unity (however, it does not have to be so large). If we consider the energy
density at the top of the ``mexican hat'' as the measure of how large is the
coupling $\lambda$ (we need this
characterization since we will soon change the shape of the potential), we see
that there is a plenty of room for play --- roughly 12 orders of magnitude in
{\em energy density} between the GUT and Planck scales, or between the scales
of spontaneous and explicit symmetry breaking.
The issue of naturalness of this scenario is not quite trivial. The first time
the possibility of inflation of this kind was investigated by Binetruy and
Gaillard in the context of superstring theory, and their conclusion was
negative,
for the reason that the typical value of the parameter $f_0$ in superstring
theory is supposed to be few times smaller than $M_{\rm P}$ \cite{BG}. Still,
the
general idea of this scenario is rather elegant. Here we would like to suggest
its generalization, which would allow us to obtain an inflationary universe
with $\Omega < 1$. We will call our scenario ``supernatural inflation'' for
its
ability to accommodate naturally low $\Omega$ universes.\footnote{For the
reason mentioned above, we do not want to imply any relation between this
scenario and supersymmetric theories.}
Our main idea is to construct models which incorporate a primary stage of
``old'' inflation in the false vacuum state near $\Phi = 0$,
and a first order transition which sets up for us the open de Sitter space as a
stage for the subsequent secondary stage of inflation where the PNGB field
$\phi$ plays the
role of the inflaton. As before, if the number of e-foldings of secondary
inflation will turn out to be just smaller 60, we will find ourselves in an
open, yet matter-rich universe today.
In order to realize this scenario one should have a potential which has a
minimum near $\Phi = 0$. A possible way to achieve it is to add to the
Lagrangian the term $-g^2\chi^2\Phi^*\Phi$
describing interaction of the field $\Phi$ with some other scalar field
$\chi$. If the coupling constant of this interaction is sufficiently large
($g^4
> 16\pi^2 \lambda$), the effective potential of the radial part $f$ of the
field $\Phi$ acquires a new minimum at $f = 0$ due to radiative corrections
\cite{MyBook}. For $g^4 = 32\pi^2 \lambda$ the minimum of the effective
potential at $f = 0$ becomes as deep as the minimum at $f = f_0$, and the
effective potential acquires the form $V(f) = {\lambda\over 2} (f_0^2-f^2)f^2
+ \lambda f^4 \ln {f\over f_0}$. Thus for $16\pi^2 \lambda< g^4 < 32\pi^2
\lambda$ the potential has a minimum at $f = 0$ which is somewhat higher than
the minimum at $f= f_0$. If $g^4$ is close to $16\pi^2 \lambda$, the potential
looks like the usual Coleman-Weinberg potential, and the phase
transition from the state $f = 0$ occurs by the Hawking-Moss mechanism.
However, one can show that if $f_0$ is not much greater than $M_{\rm P}$ and if
$g^4$
does not differ much from $32\pi^2 \lambda$ (which means that the minimum at $f
= 0$ is deep enough), then the absolute value of the mass of the field
$f$ always remains greater than the Hubble constant. In such a situation the
phase transition can be well described by the thin wall approximation
neglecting gravitational effects, and (this is important) there will be no
intermediate stage of
inflation during the rolling of the field $f$ towards $f_0$.
We do not want to discuss this issue here in great detail since radiative
corrections is just one of the reasons why the effective potential may acquire
a deep minimum at $f = 0$, and we would like to keep our discussion as general
as possible. In particular, all subsequent results will remain valid if the
potential $V(f)$ is given by the simple expression which we have used in the
previous sections, $V(f) = {m^2\over 2} f^2 - {\delta\over 3} f^3 +
{\lambda\over 4}f^4$. In order to increase the curvature of the effective
potential at $\Phi = 0$ one may also add the term $\xi R\Phi^2$ to the
Lagrangian. This term, being $U(1)$-invariant, does not affect the behavior of
the Goldstone mode, and therefore it does not modify the standard picture of
natural inflation, but it changes the curvature of the effective potential. It
also may preclude inflation at large $f \gg f_0$. This may be useful, since
otherwise inflation may begin at $f \gg f_0$, and then there will be no first
order phase transition, and the universe will be flat. On the other hand, even
if inflation begins at $f \gg f_0$, one still may experience the second stage
of inflation at $\Phi = 0$ and the subsequent bubble formation if after the end
of this stage of inflation at $f \gg f_0$ the oscillating scalar field has
enough kinetic energy to climb to the local minimum of the effective potential
at $\Phi = 0$.
An important feature of the ``supernatural'' inflation scenario is a very large
difference between the energy density at the stage of inflation at $\Phi =
0$, which has the typical energy scale $\sim {\lambda\over 4} M_{\rm P}^4$, and
the
relatively small energy density $\Lambda^4$ during the last stage of inflation.
As we have already mentioned, these two scales may differ from each other (for
large $\lambda$) by about 12 orders of magnitude.
This implies that after the tunneling there will be a long intermediate stage
of
non-exponential expansion until the kinetic energy of the radial field and the
energy density of particles produced by its oscillations becomes smaller than
$\Lambda^4$.
It takes time of the order of $H^{-1}(\Lambda)\sim M_{\rm P}/\Lambda^2$ to
complete
this intermediate stage. During the sub-luminal expansion epoch $\Omega$ stays
very small
and we can safely assume that the second inflationary stage starts at $\Omega =
0$. One can derive the adiabatic perturbation spectrum, modified by the fact
that the natural inflation starts from the curvature dominated stage.
Typically,
the modification of the density perturbation spectrum is not great (remember,
however, that the observable CMBR temperature fluctuations may differ
considerably, see Section \ref{Bubbles}).
Note that in this scenario the Hubble constant during inflation at $f = 0$ is
much greater than the Hubble constant at the second stage of inflation. In this
respect supernatural inflation resembles the simple model which we discussed in
Sect. \ref{Simplest}. The main difference is that at the first stage of
inflation in the supernatural scenario the mass of the field $f$ is supposed to
be much greater than the Hubble constant. (This condition is satisfied if the
minimum at $f = 0$ is sufficiently deep.) Therefore there are no inflationary
fluctuations of the scalar field produced outside the bubble in the
supernatural inflation scenario. Moreover, even if the mass is not much greater
than $H$, quantum fluctuations at $f = 0$ do not lead to any quantum
fluctuations of the angular field $\phi$, which is responsible for density
perturbations after inflation. Thus, in this scenario we do not have any
complications
related to perturbations penetrating the bubble from the false vacuum.
In addition to the usual density perturbations produced during the second stage
of inflation inside the bubble, there will be density perturbations induced by
the initial
inhomogeneities of the bubble. According to eq. (\ref{o14}), we can
estimate the corresponding density perturbations as follows:
\begin{equation}\label{inducedperturb}
\frac{\delta \rho}{\rho} \sim
\frac{2\sqrt {2\mu}\Lambda^2}{\sqrt{3\pi} f_0 M_{\rm P}} \sim \frac{2\sqrt
{2\mu}\Lambda^2}{\sqrt{3\pi} M_{\rm P}^2} \ ,
\end{equation}
where the last approximation assumes $f_0 \sim M_{\rm P}$. For $\Lambda \sim
10^{16}$ GeV and $\mu < 1$ these perturbations are smaller than the usual
perturbations produced during the second stage of
inflation.
If the energy in the maximum of
the radial potential is much greater than the energy of the explicit symmetry
breaking, the tunneling is likely to occur in any direction with equal
probability. If it goes towards $\phi = 0$, one obtains an empty universe with
$\Omega \ll 1$; if it goes towards $\phi \sim \pi f_0$, one obtains a universe
with $\Omega \approx 1$. Thus we may say that it is about as likely to obtain
$\Omega <
1$ as to obtain $\Omega = 1$, if we do not compare the volumes produced during
the secondary inflation (the openness of the universes which we consider makes
comparing the {\em a posteriori} volumes trickier). It is possible,
though, to construct a (fine tuned) model which has a preferred value of
$\Omega$. To do this, we should discover a reason why the phase transition
would go in a given preferred direction rather than any other.
So far we did not consider implications of the global symmetry breaking for the
structure of the potential near $\Phi = 0$. If the corresponding terms appear
due to instanton effects, which become significant only at the late stages of
the universe evolution, the shape of the effective potential at small $f$
remains unchanged. The physical reason is the infrared cutoff introduced by
the Hawking temperature $T_H = {H\over 2\pi}$ near $\Phi = 0$. (The Hawking
temperature may suppress effects induced by instantons if $H(\Phi =
0) \gg \Lambda$.) However, as we already mentioned, in addition to the
low-energy
non-perturbative effects, the high energy non-perturbative quantum
gravitational effects
may also add symmetry breaking terms of the form \cite{AW}--\cite{LLKS}
\begin{equation}\label{explicit2}
V_2(\phi(x)) = -{1 \over 2} g_{1}\, M_{\rm P}^3\, (\Phi e^{-i\theta_1} + \Phi^*
e^{i\theta_1}) = - g_{1}\, M_{\rm P}^3\, f(x)
\cos{\left(\frac{\phi(x) }{ f_0 } - {\theta_1}\right)} \ .
\end{equation}
The coupling $g_{1}$ may be strongly suppressed,
so that (\ref{explicit2}) does not change the shape of the potential
(\ref{explicit}) at $f \sim f_0$ (see \cite{LLKS} for
detailed explanations). However the linear term may play an important role at
very small $f$, where all other terms are of a higher order in $f$. It
may alter substantially the shape of the
potential near the top of the mexican hat and
determine the preferable direction of the tunneling. The phase
$\theta_1$, which determines the position of the minimum of the term
(\ref{explicit2}), does
not have to be the same as $\bar{\theta}$ from (\ref{explicit}), so in general
our potential acquires the form of a ``twisted'' mexican hat. If $\theta_1$
happens
to coincide with $\bar{\theta}$, then the tunneling typically will produce
empty universes. If $\theta_1$ differs from
$\bar{\theta}$ by $\pi$, we will obtain the bubbles with typical values of
$\Omega \sim 1$. For intermediate values of $\left|\theta_1 -
\bar{\theta}\right|$ we will obtain predominantly $\Omega < 1$ universes. We
should emphasize again that the calculation of probability of creation of
universes with any particular value of $\Omega$ is not unambiguous, see Sect.
5. It is important
that in all cases the total number of the
universes with any possible value of $\Omega$ will be infinitely large.
\section{\label{Discussion} Discussion}
In this paper we suggested several different models of a homogeneous
inflationary universe with $\Omega > 1$ and with $\Omega < 1$. At present there
is no observational evidence in favor of $\Omega > 1$. It is clear, however,
that if observational data are consistent with $\Omega = 1$, they may be
consistent with $\Omega = 1.05$ as well. Therefore it is good to have a working
model of inflationary cosmology with $\Omega > 1$.
The situation with an open universe may be much more interesting from the point
of view of observational data. That is why in this paper we concentrated on the
discussion of various models of inflationary universe with $\Omega < 1$.
We have found that in the models containing only one scalar field one
typically needs fine tuning and rather artificial bending of the effective
potential.
In the models involving two scalar fields one typically obtains infinite number
of open universes with all possible values of $\Omega < 1$ \cite{Omega}. The
simplest model of this type was discussed in Sect. \ref{Simplest}. This model
is very natural, but it has several unusual features. First of all, the
universe described by this theory is not {\it exactly} homogeneous. A habitable
part of it can be visualized as an exponentially large island inside an
expanding bubble. The difference between the local properties of the
island-like (quasiopen) universe and the homogeneous open inflationary universe
in certain cases becomes very small. In these cases one should carefully
analyse the fate of large perturbations of the scalar field which are generated
outside the bubble but may penetrate inside it. We gave some arguments
suggesting that these perturbations in this model may be harmless, but this
question requires a more detailed investigation.
In Sect. \ref{Hybrid} we proposed another model \cite{Omega}, which is based on
a certain modification of the hybrid inflation scenario. This model is also
very simple. It describe the universe which is open and homogeneous. Finally,
in Sect. \ref{Natural} we described a modified version of the natural
inflation scenario. We do not want at this
stage to discuss which of the open inflation models is better. It is clear
that many other models of this type can be proposed. However, we think that
there is a good chance that many of the qualitatively new features of the
models discussed above will appear in the new models as well.
Should we take these models seriously? Should we admit that the standard
prediction of inflationary theory that $\Omega = 1$ is not universally valid?
We are afraid that now it
is too late to discuss this question: the jinni is already out of the bottle.
We know that inflationary models describing homogeneous inflationary universes
with $\Omega \not = 1$ do exist, whether we like it or not. It is still true
that the models which lead to
$\Omega = 1$ are much more abundant and, arguably, more natural. However, in
our opinion, it is
very encouraging that
inflationary theory is versatile enough to include models with all
possible values of $\Omega$.
To make our position more clear, we would like to discuss the history of
the standard model of electroweak interactions \cite{GWS}. Even though this
model was developed by Glashow, Weinberg and Salam in the 60's, it became
popular only in 1972, when it was realized that gauge theories with
spontaneous symmetry breaking are renormalizable \cite{Hooft}. However, it
was immediately pointed out that this model is far from being perfect. In
particular, it was not based on the simple group of symmetries, and it had
anomalies. Anomalies could destroy the renormalizability, and therefore it was
necessary to invoke a mechanism of their cancellation by enlarging the fermion
sector of the theory. This did not look very natural, and therefore Georgi and
Glashow in 1972 suggested another model \cite{GG}, which at the first glance
looked much better. It was based on the simple group of symmetry $O(3)$, and it
did not have any anomalies. In the beginning it seemed that this model is a
sure winner. However, after the discovery of neutral currents which could not
be described by the Georgi-Glashow model, everybody forgot about the issues of
naturalness and simplicity and returned back to the more complicated
Glashow-Weinberg-Salam model, which gradually became the standard model of
electroweak interactions.
This model has about twenty free parameters which so far did not find an
adequate theoretical explanation. Some of these parameters may appear rather
unnatural. The best example is the coupling constant of the electron to the
Higgs field, which is $2\times 10^{-6}$. It is a pretty unnatural number which
is fine-tuned in such a way as to make the electron 2000 lighter than the
proton.
It is important, however, that all existing versions of the electroweak theory
are based on two fundamental principles: gauge invariance and spontaneous
symmetry breaking. As
far as these principles hold, we can adjust our parameters and wait until they
get their interpretation in a context of a more general theory. This is the
standard way of development of the elementary particle physics.
For a long time cosmology developed in a somewhat different way, because of the
scarcity of reliable observational data. Fifteen
years ago many different cosmological models (HDM, CDM, $\Omega = 1$, $\Omega
\ll 1$, etc.) could describe all observational data reasonably well. The main
criterion for a good theory was its beauty and naturalness. Right now it
becomes increasingly complicated to explain all observational data. Therefore
cosmology is gradually becoming a normal experimental science, where the
results of observations play a more important role than the considerations of
naturalness. However, in our search for a correct theory we cannot give up
the requirement of its internal consistency. In particle physics the two
principles which made this theory internally consistent were the gauge
invariance and spontaneous symmetry breaking. It seems that in cosmology
something like inflation is needed to make the universe large and homogeneous.
It is true that most of the inflationary models
predict a universe with $\Omega = 1$. Hopefully, several years later we
will know that our universe is flat, which will be a strong experimental
evidence in favor of inflationary cosmology in its simplest form. However, if
observational data will show,
beyond any reasonable doubt, that $\Omega \not = 1$, it will not imply that
inflationary theory is wrong, just like the discovery of neutral currents did
not disprove gauge theories of electroweak interactions. Indeed, now we know
that there is a large class of internally consistent cosmological models which
may describe
creation of large homogeneous universes with all possible values of $\Omega$,
and so far all of these models are based on inflationary cosmology.
The authors are very grateful to M. Bucher, J. Garc\'{\i}a--Bellido, L.
Kofman, V. Mukhanov, I. Tkachev and A. Vilenkin for many enlightening
discussions. This work was
supported in part by NSF grant PHY-8612280.
\
|
1,116,691,499,250 | arxiv | \section{Details of Experimental Evaluation}
\label{app:seq-experiments-details}
The following table lists the results of our experimental evaluation on the sequential programs benchmark suite.
\begin{longtable}{l@{\hskip 1.5em}rrrrrr@{\hskip 1.5em}r}
\caption{Experimental results of Horn-DT and Seahorn on the sequential programs benchmark suite. ``Rounds'' corresponds to the number of rounds of the learning process. ``Pos'', ``Neg'', and ``Horn'' refers to the number of positive, negative, and Horn examples produced during the learning, respectively. ``To'' indicates a timout after 300\,s, while``FP'' indicates a false positive. All times are given in seconds.}\\
\endfirsthead
\caption[]{continued}
\endhead
\toprule
Benchmark & \multicolumn{6}{c}{Horn-DT} & Seahorn \\
\cmidrule(l{-.5em}r{1.5em}){2-7} \cmidrule(l{-.5em}r){8-8}
& Rounds & Pos & Neg & Horn & Learner time & Total time & Time \\
\midrule
add.bpl & 8 & 1 & 6 & 1 & 0.03 & 0.8 & 4.49 \\
afnp.bpl & 13 & 1 & 3 & 9 & 0.06 & 0.81 & 0.35 \\
array2.bpl & 5 & 3 & 2 & 1 & 0.05 & 0.43 & 0.46 \\
array.bpl & 14 & 4 & 4 & 7 & 0.06 & 0.99 & 0.26 \\
array\_diff.bpl & 3 & 2 & 2 & 0 & 0.01 & 0.34 & 0.23 \\
arrayinv1.bpl & 134 & 4 & 32 & 100 & 7.44 & 11.57 & 1.34 \\
arrayinv2.bpl & 45 & 7 & 12 & 27 & 0.38 & 2.55 & 0.22 \\
bool\_dec.bpl & 3 & 2 & 1 & 0 & 0.01 & 0.31 & FP \\
bool\_inc.bpl & 3 & 1 & 1 & 1 & 0.15 & 0.45 & FP \\
cegar1.bpl & 5 & 3 & 1 & 1 & 0.02 & 0.41 & 0.28 \\
cegar2.bpl & 24 & 4 & 7 & 14 & 0.28 & 1.67 & 0.30 \\
cggmp.bpl & 67 & 1 & 13 & 53 & 0.69 & 2.11 & 0.31 \\
countud.bpl & 25 & 3 & 16 & 8 & 0.11 & 0.57 & 0.27 \\
dec.bpl & 3 & 1 & 2 & 0 & 0.01 & 0.32 & 0.31 \\
dillig01.bpl & 8 & 2 & 6 & 1 & 0.03 & 0.37 & 0.22 \\
dillig03.bpl & 9 & 2 & 4 & 3 & 0.04 & 0.38 & 0.21 \\
dillig05.bpl & 15 & 3 & 10 & 4 & 0.11 & 0.5 & TO \\
dillig07.bpl & 12 & 3 & 5 & 5 & 0.05 & 0.4 & 0.28 \\
dillig15.bpl & 12 & 2 & 3 & 7 & 0.05 & 0.41 & 0.26 \\
dillig17.bpl & 39 & 5 & 9 & 31 & 0.21 & 0.8 & 0.36 \\
dillig19.bpl & 12 & 2 & 4 & 7 & 0.05 & 0.43 & 0.31 \\
dillig24.bpl & 15 & 0 & 0 & 17 & 0.08 & 0.53 & 0.44 \\
dillig25.bpl & 47 & 1 & 25 & 36 & 0.32 & 1.1 & TO \\
dillig28.bpl & 25 & 1 & 0 & 34 & 0.12 & 0.61 & TO \\
dtuc.bpl & 28 & 6 & 15 & 14 & 0.14 & 0.74 & FP \\
ex14.bpl & 2 & 1 & 1 & 0 & 0.01 & 0.31 & 0.24 \\
ex14c.bpl & 2 & 1 & 1 & 0 & 0.01 & 0.32 & 0.22 \\
ex23.bpl & 19 & 2 & 7 & 11 & 0.1 & 0.58 & FP \\
ex7.bpl & 3 & 2 & 2 & 0 & 0.01 & 0.39 & FP \\
fig1.bpl & 5 & 2 & 3 & 1 & 0.02 & 0.34 & 0.24 \\
fig3.bpl & 3 & 2 & 2 & 0 & 0.01 & 0.34 & FP \\
fig9.bpl & 2 & 1 & 1 & 0 & 0.01 & 0.32 & 0.20 \\
formula22.bpl & 56 & 1 & 9 & 46 & 0.44 & 1.24 & 0.39 \\
formula25.bpl & 32 & 1 & 28 & 3 & 0.16 & 0.76 & 0.67 \\
formula27.bpl & 116 & 1 & 105 & 10 & 0.93 & 2.62 & 0.89 \\
inc2.bpl & 10 & 4 & 3 & 3 & 0.05 & 0.45 & 0.26 \\
inc.bpl & 102 & 1 & 1 & 100 & 1.98 & 3.14 & 0.21 \\
loops.bpl & 16 & 3 & 0 & 16 & 0.08 & 0.53 & 0.35 \\
\bottomrule
%
\pagebreak
%
\toprule
Benchmark & \multicolumn{6}{c}{Horn-DT} & Seahorn \\
\cmidrule(l{-.5em}r{1.5em}){2-7} \cmidrule(l{-.5em}r){8-8}
& Rounds & Pos & Neg & Horn & Learner time & Total time & Time \\
\midrule
matrixl1.bpl & 8 & 6 & 7 & 1 & 0.04 & 0.51 & 0.21 \\
matrixl1c.bpl & 8 & 5 & 11 & 0 & 0.04 & 0.5 & FP \\
matrixl2.bpl & 24 & 14 & 9 & 6 & 0.11 & 0.87 & 0.21 \\
matrixl2c.bpl & 36 & 15 & 18 & 6 & 0.2 & 1.31 & FP \\
nc11.bpl & 9 & 2 & 4 & 4 & 0.04 & 0.39 & 0.26 \\
nc11c.bpl & 8 & 3 & 3 & 3 & 0.04 & 0.39 & 0.27 \\
sqrt.bpl & 95 & 6 & 49 & 42 & 1.29 & 3.67 & FP \\
square.bpl & 18 & 1 & 14 & 3 & 0.08 & 0.79 & FP \\
sum1.bpl & 22 & 5 & 16 & 4 & 0.1 & 0.63 & 0.30 \\
sum3.bpl & 6 & 1 & 4 & 1 & 0.03 & 0.38 & 0.26 \\
sum4.bpl & 41 & 1 & 7 & 33 & 0.21 & 0.98 & 0.28 \\
sum4c.bpl & 19 & 5 & 11 & 4 & 0.09 & 0.59 & 0.19 \\
tacas.bpl & 32 & 15 & 9 & 11 & 0.17 & 0.69 & 0.30 \\
trex1.bpl & 4 & 2 & 3 & 0 & 0.02 & 0.36 & 0.25 \\
trex3.bpl & 10 & 4 & 7 & 2 & 0.04 & 0.46 & 0.35 \\
vsend.bpl & 2 & 1 & 1 & 0 & 0.01 & 0.34 & 0.20 \\
w1.bpl & 4 & 2 & 1 & 1 & 0.02 & 0.34 & 0.29 \\
w2.bpl & 4 & 1 & 2 & 1 & 0.02 & 0.34 & 0.27 \\
\bottomrule
\end{longtable}
\section{Node and Attribute Selection}
\label{sec:attribute-selection}
The decision tree algorithm in Section~\ref{sec:learner} returns a consistent tree irrespective of the order in which nodes of the tree are processed or the heuristic used to choose the best attribute to split nodes in the tree. If one is not careful while selecting the next node to process or one ignores the Horn constraints while choosing the attribute to split the node, seemingly good splits can turn into bad ones as data points involved in the Horn constraints get classified during the construction of the tree.
We experimented with the following strategies for node and attribute selection:
\begin{description}
\item[Node selection:] breadth-first-search; depth-first-search; random selection; selecting nodes with the maximum/minimum entropy
\item[Attribute selection:] based on a new information gain metric that penalizes node splits that cut Horn constraints; based on entropy for Horn samples obtained by assigning probabilistic likelihood values to unlabeled datapoints using model counting.
\end{description}
So as to clutter the paper not too much, we here only describe the best performing combination of strategies in detail. The experiments reported in Section~\ref{sec:experiments} have been conducted with this combination.
\subsubsection*{Choosing the next node to expand the decision tree}
We select nodes in a breadth-first search (BFS) order for building the decision tree. BFS ordering ensures that while learning multiple invariant annotations, the subtree for all invariants gets constructed simultaneously. In comparison, in depth-first ordering of the nodes, subtrees for the multiple invariants are constructed one after the other. In this case, learning a simple invariant, for e.g.\@ \emph{true}, for an annotation usually forces the invariant for a different annotation to become very complex.
\subsubsection*{Choosing attributes for splitting the given node}
Similar to ~\cite{GNMR16}, we observed that if one chooses attribute splits based on the entropy of the node that ignores Horn constraints, the invariant learning algorithm tends to produce large trees.
In the same spirit as ~\cite{GNMR16}, we penalize the information gain for attribute splits that cut Horn constraints, and choose the attribute with the highest corresponding information gain.
For a sample $S = (X, C)$ that is split with respect to attribute $a$ into subsamples $S_a$ and $S_{\neg a}$, we say that the corresponding attribute split cuts a Horn constraint $\psi \in C$ if and only if
\begin{itemize}
\item $x \in \textit{premise}(\psi)$ and $x \in S_a$ and $\textit{conclusion}(\psi) \in S_{\neg a}$; or
\item $x \in \textit{premise}(\psi)$ and $x \in S_{\neg a}$ and $\textit{conclusion}(\psi) \in S_a$.
\end{itemize}
The \emph{penalized} information gain is defined as
\[ \textit{Gain}_\textit{pen}(S, S_a, S_{\neg a}) = \textit{Gain}(S, S_a, S_{\neg a}) - \textit{Penalty}(S, S_a, S_{\neg a}, C), \]
where the penalty is proportional to the number of Horn constraints cut by the attribute split. However, we do not penalize a split when it cuts a Horn constraint such that the premise of the constraint is labeled negative and the conclusion is labeled positive. We incorporate this in the penalty function by formally defining it as
\[ \textit{Penalty}(S, S_a, S_{\neg a}, H) = \sum\limits_{\substack{\psi \in H, x \in S_a\\x \in \textit{premise}(\psi) \\ \textit{conclusion}(\psi) \in S_{\neg a}}} \big( 1 - f(S_a, S_{\neg a})\big) + \sum\limits_{\substack{\psi \in H, x \in S_{\neg a} \\x \in \textit{premise}(\psi) \\ \textit{conclusion}(\psi) \in S_{a}}} \big( 1 - f(S_{\neg a}, S_{a})\big), \]
\noindent where, for subsamples $S_1$ and $S_2$, $f(S_1, S_2)$ is the likelihood of $S_1$ being labeled negative and $S_2$ being labeled positive (i.e., $f(S_1, S_2) = \frac{N_1}{P_1 + N_1}.\frac{P_2}{P_2 + N_2}$). Here, $P_i$ and $N_i$ is the number of positive and negative datapoints respectively in the sample $S_i$.
\section{Experimental Evaluation}
\label{sec:experiments}
We have implemented a prototype, named \emph{Horn-DT} to evaluate the proposed learning framework.
The decision tree learning algorithm and the Horn solver are fresh implementations, consisting of roughly 6000 lines of C\raisebox{1pt}{\small {+\!+}} code. The teacher, on the other hand, is implemented on top of Microsoft's program verifier Boogie~\cite{DBLP:conf/fmco/BarnettCDJL05} and reuses much of the code originally developed by Garg et al.~\cite{GNMR16} for their ICE learning tool. Via C-to-Boogie conversion tools such as \textsc{Smack}~\cite{DBLP:conf/cav/RakamaricE14}, our prototype becomes a fully automatic program verifier for C programs.
We have evaluated our prototype on two suits of benchmarks. The first suite consists of 56 sequential programs of varying complexity, mainly taken the Software Verification Competition (\textsc{SV-Comp})~\cite{DBLP:conf/tacas/Beyer17}.
The second benchmark suite consists of 24 concurrent and recursive programs and includes popular concurrent protocols like Peterson's algorithm, a producer-consumer problem and complex recursive programs like McCarthy91.
A natural choice to compare Horn-DT to is \emph{Seahorn}~\cite{DBLP:conf/cav/GurfinkelKKN15}. Seahorn is a fully automated analysis framework for LLVM-based languages, which compiles C and C\raisebox{1pt}{\small {+\!+}} code via the LLVM intermediate language into verification conditions in form of constrained Horn clauses. These Horn clauses can then be solved by any suitable solver. For our experimental evaluation, we used the PDR engine~\cite{DBLP:conf/sat/HoderB12} implemented in Z3~\cite{DBLP:conf/tacas/MouraB08}, which is offered as the standard option in Seahorn.
Unfortunately, Seahorn cannot reason about concurrent programs and, thus, we can only compare to Seahorn on sequential programs (we are not aware of any other Horn-based invariant synthesis tool to which we could compare on this benchmark suite).
Moreover, it is important to emphasize that our evaluation is not a like-to-like comparison due to the following two reasons:
\begin{itemize}
\item Boogie and Seahorn produce different verification conditions: as Boogie allows very rich specifications (e.g., including quantifiers), the verification conditions produced by it are typically larger and more complex compared to those produced by Seahorn.
\item The PDR engine does not require invariant templates while Horn-DT does. More precisely, Horn-DT learns invariants that are arbitrary Boolean combinations of Boolean program variables and predicates of the form $x \pm y \leq c$ where $x,y$ are numeric program variables and $c$ is a constant determined by the decision tree learner. (Note, however, that this particular choice of template is not a general restriction of our framework and can easily be changed should the specific domain require it.)
\end{itemize}
As a consequence of these differences, our general expectation was that Seahorn with PDR to performs slightly better than Horn-DT on ``simple'' programs that do not require the overhead introduced by Boogie and permit simple invariants. Conversely, we expected a superior performance of Horn-DT on complex programs (e.g., programs working over arrays or performing nonlinear computations) due to its black box nature and the use of templates.
\pgfplotsset{runtime diagram/.style={%
xmin = 1e-2, xmax = 1e2,%
ymin = 1e-2, ymax = 1e2,%
log basis x=10,%
log basis y=10,%
enlarge x limits=false, enlarge y limits=false,%
xtickten={-1,...,1},%
ytickten={-1,...,1},%
extra x ticks={1e2}, extra x tick labels={\strut TO},%
extra y ticks={1e2}, extra y tick labels={\strut TO},%
xlabel near ticks,
ylabel near ticks,
}}
\pgfplotsset{runtime diagram marks/.style={%
only marks,%
mark=x,%
mark size=2.5,%
}}
\begin{figure}[th]
\centering
\begin{tikzpicture}
\begin{axis}[
name=plot1,
width=50mm,
y = 7.5mm,
xbar stacked,
xmin = 0, xmax = 56,
extra x ticks={56},
enlarge y limits={.75},
xlabel={Number of programs},
symbolic y coords={Horn-DT, Seahorn},
ytick=data,
legend style={at={(0,-0.6)}, anchor=north west, legend columns=1, draw=none, legend cell align=left, column sep=.75em},
cycle list={{fill=black!60,draw=black}, {fill=black!35,draw=black}, {fill=black!15,draw=black}},
]
\addplot+ [xbar] coordinates {(56,Horn-DT) (43,Seahorn) };
\addlegendentry{Verified}
\addplot+ [xbar] coordinates {(0,Horn-DT) (3,Seahorn) };
\addlegendentry{Timeout ($300$ s)}
\addplot+ [xbar] coordinates {(0,Horn-DT) (10,Seahorn) };
\addlegendentry{Error / false positive}
\end{axis}
\begin{loglogaxis}[
at={(plot1.north east)}, anchor=north west, xshift=40mm,
runtime diagram,
height=50mm,
xlabel = {Seahorn (time in s)},
ylabel = {Horn-DT (time in s)},
]
\addplot[runtime diagram marks] table[col sep=comma, x={Seahorn-Z3}, y={Horn-DT}] {horn_vs_seahorn.csv};
\draw[black!25] (rel axis cs:0, 0) -- (rel axis cs:1, 1);
\end{loglogaxis}
\end{tikzpicture}
\caption{Experimental comparison of Horn-DT with Seahorn on the the sequential programs benchmark suite. TO indicates a time out after 300 s or a false positive.} \label{fig:experimental-results-sequential}
\end{figure}
The remainder of this section presents our empirical evaluation on both benchmark suites in detail. All experiments were conducted on a Intel Core i3-4005U 4x1.70GHz CPU with 4\,GB of RAM running Ubuntu 16.04 LTS 64\,bit.
The reported results were obtained using the split and node selection strategies described in Sec.~\ref{sec:attribute-selection}.
\subsubsection*{Sequential Programs}
The first benchmark suite consists of 56 sequential programs taken \textsc{SV-Comp} as well as Garg et al.'s ICE tool~\cite{GNMR16}. These programs vary in complexity and range from simple integer manipulating programs to programs involving nonlinear computations to programs with complex array manipulations. The task for each of the benchmark programs is to find a single inductive invariant or adequate contract for a method call. We refer the reader to \textsc{SV-Comp}~\cite{DBLP:conf/tacas/Beyer17} as well as Garg et al.~\cite{GNMR16} for further details.
Figure~\ref{fig:experimental-results-sequential} summarizes the results of our experimental evaluation on the sequential programs benchmark suite (see Appendix~\ref{app:seq-experiments-details} for details). As shown on the left, Horn-DT is able to synthesize invariants and verify all 56 programs in the benchmarks suite. Seahorn, on the other hand, is able to verify 43 programs (77\,\%) and times out on 3 further programs (5\,\%). Surprisingly, Seahorn produced false positives (i.e., it reported an assertion violation) for 10 programs (18\,\%)---these programs predominantly involve arrays or nonlinear computations. After corresponding with the authors of SeaHorn, it turned out that these false positives were in fact caused by bugs in Seahorn and are now being investigated.
The right-hand-side of Figure~\ref{fig:experimental-results-sequential} shows a comparison of the runtimes of Horn-DT and Seahorn. As expected, Seahorn requires slightly less time to verify programs but frequently times out (or fails) when the programs work over arrays or perform nonlinear computations. Horn-DT, on the other hand, is able to verify the majority of programs in less than 1 s, which demonstrates the effectiveness of our technique.
\subsubsection*{Concurrent and Recursive Programs}
The second benchmark suite consists of 17 concurrent and recursive
programs selected from \textsc{SV-Comp15} \emph{pthread},
\emph{pthread-atomic}, \emph{pthread-ext}, \emph{pthread-lit},
\emph{recursive} and \emph{recursive-simple} benchmarks, as well as
popular concurrent programs from the literature~\cite{mine2014relational}.
\begin{table}[t]
\centering
\caption{Results of Horn-DT on concurrent and recursive programs. Columns show the number of invariants to be synthesized (``Inv''), total number of terms used (``Dim''), number of iterations between teacher and learner (``Rounds''), number of each kind of counter-examples generated (``Pos'', ``Neg'', and ``Horn''), and the time taken (``Time''). Benchmarks with the suffix \emph{-RG} and \emph{-OG} indicate Rely-Guarantee-style proofs and Owicki-Gries-style proofs, respectively.} \label{table:conc-rec-results}
\bigskip
\begin{tabular}{lrrrrrrr}
\toprule
\multicolumn{1}{l}{Benchmark} & \multicolumn{1}{c}{Inv} & \multicolumn{1}{c}{Dim} & \multicolumn{1}{c}{Rounds} & \multicolumn{1}{c}{Pos} & \multicolumn{1}{c}{Neg} & \multicolumn{1}{c}{Horn} & \multicolumn{1}{c}{Time (s)} \\
\midrule
Ackermann02 & 2 & 5 & 13 & 4 & 1 & 8 & 0.89 \\
Addition03 & 2 & 5 & 31 & 2 & 5 & 29 & 1.00 \\
afterrec\_2calls & 2 & 4 & 0 & 0 & 0 & 0 & 0.04 \\
BallRajamani-SPIN2000-Fig1 & 2 & 5 & 32 & 5 & 0 & 30 & 1.58 \\
fib\_bench-OG & 6 & 18 & 7 & 2 & 0 & 5 & 1.52 \\
fibo\_2calls\_5 & 4 & 6 & 105 & 1 & 3 & 105 & 3.65 \\
Fibo\_5 & 2 & 3 & 71 & 1 & 3 & 67 & 2.43 \\
id2\_b3\_o2 & 4 & 6 & 48 & 1 & 5 & 45 & 1.40 \\
id2\_i5\_o5 & 4 & 6 & 10 & 1 & 1 & 10 & 0.72 \\
McCarthy91 & 1 & 2 & 98 & 5 & 91 & 2 & 2.77 \\
peterson-OG & 8 & 32 & 370 & 3 & 91 & 277 & 65.31 \\
qw2004-OG & 13 & 20 & 24 & 2 & 1 & 23 & 3.33 \\
stateful01-OG & 6 & 12 & 220 & 2 & 0 & 218 & 9.56 \\
sum\_2x3 & 2 & 5 & 36 & 1 & 2 & 36 & 0.99 \\
sum\_non\_eq & 2 & 4 & 8 & 3 & 1 & 4 & 0.58 \\
sum\_non & 2 & 4 & 6 & 4 & 1 & 1 & 0.57 \\
18\_read\_write\_lock-OG & 8 & 16 & 16 & 2 & 1 & 13 & 2.42 \\
\midrule
Mine\_Fig\_1-OG~\cite{mine2014relational} & 10 & 20 & 35 & 2 & 0 & 41 & 2.00 \\
Mine\_Fig\_1-RG~\cite{mine2014relational} & 12 & 28 & 43 & 2 & 0 & 48 & 3.13 \\
Mine\_Fig\_4-OG~\cite{mine2014relational} & 13 & 28 & 51 & 3 & 0 & 65 & 6.03 \\
Mine\_Fig\_4-RG~\cite{mine2014relational} & 15 & 44 & 123 & 3 & 0 & 159 & 38.59 \\
pro\_cons\_atomic-OG & 8 & 16 & 32 & 2 & 0 & 30 & 2.11 \\
pro\_cons\_queue-OG & 8 & 16 & 31 & 2 & 0 & 29 & 2.09 \\
12\_hour\_clock\_serialization & 1 & 3 & 6 & 1 & 1 & 4 & 1.59 \\
\bottomrule
\end{tabular}
\end{table}
For concurrent programs we have used both
Rely-Guarantee~\cite{XuRH97} and
Owicki-Gries~\cite{owicki1976verifying} proof techniques to
verify the assertions. For recursive programs we use a modular
verification technique, in the form of function contracts for each
procedure.
All these programs were manually converted into Boogie
programs. For concurrent programs, we essentially encode the
verification conditions for Rely-Guarantee or
Owicki-Gries proofs in Boogie.
Invariants inside a loop with an empty loop-body can be removed by over constraining invariant at the loop entry. This modification will safely under approximate the search space for invariants.
We have adopted this encoding technique to verify Peterson's algorithm.
Table~\ref{table:conc-rec-results} shows the results of running
our Horn-DT tool on these programs. We note that each of these programs
produce a large number of Horn counter-examples. The Boogie Teacher
first tries to produce positive or negative counter-examples, and
produces Horn counter-examples only if these are not possible.
This indicates that Horn counter-examples are predominantly
needed for these kinds of program specifications.
Our tool successfully learns invariants for all these programs in
reasonable time, with most finishing within 10 seconds. Only one program took more than 50 s to verify.
Verification using Owicki-Gries proof rules requires
adequate invariants at each program point in each
thread. In comparison, Rely-Guarantee additionally requires two-state
invariants for each thread for the Rely/Guarantee conditions.
These additional invariants makes learning for Rely-Guarantee
proofs more difficult for the Learner.
\section{Algorithm for solving and propagating Horn constraints}
\label{sec:horn-solver}
In the decision-tree based learning approach, our aim is to construct a
decision-tree representing a Boolean combination of some base
predicates, that is consistent with a given set of Horn constraints.
A crucial step in our decision-tree algorithm is to check whether a proposed
extension of the current partial valuation maintained by the Learner,
is indeed consistent with
the given Horn constraints, and if so to determine the set of
propositional variables that are ``forced'' to be true or false.
In this section we describe an efficient algorithm to carry out
this step.
Our algorithm is an adaptation of the ``pebbling'' algorithm of
\cite{DowlingG84} for checking satisfiability of Horn formulas, to
additionally find the variables that are forced to be true or false.
\begin{algorithm}[th]
\caption{}\label{alg:horn-solver}
\begin{algorithmic}[1]
\Procedure{Horn-Solve}{}
\ \ \ \ \ \ \ \ \ \ \Require Horn constraints $C$ over $X$, partial
valuation $u$ over $X$, and $T, F \subseteq X$.
\ \ \ \ \ \ \ \ \ \ \Ensure {``Unsat'' if $u^T_F$ is inconsistent with $C$;
``Sat'', \forcedt{C}{u^T_F}, \forcedf{C}{u^T_F} otherwise.}
\State Add two new variables $\True$ and $\False$ to $X$. Let $X'
= X \union \{\True, \False\}$.
\State $C'$ $\gets$ $C$ +
clauses $\true \Implies x$ for each $x$ such
that $u(x) = \true$ or $x \in T$, and $x \Implies \false$ for each $x$
such that $u(x) = \false$ or $x \in F$.
\State Mark variable $\True$ with ``$\mymark$'', and
each variable $x \in X$ with ``$\mymark x$''.
\Repeat {
\State For each constraint $x_1 \And \cdots \And x_l \Implies y$
in $C'$:
\If {$x_1$, \ldots, $x_l$ are all marked ``$\mymark$''}
{mark $y$ with ``$\mymark$''}
\EndIf
\If {$\exists z \in X$ s.t.\@ $x_1$, \ldots,
$x_l$ are all marked ``$\mymark z$'' or ``$\mymark$''}
{mark $y$ with ``$\mymark z$''}
\EndIf
}
\Until{no new marks can be added}
\State $P \gets \{ x \in X \ | \ x \mbox{ is marked }
``\mymark\mbox{''} \}$,
$N \gets \{ x \in X \ | \ \False \mbox{ is marked }
``\mymark\!x\mbox{''} \}$
\State $P' \gets P - (T \union \dom_{\true}(u))$,
$N' \gets N - (F \union \dom_{\false}(u))$
\If {($\False$ is marked ``\mymark'')} \Return ``Unsat''
\Else{} \Return ``Sat'', $P'$, $N'$
\EndIf
\EndProcedure
\end{algorithmic}
\end{algorithm}
Procedure~\textsc{Horn-Solve} in Algorithm~\ref{alg:horn-solver}
shows our procedure for checking consistency of a partial
valuation with respect to a given set of Horn constraints $C$, as well
as identifying the subset of variables forced to true or false.
Intuitively, the standard linear-time algorithm for Horn satisfiability
in fact already identifies the \emph{minimal} $M$ set of variables that are
forced to be true in any satisfying valuation, and assures us that the
others $\vec{M}$ can be set to false. However, the other variables are not
\emph{forced} to be false. Our algorithm essentially runs another
set of SAT problems where each of the other variables are set to \emph{true}
(this is modeled by the variable being marked $*$ in the algorithm);
this returns SAT iff the variable is not forced to be \emph{false}.
The following example illustrates Algorithm~\ref{alg:horn-solver}.
\begin{figure}[th]
\centering
\begin{tikzpicture}
\node (0) at (0, 0) {$y$};
\node (1) at (0, -1.4) {$x$};
\draw[->] (0.east) -- ++(.9, -.7) coordinate (A) -- ++(.9, 0) node[anchor=west] (2) {$z$};
\draw (1.east) -- (A);
\draw[fill=white] (A) circle (1.1pt);
\draw[->] (1.north) -- (0.south);
\draw[shorten <=1pt, shorten >=0pt, <-] (1.west) -- ++(-.9, 0) node[anchor=east] (true) {$\True$};
\node (3) at (4, 0) {$b$};
\node (4) at (4, -1.4) {$a$};
\draw[->] (3.east) -- ++(.9, -.7) coordinate (B) -- ++(.9, 0) node[anchor=west] (false) {$\False$};
\draw (4.east) -- (B);
\draw[fill=white] (B) circle (1.1pt);
\draw[->] (3) edge[bend left] (4);
\draw[->] (4) edge[bend left] (3);
\begin{scope}[font=\scriptsize, inner sep=0pt, text=black!75]
\node[anchor=north, yshift=-.25ex] at (true.south) {\strut $\ast$};
\node[anchor=south west, xshift=-.25ex, yshift=-.25ex] at (0.north east) {\strut $\ast, \ast y, \ast x$};
\node[anchor=south east, xshift=.25ex, yshift=-.25ex] at (0.north west) {\strut (+)};
\node[anchor=north west, xshift=-.25ex, yshift=.25ex] at (1.south east) {\strut$\ast, \ast x$};
\node[anchor=north east, xshift=.25ex, yshift=.25ex] at (1.south west) {\strut +};
\node[anchor=north, yshift=.25ex] at (2.south) {\strut$\ast, \ast z, \ast x, \ast y$};
\node[anchor=south, yshift=-.25ex] at (2.north) {\strut (+)};
\node[anchor=south west, xshift=-.25ex, yshift=-.25ex] at (3.north east) {\strut $\ast b, \ast a$};
\node[anchor=south east, xshift=.25ex, yshift=-.25ex] at (3.north west) {\strut (-)};
\node[anchor=north west, xshift=-.25ex, yshift=.25ex] at (4.south east) {\strut $\ast a, \ast b$};
\node[anchor=north east, xshift=.25ex, yshift=.25ex] at (4.south west) {\strut (-)};
\node[anchor=north, yshift=-.25ex] at (false.south) {\strut $\ast a, \ast b$};
\end{scope}
\end{tikzpicture}
\caption{Example illustrating Algorithm~\ref{alg:horn-solver}. The given set of Horn constraints is $C = \{x \Implies y, x \And y \Implies z, a \Implies b, b \Implies a, a \And b \Implies \False\}$, $T = \{x\}$, $F = \emptyset$, and the partial valuation is empty.
The algorithm outputs ``Sat'' together with $P = \{y,z\}$ and $N = \{a,b\}$.} \label{fig:horn-algo-example}
\end{figure}
\begin{example}
Fig.~\ref{fig:horn-algo-example} illustrates the working of procedure \textsc{Horn-Solve} on an example set of Horn constraints $C = \{x \Implies y, x \And y \Implies z, a \Implies b, b \Implies a, a \And b \Implies \False\}$. The given partial valuation is empty, the set of variables $T$ initially set to $\true$ is $\{x\}$, and the set of variables $F$ set to $\false$ is $\emptyset$.
The final marking computed by the procedure is shown below each variable. Variables set to $\true$ in the partial assignment or in the input set $T$, are shown with a ``+'' above them. Variables forced to true (respectively false) are shown with a ``(+)'' (respectively ``(-)'') above them. \exampleend
\end{example}
For the remainer of this section, let us fix $X$, $C$, $u$, $T$, and $F$ to be the inputs to the
procedure, and let $X'$, $C'$, $P$, $N$, $P'$ and $N'$ be as described
in the algorithm.
It is clear that there exists an extension of $u^T_F$ satisfying
$C$ ifand only if $C'$ is satisfiable.
Furthermore, the set of variables forced true
by $C$ with respect to $u^T_F$ coincides with those forced
true in $C'$, less the variables in $\dom_\true(u) \union T$.
A similar claim holds for the variables forced to false.
In total, we obtain the following result.
\begin{theorem}
\label{thm:horn-solver}
Let $(X, C)$ be a Horn sample, $n = |X|$, and $h = |C|$. The procedure \textsc{Horn-Solve} runs in time $\mathcal O(h\cdot n^2)$ and outputs
\begin{itemize}
\item ``Sat'' and the set of variables forced to be true (in $P$) and false (in $N$), respectively,
if the given extended valuation $u^T_F$ is consistent with the Horn constraints $C$; and
\item ``Unsat'' otherwise.
\end{itemize}
\end{theorem}
To prove Theorem~\ref{thm:horn-solver}, we first introduce the notion of \emph{pebblings} (adapted from \cite{DowlingG84}) and state several straight-forward propositions.
Let $x$ be a variable in $X'$.
A $C'$-\emph{pebbling} of $(x,m)$ from $\True$ is a
sequence of markings $(x_0,m_0), \ldots, (x_k,m_k)$, such that
$x_k = x$ and $m_k = m$, and each $x_i$ and $m_i$ satisfy:
\begin{itemize}
\item $x_i \in X'$ and
$m_i \in \{*\} \union \{*x \ | \ x \in X \}$, and
\item one of the following:
\begin{itemize}
\item $x_i = \True$ and $m_i = ``*\mbox{''}$, or
\item $m_i = ``*x_i\mbox{''}$, or
\item $\exists i_1, \ldots, i_l < i$ such that each $m_{i_k} =
``*\mbox{''}$, and $m_i = ``*\mbox{''}$, and $x_{i_1} \And \cdots
\And x_{i_l} \Implies x_i \in C'$, or
\item $\exists z \in X$ and $\exists i_1, \ldots, i_l < i$ such
that each $m_{i_k} = ``*\mbox{''}$ or ``$*z$'', and $m_i = ``*\!z\mbox{''}$,
and $x_{i_1} \And \cdots \And x_{i_l} \Implies x_i \in C'$.
\end{itemize}
\end{itemize}
A $C'$-pebbling is \emph{complete} if the sequence cannot be
extended to add a new mark.
It is easy to see that each time the procedure
\textsc{Horn-Solve} marks a variable $x$ with a mark $m$, the sequence
of markings till this point forms a valid $C'$-pebbling of $(x,m)$
from $\True$, and the final sequence of markings produced does
indeed constitute a complete pebbling.
\begin{proposition}
\label{prop:peb1}
Consider a $C'$-pebbling of $(x,``*\!\mbox{''})$ from $\True$.
Let $v$ be a valuation such that $v\models C'$.
Then $v(x) = \true$.
\end{proposition}
\begin{proposition}
\label{prop:peb4}
Consider a $C'$-pebbling of $(y,``*x\!\mbox{''})$ from $\True$.
Let $v$ be a valuation such that $v(x) = \true$ and $v\models C'$.
Then $v(y) = \true$.
\end{proposition}
\begin{proposition}
\label{prop:peb2}
Consider a complete $C'$-pebbling from $\True$, in which $\False$ is
not marked ``$*$''.
Let $v$ be the valuation given by
\[
v(x) = \begin{cases} \true & \text{if $x$ is marked ``$*$''; and} \\ \false & \text{otherwise.} \end{cases}
\]
Then $v \models C'$.
\end{proposition}
\begin{proposition}
\label{prop:peb3}
Let $y \in X$.
Consider a complete $C'$-pebbling from $\True$, in which $\False$ is
not marked ``$*y$''.
Let $v$ be the valuation given by
\[
v(x) = \begin{cases} \true & \text{if $x$ is marked ``$*$'' or ``$*y$''; and} \\ \false & \text{otherwise.} \end{cases}
\]
Then $v \models C'$.
\end{proposition}
Given Propositions~\ref{prop:peb1} to \ref{prop:peb3}, we can now prove Theorem~\ref{thm:horn-solver}.
\begin{proof}[of Theorem~\ref{thm:horn-solver}]
We first observe that the procedure \textsc{Horn-Solve} clearly terminates since there is a finite number
of variables in $X$ (say $n$) and each variable can only be
marked $n+1$ times.
In fact, if $h$ is the number of clauses in $C$, the running time
of the procedure is bounded by $\mathcal O(h\cdot n^2)$.
We return now to the correctness of the procedure \textsc{Horn-Solve}.
Firstly, the procedure outputs ``Unsat'' if and only if $C'$ is unsatisfiable.
Suppose the procedure outputs ``Unsat''.
Then, $\False$ must be marked ``$*$'' by the algo.
But this means we have a $C'$-pebbling of $\False$ from $\True$.
Now, since we don't have a clause $\True \Implies \False$ in $C'$, we
must have a clause $c \in C'$ of the form $x_1 \And \cdots \And x_l
\Implies \False$ and a pebbling (by ``$*$'') of each $x_i$ from
$\True$.
Now, suppose to the contrary that there exists a valuation $v$ such
that $v \models C'$.
By Prop.~\ref{prop:peb1}, $v$ must set each $x_i$ to $\true$. But this
means that $v$ does not satisfy clause $c$, which contradicts the
assumption that $v \models C'$.
Conversely, suppose the procedure outputs ``Sat''.
Then, we have have a complete pebbling in which
$\False$ is not labelled ``$*$''.
By Prop.~\ref{prop:peb2}, we can construct a valuation $v$ which
satisfies $C'$, and hence $C'$ is satisfiable.
We further need to argue that in the latter case (when the algorithm
outputs ``Sat''), the $P$ and $N$ sets are computed correctly as the
set of variables forced to true and false respectively, in
$C'$.
We begin with the $P$ set.
Recall that $P$ is the set of variables that are marked ``$*$'' at the
end of the procedure, and hence in a complete pebbling.
It follows from Prop.~\ref{prop:peb1} that all the variables in $P$
are indeed forced true.
Conversely, suppose there is a variable $x$ that is not marked ``$*$''
by the algorithm.
Then, by Prop.~\ref{prop:peb2} there is a valuation $v$ which satisfies
$C'$, in which $x$ is set to $\false$. Hence $x$ is not forced to
true by $C'$.
Regarding the $N$ set, recall that it is the set of variables $x$
such that $\False$ is marked ``$*x$''.
Let $x \in N$, and let $v$ be a valuation that sets $x$ to $\true$,
and satisfies $C'$.
Then there must be a clause $c$ of the form $x_1 \And \cdots \And x_l
\Implies \false$ in $C'$, and each $x_i$ is marked ``$*$'' or ``$*x$''.
By Props.~\ref{prop:peb1} and \ref{prop:peb4}, $v$ must set each of
the $x_i$'s to $\true$.
But then $v$ does not satisfy clause $c$, and this contradicts the
assumption that $v \models C'$. Hence every variable in $N$ is forced
false.
Conversely, suppose $x \not\in N$.
Hence $\False$ is not marked ``$*x$''.
Then, by Prop.~\ref{prop:peb3}, there is a valuation $v$ which sets $x$
to $\true$ and satisfies $C'$.
Hence, $x$ is not forced to $\false$ by $C'$.
This completes the proof of the correctness of the procedure.
\qed
\end{proof}
\section{Introduction}
\label{sec:intro}
Synthesizing inductive invariants, including loop invariants, pre/post
contracts for functions, and rely-guarantee contracts for concurrent programs,
is one of the most important problems in program verification. In deductive
verification, this is often done by the verification engineer, and automating
invariant synthesis can significantly reduce the burden of building verified software,
allowing the
engineer to focus on the more complex specification and design
aspects of the code. In completely automated verification, where verification is typically
deployed for simple specifications, checking the validity of verification conditions of
recursion-free code has been mostly automated using constraint logic solving, and coupled
with invariant synthesis gives completely automated verification.
There are several techniques for finding inductive invariants, including
abstract interpretation~\cite{DBLP:conf/popl/CousotC77}, predicate abstraction~\cite{DBLP:conf/pldi/BallMMR01}, interpolation~\cite{mcmillan03,mcmillan_tacas06}, and IC3~\cite{DBLP:conf/vmcai/Bradley11}. These
techniques are typically \emph{white-box} techniques that carefully examine the program,
evaluating it symbolically or extracting unsatisfiable cores from proofs of unreachability
of error states in a bounded number of steps in order to synthesize
an inductive invariant
that can prove the program correct.
A new class of \emph{black-box} techniques based on learning has emerged in recent years
to synthesize inductive invariants~\cite{LMN14,GNMR16}. In this technique, there are two distinct agents, the Learner
and the Teacher. In each round the Learner proposes an invariant for the program, and the Teacher,
with access to a verification engine, checks whether the invariant
proves the program correct. If not, it synthesizes \emph{concrete} counterexamples that witness
why the invariant is inadequate and sends it back to the learner. The learner takes all such \emph{samples}
the teacher has given in all the rounds to synthesize the next proposal for the invariant.
The salient difference in the black-box approach is that the Learner synthesizes invariants from
concrete sample configurations of the program, and is otherwise oblivious to the program or its semantics.
It is tempting to think that the learner can learn invariants using positively and negatively labeled configurations,
similar to machine learning. However, Garg~et~al~\cite{LMN14} argued that we need a richer notion of samples for
robust learning of inductive invariants. Let us recall this simple argument.
Consider a system with variables $\vec{x}$, with initial states captured by a predicate $\textit{Init}(\vec{x})$,
and a transition relation captured by a predicate $\textit{Trans}(\vec{x}, \vec{x}')$, and assume we want
to prove that the system does not reach a set of bad/unsafe states captured by the predicate $\textit{Bad}(\vec{x})$.
An inductive invariant $I(\vec{s})$ that proves this property needs to satisfy three constraints:
\begin{itemize}
\item $\forall \vec{x}. \textit{Init}(\vec{x}) \Rightarrow I(\vec{x})$;
\item $\forall \vec{x}. ~\neg(I(\vec{x}) \wedge \textit{Bad}(\vec{x}))$; and
\item $\forall \vec{x}, \vec{x}'. I(\vec{x}) \wedge \textit{Trans}(\vec{x}, \vec{x}') \Rightarrow I(\vec{x}')$.
\end{itemize}
When a proposed invariant fails to satisfy the first two conditions, the verification engine can indeed come
up with configurations labeled positive and negative to indicate ways to correct the invariant. However, when
the third property above fails, it cannot come up with a single configuration labeled positive/negative; and
the most natural counterexample is a \emph{pair} of configurations $c$ and $c'$, with the instruction to the
learner that if $I(c)$ holds, then $I(c')$ must also hold. These are called \emph{implication counterexamples}
and the ICE (Implication Counter-Example) learning framework developed by Garg~et~al.\ is a robust learning
framework for synthesizing invariants~\cite{LMN14}.
Garg~et~al.\ define several learning algorithms for learning invariant synthesis, in particular learning algorithms based on decision trees that can learn Boolean combinations of Boolean predicates and inqualities that compare numerical predicates to arbitrary thresholds~\cite{GNMR16}.
Despite the argument above, it turns out that implication counterexamples are \emph{not sufficient} for learning
invariants in program verification settings. This is because reasoning in program verification is more stylized,
to deal \emph{compositionally} with the program. In particular, programs with function calls and/or
concurrency are not amenable to the above form of reasoning. In fact, it turns out that most reasoning
in program verification can be expressed in terms of \emph{Horn clauses}, where the Horn clauses have some formulas
that need to be synthesized.
For example, consider the imperative program snippet:
\[ I_\textit{pre}(\vec{x},y)~~ S(\text{mod}~\vec{x});~ y:=\textit{foo}(\vec{x}); ~ I_\textit{post}(\vec{x},y) \]
that we want to show correct, where $S$ is some straight-line program that modifies $\vec{x}$,
$I_\textit{pre}$ and $I_\textit{post}$ are some annotation (like the contract of a function we are synthesizing.
Assume that we are synthesizing the contract for $\textit{foo}$ as well,
and assume the post-condition for $\textit{foo}$ is $PostFoo(res, \vec{x})$,
where $res$ denotes the result it returns.
Then the verification condition that we want to be valid is
\[ \left( I_\textit{pre}(\vec{x}, y) \wedge \textit{Trans}_S(\vec{x}, \vec{x}') \wedge \textit{PostFoo}(y', \vec{x}') \right) \Rightarrow I_\textit{post}(\vec{x}', y'), \]
where $\textit{Trans}_S$ captures the logical semantics of the snippet $S$ in terms
of how it affects the post-state of $\vec{x}$.
In the above, all three of the predicates $I_\textit{pre}$, $I_\textit{post}$ and $\textit{PostFoo}$ need to be synthesized.
When a learner proposes concrete predicates for these, the verifier checking the above logical
formula may find it to be invalid, and find concrete valuations $v_{\vec{x}}, v_y, v_{\vec{x}'}, v_{y'}$
for $\vec{x}, y, \vec{x}', y'$ that
makes the above implication false. However, notice that the above cannot be formulated as
a simple implication constraint. The most natural constraint to return to the learner is
\[ \left( I_\textit{pre}(v_{\vec{x}}, v_{\vec{v_y}}) \wedge \textit{PostFoo}(v_{y'}, v_{\vec{x}'}) \right) \Rightarrow I_\textit{post}(v_{\vec{x}'}, v_{y'}), \]
asking the learner to meet this requirement when coming up with predicates
in the future. The above is best seen as a \emph{Horn} Implication CounterExample (Horn-ICE).
The primary goal of this paper is to build Horn-ICE (Horn implication counterexample) learners for
learning predicates that facilitate inductive invariant and contract synthesis for proving safety
properties of programs. It has been observed in the literature that most program verification mechanisms
can be stated in terms of proof rules that resemble Horn clauses~\cite{DBLP:conf/pldi/GrebenshchikovLPR12}; in fact, the formalism
of \emph{constrained Horn clauses} has emerged as a robust general mechanism for capturing program
verification problems in logic~\cite{DBLP:conf/cav/GurfinkelKKN15}. Consequently, whenever a Horn clause fails, it results in a Horn-ICE
sample that can be communicated to the learner, making Horn-ICE learners a much more general
mechanism for synthesizing invariants and contracts.
Our main technical contribution is to devise a decision-tree based Horn-ICE algorithm.
Given a set of (Boolean) predicates over configurations of programs and numerical functions
that map configurations to integers, the goal of the learning algorithm is to synthesize
predicates that are \emph{arbitrary} Boolean combinations of the Boolean predicates
and atomic predicates of the form $n \leq c$, where $n$ denotes a numerical function,
where $c$ is arbitrary. The classical decision-tree learning algorithm learns such
predicates from samples labeled $+/-$ only~\cite{Quinlan86}, and the work by Garg~et~al extends
decision-tree learning to learning from ICE-samples~\cite{GNMR16}. In this work, we extend
the latter algorithm to one that learns from Horn-ICE samples.
Extending decision-tree learning to handle Horn samples turns out to be non-trivial.
When a decision tree algorithm reaches a node that it decides to make a leaf and label it TRUE,
in the ICE-learning setting it can simply \emph{propagate} the constraints across the implication
constraints. However, it turns out that for Horn constraints,
this is much harder. Assume there is a single invariant we are synthesizing and
we have a Horn sample $(s_1 \wedge s_2) \Rightarrow s'$ and we decide
to label $s'$ false when building the decision tree. Then we must later turn \emph{at
least one of $s_1$ and $s_2$ to false}. This choice makes the algorithms and propagation
much more complex, and ensuring that the decision tree algorithm will always construct
a correct decision tree (if one exists) and work in polynomial time becomes much harder.
Furthermore, statistical measures based on entropy for choosing attributes (to split each node) get more complicated
as we have to decide on a more complex logical space of Horn constraints between samples.
The contributions of this paper are the following:
\begin{enumerate}
\item A robust decision-tree learning algorithm that learns using Horn implication counterexamples, runs in polynomial time (in the number of samples) and has a bias towards learning smaller trees (expressions) using statistical measures for choosing attributes. The algorithm also guarantees that a decision-tree
consistent with all samples is created, provided there exists one.
\item We show that we can use our learning algorithm to learn over an \emph{infinite} countable set of predicates ${\mathcal P}$, and we can ensure learning is complete (i.e., that will find an invariant if one is expressible using the predicates ${\mathcal P}$).
\item An implementation of our algorithm, extending the classical IC3 decision-tree algorithm, and an automated verification tool built with our algorithm for synthesizing invariants. We evaluate our algorithm for finding loop invariants and summaries for sequential programs and also Rely-Guarantee contracts in concurrent programs.
\end{enumerate}
The paper is structured as follows. In Section~\ref{sec:overview} we present an overview of Horn ICE invariant synthesis; in Section~\ref{sec:learner}, we describe the decision tree based algorithm for learning invariant formulas from Horn ICE samples; in Section~\ref{sec:horn-solver}, we describe the algorithm that propagates the data point classifications across Horn constraints; we describe the node/attribute selection strategies used in the decision tree based learning algorithm in Section~\ref{sec:attribute-selection} and the experimental evaluation in Section~\ref{sec:experiments}.
\section{Decision Tree Learning with Horn Constraints}
\label{sec:learner}
\subsubsection*{Valuations and Horn constraints}
We will consider propositional formulas over a fixed set of
propositional variables $X$, using the usual
Boolean connectives $\notof$, $\And$, $\Or$, $\Implies$, etc.
The \emph{data points} we introduce later will also play the role of
propositional variables.
A \emph{valuation} for $X$ is a map $v : X \rightarrow \{\true,
\false\}$.
A given formula over $X$ evaluates, in the standard way, to
either $\true$ or $\false$ under a given valuation.
We say a valuation $v$ \emph{satisfies} a
formula $\varphi$ over $X$, written $v \models \varphi$, if
$\varphi$ evaluates to $\true$ under $v$.
A \emph{partial} valuation for $X$ is a partial map
$u: X \pmap \{\true,\false\}$.
We denote by $\dom_{\true}(u)$ the set $\{x \in X \ | \ u(x) =
\true\}$ and $\dom_{\false}(u)$ the set $\{x \in X \ | \ u(x) =
\false\}$.
We say a partial valuation $u$ is \emph{consistent} with a formula
$\varphi$ over $X$, if there exists a full valuation $v$ for $X$,
which extends $u$ (in that for each $x \in X$, $u(x) = v(x)$
whenever $u$ is defined on $x$), and $v \models \varphi$.
Let $\varphi$ be a formula over $X$, and $u$ a partial valuation
over $X$ which is consistent with $\varphi$.
We say a variable $x \in X$ is \emph{forced to be true}, with
respect to $\varphi$ and $u$, if for all valuations $v$ which extend $u$,
whenever $v\models\varphi$ we have $v(x) = \true$.
Similarly we say $x$ is \emph{forced to be false}, with
respect to $\varphi$ and $u$, if for all valuations $v$ which extend $u$,
whenever $v\models\varphi$ we have $v(x) = \false$.
We denote the set of variables forced true (wrt $\varphi$ and $u$)
by $\forcedt{\varphi}{u}$, and those forced false by
$\forcedf{\varphi}{u}$.
Finally, for a partial valuation $u$ over $X$, and
subsets $T$ and $F$ of $X$, which are disjoint from each other and
from the domain of $u$, we denote by $u^T_F$ the partial valuation
which extends $u$ by mapping all variables in $T$ to $\true$, and
all variables in $F$ to $\false$.
A \emph{Horn clause} (or a \emph{Horn
constraint}) over $X$ is disjunction of literals over $X$ with \emph{at
most} one positive literal.
Without loss of generality, we will write Horn clauses in one of the
three forms:
\begin{enumerate*}[label={(\arabic*)}]
\item $\true \Implies x$,
\item $(x_1 \And \cdots \And x_k) \Implies \false$, or
\item $(x_1 \And \cdots \And x_l) \Implies y$,
\end{enumerate*}
where $l \geq 1$ and each of the $x_i$'s and $y$ belong to $X$.
\subsubsection*{Data Points}
Our decision tree learning algorithm is paired with a teacher that
refutes incorrect conjectures with positive data points, negative
data points, and, more generally, with Horn constraints
over data points. Roughly speaking, a data
point corresponds to a program configuration and contains the values
of each program variable and potentially values that are derived from
the program variables, such as $x+y$, $x^2$, or
$\mathit{is\_list}(z)$. For the sake of a simpler presentation,
however, we assume that a data point is an element $d \in \mathbb D$
of some (potentially infinite) abstract domain of data points $\mathbb D$ (we
encourage the reader to think of programs over integers where data
points correspond to vectors of integers).
\subsubsection*{Base Predicates and Decision Trees}
The aim of our learning algorithm is to construct a decision tree
representing a Boolean combination of some base predicates.
We assume a set of base predicates, each of which evaluates to $\true$
or $\false$ on a data point $d \in D$.
More precisely, a \emph{decision tree} is a binary
tree $\T$ whose nodes either have two children (\emph{internal nodes}) or no
children (\emph{leaf nodes}), whose internal nodes are labeled with base
predicates, and whose leaf nodes are labeled with $\true$ or
$\false$. The formula $\psi_{\T}$ corresponding to a decision tree
$\T$ is defined to be $\bigvee_{\pi \in \Pi_\true} \bigl( \bigwedge_{\rho \in \pi} \rho \bigr)$
where $\Pi_\true$ is the set of all paths from the root of $\T$
to a leaf node labeled $\true$, and $\rho \in \pi$
denotes that the base predicate $\rho$ occurs as a label of a node on
the path $\pi$.
Given a set of data points $X \subseteq D$, a decision tree $\T$
induces a valuation $v_{\T}$ for $X$ given by $v_{\T}(d) = \true$ iff
$d \models \psi_{\T}$.
Finally, given a set of Horn constraints $C$ over a set of data
points $X$, we say a decision tree $\T$ is \emph{consistent} with $C$
if $v_{\T} \models \bigwedge C$.
We will also deal with ``partial'' decision trees, where some of the
leaf nodes are yet unlabeled, and we define a partial valuation
$u_{\T}$ corresponding to such a tree $\T$, and the notion of
consistency, in the expected way.
\subsubsection*{Horn Samples}
In the traditional setting, the learning algorithm collects the
information returned by the
teacher as a set of \emph{samples}, comprising ``positive'' and
``negative'' data points.
In our case, the set of samples will take the form of a set of Horn
constraints $C$ over a finite set of data points $X$.
We note that a positive point $d$ (resp.\@ a negative point $e$) can be
represented as a Horn constraint $\true \Implies d$ (resp.\@ $e \Implies
\false$).
We call such a pair $(X,C)$ a \emph{Horn sample}.
In each iteration of the learning process, we require the learning
algorithm to construct a decision tree $\T$ that agrees with the
information in the current Horn sample $\S = (X, C)$, in
that $\T$ is consistent with $C$.
The learning task we address then is
\textit{``given a Horn sample $\S$, construct a decision tree
consistent with $\S$''}.
\begin{algorithm}[t]
\caption{Decision Tree Learner for Horn Samples}\label{alg:decision-tree}
\begin{algorithmic}[1]
\Procedure{Decision-Tree-Horn}{}
\ \ \ \ \ \ \ \ \ \ \Require A Horn sample $(X,C)$
\ \ \ \ \ \ \ \ \ \ \Ensure {A decision tree $\T$ consistent with $C$,
if one exists.}
\State Initialize tree $\T$ with root node $r$, with $r.dat \gets X$
\State Initialize partial valuation $u$ for $X$, with $u \gets \emptyset$
\While {($\exists$ an unlabelled leaf in $\T$)}
\State $n \gets \Call{Select-Node}{\,}$
\If {(pure($n$))} \ \ // $n.dat \intersection \dom_{\true}(u) =
\emptyset$ or $n.dat \intersection \dom_{\false}(u) = \emptyset$
\State $result = \Call{Label}{n}$
\EndIf
\If {($\notof \mbox{pure}(n) \Or \notof result$)}
\If {($n.dat$ is singleton)}
\State {print ``Unable to construct decision tree''};
\Return \label{line:dec-tree-unsuccess-1}
\EndIf
\State $result \gets \Call{Split-Node}{n}$
\If {($\notof result$)}
\State {print ``Unable to construct decision tree''};
\Return \label{line:dec-tree-unsuccess-2}
\EndIf
\EndIf
\EndWhile
\State \Return $\T$ \ \ \ \ {// decision tree constructed
successfully} \label{line:dec-tree-success}
\EndProcedure
\end{algorithmic}
\begin{minipage}[t]{6.5cm}
\begin{algorithmic}[1]
\Procedure {Label}{node $n$}
\State $Y \gets n.dat \setminus dom(u)$;
\If {($n.dat \intersection \dom_{\true}(u) \neq \emptyset$)}
\Statex \ \ \ \ \ \ \ \ \ // $n$ contains only pos/unsigned pts
\State $(res,T,F) \gets \Call{Horn-Solve}{u, Y, \emptyset}$
\If {($res$)}
\State $n.label \gets \true$
\State $u \gets u^{Y \union T}_{F}$
\State \Return $\true$
\Else \ \Return $\false$;
\EndIf
\Else \ \ldots\ \ // try to label neg
\EndIf
\EndProcedure
\end{algorithmic}
\end{minipage}
\begin{minipage}[t]{6cm}
\begin{algorithmic}[1]
\Procedure {Split-Node}{node $n$}
\State $(res, a) \gets \Call{Select-Attribute}{n}$
\If {($res$)}
\State Create new nodes $l$ and $r$
\State $l.dat \gets \{ d \in n.dat \ | \ d \models a \}$
\State $r.dat \gets \{ d \in n.dat \ | \ d \not\models a \}$
\State $n.\mathit{left} \gets l$, $n.\mathit{right} \gets r$.
\State \Return $\true$
\Else
\State \Return $\false$
\EndIf
\EndProcedure
\end{algorithmic}
\end{minipage}
\end{algorithm}
\subsubsection*{The Learning Algorithm}
Our learning algorithm, shown in Algo.~\ref{alg:decision-tree},
is an extension of Garg et al.'s \cite{GNMR16}
learning algorithm, which in turn is based on the classical decision
tree learning algorithm of Quinlan \cite{Quinlan86}.
Given a Horn sample $(X,C)$, the algorithm creates an initial
(partial) decision tree $\T$ which has a single unlabelled node, whose
associated set of data points is $X$.
As an auxiliary data structure, the learning algorithm maintains a
partial valuation
$u$, which is always an extension of the partial valuation induced by
the decision tree.
In each step, the algorithm picks an unlabelled leaf node $n$, and
checks if it is ``pure'' in that all points in the node are either
positive (i.e.\@ \true) or unsigned, or similarly neg/unsigned.
If so, it calls the procedure \textsc{Label} which
tries to label it positive if all points
are positive or unsigned, or negative otherwise.
To label it positive, the procedure first checks whether extending the
current partial valuation $u$ by making all the unsigned data points
in $n$ true results in a valuation that is consistent with the given
set of Horn constraints $C$.
It does this by calling the procedure \textsc{Horn-Solve} (described
in the next section), which not only checks whether the proposed
extension is consistent with $C$, but if so, also returns the set of
points forced to be true and false, respectively.
In this case, the node $n$ is labelled $\true$ and the partial
valuation $u$ is extended with the forced values.
If the attempt to label positive fails, it tries to label the node
negative, in a similar way.
If both these fail, it ``splits'' the node using a suitably chosen
base predicate $a$. The corresponding method \textsc{Select-Attribute},
which aims to (heuristically) obtain a small tree (i.e., a concise formula),
is described in Sec.~\ref{sec:attribute-selection}.
The crucial property of our learning algorithm is that if the given
set of constraints $C$ is satisfiable and if the data points in $X$ are
``separable,'' it will always construct a decision tree
consistent with $C$. We say that the points in $X$ are
\emph{separable} if for every pair of points $d_1$ and $d_2$ in $X$ we
have a base predicate $\rho$ which distinguishes them (i.e., either
$d_1 \models \rho$ and $d_2 \not\models \rho$, or vice-versa). This result, together with its time complexity,
is formalized in Theorem~\ref{thm:dec-tree-succ}.
\begin{theorem}
\label{thm:dec-tree-succ}
Let $(X, C)$ be a Horn sample, $n = |X|$, and $h = |C|$.
If the input set of points $X$ is separable and the input Horn
constraints $C$ are satisfiable, then Algo.~\ref{alg:decision-tree}
runs in time $\mathcal O(h\cdot n^3)$ and returns a decision tree that is consistent with the Horn sample
$(X,C)$.
\end{theorem}
\begin{proof}
At each iteration step, the algorithm maintains the invariant that the
partial valuation $u$ is an extension of the partial valuation
$u_{\T}$ induced by the current (partial) decision tree $\T$, and is
consistent with $C$. This is because each labelling step is first
checked by a call to the horn-solver, which also correctly identifies
the set of forced values, which are then used to update $u$.
It follows that if the algorithm terminates successfully in
Line~\ref{line:dec-tree-success}, then $u_{\T}$ is a full valuation
which coincides with $u$, and hence satisfies $C$.
The only way the algorithm can terminate unsuccessfully is in
Line~\ref{line:dec-tree-unsuccess-1} or
Line~\ref{line:dec-tree-unsuccess-2}.
The first case is ruled out since if $n.dat$ is singleton, and by
assumption $u_{\T}$ is consistent with $C$, we must be able to label
the single data point with either $\true$ or $\false$ in a way that is
consistent with $C$.
The second case is ruled out, since under the assumption of
separability the \textsc{Select-Attribute} procedure will always return a
non-trivial attribute (see Sec.~\ref{sec:attribute-selection}).
The learning algorithm (Algorithm~\ref{alg:decision-tree}) runs in cubic time
in the size $n$ of the input set $X$ of data points and linear in the number $h$ of Horn constraints.
To see this, observe that in each iteration of the loop the algorithm
produces a tree that is either the same as the previous step (but with a
leaf node labelled), or splits a leaf to extend the previous tree.
At each step we maintain an invariant that the collection of data points
in the leaf nodes forms a partition of the input set $X$. Thus the
number of leaf nodes is bounded by $n$, and hence each tree has a
total of at most $2n$ nodes. When the algorithm returns (successfully or
unsucessfully) each node in the final tree has been processed at
most once by calling the labelling/splitting subroutines on
it. Furthermore, the main work in the subroutines is the call to the
horn-solver, which takes $\mathcal O(h\cdot n^2)$ time (see
Sec.~\ref{sec:horn-solver}). It follows that
Algorithm.~\ref{alg:decision-tree} runs in $\mathcal O(h\cdot n^3)$ time. \qed
\end{proof}
When the points in $X$ are not separable, as done in \cite{GNMR16}, we
add ``iff'' constraints between every pair of inseparable points, and
if the resulting Horn constraints are satisfiable, our algorithm is
guaranteed to construct a decision tree consistent with the given
Horn constraints.
Furthermore, we can extend our algorithm to work on an infinite enumerable set
of predicates ${\cal P}$, and assure that the algorithm will find an invariant
if there is one, as done in~\cite{GNMR16}. We can take some finite set $X \subseteq P$,
asking whether there is some invariant over $X$ that satisfies $P$, and if not
grow $X$ by taking finitely more predicates from ${\cal P}\setminus X$. This is clearly guaranteed to converge
on an invariant if one is expressible over ${\cal P}$.
\section{Overview}
\label{sec:overview}
We give an overview of the Horn-ICE invariant synthesis framework in this section.
Fig.~\ref{fig:horn-ice-arch} shows the main components of our Horn-ICE invariant synthesis framework.
The Teacher is given a program along with a specification, and based on the style of proof the teacher wants to carry out, she determines the
kind of invariants needed (a name for each invariant, and the set of
variables or terms it may mention) and the corresponding verification
conditions they must satisfy.
The Learner conjectures a concrete invariant for each invariant name,
and communicates these to the Teacher.
The Teacher plugs in these conjectured invariants in the program and
asks a verification engine (in this case Boogie, but we could use any suitable program verifier) to check if the conjectured
invariants suffice to prove the specification.
If not, Boogie will return a counter-example showing why the
conjectured invariants do not constitute a valid proof.
The Teacher passes these counter-examples to the Learner.
The Learner now learns a new invariants that are consistent
with the set of counter-examples given by the Teacher so far.
The Learner frequently invokes the Horn Solver and Horn propagation engine to guide it
in the process of building concrete invariants that are consistent
with the set of counter-examples given by the Teacher. The Teacher and
Learner go through a number of such rounds, until the Teacher
finds that the concrete invariants supplied by the Learner constitute
a correct proof.
We illustrate the working of the tool with an example.
Fig.~\ref{fig:prog-text} shows a concurrent program adapted from
\cite{Mine14}, with two threads $T_1$ and $T_2$ that access two shared
variables \texttt{x} and \texttt{y}.
The precondition says that initially $x = y = 0$.
The postcondition asserts that when the two threads terminate, the
state satisfies $x \leq y$.
Let us say the Teacher is interested in proving this specification using
Rely-Guarantee style reasoning \cite{Jones83b,XuRH97}.
In this proof technique we need to come up with invariants associated
with the program points $P0$--$P4$ in $T_1$ and $Q0$--$Q4$ in $T_2$ (as in
the Floyd-Hoare style proofs of sequential programs), as well as
two ``two-state'' invariants $G1$ and $G2$, which act as the
``guarantee'' on the interferences caused by each thread respectively.
Symmetrically, $G2$ and $G1$ can be thought of as the conditions that
the threads $T_1$ and $T_2$ can ``rely'' on, respectively, about the
interferences from the other thread.
Fig.~\ref{fig:prog-chc} shows a partial list of the verification
conditions that the named invariants need to satisfy, in order to
constitute a valid Rely-Guarantee proof of the program.
The VCs are grouped into four categories: ``Adequacy'' and
``Inductiveness'' are similar to the requirements for sequential
programs, while ``Guarantee'' requires that the atomic statements of
each thread satisfy its promised guarantee, and ``Stability'' requires
that the invariants at each point are stable under interferences from
the other thread.
We use the notation ``$\den{x:=x+1}$'' to denote the two-state (or
``before-after'') predicate describing the semantics of the statement
``\texttt{x:=x+1}'', which in this case is the predicate
$x' = x + 1 \And y' = y$. The guarantee invariants $G1$ and $G2$ are
similar predicates over the variables $x,y,x',y'$, describing the
possible state changes that an atomic statement in a thread can
effect.
The invariants $P0$--$P4$ and $Q0$--$Q4$ are
predicates over the
variables $x,y$. We use the notation $P0'$ to denote the predicate
$P0$ applied to the variables $x',y'$.
\begin{figure}[th]
\centering
\begin{subfigure}[b]{.325\linewidth}
\begin{scriptsize}
\begin{verbatim}
Pre: x = y = 0
T1 || T2
P0 while (*) { Q0 while (*) {
P1 if (x < y) Q1 if (y < 10)
P2 x := x + 1; Q2 y := y + 3
P3 } Q3 }
P4 Q4
Post: x <= y
\end{verbatim}
\end{scriptsize}
\caption{The program} \label{fig:prog-text}
\end{subfigure}%
\hskip 2cm
\begin{subfigure}[b]{.475\linewidth}
\centering
\begin{scriptsize}
\begin{tabular}{|ll|ll|} \hline
& Adequacy & & Inductiveness \\ \hline
1.& $(x = 0 \And y = 0) \Implies P0$ & 1. & $P0 \Implies P1 \And P4$ \\
2.& $P4 \And Q4 \Implies (x \leq y)$ & 2. & $P1 \And (x < y) \Implies P2$ \\
& & 3. & $P2 \And \den{x := x + 1} \Implies P3'$ \\
& & 4. & $P3 \Implies P0$ \\
& & & $\cdots$ \\ \hline
& Stability & & Guarantee \\ \hline
1.& $P0 \And G2 \Implies P0'$ & 1.& $P2 \And \den{x := x + 1} \Implies G1$ \\
2.& $P1 \And G2 \Implies P1'$ & 2.& $Q2 \And \den{y := y + 3} \Implies G2$ \\
& $\cdots$ & & \\ \hline
\end{tabular}
\end{scriptsize}
\caption{The verification conditions}\label{fig:prog-chc}
\end{subfigure}
\caption{A concurrent program and the corresponding verification conditions for a Rely-Guarantee proof}\label{fig:1}
\end{figure}
The Teacher asks the Learner to synthesize the invariants $P0$--$P4$ and
$Q0$--$Q4$ over the variables $x,y$, and $G1$ and $G2$ over the
variables $x,y,x',y'$.
As a first cut the Learner conjectures ``$\true$'' for all these
invariants.
The Teacher encodes the VCs in Fig.~\ref{fig:prog-chc} as annotated
procedures in Boogie's programming language, plugs in $\true$ for
each invariant, and asks Boogie if the annotated program verifies.
Boogie comes back saying that the ensures clause corresponding to VC
Adequacy-1 may fail, and gives a counter-example say $\langle x
\mapsto 2, y \mapsto 1 \rangle$ which satisfies $P4$ and $Q4$, but
does not satisfy $x \leq y$.
The Teacher conveys this counter-example as a Horn sample
$d_1 \And d_2 \Implies \false$ to the Learner, where $d_1$ is the data
point $\langle P4,2,1\rangle$ and $d_2$ is the data
point $\langle Q4,2,1\rangle$.
We use the convention that the data points are vectors in which the
first component is the value of a
``location'' variable ``$l$'' which takes one of the values ``$P0$'',
``$P1$'', etc, while the second and third components are values of $x$
and $y$ respectively.
This Horn constraint is represented graphically in the bottom of
Fig.~\ref{fig:prog-hc}.
To focus on the technique used by the Learner, which is the heart of
this work, let us pan to several rounds ahead, where the Learner has
accumulated a set of counter-examples given by the Teacher, as shown
in Fig.~\ref{fig:prog-hc}.
The Learner's goal is simply to find a small (small in expression size)
invariant $\varphi$ (from a
finite class of
formulas comprising Boolean combinations of some base predicates), that
is \emph{consistent} with the given set of Horn constraints. By
``consistent'', we mean that for each Horn constraint of the form $d_1
\And \cdots \And d_k \Implies d$, whenever each of
$d_1, \ldots, d_k$ satisfy $\varphi$, it is the case that $d$ also
satisfies $\varphi$.
Our Learner uses a decision tree based learning technique. Here the
internal nodes of the decision tree are labelled by the base
predicates (or ``attributes'') and leaf-nodes are classified as
``True'', ``False'', or ``?'' (for ``Unclassified'').
Each leaf node in a decision tree represents a logical formula which
is the conjunction of the node labels along the
path from the root to the leaf node, and the whole tree represents the
disjunction of the formulas corresponding to the leaf nodes labelled
``True''.
The Learner builds a decision tree for a given set of Horn constraints
incrementally, starting from the root node.
Each leaf node in the tree has a corresponding subset of the data-points
associated with it, namely the set of points that satisfy the formula
associated with that node.
In each step the Learner can choose to mark a node as ``True'', or
``False'', or to split a node
with a chosen attribute and create two child nodes associated with
it.
Before marking a node as ``True'' or ``False'' the Learner would like
to make sure that this ``preserves'' the consistency of the decision tree
with respect to the set of Horn constraints.
For this he calls the Horn Solver/Propagation component, which reports
whether the proposed extension of the partial valuation is
indeed consistent with the given set of Horn constraints, and if so
which are the data-points which are ``forced'' to be true or false.
For example, let us say the Learner has constructed the partial
decision tree shown in Fig.~\ref{fig:prog-dec-tree}, where node $n_4$
has already been set to ``True'' and nodes $n_2$ and $n_5$ are
unclassified.
He now asks the Horn Solver if it is okay for him to turn node $n_2$
``True'', to which the Horn Solver replies ``Yes'' since this extended
valuation would still be consistent with the set of Horn constraints
in Fig.~\ref{fig:prog-hc}.
The Horn Solver also tells him that the extension would force the
data-points $d_{12}$, $d_8$, $d_7$, $d_5$, $d_4$, $d_3$, $d_1$ to be
$\true$, and the point $d_2$ to $\false$.
The Learner uses this information to go ahead and set $n_2$ to
``True'', and also to make note of the fact that $n_5$ is now a ``mixed''
node with some points that are forced to be $\true$ (like $d_1$) and
some $\false$ (like $d_2$).
Based on this information, the Learner may choose to split node $n_5$
next.
After completing the decision tree, the Learner may send the
conjecture in which $P0$--$P4$, $G1$ and $G2$ are set to $\true$, and
$Q1$--$Q4$ are set to $\false$.
The Teacher sends back another counter-example to this conjecture, and
the exchanges continue for several rounds.
Finally, our Learner eventually makes a successful conjecture like: $x
\leq y$ for $P0$, $P1$, $P3$, $P4$, and $Q0$--$Q4$; $x < y$ for $P2$;
$y = y' \And x' \leq y'$ for $G1$; and $x = x' \And y \leq y'$ for $G2$.
\begin{figure}[t]
\centering
\begin{subfigure}[b]{.53\textwidth}
\begin{tikzpicture}[font=\scriptsize]
\node[anchor=east] (true) at (-2.75, -1) {$\mathit{True}$};
\node[anchor=east] (0) at (0, 0) {$\langle P0, 0, 0 \rangle_{11}$};
\node[anchor=east] (1) at (0, -1) {$\langle G2, 0, 0 -1, 0 \rangle_{10}$};
\node[anchor=east] (2) at (0, -2) {$\langle P2, 0, 0 \rangle_{7}$};
\node[anchor=east] (3) at (0, -3) {$\langle G2, 0, 0, 1, 1 \rangle_{6}$};
\node[anchor=east] (4) at (0, -4) {$\langle P0, 2, 1 \rangle_{3}$};
\node[anchor=east] (5) at (2.5, 0) {$\langle P1, 0, 0 \rangle_{12}$};
\node[anchor=east] (6) at (2.5, -1.5) {$\langle P1, -1, 0 \rangle_{8}$};
\node[anchor=east] (7) at (2.5, -4) {$\langle P3, 2, 1 \rangle_{4}$};
\node[anchor=east] (8) at (2.5, -5) {$\langle P4, 2, 1 \rangle_{1}$};
\node[anchor=east] (9) at (2.5, -6) {$\langle Q4, 2, 1 \rangle_{2}$};
\draw[->] (0) edge (5);
\draw[->] (0.east) -- ++(.9, -.5) coordinate (A) -- ++(.9, 0) node[anchor=west] (10) {$\langle P0, -1, 0 \rangle_{9}$};
\draw (1.east) -- (A);
\draw[fill=white] (A) circle (1.1pt);
\draw[->] (10.east) -| ++(.25, -1) -- (6.east);
\draw[<-, shorten <=1pt, shorten >= 0pt] (2.west) -| ++(-.25, .5) -- (6.west);
\draw[->] (2.east) -- ++(.9, -.5) coordinate (B) -- ++(.9, 0) node[anchor=west] (11) {$\langle P2, 1, 1 \rangle_{5}$};
\draw (3.east) -- (B);
\draw[fill=white] (B) circle (1.1pt);
\draw[->] (11.east) -| ++(.25, -1.5) -- (7.east);
\draw[->] (7) edge (4);
\draw[->] (4.west) -| ++ (-.25, -1) -- (8.west);
\draw[->] (8.east) -- ++(.9, -.5) coordinate (C) -- ++(.9, 0) node[anchor=west] (false) {$\mathit{False}$};
\draw (9.east) -- (C);
\draw[fill=white] (C) circle (1.1pt);
\draw[->] (true.east) -- (1.west);
\end{tikzpicture}
\caption{Horn constraints given by the Teacher} \label{fig:prog-hc}
\end{subfigure}
\hfill
\begin{subfigure}[b]{.45\textwidth}
\begin{tikzpicture}[font=\scriptsize]
\node[draw, shape=circle] (1) at (0, 0) {$n_1$};
\node[draw, shape=circle] (2) at (-1.5, -2) {$n_2$};
\node[draw, shape=circle] (3) at (1.5, -2) {$n_3$};
\node[draw, shape=circle] (4) at (0, -4) {$n_4$};
\node[draw, shape=circle] (5) at (3, -4) {$n_5$};
\node[right=4mm] at (1) {$l = P0$};
\node[right=4mm] at (3) {$l = G2$};
\node[above left=2mm] at (2) {$3,9,11$};
\node[below=3mm] at (2) {$?$};
\node[above left=2mm] at (4) {$6,10$};
\node[below=3mm] at (4) {$+$};
\node[above right=2mm] at (5) {$1,2,4, \ldots$};
\node[below=3mm] at (5) {$?$};
\draw[shorten >=0pt] (1) -- (2) (1) -- (3) (3) -- (4) (3) -- (5);
\end{tikzpicture}
\caption{A partial decision tree built by the Learner} \label{fig:prog-dec-tree}
\end{subfigure}
\caption{Intermediate results of our Horn-ICE framework on the introductory example.} \label{fig:intermediate-results-example}
\end{figure}
The above example illustrates some of the key differences and challenges
from the classical ICE framework in \cite{LMN14,GNMR16}.
To begin with, the Learner needs to synthesize \emph{multiple}
invariants (in this case 12 different invariants) rather than a single
invariant; adapting decision-tree learning to learn multiple formulas
efficiently is challenging.
Secondly, we note that the Teacher \emph{needs} to use \emph{Horn}
counter-examples rather than simple implications, as illustrated by
the first counter-example $d_1 \And d_2 \Implies \false$, and
subsequently $d_6 \And d_7 \Implies d_5$ and $d_{10} \And d_{11}
\Implies d_{9}$.
Accordingly, the Learner's decision tree based algorithm needs to be
adapted to work robustly in the presence of Horn constraints.
\subsection*{Related Work}
Invariant synthesis is the central problem in automated program verification and, over the years, several techniques have been proposed for synthesizing invariants, including abstract interpretation~\cite{DBLP:conf/popl/CousotC77}, interpolation~\cite{mcmillan03,mcmillan_tacas06}, IC3 and PDR~\cite{DBLP:conf/vmcai/Bradley11,DBLP:conf/cav/KarbyshevBIRS15}, predicate abstraction~\cite{DBLP:conf/pldi/BallMMR01}, abductive inference~\cite{DBLP:conf/oopsla/DilligDLM13}, as well as synthesis algorithms that rely on constraint solving~\cite{DBLP:conf/pldi/gulwani08,invgen,DBLP:conf/cav/ColonSS03}.
Subsequent to~\cite{DBLP:conf/pldi/GrebenshchikovLPR12}, there has been a lot of work towards Horn-clause solving~\cite{BjornerMR13,DBLP:conf/cav/BeyenePR13}, using a combination of these techniques.
Complementing these techniques are data driven invariant synthesis techniques, the first ones to be proposed being Daikon~\cite{DBLP:conf/icse/ErnstCGN00} that learns \emph{likely} program invariants and Houdini~\cite{DBLP:conf/fm/FlanaganL01} that learns conjunctive inductive invariants.
Data-driven invariant synthesis has seen renewed interest, lately~\cite{DBLP:conf/cav/SharmaNA12,DBLP:conf/esop/0001GHALN13,DBLP:conf/sas/0001GHAN13,DBLP:conf/cav/0001LMN13,LMN14,DBLP:conf/cav/0001A14,Zhu:2015:LRT:2784731.2784766,DBLP:conf/kbse/PavlinovicLS16,DBLP:conf/pldi/PadhiSM16,DBLP:conf/icse/NguyenKWF12,GNMR16,Zhu:2016:ALS,DBLP:conf/sas/BrockschmidtCKK17,FKB17}.
When the program manipulates complex data-structures, arrays, pointers, etc., or when one needs to reason over a complicated memory model and its semantics,
the invariant for the correctness of the program might still be simple. In such a scenario, a black-box, data-driven \emph{guess and check} approach, guided by a finite set of program configurations, has been shown to be advantageous.
However, implication counter-examples proposed by Garg et al.~\cite{LMN14} are not sufficient for learning invariants in general program verification settings.
Grebenshchikov et al.~\cite{DBLP:conf/pldi/GrebenshchikovLPR12} have shown that most reasoning in program verification is expressed in terms of Horn clauses.
Subsequent to~\cite{DBLP:conf/pldi/GrebenshchikovLPR12}, there has been a lot of work towards Horn-clause solving~\cite{BjornerMR13,DBLP:conf/cav/BeyenePR13}.
SeaHorn~\cite{DBLP:conf/cav/GurfinkelKKN15} is a verification framework that translates verification conditions of a program to horn clauses that can be solved using several backend solvers.
In the context of data-driven invariant synthesis,
our work generalizes the ICE learning model~\cite{LMN14} to Horn counter-examples, and we build a decision-tree based Horn-ICE learner for learning invariant annotations in this model.
|
1,116,691,499,251 | arxiv | \section{INTRODUCTION}
Autonomous marine vehicles are important tools for many applications in both civilian and military contexts. One such application is Rapid Environmental Assessment (REA), where a vehicle provides information about the physical environment in an area of interest. This information is then used to inform future missions. In a riverine environment with unknown bathymetry, this can entail quickly identifying a channel that can provide a navigable path through the area. We refer to this as the rapid channel identification problem. This paper will focus on the development of a new algorithm to efficiently address this problem. We compare this new method to other state-of-the-art approaches and investigate the utility of using multiple vehicles in the solution.
The simplest way to survey an area is with a lawnmower search pattern. This is the most exhaustive method and will provide a comprehensive overview of the environment. However, for this problem we are interested in finding a specific feature -- a deep channel -- rather than constructing a complete map.
To speed up this process, we can employ \emph{adaptive} sampling strategies. This terminology refers to a broad class of algorithms where a robot updates its sampling strategy according to real-time observations.
\begin{figure}[ht]
\includegraphics[width=1\columnwidth]{allpaths.png}
\caption{Viable paths found through the channel in the Charles River using the PBACS algorithm. The rapid channel identification problem involves finding a deep path that connects the start (green) and goal (magenta) regions. A team of two to three Heron USVs identified these viable paths - shown as red and yellow traces - through the deep trench present in the Charles River near MIT campus. The red lines mark paths found with a minimum depth of 22 feet, and the yellow line marks paths found with a minimum depth of 23 feet. The bathymetry map data was provided by the Charles River Alliance of Boaters \cite{CRAB_Bathy}}
\label{fig:charles_plot}
\end{figure}
Using adaptive sampling and multi-vehicle task allocation techniques, we present the Proposal Based Adaptive Channel Search (PBACS) algorithm for rapid channel identification. We also compare its performance with that of three other sampling methods, and compare their performance when using different numbers of vehicles. The four approaches we tested were:
\begin{enumerate}
\item Proposal Based Adaptive Channel Search (PBACS)
\item Lawnmower survey
\item Markov decision process with Upper Confidence Bound (UCB) reward
\item Markov decision process with Maximum Value Information (MVI) reward
\end{enumerate}
We compare the methods by testing them against ten different bathymetry maps. Through this testing, we aim to identify a strategy that will work consistently well for different bathymetry configurations.
\subsection{Contributions and Overview}
The contributions of this research are the following:
\begin{itemize}
\item The PBACS algorithm, a new specialized method for solving the rapid channel identification problem
\item Application of a myopic MDP planner to the rapid channel identification problem
\item Comparison of the utility of using different amounts of vehicles in a channel finding mission for the different path planning approaches
\item Field deployment of the MDP approaches using up to three USVs, and deployment of the PBACS approach using up to four USVs.
\end{itemize}
The remainder of the paper is organized as follows. In Section \ref{sec:lit_rev}, we provide a brief overview of related work on marine adaptive sampling and multi-vehicle collaboration. In Section \ref{sec:meth}, we present our approach - the PBACS algorithm - and discuss alternatives. In Section \ref{sec:results}, we provide the experimental methods used in simulation and fieldwork studies as well as the results. Finally, in Section \ref{sec:conclusion} we summarize and conclude.
\section{LITERATURE REVIEW} \label{sec:lit_rev}
A robot is considered to be \emph{adaptively} sampling when the sampling strategy is adjusted in real time during the mission, based on the data being collected.
Adaptive sampling typically requires two main layers: field estimation, usually with Gaussian processes, and path planning. There are also at least three major classes of adaptive systems problems: source localization, front/boundary determination, and feature tracking and mapping. Furthermore, sampling with multiple vehicles often requires additional strategies for optimized performance. The remainder of this section summarizes previous research in these areas, with an emphasis on fielded systems for marine applications.
\subsection{Gaussian Processes and Path Planning}
In order to effectively make decisions and adjust its path, the vehicle needs to store past measurements and assemble a model of the environment. This can be achieved using Gaussian processes (GP) \cite{rasmussen2}, which are used to represent a probability distribution over functions. Gaussian process regression (GPR), also referred to as kriging, can be used to estimate the mean and covariance values for a set of inputs in the domain.
Gaussian processes are frequently used in marine adaptive sampling for spatial modeling of the estimated environment. Berget et al. \cite{berget} use GP methods to track suspended material plumes. The model is updated continuously throughout the mission, and the path planner uses this estimation to drive the unmanned vehicle to information-rich areas. In Fossum et al. \cite{fossum}, GP modeling is used for adaptive sampling of phytoplankton by modeling the distribution of chlorophyll-a - a common indicator of phytoplankton activity.
Yan et al. \cite{yan} use GPR analysis to guide online path planning for an AUV in an effort to locate hotspots in the field.
Another work to use GP methods is Stankiewicz et al. \cite{stankiewicz}, where an AUV uses adaptive sampling to explore an area and identify hypoxic zones. The algorithm identifies regions of interest that exhibit some local extrema and concentrates sampling there.
\subsection{Classes of Adaptive Sampling Problems}
Prior work in marine adaptive sampling can be separated into three broad categories: source localization, front/boundary determination, and feature tracking and mapping. Of these categories, the channel-finding problem explored in this paper is most closely related to the last.
Adaptive source localization, also called maximum seek and sample, is a problem adjacent to the channel-finding problem. Bayat et al. \cite{bayat} offer a survey of techniques that have been employed in solving this problem. One is gradient based, such as the approach reported by Paliotta et al. in \cite{paliotta}.
Another approach, proposed and tested by Flaspohler et al. \cite{flaspohler}, uses a partially observable Markov decision process (POMDP) with a reward based on maximum value information (MVI), originally developed by Wang et al. \cite{wang}.
The second category is front or boundary estimation. Examples of this can be found in \cite{zhang} and \cite{petillo15}, where an AUV moves in a vertical sawtooth trajectory to detect and track a front. Fiorelli et al. \cite{fiorelli2006multi} also address this type of problem using adaptive formations of multiple gliders to identify the cold boundaries in Monterey Bay.
The closest category to the problem explored in this paper is adaptive feature mapping and tracking. An example of a work in this category is Bennett et al. \cite{bennett}, where a simulated AUV is used to adaptively map bathymetric features such as trenches or specific contours.
\subsection{Multi-Vehicle Considerations}
Multi-vehicle systems provide advantages, but can incur additional costs due to increased complexity that must be addressed.
When using multiple vehicles for sampling, there are a number of considerations, including vehicle placement, information sharing, and task allocation.
\subsubsection{Vehicle placement}
One of the main considerations in multi-vehicle missions is their formation. When the vehicles all explore the entire field, they can be in fixed formations such as in a leader-follower strategy employed by Khoshrou et al. \cite{khoshrou} and Paliotta et al. \cite{paliotta}, or more flexible approaches such as generating different sailing directions for each vehicle, as proposed by Yan et al. in \cite{yan}.
When the field is explicitly divided, the divisions can either be predetermined or dynamic. An example of the latter is Kemna et al. \cite{kemna}, where dynamic Voronoi partitioning is used to divide the field among a group of AUVs.
\subsubsection{Information sharing and consensus}
\label{section:consensus}
To be effective collaborators, vehicles must fuse their information with that of their neighbors to gain a more complete understanding of the field. These communications can be range restricted, especially in the marine domain.
Consensus protocols and algorithms have been researched for multi-vehicle coordination across domains \cite{alighanbari2}, \cite{ren}. One direction proposed by Ren et al. \cite{ren2}, motivated by Kalman filters, is a Kalman consensus that accounts for the relative confidence of each vehicle's information. A modified version of this algorithm was later developed by Alighanbari et al. \cite{alighanbari} which removes potential biases that occur when the agents in the network are not fully connected.
\subsubsection{Task allocation}
\label{section:auction}
In many cases, vehicles need to coordinate new tasks/roles that arise as the mission progresses. One class of algorithms for solving this problem of task allocation is auction-based algorithms, also referred to as market-based algorithms.
The implementations of these algorithms can rely upon a centralized repository, as described by Bertsekas \cite{bertsekas}, or they can be distributed, as described by Michael et al. \cite{michael} and Zavlanos et al. \cite{zavlanos}. In the distributed case, the vehicles can perform the auction only with its neighbors, which would avoid failure if a vehicle drops out of communications range temporarily. In this case, a secondary step can be appended to the distributed auction-based algorithm, which is a consensus protocol to resolve conflicts and improve global performance \cite{brunet2008consensus,raja2021communication}.
\section{METHODOLOGY} \label{sec:meth}
This section details the architecture of our distributed autonomous multi-vehicle approach. We begin by describing how the problem is modeled, then explain how a local environmental model is generated in real-time from the sensor data. We discuss how all local models are shared and combined among the fleet using a distributed consensus protocol. Finally, we describe the decentralized PBACS algorithm which operates using the fleet-wide consensus estimate of the environment model.
We represent the bathymetry data in a discrete set of $m$ grid-cells. We define the vectors $\vec{z} \in \rm I\!R^m$ and $\vec{\sigma} \in \rm I\!R_+^m$ as the depth and the variance (respectively) of each cell. The location of the center of the $i^{th}$ cell is denoted as $\vec{x}_i\in \rm I\!R^2$
\subsection{Single-Beam Depth Sensor}
We assume the depth directly below each vehicle can be measured by a single-beam acoustic depth sensor. For field work, we used the Ping Sonar Altimeter and Echosounder made by Blue Robotics for its low cost and commercial availability. A simulated sensor with comparable accuracy and noise was used for simulation studies.
\begin{figure}[h]
\centering
\includegraphics[width=1\columnwidth]{herons.jpg}
\caption{Three Heron USVs. Altimeters were mounted onto the bottom of four vehicles. The Heron USVs are made by Clearpath Robotics}
\label{fig:vehiclesensor}
\end{figure}
The sensor was mounted onto the bottom plate of each Heron USV the lowest possible depth so that it remains submerged at all times without contacting the dock. The sensor was mounted directly below the GPS sensor in order to ensure that the distance readings are associated with the correct position. The sensor was then connected to the Heron's payload autonomy computer and integrated into the MOOS system described in Section~\ref{section:moosivp}. We set the sampling rate of the sensor to 10 Hz.
\subsection{Gaussian Process Regression}
We use GPR on each vehicle to calculate an estimate for the bathymetry field from the collection of local measurements. The GPR implementation used in this system is adapted from the "Fast GPR" developed by Das et al. \cite{das}. The Fast GPR aims to speed up computation by developing estimators on subsets of the dataset. Using the standard implementation of GPR on a dataset of size $N$ yields $O(N^3)$ time complexity. By choosing $k$ subsets of size $N_s < N$, the time complexity of Fast GPR is reduced to $O(kN_s^3)$. One limitation of this method is that the variance becomes skewed, but this can be addressed by rescaling.
We use the radial basis function kernel
\begin{equation} \label{eq:rbf}
\mathcal{K}(\vec{x}_i, \vec{x}_j) = exp \Big( - \frac{d(\vec{x}_i,\vec{x}_j)}{2 l^2} \Big),
\end{equation}
where $d(\vec{x}_i,\vec{x}_j)$ is the Euclidean distance. We determined experimentally that $l^2 = 830ft$ works well in field testing.
\subsection{Kalman Consensus}
For our multi-vehicle coordination, we use the Modified Decentralized Kalman Consensus (MDKC) reported by Alighanbari and How \cite{alighanbari}. Estimates from all vehicles are combined through iterative rounds in the consensus process. A consensus can be reached even if the group does not form a fully connected graph, which happened periodically during field operations as vehicles temporarily drop out of communication. A consensus manager was developed to manage requests, iterations, and timeouts. An instance of the manager runs on every vehicle to achieve fully decentralized estimation.
For $n$ agents $\mathcal{A} = \{\mathcal{A}_i,...,\mathcal{A}_n\}$, the solution for an agent $\mathcal{A}_i$ at time $t+1$ is given by:
\begin{align}
\bm{P_i}(t+1) =& \{[\bm{P_i}(t)+\bm{Q}(t)]^{-1} \\
&+ \sum_{j=1}^{n}(g_{ij}(t)[\mu_j(t)\bm{P_j}(t) ]^{-1})\}^{-1} \nonumber
\end{align}
\begin{align}
\vec{z}_i(t+1) &= \vec{z}_i(t) \\
& + \bm{P_i}(t+1)\sum_{j=1}^{n}\{g_{ij}(t) \cdot [\mu_j(t)\bm{P_j(t)} ]^{-1} \nonumber \\
& \cdot (\vec{z}_j(t)-\vec{z}_i(t))\} \nonumber
\end{align}
\begin{equation}
\mu_j(t) = \sum_{k=1, k \neq j}^{n}g_{kj}(t)
\end{equation}
where $\bm{P_i}$ is the covariance matrix assembled using the radius basis function kernel (\ref{eq:rbf}) and the grid variance $\vec{\sigma}$, $\bm{Q}(t)$ is the process noise (used only when a vehicle becomes completely disconnected and must complete the consensus on their own), $\vec{z}_i$ is the agent's own information, $g_{ij}$ is the adjacency matrix of the communication graph between agents $\mathcal{A}_i$ and $\mathcal{A}_j$, and $\mu_j(t)$ is a scaling factor associated with agent $\mathcal{A}_j$.
\subsection{MOOS-IvP}
\label{section:moosivp}
MOOS-IvP \cite{benjamin} is an open source C++ robotic autonomy software mainly used for marine vehicles. It consists of two parts, MOOS and the IvP Helm, and is intended to be run as backseat autonomy. The Mission Oriented Operating Suite (MOOS) is a robotic middleware that uses a publish-subscribe structure. The various processes (MOOS apps) that run in a mission publish messages to a central database process (MOOSDB) and subscribe to publications from other processes. One of these processes is the Interval Programming (IvP) Helm, which provides behavior-based autonomy.
\subsection{Proposal Based Adaptive Channel Search}
Here we present our main result, a more specialized method for solving the channel identification problem, which we refer to as the Proposal Based Adaptive Channel Search (PBACS) algorithm. There are two stages to this approach: an exploratory sweep and a search along candidate paths. A simplified version of the process is illustrated in Figure~\ref{fig:hybrid_process}, and the proposal algorithm is reproduced in Algorithm \ref{algo::PBACS}.
\subsubsection{First Stage - Exploratory Sweep}
The first stage of PBACS is an exploratory sweep of the area. Other adaptive sampling works have also employed an initial exploratory lawnmower sweep to seed the remainder of the search \cite{bennett}, \cite{paliotta}. For our problem, we can use such a sweep to help save time by eliminating areas that do not meet the necessary depth criteria, thereby directing the search toward more likely channel regions (Step 1 in Figure~\ref{fig:hybrid_process}). For a single vehicle mission we set the initial sweep to cover the start area of the grid, and for a two vehicle mission we cover the start and end goal areas. The sweep area for each subsequent vehicle is an equally spaced line in between.
\subsubsection{Second Stage - Path Exploration}
The second stage after the sweep is the path exploration stage, shown in Algorithm \ref{algo::PBACS}. Vehicles enter this stage after they complete their initial sweep, and this transition often occurs asynchronously due to the difference in transit times to the initial sweep locations.
The goal of this stage is to identify and explore candidate paths that may still be viable. Each vehicle proposes a path between the start and goal regions that may be part of a viable channel (Step 2 in Figure~\ref{fig:hybrid_process}). These candidate paths can be generated using any path planner; here we used a common variant of the A* planner which searches for a path that connects any of several start and goal locations.
The candidate path must be clear of obstacles. Grid-cells are considered obstacles only if they have a sufficiently low variance \emph{and} are too shallow. This way, candidate paths are made up of cells that are either certainly deep enough, or that we are uncertain about how deep they actually are.
The variance threshold for considering a shallow grid-cell to be an obstacle is calculated dynamically as a percentage of the range between the current minimum and maximum variance across the grid, i.e.
\begin{equation} \label{eq:threshvar}
\sigma_{th} = \sigma_{min} + \eta (\sigma_{max} - \sigma_{min}).
\end{equation}
All grid-cells whose variance is less than $\sigma_{th}$ \emph{and} whose depth estimate is shallower than the desired channel depth are considered obstacles. This threshold must be set high enough to exclude shallow grid-cells with low variance, typically those that have been directly measured by at least one vehicle. The threshold should also be low enough to not exclude shallow grid-cells with high variance, typically those that were interpolated. We experimentally determined $\eta = 0.33$ to be sufficient for both simulation and fieldwork.
\begin{figure}[h]
\centering
\includegraphics[width=\columnwidth]{figs/hybrid_fig_paper_crop.jpg}
\caption{The simplified steps of PBACS.}
\label{fig:hybrid_process}
\end{figure}
Upon completion of each GPR estimate and consensus, the path is rechecked to ensure that no new obstacles were found on it, and that it is still optimal (Steps 3,4,5 in Figure~\ref{fig:hybrid_process}). If a more optimal and obstacle-free path is found, the vehicle switches to this path (Step 4 in Figure~\ref{fig:hybrid_process}). We periodically check for the existence of a valid continuous channel, which marks the end of the mission (Step 6 in Figure~\ref{fig:hybrid_process}).
\subsubsection{Path Search Direction}
Once a vehicle has a new candidate path to explore, its first waypoint on that path is the point with the shortest Euclidean distance to its current position. Since this is most likely not an endpoint of the candidate path, we must choose the direction that the vehicle will traverse the path.
The first condition we check is whether we have an unexplored endpoint. Due to the preliminary sweep stage, this condition would only happen in the single vehicle case. The second condition is checking which side of the path has a higher variance, so that the vehicle is directed toward less explored regions. This is done by comparing the sum of the variances associated with either direction, with an exponentially increasing discount factor applied to points that are further away. The direction with the higher reward is chosen. The vehicle traverses the waypoints in this direction of the path until it reaches the endpoint. If no obstacles are found, the vehicle reverses to cover the other direction. An example of direction choosing is illustrated in Figure~\ref{fig:direction}.
\begin{figure}[ht]
\centering
\includegraphics[width=0.55\columnwidth]{figs/path_direction.png}
\caption{An example of a vehicle switching to a new candidate path, in a mission where it aims to identify a continuous deep channel. The desired channel depth is shown in red, and areas with no samples and high variance are shown in blue. The vehicle has encountered an obstacle on $\mathbf{p}_{curr}$, the path it was exploring (left path), and has found $\mathbf{p}_{prop}$, a new candidate path (right path). The blue arrow points in the direction of $p_a$, the vehicle's first waypoint on the new path. There are two choices of directions in which to traverse the path, $\mathbf{p}_{r}$ (reverse/up) or $\mathbf{p}_{f}$ (forward/down). Since $\mathbf{p}_{f}$ would pass through more high variance grid-cells, this direction is chosen. }
\label{fig:direction}
\end{figure}
\subsubsection{Proposal Bidding}
In the multi-vehicle case, vehicles allocate viable paths among themselves by sharing their locally found path with other vehicles as a proposal with an associated cost, and checking for conflicts.
When vehicles update their consensus estimate of the field, each vehicle finds the most optimal potentially viable path. The vehicle then proposes this candidate path to the other vehicles by placing a proposed bid with the cost of transiting to the path. For the cost function, we use the Euclidean distance to the closest point on the path. Each vehicle compares its own proposal to the ones it receives from the others. If there are no conflicts, the vehicle transits to this path and uses a waypoint following behavior to survey along the line. If there is a conflict, the vehicle with a lower cost bid wins. The losing vehicle must propose a new path and go through another iteration of bidding with all other vehicles that remain unassigned. The new proposal for subsequent iteration rounds takes into account the assigned paths by considering the cells in those paths to be obstacles, ensuring that new proposals do not overlap with paths assigned in the previous iteration. In the case where there are more vehicles than there are potential paths, a vehicle with no path will revert to using an MDP-based surveying mode to explore more of the field until a path becomes available. The proposal process is repeated upon each completion of a consensus.
\begin{algorithm}
\caption{Pseudocode of the main PBACS algorithm}
\label{algo::PBACS}
\begin{algorithmic}
\Global \textbf{ variables and parameters}
\Desc{$\sigma_{th}$}{variance threshold (\ref{eq:threshvar})}
\Desc{$t_{wait}$}{time to wait for other proposals}
\Desc{$\mathbf{p}_{curr}$}{\parbox[t]{.7\columnwidth}{vector of waypoints of current path $[p_1,p_2,\hdots,p_N]$}}
\vspace{1mm}
\Desc{$\mathbf{N}_{\mathbf{p}}$}{number of waypoints/vertices in path $\mathbf{p}$}
\Desc{$\mathbf{P}$}{\parbox[t]{.7\columnwidth}{vector of tuples $(\mathbf{p}_i,c_i,t_i)$ containing received proposals, costs, and time received}}
\EndGlobal
\end{algorithmic}
\begin{algorithmic}[1]
\Procedure {PBACS}{}
\State initialize $won \leftarrow false$
\Repeat
\State \parbox[t]{.8\columnwidth}{Handle incoming messages, checking for new consensus data and proposals}
\vspace{1mm}
\If {new consensus data received}
\State $chnl\_fnd \leftarrow$ check for channel \Comment{A*}
\State $waited \leftarrow time\_since\_last\_prop > t_{wait}$
\If{$\neg chnl\_fnd$ \textbf{and} $\neg won$ \textbf{and} $waited$}
\State $won = \textsc{checkProposals}(\mathbf{P})$
\If {$\neg won$}
\State $\mathbf{p}_{new} \leftarrow $ find candidate path \Comment{A*}
\If {$\mathbf{p}_{new} \not= [~]$}
\State $cond1 = \mathbf{p}_{curr}$~has new obstacles
\State $cond2 = N_{\mathbf{p}_{curr}} > N_{\mathbf{p}_{new}}$
\If{$cond1$~\textbf{or}~$cond2$}
\State $\mathbf{p}_{prop} \leftarrow \mathbf{p}_{new}$
\Else
\State $\mathbf{p}_{prop} \leftarrow \mathbf{p}_{curr}$
\EndIf
\Else
\State \textbf{exit}, use MDP planning
\EndIf
\State $c_{prop} \leftarrow$ min distance to $\mathbf{p}_{prop}$
\State send $\mathbf{p}_{prop}$ and ${c}_{prop}$ to other vehicles
\ElsIf{$won~\textbf{and}~ \mathbf{p}_{prop} \not= \mathbf{p}_{curr}$}
\State $p_a \leftarrow$ get closest waypoint in $\mathbf{p}_{prop}$
\State $\mathbf{p}_{curr} = \textsc{getDirection} (\mathbf{p}_{prop},p_a)$
\State publish $\mathbf{p}_{curr}$ to waypoint behavior
\EndIf
\EndIf
\EndIf
\Until{$channel\_found$}
\EndProcedure
\end{algorithmic}
\end{algorithm}
\subsection{Existing Methods for Comparison}
\subsubsection{Lawnmower Survey}
The most common method of surveying a region is with a lawnmower pattern, also referred to as a boustrophedon in coverage path planning \cite{choset}.
However, the success of lawnmower surveys in quickly identifying a channel is highly dependent on the (lucky) choice of starting locations and distributions of vehicles that fit the underlying map. In this work, we are interested in approaches that do not rely on luck and perform better in aggregate.
\subsubsection{Markov decision processes}
\label{section:mdp}
We can formulate the problem as an MDP \cite{bellman}.
An MDP consists of the following: a set of states $S$ with an initial state $s_0$, a set of actions in each state $A$, a transition model $P(s'|s,a)$ giving the probability that an action $a$ in state $s$ will lead to state $s'$, and a reward function $R(s)$.
A solution to the MDP is in the form of a policy $\pi$, where $\pi(s)$ gives the recommended action at state $s$. The optimal policy for a state $\pi^*(s)$ is given by:
\begin{equation}
\pi^*(s) = \underset{{a \in A(s)}}{\mathrm{argmax}}\, \sum_{s'} P(s'|s,a) U(s')
\end{equation}
where U(s) is a utility function. This utility function can be estimated using value iteration, which is computed by iteratively applying a Bellman update.
For our implementation, we define the root state to be the grid-cell corresponding to the vehicle's current position and heading. For the actions, to account for the underactuated vehicle's maneuverability, we define three possible next grid-cells based on the vehicle's heading: ahead, ahead-left, or ahead-right. These heading-based grid-cell choices are shown in Figure~\ref{fig:mdp_grid} below. If the action is to maintain heading, meaning transiting to the cell straight ahead, the probability is 0.82 that this state transition is achieved. The probability that the vehicle turns and goes ahead-left or ahead-right instead is 0.09 for each turn. If the action is to change heading, meaning transiting to the ahead-left or ahead-right cell, the probability that this is achieved is 0.7. The probability that the vehicle maintains heading instead is 0.2, and the probability of turning the other direction is 0.1. These probabilities are normalized based on the total probability of all feasible actions.
\begin{figure}[ht]
\centering
\includegraphics[width=0.17\textwidth]{figs/mdp_grid_cropped.png}
\caption{Examples of possible neighbor cells to which a vehicle can transit are highlighted based on current heading, represented by an arrow direction.}
\label{fig:mdp_grid}
\end{figure}
An MDP can be implemented with different reward functions. We compare the performance of two different functions, UCB and MVI.
UCB as a reward criterion is a common, state-of-the-art choice in online decision making \cite{srinivas} \cite{sun}.
An alternative to UCB reward calculation is MVI, defined by Flaspohler et al. \cite{flaspohler}, based on the Max-value Entropy Search (MES) criterion defined by Wang et al. \cite{wang}. As shown by Flaspohler et al., over time the MVI reward converges to the global maximum, while UCB rewards high-value regions more uniformly.
Due to computational constraints of fieldwork hardware, we implement the MDP as a myopic planner with a limited look-ahead depth. Based on this information, the vehicle determines the current optimal path to take. The utility function is recalculated once it reaches the next grid-cell.
\section{RESULTS} \label{sec:results}
\subsection{Simulation Experiments}
To test algorithm performance in simulation, we generated six possible bathymetry scenario files. For non-symmetric bathymetry configurations, we also used mirrored versions to check for biases that different orientations can introduce. Including these mirrored versions, we tested a total of ten scenarios. The ten bathymetry scenarios are shown in Figure~\ref{fig:sim_gt} below. Each area represents a 500 meter by 750 meter rectangle with 20 meter by 20 meter grid-cells. The grids are tilted to represent the path planning algorithms not being dependent on a perfectly horizontal or vertical grid, to account for possible shoreline bounds. The depth values over all scenarios range from 6 feet to 26 feet.
\begin{figure}[h]
\includesvg[width=\columnwidth]{ground_truth_all}
\caption{The ten bathymetry grid scenarios for simulation testing. The x and y axis markers represent distances in meters.}
\label{fig:sim_gt}
\end{figure}
We define the depth threshold to be 20 feet. Each vehicle's speed is set to 2.4 m/s. For the two MDP approaches, we use a look-ahead depth of 6 cells, recalculated every time a vehicle enters a cell.
\subsubsection{Simulation Results}
We only provide a general summary of performance due to page length constraints. Detailed results from each scenario can be found in \cite{gershfeld}.
In general, the PBACS algorithm performs better than the lawnmower survey pattern. As shown in Figure~\ref{fig:times_allsims}, the PBACS mission durations are shorter than the lawnmower mission durations on average, but there are some outlier cases of long PBACS missions. The single vehicle case of PBACS times out or is close to timing out for the very curved channel in the Bathymetry 6 scenario and for one orientation of the "dead end" channel in the Bathymetry 4 scenario. The two vehicle case of PBACS also has a long mission duration for the Bathymetry 6 scenario. The three and four vehicle cases of PBACS consistently perform better than the three and four vehicle lawnmower surveys, except when a vehicle's lawnmower start location aligns with a straight channel.
Overall, the results shown here indicate that the optimal amount of vehicles for the scenarios used here is three when using PBACS. Using three vehicles avoids the possible long mission durations of the one and two vehicle cases. Adding a fourth vehicle does not offer much improvement in mission duration, and could potentially slightly increase it as well. These numbers are based on a maximum of one bend in the channel (scenario 6). If more bends were to be added, it is possible that four or more vehicles would be optimal instead.
\begin{figure}[h]
\centering
\includesvg[width=\columnwidth]{times_all}
\caption{Means and ranges of mission times across all 10 simulated bathymetry scenarios for the lawnmower and PBACS approaches. For each vehicle number, there are 10 lawnmower data points and 50 PBACS data points. The dashed black line shows the duration of a theoretical mission where the lawnmower covers the entire field. This is not linear due to the field not being evenly divisible by all of the vehicle amounts, and to account for consensus times.}
\label{fig:times_allsims}
\end{figure}
\subsection{Field Experiments}
From April to June 2022, we performed fieldwork trials of the four path planning approaches on the Charles River near the MIT Sailing Pavilion. There were two sets of fieldwork trials. We first performed a comparison study of the four methods in a section of the river adjacent the MIT Sailing Pavilion. The details of these experiments are provided in the remainder of this section. We also repeatedly demonstrated the performance of the PBACS approach to quickly identify the well-defined channel that was dredged in the Charles River on the side opposite the MIT Sailing Pavilion. The results of these demonstrations are shown in Figure \ref{fig:charles_plot}.
For the comparison study, the survey area is a 170 meter by 260 meter rectangular grid, with 10 meter by 10 meter grid-cells. This area, chosen for its proximity to the lab space, does not have a clearly defined channel. However, it contains some straight paths down through deep areas as well as curved paths.
\subsubsection{Field Results}
In this section, we describe the results of 28 comparison field trials. As in the simulations, the results for the lawnmower survey are taken from one run. The results of the PBACS algorithm are taken from four runs of each vehicle number. We only have data for the two and three vehicle cases of both MDP approaches; these results are taken from two runs each.
Figures~\ref{fig:ranges_field} and \ref{fig:paths_field} below show the mission times and paths found. The ground truth grid displayed in Figure~\ref{fig:paths_field} was averaged from three complete lawnmower surveys, but there is still some variability, $\pm 1$ foot, in the water level and surface conditions of each field mission. This would account for some of the paths passing through grid-cells that appear shallow on the plotted grid.
\begin{figure}[ht]
\centering
\includesvg[width=\columnwidth]{ranges_field}
\caption{Field mission time ranges for the MDP and PBACS. The means are marked with dashed lines.}
\label{fig:ranges_field}
\end{figure}
\begin{figure}[ht]
\centering
\includesvg[width=\columnwidth]{paths_field}
\caption{All final paths found during field missions, separated by path planning approach. The x and y axis markers represent distances in meters.}
\label{fig:paths_field}
\end{figure}
The PBACS mission duration decreases when the number of vehicles is increased, as does the variation in the mission time. As was found in simulations, when a relatively straight channel path exists, the largest improvement in time comes from increasing the vehicle amount from one to two.
Because the channel is less defined in our field area than it is in the simulation scenarios, there is more variation between approaches in the types of paths found. In all of the simulation runs, the channel paths are all concentrated in the same general areas and have the same general shapes. Here, the lawnmower survey and PBACS both have generally straighter paths than the two MDP approaches, since the lawnmower survey runs parallel to this direction and since PBACS checks for the shortest straight paths first.
\section{CONCLUSIONS} \label{sec:conclusion}
To the best of our knowledge, this paper describes the first formal investigation into the rapid channel identification problem. The rapid channel identification problem is well suited to the use of adaptive sampling. However, our results suggest that the problem requires fundamentally different approaches than those used in other adaptive sampling problems. Through our simulation testing and field testing with USVs and single beam altimeters, we found that the PBACS algorithm generally outperforms the lawnmower and both MDP reward functions in multi-vehicle cases. There are some exceptions, namely when the lawnmower survey start is aligned with a straight channel. Another notable exception is when the channel is very curved. In this case, a two vehicle mission may have poor performance, but on average three and four vehicle PBACS missions were still faster than three and four lawnmower or MDP missions.
Potential areas of improvement for the PBACS algorithm include implementing a more sophisticated auction-based approach to allocating proposed paths among vehicles. Another potential improvement is extending the proposal cost function beyond the Euclidean distance, perhaps including penalties for heading changes.
\addtolength{\textheight}{-5cm}
\pagestyle{fancy}
\bibliographystyle{IEEEtran}
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1,116,691,499,252 | arxiv | \section{Introduction}
Quantum phase transitions and quantum criticality are among the central concepts in the physics of correlated electrons \cite{Sachdev-Book,Vojta-Review}. In general, quantum fluctuations (QF) near magnetic e.g. spin-density-wave (SDW) quantum critical point (QCP) give rise to non-Fermi liquid behavior that manifests in singularities and nonanalyticity of various electronic characteristics \cite{Stewart,Abanov,Metlitski,Sur-Lee}. This problem is further enriched in the situations when magnetic instability competes with superconductivity (SC) \cite{Pelissetto,Basov,Chubukov-Review,Efetov,Chubukov-Eliashberg}, see Fig. \ref{Fig-QCP-PD} for the exemplary phase diagram. This is the case in the context of cuprate- and iron-based superconductors where interest in the topic is constantly fueled by a multitude of experimental activities (for the recent detailed reviews see, e.g., Refs. \cite{Shibauchi-Review,Taillefer-Review} and references herein).
The key signatures of QCP behavior in SCs include correlated anomalies in both transport coefficients and thermodynamic properties, which emerge in different temperature regimes of the phase diagram when the system is tuned by an external control parameter (e.g., doping $x$) to an optimal composition $x_c$. Indeed, some of these anomalies persist in the normal state such as linear-in-$T$ resistivity observed in various materials at $x_c$ \cite{Mackenzie-Planckian,Shekhter,Analytis-Planckian,Taillefer-Planckian,Paglione-Planckian}. It is typically accompanied by anomalies in the transverse Hall and thermoelectric responses \cite{Analytis-Hall,Taillefer-ThHall,Canfield-TEP}. When system is brought to the proximity of the phase transition, then thermally activated fluctuations of magnetic and superconducting orders start to play a dominant role. This translates into the nonmonotonic discontinuity of the specific heat jump which also peaks at $x_c$ \cite{Hardy,Carrington,Klauss}. Further, when the system is cooled into the superconducting state, quantum fluctuations proliferate and their effect becomes most pronounced near the transition line that separates pure superconducting and mixed phase coexisting with magnetism that ultimately terminates at the QCP. Near that region one often detects enhanced critical supercurrents \cite{Eisterer} and observes the apparent sharp peak in the magnetic penetration depth \cite{Matsuda,Auslaender-1,Auslaender-2,Zheng,Prozorov}.
In part motivated by these results, the interplay of possible magnetic and structural quantum phase transitions shielded by the superconductivity was a subject of an immediate scrutiny \cite{Fernandes-QCP}. In a parallel vein of studies, various models of Planckian resistivity were proposed \cite{Efetov-T-linear,Balents,Senthil,Patel}, thermal and electrical transport properties across antiferromagnetic quantum transition were considered \cite{Eberlein}, and further extensions to anomalous Hall phenomena were developed \cite{Vafek,SLAL}. Thermodynamic signatures of QCP were analyzed theoretically in the context of the specific heat \cite{Vavilov,Kuzmanovski,Carvalho} and Josephson effect \cite{Dzero-Josephson,Kirmani}. However, despite much of the efforts \cite{LVKC,Chowdhury-PRL,Nomoto-Ikeda,Chowdhury-PRB,Dzero,Zhang} there is no consensus on the explanation of the observed peak in the London penetration depth.
In this work we show that finite-temperature effects of quantum spin-density-wave fluctuations yield anomalous thermodynamic properties of gapped fermions with pronounced power-law dependencies in both specific heat and London penetration depth which is reminiscent of that of nodal superconductors. We also demonstrate generality of these results, in particular robustness to effects of disorder.
The rest of the paper is organized as follows. In Sec. \ref{Sec:Model} we present a disorder model of SC-SDW coexistence, develop its mean-field description, and establish a phase diagram. In Sec. \ref{Sec:Applications} we apply this model to calculate quantum fluctuation corrections to heat capacity and penetration depth near magnetic QCP hidden beneath the SC dome. We summarize our main results in Sec. \ref{Sec:Summary}. The main sections of the paper are accompanied by several Appendices \ref{App:QuasiClass}--\ref{App:Fluctuations}, where an in-depth technical discussion is provided and supporting detailed calculations are carried out. Throughout the paper we work in the units $\hbar=k_B=c=1$.
\begin{figure*}
\centering
\includegraphics[width=7in]{Fig1ab.pdf}
\caption{Phase diagram of the (a) magnetic SDW quantum criticality without SC and (b) with SC coexistence computed numerically from the so-called disorder model presented in Sec \ref{Sec:Model}. In the superconducting case the fan region extending away from the QCP represents an anomalous part of the phase diagram where nodal-like behavior of gapped fermions emerges.}\label{Fig-QCP-PD}
\end{figure*}
\section{Disorder model of SC-SDW coexistence}\label{Sec:Model}
\subsection{Mean-field description}
We adopt the two-band model which is defined by the Hamiltonian \cite{Vavilov,Dzero}:
\begin{equation}\label{Eq:H}
\hat{H}=\hat{H}_0+\hat{H}_{\textrm{sdw}}+\hat{H}_{\textrm{sc}}+\hat{H}_{\textrm{dis}}.
\end{equation}
The first term describes noninteracting fermions occupying two (one electron- and one hole-like) bands:
\begin{equation}\label{Eq:H0}
\hat{H}_0=\sum_{\bm{k}}\xi_{\bm{k}}\Psi_{\bm{k}}^\dagger (\hat{\tau}_3 \hat{\rho}_3\hat{\sigma}_0)\Psi_{\bm{k}},
\end{equation}
where we take simple parabolic band dispersion $\xi_{\bm{k}}=k^2/2m-\mu$, defined relative to the chemical potential $\mu$, and all momenta are counted relative to the center of the corresponding pocket. In the Balian-Werthammer representation \cite{BW}, the eight component spinor
\begin{equation}\label{Eq:Psi}
\Psi^\dag_{\bm{k}}=(\hat{c}_{\bm{k}\uparrow}^\dag, ~\hat{c}_{\bm{k}\downarrow}^\dag,~\textrm{-}\hat{c}_{-\bm{k}\downarrow}, ~\hat{c}_{-\bm{k}\uparrow},
~\hat{f}_{\bm{k}\uparrow}^\dag,~\hat{f}_{\bm{k}\downarrow}^\dag,~\textrm{-}\hat{f}_{-\bm{k}\downarrow}, ~\hat{f}_{-\bm{k}\uparrow})
\end{equation}
is composed of electron-$c$ and hole-$f$ creation and annihilation operators with spin projections $\uparrow\downarrow$. Three sets of Pauli matrices $(\hat{\tau},\hat{\rho},\hat{\sigma})$ operate in the band, Nambu, and spin spaces, respectively. The second term in Eq. \eqref{Eq:H} describes magnetic interpocket interaction between fermions
\begin{equation}\label{Eq:Hsdw}
\hat{H}_{\textrm{sdw}}=-\frac{g_{\textrm{sdw}}}{2}\sum_{\bm{Q}}\bm{S}_{\bm{Q}}\bm{S}_{-\bm{Q}},
\end{equation}
where the magnetization fluctuation at momentum $\bm{Q}$ is $\bm{S}_{\bm{Q}}=(1/2)\sum_{\bm{k}}\Psi^\dag_{\bm{k}+\bm{Q}}\hat{\bm{\Xi}}\Psi_{\bm{k}}$, $\hat{\bm{\Xi}}= \hat{\tau}_1\hat{\rho}_0\hat{\bm{\sigma}}$.
The third term in Eq. \eqref{Eq:H} captures pairing interaction and in the model of $s^{\pm}$ order parameter changing sign between the hole and electron pockets takes the form
\begin{equation}\label{Eq:Hsc}
\hat{H}_{\textrm{sc}}=-\frac{g_{\textrm{sc}}}{2}\sum_{\bm{k}\bm{k}'}B_{\bm{k}}B_{\bm{k}'},
\end{equation}
where the fermion bi-linear is defined as $B_{\bm{k}}=\Psi^\dag_{\bm{k}}(\hat{\tau}_3\hat{\rho}_1\hat{\sigma}_0)\Psi_{\bm{k}}$.
In this low-energy description we impose high-energy cutoff $\Lambda$, and consider only angle-independent interactions in the SDW channel and in the $s^{\pm}$ SC channel with the couplings $g_{\textrm{sdw}}$ and $g_{\textrm{sc}}$. With the last term in Eq. \eqref{Eq:H} we introduced a disorder potential into the problem. We account for two types of scattering processes: the intraband disorder with potential $U_0$, which scatters quasiparticles within the same band, and interband scattering between the Fermi pockets mediated by the potential $U_\pi$.
In the basis of spinors $\Psi_{\bm{k}}$ the disorder term reads
\begin{equation}\label{Eq:Hdis}
\hat{H}_{\text{dis}}=\!\!\sum_{\bm{k}\bm{k}'\bm{R}_j}\!\! \Psi^\dag_{\bm{k}}\left[U_0(\hat{\tau}_0\hat{\rho}_3\hat{\sigma}_0)+U_\pi(\hat{\tau}_1 \hat{\rho}_3\hat{\sigma}_0)\right]\Psi_{\bm{k}'}e^{i(\bm{k}-\bm{k}')\bm{R}_j}
\end{equation}
where summation goes over the random locations $\bm{R}_j$ of individual impurities. When performing disorder averaging within the self-consistent Born approximation we assume that concentration of impurities is $n_{\text{imp}}$. This naturally introduces two scattering rates into the model $\Gamma_{0,\pi}={\pi\nu_F n_{\text{imp}}}|U_{0,\pi}|^2/4$, where $\nu_F$ is the total quasiparticle density of states at the Fermi energy.
The mean-field (MF) analysis of this model proceeds in a standard way by decoupling interaction terms via Hubbard-Stratonovich transformation with magnetic $M$ and superconducting $\Delta$ order parameters, and integrating out fermions \cite{Vavilov,Dzero}. This approach naturally leads to the semiclassical description based on the Eilenberger equation, which is further elaborated on in Appendix \ref{App:QuasiClass}.
In this treatment, the pure SC transition temperature $T_c$ is suppressed only by the interband scattering as described in accordance with the Abrikosov-Gor'kov scenario
\begin{equation}
\ln\left(\frac{T_{c0}}{T_c}\right)=\psi\left(\frac{1}{2}+\frac{\Gamma_\pi}{\pi T_c}\right)-\psi\left(\frac{1}{2}\right),
\end{equation}
where $T_{c0}\simeq \Lambda e^{-2/\nu_Fg_{\text{sc}}}$ and $\psi(x)$ is the digamma function. This is similar to the equation for $T_c$ in conventional single-band $s$-wave superconductors with magnetic impurities, and in the unconventional $d$-wave superconductors with potential impurities. In contrast, pure SDW transition temperature $T_s$ is suppressed by the total scattering rate,
\begin{equation}
\ln\left(\frac{T_{s0}}{T_s}\right)=\psi\left(\frac{1}{2}+\frac{\Gamma_0+\Gamma_\pi}{\pi T_s}\right)-\psi\left(\frac{1}{2}\right),
\end{equation}
where $T_{s0}\simeq \Lambda e^{-2/\nu_Fg_{\text{sdw}}}$. As a result of different sensitivity to disorder, there exists a finite parameter range in $\Gamma_{0,\pi}$ where both orders $M$ and $\Delta$ can coexist simultaneously. The magnetic QCP is defined by the condition $T_s(\Delta)=0$, which corresponds to $M=0$ for certain values of $\Gamma_{0,\pi}$, see Fig. \ref{Fig-QCP-PD}(b) for an example. We note that this phase diagram was calculated numerically for the choice of parameters when $\Gamma_\pi/\Gamma_0=0.325$ and $T_{s0}/T_{c0}=1.7$.
At this point, it is worth commenting that usually stability of QCP in disordered systems is analyzed through the prism of the Harris criterion \cite{Harris,Chayes}, namely when disorder is introduced on top of the control parameter that defines QCP. In the model considered here, it is disorder itself that defines QCP and, as we show below, controls fluctuations around it.
\subsection{SDW fluctuation propagator in SC state}
Extending theory beyond the mean field description we consider the critical fluctuations that mediate an effective interaction in the spin channel $(S_z)$ represented by the propagator
\begin{equation}\label{Eq:L}
L_{Q,\Omega_m}=\left( g_{\text{sdw}}^{-1}+\Pi^{z}_{Q,\Omega_m}\right)^{-1}.
\end{equation}
The disorder-averaged polarization operator $\Pi^z_{Q,\Omega_m}$ needs to be calculated by resumming the whole sequence of ladder-type diagrams with impurity line insertions into the fermionic loop. It can be shown that, in the proper matrix basis representation, this averaging can be reduced to a geometric series that sums to
\begin{equation}\label{Eq:Pi}
\Pi^z_{Q,\Omega_m}=T\sum_{\omega_n}\left[\hat{P}_{Q,\Omega_m}\left(1-\hat{\Gamma}\circ\hat{P}_{Q,\Omega_m}\right)^{-1}\right]_z,
\end{equation}
where the notation with subscript $z$ implies a specific matrix element. The convolution in Eq. \eqref{Eq:Pi} between the disorder matrix ($\hat{\Gamma}$) and matrix polarization function ($\hat{P}_{Q,\Omega_m}$) requires a specification of the basis representation which is explained in Appendix \ref{App:Disorder} [see Eqs. \eqref{dec17a} and \eqref{Eq:Pz-vs-P103} for definitions]. For instance, the diagonal matrix element of the bare fermionic loop, namely the polarization operator without vertical impurity lines, is given by
\begin{equation}\label{Eq:Pi-bare}
P^z_{Q,\Omega_m} (\omega_n)= \sum_k\mathop{\rm tr}\nolimits\left[\hat{\Xi}^z \hat{G}_{k_+,\omega_+} \hat{\Xi}^z \hat{G}_{k_-,\omega_-}\right],
\end{equation}
with $\hat{\Xi}^z=\hat{\tau}_1\hat{\rho}_0\hat{\sigma}_3$, $k_{\pm}=k\pm Q/2$, and $\omega_\pm=\omega_n\pm\Omega_m/2$.
The latter is defined via the disorder averaged single-particle propagator Green's function
\begin{equation}
[\hat{G}_{k,\omega_n}]_{\alpha \beta} = -\int_0^{T^{-1}} d \tau e^{ i \omega_n \tau} \langle \Psi_{k\alpha}(\tau) \Psi^{\dag}_{k\beta}\rangle.
\end{equation}
The trace in Eq. \eqref{Eq:Pi-bare} assumes reduction over all the matrix indices. Later in the text we will use a global trace that in addition includes summation over the Matsubara frequency, $\omega_n=\pi T(2n+1)$, and momenta, thus introducing a notation $\mathop{\rm Tr}\nolimits[\ldots]=T\sum_{k,\omega}\mathop{\rm tr}\nolimits[\ldots]$.
The critical paramagnon described by the spin correlation function in Eq. \eqref{Eq:L} with $\Omega_m=2\pi mT$ softens towards the QCP,
\begin{equation}
g_{\text{sdw}}^{-1}+\Pi^z_{Q,\Omega_m} =\pi \nu_F \left(\gamma + Q^2/Q_c^2 + \Omega_m^2/\Omega_c^2 \right),
\end{equation}
reached at $\Gamma_0 = \Gamma_c $ such that,
$\gamma(\Gamma) \approx \gamma'_{\pm}|\Gamma - \Gamma_c|$, $\gamma'_{\pm} = |d \gamma/d\Gamma|$
taken at $\Gamma = \Gamma_c \pm 0^+$. It should be noted that in a SC state, the dynamical exponent due to the exchange of near-critical SDW fluctuations changes from $z=2$ to $z=1$ as compared to the normal case, because fermions which contribute to bosonic dynamics become massive excitations protected by a gap. This is reflected in the asymptotic expansion of $\Pi^z_{Q,\Omega_m}$ having a $\Omega^2_m$ term, which is valid at low energies $\{v_FQ,\Omega_m\}\ll\Delta$.
We find in this model rather generally that the QCP is located at $\Gamma_c=2\pi aT_{c0}$ where a precise numerical value of the parameter $a(\Gamma_\pi/\Gamma_0,T_{c0}/T_{s0})$ depends on the choice of two ratios between scattering rates and bare interaction parameters (or alternatively bare transition temperatures). Furthermore, while the ratio $\gamma'_{+}/\gamma'_-$ can be arbitrary, the low-energy expansion coefficients $Q_c$ and $\Omega_c$ may be computed right at the QCP. Further details and generalities of calculation of $L_{Q,\Omega_m}$ are relegated to Appendices \ref{App:Disorder} and \ref{App:Propagator}, where in particular we discuss separately disorder renormalizations of vertex functions and impurity ladders in the low-energy expansion of the polarization operator defined by Eq. \eqref{Eq:Pi}.
\section{Applications}\label{Sec:Applications}
\subsection{Specific heat near QCP}
We now focus on the impact of quantum SDW fluctuations on the low-temperature behavior of the specific heat inside the dome of $s^{\pm}$ superconductivity. Recall that at the level of the mean-field analysis, the low-$T$ asymptotic behavior of the specific heat in a fully gapped SC state is exponential $C_{\text{MF}}\propto (\Delta/T)^{3/2}e^{-\Delta/T}$ for $T\ll\Delta$. Our intent is to investigate the fate of this result as the system approaches a QCP by accounting for the extra contribution of the spin fluctuations. Following the standard procedure, we integrate out these soft magnetic modes from the action. At the Gaussian level we thus get a renormalized free energy of a superconductor per unit layer area $F=F_{\text{SC}}(\Delta,M)+\delta F_{\text{QF}}$ that can be expressed in terms of the SDW propagator from Eq. \eqref{Eq:L} as follows
\begin{equation}\label{Eq:FQF}
\frac{\delta F_{\text{QF}}}{T}=\frac{\mathcal{N}}{2}\mathop{\rm Tr}\nolimits\ln\left[L^{-1}_{Q,\Omega_m}\right],
\end{equation}
where $\mathcal{N}$ counts the number of soft modes.
In our model $\mathcal{N}=3$ at $x>x_c$ and $\mathcal{N}=1$ at $x<x_c$ as only the longitudinal mode has a mass changing with $\Gamma$. The factor $1/2$ is present because the paramagnons commensurate with the lattice are represented by a real boson field. Next, performing the Matsubara sum, and separating the temperature independent term, one can easily analyze limiting cases for the corresponding specific heat correction $\delta C_{\text{QF}}=-T\partial^2_T \delta F_{\text{QF}}$ (see Appendix \ref{App:Fluctuations} for further technical details). We thus find in a broad regime of temperatures $\Delta_{\text{QCP}}<T<\Delta$,
\begin{equation}\label{C-QF}
\delta C_{\text{QF}}=\frac{9\zeta(3)}{\pi}\left(\frac{v_FQ_c}{\Omega_c}\right)^2\left(\frac{T}{v_F}\right)^2,
\end{equation}
where we introduced a gap to QCP as $\Delta_{\text{QCP}}(\Gamma)=\sqrt{\gamma(\Gamma)}\Omega_c$. The most striking feature of this result is that proliferation of quantum fluctuations to finite temperatures gives a power law instead of exponential behavior even in the presence of a full SC gap. As is known, a power law in the specific heat occurs only in the unconventional superconductors having nodal structure of the gap. In particular $\propto T^2$ is a characteristic of a gap structure with first-order nodes at isolated points.
We note that the details of the microscopic model enter Eq. \eqref{C-QF} only via the ratio $v_F Q_c/\Omega_c$ so that $T^2$ dependence is a model independent result.
Furthermore, as $\Gamma_c/2\pi T_{c0}\ll1$ for a broad range of parameters, then to a good approximation $v_F Q_c = \Delta \sqrt{\pi}$ and $\Omega_c = \Delta \sqrt{\pi/2}$ leading to a universal expression, $\delta C_{\text{QF}}=(18\zeta(3)/\pi)(T/v_F)^2$.
Ultimately, at the lowest temperatures, $T<\Delta_{\text{QCP}}$, specific heat crosses over to exponential dependence, $\delta C_{\text{QF}}\propto (\Delta_{\text{QCP}}/T) e^{-\Delta_{\text{QCP}}/T}$.
We note that the same conclusion has been reached independently in the considerations of a different model \cite{Carvalho}.
\begin{figure*}[t!]
\centering
\includegraphics[width=7in]{Fig2ab.pdf}
\caption{(a) Contour plot of the London penetration depth
$\lambda^{-2}(\Gamma,T)$ (arb. units) calculated within the MF theory approximation as a function of temperature and disorder scattering rate $\Gamma_0$ assuming $\Gamma_\pi=0.4\Gamma_0$. We note that already at the MF the width of the region in which $\lambda^{-2}$ has maximum value narrows upon an increase in temperature. (b) Normalized quantum fluctuation correction to the electromagnetic response kernel [Eq. \eqref{K-SDW}] as a function of the proximity to the QCP gap $\gamma$, showing an emergent peak in a color plot.}
\label{lambda-3D-Visual}
\end{figure*}
\subsection{Penetration depth near QCP}
We turn our attention to the anomalies in the magnetic penetration depth $\lambda(T,x)$, where numerous recent measurements \cite{Matsuda,Auslaender-1,Auslaender-2,Zheng,Prozorov} revealed a distinct peak in the low-temperature limit $T\ll\Delta$ concentrated around the putative QCP $x\to x_c$. The model we explore in this study with $x=\Gamma$ is perhaps best suited to experiments of Ref. \cite{Prozorov} on Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$. This material is in the disordered limit with fully gapped Fermi surface as opposed to BaFe$_2$(As$_{1-x}$P$_x$)$_2$, which is a rather clean system that displays nodal superconductivity. However, the arguments we put forward are rather generic and in fact apply to both compounds.
It is natural to account for soft bosonic modes in the fermionic electromagnetic response function that defines $\lambda(x,T)$. However, the one-loop fluctuation correction while giving a good approximation outside the critical region is inapplicable inside this region.
In the present context, this implies that as a matter of principle, the character of the $\lambda(x)$ singularity cannot be determined on the level of a one-loop approximation. Indeed, the mean field theory predicts a deep in $\lambda(x)$ \cite{Dzero,Fernandes-London,Kuzmanovski-London}, see also Fig. \ref{lambda-3D-Visual} for the further illustration. Therefore, in order to turn the deep into a peak the fluctuation correction must exceed the mean field value. According to the Ginzburg criterion, however, this cannot happen in the region of validity of one-loop approximation. For this reason, the problem has to be solved inside a critical region and is essentially nonperturbative.
Such a solution valid in critical region is possible at $T=0$ for the model of electrons coupled to critical bosons with the mass term $\propto (x- x_c)$ \cite{Chowdhury-PRL}. In this model there is a universal relation between the critical scaling of $\lambda$ at $x$ above and below $x_c$. When the bosons are viewed as collective fermion excitations as captured by $L_{Q,\Omega_m}$ such a universal relation is lost as the ratio of the paramagnon masses at $x = x_c \pm \delta$ is a model dependent number, while in, e.g., Ising boson theory it is $2$. In our specific model this number $\gamma'_+/\gamma'_-$ is a nonuniversal function of $\Gamma_{0,\pi}$. Despite this discrepancy with the purely bosonic approach, the $x$ dependence of $\lambda$ established in Ref.~\cite{Chowdhury-PRL} remains monotonic in our model as well.
This leaves us with the unresolved puzzle of the peak in $\lambda(x)$.
Our solution to this problem builds on the strong $x$ dependence of the $T$-dependent part of the penetration depth, $\lambda(T)-\lambda(T=0)$.
In distinction with the $T=0$ case, at the mean field level $\delta\lambda(T)=\lambda(T) - \lambda(0)\propto e^{-\Delta/T}$ is suppressed exponentially at $T\ll\Delta$.
Therefore, just outside the critical region the one-loop correction gives a reliable estimate of fluctuation correction to $\delta\lambda(T)$.
This correction yields the peak in $\lambda(T)$ at the temperatures
$T\gtrsim \Delta^2/E_F$ much smaller than $\Delta$.
To quantify these statements we express the fluctuation correction to $\lambda=\lambda_0+\delta\lambda_{\text{QF}}$ through the correction to the static, long wave length limit of the current correlation function $K=K_0+\delta K_{\text{QF}}$ (see Appendix \ref{App:Fluctuations}):
\begin{equation}
\frac{\delta\lambda_{\text{QF}}}{\lambda_0}=-\frac{\delta K_{\text{QF}}}{2K_0}, \quad K_0=\frac{1}{2}\nu_Fe^2v^2_F,\quad \lambda^{-2}_{0}=\frac{4\pi}{w} K_0,
\end{equation}
where $w$ is the interlayer separation as appropriate to the quasi-2D systems.
The one-loop correction of the electromagnetic kernel $K_{\text{QF}}$ contains both effective mass renormalization, captured by the density of states (DOS) type diagrams, and vertex renormalization expressed by the Maki-Thompson (MT) type quantum interference processes. The Aslamazov-Larkin vertex corrections cancel for the case when the gaps on hole and electron Fermi surfaces are of equal magnitude (and opposite sign), which is implicit in our model. The cancellation occurs at the level of fermionic triangular blocks as for each block there are two ways to arrange electron and hole Green's function lines and their corresponding momenta which cancel each other. We thus have
\begin{equation}\label{dgamma}
\partial_{\gamma}\delta\! K_{\text{QF}}\!=\!\frac{\mathcal{N}}{2}\!e^2 v_F^2 \mathop{\rm Tr}\nolimits\left[\partial_\gamma L_{Q,\Omega_m}\right]F_{\text{l}},\, \, F_{\text{l}}=F_{\text{DOS}}\!+\!F_{\text{MT}}.
\end{equation}
Apart from excluding transverse spin excitations, taking the derivative of $\delta K_{\text{QF}}$ makes the integration over the boson energies and momenta convergent at the ultraviolet.
This means that at $\gamma \ll 1$ the important values of $Q$ and $\Omega$ are within the region of applicability of low-energy expansion of $L_{Q,\Omega_m}$ assumed above.
At the same time the integrations over fermion and boson energies and momenta in Eq.~\eqref{dgamma} factorize,
and the fermionic loop $F_{\text{l}}$ can be taken at zero boson energy and momentum $(Q,\Omega_m)\to0$.
The factorization in Eq.~\eqref{dgamma} is possible thanks to the energy scale separation of fermions $\Delta$ and bosons $\Delta_{\text{QCP}}\ll \Delta$.
The individual terms are $F_{\text{DOS}}=2 \mathop{\rm Tr}\nolimits\left[V_S^2 \hat{G}\hat{\tau}_3 \hat{G}\hat{\Xi}^z \hat{G} \hat{\Xi}^z \hat{G}\hat{\tau}_3\right]$ and $F_{\text{MT}}=\mathop{\rm Tr}\nolimits\left[V_S^2 \hat{G}\hat{\tau}_3 \hat{G}\hat{\Xi}^z \hat{G}
\hat{\tau}_3 \hat{G} \hat{\Xi}^z\right]$,
where the spin vertex renormalization $V_S$ can be evaluated at $(Q,\Omega_m)\to0$ taken in any order due to the non-conservation of the magnetization (see Appendix \ref{App:Disorder} for exhaustive details).
For $\Gamma_{\pi}=0$, $V_S = (1 + \Gamma_0/\sqrt{\Delta^2 + \omega^2_n})^{-1}$, where $\omega_n$ is a frequency argument of Green functions. In the wide range of parameters $(T,\Gamma_c) \ll \Delta$, $F_{\text{l}}\simeq \nu_F/\Delta^2$.
We further separate $\delta K_{\text{QF}}=\delta K_{\text{QCP}}+\delta K_{\text{SDW}}$ into zero-temperature $(\delta K_{\text{QCP}})$ and finite-temperature $(\delta K_{\text{SDW}})$ terms. For the former we straightforwardly find in a limit, $\Gamma_{c} \ll \Delta$
\begin{equation}\label{K-QCP}
\frac{\delta K_{\text{QCP}}}{K_0}=- \sqrt{\frac{\pi}{2}}\frac{3 \pi \mathcal{N}}{16} \frac{\Delta}{E_F}\sqrt{\gamma}
\end{equation}
up to a constant with the Fermi energy, $E_F = \pi \nu_F v_F^2/4$.
This result applies at both sides of QCP, and complements a similar calculation in a paramagnetic phase done in a band model of the QCP \cite{LVKC}.
This result gives positive correction to the penetration depth, however as we discussed above, is insufficient to turn it into a peak within the validity of perturbative analysis.
To elucidate this point, we introduce the Ginzburg parameter $\mathrm{Gi}$ given by a $\sqrt{\gamma}$ such that the fluctuation correction, $\partial_{\gamma}\delta K_{\text{QCP}}$ becomes comparable to the mean field value. It follows that the loop expansion is a series in powers of $\mathrm{Gi}/\gamma$. For instance, the two-loop contribution can be estimated to give a correction to $\delta K_{\text{QCP}}/K_0 $ of the form, $\propto\mathrm{Gi}^2\gamma^{-1/2}$ (see Appendix \ref{App:Fluctuations} for extensive details). From the comparison to Eq. \eqref{K-QCP} we then conclude that $\mathrm{Gi}=\Delta/E_F$.
We proceed to analyze the temperature dependent part of the response kernel. After the Matsubara sum we arrive at
\begin{equation}\label{K-SDW}
\frac{\partial_\gamma\delta K_{\text{SDW}}}{K_0}=-\frac{\mathcal{N}}{2\pi}\frac{\Omega^4_c F_{\text{l}}}{\nu^2_F}\int_Q\frac{f(E_Q/2T)}{E^3_Q},
\end{equation}
where $E_Q=\Omega_c\sqrt{\gamma+(Q/Q_c)^2}$ and $f(z)=\coth(z)-1+z/\sinh^2(z)$. In the temperature range above the QCP gap, $\Delta_{\text{QCP}}<T<\Delta$, we find for the penetration depth
\begin{equation}\label{lambda-QF}
\frac{\delta\lambda_{\text{QF}}}{\lambda_0}=\frac{\mathcal{N}}{8}\frac{T}{E_F}\ln\left(\frac{1}{\gamma}\right),
\end{equation}
so that the peak height is estimated as $\delta\lambda^{\text{max}}_{\text{QF}}/\lambda_0\simeq(T/\Delta)\text{Gi}\ln(1/\text{Gi})$. At temperatures within the QCP gap, $T<\Delta_{\text{QCP}}$, we instead find an exponential dependence $\delta\lambda_{\text{QF}}/\lambda_0\propto e^{-\Delta_{\text{QCP}}/T}$.
The linear in $T$ result holds in both paramagnetic and magnetically ordered phases.
The only difference originates from the difference in the coefficients $\gamma'_{\pm}$ describing the paramagnon softening in the two phases as introduced above.
\section{Discussion, summary, and outlook}\label{Sec:Summary}
To address implications of these results in light of experiments we stress that measurements of Ref. \cite{Prozorov} on Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$ were carried out at $T\sim4.5$K (with maximal $T_c\sim 25$K), whereas measurements of Ref. \cite{Matsuda} on BaFe$_2$(As$_{1-x}$P$_x$)$_2$ were done at $T=1.2$K (with maximal $T_c\sim 30$K). In the Co-doped case we interpret the emergence of the peak as due to SDW fluctuations at finite temperature once the system is tuned into the anomalous region by doping so that $\lambda_{\text{QF}}$ from Eq. \eqref{lambda-QF} dominates over suppressed mean field behavior $\delta\lambda_{\text{MF}}\propto e^{-\Delta/T}$. This is exemplified in Fig. \ref{Fig-QCP-PD}(b) and further in Fig. \ref{lambda-3D-Visual}(b). In addition, due to renormalization of fluctuations by finite $M$ in the phase of coexistence the structure of the peak should be nonsymmetric from both sides of QCP. This is supported by our model analysis and is in qualitative agreement with observations.
In contrast, in the P-doped case a quasi-linear-$T$ dependence of $\lambda$ was seen and attributed to the nodal structure of the gap. However, it is crucial to point out that the slope of this linear behavior was changing with doping attaining a maximum at QCP (see Fig. 3 of Ref. \cite{Matsuda}). We attribute this enhancement to SDW fluctuations which also result in linear-in-$T$ penetration depth as we show in Eq. \eqref{lambda-QF}. Indeed, this statement can be made more precise as in the case of a SC with simple isolated nodes, Eq. \eqref{lambda-QF} defines the renormalization of the relative slope in $T$-linear behavior of the penetration depth, $\delta\lambda(T)/\lambda_0=s(T/\Delta)$, so that slope receives a correction $\delta s\simeq \text{Gi}\ln |x-x_c|^{-1}$, which becomes progressively steeper towards a QCP. This prediction is thus consistent with corresponding behavior seen in experiments of Ref. \cite{Matsuda}.
A signature of the peak was also detected in (Ba$_{1-x}$K$_x$)Fe$_2$As$_2$ \cite{Auslaender-2}
concomitant with nonmonotonic doping dependence and change in $\delta \lambda\propto T^n$ power law \cite{Cho}. While SDW fluctuations certainly play an important role near QCP, interpretation of the data in the whole range is difficult as K-doped system displays a series of Lifshitz topological phase transitions resulting in gaped-to-nodal change of the pairing gap. Additional complications come from the apparent narrow dome of $s+is'$ superconductivity separating gapped and nodal regions \cite{Klauss} capturing which is beyond our two-band model.
In summary, in this work we studied the interplay of magnetism and superconductivity in the context of iron pnictides. Our principal results are Eqs. \eqref{C-QF} and \eqref{lambda-QF} for the temperature dependence of the specific heat and the London penetration depth, respectively, due to physics associated with the QCP. These results are significant as power-law $T$ dependence of thermodynamic properties of SCs is used as a hallmark diagnostic for their unconventional character, namely determination of the types of the nodes of superconducting order parameter. Yet we demonstrate that even in the presence of the full gap such behavior can be promoted by the quantum criticality under the dome of superconductivity.
We further comment that there remain some unresolved issues that warrant additional studies. In particular, a double-peak structure was detected in the penetration depth measurements in NaFe$_{1-x}$Co$_x$As \cite{Zheng}. This remarkable feature was attributed to the second putative QCP of nematic origin. However, the mere statement of multiple possible QCPs under the SC dome is at odds with the present state of the theory \cite{Fernandes-QCP} that predicts that magnetic and nematic transitions merge together into the weak first-order quantum critical line that thus terminates at a single QCP.
In closing, we mention that our results open interesting perspectives for the studies of transport properties due to QCP, specifically for the optical conductivity and thermoelectric effects, where one may hope to obtain anomalous frequency and temperature dependencies due to quantum fluctuations. It is also of special interest to investigate the QCP behavior due to the interplay of charge/pair-density order and superconductivity, which is highly relevant topic in the physics of cuprates.
\section*{Acknowledgments}
We thank E. Berg, V. S. de Carvalho, A. Chubukov, R. Fernandes, S. Gazit, Y. Matsuda, D. Orgad, R. Prozorov, S. Sachdev, J. Schmalian, and T. Shibauchi for important discussions that shaped this study. This work was supported in part by BSF Grant No. 2016317, ISF Grant No. 2665/20 (M.K.), NSF-DMR-BSF-2002795 (M.D. and M.K.), and U. S. Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES) Program for Materials and Chemistry Research in Quantum Information Science under Award No. DE-SC0020313 (A.L.) and DE-SC0016481 (M.D.). This work was performed in part at the Landau Institute for Theoretical Physics, Max Planck Institute for the Physics of Complex Systems, and at the Aspen Center for Physics, which is supported by National Science Foundation Grant No. PHY-1607611.
|
1,116,691,499,253 | arxiv | \section{Conclusion}\label{conclusion}
In this paper, we have proposed and evaluated a
novel Q-learning based
online checkpointing policy
for secure, intermittently powered IoT devices.
Compared to the current state-of-the-art, the proposed policy is the first
to take into account multiple factors, including forward progress, distance
to previous checkpoint and current battery level
to judiciously determine when to checkpoint.
A detailed evaluation of the scheme compared to the state-of-the-art
demonstrates up to
$1.4\times$ reduction in execution time. We examine the effects of varying model parameters such as battery level and interval size on performance and the resulting trade-offs between model complexity and performance and memory storage requirements.
Our
future work involves incorporating other run-time information
in the framework, and experimenting with an energy harvester and a lightweight processor.
\section{Experimental Results}\label{experiments}
\subsection{System Setup}
For experimental evaluation, we used the TigerMIPS processor,
a 32-bit 5-stage implementation of the MIPS ISA which includes 8KB L1
instruction and data caches~\cite{tiger}.
The light-weight PRINCE block cipher
is used to encrypt and decrypt data written to and read from the non-volatile main memory.
In order to reduce the overhead of memory writes, a TigerMIPS' write-through cache is
replaced with a write-back cache.
An RRAM based non-volatile main memory is assumed, and NVSim~\cite{nvsim} is used
to derive power and performance parameters for an 8 MB non-volatile memory.
The TigerMIPS RTL was modified to incorporate checkpointing and roll-back operations.
On a checkpoint, the fetch stage is stalled until all instructions in the pipeline retire.
The contents of PC, RF and dirty cache lines are encrypted and written back to main memory.
The effect of energy failure is simulated by flushing the processor's pipeline, discarding the data in the instruction and data caches,
and resetting the RF and PC.
When energy is restored, the checkpointed state is read from main memory, decrypted and restored in the processor.
The modified TigerMIPS processor and
PRINCE block cipher are synthesized using Cadence RTL Compiler~\cite{rc} with a 45 $nm$ technology library and
a target 100 KHz clock.
The $V_{dd}$ was set at $1.1V$.
\begin{figure}
\centering
\subfloat[]{%
\includegraphics[width=0.5\textwidth]{./p.png}%
\label{fig:synpa}
}\par
\subfloat[]{%
\includegraphics[width=0.5\textwidth]{./pq.png}%
\label{fig:synpb}
}\par
\subfloat[]{%
\includegraphics[width=0.5\textwidth]{./gp.png}%
\label{fig:synpc}
}
\caption{Generating synthetic power traces based on measured harvested power from radio frequency. (a) measured radio frequency power trace from~\cite{ma:hpca}, (b) quantized measured power to six power levels, and (c) the synthetic power trace with transition probabilities extracted from the quantized measured trace.}
\label{fig:synp}
\end{figure}
For Q-learning we set the discount factor $\gamma = 1$, and $\epsilon$ (i.e., the rate at which
random actions are picked instead of greedy ones) was reduced from $0.9$ to $0.1$ so as to perform more action exploration at the beginning.
The size of an interval $N$ is set to 500 instructions, and a super-interval $S$
contains 100 intervals (i.e., a super-interval has 50K instructions).
Thus, the Progress Counter and Checkpoint Counter are small (9 bits each).
For a super interval containing 100 intervals and 20 battery levels, the state-space has $100\times 100 \times 20 = 200K$ states. For each state, one action bit needs to be stored in memory, indicating whether a checkpoint has to be placed or not. Therefore, a memory size of $25KB$ would be sufficient to store the Q-table information. At run time, the action bit corresponding to the current state is read and the appropriate action is performed by the scheduler.
\begin{comment}
Alternatively, to eliminate the memory cost of storing the Q-table information, we approximate the Q-values using a NN
with two hidden layers of $5$ neurons each, $3$ neurons in the input corresponding to $3$ state elements, and $2$ neurons in the output corresponding to two possible actions. Note that each of the three inputs to the NN can be represented using
9 bits or less (the battery level needs only 5 bits).
\end{comment}
\begin{bf}{Energy Penalty Analysis}\end{bf}:
The energy penalty of storing the action bits in NVM and reading is one read from the non-volatile memory every interval which adds up to $65.16pJ\times 100 = 6.5nJ$ per super-interval. This is only $0.002\%$ of the energy consumed by the processor in a super-interval.
\begin{comment}
The second method approximates the Q-values using a neural network. The forward propagation computation of this network requires: $3\times 5 + 5\times 5 + 5 \times 2 = 50$ multiply accumulate (MAC) operations and $10$ times tansig function calculations (only used for hidden layers). Based on the data presented in~\cite{hoang2010high}, we estimate the energy consumption of the neural network as: $50\times 10.7pJ = 535pJ$. Therefore the energy penalty of this scheme is $535pJ\times 100 = 53nJ$ per super-interval, which is only $0.017\%$ of the average processor energy consumption in a super-interval (more detailed investigation of the energy cost of an on-chip NN implementation is an on-going effort).
\end{comment}
\begin{comment}
Once the Q-table is learned, we approximate it using a 2-layer NN
with 4 perceptrons in the first layer, and 2 perceptrons in the second layer.
Note that each of the three inputs to the NN can be represented using
9 bits or less (the battery level needs only 5 bits).
To keep the area overhead low, the
hardware implementation consists of a single perceptron (essentially an 8-bit multiply and accumulate unit)
re-used over multiple clock cycles.
The NN hardware module also
has dedicated registers to store weights
that are loaded from memory before the benchmark executes. \todo{fix}
Table~\ref{tb:pars} notes the estimated energy consumption of
a single execution of the NN,
which is negligible compared to the
energy required to process an instruction or encrypt/decrypt data.
\end{comment}
We model a radio frequency based energy harvesting source as described by~\cite{ma:hpca}. Unfortunately, the sample power traces
provided are small, while we need large traces both for training and validation.
Modeling the harvested power as a Markov chain is common in literature~\cite{ho2010markovian}.
Thus, we
developed
a first-order Markov chain based model for the harvested
power that transitions
between six discrete power levels. To obtain the transition probabilities, we quantized the
measured trace to six power levels, $L=\{0, 5, 10, 15, 20, 25\}$ (power levels in $mW$). We construct a transition matrix $T$, where $T(i,j)$ indicates the number of power level transitions from $p_i$ to $p_j$ within the trace, where $p_i, p_j \in L$. Each row $i$ of the transition matrix keeps track of the number of transitions from power level $p_i$ to all other power levels. Therefore, we can estimate the transition probabilities between power levels from the transition matrix. Fig~\ref{fig:synpa} shows the measured power from a radio frequency source. The quantized power over the six levels is shown in Fig~\ref{fig:synpb}. Once the transition matrix and the transition probability matrix is obtained, we can use the Markov chain model to obtain new synthetic power traces. An example of a synthetic power trace generated using this model is shown in Fig~\ref{fig:synpc}.
\begin{figure}
\centering
\includegraphics[width=0.4\textwidth]{./toolflow2.png}
\caption{In-house toolflow developed for experimental validation of the proposed approach.}
\label{fig:toolflow}
\end{figure}
\begin{comment}
The transition probabilities
were set to obtain energy traces that were
qualitatively similar to the provided energy trace with the
average energy close to the average processor energy consumption.
\end{comment}
We assume a battery with a 2 $\mu J$ capacity which is smaller than
the $\approx 20 \mu J$ battery assumed in
QuickRecall \cite{quick} but larger than
the $\approx 0.5 \mu J$ capacity in \cite{ma:hpca}.
Table~\ref{tb:pars} shows the estimated
energy consumption
for
different components of our system, the average incoming energy, and battery parameters.
\begin{table}[h!]
\centering
\resizebox{0.45\textwidth}{!} {
\begin{tabular} { | l | l | l | l |}
\hline
\bf{Parameter} & \bf{Value} & \bf{Parameter} & \bf{Value} \\ \hline \hline
Processor & $6.3$ nJ/inst & Action Bit Memory & 25KB\\ \hline
PRINCE & $1.6$ nJ/8B block & Harvested (Avg) & 6 nJ/clock \\ \hline
NV Read & $65.16$ pJ/4B line & Battery Capacity & 2 $\mu$J \\ \hline
NV Write & $71.78$ pJ/4B line & Battery Levels & $20$\\ \hline
\end{tabular}
}
\caption{Experimental parameters. The processor and PRINCE cipher energy consumptions
are obtained from RTL synthesis. The data for non-volatile memory is from NVSim~\cite{nvsim}.}
\label{tb:pars}
\end{table}
\begin{comment}
Intermittent execution can pose threats to memory consistency \cite{ransford:atomic}. Between two checkpoints, any write to nonvolatile main memory (NVMM) alters the checkpointed state and therefore, rolling back by restoring only the volatile memory of the processor will leave the processor in an incorrect state. To address this issue, we dedicate one bit called \textit{changed bit} for each block in memory which is set to one if there is a write to that block between two checkpoints. Before writing to memory, the controller copies the previous value of the block (which is the correct value at checkpoint) into a \textit{shadow memory} and sets the changed bit to one. When rolling back, for memory locations with changed bit set to one, the values are restored from the shadow memory. At checkpoint or rollback, all changed bits are set to zero.
\end{comment}
\subsection{Tool Flow}
Our validation tool flow is shown
in Fig~\ref{fig:toolflow} and is based on two components: an offline component that
learns the optimal policy
and obtains the action bit table based on the learned Q-table in \textsc{Matlab}\xspace,
and an on-line component that uses
detailed RTL simulations to measure execution time using our proposed and other state-of-the-art checkpointing
policies.
The offline Q-learning phase takes as input the energy harvesting traces and average of
processor power consumption for each benchmark.
Once the optimal policy is
learned offline, the corresponding action bit table is fed
to the RTL simulator along with dynamic traces of processor power consumption to estimate
the program execution time with the learned policy. The power traces are generated by feeding the value change dump (VCD) files from each
benchmark's execution to Cadence RTL Compiler.
\begin{comment}
We implemented a behavioral model of the NN in hardware for online decision making and used the energy numbers derived in previous section to account for the energy consumption of the network.
\end{comment}
The power traces used for
offline training and online validation are different, obtained from the same Markov chain model. Our results are presented on several benchmarks (CHStone~\cite{hara2009proposal} and FFT~\cite{legup}).
\begin{comment}
The weights extracted from the neural network are plugged in the neural network implemented in hardware as the checkpoint scheduler.
each testbench program was simulated on TigerMIPS and the execution was divided into several time frames. A Value Change Dump (VCD) file was generated for each of these time frames. These VCD files were then fed into Cadence RTL-compiler (RC) for dynamic power computation. Using this method, we created the processor power profile for each of the testbench programs. We also extracted program characteristics (i.e. rate of writes to cache) from simulation of programs on TigerMIPS. The policy is obtained in the form of a Q-table for each state-action pair and the neural network is trained on the table.
\end{comment}
\begin{figure}
\centering
\subfloat[]{\includegraphics[width=0.25\textwidth]{./Q1.png}\label{fig:qvalue1}}
\subfloat[]{\includegraphics[width=0.25\textwidth]{./Q2.png}\label{fig:qvalue2}}
\caption{Converged Q-values for $a_{chpt}$ and $a_{proc}$ as a function of battery level for
(a) $PrC=10$ and $CC=9$, i.e., at the $10^{th}$ interval with the most recent checkpoint at the $9^{th}$ interval, and
(b) $PrC=10$ and $CC=4$.}
\label{fig:qvalue}
\end{figure}
\subsection{Comparison with Prior Art}
The Q-learning based approach is compared with
two techniques from prior art.
The \emph{conservative} policy is based on the work in
\cite{hibernus,ma:hpca}, where the checkpointing decisions are made \emph{only} on the current
battery level.
The conservative policy checkpoints
any time the energy level in the battery
falls below a threshold, that is equal to the energy required
to checkpoint the PC, RF contents and the data cache.
The processor is then turned off and restored only when the
battery level exceeds another threshold, which
is at least the amount of energy required to
read and decrypt data from the nonvolatile memory plus the energy to perform a checkpoint.
The policy is conservative in that it
accumulates the energy required to perform a full checkpoint in battery before starting the program execution, guaranteeing that it \emph{never} incurs any roll-backs.
The second technique is a \emph{periodic}
checkpointing policy~\cite{liu2016lightweight}. A running instruction counter is maintained, and
a checkpoint is performed any time the counter exceeds a threshold. For fairness, we
pick the best threshold value averaged over all benchmarks (1000 instructions).
The learned Q-values as a function of battery level are shown in Fig~\ref{fig:qvalue} for checkpoint ($a_{chpt}$) and proceed with no checkpoint ($a_{proc}$) actions. In Fig~\ref{fig:qvalue1} and Fig~\ref{fig:qvalue2}, the previous checkpoint was taken one and six intervals before the current interval, respectively.
\begin{comment}
Fig~\ref{fig:qvalue}(a) and Fig~\ref{fig:qvalue}(b) show the learned Q-values as a function of battery level for
the checkpoint ($a_{chpt}$) and proceed with no checkpoint actions ($a_{proc}$) for
cases when the previous checkpoint was taken one and six intervals before the
current interval,
respectively.
\end{comment}
The Q-values represent expected cost of the action,
thus the optimal policy chooses the action with lower Q-value. We make two observations from this figure:
(i) in both cases,
as the battery level reduces, the checkpointing action becomes preferred to the no checkpointing action;
(ii) however, the threshold below which the checkpointing is preferred is lower when the previous checkpoint
was taken recently.
In other words, policies based on a \emph{static} battery level threshold~\cite{ hibernus,ma:hpca}
are sub-optimal.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{./runningtime.png}
\caption{Comparison of normalized execution times: Q-Learning based checkpointing policy
vs periodic~\cite{liu2016lightweight} and conservative policies~\cite{hibernus,ma:hpca}. For each testbench, runtime is normalized with respect to the baseline execution time (without energy failure) for that testbench.}
\label{fig:results}
\end{figure}
\begin{table}[h!]
\centering
\resizebox{0.45\textwidth}{!} {
\begin{tabular} { | l | c | c | c | c |}
\hline
\multirow{3}{*}{\bf{Benchmark}}& \multicolumn{2}{|c|}{\bf{Periodic}} & \multicolumn{2}{|c|}{\bf{Q-learning}}\\\cline{2-5}
& \multirow{2}{*}{\bf{\#CPs}} & \bf{Total RB}
& \multirow{2}{*}{\bf{\#CPs}} & \bf{Total RB} \\
& & \bf{Cost (s)} && \bf{Cost (s)} \\ \hline
DFADD & 12 & 0.302 & 7 & 0.290 \\ \hline
MIPS & 69 & 0.767 & 38 & 0.632 \\ \hline
ADPCM & 237 & 6.13 & 139 & 5.36 \\ \hline
GSM & 26 & 0.629 & 20 & 0.462 \\ \hline
MOTION & 58 & 0.674 & 42 & 0.769 \\ \hline
AES & 69 & 2.04 & 48 & 2.01 \\ \hline
FFT & 17 & 0.391 & 11 & 0.250 \\ \hline
\end{tabular}
}
\caption{Comparison of Q-learning based checkpointing with periodic checkpointing in terms of the number of checkpoints (CP) and roll-back (RB) cost corresponding to re-execution for various benchmarks.}
\label{tb:statistics}
\end{table}
The normalized execution time of the
proposed Q-learning based dynamic checkpointing policy to the periodic and conservative policies are compared in Fig~\ref{fig:results}. Normalization is done with respect to the baseline execution time with no energy failure.
For all the benchmarks, Q-learning results in the lowest execution time followed by periodic and finally
conservative. Specifically, Q-learning is on average $1.3\times$ faster than periodic checkpointing.
The improvements are even greater when compared against the conservative policy.
From Fig~\ref{fig:results} we can see that the conservative policy performs poorly compared to the other methods. Although in this method no overhead is incurred from roll-backs (and consequently lost computation), significant amount of time is lost
waiting for the battery to recharge. On the other hand, the periodic policy does not \emph{adaptively} determine
when to checkpoint. Table~\ref{tb:statistics} shows the number of checkpoints placed by the periodic and Q-learning policies during runtime. We can see that the Q-learning based policy places fewer checkpoints compared to the periodic policy, thus incurring smaller overhead due to checkpointing. The other observation is that the re-execution cost due to roll-backs is smaller for the Q-learning policy, proving that the checkpoints are placed close to energy failure points, thus reducing the re-execution cost.
\begin{comment}
\begin{figure}
\centering
\subfloat[]{\includegraphics[width=0.4\textwidth]{./fb0.png}}\\
\subfloat[]{\includegraphics[width=0.4\textwidth]{./fb1.png}}
\caption{Progress plots for (a) Q-learning based policy, and (b) the
periodic policy. Battery levels are shown with time and the locations where checkpoints are
taken (vertical lines).}
\label{fig:prog}
\end{figure}
\end{comment}
The execution progress of the FFT benchmark is shown in Fig~\ref{fig:progress}. This figure plots the number of (useful) instructions executed, battery level and checkpoint locations for the proposed Q-learning based checkpointing policy.
\begin{comment}
Fig~\ref{fig:progress} plots the execution progress
in terms of number of (useful)
instructions executed, battery level and checkpoint locations for the proposed Q-learning
based checkpointing during the execution of the FFT benchmark.
\end{comment}
Observe that
checkpoints can be triggered by low battery level (CP1, CP2, CP3, CP5 and CP6), or
when a long time has
passed since the previous checkpoint even if the battery level is high (CP4).
\begin{comment}
As can be observed from this figure, the Q-learning based policy makes faster forward progress than the periodic policy and therefore finishes the execution of the program faster. Since the periodic checkpointing is oblivious to the battery state, it performs a lot of redundant checkpoints even when the battery is fully charged, whereas the Q-learning based method performs only one when it's close to energy failure. In spite of performing more checkpoints in the periodic method, because the checkpoints are poorly placed, the roll-back cost is higher compared to the Q-learning based method.
\end{comment}
\begin{figure}
\centering
\includegraphics[width=0.4\textwidth]{./progress1.png}
\caption{Progress plot for the FFT benchmark using Q-learning based checkpointing. Vertical lines represent checkpoint placements. Checkpoints are triggered either by a low battery level or when a long time has passed since the previous checkpoint.}
\label{fig:progress}
\end{figure}
\subsection{Sensitivity and Security Analysis}
We explored the effects of varying the number of possible battery levels $(B)$ and super interval size $(S)$. A simulator in \textsc{Matlab}\xspace was developed, implementing the program execution with checkpointing and roll-back.
The simulator estimates the running time of the program by keeping track of the available energy in the battery based on the incoming harvested energy (synthetic traces) and consumed energy by the processor.
In the first experiment, we fixed the size of the super interval, $S=100$ and trained different models by varying $B$. The running time of the program when the Q-learning policy is used for checkpointing was estimated. Fig~\ref{fig:sens_bl} shows the relative speedup with respect to the performance of the model with $B=5$. We can see that training on higher granularity for battery level results in better performance of the Q-learning policy. Training a model with $B=160$ resulted in more than $18\%$ speedup of program execution time. However, the size of state space grows linearly with the number of battery levels, which is directly proportional to the size of memory required to store the Q-values and also results in higher training times. Note that the performance gain from increasing the battery levels begins to converge after which higher granularity would not gain much in terms of speedup.
In the second experiment, the effect of changing $S$ was examined.
We trained various models by varying $S$ and fixing $B=20$. Fig~\ref{fig:sens_si} shows the relative speedup of models based on Q-learning policy with different values of $S$ compared to the model with $S=50$. We can see that by increasing the super interval size to $800$, we get up to $7.2\%$ speedup in program execution time. Note that the size of state space and hence the memory required to store Q-values grows quadratically with the size of the super interval.
\begin{figure}
\centering
\subfloat[]{\includegraphics[width=0.17\textwidth, angle=-90]{./sens_bl.eps}\label{fig:sens_bl}}
\subfloat[]{\includegraphics[width=0.17\textwidth, angle=-90]{./sens_si.eps}\label{fig:sens_si}}
\caption{Sensitivity analysis of the Q-learning checkpointing policy to (a) battery level B and (b) size of the super interval S. The relative speedups compared to a model with $B=5$ and $S=50$ are illustrated. With higher granularity of the battery level and a larger super interval size, the Q-learning policy performs better and program execution time decreases.}
\label{fig:sens}
\end{figure}
As was mentioned before, the non-volatile memory has to be encrypted at all times to prevent an attacker from reading out potentially sensitive data. However, once an attacker physically captures an IoT device, they can read out an encrypted
image of main memory and checkpointed state. Then, by
executing their
own code on the device and observing the corresponding encrypted data, the attacker might
be able to carry out a
chosen plain-text attack to recover the on-chip encryption key.
Although security analysis of
PRINCE suggests that such an attack is
impractical (requiring at least $2^{63}$ plaintexts~\cite{princean}), a cautious defender might still wish to
to use stronger block ciphers like AES. However, AES encryption further increases the
energy overhead of checkpointing.
Our experiments for the FFT benchmark showed $22\%$ and $124\%$ increase in runtime when PRINCE and AES were used, respectively, for encrypting the checkpoints.
In addition to chosen plaintext attacks, the attacker might be able to launch a denial of service attack if they can tamper
with the energy source. While both the original and secure schemes are susceptible to such attacks, the energy overhead of
secure checkpointing might increase the vulnerability to denial of service.
\begin{comment}
Table~\ref{} notes the execution latency of the xxx benchmark with no encryption,
PRINCE, and AES encryption, clearly illustrating the cost versus security trade-offs for energy-scavenging IoT devices.
\end{comment}
\begin{comment}
To evaluate the overhead of secure checkpointing, we compute the time and energy cost of performing checkpoints for each of the testbenches. We then compare these costs to the baseline execution time and energy consumption of each testbench, assuming no energy failure. Using the fully unrolled implementation of PRINCE block cipher \footnote{PRINCE uses about $4-5$ times less area than other ciphers in literature~\cite{prince}.} reduces the encryption/decryption cost of a $64$-bit block to one clock cycle. By pipelining the cipher operation and memory access, we can hide the latency of encryption/decryption operations. Therefore, the time overhead would be equal to writing the checkpoints to off-chip NVM. Table~\ref{tb:overhead} shows the runtime overhead of checkpointing for various testbenches. The checkpoints incur a time overhead of less than $10\%$. The energy cost of a checkpoint is the sum of cipher energy consumption and the write operation on NVM. From table~\ref{tb:overhead}, we can see that the energy overhead, ratio of the total energy for checkpoints to the computation energy, is less than $1.3\%$. The results verify the applicability of our proposed methodology.
\begin{table}[h!]
\centering
\resizebox{0.45\textwidth}{!} {
\begin{tabular} { | l | c | c |}
\hline
\bf{Benchmark} & \bf{Time Overhead(\%)} & \bf{Energy Overhead(\%)} \\ \hline
DFADD & 2.28 & 0.296 \\ \hline
MIPS & 5.03 & 0.621 \\ \hline
ADPCM & 9.962 & 1.29 \\ \hline
GSM & 2.54 & 0.328 \\ \hline
MOTION & 6.62 & 0.856 \\ \hline
AES & 6.73 & 0.871 \\ \hline
FFT & 6.02 & 0.778 \\ \hline
\end{tabular}
}
\caption{Runtime and energy overheads of the Q-learning based secure checkpointing.}
\label{tb:overhead}
\end{table}
\end{comment}
\section{Introduction}\label{intro}
The Internet of Things (IoT) is envisioned as a network of devices that operate anytime from anywhere. IoT devices are expected to be small and guarantee perpetual and autonomous operation even in hard
to reach places.
Energy harvesting has been proposed as the source of
energy for such IoT devices \cite{liu:dac,eh}.
Ambient energy can be harvested from many sources, including solar, wind, thermal, WiFi, radio frequency, biological and chemical~\cite{h2,h1, h3}.
The amount of energy harvested is highly time varying and the energy supply is intermittent. For this reason, processors with off-chip non-volatile main memory (and potentially even on-chip caches and register files)
have been proposed \cite{ liu:dac,ma:hpca} to guarantee
\emph{forward progress} of program execution. In these approaches, the processor saves its intermediate results in non-volatile memory when the energy supply is too low for the processor to operate.
Security is another critical concern with IoT devices.
An attacker with physical access to an IoT
device might be able to read out potentially sensitive data from the non-volatile memory~\cite{nvmat}.
Prior work secures data in IoT devices using non-volatile main memory, i.e. by
encrypting and decrypting every write to and read from the non-volatile memory~\cite{Chhabra:i-nvmm,Enck:securingnvmm}. However, simply encrypting main memory is not sufficient. An attacker might capture an IoT device
in the middle of program execution and recover a checkpoint of architectural state. In fact, prior work has shown that unencrypted intermediate
program state is an even greater security threat~\cite{Halderman}. Thus, we propose to encrypt the intermediate checkpoints of program state to guarantee confidentiality at all times during program execution.
\begin{comment}
(indeed, prior
work has shown that unencrypted
intermediate program state is an even greater security threat~\cite{Halderman});
but this increases
the energy and time overhead of checkpointing.
\end{comment}
\newcommand{\myt}{
\begin{tabular}{ |c|c|c| }
\hline
Cipher & CP Energy (nJ) & CP Latency (cycles) \\ \hline
None & 5.75 & 82 \\
Prince & 66.90 & 164 \\
AES & 410.45 & 164 \\
\hline
\end{tabular}
}
\begin{figure}
\centering
\subfloat[][]{%
\includegraphics[width=0.48\textwidth]{./prog1.png}%
\label{fig:motive1}
}\par
\subfloat[][]{%
\includegraphics[width=0.48\textwidth]{./prog2.png}%
\label{fig:motive2}
}
\caption{Runtime execution progress of the FFT program running on harvested energy with (a) no encryption and (b) PRINCE~\cite{prince} as the block cipher. Vertical lines represent checkpoints and arrows show roll-backs due to energy failure. Encryption increases the average energy and latency of each checkpoint. Due to this overhead, the total number of roll-backs, time spent for each checkpoint and therefore the program execution time increases.}
\label{fig:motive}
\end{figure}
\begin{figure*}[t]
\centering
\includegraphics[width=1.0\textwidth]{./system2.png}
\caption{Optimal checkpointing of secure IoT processors: System model with a finite-sized battery with capacity $B_{max}$ and random energy arrivals. At a checkpoint, the volatile IoT system state (Program Counter (PC), Register File (RF) and dirty cache lines) is encrypted and saved into NVM.}
\label{fig:model}
\end{figure*}
Encryption increases the energy and latency overhead of checkpoints, by $11.6\%$ and $200\%$ respectively for PRINCE~\cite{prince} block cipher.
Finding the optimum checkpoint placement policy when the energy source is intermittent is crucial, and becomes even more significant when checkpoints have to be encrypted for security. Naively replacing unencrypted
checkpoints (Fig~\ref{fig:motive1})
with encrypted checkpoints (Fig~\ref{fig:motive2})
can significantly
increase the number of roll-backs and total program execution time.
This motivates the need for a
smart checkpointing policy that
places checkpoints judiciously.
The checkpointing policy must
account for multiple
factors including
the energy level of the battery, the stochastic behaviour of the
harvested energy (which differ from one source to another), when
the previous checkpoint was taken, and the
expected time to program completion. Prior checkpointing policies account for some of these effects,
for instance, periodic checkpointing~\cite{liu2016lightweight}
or checkpointing when the available energy in the battery falls below a threshold~\cite{hibernus,quick}.
We formalize the online checkpointing problem as a
Markov decision process, and compute the optimal checkpointing
policy offline using Q-learning \cite{Watkins1992}.
Online decisions are made using a table including the optimum action which is obtained from the Q-table (stored in memory).
Our solution: (i) is the first approach that simultaneously
(and explicitly) accounts for: the
current battery level, previous time a checkpoint was taken, and
the program's forward progress in informing checkpointing decisions;
(ii) accounts for the stochastic nature of the energy source,
checkpointing overheads, and processor energy consumption;
(iii) uses a model-free approach to solving the Markov decision process that can be trained using empirically obtained and synthetically generated traces of harvested energy;
and (iv) offers computational and energy efficient hardware solutions to make online decisions.
The rest of the paper is organized as follows: Section~\ref{related}
reviews the recent work in this area, while Section~\ref{method}
explains system model and the proposed Q-learning based
online checkpointing policy.
Section~\ref{experiments} describes our experimental setup, results and
compares the proposed approach to prior art. We conclude in Section~\ref{conclusion}
to future work.
\section{Online Checkpointing}\label{method}
We describe an online checkpointing framework for secure IoT processors starting with the system model, a mathematical formulation
of the optimal checkpointing problem, and a learning based solution.
\subsection{System Model}
Our target IoT system shown in Fig~\ref{fig:model},
includes an in-order processor with conventional volatile caches and registers running on harvested energy and an off-chip non-volatile main memory. Although the proposed techniques are agnostic to the NVM technology, in Section~\ref{experiments} we empirically evaluate an RRAM-based NVM. For securing the checkpoints, our system encrypts main memory similar to prior work~\cite{Chhabra:i-nvmm,Enck:securingnvmm}.
Data blocks written to and read from main memory are encrypted and decrypted, respectively,
using a light-weight block cipher PRINCE~\cite{prince}.
The system is powered by harvested energy which is highly intermittent. To smooth out large temporal variations in harvested energy, the IoT device has on-board energy storage\footnote{On-board energy storage can range from a simple super-capacitor to a
battery management IC with an in-built battery tailored for ultra-low power energy harvesting devices such as the TI BQ25504~\cite{TIBQ}.}. To guarantee forward progress of program execution, the processor state is checkpointed in NVM, and to guarantee security, all checkpoints are encrypted before being written to the NVM.
Shown as shaded in Fig \ref{fig:model}, a checkpointed IoT system state consists of:
(i) the program counter (PC),
(ii) the contents of the register file (RF);
and (iii) all dirty cache lines in the L1 data cache\footnote{We assume a single-level cache hierarchy.}.
The NVM has a \emph{shadow memory} to store a checkpoint.
When rolling back to the last checkpoint, the state of the system consisting of volatile and non-volatile states should be maintained consistently. The volatile states are stored and recovered from the shadow memory, therefore they remain consistent. Ransford et. al.~\cite{ransford2014nonvolatile} showed that checkpointing and recovering the program state can lead to inconsistency in the NVM. All writes to NVM that happen after a checkpoint change the non-volatile state of the program. In case of an energy failure, the processor rolls back to the checkpointed state and the volatile states are recovered correctly, however, the non-volatile state has changed since the checkpoint and the NVM will be inconsistent.
To address this issue, previous work has proposed enforcing a checkpoint between every non-idempotent memory access pairs~\cite{xie2015fixing}, or versioning the inconsistent data~\cite{ransford:atomic} so that every non-idempotent memory access pair is eliminated. Liu et. al.~\cite{liu2016lightweight} propose a hardware solution in which all stores are regarded as speculative and are delayed. These stores are written back to non-volatile memory only when the program reaches the next checkpoint. To keep the overhead of processing low and avoid placing an excessive number of checkpoints, we simply expand the shadow memory to keep track of the memory locations which are written to between two checkpoints. For each write to the non-volatile memory, the previous value in the memory is copied to the shadow memory. If the next checkpoint is performed successfully, these values are discarded and otherwise they are restored along with the saved volatile state.
Finally, the IoT processor is provisioned with the scheduler module (Fig~\ref{fig:model}),
that determines \emph{when} checkpoints are performed. The design and evaluation of a security-aware checkpoint scheduler is the primary contribution of this work.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{./mdp.png}
\caption{State transition diagram for actions $a_{chpt}$ (checkpoint) and $a_{proc}$ (no checkpoint) with and without an energy failure event. Each state is defined as $s_{i}= (p_i, c_i, b_i)$ where $p_i$ represents the progress made so far,
$c_i$ is where the last checkpoint was performed and $b_i$ is the current battery level. For each action, $C$ indicates the immediate cost incurred from that action.}
\label{fig:mdp}
\end{figure}
\subsection{Problem Formulation}
The checkpoint scheduler
decides whether to checkpoint or not every $N$
instructions. Every block of $N$ executed instructions is
referred to as an \emph{interval}.
To track forward progress, the
scheduler maintains two
run-time counters:
(i)
a \textit{Progress Counter (PrC)} which counts the number of intervals of actual
forward progress that the program has made; and
(ii) a \textit{Checkpoint Counter (CC)}
which records when the last checkpoint was taken. Both
counters are initialized to zero when the
program begins.
PrC is incremented every time $N$
instructions execute successfully.
If a checkpoint is made, CC is updated to the current PrC. Finally,
on energy failure, the processor rolls back to the last checkpointed state and this resets PrC to the current value of CC.
We formulate the execution of a
program with online checkpointing
as an instance of a
Markov decision process (MDP) consisting of
a set of states $S$, a set of actions $A$, a stochastic transition function $T:S\times A \times S \rightarrow [0,1]$ which gives the probability of moving from one state to another for each action, and an immediate cost function $C:S \times A \times S \rightarrow \mathbb{R}$.
In our formulation, the state of the Markov decision process is given by the current values of the progress counter (PrC), the checkpoint counter (CC) and the battery level. Specifically, the state at the end of each control interval $i$
is given by
$s_{i}= (p_i, c_i, b_i)$
where $p_i$ is the PrC value (representing the progress made so far),
$c_i$ is CC (when the last checkpoint was performed) and $b_i$ is the current battery level.
The set of actions is $A \in \{a_{chpt},a_{proc}\}$, where the action $a_{chpt}$ represents taking a checkpoint while the action $a_{proc}$ represents
proceeding without a checkpoint.
If the scheduler decides to checkpoint, a cost $t_{cp}$
corresponding to the latency of checkpointing is incurred. Assuming no energy failure during the next interval, the next state is $s_{i+1} = \{p_{i}+1, p_{i}, b_{i+1}\}$. On the other hand, if there is an energy failure in the next interval, the next
state is $s_{i+1} = \{p_{i}, p_{i}, b'_{i+1}\}$. In both cases, $b_{i+1}$ and $b'_{i+1}$ are the
new battery levels accounting for the energy cost of checkpointing, the net energy received/consumed in interval $i+1$.
On the other hand, if the scheduler decides \emph{not} to checkpoint, there is no immediate cost. Assuming no energy failure during the next interval, the next
state is $s_{i+1} = \{p_{i}+1, c_{i}, b_{i+1}\}$.
However, if there is an energy failure in the next interval, the next state is $s_{i+1} = \{c_{i}, c_{i}, b'_{i+1}\}$, incurring a cost of $(p_i-c_i) \times N$ representing the lost progress. The state transition diagram is shown in Fig~\ref{fig:mdp}.
The optimal policy balances the cost of checkpointing with the cost of recovering from an energy failure by rolling back to the last checkpoint. The total cost can be defined as the time overhead of checkpointing plus the re-execution time due to roll-backs. Given the Markov decision process specification, our
goal is to find an optimal policy
$\pi^*:S \rightarrow A$ which
minimizes the total expected cost as defined above, i.e., the expected overhead of the checkpointing policy relative to the baseline uninterrupted execution time.
The optimal policy can be obtained by solving for a fixed-point of the Markov decision process
given the stochastic model for the harvested energy and system statistics i.e., the probability of energy failure in future control intervals and a distribution of next state battery levels. However, this model would be hard to obtain and can be highly inaccurate.
Hence, the proposed system learns an optimal policy using experimentally obtained traces of harvested energy and system statistics.
\subsection{Offline Learning of Checkpointing Policy}\label{model}
The system uses the Q-learning algorithm to find the optimal policy. This algorithm assigns a
Q-value to each state-action pair
$Q: S \times A \rightarrow \mathbb{R}$ that, once the algorithm converges, represents the expected total cost of executing action $a$ in state $s$ and choosing greedy actions afterwards. The algorithm starts by
initializing all Q-values arbitrarily, and iteratively updates by simulating the system. In iteration $i$,
the algorithm chooses the action $a$ that results in the smallest Q-value for the current state, simulates the system, and observes the next state and corresponding cost. It then updates the Q-values
as follows:
\begin{equation}
\begin{aligned}
Q(s_i,a) & \leftarrow Q_i(s_i,a) + \\
& \alpha ( C(s_{i},a) + \gamma \min_{a' \in A} Q_i(s_{i+1},a') - Q_i(s_{i},a)).
\end{aligned}
\label{eq:q}
\end{equation}
Here,
$C(s,a)$ is the checkpointing cost $t_{cp}$
for checkpointing and
roll-back cost $(p_{i} - c_{i})\times N$ in the event of an energy failure for no checkpointing.
The parameter
$\gamma$ discounts future costs,
and $\alpha$ determines the learning rate, i.e.,
how strongly the Q-values are overwritten by new ones after each iteration. With certain constraints on the learning rate \cite{Tsitsiklis1994}, the
Q-values have been proven to converge to
those corresponding to the optimal policy $\pi^{*}$. Empirically, we found that
using a variable learning rate worked best in practice.
\begin{equation}
\alpha_{(s,a)} = \frac{1}{n(s,a)}
\end{equation}
where $n(s,a)$ is the number of times a state-action pair $(s,a)$ is visited.
Further, we use the $\epsilon-greedy$ algorithm which chooses an
action at random with probability $\epsilon$ and follows the greedy strategy with probability $1-\epsilon$ where $0<\epsilon<1$.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{./chpt2.png}
\caption{Illustration of the checkpointing policy. A mandatory checkpoint is taken after every super interval ($S$ intervals).
Within an interval, the checkpointing decisions are made based on the learned Q-table.}
\label{fig:cprb}
\end{figure}
\subsection{Online Decision Making}
After the Q table is learned, it is used online to derive the optimal policy for any state by picking the
action with smallest Q-value. Therefore, it is sufficient to store one \emph{action bit} corresponding to the optimum action for each state. For a small state-space, this approach entails small memory overhead and has the benefit of low energy consumption since it only requires one read from memory at the end of each interval.
\begin{comment}
As an alternative, prior work has
proposed to store a function
$\tilde{Q}: S \times A \rightarrow \mathbb{R}$,
for instance a
neural network (NN), that approximates
the learned Q-values. The NN weights are trained offline.
Even a \emph{small} NN with two hidden layers of $5$ neurons each, accurately represents the Q-table.
The energy overheads of the NN implementation are quantified in Section~\ref{experiments}.
\end{comment}
The state-space of the Markov decision process grows quadratically with
the number of intervals and hence with the dynamic execution length of the program.
This can result in large offline training times for the Q-learning to converge, as well as large storage requirements for action bit values. We use a hybrid policy as shown in Fig~\ref{fig:cprb} that checkpoints every $S$ intervals (the super-interval). The optimal checkpointing locations \emph{within} a super-interval are determined by the proposed Q-learning approach.
The hybrid approach limits the size of the Markov decision process state-space to $S^{2}B$
states, where $B$ is the number of possible battery levels. As a result, we are able to limit not only the training time, but the maximum amount of space required to store the Q-table in non-volatile main memory.
\subsection{Related Work}\label{related}
Checkpointing has been used for fault tolerance
in computing systems~\cite{cp1,cp2}, allowing the
processor to roll back to the last valid checkpoint and recover the processor
state in case of failure.
Okamura et al.~\cite{okamura2004dynamic} propose a dynamic checkpointing scheme based
on Q-learning for fault-tolerance,
but assume random failures unrelated to energy harvesting or power failures.
Checkpointing has also been used to guarantee forward progress
in intermittently powered
devices \cite{hibernus,quick,liu2016lightweight,ma:hpca,tpc,mementos}.
Mementos \cite{mementos}
inserts trigger points in the software
at compile time and, at run time, checkpoints when
the stored energy level falls below a threshold at a trigger point.
QuickRecall~\cite{quick}
and Hibernus~\cite{hibernus} propose a policy based on two thresholds, one that triggers checkpoints
(when the energy level falls below a \emph{low} threshold)
and one that determines when to start re-execution (when the energy level rises
above a \emph{high} threshold).
A similar approach is used in \cite{ma:hpca}, except that
they assume access to a processor with on-chip non-volatile state.
Hardware support for checkpointing that uses two counters for number of instructions and number of stores is proposed in~\cite{liu2016lightweight}. A checkpoint is performed if either of the two counters exceeds its threshold.
From a security stand-point, several proposals exist on main memory encryption for security, \cite{Chhabra:i-nvmm,Enck:securingnvmm} but they do not
consider battery-operated intermittently powered processors. Finally, Q-learning has been used in other online decision making contexts such as communications and power management \cite{1581377,790549,Tan:2009}.
|
1,116,691,499,254 | arxiv |
\section{Introduction}
Being able to model a dynamical system is fundamental in many engineering applications. Dynamical models can be constructed with first principles laws and/or data.
In the first case, physics laws, such as conservation of energy, matter and thermodynamics laws, are used. A relatively limited amount of data might also be used to fit the parameters of the model. These parameters usually have a physical meaning (e.g., heat transfer coefficients, friction coefficients). If only data is used, we generally talk about machine learning models. Generally, the fitted parameters do not have a physical meaning in these cases. Machine learning has recently obtained much attention since it allows to model complex or not well-understood systems, provided that a sufficiently large amount of data is available. For many decades the scarcity of data, features extraction and computational power have been the main bottlenecks to the widespread use of machine learning models. The situation is rapidly changing thanks to digitization (e.g. internet-of-things and industry 4.0 initiatives), deep learning and cloud computing. Hence, machine learning is finding more and more applications, not only in research but in our everyday life. In recent years, the number of machine learning applications, publications, new methods and theoretical findings has increased rapidly. Furthermore, open-source libraries for machine learning such as TensorFlow \citep{tensorflow2015} and PyTorch \citep{pytorch} offer a low barrier to entry to this technology and allow fast development of machine learning solutions. The combination of these factors boosted the quality and quantity of results in many fields such as medicine \cite{Shen2017}, autonomous driving \cite{Arnold2019}, transportation \cite{Nguyen2018}, internet-of-things \cite{Mohammadi2018}, image recognition \cite{Voulodimos2018}, and many others. The control and optimization community also recognized the usefulness of machine learning for control applications, e.g., for optimal control and model predictive control (MPC), motivating a large number of contributions (cf. \cite{Hewing2020} and references therein).
Machine learning libraries help the control practitioners to build machine learning models, but using these models in a control strategy requires ad-hoc solutions, which are complicated, prone to bugs and not easily transferable to other applications. HILO-MPC is an open-source Python library that aims at providing a way to easily implement machine learning models into a wide variety of control, optimization and estimation problems. HILO-MPC interfaces with PyTorch and TensorFlow to train neural networks and uses in-house code to train Gaussian processes. Once the models are trained, they can be used in a large variety of control or estimation problems. Furthermore, to speed up the development time, HILO-MPC provides a way to define and solve a broad spectrum of predefined optimal control and estimation problems quickly and with minimum effort. Tab.~\ref{tab:problems-hilo} summarizes the different problems that can be solved with version {1.0}{} of HILO-MPC. The philosophy that HILO-MPC follows is ease of use and modularity. Easy of use is achieved by providing an intuitive high-level interface for the problem definition. At the same time, modularity is obtained by ensuring effortless interfaces between the different tools present in the models (see Fig. \ref{fig:scheme}). This allows using the toolbox both for research and for teaching.
Comparing HILO-MPC with other similar toolboxes is challenging since it can solve a wide variety of control and estimation problems. Here, we consider only open-source toolboxes that can solve model predictive control problems for simplicity. The software ACADO \citep{Houska2011a} and ACADOS \citep{Verschueren2018} aim at efficiency and fast solution keeping embedded applications in mind. MPT3 \citep{MPT3} focuses on computational geometry, which is particularly useful for some classes of robust MPC approaches. YALMIP \citep{YALMIP} instead offers a broad spectrum of optimization problems, such as bilevel optimization and mixed-integer programming and some MPC schemes. The toolbox do-MPC \citep{LUCIA201751} implements robust multi-stage MPC and Moving Horizon Estimation. MPC-Tools \citep{risbeck2015nonlinear} offers some interfaces that help build MPC and MHE control problems in CasADi. In Tab.~\ref{tab:comparison-toolboxes} we summarize the main differences between the toolboxes.
Using machine learning models in the previously mentioned toolboxes is not straightforward. HILO-MPC solves this problem by providing wrappers with machine learning libraries and some in-house learning libraries. Furthermore, HILO-MPC focuses on a light weight and intuitive interface that makes it easy to use also for beginners.
The backbone of HILO-MPC is CasADi \citep{Andersson2019}. CasADi is a tool for algorithmic differentiation and optimization, and it is used to build the models, optimization and estimation problems. When TensorFlow or PyTorch are used, HILO-MPC offers an easy way to interface with those libraries and automatically translate the models into CasADi-compatible structures. Hence, the CasADi problem is defined and solved. CasADi has interfaces with a wide range of solvers for linear and quadratic programming (CLP \citep{johnjforrest2020}, qpOASES \citep{Ferreau2014}), nonlinear programming such as IPOPT, \citep{wachter2006implementation}, quadratic mixed-integer programming (CPLEX \citep{cplex2009v12}, GUROBI \citep{gurobi}), for non quadratic nonlinear mixed-integer problems (BONMIN \citep{bonami2007bonmin}, KNITRO \citep{nocedal2006knitro}) and large nonlinear problems WORHP \citep{nikolayzik2010nonlinear}.
Note that giving a comprehensive description of the toolbox is outside the scope of this article. For this, the reader is referred to the documentation. Here, we rather want to give a glimpse of the different problems HILO-MPC can solve and give an idea of the workflow and syntax with some examples.
The rest of this paper is structured as follows: in Section~\ref{sec:hilo-mpc} the different modules of the toolbox are described. Furthermore, we show example code snippets to illustrate the HILO-MPC syntax. In Section~\ref{sec:examples} we apply the toolbox in three examples. Finally, in Section~\ref{sec:conclusions} we conclude and give an outlook on current and future developments of the toolbox.
\begin{figure*}[bt]
\centering
\input{figures/overview.tex}
\caption{Overview HILO-MPC}
\label{fig:overview}
\end{figure*}
\begin{table}[bt]
\caption{Current tools implemented in HILO-MPC.}
\label{tab:problems-hilo}
\begin{threeparttable}
\begin{tabular}{llll}
\headrow
Models & Controllers & Machine Learning & Observers \\
\parbox[c][2.5cm][c]{3cm}{Linear/nonlinear\\Time-invariant/variant\\Continuous and discrete\\ODEs, DAEs} & \parbox[c][2.5cm][c]{3cm}{Nonlinear MPC\\Trajectory tracking MPC\\Path-following MPC\\PID} & \parbox[c][2.5cm][c]{3cm}{Neural networks\\Gaussian processes} & \parbox[c][2.5cm][c]{3.5cm}{Moving horizon estimation\\Kalman filter\\Extended Kalman filter\\Unscented Kalman filter\\Particle filter} \\
\hline
\end{tabular}
\end{threeparttable}
\end{table}
\begin{table}[bt]
\caption{Comparison overview on the different open-source MPC software. ML: machine learning, MPC: model predictive control, MHE: moving horizon estimator, LMPC: linear MPC, PFMPC: path-following MPC, KF: Kalman filter, EKF: extended Kalman filter, UKF: unscented Kalman filter.}
\label{tab:comparison-toolboxes}
\begin{threeparttable}
\begin{tabular}{llclll}
\headrow
& Problems & ML & Interface & Focus & Article \\
HILO-MPC & \parbox[c][1cm][c]{2.4cm}{MPC, PFMPC,\\MHE, KF, EKF, UKF} & Yes & Python & Machine Learning & \\
do-mpc & \parbox[c][1cm][c]{2.4cm}{MPC, MHE,\\Multistage MPC} & No & Python & Multistage MPC & \cite{LUCIA201751} \\
ACADO & MPC & No & C++, MATLAB & Embedded & \cite{Houska2011a} \\
ACADOS & MPC, MHE & No & \parbox[c][1cm][c]{2.5cm}{C, MATLAB, Octave,\\Python, Simulink} & Embedded & \cite{Verschueren2018, Verschueren2019} \\
NMPC tools & MPC, MHE & No & Octave, Python & MPC, MHe & \cite{risbeck2015nonlinear} \\
MPT3 Toolbox & LMPC & No & MATLAB & Comp. Geometry & \cite{MPT3} \\
YALMIP & MPC, MHE & No & MATLAB & Optimization & \cite{YALMIP} \\
\hline
\end{tabular}
\end{threeparttable}
\end{table}
\section{HILO-MPC modules}
\label{sec:hilo-mpc}
The toolbox can be divided into four modules: model, machine learning, control, and observer. Each of these contains a set of tools, each of which solves a different problem. The model module is used to generate dynamic models. The control module uses the model module to generate, e.g., optimal control and predictive control problems. The observer module also uses the model module to generate state and parameter estimators. Finally, the machine learning model is responsible for defining and training machine learning models that can be used in any of the previous modules. In the next sections, we will briefly describe the different modules in more detail \footnote{Note that since HILO-MPC is currently in development, new tools will be added, and current tools could be modified. Refer always to the documentation to get an update on the latest version of HILO-MPC.}.
\subsection{Modeling module}
\label{sec:modeling_module}
At the core of HILO-MPC sits the modeling module. It is a high-level interface to generate representations of dynamical systems that can be used for model-based controllers and observers, like MPC or MHE, or inside a control loop to simulate the behavior of a plant. The system properties supported by HILO-MPC are reported in column ``Models'' of Tab.~\ref{tab:problems-hilo}.
A general time-variant continuous-time nonlinear system can be modeled using the following DAEs
\begin{align}
\dot{x}(t)&=f(t,x(t),z(t),u(t),p),\nonumber\\
0&=q(t,x(t),z(t),u(t),p),\label{eq:nonlin_model}\\
y(t)&=h(t,x(t),z(t),u(t),p)\nonumber,
\end{align}
where $x(t)\in\mathbb{R}^{n_x}$ is the differential state vector, $z(t)\in\mathbb{R}^{n_z}$ the vector of algebraic states, $u(t)\in\mathbb{R}^{n_u}$ is the input vector and $p\in\mathbb{R}^{n_p}$ is a vector of parameters. The function $f\colon\mathbb{R}\times\mathbb{R}^{n_x}\times\mathbb{R}^{n_z}\times\mathbb{R}^{n_u}\times\mathbb{R}^{n_p}\rightarrow\mathbb{R}^{n_x}$ represents the ODEs of the model and the function $q\colon\mathbb{R}\times\mathbb{R}^{n_x}\times\mathbb{R}^{n_z}\times\mathbb{R}^{n_u}\times\mathbb{R}^{n_p}\rightarrow\mathbb{R}^{n_z}$ describes the algebraic equations forming a semi-explicit DAE system overall \footnote{ Note that HILO-MPC only supports semi-explicit DAE systems of index 1. The index 1 indicates that the corresponding DAE system can be transformed into a pure ODE system by differentiating the algebraic equations once.}. The function $h\colon\mathbb{R}\times\mathbb{R}^{n_x}\times\mathbb{R}^{n_z}\times\mathbb{R}^{n_u}\times\mathbb{R}^{n_p}\rightarrow\mathbb{R}^{n_y}$ describes the measurement equations mapping to some measurable quantity of the system. If these measurement equations are not defined during the setup of the model, the controllers and observers using the model assume that all states are measured.
Furthermore, HILO-MPC also supports the modeling of discrete-time systems. When necessary, continuous-time models can be easily discretized using the \mintinline{python}{model.discretize(...)} method. Available discretization methods are explicit Runge-Kutta methods up to the 4th order and implicit collocation schemes \citep{Biegler2010}. For the explicit Runge-Kutta methods, a series of Butcher tableaus is available, while the implicit collocation schemes only support Gauss-Legendre polynomials and Gauss-Radau polynomials for the calculation of the collocation points.
For some applications, it might be necessary to use a linear model. If only a nonlinear model is available, this can be easily linearized with respect to an equilibrium point using the \mintinline{python}{model.linearize(...)} method. In HILO-MPC, it is also possible to directly create linear models by supplying the required matrices during model setup.
As an example, we show how to set up a simple bike model, which will also be used as an example model in the following sections
\begin{align*}
\dot{p}_\text{x} &= v \cos(\phi(t) + \beta), \\
\dot{p}_\text{y} &= v \sin(\phi(t) + \beta), \\
\dot{v} &= a, \\
\dot{\phi} &= v/l_\text{r} \sin(\beta), \\
\beta & = \arctan\left(\frac{l_r}{l_\text{r} + l_\text{f}}\tan(\delta)\right).
\end{align*}
Here, $p_{\text{x}}$ and $p_{\text{y}}$ are the $x$- and $y$-coordinates of the bike's center of mass, $v$ is the norm of the velocity of the center of mass, $\phi$ is the orientation angle of the vehicle with respect to the $x$-coordinate. The inputs are the acceleration of the center of mass $a$ and the steering angle $\delta$. The parameter $l_\text{r}$ is the distance between the center of mass and the rear wheel, and $l_\text{f}$ is the distance between the center of mass and front wheel. One way to set up the model looks as follows
\begin{minted}[bgcolor=bg]{python}
from hilo_mpc import Model
# Initialize empty model
model = Model(name='Bike')
# Define model equations
equations = """
# ODEs
dpx/dt = v(t)*cos(phi(t) + beta)
dpy/dt = v(t)*sin(phi(t) + beta)
dv/dt = a(k)
dphi/dt = v(t)/lr*sin(beta)
# Algebraic equations
beta = arctan(lr/(lr + lf)*tan(delta(k)))
"""
model.set_equations(equations=equations)
# Sampling time in seconds
dt = 0.01
model.setup(dt=dt)
model.set_initial_conditions(x0=[0,0,0,0])
\end{minted}
where the whole model equations are supplied in form of a string. HILO-MPC is able to parse the most common functions and operations in this format. The model can be also defined using the model variables. This reads
\begin{minted}[bgcolor=bg]{python}
# Initialize empty model
model = Model(name='Bike')
# Set the variables
inputs = model.set_inputs(['a', 'delta'])
states = model.set_states(['px','py','v','phi']
# Unwrap states
px = states[0]
py = states[1]
v = states[2]
phi = states[3]
# Unwrap states
a = inputs[0]
delta = inputs[1]
# Parameters
lr = 1.4 # [m]
lf = 1.8 # [m]
beta = ca.arctan(lr / (lr + lf) * ca.tan(delta))
# ODE
dpx = v * ca.cos(phi + beta)
dpy = v * ca.sin(phi + beta)
dv = a
dphi = v / lr * ca.sin(beta)
# Pass the differential equations to the model
model.set_differential_equations([dpx, dpy, dv, dphi])
# Sampling time in seconds
dt = 0.01
# Setup the model
model.setup(dt=dt)
# Pass the initial conditions
model.set_initial_conditions(x0=[0,0,0,0])
\end{minted}
Units, labels and descriptions of all variables can also be set for convenience and for use in the plotting functionality. For a more in-depth description of available functionalities, refer to the documentation {\color{red}[REFERENCE TO LINK HERE]}.
Once the initial conditions are set, the model can be used for simulation by running \mintinline{python}{model.simulate(u=u,p=p)}. Note that, for every integration interval $dt$ the input is kept constant. This is a common assumption that reduces the dimensionality of the optimal control problems we will define in the next sections.
Finally, HILO-MPC makes it easy to store, access and plot solutions generated by simulations. Every time the method \mintinline{python}{model.simulate(...)} is run, the solution is saved in the \mintinline{python}{model.solution} object. This can be accessed similarly to a Python dictionary. To plot the solution, i.e., the evolution of states, inputs and parameters with time, the user can use the method \mintinline{python}{model.solution.plot()}.
\subsection{Machine Learning module}
This module is responsible for the definition and training of machine learning models. Machine learning models are data-driven models that are used to map selected inputs $v \in \mathbb{R}^{n_v}$, also called features, to outputs $l \in \mathbb{R}^{n_l}$, also called labels. The current version ({1.0}) contains two approaches: artificial feedforward fully connected neural networks and Gaussian processes.
\subsubsection{Artificial Neural Network}
\begin{figure}[bt]
\centering
\input{figures/ann.tex}
\caption{Simple feedforward neural network with four features (inputs), one hidden layer and one label (output)}
\label{fig:ann}
\end{figure}
Inspired by the neural circuits in brains \citep{McCulloch1943}, an artificial neural network (ANN) can be represented as a collection of interconnected nodes, the so-called neurons, that can be organized into layers. Here we focus on feedforward fully connected ANNs (Fig.~\ref{fig:ann}). The neurons are just a graphic representation of mathematical operations. In a layer, the operations are linear regressions propagated through some functions called \emph{activation functions}
\begin{equation}
z_i=\sigma_i\left(W_iz_{i-1}+b_i\right),
\end{equation}
where $z_i\in\mathbb{R}^{n_{z_i}}$ is the output vector of layer $i$ with the features $v$ being the input to the first layer, i.e., $x=z_0$, and the labels $l$ being the output of the last layer. The function $\sigma_i\colon\mathbb{R}^{n_{z_i}}\rightarrow\mathbb{R}^{n_{z_i}}$ is the activation function. To learn nonlinear maps it is fundamental that activation functions of the hidden layers, i.e., the layers between input and output layer, contain some nonlinearities. Common nonlinear activation functions are the sigmoid function, hyperbolic tangent or the ReLU function \citep{Ding2018}. The training of the ANN is achieved by iteratively adapting the weight matrices $W_i\in\mathbb{R}^{n_{z_i}\times n_{z_{i-1}}}$ and the bias vectors $b_i\in\mathbb{R}^{n_{z_i}}$, so that a user-defined loss function is minimized. For regression problems, one of the most commonly used loss functions is the mean squared error (MSE)
\begin{equation*}
\mathrm{MSE}=\dfrac{1}{n_l}\sum_{i=1}^{n_l}(l_i-\hat{l}_i)^2,
\end{equation*}
where $\hat{l}_i$ are the values of the labels predicted by the ANN. Other loss functions supported by HILO-MPC include the mean absolute error (MAE) or the Huber loss. Optimization algorithms like gradient descent or stochastic gradient descent are used to minimize the loss. A number of minimization algorithms can be interfaced in HILO-MPC, e.g., the Adam algorithm, a modification of stochastic gradient descent. These minimization algorithms return a gradient vector of the loss with respect to the output $l$ of the ANN. By applying the backpropagation algorithm \citep{Rumelhart1986} this gradient vector can then be used to update the weight matrices $W_i$ and the bias vectors $b_i$ throughout the ANN. The training is stopped when there is no significant change in the loss over a defined number of iterations or if the number of maximum iterations is reached. At the moment, HILO-MPC supports feedforward fully connected ANNs.
In the following example, we show how to define an ANN consisting of two layers with 10 neurons each, using a sigmoid function as activation function
\begin{minted}[bgcolor=bg]{python}
from hilo_mpc import ANN, Dense
# Initialize NN
ann = ANN(features, labels)
ann.add_layers(Dense(10, activation='sigmoid'))
ann.add_layers(Dense(10, activation='sigmoid'))
# Add training data set
ann.add_data_set(df)
# Set up NN
ann.setup()
# Train the NN
batch_size = 64
epochs = 1000
ann.train(batch_size, epochs, validation_split=.2, patience=100, scale_data=True)
\end{minted}
where \mintinline{python}{Dense(...)} is the command for creating a fully connected layer. To replace specific variables of the model with the predictions of the ANN, simply run \mintinline{python}{model.substitute_from(ann)}. Here, the variables to be replaced will be automatically detected and exchanged with the corresponding predictions of the ANN. Alternatively, it is also possible to directly add the ANN to the model via summation or subtraction.
\subsubsection{Gaussian Process}
A Gaussian process (GP) is a probability distribution over a function in which any finite subset of random variables has a joint Gaussian distribution \citep{Rasmussen2006}. An advantage of using a GP is that it naturally measures the output uncertainty by providing the variance of the predictions. For regression problems, GPs can be used to approximate the input-output mapping
\begin{equation*}
l=f(v)+\varepsilon,
\end{equation*}
where the output $l\in\mathbb{R}$ is affected by some zero-mean normally distributed measurement noise $\varepsilon\sim\mathcal{N}(0,\sigma_n^2)$ with variance $\sigma_n^2$.
Starting from a given prior distribution, described by a mean function and a covariance function, the GP is trained on observed data. Both, the prior mean function and the prior covariance function, in general depend on the so-called hyperparameters. During the training step an optimization problem is solved, where these hyperparameters act as decision variables, i.e. the hyperparameters are chosen such that the resulting GP can be used to infer the input-output mapping given available training data. Usually the prior mean function is assumed to be zero \citep{Kocijan2003}, but other user-defined prior mean functions are also supported by HILO-MPC. The covariance function, or kernel, is often assumed to be a smooth function that is infinitely differentiable. A number of covariance functions that meet this criterion are provided in HILO-MPC, like e.g. the stationary squared exponential and rational quadratic kernels. These kernels can be combined with each other through summation or multiplication. The objective of the optimization problem that underlies the training step is usually to maximize the log marginal likelihood \citep{Rasmussen2006}
\begin{equation*}
\log\left(p(l|V)\right)=-\dfrac{1}{2}l^{\mkern-1.5mu\mathsf{T}}\left(K+\sigma_n^2I\right)^{-1}l-\dfrac{1}{2}\log\left(\left|K+\sigma_n^2I\right|\right)-\dfrac{n}{2}\log\left(2\pi\right),
\end{equation*}
where $V\in\mathbb{R}^{n_v\times n_d}$ describes the matrix containing the $n_d$ training inputs, $K\in\mathbb{R}^{n_d\times n_d}$ is the covariance matrix resulting from the used kernel and the supplied training data and $I$ is the identity matrix of corresponding size. The element $(i,j)$ of $K$ is $k(v_i,v_j)$, where $k\colon\mathbb{R}^{n_v}\times\mathbb{R}^{n_v}\rightarrow\mathbb{R}$ is the kernel. The predictive mean $\bar{f}$ and variance $\sigma^2$ of the trained GP can be obtained for test data $w$ as follows
\begin{align*}
\bar{f}(w)&=\tilde{k}^{\mkern-1.5mu\mathsf{T}}(K+\sigma_n^2I)^{-1}l,\\
\sigma^2(w)&=k(w,w)-\tilde{k}^{\mkern-1.5mu\mathsf{T}}(K+\sigma_n^2I)^{-1}\tilde{k},
\end{align*}
where the $i$-th element of $\tilde{k}\in\mathbb{R}^{n_d}$ is $k(v_i,w)$. Here is an example of how to set up a GP in HILO-MPC
\begin{minted}[bgcolor=bg]{python}
from hilo_mpc import GPR, SquaredExponential
# Initialize kernel
kernel = SquaredExponential(variance=0.002,
bounds={'length_scales': (0.0001, 10), 'variance': (0.0001 10)})
# Initialize GP
gpr = GPR(features, labels, prior_mean=0, kernel=kernel)
# Add training data set
gpr.set_training_data(train_in, train_out)
# Set up GP
gpr.setup()
# Fit the GP
gpr.fit_model()
\end{minted}
Note how bounds on the hyperparamters can be given. Finally, if necessary, the GP can be integrated in the model the same way as the ANN.
\subsection{Control module}
The control module contains the MPC and a proportional–integral–derivative (PID) controller. Here we describe only the MPC for brevity.
\subsubsection{Model Predictive Control}
Model predictive control solves repeatedly a finite-horizon optimal control problem (OCP). We here consider a sampled-data nonlinear MPC problem with continuous time-varying DAE system. A similar formulation can be obtained for discrete-time nonlinear system. At the sampling time $t_0$ the optimal control problem to solve reads
\begin{mini!}
{u(\cdot)}{\int^{t_0+T}_{t_0} l(t,x(t),z(t),u(t),p) \mathrm{d}t + e(t_0+T,x(t_0+T),z(t_0+T),p),\label{subeq:obj1}}{\label{eq:problem-mpc}}{}
\addConstraint{\dot{x}(t)}{=f(t,x(t),z(t),u(t),p),\quad\label{subeq:model1}}{x(t_0)=\hat{x}(t_0)}
\addConstraint{0}{=q(t,x(t),z(t),u(t),p)}{}
\addConstraint{0}{\geq g(t,x(t),z(t),u(t),p)\label{subeq:path-con}}{}
\addConstraint{0}{\geq g_T(t_0+T,x(t_0+T),z(t_0+T),p)\label{subeq:terminal-con}}{}
\addConstraint{y(t)}{=h(t, x(t),z(t),u(t),p)\label{subeq:meas-eq}}{}
\addConstraint{u(t)}{=u(t_0 + T_c)\,\ \text{for} \,\ t\geq t_0 + T_c}{}
\addConstraint{}{\text{for} \quad t \in [t_0, t_0+T] \subset \mathbb{R}.}{}
\end{mini!}
\begin{figure}[bt]
\centering
\input{figures/scheme1}
\caption{Schematic representation of machine learning supported open-loop optimization or model predictive control. (ML: machine learning)}
\label{fig:scheme}
\end{figure}
Here, $u(\cdot)$ is the optimal input function, $l:\mathbb{R}\times\mathbb{R}^{n_x}\times\mathbb{R}^{n_z}\times\mathbb{R}^{n_u} \times \mathbb{R}^{n_p} \rightarrow \mathbb{R}$ is the stage cost, $e:\mathbb{R}\times\mathbb{R}^{n_x}\times\mathbb{R}^{n_z} \times \mathbb{R}^{n_p} \rightarrow \mathbb{R}$ the terminal cost, $T$ is the prediction horizon and $T_c$ the control horizon. $\hat{x}(t_0)$ is the measured or estimated state vector at the current sampling time $t_0$ and $t$ is the time variable. The equation $g:\mathbb{R}\times\mathbb{R}^{n_x}\times\mathbb{R}^{n_z}\times\mathbb{R}^{n_u} \times \mathbb{R}^{n_p} \rightarrow \mathbb{R}^{n_g}$ are the path constraints and $g_T:\mathbb{R}\times\mathbb{R}^{n_x}\times\mathbb{R}^{n_z} \times \mathbb{R}^{n_p} \rightarrow \mathbb{R}^{n_{g,\text{T}}}$ are terminal constraints.
\begin{remark}
Any of the equations in \eqref{eq:problem-mpc} could be at least partially learned from data (see Fig.~ \ref{fig:scheme}). For example in \cite{matschek2019learning} the output function \label{subeq:meas-eq} is learned, in \cite{rosolia2017autonomous,rosolia2017learning,brunner2017repetitive} the terminal constraints \eqref{subeq:terminal-con} and in \cite{bujarbaruah2020learning,Leopoldo2017} the path constraints. Also the objective function can be learned, see e.g. \cite{tamar2017learning,BECKENBACH201860,Bradford2018}. HILO-MPC allows to use machine learning models in any of these components.
\end{remark}
Problem \eqref{eq:problem-mpc} cannot be solved directly since $u(\cdot)$ and the constraints are infinite dimensional. HILO-MPC uses \emph{direct approaches} to transform the problem to an equivalent finite-dimensional optimization problem. These approaches parametrize the input with a finite number of parameters, for example, using piece-wise constant inputs, and force the nonlinear constraints only on a finite number of points. This reads as follows
\begin{mini!}
{\mathbf{u}}{\sum_{i = k}^{N} \int_{t_i}^{t_{i} + \Delta t} l(t,x(t),z(t),u_i,p) \mathrm{d}t + e(t_k+T,x(t_k+T),z(t_k+T),p),}{\label{eq:disc-mpc}}{}
\addConstraint{x(t_{i+1})}{= x(t_{i}) + \int_{t_i}^{t_{i} + \Delta t} f(t,x(t),z(t),u_i,p) \mathrm{d}t}{}
\addConstraint{x(t_k)}{=\hat{x}(t_k)}{}
\addConstraint{0}{=q(t_i,x(t_i),z(t_i),u_i,p)}{}
\addConstraint{0}{\geq g(t_i,x(t_i),z(t_i),u_i,p)}{}
\addConstraint{0}{\geq g_T(t_k+T,x(t_k+T),z(t_k+T),p)}{}
\addConstraint{y(t_i)}{=h(t_i, x(t_i),z(t_i),u_i,p)}{}
\addConstraint{u_i}{=u_{N_c},\,\ \text{for} \,\ i\geq N_c}{}
\addConstraint{}{\text{for} \,\ i \in [k, N] \subset \mathbb{N}_0,}{}
\end{mini!}
where $\Delta t$ is the sampling time, $N_c = \text{ceil}(T_c/\Delta t)$ is the control horizon and $\mathbf{u}=[u_k,...,u_{N_c}]$ is sequence of piece-wise constant inputs applied to the plant, i.e. $u(t) = u_i, \,\ \forall t \in [t_i,t_i + \Delta t) \subset \mathbb{R}$.
HILO-MPC implements the multiple shooting approach \citep{BOCK19841603}. The initegration of the system model in between the shooting points can be done with Runge-Kutta methods of various orders, orthogonal collocation \citep{Oh1977} or the dynamics can be integrated with CVODES or IDAS solvers \cite{Hindmarsh2005}. The default method is orthogonal collocation and the user can select any other method if needed. Piece-wise constant input is assumed at every control interval.
Hence, the real formulation of Problem \eqref{eq:problem-mpc} depends on the which method is used for the discretization. Here, we do not go into the details of the formulation of the approximated problem. The reader is invited to refer to the previously cited papers for more details on the formulation of the OCP for the single cases.
\begin{remark}
HILO-MPC can also optimize the sampling intervals. This will add to Problem \eqref{eq:disc-mpc} the vector $[\Delta t_k,..., \Delta t_{N}]$ as optimization variable. This allows the solution of minimum time problems, and optimal sampling time problems.
\end{remark}
The stage cost $l(\cdot)$ arrival cost $e(\cdot)$ depend on the type of MPC implemented. In the next sections we will go through the different MPC problems HILO-MPC can solve. For simplicity of explanation, we show only quadratic cost functions. Note however that HILO-MPC can use any type of continuous cost functions.
\subsubsection{MPC for set-point tracking}
In set-point tracking MPC the variables $x,u$ and $y$ need to track a given reference.
\begin{align*}
l(x(t),u(t),y(t)) &= \Vert x(t) - x_r\Vert^2_Q + \Vert u(t) - u_r\Vert^2_R + \Vert y(t) - y_r\Vert^2_P, \\
e(x(t_0+T),y(t_0+T)) &= \Vert x(t_0+T) - x_r\Vert^2_{Q_{\text{T}}} + \Vert y(t_0+T) - y_r\Vert^2_{P_{\text{T}}},
\end{align*}
where $x_r,u_r,y_r$ are fixed references, and $Q \in \mathbb{R}^{n_x \times n_x},\,\ R \in \mathbb{R}^{n_u \times n_u} \,\ P \in \mathbb{R}^{n_y \times n_y} \,\ P_{\text{T}} \in \mathbb{R}^{n_y \times n_y}$ and $ Q_{\text{T}} \in \mathbb{R}^{n_x \times n_x}$ are weighting matrices.
For example, once we have defined the bike model like in section \ref{sec:modeling_module}. A possible set-point tracking MPC that tracks a reference speed of $v_{\text{r}} = 2 \,\ \mathrm{m}/\mathrm{s}$, with a prediction horizon of 20 steps, can be defined as follows
\begin{minted}[bgcolor=bg]{python}
from hilo_mpc import NMPC
nmpc = NMPC(model)
nmpc.horizon = 20
nmpc.quad_stage_cost.add_states(names=['v'], ref=2, weights=10)
nmpc.quad_term_cost.add_states(names=['v'], ref=2, weights=10)
nmpc.setup()
\end{minted}
After the MPC has been setup, it can be used in a control loop as follows
\begin{minted}[bgcolor=bg]{python}
# Choose number of steps
n_steps = 100
# Begin simulation loop
for step in range(n_steps):
# find optimal input
u = nmpc.optimize(x0)
# Simulate the plant
model.simulate(u=u)
# Get the current states
x0 = sol['xf']
\end{minted}
As for the the other HILO-MPC objects, the solution of the MPC (e.g., input and state sequences) are stored in the \mintinline{python}{mpc.solution()} object.
\subsubsection{MPC for trajectory-tracking}
\label{sec:mpc-traj-tracking}
For trajectory tracking problems the variables need to track a time-varying reference. For quadratic cost function, this looks like the following
\begin{align*}
\label{eq:mpc-traj-tracking}
l(x(t),u(t),y(t)) &= \Vert x(t) - x_r(t)\Vert^2_Q + \Vert u(t) - u_r(t)\Vert^2_R + \Vert y(t) - y_r(t_i)\Vert^2_P, \\
e(x(t_0+T),y(t_0+T)) &= \Vert x(t_0+T) - x_r(t_0+T) \Vert^2_{Q_{\text{T}}} + \Vert y(t_0+T) - y_r(t_0+T) \Vert^2_{P_{\text{T}}},
\end{align*}
where $x_r : \mathbb{R} \rightarrow \mathbb{R}^{n_x},u_r: \mathbb{R} \rightarrow \mathbb{R}^{n_u}, y_r: \mathbb{R} \rightarrow \mathbb{R}^{n_y}$ are time-varying references. For example, a trajectory tracking problem for the racing car problem can be defined as follows
\begin{minted}[bgcolor=bg]{python}
from hilo_mpc import NMPC
import casadi as ca
nmpc = NMPC(model)
nmpc.horizon = 20
t = nmpc.create_time_variable()
traj_x = 30 - 14 * ca.cos(t)
traj_y = 30 - 16 * ca.sin(t)
nmpc.quad_stage_cost.add_states(names=['px', 'py'], ref=[traj_x, traj_y],
weights=[10, 10], trajectory_tracking=True)
nmpc.quad_term_cost.add_states(names=['px', 'py'], ref=[traj_x, traj_y],
weights=[10, 10], trajectory_tracking=True)
nmpc.setup()
\end{minted}
\subsubsection{MPC for path-following}
While in trajectory tracking MPC, the both \emph{value of the reference and time are fixed simultaneously} in path-following MPC, the controller has more degrees of freedom in choosing \emph{when} to be on the path \citep{Matschek2019,faulwasser2016}. In this case the model is augmented with a \emph{virtual path state} as follows
\begin{mini!}
{u(\cdot),u_{\text{pf}}(\cdot)}{\int_{t_0}^{t_0+T} l(t,x(t),z(t),u(t),p) \mathrm{d}t + e(t_0+T,x(t_0+T),z(t_0+T),p),}{}{}
\addConstraint{\dot{x}(t)}{=f(t, x(t),z(t),u(t),p),\quad}{x(t_0)=\hat{x}(t_0)}
\addConstraint{0}{=q(t, x(t),z(t),u(t),p)}{}
\addConstraint{\dot{\theta}}{=u_{\text{pf}},\quad}{\theta(t_0)=0}
\addConstraint{0}{\geq g(t,x(t),z(t),u(t),p, \epsilon)}{}
\addConstraint{0}{\geq g_T(t_0+T,x(t_0+T),z(t_0+T),p)}{}
\addConstraint{y(t)}{=h(t, x(t), z(t), u(t), p)}{}
\addConstraint{}{\text{for} \,\ t \in [t_0, t_0+T] \subset \mathbb{R},}{}
\end{mini!}
where $\theta \in \mathbb{R}^{n_\theta}$ is a virtual path state vector and $u_{\text{pf}} \in \mathbb{R}^{n_{u,\theta}}$ is the virtual input that can controller can choose. Hence, the objective function looks like
\begin{align*}
l(x(t),u(t),y(t)) &= \Vert x(t) - x_r(\theta(t))\Vert^2_Q + \Vert u(t) - u_r(\theta(t))\Vert^2_R + \Vert y(t) - y_r(\theta(t))\Vert^2_P, \\
e(x(t_0+T),y(t_0+T)) &= \Vert x(t_0+T) - x_r(\theta(t_0+T)) \Vert^2_{Q_{\text{T}}} + \Vert y(t_0+T) - y_r(\theta(t_0+T)) \Vert^2_{P_{\text{T}}}.
\end{align*}
Usually, to force the controller to move into only one direction along the path usually a lower bound on $u_{\text{pf}}$ is added, i.e. $u_{\text{pf}}>u_{\text{pf,min}}$ with $u_{\text{pf,min}} \in \mathbb{R}^{n_{u,\theta}}_{+}$.
HILO-MPC also allows to track a constant $u_{\text{pf}}$, so that the MPC tries to maintain a constant \emph{speed} of the virtual state
\begin{align*}
l(x(t),u(t),y(t)) &= \Vert x(t) - x_r(\theta(t))\Vert^2_Q + \Vert u(t_i) - u_r(\theta(t))\Vert^2_R + \Vert y(t) - y_r(\theta)\Vert^2_P + \Vert u_{\text{pf}} - u_{\text{pf,ref}}\Vert_{R_{\text{pf}}}^2, \\
e(x(t_0+T),y(t_0+T)) &= \Vert x(t_0+T) - x_r(\theta(t_0+T))\Vert^2_Q + \Vert y(t_0+T) - y_r(\theta(t_0+T))\Vert^2_P.
\end{align*}
Contrary to the other available toolboxes, path-following MPC problems are automatically generated. The user needs just to activate the path following mode for the desired variables. Here, we show how a path-following problem can be defined using HILO-MPC
\begin{minted}[bgcolor=bg]{python}
from hilo_mpc import NMPC
import casadi as ca
nmpc = NMPC(model)
nmpc.horizon = 20
theta = nmpc.create_path_variable()
traj_x = 30 - 14 * ca.cos(theta)
traj_y = 30 - 16 * ca.sin(theta)
nmpc.quad_stage_cost.add_states(names=['px', 'py'], ref=[traj_x, traj_y],
weights=[10, 10], path_following=True)
nmpc.quad_term_cost.add_states(names=['px', 'py'], ref=[traj_x, traj_y],
weights=[10, 10], path_following=True)
nmpc.setup()
\end{minted}
Note that HILO-MPC allows to mix the previous problem formulations together, with minimum effort: e.g. the user can track some constant references for some of the variables, while tracking a path with other variables.
\subsubsection{Soft constraints}
Soft constraints can be easily implemented. When soft constraints are selected, HILO-MPC automatically adds the slack variables $\epsilon_{\text{p}} \in \mathbb{R}^{n_g}$ and $\epsilon_{\text{T}} \in \mathbb{R}^{n_{g,\text{T}}} $ to the path and terminal constraints respectively as follows
\begin{mini!}
{u(\cdot), \epsilon_{\text{T}}, \epsilon_{\text{p}}}{\int_{t_0}^{t_0+T} l(\cdot) \mathrm{d}t + e(\cdot) + \Vert \epsilon_{\text{s}} \Vert^2_{E_{\text{p}}} + \Vert \epsilon_{\text{T}} \Vert^2_{E_{\text{T}}},}{\label{eq:problem-mpc-soft}}{}
\addConstraint{\dot{x}(t)}{=f(t,x(t),z(t),u(t), p),\quad}{x(0)=\hat{x}_0}
\addConstraint{0}{=q(t, x(t),z(t),u(t), p)}{}
\addConstraint{\epsilon_{\text{p}}}{\geq g(t,x(t),z(t),u(t), p)}{}
\addConstraint{\epsilon_{\text{T}}}{\geq g_T(t_0+T,x(t_0+T),z(t_0+T),p)}{}
\addConstraint{y(t)}{=h(t, x(t),z(t), u(t),p)}{}
\addConstraint{0}{\leq\epsilon_{\text{p}} \leq \epsilon_{\text{p,max}}, \quad 0 \leq \epsilon_{\text{T}} \leq \epsilon_{\text{T, max}}}{}
\addConstraint{}{\text{for} \,\ t \in [t_0, t_0+T] \subset \mathbb{R},}{}
\end{mini!}
where and $E_{\text{p}}\in \mathbb{R}^{n_g \times n_{g}}$ and $E_{\text{T}}\in \mathbb{R}^{n_{g,\text{T}} \times n_{g,\text{T}}}$ are weighting matrices that limit the increase of the the slack variables and can be chosen by the user, and $\epsilon_{\text{p,max}}$ and $\epsilon_{\text{T,max}}$ are the maximum constraint violations of path and terminal constraints respectively.
\subsection{Observer module}
The observer model contains the Kalman filters, moving horizon estimators and particle filter.
\subsubsection{Moving Horizon Estimation}
The Moving Horizon Estimator (MHE) is an observer based on the solution of an optimization problem similar to the MPC problem \citep{rawlings2017model}. We consider a sample-data problem with equidistant sampling times as it is more common in practice. At these sampling times, measurements are taken. Since measurements are discontinuous in MHE the objective function is usually in discrete form. For simplicity we indicate with $(\cdot)_{i|k}$ the variable at time $t_i$, i.e. for the MHE problem solved at time $t_k$ we have
\begin{mini!}
{x_{k-N|k},p_k, w_{(\cdot)|k},z_{(\cdot)|k}}{\left\Vert \begin{bmatrix} x_{k-\bar{N}|k} - \hat{x}_{k-N|k} \\ p_{k} - \hat{p}_{k} \end{bmatrix} \right\Vert^2_{P_k} + \sum^k_{i=k-N} \Vert \hat{y}_i - y_{i|k} \Vert^2_R + \Vert w_{i|k}\Vert^2_W,}{\label{eq:problem-mhe}}{}
\addConstraint{x_{i+1|k}}{=x_{i|k} + \int_{t_i}^{t_i + \Delta t} \left( f(t,x(t),z(t),\hat{u}(t), p_k) + w(t) \right) \mathrm{d}t\label{subeq:mhe-model1}}{}
\addConstraint{y_{i|k}}{=h(t_i,x_{i|k},z_{i|k},\hat{u}_i,p_k) + v_{i|k}}{}
\addConstraint{0}{\geq g(t_i,x_{i|k},u_i)}{}
\addConstraint{}{\text{for} \,\ i \in [k-\bar{N}, k], \,\ k,\bar{N} \in \mathbb{N},}{}
\end{mini!}
where $\bar{N}$ is the horizon length, $\hat{y}_i$ the output measurements and $\hat{u}_i$ the input measurements at time $t_i$. The first term of the objective function is the \emph{arrival cost}, while the second and third weight the measurements and state noise respectively \cite{rawlings2017model,allgower1999nonlinear}. $R \in \mathbb{R}^{n_y \times n_y}$, $W \in \mathbb{R}^{n_x \times n_x}$ and $P+k \in \mathbb{R}^{(n_x+n_p) \times (n_x + n_p)}$ are the weighting matrix for the outputs, state noise and arrival cost. The optimization variables are: the state $x_{k-\bar{N}|k}$, i.e, the state at the beginning of the horizon, the state noise $w_{(\cdot)|k} = \{ w_{i|k}, \,\ \forall i \in [k-\bar{N},k] \}$, the algebraic states (for DAE systems) $z_{(\cdot)|k} = \{ z_{i|k}, \,\ \forall i \in [k-\bar{N},k] \}$, and the system parameters $p_k$. Note that the parameters are considered constant in the horizon, but can be every time the optimization is run, to adapt to the new measurements.
Also in this case we use direct approaches to solve \eqref{eq:problem-mhe} as for the MPC case. For the bike model, an MHE model with $R=W=P_{k}=10 I$ (where $I$ is the identity matrix of appropriate dimensions) can be easily defined as follows
\begin{minted}[bgcolor=bg]{python}
from hilo_mpc import MHE
mhe = MHE(model)
mhe.horizon = 20
mhe.quad_arrival_cost.add_states(weights=[10,10,10,10], guess=x0_est)
mhe.quad_stage_cost.add_measurements(weights=[10,10,10,10])
mhe.quad_stage_cost.add_state_noise(weights=[10,10,10,10]])
mhe.setup()
\end{minted}
After the MHE has been setup, it can be deployed for estimation, for example, as follows
\begin{minted}[bgcolor=bg]{python}
# Set number of steps
n_steps = 100
# Simulation/estimation loop
for step in range(n_steps):
# Simulate plant with input
model.simulate(u=u)
# Get measurements
y_meas = model.solution['y'][:, -2]
# Pass the measurements to the MHE
mhe.add_measurements(y_meas, u_meas=u)
# Estimate states
x0 = mhe.estimate()
\end{minted}
As for the the other HILO-MPC objects, the solution of the MHE are stored in the \mintinline{python}{mhe.solution()} object.
\subsubsection{Kalman filters}
The Kalman filter (KF) is an algorithm that allows for estimation of observable states via available measurement data. In general it consists of two steps, the prediction step, where the estimated states from the previous iteration are propagated through the model dynamics to obtain preliminary values for the states at the current time step, the so-called a priori estimates, and the update step, where these a priori estimates are updated using the measurement data to obtain the a posteriori estimates. The original formulation of the KF was developed for linear discrete-time systems \citep{kalman1960}
\begin{align*}
x_k&=A_kx_{k-1}+B_ku_k+w_k,\\
y_k&=C_kx_k+v_k,
\end{align*}
where $w_k$ is the process noise and $v_k$ is the measurement noise. The process noise $w_k\sim\mathcal{N}(0,Q_k)$ and the measurement noise $v_k\sim\mathcal{N}(0,R_k)$ are assumed to be zero mean normal distributions with the covariance matrices $Q_k$ and $R_k$, respectively. Accordingly, the prediction step and update step of the KF are as follows
\begin{align}
\hat{x}_{k|k-1}&=A_k\hat{x}_{k-1}+B_ku_k,\\
P_{k|k-1}&=A_kP_{k-1}A_k^{\mkern-1.5mu\mathsf{T}}+Q_k,\label{eq:kf_prop}\\
\hat{x}_k&=\hat{x}_{k|k-1}+K(y_k-\hat{y}_k),\\
P_k&=P_{k|k-1}-KP_{y_ky_k}K^{\mkern-1.5mu\mathsf{T}},
\end{align}
with
\begin{align*}
K&=P_{x_ky_k}P_{y_ky_k}^{-1},\\
\hat{y}_k&=C_k\hat{x}_{k|k-1},\\
P_{x_ky_k}&=P_{k|k-1}C_k^{\mkern-1.5mu\mathsf{T}},\\
P_{y_ky_k}&=C_kP_{k|k-1}C_k^{\mkern-1.5mu\mathsf{T}}+R_k,
\end{align*}
where $P_k$ is the error covariance matrix, $P_{x_ky_k}$ is the state-measurement cross-covariance matrix, $P_{y_ky_k}$ is the measurement covariance matrix, $K$ is called the Kalman gain and $\hat{y}_k$ is the predicted output. The notation $\hat{x}_{k|k-1}$ represents the estimated state $\hat{x}$ at time step $k$ given observations up to and including time step $k-1$. The covariance matrix $P_k$ is a measure of the estimation accuracy of the estimated states, i.e. the lower the value the closer the estimated state is to the actual one assuming normally distributed noise.
There are also extensions to the original KF formulation for dealing with nonlinear models. The two most known extensions are the extended Kalman filter (EKF) \citep{Jazwinski1970} and the unscented Kalman filter (UKF) \citep{Wan2000}. The EKF differs from the original formulation in that the propagation of the error covariance matrix and the consecutive update step are not as straightforward. The a priori estimates of the states can be easily obtained by propagating through the nonlinear model dynamics. The propagation of the error covariance matrix is approximated by calculating the Jacobi matrix
\begin{equation*}
A_k=\left.\dfrac{\partial f}{\partial x}\right|_{\hat{x}_{k-1},u_k},
\end{equation*}
and using Eq.~\eqref{eq:kf_prop}. The output matrix $C_k$ is approximated analogously for the update step. A drawback is that the approximation via Jacobi matrices can produce highly unstable Kalman filters \citep{Wan2000}. Another nonlinear extension is the UKF. It approximates the nonlinear problem statistically by a linear one. The UKF makes use of the unscented transform to generate a set of samples, the so-called sigma points, that represent the initial probability distribution. These sigma points are then propagated through the nonlinear system dynamics and afterwards used to approximate the a priori estimates as well as the predicted output and the various covariance matrices. The update step is then the same as for the KF and EKF.
The setup of a Kalman filter in HILO-MPC can be easily set up as
\begin{minted}[bgcolor=bg]{python}
from hilo_mpc import UKF
ukf = UKF(model)
ukf.setup()
\end{minted}
\subsubsection{Particle filter}
The particle filter (PF) \citep{Simon2006} is another algorithm that can be used for estimation. It works similarly to the UKF in that a set of samples, the ``particles'', is propagated through the nonlinear system dynamics. This set of samples is drawn from a probability density function that is assumed to be known. Next, the relative likelihood of each propagated particle is determined
\begin{equation*}
q_i^*=\dfrac{q_i}{\sum_{i=1}^{n_p}q_i},
\end{equation*}
with
\begin{equation*}
q_i\sim\mathrm{pdf}(y_k|\hat{y}_k),
\end{equation*}
where $n_p$ is the number of particles in the set. This relative likelihood acts as a measure of how good a particle matches the measurement $y_k$ and is used in the resampling step to generate the set of a posteriori particles. This resampling step is analogous to the update step of the Kalman filters. The resampling can be achieved by using one of the several available sampling strategies, like e.g. survival of the fittest \citep{Kanazawa1995} or regularization \citep{Musso2001}. Similarly to the UKF, the a posteriori particles can now be used to determine the estimated states via computation of the mean of the distribution. For highly nonlinear system dynamics or non-Gaussian probability density functions the PF usually outperforms the Kalman filter algorithms \citep{Simon2006}.
The setup of a PF in HILO-MPC is similar to that of the Kalman filters
\begin{minted}[bgcolor=bg]{python}
from hilo_mpc import PF
pf = PF(model)
pf.setup()
\end{minted}
\section{Examples}
\label{sec:examples}
In the next section, we present some examples that use HILO-MPC. These examples can also be found in the examples repository, as Python files or Jupyter notebooks. For these reasons, here we do not go into the implementation details but give just a high-level description of the problems and present the results. The reader is invited to read the documentation and the examples codes for details.
\subsection{Learning the dynamics - Race car}
We use HILO-MPC for a realistic example of an autonomous racing minicar. The goal is to follow a complex track using a path-following formulation in presence of disturbances. We use a bike model with nonlinear tire models and drive train model (see Fig.~\ref{fig:bike}) that has been identified and validated in experiments by \cite{Liniger2015}. To this model we add a component describing the effect of lateral drag forces due to, for example, a strong wind. The model is represented by the following system of nonlinear differential equations
\begin{align}
\dot{p}_x &= v_x \cos (\psi) - v_y \sin(\psi), \\
\dot{p}_y &= v_x \sin (\psi) + v_y \cos(\psi), \\
\dot{\psi} &= w, \\
\dot{v}_x &= \frac{1}{m}\left( F_{r,x}-F_{a,x}- (F_{f,y}-F_{a,y}) \sin(\delta) + m v_y \omega \right), \\
\dot{v}_y &= \frac{1}{m}\left( F_{r,y}-F_{a,y}- (F_{f,y}-F_{a,y}) \cos(\delta) - m v_x \omega \right), \\
\dot{\omega} &= \frac{1}{I_z} \left( F_{f,y}l_f \cos(\delta) - F_{r,y}l_r\right),
\end{align}
where $p_x$ and $p_y$ are the coordinates of the center of gravity (CoG), $v_x$ and $v_y$ are longitudinal and later velocities of the CoG. The orientation is denoted by $\psi$ and the yaw rate by $\omega$. The control inputs are the motor duty cycle $d$ and steering angle $\delta$. The two parameters $l_f$ and $l_r$ are the distances from the CoG to the front axle and rear axle respectively, $m$ is the mass of the car and $I_z$ is the inertia. The path to follow is the center of a racing track which has been interpolated using splines.
The tire forces are modeled with a simplified Pacejka Tire Model \citep{bakker1987tyre}
\begin{subequations}
\begin{align}
F_{f,y} &= D_f \sin(C_f \arctan(B_f \alpha_f)), \\
F_{r,y} &= D_r \sin(C_r \arctan(B_r \alpha_r)), \\
F_{r,x} &= (C_{m1}-C_{m2}v_x)d - C_r - C_d v^2_{x}, \\
\alpha_f &= - \arctan\left( \frac{w l_f + v_y}{v_x} \right) + \delta,\\
\alpha_r &= \arctan\left( \frac{w l_r - v_y}{v_x} \right),
\end{align}
\end{subequations}
where $D_f,D_r,B_f,B_r, C_{m1},C_{m2},C_r$ and $C_d$ are parameters. The longitudinal and lateral drag forces are defined respectively as
\begin{align*}
F_{a,x} = 0.5 \,\ c_w \rho A v_x, \\
F_{a,y} = 0.5 \,\ c_w \rho A v_{y,\text{wind}},
\end{align*}
where $c_w$ is the drag coefficient, $\rho$ is the air density, $A$ the effective flow surface and $v_{y,wind}$ is the lateral wind velocity.
The model used by the MPC does not have the drag effect. The goal is learning this effect from data using a Neural Network, and then augment the first principle model with a machine learning component that models the drag effect. After discretization, the hybrid model can be written as:
\begin{equation}
x_{k+1} = f(x_k,u_k) + B^T m(x_k,u_k),
\end{equation}
where $m(x_k,u_k)$ is an NN model and
\begin{equation}
B = \begin{bmatrix}
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{bmatrix}.
\end{equation}
The features of the NN are the $v_{y,\text{wind}}$ and $v_x$, the labels are correction terms on $\phi,v_x,v_y$ and $\omega$. To show the effectiveness of the learning, we compare the results of MPC using the perfect model (i.e. with known drag force effects), the hybrid model (using the NN) and the model without drag forces. Furthermore, the measurements of position, velocity and directions are effected by Gaussian noise, and estimated using an Unscented Kalman Filter.
Figure \ref{fig:race_car} shows the results of the simulation. While the hybrid model has similar results as the perfectly-known model, the model without drag exits the race area after the fifth curve. The complete code can be found in the HILO-MPC repository.
\begin{figure*}[bt]
\captionsetup[subfigure]{justification=centering}
\centering
\begin{subfigure}[b]{.49\linewidth}
\input{figures/bike}
\caption{Model used for the racing car.}
\label{fig:bike}
\end{subfigure}
\begin{subfigure}[b]{.49\linewidth}
\input{figures/racing}
\caption{Results of the path-following problem}
\label{fig:race_car}
\end{subfigure}
\caption{Example of a racing car driving on a racetrack following a given path.}
\end{figure*}
\subsection{Learning a reference - Cooperative robots}
In this example, inspired by \cite{Matschek2021a}, a follower robot has to track the position of a leader robot. The leader moves in a periodic but unknown trajectory. The objective is learning the trajectory of the leader with GPs and pass the learned trajectory to the follower. Hence, in this case the machine learning model enters the reference function (see Fig.~\ref{fig:mobile-robots}). The nonlinear dynamics of the robots is described with the following ODE
\begin{subequations}
\begin{align}
\dot{x}_1 &= u_1 \sin(x_3), \\
\dot{x}_2 &= u_1 \cos(x_3), \\
\dot{x}_3 &= u_2 ,
\end{align}
\end{subequations}
where $x_1$ and $x_2$ are the horizontal and vertical position of the root and $x_3$ its heading angle, $u_1$ is the speed and $u_2$ the turning rate.
The problem we want to solve is of the form \eqref{eq:problem-mpc} with objective function \eqref{eq:mpc-traj-tracking} where $x_r(\cdot)$ is the mean function of a GP trained on the data set collected from the position of the leader.
The trajectory generated by the leader, results from applying the following time-varying forces
\begin{subequations}
\begin{align}
u_1(t) &= 2 + 0.1 \sin(t), \\
u_2(t) &= 0.5 + \sin(t).
\end{align}
\end{subequations}
Figures \ref{fig:mobile-robots-results} show the results of the reference learning and closed-loop simulation.
\begin{figure*}[bt]
\centering
\input{figures/mobile_robots}
\caption{Cooperative robot example. The follower needs to track the trajectory of the leader. The trajectory is learned from data using a GP regressor.}
\label{fig:mobile-robots}
\end{figure*}
\begin{figure*}[bt]
\centering
\input{figures/tot_mobile_robots}
\caption{Left column: learned GP. Right column: states of the follower robot following the learned references.}
\label{fig:mobile-robots-results}
\end{figure*}
\subsection{Learning the controller - String damper system}
Solving a MPC requires the solution of a (nonlinear) optimization problem online. This is possible only for applications where the available computational power and energy are sufficient to guarantee a solution within the sampling times. For embedded applications, with low computational power or that can use only a limited amount of energy (for example battery powered systems) this is often not possible. Hence, methods that provide at least a close-to-optimal solution without solving the optimization approach at every time step are necessary. This can be done using \emph{explicit} MPC approaches. Some of these approaches are based on learning the solution of an MPC offline, i.e., the map $x \mapsto \rho_{\theta}(x)$ that approximate the implicit MPC control law, and then using the learned controller online \citep{Parisini1995, Karg2020, Maddalena2020, Cao2020, Chen2018, Cseko2015, Ponkumar2018}, (cf. Fig. \ref{fig:ml-controller}). In this way, the control action can be found with a simple and fast function evaluation.
In this example, we want to control a mass-spring-damper system using a learned controller (Fig. \ref{fig:ml-controller}). The models is
\begin{subequations}
\begin{align}
& \dot{x}_1 = x_2, \\
& \dot{x}_2 = \frac{1}{m} (u - k x_1 - d x_2) ,
\end{align}
\end{subequations}
where $x_1$ is the vertical position, $x_2$ the vertical velocity, $u$ the vertical force and $k,d$ the system parameters. The equilibrium point is $x=(0,0), u=0$. The objective is to maintain the reference $x_{\text{ref}}=[1,0]$ using a learned MPC. To do so we use the results of just one closed-loop MPC simulation starting from the initial conditions $x(0)=(12,0)$. In total 667 datapoints are collected. We use the data collected to train a NN with three fully-connected layers, with 10 neurons each. The features of the NN are $x_1$ and $x_2$, the labels is the input $u$. We test the learned controller starting from a different initial condition $x(0)=(10,0)$. In Fig.~\ref{fig:ml-controller-results} the simulation results are shown. The learned controller is able to bring the system to the reference as the original controller.
\begin{figure*}[bt]
\centering
\input{figures/machine_learning_controller}
\caption{Scheme of a machine learning controller.}
\label{fig:ml-controller}
\end{figure*}
\begin{figure*}[bt]
\centering
\input{figures/learned_mpc}
\caption{Results of the learned MPC.}
\label{fig:ml-controller-results}
\end{figure*}
\section{Conclusions and outlook}
\label{sec:conclusions}
We introduced HILO-MPC, a toolbox for fast development of predictive control and estimation problems that facilitates the use of machine learning models trained using PyTorch and TensorFlow. We showed three applications of the toolbox, the code of the presented applications is openly available in the HILO-MPC repository.
The toolbox can solve many problems, such as model predictive control, moving horizon estimation, Kalman filters, and particle filters. Its flexibility and simplicity can make it a valuable tool for research and teaching.
We are currently working on a fifth module that allows solving optimization problems efficiently and fast, aiming at embedded applications such as embedded MPC. Also, future versions will allow using other machine learning models such as recurrent neural networks and reinforcement learning. Furthermore, other MPC formulations, such as tube-based MPC \citep{Mayne2005} and multi-mode MPC \citep{bethge2018multi,morabito2019multi} will be available.
\section*{acknowledgements}
We would like to thank Michael Maiworm, Tim Zieger, Johanna Bethge, Maik Pfefferkorn, Rudolph Kok, Sebasti{\'a}n Espinel R{\'i}os and Janine Matschek for providing feedback and examples for HILO-MPC.
\section*{conflict of interest}
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\section*{Supporting Information}
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\printendnotes
\section{Introduction}
Being able to model a dynamical system is fundamental in many engineering applications. Dynamical models can be constructed with first principles laws and/or data.
In the first case, physics laws, such as conservation of energy, matter and thermodynamics laws, are used. A relatively limited amount of data might also be used to fit the parameters of the model. These parameters usually have a physical meaning (e.g., heat transfer coefficients, friction coefficients). If only data is used, we generally talk about machine learning models. Generally, the fitted parameters do not have a physical meaning in these cases. Machine learning has recently obtained much attention since it allows to model complex or not well-understood systems, provided that a sufficiently large amount of data is available. For many decades the scarcity of data, features extraction and computational power have been the main bottlenecks to the widespread use of machine learning models. The situation is rapidly changing thanks to digitization (e.g. internet-of-things and industry 4.0 initiatives), deep learning and cloud computing. Hence, machine learning is finding more and more applications, not only in research but in our everyday life. In recent years, the number of machine learning applications, publications, new methods and theoretical findings has increased rapidly. Furthermore, open-source libraries for machine learning such as TensorFlow \citep{tensorflow2015} and PyTorch \citep{pytorch} offer a low barrier to entry to this technology and allow fast development of machine learning solutions. The combination of these factors boosted the quality and quantity of results in many fields such as medicine \cite{Shen2017}, autonomous driving \cite{Arnold2019}, transportation \cite{Nguyen2018}, internet-of-things \cite{Mohammadi2018}, image recognition \cite{Voulodimos2018}, and many others. The control and optimization community also recognized the usefulness of machine learning for control applications, e.g., for optimal control and model predictive control (MPC), motivating a large number of contributions (cf. \cite{Hewing2020} and references therein).
Machine learning libraries help the control practitioners to build machine learning models, but using these models in a control strategy requires ad-hoc solutions, which are complicated, prone to bugs and not easily transferable to other applications. HILO-MPC is an open-source Python library that aims at providing a way to easily implement machine learning models into a wide variety of control, optimization and estimation problems. HILO-MPC interfaces with PyTorch and TensorFlow to train neural networks and uses in-house code to train Gaussian processes. Once the models are trained, they can be used in a large variety of control or estimation problems. Furthermore, to speed up the development time, HILO-MPC provides a way to define and solve a broad spectrum of predefined optimal control and estimation problems quickly and with minimum effort. Tab.~\ref{tab:problems-hilo} summarizes the different problems that can be solved with version {1.0}{} of HILO-MPC. The philosophy that HILO-MPC follows is ease of use and modularity. Easy of use is achieved by providing an intuitive high-level interface for the problem definition. At the same time, modularity is obtained by ensuring effortless interfaces between the different tools present in the models (see Fig. \ref{fig:scheme}). This allows using the toolbox both for research and for teaching.
Comparing HILO-MPC with other similar toolboxes is challenging since it can solve a wide variety of control and estimation problems. Here, we consider only open-source toolboxes that can solve model predictive control problems for simplicity. The software ACADO \citep{Houska2011a} and ACADOS \citep{Verschueren2018} aim at efficiency and fast solution keeping embedded applications in mind. MPT3 \citep{MPT3} focuses on computational geometry, which is particularly useful for some classes of robust MPC approaches. YALMIP \citep{YALMIP} instead offers a broad spectrum of optimization problems, such as bilevel optimization and mixed-integer programming and some MPC schemes. The toolbox do-MPC \citep{LUCIA201751} implements robust multi-stage MPC and Moving Horizon Estimation. MPC-Tools \citep{risbeck2015nonlinear} offers some interfaces that help build MPC and MHE control problems in CasADi. In Tab.~\ref{tab:comparison-toolboxes} we summarize the main differences between the toolboxes.
Using machine learning models in the previously mentioned toolboxes is not straightforward. HILO-MPC solves this problem by providing wrappers with machine learning libraries and some in-house learning libraries. Furthermore, HILO-MPC focuses on a light weight and intuitive interface that makes it easy to use also for beginners.
The backbone of HILO-MPC is CasADi \citep{Andersson2019}. CasADi is a tool for algorithmic differentiation and optimization, and it is used to build the models, optimization and estimation problems. When TensorFlow or PyTorch are used, HILO-MPC offers an easy way to interface with those libraries and automatically translate the models into CasADi-compatible structures. Hence, the CasADi problem is defined and solved. CasADi has interfaces with a wide range of solvers for linear and quadratic programming (CLP \citep{johnjforrest2020}, qpOASES \citep{Ferreau2014}), nonlinear programming such as IPOPT, \citep{wachter2006implementation}, quadratic mixed-integer programming (CPLEX \citep{cplex2009v12}, GUROBI \citep{gurobi}), for non quadratic nonlinear mixed-integer problems (BONMIN \citep{bonami2007bonmin}, KNITRO \citep{nocedal2006knitro}) and large nonlinear problems WORHP \citep{nikolayzik2010nonlinear}.
Note that giving a comprehensive description of the toolbox is outside the scope of this article. For this, the reader is referred to the documentation. Here, we rather want to give a glimpse of the different problems HILO-MPC can solve and give an idea of the workflow and syntax with some examples.
The rest of this paper is structured as follows: in Section~\ref{sec:hilo-mpc} the different modules of the toolbox are described. Furthermore, we show example code snippets to illustrate the HILO-MPC syntax. In Section~\ref{sec:examples} we apply the toolbox in three examples. Finally, in Section~\ref{sec:conclusions} we conclude and give an outlook on current and future developments of the toolbox.
\begin{figure*}
\centering
\input{figures/overview.tex}
\caption{Overview HILO-MPC}
\label{fig:overview}
\end{figure*}
\begin{table*}[]
\centering
\caption{Current tools implemented in HILO-MPC.}
\label{tab:problems-hilo}
\begin{tabular}{llll}
\toprule
Models & Controllers & Machine Learning & Observers \\
\midrule
\makecell[l]{Linear/nonlinear \\ Time-invariant/variant \\ Continuous and discrete \\ ODEs, DAEs}& \makecell[l]{Nonlinear MPC \\ Trajectory tracking MPC \\ Path-following MPC \\ PID} &
\makecell[l]{Neural networks \\ Gaussian processes} & \makecell[l]{Moving horizon estimation \\ Kalman filter \\ Extended Kalman filter \\ Unscented Kalman filter \\ Particle filter} \\
\bottomrule
\end{tabular}
\end{table*}
\begin{table}[]
\begin{footnotesize}
\caption{Comparison overview on the different open-source MPC software. ML: machine learning, MPC: model predictive control, MHE: moving horizon estimator, LMPC: linear MPC, PFMPC: path-following MPC, KF: Kalman filter, EKF: extended Kalman filter, UKF: unscented Kalman filter.}
\label{tab:comparison-toolboxes}
\begin{tabular}{llclll}
\toprule
& Problems & ML & Interface & Focus & Article\\
\midrule
HILO-MPC & \makecell[l]{MPC, PFMPC \\ MHE,KF,EKF,UKF} & Yes & Python & Machine Learning &\\
do-mpc & \makecell[l]{MPC, MHE \\ Multistage MPC} & No & Python & Multistage MPC & \cite{LUCIA201751}\\
ACADO & MPC & No & C++, Matlab &Embedded & \cite{Houska2011a}\\
ACADOS & MPC, MHE & No & \makecell[l]{C, Matlab, Octave,\\ Python, Simulink} &Embedded & \cite{Verschueren2018, Verschueren2019}\\
NMPC tools & MPC, MHE & No & Octave, Python. &MPC,MHE & \cite{risbeck2015nonlinear}\\
MPT3 Toolbox & LMPC & No & Matlab & Comp. Geometry &\cite{MPT3}\\
YALMIP & MPC, MHE & No & Matlab & Optimization & \cite{YALMIP} \\
\bottomrule
\end{tabular}
\end{footnotesize}
\end{table}
\section{HILO-MPC modules}
\label{sec:hilo-mpc}
The toolbox can be divided into four modules: model, machine learning, control, and observer. Each of these contains a set of tools, each of which solves a different problem. The model module is used to generate dynamic models. The control module uses the model module to generate, e.g., optimal control and predictive control problems. The observer module also uses the model module to generate state and parameter estimators. Finally, the machine learning model is responsible for defining and training machine learning models that can be used in any of the previous modules. In the next sections, we will briefly describe the different modules in more detail \footnote{Note that since HILO-MPC is currently in development, new tools will be added, and current tools could be modified. Refer always to the documentation to get an update on the latest version of HILO-MPC.}.
\subsection{Modeling module}
\label{sec:modeling_module}
At the core of HILO-MPC sits the modeling module. It is a high-level interface to generate representations of dynamical systems that can be used for model-based controllers and observers, like MPC or MHE, or inside a control loop to simulate the behavior of a plant. The system properties supported by HILO-MPC are reported in column ``Models'' of Tab.~\ref{tab:problems-hilo}.
A general time-variant continuous-time nonlinear system can be modeled using the following DAEs
\begin{align}
\dot{x}(t)&=f(t,x(t),z(t),u(t),p),\nonumber\\
0&=q(t,x(t),z(t),u(t),p),\label{eq:nonlin_model}\\
y(t)&=h(t,x(t),z(t),u(t),p)\nonumber,
\end{align}
where $x(t)\in\mathbb{R}^{n_x}$ is the differential state vector, $z(t)\in\mathbb{R}^{n_z}$ the vector of algebraic states, $u(t)\in\mathbb{R}^{n_u}$ is the input vector and $p\in\mathbb{R}^{n_p}$ is a vector of parameters. The function $f\colon\mathbb{R}\times\mathbb{R}^{n_x}\times\mathbb{R}^{n_z}\times\mathbb{R}^{n_u}\times\mathbb{R}^{n_p}\rightarrow\mathbb{R}^{n_x}$ represents the ODEs of the model and the function $q\colon\mathbb{R}\times\mathbb{R}^{n_x}\times\mathbb{R}^{n_z}\times\mathbb{R}^{n_u}\times\mathbb{R}^{n_p}\rightarrow\mathbb{R}^{n_z}$ describes the algebraic equations forming a semi-explicit DAE system overall \footnote{ Note that HILO-MPC only supports semi-explicit DAE systems of index 1. The index 1 indicates that the corresponding DAE system can be transformed into a pure ODE system by differentiating the algebraic equations once.}. The function $h\colon\mathbb{R}\times\mathbb{R}^{n_x}\times\mathbb{R}^{n_z}\times\mathbb{R}^{n_u}\times\mathbb{R}^{n_p}\rightarrow\mathbb{R}^{n_y}$ describes the measurement equations mapping to some measurable quantity of the system. If these measurement equations are not defined during the setup of the model, the controllers and observers using the model assume that all states are measured.
Furthermore, HILO-MPC also supports the modeling of discrete-time systems. When necessary, continuous-time models can be easily discretized using the \mintinline{python}{model.discretize(...)} method. Available discretization methods are explicit Runge-Kutta methods up to the 4th order and implicit collocation schemes \citep{Biegler2010}. For the explicit Runge-Kutta methods, a series of Butcher tableaus is available, while the implicit collocation schemes only support Gauss-Legendre polynomials and Gauss-Radau polynomials for the calculation of the collocation points.
For some applications, it might be necessary to use a linear model. If only a nonlinear model is available, this can be easily linearized with respect to an equilibrium point using the \mintinline{python}{model.linearize(...)} method.
In HILO-MPC, it is also possible to directly create linear models by supplying the required matrices during model setup.
As an example, we show how to set up a simple bike model, which will also be used as an example model in the following sections
\begin{align*}
\dot{p}_\text{x} &= v \cos(\phi(t) + \beta), \\
\dot{p}_\text{y} &= v \sin(\phi(t) + \beta), \\
\dot{v} &= a, \\
\dot{\phi} &= v/l_\text{r} \sin(\beta), \\
\beta & = \arctan\left(\frac{l_r}{l_\text{r} + l_\text{f}}\tan(\delta)\right).
\end{align*}
Here, $p_{\text{x}}$ and $p_{\text{y}}$ are the $x$- and $y$-coordinates of the bike's center of mass, $v$ is the norm of the velocity of the center of mass, $\phi$ is the orientation angle of the vehicle with respect to the $x$-coordinate. The inputs are the acceleration of the center of mass $a$ and the steering angle $\delta$. The parameter $l_\text{r}$ is the distance between the center of mass and the rear wheel, and $l_\text{f}$ is the distance between the center of mass and front wheel. One way to set up the model looks as follows
\begin{minted}[fontsize=\footnotesize, bgcolor=bg]{python}
from hilo_mpc import Model
# Initialize empty model
model = Model(name='Bike')
# Define model equations
equations = """
# ODEs
dpx/dt = v(t)*cos(phi(t) + beta)
dpy/dt = v(t)*sin(phi(t) + beta)
dv/dt = a(k)
dphi/dt = v(t)/lr*sin(beta)
# Algebraic equations
beta = arctan(lr/(lr + lf)*tan(delta(k)))
"""
model.set_equations(equations=equations)
# Sampling time in seconds
dt = 0.01
model.setup(dt=dt)
model.set_initial_conditions(x0=[0,0,0,0])
\end{minted}
where the whole model equations are supplied in form of a string. HILO-MPC is able to parse the most common functions and operations in this format. The model can be also defined using the model variables. This reads
\begin{minted}[fontsize=\footnotesize, bgcolor=bg]{python}
# Initialize empty model
model = Model(name='Bike')
# Set the variables
inputs = model.set_inputs(['a', 'delta'])
states = model.set_states(['px','py','v','phi']
# Unwrap states
px = states[0]
py = states[1]
v = states[2]
phi = states[3]
# Unwrap states
a = inputs[0]
delta = inputs[1]
# Parameters
lr = 1.4 # [m]
lf = 1.8 # [m]
beta = ca.arctan(lr / (lr + lf) * ca.tan(delta))
# ODE
dpx = v * ca.cos(phi + beta)
dpy = v * ca.sin(phi + beta)
dv = a
dphi = v / lr * ca.sin(beta)
# Pass the differential equations to the model
model.set_differential_equations([dpx, dpy, dv, dphi])
# Sampling time in seconds
dt = 0.01
# Setup the model
model.setup(dt=dt)
# Pass the initial conditions
model.set_initial_conditions(x0=[0,0,0,0])
\end{minted}
Units, labels and descriptions of all variables can also be set for convenience and for use in the plotting functionality. For a more in-depth description of available functionalities, refer to the documentation {\color{red}[REFERENCE TO LINK HERE]}.
Once the initial conditions are set, the model can be used for simulation by running \mintinline{python}{model.simulate(u=u,p=p)}. Note that, for every integration interval $dt$ the input is kept constant. This is a common assumption that reduces the dimensionality of the optimal control problems we will define in the next sections.
Finally, HILO-MPC makes it easy to store, access and plot solutions generated by simulations. Every time \mintinline{python}{model.simulate(...)} is run, the solution is saved in the \mintinline{python}{model.solution} object. This can be accessed similarly to a Python dictionary. To plot the solution, i.e., the evolution of states, inputs and parameters with time, the user can use the method \mintinline{python}{model.solution.plot()}.
\subsection{Machine Learning module}
This module is responsible for the definition and training of machine learning models. Machine learning models are data-driven models that are used to map selected inputs $v \in \mathbb{R}^{n_v}$, also called features, to outputs $l \in \mathbb{R}^{n_l}$, also called labels. The current version ({1.0}) contains two approaches: artificial feedforward fully connected neural networks and Gaussian processes.
\subsubsection{Artificial Neural Network}
\begin{figure}
\centering
\input{figures/ann.tex}
\caption{Simple feed-forward neural network with four features (inputs), one hidden layer and one label (output)}
\label{fig:ann}
\end{figure}
Inspired by the neural circuits in brains \citep{McCulloch1943}, an artificial neural network (ANN) can be represented as a collection of interconnected nodes, the so-called neurons, that can be organized into layers. Here we focus on feedforward fully connected ANNs (Fig.~\ref{fig:ann}). The neurons are just a graphic representation of mathematical operations.
In a layer, the operations are linear regressions propagated through some functions called \emph{activation functions}
\begin{equation}
z_i=\sigma_i\left(W_iz_{i-1}+b_i\right),
\end{equation}
where $z_i\in\mathbb{R}^{n_{z_i}}$ is the output vector of layer $i$ with the features $v$ being the input to the first layer, i.e., $x=z_0$, and the labels $l$ being the output of the last layer. The function $\sigma_i\colon\mathbb{R}^{n_{z_i}}\rightarrow\mathbb{R}^{n_{z_i}}$ is the activation function. To learn nonlinear maps it is fundamental that activation functions of the hidden layers, i.e., the layers between input and output layer, contain some nonlinearities. Common nonlinear activation functions are the sigmoid function, hyperbolic tangent or the ReLU function \citep{Ding2018}. The training of the ANN is achieved by iteratively adapting the weight matrices $W_i\in\mathbb{R}^{n_{z_i}\times n_{z_{i-1}}}$ and the bias vectors $b_i\in\mathbb{R}^{n_{z_i}}$, so that a user-defined loss function is minimized. For regression problems, one of the most commonly used loss functions is the mean squared error (MSE)
\begin{equation*}
\mathrm{MSE}=\dfrac{1}{n_l}\sum_{i=1}^{n_l}(l_i-\hat{l}_i)^2,
\end{equation*}
where $\hat{l}_i$ are the values of the labels predicted by the ANN. Other loss functions supported by HILO-MPC include the mean absolute error (MAE) or the Huber loss. Optimization algorithms like gradient descent or stochastic gradient descent are used to minimize the loss. A number of minimization algorithms can be interfaced in HILO-MPC, e.g., the Adam algorithm, a modification of stochastic gradient descent. These minimization algorithms return a gradient vector of the loss with respect to the output $l$ of the ANN. By applying the backpropagation algorithm \citep{Rumelhart1986} this gradient vector can then be used to update the weight matrices $W_i$ and the bias vectors $b_i$ throughout the ANN. The training is stopped when there is no significant change in the loss over a defined number of iterations or if the number of maximum iterations is reached. At the moment, HILO-MPC supports feedforward fully connected ANNs.
In the following example, we show how to define an ANN consisting of two layers with 10 neurons each, using a sigmoid function as activation function
\begin{minted}[fontsize=\footnotesize, bgcolor=bg]{python}
from hilo_mpc import ANN, Dense
# Initialize NN
ann = ANN(features, labels)
ann.add_layers(Dense(10, activation='sigmoid'))
ann.add_layers(Dense(10, activation='sigmoid'))
# Add training data set
ann.add_data_set(df)
# Set up NN
ann.setup()
# Train the NN
batch_size = 64
epochs = 1000
ann.train(batch_size, epochs, validation_split=.2, patience=100, scale_data=True)
\end{minted}
where \mintinline{python}{Dense(...)} is the command for creating a fully connected layer. To replace specific variables of the model with the predictions of the ANN, simply run \mintinline{python}{model.substitute_from(ann)}. Here, the variables to be replaced will be automatically detected and exchanged with the corresponding predictions of the ANN. Alternatively, it is also possible to directly add the ANN to the model via summation or subtraction.
\subsubsection{Gaussian Process}
A Gaussian process (GP) is a probability distribution over a function in which any finite subset of random variables has a joint Gaussian distribution \citep{Rasmussen2006}. An advantage of using a GP is that it naturally measures the output uncertainty by providing the variance of the predictions. For regression problems, GPs can be used to approximate the input-output mapping
\begin{equation*}
l=f(v)+\varepsilon,
\end{equation*}
where the output $l\in\mathbb{R}$ is affected by some zero-mean normally distributed measurement noise $\varepsilon\sim\mathcal{N}(0,\sigma_n^2)$ with variance $\sigma_n^2$.
Starting from a given prior distribution, described by a mean function and a covariance function, the GP is trained on observed data. Both, the prior mean function and the prior covariance function, in general depend on the so-called hyperparameters. During the training step an optimization problem is solved, where these hyperparameters act as decision variables, i.e. the hyperparameters are chosen such that the resulting GP can be used to infer the input-output mapping given available training data. Usually the prior mean function is assumed to be zero \citep{Kocijan2003}, but other user-defined prior mean functions are also supported by HILO-MPC. The covariance function, or kernel, is often assumed to be a smooth function that is infinitely differentiable. A number of covariance functions that meet this criterion are provided in HILO-MPC, like e.g. the stationary squared exponential and rational quadratic kernels. The objective of the optimization problem that underlies the training step is usually to maximize the log marginal likelihood \citep{Rasmussen2006}
\begin{equation*}
\log\left(p(l|V)\right)=-\dfrac{1}{2}l^{\mkern-1.5mu\mathsf{T}}\left(K+\sigma_n^2I\right)^{-1}l-\dfrac{1}{2}\log\left(\left|K+\sigma_n^2I\right|\right)-\dfrac{n}{2}\log\left(2\pi\right),
\end{equation*}
where $V\in\mathbb{R}^{n_v\times n_d}$ describes the matrix containing the $n_d$ training inputs, $K\in\mathbb{R}^{n_d\times n_d}$ is the covariance matrix resulting from the used kernel and the supplied training data and $I$ is the identity matrix of corresponding size. The element $(i,j)$ of $K$ is $k(v_i,v_j)$, where $k\colon\mathbb{R}^{n_v}\times\mathbb{R}^{n_v}\rightarrow\mathbb{R}$ is the kernel. The predictive mean $\bar{f}$ and variance $\sigma^2$ of the trained GP can be obtained for test data $w$ as follows
\begin{align*}
\bar{f}(w)&=\tilde{k}^{\mkern-1.5mu\mathsf{T}}(K+\sigma_n^2I)^{-1}l,\\
\sigma^2(w)&=k(w,w)-\tilde{k}^{\mkern-1.5mu\mathsf{T}}(K+\sigma_n^2I)^{-1}\tilde{k},
\end{align*}
where the $i$-th element of $\tilde{k}\in\mathbb{R}^{n_d}$ is $k(v_i,w)$. The kernels currently available in HILO-MPC: Rational quadratic, exponential sine, gamma exponential, polynomial and squared exponential. These kernels can be combined with eachother through summation or multiplication. Here an example of how to set-up a GP in HILO-MPC:
\begin{minted}[fontsize=\footnotesize, bgcolor=bg]{python}
from hilo_mpc import GPR, SquaredExponential
# Initialize kernel
kernel = SquaredExponential(variance=0.002,
bounds={'length_scales': (0.0001, 10), 'variance': (0.0001 10)})
# Initialize GP
gpr = GPR(features, labels, prior_mean=0, kernel=kernel)
# Add training data set
gpr.set_training_data(train_in, train_out)
# Set up GP
gpr.setup()
# Fit the GP
gpr.fit_model()
\end{minted}
Note how bounds on the hyperparamters can be given. The hyperparameters optimization is carried out using IPOPT. Finally, if necessary, the GP can be integrated in the model the same way as the ANN.
\subsection{Control module}
The control module contains the MPC and a proportional–integral–derivative (PID) controller. Here we describe only the MPC for brevity.
\subsection{Model Predictive Control}
Model Predictive Control solves repeatedly a finite-horizon optimal control problem (OCP). We here consider a sampled-data nonlinear MPC problem with continuous time-varying DAE system. A similar formulation can be obtained for discrete-time nonlinear system. At the sampling time $t_0$ the optimal control problem to solve reads
\begin{mini!}
{u(\cdot)}{\int^{t_0+T}_{t_0} l(t,x(t),z(t),u(t),p) \mathrm{d}t + e(t_0+T,x(t_0+T),z(t_0+T),p),\label{subeq:obj1}}{\label{eq:problem-mpc}}{}
\addConstraint{\dot{x}(t)}{=f(t,x(t),z(t),u(t),p),\quad\label{subeq:model1}}{x(t_0)=\hat{x}(t_0)}
\addConstraint{0}{=q(t,x(t),z(t),u(t),p)}{}
\addConstraint{0}{\geq g(t,x(t),z(t),u(t),p)\label{subeq:path-con}}{}
\addConstraint{0}{\geq g_T(t_0+T,x(t_0+T),z(t_0+T),p)\label{subeq:terminal-con}}{}
\addConstraint{y(t)}{=h(t, x(t),z(t),u(t),p)\label{subeq:meas-eq}}{}
\addConstraint{u(t)}{=u(t_0 + T_c)\,\ \text{for} \,\ t\geq t_0 + T_c}{}
\addConstraint{}{\text{for} \quad t \in [t_0, t_0+T] \subset \mathbb{R}.}{}
\end{mini!}
\begin{figure}
\centering
\input{figures/scheme1}
\caption{Schematic representation of machine learning supported open-loop optimization or model predictive control. (ML: machine learning)}
\label{fig:scheme}
\end{figure}
Here, $u(\cdot)$ is the optimal input function, $l:\mathbb{R}\times\mathbb{R}^{n_x}\times\mathbb{R}^{n_z}\times\mathbb{R}^{n_u} \times \mathbb{R}^{n_p} \rightarrow \mathbb{R}$ is the stage cost, $e:\mathbb{R}\times\mathbb{R}^{n_x}\times\mathbb{R}^{n_z} \times \mathbb{R}^{n_p} \rightarrow \mathbb{R}$ the terminal cost, $T$ is the prediction horizon and $T_c$ the control horizon. $\hat{x}(t_0)$ is the measured or estimated state vector at the current sampling time $t_0$ and $t$ is the time variable. The equation $g:\mathbb{R}\times\mathbb{R}^{n_x}\times\mathbb{R}^{n_z}\times\mathbb{R}^{n_u} \times \mathbb{R}^{n_p} \rightarrow \mathbb{R}^{n_g}$ are the path constraints and $g_T:\mathbb{R}\times\mathbb{R}^{n_x}\times\mathbb{R}^{n_z} \times \mathbb{R}^{n_p} \rightarrow \mathbb{R}^{n_{g,\text{T}}}$ are terminal constraints.
\begin{remark}
Any of the equations in \eqref{eq:problem-mpc} could be at least partially learned from data (see Fig.~ \ref{fig:scheme}). For example in \cite{matschek2019learning} the output function \label{subeq:meas-eq} is learned, in \cite{rosolia2017autonomous,rosolia2017learning,brunner2017repetitive} the terminal constraints \eqref{subeq:terminal-con} and in \cite{bujarbaruah2020learning,Leopoldo2017} the path constraints. Also the objective function can be learned, see e.g. \cite{tamar2017learning,BECKENBACH201860,Bradford2018}. HILO-MPC allows to use machine learning models in any of these components.
\end{remark}
Problem \eqref{eq:problem-mpc} cannot be solved directly since $u(\cdot)$ and the constraints are infinite dimensional. HILO-MPC uses \emph{direct approaches} to transform the problem to an equivalent finite-dimensional optimization problem. These approaches parametrize the input with a finite number of parameters, for example, using piece-wise constant inputs, and force the nonlinear constraints only on a finite number of points. This reads as follows
\begin{mini!}
{\mathbf{u}}{\sum_{i = k}^{N} \int_{t_i}^{t_{i} + \Delta t} l(t,x(t),z(t),u_i,p) \mathrm{d}t + e(t_k+T,x(t_k+T),z(t_k+T),p),}{\label{eq:disc-mpc}}{}
\addConstraint{x(t_{i+1})}{= x(t_{i}) + \int_{t_i}^{t_{i} + \Delta t} f(t,x(t),z(t),u_i,p) \mathrm{d}t}{}
\addConstraint{x(t_k)}{=\hat{x}(t_k)}{}
\addConstraint{0}{=q(t_i,x(t_i),z(t_i),u_i,p)}{}
\addConstraint{0}{\geq g(t_i,x(t_i),z(t_i),u_i,p)}{}
\addConstraint{0}{\geq g_T(t_k+T,x(t_k+T),z(t_k+T),p)}{}
\addConstraint{y(t_i)}{=h(t_i, x(t_i),z(t_i),u_i,p)}{}
\addConstraint{u_i}{=u_{N_c},\,\ \text{for} \,\ i\geq N_c}{}
\addConstraint{}{\text{for} \,\ i \in [k, N] \subset \mathbb{N}_0,}{}
\end{mini!}
where $\Delta t$ is the sampling time, $N_c = \text{ceil}(T_c/\Delta t)$ is the control horizon and $\mathbf{u}=[u_k,...,u_{N_c}]$ is sequence of piece-wise constant inputs applied to the plant, i.e. $u(t) = u_i, \,\ \forall t \in [t_i,t_i + \Delta t) \subset \mathbb{R}$.
HILO-MPC implements the multiple shooting approach \citep{BOCK19841603}. The initegration of the system model in between the shooting points can be done with Runge-Kutta methods of various orders, orthogonal collocation \citep{Oh1977} or the dynamics can be integrated with CVODES or IDAS solvers \cite{Hindmarsh2005}. The default method is orthogonal collocation and the user can select any other method if needed. Piece-wise constant input is assumed at every control interval.
Hence, the real formulation of Problem \eqref{eq:problem-mpc} depends on the which method is used for the discretization. Here, we do not go into the details of the formulation of the approximated problem. The reader is invited to refer to the previously cited papers for more details on the formulation of the OCP for the single cases.
\begin{remark}
HILO-MPC can also optimize the sampling intervals. This will add to Problem \eqref{eq:disc-mpc} the vector $[\Delta t_k,..., \Delta t_{N}]$ as optimization variable. This allows the solution of minimum time problems, and optimal sampling time problems.
\end{remark}
The stage cost $l(\cdot)$ arrival cost $e(\cdot)$ depend on the type of MPC implemented. In the next sections we will go through the different MPC problems HILO-MPC can solve. For simplicity of explanation, we show only quadratic cost functions. Note however that HILO-MPC can use any type of continuous cost functions.
\subsubsection{MPC for set-point tracking}
In set-point tracking MPC the variables $x,u$ and $y$ need to track a given reference.
\begin{align*}
l(x(t),u(t),y(t)) &= \Vert x(t) - x_r\Vert^2_Q + \Vert u(t) - u_r\Vert^2_R + \Vert y(t) - y_r\Vert^2_P, \\
e(x(t_0+T),y(t_0+T)) &= \Vert x(t_0+T) - x_r\Vert^2_{Q_{\text{T}}} + \Vert y(t_0+T) - y_r\Vert^2_{P_{\text{T}}},
\end{align*}
where $x_r,u_r,y_r$ are fixed references, and $Q \in \mathbb{R}^{n_x \times n_x},\,\ R \in \mathbb{R}^{n_u \times n_u} \,\ P \in \mathbb{R}^{n_y \times n_y} \,\ P_{\text{T}} \in \mathbb{R}^{n_y \times n_y}$ and $ Q_{\text{T}} \in \mathbb{R}^{n_x \times n_x}$ are weighting matrices.
For example, once we have defined the bike model like in section \ref{sec:modeling_module}. A possible set-point tracking MPC that tracks a reference speed of $v_{\text{r}} = 2 \,\ m/s$, with a prediction horizon of 20 steps, can be defined as follows
\begin{minted}[fontsize=\footnotesize, bgcolor=bg]{python}
from hilo_mpc import NMPC
nmpc = NMPC(model)
nmpc.horizon = 20
nmpc.quad_stage_cost.add_states(names=['v'], ref=2, weights=10)
nmpc.quad_term_cost.add_states(names=['v'], ref=2, weights=10)
nmpc.setup()
\end{minted}
After the MPC has been setup, it can be used in a control loop as follows
\begin{minted}[fontsize=\footnotesize, bgcolor=bg]{python}
# Choose number of steps
n_steps = 100
# Begin simulation loop
for step in range(n_steps):
# find optimal input
u = nmpc.optimize(x0)
# Simulate the plant
model.simulate(u=u)
# Get the current states
x0 = sol['xf']
\end{minted}
As for the the other HILO-MPC objects, the solution of the MPC (e.g., input and state sequences) are stored in the \mintinline{python}{mpc.solution()} object.
\subsubsection{MPC for trajectory-tracking}
\label{sec:mpc-traj-tracking}
For trajectory tracking problems the variables need to track a time-varying reference. For quadratic cost function, this looks like the following:
\begin{align*}
\label{eq:mpc-traj-tracking}
l(x(t),u(t),y(t)) &= \Vert x(t) - x_r(t)\Vert^2_Q + \Vert u(t) - u_r(t)\Vert^2_R + \Vert y(t) - y_r(t_i)\Vert^2_P, \\
e(x(t_0+T),y(t_0+T)) &= \Vert x(t_0+T) - x_r(t_0+T) \Vert^2_{Q_{\text{T}}} + \Vert y(t_0+T) - y_r(t_0+T) \Vert^2_{P_{\text{T}}},
\end{align*}
where $x_r : \mathbb{R} \rightarrow \mathbb{R}^{n_x},u_r: \mathbb{R} \rightarrow \mathbb{R}^{n_u}, y_r: \mathbb{R} \rightarrow \mathbb{R}^{n_y}$ are time-varying references. For example, a trajectory tracking problem for the racing car problem can be defined as follows
\begin{minted}[fontsize=\footnotesize, bgcolor=bg]{python}
from hilo_mpc import NMPC
import casadi as ca
nmpc = NMPC(model)
nmpc.horizon = 20
t = nmpc.create_time_variable()
traj_x = 30 - 14 * ca.cos(t)
traj_y = 30 - 16 * ca.sin(t)
nmpc.quad_stage_cost.add_states(names=['px', 'py'], ref=[traj_x, traj_y],
weights=[10, 10], trajectory_tracking=True)
nmpc.quad_term_cost.add_states(names=['px', 'py'], ref=[traj_x, traj_y],
weights=[10, 10], trajectory_tracking=True)
nmpc.setup()
\end{minted}
\subsubsection{MPC for path-following}
While in trajectory tracking MPC, the both \emph{value of the reference and time are fixed simultaneously} in path-following MPC, the controller has more degrees of freedom in choosing \emph{when} to be on the path \citep{Matschek2019,faulwasser2016}. In this case the model is augmented with a \emph{virtual path state} as follows
\begin{mini!}
{u(\cdot),u_{\text{pf}}(\cdot)}{\int_{t_0}^{t_0+T} l(t,x(t),z(t),u(t),p) \mathrm{d}t + e(t_0+T,x(t_0+T),z(t_0+T),p),}{}{}
\addConstraint{\dot{x}(t)}{=f(t, x(t),z(t),u(t),p),\quad}{x(t_0)=\hat{x}(t_0)}
\addConstraint{0}{=q(t, x(t),z(t),u(t),p)}{}
\addConstraint{\dot{\theta}}{=u_{\text{pf}},\quad}{\theta(t_0)=0}
\addConstraint{0}{\geq g(t,x(t),z(t),u(t),p, \epsilon)}{}
\addConstraint{0}{\geq g_T(t_0+T,x(t_0+T),z(t_0+T),p)}{}
\addConstraint{y(t)}{=h(t, x(t), z(t), u(t), p)}{}
\addConstraint{}{\text{for} \,\ t \in [t_0, t_0+T] \subset \mathbb{R},}{}
\end{mini!}
where $\theta \in \mathbb{R}^{n_\theta}$ is a virtual path state vector and $u_{\text{pf}} \in \mathbb{R}^{n_{u,\theta}}$ is the virtual input that can controller can choose. Hence, the objective function looks like
\begin{align*}
l(x(t),u(t),y(t)) &= \Vert x(t) - x_r(\theta(t))\Vert^2_Q + \Vert u(t) - u_r(\theta(t))\Vert^2_R + \Vert y(t) - y_r(\theta(t))\Vert^2_P, \\
e(x(t_0+T),y(t_0+T)) &= \Vert x(t_0+T) - x_r(\theta(t_0+T)) \Vert^2_{Q_{\text{T}}} + \Vert y(t_0+T) - y_r(\theta(t_0+T)) \Vert^2_{P_{\text{T}}}.\end{align*}
Usually, to force the controller to move into only one direction along the path usually a lower bound on $u_{\text{pf}}$ is added, i.e. $u_{\text{pf}}>u_{\text{pf,min}}$ with $u_{\text{pf,min}} \in \mathbb{R}^{n_{u,\theta}}_{+}$.
HILO-MPC also allows to track a constant $u_{\text{pf}}$, so that the MPC tries to maintain a constant \emph{speed} of the virtual state
\begin{align*}
l(x(t),u(t),y(t)) &= \Vert x(t) - x_r(\theta(t))\Vert^2_Q + \Vert u(t_i) - u_r(\theta(t))\Vert^2_R + \Vert y(t) - y_r(\theta)\Vert^2_P + \Vert u_{\text{pf}} - u_{\text{pf,ref}}\Vert_{R_{\text{pf}}}^2, \\
e(x(t_0+T),y(t_0+T)) &= \Vert x(t_0+T) - x_r(\theta(t_0+T))\Vert^2_Q + \Vert y(t_0+T) - y_r(\theta(t_0+T))\Vert^2_P.
\end{align*}
Contrary to the other available toolboxes, path-following MPC problems are automatically generated. The user needs just to activate the path following mode for the desired variables. Here, we show how a path-following problem can be defined using HILO-MPC:
\begin{minted}[fontsize=\footnotesize, bgcolor=bg]{python}
from hilo_mpc import NMPC
import casadi as ca
nmpc = NMPC(model)
nmpc.horizon = 20
theta = nmpc.create_path_variable()
traj_x = 30 - 14 * ca.cos(theta)
traj_y = 30 - 16 * ca.sin(theta)
nmpc.quad_stage_cost.add_states(names=['px', 'py'], ref=[traj_x, traj_y],
weights=[10, 10], path_following=True)
nmpc.quad_term_cost.add_states(names=['px', 'py'], ref=[traj_x, traj_y],
weights=[10, 10], path_following=True)
nmpc.setup()
\end{minted}
Note that HILO-MPC allows to mix the previous problem formulations together, with minimum effort: e.g. the user can track some constant references for some of the variables, while tracking a path with other variables.
\subsubsection{Soft constraints}
Soft constraints can be easily implemented. When soft constraints are selected, HILO-MPC automatically adds the slack variables $\epsilon_{\text{p}} \in \mathbb{R}^{n_g}$ and $\epsilon_{\text{T}} \in \mathbb{R}^{n_{g,\text{T}}} $ to the path and terminal constraints respectively as follows
\begin{mini!}
{u(\cdot), \epsilon_{\text{T}}, \epsilon_{\text{p}}}{\int_{t_0}^{t_0+T} l(\cdot) \mathrm{d}t + e(\cdot) + \Vert \epsilon_{\text{s}} \Vert^2_{E_{\text{p}}} + \Vert \epsilon_{\text{T}} \Vert^2_{E_{\text{T}}},}{\label{eq:problem-mpc-soft}}{}
\addConstraint{\dot{x}(t)}{=f(t,x(t),z(t),u(t), p),\quad}{x(0)=\hat{x}_0}
\addConstraint{0}{=q(t, x(t),z(t),u(t), p)}{}
\addConstraint{\epsilon_{\text{p}}}{\geq g(t,x(t),z(t),u(t), p)}{}
\addConstraint{\epsilon_{\text{T}}}{\geq g_T(t_0+T,x(t_0+T),z(t_0+T),p)}{}
\addConstraint{y(t)}{=h(t, x(t),z(t), u(t),p)}{}
\addConstraint{0}{\leq\epsilon_{\text{p}} \leq \epsilon_{\text{p,max}}, \quad 0 \leq \epsilon_{\text{T}} \leq \epsilon_{\text{T, max}}}{}
\addConstraint{}{\text{for} \,\ t \in [t_0, t_0+T] \subset \mathbb{R},}{}
\end{mini!}
where and $E_{\text{p}}\in \mathbb{R}^{n_g \times n_{g}}$ and $E_{\text{T}}\in \mathbb{R}^{n_{g,\text{T}} \times n_{g,\text{T}}}$ are weighting matrices that limit the increase of the the slack variables and can be chosen by the user,
and $\epsilon_{\text{p,max}}$ and $\epsilon_{\text{T,max}}$ are the maximum constraint violations of path and terminal constraints respectively.
\subsection{Observer module}
The observer model contains the Kalman filters, moving horizon estimators and particle filter.
\subsection{Moving Horizon Estimation}
The Moving Horizon Estimator (MHE) is an observer based on the solution of an optimization problem similar to the MPC problem \citep{rawlings2017model}. We consider a sample-data problem with equidistant sampling times as it is more common in practice. At these sampling times, measurements are taken. Since measurements are discontinuous in MHE the objective function is usually in discrete form. For simplicity we indicate with $(\cdot)_{i|k}$ the variable at time $t_i$, i.e. for the MHE problem solved at time $t_k$ we have
\begin{mini!}
{x_{k-N|k},p_k, w_{(\cdot)|k},z_{(\cdot)|k}}{\left\Vert \begin{bmatrix} x_{k-\bar{N}|k} - \hat{x}_{k-N|k} \\ p_{k} - \hat{p}_{k} \end{bmatrix} \right\Vert^2_{P_k} + \sum^k_{i=k-N} \Vert \hat{y}_i - y_{i|k} \Vert^2_R + \Vert w_{i|k}\Vert^2_W,}{\label{eq:problem-mhe}}{}
\addConstraint{x_{i+1|k}}{=x_{i|k} + \int_{t_i}^{t_i + \Delta t} \left( f(t,x(t),z(t),\hat{u}(t), p_k) + w(t) \right) \mathrm{d}t\label{subeq:mhe-model1}}{}
\addConstraint{y_{i|k}}{=h(t_i,x_{i|k},z_{i|k},\hat{u}_i,p_k) + v_{i|k}}{}
\addConstraint{0}{\geq g(t_i,x_{i|k},u_i)}{}
\addConstraint{}{\text{for} \,\ i \in [k-\bar{N}, k], \,\ k,\bar{N} \in \mathbb{N},}{}
\end{mini!}
where $\bar{N}$ is the horizon length, $\hat{y}_i$ the output measurements and $\hat{u}_i$ the input measurements at time $t_i$. The first term of the objective function is the \emph{arrival cost}, while the second and third weight the measurements and state noise respectively \cite{rawlings2017model,allgower1999nonlinear}. $R \in \mathbb{R}^{n_y \times n_y}$, $W \in \mathbb{R}^{n_x \times n_x}$ and $P+k \in \mathbb{R}^{(n_x+n_p) \times (n_x + n_p)}$ are the weighting matrix for the outputs, state noise and arrival cost. The optimization variables are: the state $x_{k-\bar{N}|k}$, i.e, the state at the beginning of the horizon, the state noise $w_{(\cdot)|k} = \{ w_{i|k}, \,\ \forall i \in [k-\bar{N},k] \}$, the algebraic states (for DAE systems) $z_{(\cdot)|k} = \{ z_{i|k}, \,\ \forall i \in [k-\bar{N},k] \}$, and the system parameters $p_k$. Note that the parameters are considered constant in the horizon, but can be every time the optimization is run, to adapt to the new measurements.
Also in this case we use direct approaches to solve \eqref{eq:problem-mhe} as for the MPC case. For the bike model, an MHE model with $R=W=P_{k}=10 I$ (where $I$ is the identity matrix of appropriate dimensions) can be easily defined as follows:
\begin{minted}[fontsize=\footnotesize, bgcolor=bg]{python}
from hilo_mpc import MHE
mhe = MHE(model)
mhe.horizon = 20
mhe.quad_arrival_cost.add_states(weights=[10,10,10,10], guess=x0_est)
mhe.quad_stage_cost.add_measurements(weights=[10,10,10,10])
mhe.quad_stage_cost.add_state_noise(weights=[10,10,10,10]])
mhe.setup()
\end{minted}
After the MHE has been setup, it can be deployed for estimation, for example, as follows
\begin{minted}[fontsize=\footnotesize, bgcolor=bg]{python}
# Set number of steps
n_steps = 100
# Simulation/estimation loop
for step in range(n_steps):
# Simulate plant with input
model.simulate(u=u)
# Get measurements
y_meas = model.solution['y'][:, -2]
# Pass the measurements to the MHE
mhe.add_measurements(y_meas, u_meas=u)
# Estimate states
x0 = mhe.estimate()
\end{minted}
As for the the other HILO-MPC objects, the solution of the MHE are stored in the \mintinline{python}{mhe.solution()} object.
\subsubsection{Kalman filters}
The Kalman filter (KF) is an algorithm that allows for estimation of observable states via available measurement data. In general it consists of two steps, the prediction step, where the estimated states from the previous iteration are propagated through the model dynamics to obtain preliminary values for the states at the current time step, the so-called a priori estimates, and the update step, where these a priori estimates are updated using the measurement data to obtain the a posteriori estimates. The original formulation of the KF was developed for linear discrete-time systems \citep{kalman1960}
\begin{align*}
x_k&=A_kx_{k-1}+B_ku_k+w_k,\\
y_k&=C_kx_k+v_k,
\end{align*}
where $w_k$ is the process noise and $v_k$ is the measurement noise. The process noise $w_k\sim\mathcal{N}(0,Q_k)$ and the measurement noise $v_k\sim\mathcal{N}(0,R_k)$ are assumed to be zero mean normal distributions with the covariance matrices $Q_k$ and $R_k$, respectively. Accordingly, the prediction step and update step of the KF are as follows
\begin{align}
\hat{x}_{k|k-1}&=A_k\hat{x}_{k-1}+B_ku_k,\\
P_{k|k-1}&=A_kP_{k-1}A_k^{\mkern-1.5mu\mathsf{T}}+Q_k,\label{eq:kf_prop}\\
\hat{x}_k&=\hat{x}_{k|k-1}+K(y_k-\hat{y}_k),\\
P_k&=P_{k|k-1}-KP_{y_ky_k}K^{\mkern-1.5mu\mathsf{T}},
\end{align}
with
\begin{align*}
K&=P_{x_ky_k}P_{y_ky_k}^{-1},\\
\hat{y}_k&=C_k\hat{x}_{k|k-1},\\
P_{x_ky_k}&=P_{k|k-1}C_k^{\mkern-1.5mu\mathsf{T}},\\
P_{y_ky_k}&=C_kP_{k|k-1}C_k^{\mkern-1.5mu\mathsf{T}}+R_k,
\end{align*}
where $P_k$ is the error covariance matrix, $P_{x_ky_k}$ is the state-measurement cross-covariance matrix, $P_{y_ky_k}$ is the measurement covariance matrix, $K$ is called the Kalman gain and $\hat{y}_k$ is the predicted output. The notation $\hat{x}_{k|k-1}$ represents the estimated state $\hat{x}$ at time step $k$ given observations up to and including time step $k-1$. The covariance matrix $P_k$ is a measure of the estimation accuracy of the estimated states, i.e. the lower the value the closer the estimated state is to the actual one assuming normally distributed noise.
There are also extensions to the original KF formulation for dealing with nonlinear models. The two most known extensions are the extended Kalman filter (EKF) \citep{Jazwinski1970} and the unscented Kalman filter (UKF) \citep{Wan2000}. The EKF differs from the original formulation in that the propagation of the error covariance matrix and the consecutive update step are not as straightforward. The a priori estimates of the states can be easily obtained by propagating through the nonlinear model dynamics. The propagation of the error covariance matrix is approximated by calculating the Jacobi matrix
\begin{equation*}
A_k=\left.\dfrac{\partial f}{\partial x}\right|_{\hat{x}_{k-1},u_k},
\end{equation*}
and using Eq.~\eqref{eq:kf_prop}. The output matrix $C_k$ is approximated analogously for the update step. A drawback is that the approximation via Jacobi matrices can produce highly unstable Kalman filters \citep{Wan2000}. Another nonlinear extension is the UKF. It approximates the nonlinear problem statistically by a linear one. The UKF makes use of the unscented transform to generate a set of samples, the so-called sigma points, that represent the initial probability distribution. These sigma points are then propagated through the nonlinear system dynamics and afterwards used to approximate the a priori estimates as well as the predicted output and the various covariance matrices. The update step is then the same as for the KF and EKF.
The setup of a Kalman filter in HILO-MPC can be easily set up as
\begin{minted}[fontsize=\footnotesize,bgcolor=bg]{python}
from hilo_mpc import UKF
ukf = UKF(model)
ukf.setup()
\end{minted}
\subsubsection{Particle filter}
The particle filter (PF) \citep{Simon2006} is another algorithm that can be used for estimation. It works similarly to the UKF in that a set of samples, the ``particles'', is propagated through the nonlinear system dynamics. This set of samples is drawn from a probability density function that is assumed to be known. Next, the relative likelihood of each propagated particle is determined
\begin{equation*}
q_i^*=\dfrac{q_i}{\sum_{i=1}^{n_p}q_i},
\end{equation*}
with
\begin{equation*}
q_i\sim\mathrm{pdf}(y_k|\hat{y}_k),
\end{equation*}
where $n_p$ is the number of particles in the set. This relative likelihood acts as a measure of how good a particle matches the measurement $y_k$ and is used in the resampling step to generate the set of a posteriori particles. This resampling step is analogous to the update step of the Kalman filters. The resampling can be achieved by using one of the several available sampling strategies, like e.g. survival of the fittest \citep{Kanazawa1995} or regularization \citep{Musso2001}. Similarly to the UKF, the a posteriori particles can now be used to determine the estimated states via computation of the mean of the distribution. For highly nonlinear system dynamics or non-Gaussian probability density functions the PF usually outperforms the Kalman filter algorithms \citep{Simon2006}.
The setup of a PF in HILO-MPC is similar to that of the Kalman filters
\begin{minted}[fontsize=\footnotesize,bgcolor=bg]{python}
from hilo_mpc import PF
pf = PF(model)
pf.setup()
\end{minted}
\section{Examples}
\label{sec:examples}
In the next section, we present some examples that use HILO-MPC. These examples can also be found in the examples repository, as Python files or Jupyter notebooks. For these reasons, here we do not go into the implementation details but give just a high-level description of the problems and present the results. The reader is invited to read the documentation and the examples codes for details.
\subsection{Learning the dynamics - Race car}
We use HILO-MPC for a realistic example of an autonomous racing minicar. The goal is to follow a complex track using a path-following formulation in presence of disturbances. We use a bike model with nonlinear tire models and drive train model (cf. Fig.~\ref{fig:bike}) that has been identified and validated in experiments by \cite{Liniger2015}. To this model we add a component describing the effect of lateral drag forces due to, for example, a strong wind. The model is represented by the following system of nonlinear differential equations
\begin{align}
\dot{p}_x &= v_x \cos (\psi) - v_y \sin(\psi), \\
\dot{p}_y &= v_x \sin (\psi) + v_y \cos(\psi), \\
\dot{\psi} &= w, \\
\dot{v}_x &= \frac{1}{m}\left( F_{r,x}-F_{a,x}- (F_{f,y}-F_{a,y}) \sin(\delta) + m v_y \omega \right), \\
\dot{v}_y &= \frac{1}{m}\left( F_{r,y}-F_{a,y}- (F_{f,y}-F_{a,y}) \cos(\delta) - m v_x \omega \right), \\
\dot{\omega} &= \frac{1}{I_z} \left( F_{f,y}l_f \cos(\delta) - F_{r,y}l_r\right),
\end{align}
where $p_x$ and $p_y$ are the coordinates of the center of gravity (CoG), $v_x$ and $v_y$ are longitudinal and later velocities of the CoG. The orientation is denoted by $\psi$ and the yaw rate by $\omega$. The control inputs are the motor duty cycle $d$ and steering angle $\delta$. The two parameters $l_f$ and $l_r$ are the distances from the CoG to the front axle and rear axle respectively, $m$ is the mass of the car and $I_z$ is the inertia. The path to follow is the center of a racing track which has been interpolated using splines.
The tire forces are modeled with a simplified Pacejka Tire Model \citep{bakker1987tyre}
\begin{subequations}
\begin{align}
F_{f,y} &= D_f \sin(C_f \arctan(B_f \alpha_f)), \\
F_{r,y} &= D_r \sin(C_r \arctan(B_r \alpha_r)), \\
F_{r,x} &= (C_{m1}-C_{m2}v_x)d - C_r - C_d v^2_{x}, \\
\alpha_f &= - \arctan\left( \frac{w l_f + v_y}{v_x} \right) + \delta,\\
\alpha_r &= \arctan\left( \frac{w l_r - v_y}{v_x} \right),
\end{align}
\end{subequations}
where $D_f,D_r,B_f,B_r, C_{m1},C_{m2},C_r$ and $C_d$ are parameters. The longitudinal and lateral drag forces are defined respectively as
\begin{align*}
F_{a,x} = 0.5 \,\ c_w \rho A v_x, \\
F_{a,y} = 0.5 \,\ c_w \rho A v_{y,\text{wind}},
\end{align*}
where $c_w$ is the drag coefficient, $\rho$ is the air density, $A$ the effective flow surface and $v_{y,wind}$ is the lateral wind velocity.
The model used by the MPC does not have the drag effect. The goal is learning this effect from data using a Neural Network, and then augment the first principle model with a machine learning component that models the drag effect. After discretization, the hybrid model can be written as:
\begin{equation}
x_{k+1} = f(x_k,u_k) + B^T m(x_k,u_k),
\end{equation}
where $m(x_k,u_k)$ is an NN model and
\begin{equation}
B =
\begin{bmatrix}
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{bmatrix}.
\end{equation}
The features of the NN are the $v_{y,\text{wind}}$ and $v_x$, the labels are correction terms on $\phi,v_x,v_y$ and $\omega$. To show the effectiveness of the learning, we compare the results of MPC using the perfect model (i.e. with known drag force effects), the hybrid model (using the NN) and the model without drag forces. Furthermore, the measurements of position, velocity and directions are effected by Gaussian noise, and estimated using an Unscented Kalman Filter.
Figure \ref{fig:race_car} shows the results of the simulation. While the hybrid model has similar results as the perfectly-known model, the model without drag exits the race area after the fifth curve. The complete code can be found in the HILO-MPC repository.
\begin{figure*}
\begin{subfigure}[b]{.49\linewidth}
\input{figures/bike}
\caption{Model used for the racing car.}
\label{fig:bike}
\end{subfigure}
\begin{subfigure}[b]{.49\linewidth}
\input{figures/racing
\caption{Results of the path-following problem}
\label{fig:race_car}
\end{subfigure}
\end{figure*}
\subsection{Learning a reference - Cooperative robots}
In this example, inspired by \cite{Matschek2021a}, a follower robot has to track the position of a leader robot. The leader moves in a periodic but unknown trajectory. The objective is learning the trajectory of the leader with GPs and pass the learned trajectory to the follower. Hence, in this case the machine learning model enters the reference function (cf. Fig.~\ref{fig:mobile-robots}). The nonlinear dynamics of the robots is described with the following ODE
\begin{subequations}
\begin{align}
\dot{x}_1 &= u_1 \sin(x_3), \\
\dot{x}_2 &= u_1 \cos(x_3), \\
\dot{x}_3 &= u_2 ,
\end{align}
\end{subequations}
where $x_1$ and $x_2$ are the horizontal and vertical position of the root and $x_3$ its heading angle, $u_1$ is the speed and $u_2$ the turning rate.
The problem we want to solve is of the form \eqref{eq:problem-mpc} with objective function \eqref{eq:mpc-traj-tracking} where $x_r(\cdot)$ is the mean function of a GP trained on the data set collected from the position of the leader.
The trajectory generated by the leader, results from applying the following time-varying forces
\begin{subequations}
\begin{align}
u_1(t) &= 2 + 0.1 \sin(t), \\
u_2(t) &= 0.5 + \sin(t).
\end{align}
\end{subequations}
Figures \ref{fig:mobile-robots-results} show the results of the reference learning and closed-loop simulation.
\begin{figure*}
\centering
\input{figures/mobile_robots}
\caption{Cooperative robot example. The follower needs to track the trajectory of the leader. The trajectory is learned from data using a GP regressor.}
\label{fig:mobile-robots}
\end{figure*}
\begin{figure*}
\input{figures/tot_mobile_robots}
\caption{Left column: learned GP. Right column: states of the follower robot following the learned references. }
\label{fig:mobile-robots-results}
\end{figure*}
\subsection{Learning the controller - String damper system}
Solving a MPC requires the solution of a (nonlinear) optimization problem online. This is possible only for applications where the available computational power and energy are sufficient to guarantee a solution within the sampling times. For embedded applications, with low computational power or that can use only a limited amount of energy (for example battery powered systems) this is often not possible. Hence, methods that provide at least a close-to-optimal solution without solving the optimization approach at every time step are necessary. This can be done using \emph{explicit} MPC approaches. Some of these approaches are based on learning the solution of an MPC offline, i.e., the map $x \mapsto \rho_{\theta}(x)$ that approximate the implicit MPC control law, and then using the learned controller online \citep{Parisini1995, Karg2020, Maddalena2020, Cao2020, Chen2018, Cseko2015, Ponkumar2018}, (cf. Fig. \ref{fig:ml-controller}). In this way, the control action can be found with a simple and fast function evaluation.
In this example, we want to control a mass-spring-damper system using a learned controller (Fig. \ref{fig:ml-controller}. The models is
\begin{subequations}
\begin{align}
& \dot{x}_1 = x_2, \\
& \dot{x}_2 = \frac{1}{m} (u - k x_1 - d x_2) ,
\end{align}
\end{subequations}
where $x_1$ is the vertical position, $x_2$ the vertical velocity, $u$ the vertical force and $k,d$ the system parameters. The equilibrium point is $x=(0,0), u=0$. The objective is to maintain the reference $x_{\text{ref}}=[1,0]$ using a learned MPC. To do so we use the results of just one closed-loop MPC simulation starting from the initial conditions $x(0)=(12,0)$. In total 667 datapoints are collected. We use the data collected to train a NN with three fully-connected layers, with 10 neurons each. The features of the NN are $x_1$ and $x_2$, the labels is the input $u$. We test the learned controller starting from a different initial condition $x(0)=(10,0)$. In Fig.~\ref{fig:ml-controller-results} the simulation results are shown. The learned controller is able to bring the system to the reference as the original controller
\begin{figure*}
\centering
\input{figures/machine_learning_controller}
\caption{Scheme of a machine learning controller.}
\label{fig:ml-controller}
\end{figure*}
\begin{figure*}
\centering
\input{figures/learned_mpc}
\caption{Results of the learned MPC.}
\label{fig:ml-controller-results}
\end{figure*}
\section{Conclusions and outlook}
\label{sec:conclusions}
We introduced HILO-MPC, a toolbox for fast development of predictive control and estimation problems that facilitates the use of machine learning models trained using PyTorch and TensorFlow. We showed three applications of the toolbox, the code of the presented applications is openly available in the HILO-MPC repository.
The toolbox can solve many problems, such as model predictive control, moving horizon estimation, Kalman filters, and particle filters. Its flexibility and simplicity can make it a valuable tool for research and teaching.
We are currently working on a fifth module that allows solving optimization problems efficiently and fast, aiming at embedded applications such as embedded MPC. Also, future versions will allow using other machine learning models such as recurrent neural networks and reinforcement learning. Furthermore, other MPC formulations, such as tube-based MPC \citep{Mayne2005} and multi-mode MPC \citep{bethge2018multi, morabito2019multi} will be available.
\section*{Acknowledgments}
We would like to thank Michael Maiworm, Tim Zieger, Johanna Bethge, Maik Pfefferkorn, Rudolph Kok and Janine Matschek for providing feedback and examples for HILO-MPC.
\bibliographystyle{unsrtnat}
\section{Introduction}
Advanced optimal control and estimation problems such as model predictive control, moving horizon estimation, and Kalman filters require models representing, e.g., the system dynamics, constraints, reference values, and objective functions. Based on these models, the inputs, the estimates of the states, or parameters are found based on the minimization of an objective function \cite{brunton2019data,hou2013model}
Often obtaining accurate models is challenging. For example, dynamical models can be constructed with first principles, such as conservation of energy, matter, and thermodynamics laws.
However, underlying unknown nonlinearities, parameters, or coupling effects, often results in complex dynamics, which is challenging to model, thus rendering it difficult to employ such model-based methods for process control or numerical simulations.
Alternatively, if only data is used to derive a model, one generally talks about data-driven or machine learning models.\cite{brunton2019data,glassey2018hybrid} Machine learning has recently obtained much attention since it allows to model and to simulate complex, not well-understood phenomena or efficiently simulate complex, large-scale systems, provided that a sufficiently large amount of data is available. For many decades the scarcity of data, features selection, and computational power have been the main bottlenecks to the widespread use of machine learning for modeling and control. The situation is rapidly changing due to the rapid digitization and interconnection (e.g., Internet of Things and Industry 4.0 initiatives), advances in machine learning, and cloud computing.\cite{baheti2011cyber,lucia2016predictive} Hence, machine learning is finding more and more applications, not only in research but in everyday life. In recent years, the number of machine learning applications, publications, new methods, and theoretical findings has increased rapidly. Furthermore, open-source libraries for machine learning such as TensorFlow\cite{tensorflow2015} and PyTorch\cite{pytorch} offer a low entry barrier to this technology and allow fast development of machine learning solutions. The combination of these factors boosted the quality and quantity of results in many fields such as medicine,\cite{Shen2017} autonomous driving,\cite{Arnold2019} transportation,\cite{Nguyen2018} Internet of Things,\cite{Mohammadi2018} image recognition\cite{Voulodimos2018} and many others. The control and optimization community also recognized the usefulness of machine learning for control, e.g., for efficient and safe optimal control and model predictive control (MPC), motivating many contributions.\cite{hou2013model,Hewing2020,maiworm2021online,aswani2013provably}
While machine learning libraries are powerful tools to build machine learning models, using these models in a control strategy requires typically ad-hoc solutions, which are complicated, prone to bugs, and not easily transferable to other applications. In the frame of this work, we outline an approach to interconnect machine learning methods and tools with methods from optimization-based control and estimation using Python. The resulting open-source Python library, named HILO-MPC \footnote{HILO-MPC stands for mac{\bf HI}ne {\bf L}earning and {\bf O}ptimization for {\bf M}odeling, {\bf P}rediction and {\bf C}ontrol.} aims at providing a way to easily implement machine learning models into a wide variety of control, optimization and estimation problems, tailored towards fundamental research and basic development tasks (see Fig.~\ref{fig:overview}). It interfaces with PyTorch and TensorFlow to train neural networks and uses in-house code to train Gaussian processes. The trained models can be used in a large variety of control or estimation problems for example, in the system's dynamics, the constraints, the cost functions, or even the controller itself (see Figure~\ref{fig:scheme}).
To speed up the development time, HILO-MPC provides a way to define and solve a broad spectrum of optimal control and estimation problems quickly and with minimum effort. Table~\ref{tab:problems-hilo} summarizes the different problems that can be solved. The philosophy followed is the ease of use and modularity. Ease of use is achieved by providing an intuitive high-level interface for the problem definition. At the same time, modularity is obtained by ensuring effortless interfaces between the different tools (see Figure~\ref{fig:scheme}). This allows using the toolbox both for research and for teaching. Table \ref{tab:problems-hilo} summarizes the main tools and methods available.
\begin{figure*}[h]
\centering
\inputpdf{overview}
\caption{Core elements of HILO-MPC.}
\label{fig:overview}
\end{figure*}
The main contribution of this paper is the presentation of a modular way to fuse modeling, simulation, estimation, control, machine learning approaches, and optimization, resulting in a flexible yet easy-to-use open-source toolbox. The usability is underlined, presenting a series of application examples, spanning from the problem to learn the dynamic model of a car for racing, learning references to be tracked, learning the solution of an MPC controller, and the embedded implementation of a MPC controller. Note that, while we focus on the integration of machine learning into control and estimation problems, most of the parts can also be used without any machine learning component.
The remainder of this paper is structured as follows. Section~\ref{sec:hilo-mpc} provides a brief overview of toolboxes available for optimal and predictive control, with a specific focus on open-source toolboxes.
In Section~\ref{sec:hilo-mpc} the different modules of the toolbox are described. Furthermore, we show example code snippets to illustrate the required syntax. In Section~\ref{sec:examples} we apply the toolbox in four examples. Finally, in Section~\ref{sec:conclusions} we conclude and give an outlook on current and future developments
Note that this paper only provides a glimpse of the principles and main ideas and the usability via some example applications. It is not a manual, as providing a comprehensive description of the toolbox is outside the scope of this work. For this, the reader is referred to the documentation.\footnote{The full documentation and toolbox are available under \url{http://www.ccps.tu-darmstadt.de/hilo_mpc}.}
\begin{table*}[b]
\caption{Components and formulations imtegrated in HILO-MPC.}
\label{tab:problems-hilo}
\centering
\begin{tabular}{lllll}
\toprule
Models & Controllers & Machine Learning & Estimators & Embedded \\
\midrule
\parbox[c][2.5cm][c]{3.5cm}{Linear/nonlinear\\Time-invariant/variant\\Continuous/discrete\\ODEs, DAEs} & \parbox[c][2.5cm][c]{3.5cm}{Nonlinear MPC\\Trajectory tracking MPC\\Path-following MPC\\PID} & \parbox[c][2.5cm][c]{3cm}{Neural networks\\Gaussian processes} & \parbox[c][2.5cm][c]{3.9cm}{Moving horizon estimation\\Kalman filter\\Extended Kalman filter\\Unscented Kalman filter\\Particle filter} & \ensuremath{\mu}\emph{AO-MPC}\cite{zometa2013} \\
\bottomrule
\end{tabular}
\end{table*}
\begin{figure*}[htb]
\centering
\include{figures/control_loop}
\caption{Schematic diagram of the interactions between the components of the toolbox. The machine learning component can be used to learn model parts (see Section~\ref{sec:example-race-car}), cost functions, constraints, or references/disturbances (see Section~\ref{sec:cooperative-robots}) from real or simulation data, that can then be used in the controller or estimator. It can also be used to learn an approximated solution of an optimal controller/MPC controller (see Section~\ref{sec:spring-damper}). It is also possible to generate embedded code for the controller (see Section~\ref{sec:embedded}). In principle, similarly, the same holds for learning components of the estimator.}
\label{fig:scheme}
\end{figure*}
\section{A Glimpse Review of Open-source MPC tools}
\label{sec:OSMPC}
\begin{table*}[htb]
\caption{Comparison of different freely accessible MPC tools. ML: machine learning, MHE: moving horizon estimator, LMPC: linear MPC, PFMPC: path-following MPC, KF: Kalman filters, PF: particle filters}
\label{tab:comparison-toolboxes}
\centering
\begin{tabular}{llclll}
\toprule
& Problems & ML & Interface & Focus & Article \\
\midrule
ACADO & MPC & No & C++, MATLAB & Embedded & Houska et al.\cite{Houska2011a} \\
ACADOS & MPC, MHE & No & \parbox[c][1cm][c]{3.2cm}{C, MATLAB, Octave,\\Python, Simulink} & Embedded & \parbox[c][1cm][c]{2.8cm}{Verschueren et al.\cite{Verschueren2018}\\Verschueren et al.\cite{Verschueren2019}} \\
do-mpc & \parbox[c][1cm][c]{3.2cm}{MPC, MHE,\\Multi-stage MPC} & No & Python & Multi-stage MPC & Lucia et al.\cite{LUCIA201751} \\
HILO-MPC & \parbox[c][1cm][c]{3.2cm}{MPC, PFMPC, LMPC\\MHE, KF, PF} & Yes & Python & Machine Learning & \\
MPCTools & MPC, MHE & No & \parbox[c][1cm][c]{3.2cm}{Python,\\Octave (MATLAB)} & MPC, MHE & \parbox[c][1cm][c]{3.4cm}{Risbeck and Rawlings\cite{risbeck2015nonlinear}\\Risbeck and Rawlings\cite{risbeck2016nonlinear}} \\
MPT3 Toolbox & LMPC & No & MATLAB & Comp. Geometry & Herceg et al.\cite{MPT3} \\
YALMIP & MPC, MHE & No & MATLAB & Optimization & L\"{o}fberg\cite{YALMIP} \\
\bottomrule
\end{tabular}
\end{table*}
By now a wide variety of toolboxes for MPC exist. We ocus on freely available, open-source toolboxes. The software ACADO\cite{Houska2011a} and ACADOS\cite{Verschueren2018} aim at efficiency and fast solution keeping embedded applications in mind. Note that, although interfaces to embedded MPC libraries are offered, embedded MPC is not the main focus of the toolbox. MPT3 focuses on computational geometry, which is particularly useful for some classes of robust MPC approaches.\cite{MPT3} YALMIP instead offers a broad spectrum of optimization problems, such as bilevel optimization and mixed-integer programming, and some MPC schemes.\cite{YALMIP} The toolbox do-mpc implements robust multi-stage MPC and moving horizon estimation (MHE).\cite{LUCIA201751} MPCTools offers some interfaces that help build MPC and MHE control problems in CasADi.\cite{risbeck2015nonlinear,risbeck2016nonlinear} Table~\ref{tab:comparison-toolboxes} summarizes some of the main differences between existing open-source toolboxes.
Using machine learning models in the previously mentioned toolboxes is often not straightforward. We tackle this challenge by providing wrappers for machine learning libraries and some in-house learning libraries. Furthermore, compared to the other toolboxes, it focuses on a simple and intuitive interface that makes it easy to use also for beginners.
\section{Basic Concepts and Components}
\label{sec:hilo-mpc}
To provide flexibility five main components are tightly integrated --- in the following denoted as \emph{modules} named \emph{dynamic model}, \emph{control}, \emph{estimator}, \emph{machine learning}, and \emph{embedded} (see Figure~\ref{fig:overview}). The dynamic model module is used to generate dynamic models which are used in all other modules. The control module allows to setup and formulate controllers, solve and apply them. It focuses, but is not limited to predictive and optimal controllers. The estimator module focuses on state and parameter estimation using the dynamic models. The machine learning module is responsible for defining and training machine learning models that can be used in any of the previous modules. Finally, the embedded module currently interfaces with embedded linear MPC software (\ensuremath{\mu}\emph{AO-MPC}\cite{zometa2013}) that allows to easily deploy the code for embedded applications. In the next sections, we will briefly describe the different modules in more detail.\footnote{New tools will be added, and current tools might be modified, please refer always to the current documentation for the newest features..}
The backbone of HILO-MPC is CasADi.\cite{Andersson2019} CasADi is a flexible and widely used tool for algorithmic differentiation and optimization. It has interfaces with a wide range of solvers for linear and quadratic programming (CLP,\cite{john_forrest_2022_5839302} qpOASES\cite{Ferreau2014}), nonlinear programming such as IPOPT,\cite{wachter2006implementation} quadratic mixed-integer programming (CPLEX,\cite{cplex2009v12} GUROBI\cite{gurobi}), for non quadratic nonlinear mixed-integer problems (BONMIN,\cite{bonami2007bonmin} KNITRO\cite{nocedal2006knitro}) and large nonlinear problems WORHP.\cite{nikolayzik2010nonlinear}
CasADI is used to build the models, optimization, and estimation problems. TensorFlow or PyTorch are interfaced by automatically translating the models into CasADi-compatible structures. Hence, a CasADi problem is defined and solved.
In the following, we briefly outline the core modules and provide insights in the formulations and solution approaches.
\subsection{Dynamic Model Module}
\label{sec:modeling_module}
The dynamic model module forms the foundation for all other components. It is a high-level interface to generate representations of dynamical systems that can be used for model-based controllers and estimators, like MPC or MHE, or inside simulations to reproduce the behavior of a plant (or a controller) in a control loop. The module currently supports linear and nonlinear, time-invariant and time-variant, discrete-time systems described in discrete time by difference equations, continuous-time systems described by ordinary differential equations (ODEs), and differential algebraic equations (DAEs).
For example, the module allows to describe a general time-variant continuous-time nonlinear model given by a DAE of the form
\begin{align}
\dot{x}(t)&=f(t,x(t),z(t),u(t),p),\nonumber\\
0&=q(t,x(t),z(t),u(t),p),\label{eq:nonlin_model}\\
y(t)&=h(t,x(t),z(t),u(t),p)\nonumber.
\end{align}
Here $x(t)\in\mathbb{R}^{n_x}$ is the differential state vector, $z(t)\in\mathbb{R}^{n_z}$ the vector of algebraic states, $u(t)\in\mathbb{R}^{n_u}$ is the input vector and $p\in\mathbb{R}^{n_p}$ is a vector of parameters. The function $f\colon\mathbb{R}\times\mathbb{R}^{n_x}\times\mathbb{R}^{n_z}\times\mathbb{R}^{n_u}\times\mathbb{R}^{n_p}\rightarrow\mathbb{R}^{n_x}$ represents the ODEs of the model, and the function $q\colon\mathbb{R}\times\mathbb{R}^{n_x}\times\mathbb{R}^{n_z}\times\mathbb{R}^{n_u}\times\mathbb{R}^{n_p}\rightarrow\mathbb{R}^{n_z}$ describes the algebraic equations forming a semi-explicit DAE system overall.\footnote{Note that HILO-MPC only supports semi-explicit index 1 DAE systems.\cite{kunkel2006differential} Index 1 indicates DAE systems that can be transformed into a pure ODE system by differentiating the algebraic equations once.} The function $h\colon\mathbb{R}\times\mathbb{R}^{n_x}\times\mathbb{R}^{n_z}\times\mathbb{R}^{n_u}\times\mathbb{R}^{n_p}\rightarrow\mathbb{R}^{n_y}$ describes the measurement equations mapping to some measurable quantity of the system. If these measurement equations are not defined during the setup of the model, the controllers and estimators using the model assume that all states are measured.
When necessary, continuous-time models can be easily discretized using the \mintinline{python}{model.discretize(...)} method. Available discretization methods are explicit Runge-Kutta methods up to the 4th order and implicit collocation schemes.\cite{Biegler2010} For the explicit Runge-Kutta methods, a series of Butcher tableaus is available, while the implicit collocation schemes only support Gauss-Legendre polynomials and Gauss-Radau polynomials for the calculation of the collocation points.
For some applications and control and estimation methods a linear model might be necessary. The direct creation of linear models is also possible by supplying the required matrices during model setup. If only a nonlinear model is available, this can be easily linearized with respect to an equilibrium point using \mintinline{python}{model.linearize(...)}.
\begin{figure*}[htb]
\captionsetup[figure]{justification=centering,font=large}
\centering
\inputpdf{bike_simple}
\caption{Simple single-track/bicycle car model.}
\label{fig:bike_simple}
\end{figure*}
To underline the simplicity of defining a model, we consider a simple single-track car/bicycle (see Fig.~\ref{fig:bike_simple}) model which will be used as an example model in the following sections
\begin{align*}
\dot{p}_\text{x} &= v \cos(\phi(t) + \beta), \\
\dot{p}_\text{y} &= v \sin(\phi(t) + \beta), \\
\dot{v} &= a, \\
\dot{\phi} &= v/l_\text{r} \sin(\beta), \\
\beta & = \arctan\left(\frac{l_r}{l_\text{r} + l_\text{f}}\tan(\delta)\right).
\end{align*}
Here, $p_{\text{x}}$ and $p_{\text{y}}$ are the $x$- and $y$-coordinates of the vehicle center of mass, $v$ is the norm of the velocity of the center of mass, and $\phi$ is the orientation angle of the vehicle with respect to the $x$-coordinate. The inputs are the acceleration of the center of mass $a$ and the steering angle $\delta$. The parameter $l_\text{r}$ is the distance between the center of mass and the rear wheel, and $l_\text{f}$ is the distance between the center of mass and front wheel. The corresponding model can be set up as follows
\begin{minted}[bgcolor=bg,breaklines]{python}
from hilo_mpc import Model
# Initialize empty model
model = Model(name='Bike')
# Define model equations
equations = """
# ODEs
dpx/dt = v(t)*cos(phi(t) + beta)
dpy/dt = v(t)*sin(phi(t) + beta)
dv/dt = a(k)
dphi/dt = v(t)/lr*sin(beta)
# Algebraic equations
beta = arctan(lr/(lr + lf)*tan(delta(k)))
"""
model.set_equations(equations=equations)
# Sampling time in seconds
dt = 0.01
model.setup(dt=dt)
model.set_initial_conditions(x0=[0, 0, 0, 0])
\end{minted}
Note that the model equations are supplied in the form of a string. HILO-MPC is able to parse the most common functions and operations in this format, while it also provides the flexibility to use other input formats, e.g., by explicitly defining the inputs and states of the system.
Units, labels and descriptions of all variables can also be set for convenience and for use in the plotting functionality. For a more in-depth description of available functionalities, refer to the documentation.
Once the initial conditions are set, the model can be used for simulation by running \mintinline{python}{model.simulate(u=u, p=p)}. Note that, for every integration interval given by the sampling time $\mathrm{d}t$ the input for the continuous times system is kept constant, as it is commonly done.
One can easily store, access, and plot solutions generated by simulations. Every time the method \mintinline{python}{model.simulate(...)} is executed, the solution is saved in the \mintinline{python}{model.solution} object. This can be accessed similarly to a Python dictionary. To plot the solution, i.e., the evolution of states, inputs, and parameters with time, the user can use the method \mintinline{python}{model.solution.plot()}.
\subsection{Machine Learning Module}\label{sec:learning}
The machine learning module is responsible for defining and training machine learning models. Machine learning models are data-driven models that are used to map selected inputs $v \in \mathbb{R}^{n_v}$, also called features, to outputs $l \in \mathbb{R}^{n_l}$, also called labels. The machine learning models can be used to describe dynamical systems, unknown functions, constraints, cost functions, or a controller (see Fig.~\ref{fig:scheme}).
Two machine learning approaches/models are currently integrated: artificial feedforward fully connected neural networks and Gaussian processes. Further approaches are under development.
\subsubsection{Artificial Neural Networks}
We briefly review artificial neural networks (ANNs) to introduce the required notation and concepts.
\begin{figure}
\centering
\inputpdf{ann}
\caption{Simple feedforward neural network with four features (inputs), one hidden layer and one label (output).}
\label{fig:ann}
\end{figure}
Inspired by the neural circuits in brains,\cite{McCulloch1943} ANNs can be represented as a collection of interconnected nodes, the so-called neurons, that are usually organized into layers. We focus on feedforward fully connected ANNs (Figure~\ref{fig:ann}). Each layer of the ANN is mathematically represented b
\begin{equation}
z_i=\sigma_i\left(W_iz_{i-1}+b_i\right),
\end{equation}
where $z_i\in\mathbb{R}^{n_{z_i}}$ is the output vector of layer $i$ with the features $v$ being the input to the first layer (i.e., $v=z_0$) and the labels $l$ being the output of the last layer. The function $\sigma_i\colon\mathbb{R}^{n_{z_i}}\rightarrow\mathbb{R}^{n_{z_i}}$ is the activation function.
Common nonlinear activation functions are the sigmoid function, hyperbolic tangent, or the ReLU function.\cite{Ding2018} The training of the ANN is achieved by iteratively adapting the weight matrices $W_i\in\mathbb{R}^{n_{z_i}\times n_{z_{i-1}}}$ and the bias vectors $b_i\in\mathbb{R}^{n_{z_i}}$, so that a user-defined loss function is minimized. For regression problems, one of the most commonly used loss functions is the mean squared error (MSE)
\begin{equation*}
\mathrm{MSE}=\dfrac{1}{n_l}\sum_{i=1}^{n_l}(l_i-\hat{l}_i)^2,
\end{equation*}
where $\hat{l}_i$ are the values of the labels predicted by the ANN. Other loss functions supported by HILO-MPC include the mean absolute error (MAE) or the Huber loss. Optimization algorithms like gradient descent or stochastic gradient descent are used to minimize the loss. A number of minimization algorithms can be interfaced in HILO-MPC, e.g., the Adam algorithm, a modification of stochastic gradient descent. These minimization algorithms return a gradient vector of the loss with respect to the output $l$ of the ANN. By applying the backpropagation algorithm, this gradient vector can then be used to update the weight matrices $W_i$ and the bias vectors $b_i$ throughout the ANN.\cite{Rumelhart1986} The training is stopped when there is no significant change in the loss over a defined number of iterations or if the number of maximum iterations is reached.
In HILO-MPC, the training of the ANNs itself is performed by the respective machine learning libraries, i.e., currently TensorFlow or PyTorch. The specific parameters and characteristics for the training of the ANN (e.g., epochs, batch size, loss function, etc.) as well as the training data are automatically processed and forwarded to the chosen machine learning library. After the training of the ANN is completed, all relevant information is extracted, and an equivalent CasADi graph is created for later use in the model or model-based controllers and estimators.
The following example shows how to define an ANN consisting of two layers with 10 neurons each, using a sigmoid function as activation function, and how to train the resulting model
\begin{minted}[bgcolor=bg,breaklines]{python}
from hilo_mpc import ANN, Dense
# Initialize NN
ann = ANN(features, labels)
ann.add_layers(Dense(10, activation='sigmoid'))
ann.add_layers(Dense(10, activation='sigmoid'))
# Add training data set
ann.add_data_set(df)
# Set up NN
ann.setup()
# Train the NN
batch_size = 64
epochs = 1000
ann.train(batch_size, epochs, validation_split=.2, patience=100, scale_data=True)
\end{minted}
where \mintinline{python}{Dense(...)} creates a fully connected layer. One can simply replace specific variables of a model with the predictions of the ANN by the command \mintinline{python}{model.substitute_from(ann)}. The variables to be replaced will be automatically detected and exchanged with the corresponding predictions of the ANN. Alternatively, it is also possible to directly add the ANN to the model via summation or subtraction.
\subsubsection{Gaussian Processes}
A Gaussian process (GP) is a probability distribution over a function in which any finite subset of random variables has a joint Gaussian distribution.\cite{Rasmussen2006}
Besides being less prone to overfitting, one key advantage of using a GP is that it naturally measures the output uncertainty by estimating the variance of the predictions. GPs can be used to approximate input-output mappings
\begin{equation*}
l=f(v)+\varepsilon,
\end{equation*}
where the output $l\in\mathbb{R}$ is affected by some zero-mean normally distributed measurement noise $\varepsilon\sim\mathcal{N}(0,\sigma_n^2)$ with variance $\sigma_n^2$.
Starting from a given prior distribution, described by a mean function and a covariance function, the GP is trained on observed data (see Figure~\ref{fig:gp_training}).
\begin{figure}
\centering
\inputpdf{gp}
\caption{Simple example of a GP fitting to a sinusoidal function. The left figure shows a non-zero prior mean with $1\sigma$ standard deviation. The right figure shows the posterior of the GP conditioned on the measurements with optimal hyperparameters. The measurements are affected by a zero-mean white noise.}
\label{fig:gp_training}
\end{figure}
In general, both the prior mean function and the prior covariance function depend on so-called hyperparameters. During the training, the hyperparameters are obtained via optimization, i.e., the hyperparameters are chosen such that the resulting GP can be used to infer the input-output mapping given available training data. Typically, the prior mean function is assumed to be zero,\cite{Kocijan2003} but HILO-MPC also supports other prior mean functions. The covariance function, or kernel, is often assumed to be a smooth function that is infinitely differentiable. A number of covariance functions that meet this criterion are provided, e.g., the stationary squared exponential and rational quadratic kernels. The prior means, as well as the kernels, can be combined with each other through summation or multiplication.
The objective of the optimization problem that underlies the training step is usually to maximize the log marginal likelihood\cite{Rasmussen2006}
\begin{equation*}
\log\left(p(l|V)\right)=-\dfrac{1}{2}l^\Tr\left(K+\sigma_n^2I\right)^{-1}l-\dfrac{1}{2}\log\left(\left|K+\sigma_n^2I\right|\right)-\dfrac{n}{2}\log\left(2\pi\right),
\end{equation*}
where $V\in\mathbb{R}^{n_v\times n_d}$ describes the matrix containing the $n_d$ training inputs, $K\in\mathbb{R}^{n_d\times n_d}$ is the covariance matrix resulting from the used kernel and the supplied training data, and $I$ is the identity matrix of corresponding size. The element $(i,j)$ of $K$ is $k(v_i,v_j)$, where $k\colon\mathbb{R}^{n_v}\times\mathbb{R}^{n_v}\rightarrow\mathbb{R}$ is the kernel. The predictive mean $\bar{f}$ and variance $\sigma^2$ of the trained GP can be obtained for test data $w$ as follows
\begin{align*}
\bar{f}(w)&=\tilde{k}^\Tr(K+\sigma_n^2I)^{-1}l,\\
\sigma^2(w)&=k(w,w)-\tilde{k}^\Tr(K+\sigma_n^2I)^{-1}\tilde{k},
\end{align*}
where the $i$-th element of $\tilde{k}\in\mathbb{R}^{n_d}$ is $k(v_i,w)$. The following example outlines how GPs can be setup
\begin{minted}[bgcolor=bg,breaklines]{python}
from hilo_mpc import GP, SquaredExponentialKernel
# Initialize kernel
kernel = SquaredExponentialKernel(variance=0.002)
# Initialize GP
gp = GP(features, labels, kernel=kernel)
# Add training data set
gp.set_training_data(train_in, train_out)
# Set up GP
gp.setup()
# Fit the GP
gp.fit_model()
\end{minted}
Similarly to ANNs, the GPs can be integrated in models.
\subsection{Control Module}
The control module realizes feedback controllers of proportional–integral–derivative (PID) type, linear quadratic regulators (LQR), as well as optimal and MPC controllers. The latter are outlined in the following.
\subsubsection{Model Predictive Control}
Model predictive control repeatedly solves a finite-horizon optimal control problem (OCP).\cite{grune2017nonlinear,rawlings2017model,findeisen2002introduction} Optimal control problems and MPC problems for continuous-time systems are realized in sampled-data fashion in HILO-MPC as outlined in the following.\cite{findeisen2003towards} Equivalent formulations for discrete-time nonlinear system are also possible. At a sampling time $t_0$ the optimal control problem to solve reads
\begin{mini!}
{u(\cdot)}{\int^{t_0+T}_{t_0} l(t,x(t),z(t),u(t),p) \mathrm{d}t + e(t_0+T,x(t_0+T),z(t_0+T),p),\label{subeq:obj1}}{\label{eq:problem-mpc}}{}
\addConstraint{\dot{x}(t)}{=f(t,x(t),z(t),u(t),p),\quad\label{subeq:model1}}{x(t_0)=\hat{x}(t_0)}
\addConstraint{0}{=q(t,x(t),z(t),u(t),p)}{}
\addConstraint{0}{\geq g(t,x(t),z(t),u(t),p)\label{subeq:path-con}}{}
\addConstraint{0}{\geq g_T(t_0+T,x(t_0+T),z(t_0+T),p)\label{subeq:terminal-con}}{}
\addConstraint{y(t)}{=h(t, x(t),z(t),u(t),p)\label{subeq:meas-eq}}{}
\addConstraint{u(t)}{=u(t_0 + T_c)\,\ \text{for} \,\ t\geq t_0 + T_c}{}
\addConstraint{}{\text{for} \quad t \in [t_0, t_0+T] \subset \mathbb{R}.}{}
\end{mini!}
Here, $u(\cdot)$ is the optimal input function, $l:\mathbb{R}\times\mathbb{R}^{n_x}\times\mathbb{R}^{n_z}\times\mathbb{R}^{n_u} \times \mathbb{R}^{n_p} \rightarrow \mathbb{R}$ is the stage cost, $e:\mathbb{R}\times\mathbb{R}^{n_x}\times\mathbb{R}^{n_z} \times \mathbb{R}^{n_p} \rightarrow \mathbb{R}$ the terminal cost, $T$ is the prediction horizon, and $T_c$ the control horizon. The vector $\hat{x}(t_0)$ contains the measured or estimated states at the current sampling time $t_0$ and $t$ is the time variable. The equation $g:\mathbb{R}\times\mathbb{R}^{n_x}\times\mathbb{R}^{n_z}\times\mathbb{R}^{n_u} \times \mathbb{R}^{n_p} \rightarrow \mathbb{R}^{n_g}$ represents path constraints and $g_T:\mathbb{R}\times\mathbb{R}^{n_x}\times\mathbb{R}^{n_z} \times \mathbb{R}^{n_p} \rightarrow \mathbb{R}^{n_{g,\text{T}}}$ are terminal constraints.
\begin{remark}
In principle any of the equations in \eqref{eq:problem-mpc} could be completely or partially learned from data (see Figure~ \ref{fig:scheme}), using the learning approaches outlined in Section \ref{sec:learning}. Table \ref{tab:what_is_learned} gives an overview of other works where different components are learned. These cases can be handled using the \emph{machine learning} module.
\end{remark}
\begin{table*}[htb]
\caption{Selected, non-extensive overview of learned components of MPC and optimal control problems in the literature.}
\label{tab:what_is_learned}
\centering
\begin{tabular}{ll}
\toprule
What is learned? & Related work \\
\midrule
reference/desired output function \eqref{subeq:meas-eq} & Matschek et al.\cite{matschek2019learning}, Carron et al. \cite{carron2019data}, \dots \\
terminal constraints \eqref{subeq:terminal-con} & Rosolia et al.\cite{rosolia2017autonomous}, Rosolia and Borrelli\cite{rosolia2017learning}, Brunner et al.\cite{brunner2017repetitive}, \dots \\
path constraints \eqref{subeq:path-con} & Bujarbaruah et al.\cite{bujarbaruah2020learning}, Armesto et al.\cite{Leopoldo2017}, Holzmann et al. \cite{holzmann2022} , \dots\\
objective function \eqref{subeq:obj1} & Tamar et al.\cite{tamar2017learning}, Beckenbach et al.\cite{BECKENBACH201860}, Bradford and Imsland\cite{Bradford2018}, \dots \\
\bottomrule
\end{tabular}
\end{table*}
Solving Problem \eqref{eq:problem-mpc} directly is challenging, since $u(\cdot)$ and the constraints are infinite-dimensional. HILO-MPC uses \emph{direct approaches} to transform the problem into an equivalent finite-dimensional optimization problem. These approaches parameterize the input with a finite number of parameters, for example, using piece-wise constant inputs, and enforce the constraints only on a finite number of points. The resulting problem becomes
\begin{mini!}
{\mathbf{u}}{\sum_{i = k}^{N} \int_{t_i}^{t_{i} + \Delta t} l(t,x(t),z(t),u_i,p) \mathrm{d}t + e(t_k+T,x(t_k+T),z(t_k+T),p),}{\label{eq:disc-mpc}}{}
\addConstraint{x(t_{i+1})}{= x(t_{i}) + \int_{t_i}^{t_{i} + \Delta t} f(t,x(t),z(t),u_i,p) \mathrm{d}t}{}
\addConstraint{x(t_k)}{=\hat{x}(t_k)}{}
\addConstraint{0}{=q(t_i,x(t_i),z(t_i),u_i,p)}{}
\addConstraint{0}{\geq g(t_i,x(t_i),z(t_i),u_i,p)}{}
\addConstraint{0}{\geq g_T(t_k+T,x(t_k+T),z(t_k+T),p)}{}
\addConstraint{y(t_i)}{=h(t_i, x(t_i),z(t_i),u_i,p)}{}
\addConstraint{u_i}{=u_{N_c},\,\ \text{for} \,\ i\geq N_c}{}
\addConstraint{}{\text{for} \,\ i \in [k, N] \subset \mathbb{N}_0,}{}
\end{mini!}
where $\Delta t$ is the sampling time, $N_c = \text{ceil}(T_c/\Delta t)$ is the control horizon and $\mathbf{u}=\begin{bmatrix}u_k&\dots&u_{N_c}\end{bmatrix}$ is the sequence of piece-wise constant inputs applied to the plant, i.e., $u(t) = u_i, \,\ \forall t \in [t_i,\,t_i + \Delta t) \subset \mathbb{R}$. The function $\text{ceil}\colon\mathbb{R}\rightarrow\mathbb{Z}$ is the ceiling function, i.e., a function that returns the smallest integer that is larger than a given real number.
HILO-MPC also implements the multiple shooting approach.\cite{BOCK19841603} The integration of the system model in between the shooting points can be done with Runge-Kutta methods of various order, orthogonal collocation,\cite{Oh1977} or the dynamics can be integrated with CVODES or IDAS solvers.\cite{Hindmarsh2005} The default method is orthogonal collocation, while any other method can be selected if needed. Note that for simplicity piece-wise, constant inputs are assumed at every control interval, which can be generalized to introducing virtual inputs.
Hence, the real formulation of Problem \eqref{eq:problem-mpc} depends on which method is used for the discretization. Here, no further details of the formulation of the approximated problem are provided. See the cited papers for more information.
\begin{remark}
Note that it is also possible to optimize the sampling intervals. This will add to Problem \eqref{eq:disc-mpc} the vector $\begin{bmatrix}\Delta t_k&\dots&\Delta t_{N}\end{bmatrix}$ as optimization variable. This allows the solution of minimum time problems, and optimal sampling time problems. This frequently appears for example in network controlled systems.\cite{heemels2010networked,lucia2016predictive}
\end{remark}
The stage cost $l(\cdot)$ and arrival cost $e(\cdot)$ depend on the type of optimal control formulation/MPC formulation implemented. In the following we sketch different MPC formulations contained in the toolbox. For simplicity of explanation, we focus on quadratic cost functions; however, HILO-MPC is not limited to those.
\paragraph{MPC for set point tracking}
In set point tracking MPC\cite{Matschek2019} the variables $x$, $u$ and $y$ are desired to track a known reference, which can be formulated in terms of the cost function and terminal cost, for example, as
\begin{align*}
l(x(t),u(t),y(t)) &= \Vert x(t) - x_r\Vert^2_Q + \Vert u(t) - u_r\Vert^2_R + \Vert y(t) - y_r\Vert^2_P, \\
e(x(t_0+T),y(t_0+T)) &= \Vert x(t_0+T) - x_r\Vert^2_{Q_{\text{T}}} + \Vert y(t_0+T) - y_r\Vert^2_{P_{\text{T}}}.
\end{align*}
Here $x_r$, $u_r$, and $y_r$ are fixed references and $Q \in \mathbb{R}^{n_x \times n_x}$, $R \in \mathbb{R}^{n_u \times n_u}$, $P \in \mathbb{R}^{n_y \times n_y}$, $P_{\text{T}} \in \mathbb{R}^{n_y \times n_y}$, and $ Q_{\text{T}} \in \mathbb{R}^{n_x \times n_x}$ are weighting matrices.
To outline the simplicity to formulate MPC problems, consider the one-track car model as discussed in section \ref{sec:modeling_module}. The goal is that the car should track a set point reference speed $v_{\text{r}} = 2 \,\ \mathrm{m}/\mathrm{s}$. We consider a prediction horizon of 20 steps. The corresponding MPC controller can be formulated as
\begin{minted}[bgcolor=bg,breaklines]{python}
from hilo_mpc import NMPC
nmpc = NMPC(model)
nmpc.horizon = 20
nmpc.quad_stage_cost.add_states(names=['v'], ref=2, weights=10)
nmpc.quad_term_cost.add_states(names=['v'], ref=2, weights=10)
nmpc.setup()
\end{minted}
The resulting MPC controller can be easily used in a control loop
\begin{minted}[bgcolor=bg,breaklines]{python}
# Choose number of steps
n_steps = 100
# Begin simulation loop
for step in range(n_steps):
# find optimal input
u = nmpc.optimize(x0)
# Simulate the plant
model.simulate(u=u)
# Get the current states
x0 = sol['xf']
\end{minted}
The solution of the MPC (e.g., input and state sequences) are stored in the \mintinline{python}{mpc.solution()} object.
\paragraph{MPC for trajectory-tracking}
For trajectory tracking problems the variables need to track a time-varying reference.\cite{Matschek2019} Considering a quadratic cost function, this can be formulated as
\begin{align*}
\label{eq:mpc-traj-tracking}
l(x(t),u(t),y(t)) &= \Vert x(t) - x_r(t)\Vert^2_Q + \Vert u(t) - u_r(t)\Vert^2_R + \Vert y(t) - y_r(t_i)\Vert^2_P, \\
e(x(t_0+T),y(t_0+T)) &= \Vert x(t_0+T) - x_r(t_0+T) \Vert^2_{Q_{\text{T}}} + \Vert y(t_0+T) - y_r(t_0+T) \Vert^2_{P_{\text{T}}}.
\end{align*}
Here $x_r : \mathbb{R} \rightarrow \mathbb{R}^{n_x}$, $u_r: \mathbb{R} \rightarrow \mathbb{R}^{n_u}$, and $y_r: \mathbb{R} \rightarrow \mathbb{R}^{n_y}$ are time-varying references. For example, a trajectory tracking problem for a single-track car model can be defined by
\begin{minted}[bgcolor=bg,breaklines]{python}
from hilo_mpc import NMPC
nmpc = NMPC(model)
nmpc.horizon = 20
t = nmpc.create_time_variable()
traj_x = 30 - 14 * cos(t)
traj_y = 30 - 16 * sin(t)
nmpc.quad_stage_cost.add_states(names=['px', 'py'], ref=[traj_x, traj_y], weights=[10, 10], trajectory_tracking=True)
nmpc.quad_term_cost.add_states(names=['px', 'py'], ref=[traj_x, traj_y], weights=[10, 10], trajectory_tracking=True)
nmpc.setup()
\end{minted}
\paragraph{MPC for path following}
While in the MPC for trajectory tracking both the \emph{value of the reference and time are fixed simultaneously}, MPC for path following exploits additional degrees of freedom by choosing \emph{when} to be where on the path.\cite{Matschek2019,faulwasser2016} Do do so, we augment the model with a \emph{virtual path state}
\begin{mini!}
{u(\cdot),u_{\text{pf}}(\cdot)}{\int_{t_0}^{t_0+T} l(t,x(t),z(t),u(t),p) \mathrm{d}t + e(t_0+T,x(t_0+T),z(t_0+T),p),}{}{}
\addConstraint{\dot{x}(t)}{=f(t, x(t),z(t),u(t),p),\quad}{x(t_0)=\hat{x}(t_0)}
\addConstraint{0}{=q(t, x(t),z(t),u(t),p)}{}
\addConstraint{\dot{\theta}}{=u_{\text{pf}},\quad}{\theta(t_0)=0}
\addConstraint{0}{\geq g(t,x(t),z(t),u(t),p, \epsilon)}{}
\addConstraint{0}{\geq g_T(t_0+T,x(t_0+T),z(t_0+T),p)}{}
\addConstraint{y(t)}{=h(t, x(t), z(t), u(t), p)}{}
\addConstraint{}{\text{for} \,\ t \in [t_0, t_0+T] \subset \mathbb{R}.}{}
\end{mini!}
Here $\theta \in \mathbb{R}^{n_\theta}$ is a virtual path state vector and $u_{\text{pf}} \in \mathbb{R}^{n_{u,\theta}}$ is the virtual input that the controller can choose. Hence, the objective function becomes
\begin{align*}
l(x(t),u(t),y(t)) &= \Vert x(t) - x_r(\theta(t))\Vert^2_Q + \Vert u(t) - u_r(\theta(t))\Vert^2_R + \Vert y(t) - y_r(\theta(t))\Vert^2_P, \\
e(x(t_0+T),y(t_0+T)) &= \Vert x(t_0+T) - x_r(\theta(t_0+T)) \Vert^2_{Q_{\text{T}}} + \Vert y(t_0+T) - y_r(\theta(t_0+T)) \Vert^2_{P_{\text{T}}}.
\end{align*}
To force the controller to move only in the direction of the path usually a lower bound on $u_{\text{pf}}$ is added, i.e., $u_{\text{pf}}>u_{\text{pf,min}}$ with $u_{\text{pf,min}} \in \mathbb{R}^{n_{u,\theta}}_{+}$.
It is also possible to track a constant $u_{\text{pf}}$, so that the MPC tries to maintain a constant \emph{speed} of the virtual state
\begin{align*}
l(x(t),u(t),y(t)) &= \Vert x(t) - x_r(\theta(t))\Vert^2_Q + \Vert u(t_i) - u_r(\theta(t))\Vert^2_R + \Vert y(t) - y_r(\theta)\Vert^2_P + \Vert u_{\text{pf}} - u_{\text{pf,ref}}\Vert_{R_{\text{pf}}}^2, \\
e(x(t_0+T),y(t_0+T)) &= \Vert x(t_0+T) - x_r(\theta(t_0+T))\Vert^2_Q + \Vert y(t_0+T) - y_r(\theta(t_0+T))\Vert^2_P.
\end{align*}
In comparison to other toolboxes, path following MPC/optimal control problems can be easily formulated. The user needs to activate the path following mode for the desired variables, as shown in the following
\begin{minted}[bgcolor=bg,breaklines]{python}
from hilo_mpc import NMPC
nmpc = NMPC(model)
nmpc.horizon = 20
theta = nmpc.create_path_variable()
traj_x = 30 - 14 * cos(theta)
traj_y = 30 - 16 * sin(theta)
nmpc.quad_stage_cost.add_states(names=['px', 'py'], ref=[traj_x, traj_y], weights=[10, 10], path_following=True)
nmpc.quad_term_cost.add_states(names=['px', 'py'], ref=[traj_x, traj_y], weights=[10, 10], path_following=True)
nmpc.setup()
\end{minted}
Note that the toolbox allows to mix the previous problem formulations with minimum effort, e.g., the user can track some constant references for some of the variables, while performing path tracking for the other variables.
\paragraph{Soft constraints}
Soft constraints are typically used in optimal control/MPC to avoid infeasibility, conservative solutions due to active constraints, or to speed up computations.\cite{kerrigan2000soft,richards2015fast} When soft constraints are selected, slack variables $\epsilon_{\text{p}} \in \mathbb{R}^{n_g}$ and $\epsilon_{\text{T}} \in \mathbb{R}^{n_{g,\text{T}}} $ are automatically added to the path and terminal constraints, respectively, as follows
\begin{mini!}
{u(\cdot), \epsilon_{\text{T}}, \epsilon_{\text{p}}}{\int_{t_0}^{t_0+T} l(\cdot) \mathrm{d}t + e(\cdot) + \Vert \epsilon_{\text{s}} \Vert^2_{E_{\text{p}}} + \Vert \epsilon_{\text{T}} \Vert^2_{E_{\text{T}}},}{\label{eq:problem-mpc-soft}}{}
\addConstraint{\dot{x}(t)}{=f(t,x(t),z(t),u(t), p),\quad}{x(0)=\hat{x}_0}
\addConstraint{0}{=q(t, x(t),z(t),u(t), p)}{}
\addConstraint{\epsilon_{\text{p}}}{\geq g(t,x(t),z(t),u(t), p)}{}
\addConstraint{\epsilon_{\text{T}}}{\geq g_T(t_0+T,x(t_0+T),z(t_0+T),p)}{}
\addConstraint{y(t)}{=h(t, x(t),z(t), u(t),p)}{}
\addConstraint{0}{\leq\epsilon_{\text{p}} \leq \epsilon_{\text{p,max}}, \quad 0 \leq \epsilon_{\text{T}} \leq \epsilon_{\text{T, max}}}{}
\addConstraint{}{\text{for} \,\ t \in [t_0, t_0+T] \subset \mathbb{R},}{}
\end{mini!}
where and $E_{\text{p}}\in \mathbb{R}^{n_g \times n_{g}}$ and $E_{\text{T}}\in \mathbb{R}^{n_{g,\text{T}} \times n_{g,\text{T}}}$ are weighting matrices that limit the increase of the slack variables and can be chosen by the user, and $\epsilon_{\text{p,max}}$ and $\epsilon_{\text{T,max}}$ are the maximum constraint violations of path and terminal constraints, respectively.
\subsection{Estimator Module}
Estimating states and parameters from measurements is indispensable in control if not all variables can be accessed directly. The estimator module provides access to a moving horizon estimator, Kalman filters, and particle filters.
\subsubsection{Moving Horizon Estimation}
Moving horizon estimators (MHE) obtain state estimates based on the solution of an optimization problem similar to the MPC problem.\cite{rawlings2017model,johansen2011introduction,findeisen2003state} We focus on moving horizon estimators operating on equidistant sampling times. At these sampling times measurements are taken. Due to the discrete measurements, the objective function in an MHE is usually given in discrete form. For simplicity we indicate with $(\cdot)_{i|k}$ the variable at time $t_i$, i.e., for the MHE problem solved at time $t_k$ we obtain
\begin{mini!}
{x_{k-N|k},p_k, w_{(\cdot)|k},z_{(\cdot)|k}}{\left\Vert \begin{bmatrix} x_{k-\bar{N}|k} - \hat{x}_{k-N|k} \\ p_{k} - \hat{p}_{k} \end{bmatrix} \right\Vert^2_{P_k} + \sum^k_{i=k-\bar N} \Vert \hat{y}_i - y_{i|k} \Vert^2_R + \Vert w_{i|k}\Vert^2_W,}{\label{eq:problem-mhe}}{}
\addConstraint{x_{i+1|k}}{=x_{i|k} + \int_{t_i}^{t_i + \Delta t} \left( f(t,x(t),z(t),\hat{u}(t), p_k) \right) \mathrm{d}t + w_{i|k} \label{subeq:mhe-model1}}{}
\addConstraint{y_{i|k}}{=h(t_i,x_{i|k},z_{i|k},\hat{u}_i,p_k) + v_{i|k}}{}
\addConstraint{0}{\geq g(t_i,x_{i|k},u_i)}{}
\addConstraint{}{\text{for} \,\ i \in [k-\bar{N}, k], \,\ k,\bar{N} \in \mathbb{N},}{}
\end{mini!}
where $\bar{N}$ is the horizon length, $\hat{y}_i$ the output measurements, and $\hat{u}_i$ the input measurements at time $t_i$. The first term of the objective function is the \emph{arrival cost}, while the second and third weigh the measurement and state noise, respectively.\cite{rawlings2017model,allgower1999nonlinear} The matrices $R \in \mathbb{R}^{n_y \times n_y}$, $W \in \mathbb{R}^{n_x \times n_x}$ and $P_k \in \mathbb{R}^{(n_x+n_p) \times (n_x + n_p)}$ are the weighting matrices for the outputs, state noise, and arrival cost. The optimization variables are: the state $x_{k-\bar{N}|k}$, i.e., the state at the beginning of the horizon, the state noise $w_{(\cdot)|k} = \{ w_{i|k}, \,\ \forall i \in [k-\bar{N},k] \}$, the algebraic states (for DAE systems) $z_{(\cdot)|k} = \{ z_{i|k}, \,\ \forall i \in [k-\bar{N},k] \}$ and the system parameters $p_k$. Note that the parameters are considered constant in the horizon, but can be updated every time the optimization is run, to adapt to the new measurements.
For the single-track car model, an MHE with $R=W=P_{k}=10 I$ (where $I$ is the identity matrix of appropriate dimensions) can be easily defined as follows
\begin{minted}[bgcolor=bg,breaklines]{python}
from hilo_mpc import MHE
mhe = MHE(model)
mhe.horizon = 20
mhe.quad_arrival_cost.add_states(weights=[10,10,10,10], guess=x0_est)
mhe.quad_stage_cost.add_measurements(weights=[10,10,10,10])
mhe.quad_stage_cost.add_state_noise(weights=[10,10,10,10]])
mhe.setup()
\end{minted}
After this setup, it can be deployed for estimation simply by
\begin{minted}[bgcolor=bg,breaklines]{python}
# Set number of steps
n_steps = 100
# Simulation/estimation loop
for step in range(n_steps):
# Simulate plant with input
model.simulate(u=u)
# Get measurements
y_meas = model.solution['y'][:, -2]
# Pass the measurements to the MHE
mhe.add_measurements(y_meas, u_meas=u)
# Estimate states
x0 = mhe.estimate()
\end{minted}
Similar to other objects, the solution of the MHE is stored in \mintinline{python}{mhe.solution()}.
\subsubsection{Kalman Filters and Particle Filters}
Besides an MHE, Kalman filters (KFs) and particle filters (PFs) are provided for estimation. KFs allow for the estimation of observable states via available measurement data. Generally, a KF consists of two steps: the prediction step and the update step. In the prediction step, the estimated states from the previous iteration are propagated through the model dynamics to obtain preliminary values for the states at the current time step, the so-called \emph{a priori} estimates. These \emph{a priori} estimates are then updated in the update step using the measurement data to obtain \emph{a posteriori} estimates.
HILO-MPC contains KFs in various formulations. An unscented KF\cite{Wan2000} can simply be integrated b
\begin{minted}[bgcolor=bg,breaklines]{python}
from hilo_mpc import UKF
ukf = UKF(model)
ukf.setup()
\end{minted}
In a similar fashion particle filters can be formulated and used.
\subsection{Embedded Module}
\label{sec:emb-mod}
Implementing optimization-based controllers/MPC controllers on embedded systems requires a tailored implementation. By now, a series of tailored automatic code generation tools for MPC exist.\cite{Houska2011a,Verschueren2019,zometa2013,domahidi2014forces}
HILO-MPC provides an interface for the code generation of MPC for linear time-invariant discrete-time model
predictive control, tailored for small embedded systems,
like microcontrollers.\cite{zometa2013}
The code generation is limited to discrete-time linear time-invariant systems of the form
\begin{equation}
\label{eqn:system}
x^+ = Ax + Bu
\end{equation}
subject to input and state constraints
$\underline{u} \leq u \leq \overline{u}$, $\underline{x} \leq x \leq \overline{x}$,
where $x$ and $u$ denote the state and input at the current sampling time,
and $x^+$ denotes the state at the next sampling time.
The considered linear MPC problem formulation is
\begin{mini}
{\mathbf{u}}{\frac{1}{2} \sum\limits_{j=0}^{N-1} (\|x_j\|^2_Q + \|u_j\|^2_R)
+ \frac{1}{2} \|x_N\|^2_P,}{\label{eqn:mpcproblem}}{}
\addConstraint{x_{j+1}}{= A x_j + B u_j,\qquad}{j = 0,\dots, N-1}
\addConstraint{\underline{u}}{\leq u_j \leq \overline{u},}{j = 0,\dots, N-1}
\addConstraint{\underline{x}}{\leq x_j \leq \overline{x},}{j = 0,\dots, N-1}
\addConstraint{x_{0}}{=x,}{}
\end{mini}
where the integer $ N \geq 2$ is the prediction horizon, the matrices $Q \inRpow{n_x
\times n_x}$, $R \inRpow{n_u \times n_u}$ and $P \inRpow{n_x \times n_x}$ are the state,
input, and terminal weighting matrix, respectively. The input sequence
$\vb{u} = \begin{bmatrix}u_{0}&\dots&u_{N-1}\end{bmatrix}$ is the optimization variable.
For automatic code generation of the problem \eqref{eqn:mpcproblem} tailored towards microcontrollers, we rely on \ensuremath{\mu}\emph{AO-MPC},\cite{zometa2013} a code generation tool
that uses an augmented Lagrangian and Nesterov's fast gradient method.\cite{zometa2012, koegel2011}
By design, the algorithm's implementation is simple
and relies only on matrix-vector operations (i.e., no divisions),
it can easily be warm started and computes good MPC inputs
in very few iterations of the algorithm.
For well-behaved problems, the computations can be made more efficient by using
fixed-point arithmetic (instead of
floating-point).
The code generation reformulates the MPC optimization problem (\ref{eqn:mpcproblem})
as a condensed parametric QP $\mathcal{P}(x)$
\begin{mini}
{\vb{u} \in \mathcal{U}}{\frac{1}{2} \vb{u}^\Tr H \vb{u} + \vb{u}^\Tr g(x),}{\label{eqn:qp}}{}
\addConstraint{\underline{\vb{z}}(x)}{\leq E \vb{u} \leq \overline{\vb{z}}(x),}{}
\end{mini}
where the constant Hessian matrix $H^\Tr\!=\!H > 0 \inRpow{Nn_u \times Nn_u}$ and the gradient vector $g(x)$
depend on the system (\ref{eqn:system}) and the objective function
in (\ref{eqn:mpcproblem}). Additionally, $g(x)$ depends on the current state $x$ (i.e., the parameter).
The set $\mathcal{U} = \{ \vb{u} \; | \; \underline{u} \leq u_j \leq \overline{u}, \; j\!=\!0,...,N-1\}$ defines the
input constraints.
The constant matrix $E$ and the constraint vectors $\underline{\vb{z}}(x)$ and
$\overline{\vb{z}}(x)$
depend on the
constraints in (\ref{eqn:mpcproblem}).
The generated code is highly portable C-code.
It is split into the data for the QP $\mathcal{P}(x)$ (e.g., the constant matrix $H$
and the parametric vector $g(x)$)
and the implementation of the optimization
algorithm to solve the QP.\cite{zometa2012, koegel2011}
\begin{figure*}
\centering
\begin{subfigure}{0.45\textwidth}
\inputpdf{codegen}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\inputpdf{hilsim}
\end{subfigure}
\caption{A scheme of the Embedded module. Left, the typical used of the module, including automatic code generation provided by \ensuremath{\mu}\emph{AO-MPC}. Right, a hardware-in-the-loop simulation used as example.}
\label{fig:hilo-muaompc}
\end{figure*}
Figure~\ref{fig:hilo-muaompc} depicts an overview of the embedded module.
Similarly to other modules, the \mintinline{python}{LMPC} class is initialized
using a \mintinline{python}{Model} object. In this case, a linear time-invariant discrete-time
model is used. Calling the \mintinline{python}{LMPC.setup()} method, specifying the solver
\mintinline{python}{'muaompc'}, will trigger the code generation procedure.
The code generation automatically creates C-code which can be compiled
to run on an embedded target. Additionally, a Python interface to the
generated C-code is also created. This Python interface is used by the \mintinline{python}{LMPC.optimize()} method to obtain the input that minimizes \ref{eqn:mpcproblem}. The basic use of the Embedded Module is exemplified in the following Python code
\begin{minted}[bgcolor=bg]{python}
from hilo_mpc import LMPC, Model
# The Model class uses the linear system matrices A, B, and
# the LMPC class uses the weighting matrices Q, R, P, constraints, etc.,
# to create the mpc object. Using the solver muaompc triggers the code generation
mpc.setup(nlp_solver='muaompc')
# the generated MPC code can be used directly on an embedded target using C,
# or via the Python interface as HILO-MPC does on the next line
u = mpc.optimize(x0=x)
\end{minted}
\section{Examples}
\label{sec:examples}
We underline the flexibility of the toolbox considering a series of examples, which can be also found in the repository, as Python files or Jupyter notebooks. Thus, we do not provide implementation details, just a high-level description of the problems and outline some of the results.
\subsection{Learning the System Dynamics - Racing a Minicar}
\label{sec:example-race-car}
We consider autonomous racing of a minicar, based on the results and models presented in Liniger~et~al.\cite{Liniger2015} The goal is to use model predictive control to follow a complex track using a path following formulation in presence of disturbances (see Figure \ref{fig:bike1}). An extended single-track model of the car, see \ref{fig:bike1} and section \ref{sec:modeling_module}, including nonlinear tire models and drive train models as identified and validated in experiments in Liniger~et~al.\cite{Liniger2015} is used. We extend the model adding a component describing the effect of lateral drag forces due to, for example, a strong wind. The overall model is represented by the following system of nonlinear differential equations
\begin{align*}
\dot{p}_x &= v_x \cos (\psi) - v_y \sin(\psi), \qua
\dot{p}_y = v_x \sin (\psi) + v_y \cos(\psi), \quad
\dot{\psi} = w, \\
\dot{v}_x &= \frac{1}{m}\left( F_{r,x}-F_{a,x}- (F_{f,y}-F_{a,y}) \sin(\delta) + m v_y \omega \right), \quad
\dot{v}_y = \frac{1}{m}\left( F_{r,y}-F_{a,y}- (F_{f,y}-F_{a,y}) \cos(\delta) - m v_x \omega \right), \\
\dot{\omega} &= \frac{1}{I_z} \left( F_{f,y}l_f \cos(\delta) - F_{r,y}l_r\right),
\end{align*}
where $p_x$ and $p_y$ are the coordinates of the center of gravity, $v_x$ and $v_y$ are longitudinal and lateral velocities of the center of gravity. The orientation is denoted by $\psi$ and the yaw rate by $\omega$. The control inputs are the motor duty cycle $d$ and steering angle $\delta$. The two parameters $l_f$ and $l_r$ are the distances from the center of gravity to the front axle and rear axle, respectively, $m$ is the mass of the car, and $I_z$ is the inertia. Figure~\ref{fig:bike1} shows the single-track car model. The path is the center of a racing track that has been interpolated using splines.
The tire forces are modeled with a simplified Pacejka Tire Model\cite{bakker1987tyre}
\begin{align*}
F_{f,y} &= D_f \sin(C_f \arctan(B_f \alpha_f)),\quad
F_{r,y} = D_r \sin(C_r \arctan(B_r \alpha_r)), \quad
F_{r,x} = (C_{m1}-C_{m2}v_x)d - C_r - C_d v^2_{x}, \\
\alpha_f &= - \arctan\left( \frac{w l_f + v_y}{v_x} \right) + \delta,\quad
\alpha_r = \arctan\left( \frac{w l_r - v_y}{v_x} \right)
\end{align*}
where $D_f$, $D_r$, $B_f$, $B_r$, $C_{m1}$, $C_{m2}$, $C_r$, and $C_d$ are parameters. The longitudinal and lateral drag forces are defined, respectively, as
\begin{align*}
F_{a,x} = 0.5 \,\ c_w \rho A v_x, \quad
F_{a,y} = 0.5 \,\ c_w \rho A v_{y,\text{wind}},
\end{align*}
where $c_w$ is the drag coefficient, $\rho$ is the air density, $A$ is the effective flow surface, and $v_{y,wind}$ is the lateral wind velocity.
The model used by the MPC does not have the drag effect. The goal is to learn this effect from data using a neural network and then augment the first principles model with a machine learning component that models the drag effect. After discretization, the hybrid model can be written as
\begin{equation}
x_{k+1} = f(x_k,u_k) + B^\Tr m(x_k,u_k),
\end{equation}
where $m(x_k,u_k)$ is an NN model. The features of the NN are $v_{y,\text{wind}}$ and $v_x$, the labels are correction terms for $\phi$, $v_x$, $v_y$, and $\omega$ via the weighting matrix $B^\Tr$. To show the effectiveness of the learning, we compare the results of MPC using the perfect model (i.e., with known drag force effects), the hybrid model (using the NN), and the model without drag forces. Furthermore, the measurements of position, velocity, and directions are affected by Gaussian noise and estimated using an unscented KF.
Figure \ref{fig:race_car} shows the results of the simulation. While the hybrid model has similar results as the perfectly-known model, the model without drag exits the race area after the fifth curve.
\begin{figure*}[htb
\captionsetup[sub]{justification=centering,font=large}
\centering
\begin{subfigure}[t]{.49\linewidth}
\inputpdf{bike}
\caption{Complex single-track/bicycle car model.}
\label{fig:bike1}
\end{subfigure}
\begin{subfigure}[t]{.49\linewidth}
\inputpdf{racing}
\caption{Race track and results.}
\label{fig:race_car}
\end{subfigure}
\caption{Example of a racing car driving on a racetrack following a given path.}
\end{figure*}
\subsection{Learning a Reference - Cooperative Robots}
\label{sec:cooperative-robots}
Inspired by\cite{Matschek2021a} we consider that a follower robot has to track the position of a leader robot, where the leader moves along a periodic but unknown trajectory (see Figure~\ref{fig:mobile-robots}).
\begin{figure*}[htb]
\centering
\inputpdf{mobile_robots}
\caption{Cooperative robot example. The follower should track the leader, who follows a periodic, yet unknown trajectory. The trajectory is learned from data using a GP.}
\label{fig:mobile-robots}
\end{figure*}
The objective is to learn the trajectory of the leader with GPs, which are used in the follower. Hence, in this case the machine learning model enters the cost function. The nonlinear dynamics of the robots are described with the following simple dynamics ODE
\begin{align*}
\dot{x}_1 &= u_1 \sin(x_3), \\
\dot{x}_2 &= u_1 \cos(x_3), \\
\dot{x}_3 &= u_2.
\end{align*}
Here $x_1$ and $x_2$ are the horizontal and vertical position of the root, $x_3$ its heading angle, $u_1$ is the speed, and $u_2$ the turning rate.
The resulting tracking problem takes the form \eqref{eq:problem-mpc} with the objective function \eqref{eq:mpc-traj-tracking}, where $x_r(\cdot)$ is the mean function of a GP trained on the data set collected from the position of the leader.
The trajectory generated by the leader results from applying the time-varying forces
$u_1(t) = 2 + 0.1 \sin(t)$ and $u_2(t) = 0.5 + \sin(t)$.
Figure \ref{fig:mobile-robots-results} shows the results of the reference learning and closed-loop simulations.
\begin{figure*}[htb]
\centering
\inputpdf{tot_mobile_robots}
\caption{Left column: learned GP. Right column: states of the follower robot following the learned references.}
\label{fig:mobile-robots-results}
\end{figure*}
\subsection{Learning the Controller - Spring Damper System}
\label{sec:spring-damper}
Solving an MPC requires the solution of a (nonlinear) optimization problem online.
For embedded applications with low computational power or applications that can use only a limited amount of energy (for example battery powered systems) this is often not possible. Hence, methods that provide at least a close-to-optimal solution without solving the optimization problem at every time step are necessary. This can be done using \emph{explicit} MPC approaches.\cite{alessio2009survey,besselmann2012explicit} One way to avoid the explicit solution is to learn the solution of an MPC offline, i.e., to learn the map $x \mapsto \rho_{\theta}(x)$ that (approximates) the implicit MPC control law, and then using the learned controller online (see Figure~\ref{fig:scheme}).\cite{Parisini1995,Karg2020,Maddalena2020,Cao2020,Chen2018,Cseko2015,Ponkumar2018}
In this way, the control action can be found with a simple and fast function evaluation.
We consider the control of a mass-spring-damper system using a learned controller. The models is given by
\begin{subequations}
\begin{align}
& \dot{x}_1 = x_2, \\
& \dot{x}_2 = \frac{1}{m} (u - k x_1 - d x_2) ,
\end{align}
\end{subequations}
where $x_1$ is the vertical position, $x_2$ the vertical velocity, $u$ the vertical force, and $k$, $d$ the system parameters. The equilibrium point is $x=(0,0)$, $u=0$. The objective is to maintain the reference $x_{\text{ref}}=[1,0]$ using a learned MPC.
In total 667 data points, obtained by simulations, are collected. We use the data to train an NN via HILO-MPC with three fully-connected layers, with 10 neurons each. The features of the NN are $x_1$ and $x_2$, and the label is the input $u$. We test the learned controller starting from a different initial condition $x(0)=(10,0)$. In Figure~\ref{fig:ml-controller-results} the simulation results are shown. The learned controller is able to bring the system to the reference like the original controller.
\begin{figure*}[htb]
\centering
\inputpdf{learned_mpc}
\caption{Comparison of the learned and exact MPC.}
\label{fig:ml-controller-results}
\end{figure*}
\subsection{Embedded MPC - HIL Simulation using an Arduino}
\label{sec:embedded}
We conclude by presenting a hardware-in-the-loop (HIL) simulation
based on the code generated by the embedded module (see
Subsection~\ref{sec:emb-mod}).
Figure~\ref{fig:hilo-muaompc} depicts the code generation and HIL simulation setup.
The generated C-code is cross compiled to run on an Arduino
microcontroller.
The system dynamics are simulated on a laptop computer using the nominal linear model $x^+=Ax+Bu$.
Using a serial interface (i.e. USB to UART), the state $x^+$
is transferred to the Arduino,
to compute a new sequence $\vb{u}$
by solving the parametric QP $\mathcal{P}(x)$.
Only the first part $u_0$ of this sequence is transferred by the Arduino to the computer via the serial interface
The system to be controlled is the
linearized
lateral dynamics of the race car.
The input is the steering angle $u=\delta$.
The state is defined as $x = [p_y \ \ \psi \ \ \omega]^Tr$,
where $p_y$ is the lateral position of the center of mass of the car,
and $\psi$ is the orientation
of the car with respect to the $X$-axis and $\omega = \dot \psi$
(refer to Figure~\ref{fig:bike1}).
The input is constrained to $-0.1 \leq u \leq 0.1$,
and the state $x_1 = \psi$ must satisfy $-0.1 \leq x_1 \leq 0.1$.
For the HIL simulation, we used an Arduino Nano 33 BLE.\cite{arduinonano}
\footnote{It has a 32-bit Cortex-M4F microcontroller core (with single precision floating-point unit)
running at 64 MHz.}
We use a discretization time of $15\,\text{ms}$, and a horizon length of $20$ steps.
The Arduino requires around $10\,\text{ms}$ to find (an approximate) solution
to the MPC problem on each iteration. The compiled C-code takes about
$20\,\text{kB}$ of flash memory.
Figure~\ref{fig:hil-sim-plot} shows the compared trajectories of states and input
for the HIL simulation (approximate solution on Arduino using single precision float)
and the exact solution computed by a laptop using HILO-MPC's
interface to CPLEX.
Note that neither the input nor the state constraints are violated.
The main differences are due to the early
termination of the embedded algorithm, and the use of
single precision float.
\begin{figure*}[htb]
\centering
\inputpdf{plot_embmpc}
\caption{Comparison of HIL simulation and exact simulation}
\label{fig:hil-sim-plot}
\end{figure*}
\section{Conclusions and Outlook}
\label{sec:conclusions}
While the use of machine learning approaches for model-based optimal control, such as predictive control and estimation are becoming increasingly important, currently there is no easy-to-use open-source tool available to support the researcher and developer. We outlined the concept and ideas behind HILO-MPC, a toolbox for fast development of optimal and predictive control and estimation methods that facilitates the use of machine learning models trained using PyTorch and TensorFlow for simulations and embedded applications. The concept is easily expandable and we underlined the simple use considering four examples.
The toolbox allows to consider different methods, spanning from model predictive and optimal control, moving horizon estimation, and Kalman filters to particle filters. We believe that it is a flexible, yet easy-to-use tool for research and teaching.
Future developments will allow using extended machine learning models such as recurrent neural networks, Bayesian neural networks and reinforcement learning, as well as extended control formulations, such as tube-based MPC, stochastic MPC, \cite{Mayne2005} and multi-mode MPC.\cite{bethge2018multi, morabito2019multi} Furthermore, we are expanding the toolbox towards code generation for embedded nonlinear MPC using the Al'brekht's method.\cite{Kallies2020a,Kallies2020b,Kallies2021}
\section*{Acknowledgements}
The authors thank Johanna Bethge, Michael Maiworm, Tim Zieger, Maik Pfefferkorn, Rudolph Kok, Sebasti{\'a}n Espinel-R{\'i}os and Janine Matschek for providing feedback and examples for HILO-MPC. Special thanks go to Lena Kranert for the cricket.
\section{Introduction}
Being able to model a dynamical system is fundamental in many engineering applications. Dynamical models can be constructed with first principles laws and/or data.
In the first case, physical laws, such as conservation of energy, matter and thermodynamics laws, are used. A relatively limited amount of data might also be used to fit the parameters of the model. These parameters usually have a physical meaning (e.g., heat transfer coefficients, friction coefficients, ...).
If only data is used, we generally talk about machine learning models. Generally, the fitted parameters do not have a physical meaning in these cases. Machine learning has recently obtained much attention since it allows to model complex or not well-understood systems, provided that a sufficiently large amount of data is available. For many decades the scarcity of data, feature extraction and computational power have been the main bottlenecks to the widespread use of machine learning models. The situation is rapidly changing thanks to digitization (e.g. internet-of-things and industry 4.0 initiatives), deep learning and cloud computing. Hence, machine learning is finding more and more applications, not only in research but in our everyday life. In recent years, the number of machine learning applications, publications, new methods and theoretical findings has increased rapidly. Furthermore, open-source libraries for machine learning such as TensorFlow\cite{tensorflow2015} and PyTorch\cite{pytorch} offer a low entry barrier to this technology and allow fast development of machine learning solutions. The combination of these factors boosted the quality and quantity of results in many fields such as medicine,\cite{Shen2017} autonomous driving,\cite{Arnold2019} transportation,\cite{Nguyen2018} internet-of-things,\cite{Mohammadi2018} image recognition\cite{Voulodimos2018} and many others. The control and optimization community also recognized the usefulness of machine learning for control applications, e.g., for optimal control and model predictive control (MPC), motivating a large number of contributions (see Hewing~et~al.\cite{Hewing2020} and references therein).
Machine learning libraries help the control practitioners to build machine learning models, but using these models in a control strategy requires ad-hoc solutions, which are complicated, prone to bugs and not easily transferable to other applications. HILO-MPC is an open-source Python library that aims at providing a way to easily implement machine learning models into a wide variety of control, optimization and estimation problems. HILO-MPC interfaces with PyTorch and TensorFlow to train neural networks and uses in-house code to train Gaussian processes. Once the models are trained, they can be used in a large variety of control or estimation problems. Furthermore, to speed up the development time, HILO-MPC provides a way to define and solve a broad spectrum of predefined optimal control and estimation problems quickly and with minimum effort. Table~\ref{tab:problems-hilo} summarizes the different problems that can be solved with version {1.0}{} of HILO-MPC. The philosophy that HILO-MPC follows is ease of use and modularity. Easy of use is achieved by providing an intuitive high-level interface for the problem definition. At the same time, modularity is obtained by ensuring effortless interfaces between the different tools present in the models (see Figure~\ref{fig:scheme}). This allows using the toolbox both for research and for teaching.
Comparing HILO-MPC with other similar toolboxes is challenging since it can solve a wide variety of control and estimation problems. Here, we consider only open-source toolboxes that can solve model predictive control problems for simplicity. The software ACADO\cite{Houska2011a} and ACADOS\cite{Verschueren2018} aim at efficiency and fast solution keeping embedded applications in mind. MPT3 focuses on computational geometry, which is particularly useful for some classes of robust MPC approaches.\cite{MPT3} YALMIP instead offers a broad spectrum of optimization problems, such as bilevel optimization and mixed-integer programming and some MPC schemes.\cite{YALMIP} The toolbox do-mpc implements robust multi-stage MPC and moving horizon estimation (MHE).\cite{LUCIA201751} MPCTools offers some interfaces that help build MPC and MHE control problems in CasADi.\cite{risbeck2015nonlinear,risbeck2016nonlinear} In Table~\ref{tab:comparison-toolboxes} we summarize the main differences between the toolboxes.
Using machine learning models in the previously mentioned toolboxes is not straightforward. HILO-MPC solves this problem by providing wrappers for machine learning libraries and some in-house learning libraries. Furthermore, HILO-MPC focuses on a light weight and intuitive interface that makes it easy to use also for beginners.
The backbone of HILO-MPC is CasADi.\cite{Andersson2019} CasADi is a tool for algorithmic differentiation and optimization, and it is used to build the models, optimization and estimation problems. When TensorFlow or PyTorch are used, HILO-MPC offers an easy way to interface with those libraries and automatically translate the models into CasADi-compatible structures. Hence, the CasADi problem is defined and solved. CasADi has interfaces with a wide range of solvers for linear and quadratic programming (CLP,\cite{john_forrest_2022_5839302} qpOASES\cite{Ferreau2014}), nonlinear programming such as IPOPT,\cite{wachter2006implementation} quadratic mixed-integer programming (CPLEX,\cite{cplex2009v12} GUROBI\cite{gurobi}), for non quadratic nonlinear mixed-integer problems (BONMIN,\cite{bonami2007bonmin} KNITRO\cite{nocedal2006knitro}) and large nonlinear problems WORHP.\cite{nikolayzik2010nonlinear}
Note that giving a comprehensive description of the toolbox is outside the scope of this article. For this, the reader is referred to the documentation.\footnote{link to webpage here} Here, we rather want to give a glimpse of the different problems HILO-MPC can solve and give an idea of the workflow and syntax with some examples.
The rest of this paper is structured as follows. In Section~\ref{sec:hilo-mpc} the different modules of the toolbox are described. Furthermore, we show example code snippets to illustrate the HILO-MPC syntax. In Section~\ref{sec:examples} we apply the toolbox in four examples. Finally, in Section~\ref{sec:conclusions} we conclude and give an outlook on current and future developments of the toolbox.
\begin{figure*}
\centering
\input{figures/overview}
\caption{Overview of the modules of HILO-MPC}
\label{fig:overview}
\end{figure*}
\begin{table*}
\caption{Current tools implemented in HILO-MPC.}
\label{tab:problems-hilo}
\centering
\begin{tabular}{lllll}
\toprule
Models & Controllers & Machine Learning & Observers & Embedded \\
\midrule
\parbox[c][2.5cm][c]{3.5cm}{Linear/nonlinear\\Time-invariant/variant\\Continuous and discrete\\ODEs, DAEs} & \parbox[c][2.5cm][c]{3.5cm}{Nonlinear MPC\\Trajectory tracking MPC\\Path-following MPC\\PID} & \parbox[c][2.5cm][c]{3cm}{Neural networks\\Gaussian processes} & \parbox[c][2.5cm][c]{3.9cm}{Moving horizon estimation\\Kalman filter\\Extended Kalman filter\\Unscented Kalman filter\\Particle filter} & \ensuremath{\mu}\emph{AO-MPC}\cite{zometa2013} \\
\bottomrule
\end{tabular}
\end{table*}
\begin{table*}
\caption{Comparison overview on the different open-source MPC software. ML: machine learning, MPC: model predictive control, MHE: moving horizon estimator, LMPC: linear MPC, PFMPC: path-following MPC, KF: Kalman filter, EKF: extended Kalman filter, UKF: unscented Kalman filter.}
\label{tab:comparison-toolboxes}
\centering
\begin{tabular}{llclll}
\toprule
& Problems & ML & Interface & Focus & Article \\
\midrule
HILO-MPC & \parbox[c][1cm][c]{3.2cm}{MPC, PFMPC,\\MHE, KF, EKF, UKF} & Yes & Python & Machine Learning & \\
do-mpc & \parbox[c][1cm][c]{3.2cm}{MPC, MHE,\\Multistage MPC} & No & Python & Multistage MPC & Lucia et al.\cite{LUCIA201751} \\
ACADO & MPC & No & C++, MATLAB & Embedded & Houska et al.\cite{Houska2011a} \\
ACADOS & MPC, MHE & No & \parbox[c][1cm][c]{3.2cm}{C, MATLAB, Octave,\\Python, Simulink} & Embedded & \parbox[c][1cm][c]{2.8cm}{Verschueren et al.\cite{Verschueren2018}\\Verschueren et al.\cite{Verschueren2019}} \\
MPCTools & MPC, MHE & No & \parbox[c][1cm][c]{3.2cm}{Python,\\Octave (MATLAB)} & MPC, MHE & \parbox[c][1cm][c]{3.4cm}{Risbeck and Rawlings\cite{risbeck2015nonlinear}\\Risbeck and Rawlings\cite{risbeck2016nonlinear}} \\
MPT3 Toolbox & LMPC & No & MATLAB & Comp. Geometry & Herceg et al.\cite{MPT3} \\
YALMIP & MPC, MHE & No & MATLAB & Optimization & L\"{o}fberg\cite{YALMIP} \\
\bottomrule
\end{tabular}
\end{table*}
\section{HILO-MPC Modules}
\label{sec:hilo-mpc}
The toolbox can be divided into five modules: model, control, observer, machine learning and embedded (see Figure~\ref{fig:overview}). Each of these contains a set of tools, each of which solves a different problem. The model module is used to generate dynamic models. The control module uses the model module to generate, e.g., optimal control and predictive control problems. The observer module also uses the model module to generate state and parameter estimators. The machine learning model is responsible for defining and training machine learning models that can be used in any of the previous modules. Finally, the embedded module currently interfaces with embedded linear MPC software (\ensuremath{\mu}\emph{AO-MPC}\cite{zometa2013}) that allows to easily deploy the code for embedded applications. In the next sections, we will briefly describe the different modules in more detail.\footnote{Note that since HILO-MPC is currently in development, new tools will be added, and current tools could be modified. Refer always to the documentation to get an update on the latest version of HILO-MPC.}
\subsection{Modeling Module}
\label{sec:modeling_module}
At the core of HILO-MPC sits the modeling module. It is a high-level interface to generate representations of dynamical systems that can be used for model-based controllers and observers, like MPC or MHE, or inside a control loop to simulate the behavior of a plant. The system properties supported by HILO-MPC are reported in column ``Models'' of Table~\ref{tab:problems-hilo}.
A general time-variant continuous-time nonlinear system can be modeled using the following system of differential algebraic equations (DAEs)
\begin{align}
\dot{x}(t)&=f(t,x(t),z(t),u(t),p),\nonumber\\
0&=q(t,x(t),z(t),u(t),p),\label{eq:nonlin_model}\\
y(t)&=h(t,x(t),z(t),u(t),p)\nonumber,
\end{align}
where $x(t)\in\mathbb{R}^{n_x}$ is the differential state vector, $z(t)\in\mathbb{R}^{n_z}$ the vector of algebraic states, $u(t)\in\mathbb{R}^{n_u}$ is the input vector and $p\in\mathbb{R}^{n_p}$ is a vector of parameters. The function $f\colon\mathbb{R}\times\mathbb{R}^{n_x}\times\mathbb{R}^{n_z}\times\mathbb{R}^{n_u}\times\mathbb{R}^{n_p}\rightarrow\mathbb{R}^{n_x}$ represents the ordinary differential equations (ODEs) of the model and the function $q\colon\mathbb{R}\times\mathbb{R}^{n_x}\times\mathbb{R}^{n_z}\times\mathbb{R}^{n_u}\times\mathbb{R}^{n_p}\rightarrow\mathbb{R}^{n_z}$ describes the algebraic equations forming a semi-explicit DAE system overall.\footnote{Note that HILO-MPC only supports semi-explicit DAE systems of index 1. The index 1 indicates that the corresponding DAE system can be transformed into a pure ODE system by differentiating the algebraic equations once.} The function $h\colon\mathbb{R}\times\mathbb{R}^{n_x}\times\mathbb{R}^{n_z}\times\mathbb{R}^{n_u}\times\mathbb{R}^{n_p}\rightarrow\mathbb{R}^{n_y}$ describes the measurement equations mapping to some measurable quantity of the system. If these measurement equations are not defined during the setup of the model, the controllers and observers using the model assume that all states are measured.
Furthermore, HILO-MPC also supports the modeling of discrete-time systems. When necessary, continuous-time models can be easily discretized using the \mintinline{python}{model.discretize(...)} method. Available discretization methods are explicit Runge-Kutta methods up to the 4th order and implicit collocation schemes.\cite{Biegler2010} For the explicit Runge-Kutta methods, a series of Butcher tableaus is available, while the implicit collocation schemes only support Gauss-Legendre polynomials and Gauss-Radau polynomials for the calculation of the collocation points.
For some applications, it might be necessary to use a linear model. If only a nonlinear model is available, this can be easily linearized with respect to an equilibrium point using the \mintinline{python}{model.linearize(...)} method. In HILO-MPC, it is also possible to directly create linear models by supplying the required matrices during model setup.
As an example, we show how to set up a simple bike model, which will also be used as an example model in the following sections
\begin{align*}
\dot{p}_\text{x} &= v \cos(\phi(t) + \beta), \\
\dot{p}_\text{y} &= v \sin(\phi(t) + \beta), \\
\dot{v} &= a, \\
\dot{\phi} &= v/l_\text{r} \sin(\beta), \\
\beta & = \arctan\left(\frac{l_r}{l_\text{r} + l_\text{f}}\tan(\delta)\right).
\end{align*}
Here, $p_{\text{x}}$ and $p_{\text{y}}$ are the $x$- and $y$-coordinates of the bike's center of mass, $v$ is the norm of the velocity of the center of mass, $\phi$ is the orientation angle of the vehicle with respect to the $x$-coordinate. The inputs are the acceleration of the center of mass $a$ and the steering angle $\delta$. The parameter $l_\text{r}$ is the distance between the center of mass and the rear wheel, and $l_\text{f}$ is the distance between the center of mass and front wheel. One way to set up the model looks as follows
\begin{minted}[bgcolor=bg,breaklines]{python}
from hilo_mpc import Model
# Initialize empty model
model = Model(name='Bike')
# Define model equations
equations = """
# ODEs
dpx/dt = v(t)*cos(phi(t) + beta)
dpy/dt = v(t)*sin(phi(t) + beta)
dv/dt = a(k)
dphi/dt = v(t)/lr*sin(beta)
# Algebraic equations
beta = arctan(lr/(lr + lf)*tan(delta(k)))
"""
model.set_equations(equations=equations)
# Sampling time in seconds
dt = 0.01
model.setup(dt=dt)
model.set_initial_conditions(x0=[0,0,0,0])
\end{minted}
where the whole model equations are supplied in form of a string. HILO-MPC is able to parse the most common functions and operations in this format. Other input formats are also possible.
Units, labels and descriptions of all variables can also be set for convenience and for use in the plotting functionality. For a more in-depth description of available functionalities, refer to the documentation {\color{red}[REFERENCE TO LINK HERE]}.
Once the initial conditions are set, the model can be used for simulation by running \mintinline{python}{model.simulate(u=u,p=p)}. Note that, for every integration interval $\mathrm{d}t$ the input is kept constant. This is a common assumption that reduces the dimensionality of the optimal control problems we will define in the next sections.
Finally, HILO-MPC makes it easy to store, access and plot solutions generated by simulations. Every time the method \mintinline{python}{model.simulate(...)} is executed, the solution is saved in the \mintinline{python}{model.solution} object. This can be accessed similarly to a Python dictionary. To plot the solution, i.e., the evolution of states, inputs and parameters with time, the user can use the method \mintinline{python}{model.solution.plot()}.
\subsection{Machine Learning Module}
This module is responsible for the definition and training of machine learning models. Machine learning models are data-driven models that are used to map selected inputs $v \in \mathbb{R}^{n_v}$, also called features, to outputs $l \in \mathbb{R}^{n_l}$, also called labels. The current version ({1.0}) contains two approaches: artificial feedforward fully connected neural networks and Gaussian processes.
\subsubsection{Artificial Neural Network}
\begin{figure}
\centering
\input{figures/ann}
\caption{Simple feedforward neural network with four features (inputs), one hidden layer and one label (output)}
\label{fig:ann}
\end{figure}
Inspired by the neural circuits in brains,\cite{McCulloch1943} an artificial neural network (ANN) can be represented as a collection of interconnected nodes, the so-called neurons, that are usually organized into layers. Here, we focus on feedforward fully connected ANNs (Figure~\ref{fig:ann}). Each layer of the ANN is mathematically represented by a linear regression propagated through some function called \emph{activation function}
\begin{equation}
z_i=\sigma_i\left(W_iz_{i-1}+b_i\right),
\end{equation}
where $z_i\in\mathbb{R}^{n_{z_i}}$ is the output vector of layer $i$ with the features $v$ being the input to the first layer, i.e., $v=z_0$ and the labels $l$ being the output of the last layer. The function $\sigma_i\colon\mathbb{R}^{n_{z_i}}\rightarrow\mathbb{R}^{n_{z_i}}$ is the activation function. To learn nonlinear maps it is fundamental that activation functions of the hidden layers, i.e., the layers between input and output layer, contain some nonlinearities. Common nonlinear activation functions are the sigmoid function, hyperbolic tangent or the ReLU function.\cite{Ding2018} The training of the ANN is achieved by iteratively adapting the weight matrices $W_i\in\mathbb{R}^{n_{z_i}\times n_{z_{i-1}}}$ and the bias vectors $b_i\in\mathbb{R}^{n_{z_i}}$, so that a user-defined loss function is minimized. For regression problems, one of the most commonly used loss functions is the mean squared error (MSE)
\begin{equation*}
\mathrm{MSE}=\dfrac{1}{n_l}\sum_{i=1}^{n_l}(l_i-\hat{l}_i)^2,
\end{equation*}
where $\hat{l}_i$ are the values of the labels predicted by the ANN. Other loss functions supported by HILO-MPC include the mean absolute error (MAE) or the Huber loss. Optimization algorithms like gradient descent or stochastic gradient descent are used to minimize the loss. A number of minimization algorithms can be interfaced in HILO-MPC, e.g., the Adam algorithm, a modification of stochastic gradient descent. These minimization algorithms return a gradient vector of the loss with respect to the output $l$ of the ANN. By applying the backpropagation algorithm this gradient vector can then be used to update the weight matrices $W_i$ and the bias vectors $b_i$ throughout the ANN.\cite{Rumelhart1986} The training is stopped when there is no significant change in the loss over a defined number of iterations or if the number of maximum iterations is reached. At the moment, HILO-MPC supports feedforward fully connected ANNs.
In the following example, we show how to define an ANN consisting of two layers with 10 neurons each, using a sigmoid function as activation function
\begin{minted}[bgcolor=bg,breaklines]{python}
from hilo_mpc import ANN, Dense
# Initialize NN
ann = ANN(features, labels)
ann.add_layers(Dense(10, activation='sigmoid'))
ann.add_layers(Dense(10, activation='sigmoid'))
# Add training data set
ann.add_data_set(df)
# Set up NN
ann.setup()
# Train the NN
batch_size = 64
epochs = 1000
ann.train(batch_size, epochs, validation_split=.2, patience=100, scale_data=True)
\end{minted}
where \mintinline{python}{Dense(...)} is the command for creating a fully connected layer. To replace specific variables of the model with the predictions of the ANN, simply run \mintinline{python}{model.substitute_from(ann)}. Here, the variables to be replaced will be automatically detected and exchanged with the corresponding predictions of the ANN. Alternatively, it is also possible to directly add the ANN to the model via summation or subtraction.
\subsubsection{Gaussian Process}
A Gaussian process (GP) is a probability distribution over a function in which any finite subset of random variables has a joint Gaussian distribution.\cite{Rasmussen2006} An advantage of using a GP is that it naturally measures the output uncertainty by providing the variance of the predictions. For regression problems, GPs can be used to approximate the input-output mapping
\begin{equation*}
l=f(v)+\varepsilon,
\end{equation*}
where the output $l\in\mathbb{R}$ is affected by some zero-mean normally distributed measurement noise $\varepsilon\sim\mathcal{N}(0,\sigma_n^2)$ with variance $\sigma_n^2$.
Starting from a given prior distribution, described by a mean function and a covariance function, the GP is trained on observed data. Both, the prior mean function and the prior covariance function, in general depend on the so-called hyperparameters. During the training step an optimization problem is solved, where these hyperparameters act as decision variables, i.e., the hyperparameters are chosen such that the resulting GP can be used to infer the input-output mapping given available training data. Usually the prior mean function is assumed to be zero,\cite{Kocijan2003} but other prior mean functions are also supported by HILO-MPC. The covariance function, or kernel, is often assumed to be a smooth function that is infinitely differentiable. A number of covariance functions that meet this criterion are provided in HILO-MPC, like e.g., the stationary squared exponential and rational quadratic kernels. The prior means as well as the kernels can be combined with each other through summation or multiplication. The objective of the optimization problem that underlies the training step is usually to maximize the log marginal likelihood\cite{Rasmussen2006}
\begin{equation*}
\log\left(p(l|V)\right)=-\dfrac{1}{2}l^\Tr\left(K+\sigma_n^2I\right)^{-1}l-\dfrac{1}{2}\log\left(\left|K+\sigma_n^2I\right|\right)-\dfrac{n}{2}\log\left(2\pi\right),
\end{equation*}
where $V\in\mathbb{R}^{n_v\times n_d}$ describes the matrix containing the $n_d$ training inputs, $K\in\mathbb{R}^{n_d\times n_d}$ is the covariance matrix resulting from the used kernel and the supplied training data and $I$ is the identity matrix of corresponding size. The element $(i,j)$ of $K$ is $k(v_i,v_j)$, where $k\colon\mathbb{R}^{n_v}\times\mathbb{R}^{n_v}\rightarrow\mathbb{R}$ is the kernel. The predictive mean $\bar{f}$ and variance $\sigma^2$ of the trained GP can be obtained for test data $w$ as follows
\begin{align*}
\bar{f}(w)&=\tilde{k}^\Tr(K+\sigma_n^2I)^{-1}l,\\
\sigma^2(w)&=k(w,w)-\tilde{k}^\Tr(K+\sigma_n^2I)^{-1}\tilde{k},
\end{align*}
where the $i$-th element of $\tilde{k}\in\mathbb{R}^{n_d}$ is $k(v_i,w)$. Here is an example of how to set up a GP in HILO-MPC
\begin{minted}[bgcolor=bg,breaklines]{python}
from hilo_mpc import GP, SquaredExponentialKernel
# Initialize kernel
kernel = SquaredExponentialKernel(variance=0.002)
# Initialize GP
gp = GP(features, labels, kernel=kernel)
# Add training data set
gp.set_training_data(train_in, train_out)
# Set up GP
gp.setup()
# Fit the GP
gp.fit_model()
\end{minted}
Finally, if necessary, the GP can be integrated in the model the same way as the ANN.
\subsection{Control Module}
The control module contains the MPC and the proportional–integral–derivative (PID) controller. Here we describe only the MPC for brevity.
\subsubsection{Model Predictive Control}
Model predictive control solves repeatedly a finite-horizon optimal control problem (OCP). We here consider a sampled data nonlinear MPC problem with a continuous time-varying DAE system. A similar formulation can be obtained for discrete-time nonlinear system. At the sampling time $t_0$ the optimal control problem to solve reads
\begin{mini!}
{u(\cdot)}{\int^{t_0+T}_{t_0} l(t,x(t),z(t),u(t),p) \mathrm{d}t + e(t_0+T,x(t_0+T),z(t_0+T),p),\label{subeq:obj1}}{\label{eq:problem-mpc}}{}
\addConstraint{\dot{x}(t)}{=f(t,x(t),z(t),u(t),p),\quad\label{subeq:model1}}{x(t_0)=\hat{x}(t_0)}
\addConstraint{0}{=q(t,x(t),z(t),u(t),p)}{}
\addConstraint{0}{\geq g(t,x(t),z(t),u(t),p)\label{subeq:path-con}}{}
\addConstraint{0}{\geq g_T(t_0+T,x(t_0+T),z(t_0+T),p)\label{subeq:terminal-con}}{}
\addConstraint{y(t)}{=h(t, x(t),z(t),u(t),p)\label{subeq:meas-eq}}{}
\addConstraint{u(t)}{=u(t_0 + T_c)\,\ \text{for} \,\ t\geq t_0 + T_c}{}
\addConstraint{}{\text{for} \quad t \in [t_0, t_0+T] \subset \mathbb{R}.}{}
\end{mini!}
\begin{figure*}
\centering
\input{figures/scheme1}
\caption{Schematic representation of machine learning supported open-loop optimization or model predictive control. (ML: machine learning)}
\label{fig:scheme}
\end{figure*}
Here, $u(\cdot)$ is the optimal input function, $l:\mathbb{R}\times\mathbb{R}^{n_x}\times\mathbb{R}^{n_z}\times\mathbb{R}^{n_u} \times \mathbb{R}^{n_p} \rightarrow \mathbb{R}$ is the stage cost, $e:\mathbb{R}\times\mathbb{R}^{n_x}\times\mathbb{R}^{n_z} \times \mathbb{R}^{n_p} \rightarrow \mathbb{R}$ the terminal cost, $T$ is the prediction horizon and $T_c$ the control horizon. The vector $\hat{x}(t_0)$ is the measured or estimated state vector at the current sampling time $t_0$ and $t$ is the time variable. The equation $g:\mathbb{R}\times\mathbb{R}^{n_x}\times\mathbb{R}^{n_z}\times\mathbb{R}^{n_u} \times \mathbb{R}^{n_p} \rightarrow \mathbb{R}^{n_g}$ are the path constraints and $g_T:\mathbb{R}\times\mathbb{R}^{n_x}\times\mathbb{R}^{n_z} \times \mathbb{R}^{n_p} \rightarrow \mathbb{R}^{n_{g,\text{T}}}$ are terminal constraints.
\begin{remark}
Any of the equations in \eqref{eq:problem-mpc} could be at least partially learned from data (see Figure~ \ref{fig:scheme}). Table \ref{tab:what_is_learned} gives an overview of equations that have been learned in other works. HILO-MPC allows to use machine learning models in any of these components.
\end{remark}
\begin{table*}
\caption{Overview of already learned equations in literature.}
\label{tab:what_is_learned}
\centering
\begin{tabular}{ll}
\toprule
What is learned? & Article \\
\midrule
output function \eqref{subeq:meas-eq} & Matschek et al.\cite{matschek2019learning} \\
terminal constraints \eqref{subeq:terminal-con} & Rosolia et al.\cite{rosolia2017autonomous}, Rosolia and Borrelli\cite{rosolia2017learning}, Brunner et al.\cite{brunner2017repetitive} \\
path constraints \eqref{subeq:path-con} & Bujarbaruah et al.\cite{bujarbaruah2020learning}, Armesto et al.\cite{Leopoldo2017} \\
objective function \eqref{subeq:obj1} & Tamar et al.\cite{tamar2017learning}, Beckenbach et al.\cite{BECKENBACH201860}, Bradford and Imsland\cite{Bradford2018} \\
\bottomrule
\end{tabular}
\end{table*}
Problem \eqref{eq:problem-mpc} cannot be solved directly since $u(\cdot)$ and the constraints are infinite dimensional. HILO-MPC uses \emph{direct approaches} to transform the problem to an equivalent finite-dimensional optimization problem. These approaches parameterize the input with a finite number of parameters, for example, using piece-wise constant inputs, and force the nonlinear constraints only on a finite number of points. This reads as follows
\begin{mini!}
{\mathbf{u}}{\sum_{i = k}^{N} \int_{t_i}^{t_{i} + \Delta t} l(t,x(t),z(t),u_i,p) \mathrm{d}t + e(t_k+T,x(t_k+T),z(t_k+T),p),}{\label{eq:disc-mpc}}{}
\addConstraint{x(t_{i+1})}{= x(t_{i}) + \int_{t_i}^{t_{i} + \Delta t} f(t,x(t),z(t),u_i,p) \mathrm{d}t}{}
\addConstraint{x(t_k)}{=\hat{x}(t_k)}{}
\addConstraint{0}{=q(t_i,x(t_i),z(t_i),u_i,p)}{}
\addConstraint{0}{\geq g(t_i,x(t_i),z(t_i),u_i,p)}{}
\addConstraint{0}{\geq g_T(t_k+T,x(t_k+T),z(t_k+T),p)}{}
\addConstraint{y(t_i)}{=h(t_i, x(t_i),z(t_i),u_i,p)}{}
\addConstraint{u_i}{=u_{N_c},\,\ \text{for} \,\ i\geq N_c}{}
\addConstraint{}{\text{for} \,\ i \in [k, N] \subset \mathbb{N}_0,}{}
\end{mini!}
where $\Delta t$ is the sampling time, $N_c = \text{ceil}(T_c/\Delta t)$ is the control horizon and $\mathbf{u}=\begin{bmatrix}u_k&...&u_{N_c}\end{bmatrix}$ is sequence of piece-wise constant inputs applied to the plant, i.e., $u(t) = u_i, \,\ \forall t \in [t_i,\,t_i + \Delta t) \subset \mathbb{R}$.
HILO-MPC implements the multiple shooting approach.\cite{BOCK19841603} The integration of the system model in between the shooting points can be done with Runge-Kutta methods of various orders, orthogonal collocation\cite{Oh1977} or the dynamics can be integrated with CVODES or IDAS solvers.\cite{Hindmarsh2005} The default method is orthogonal collocation and the user can select any other method if needed. Piece-wise constant input is assumed at every control interval.
Hence, the real formulation of Problem \eqref{eq:problem-mpc} depends on which method is used for the discretization. Here, we do not go into the details of the formulation of the approximated problem. The reader is invited to refer to the previously cited papers for more details on the formulation of the OCP for the single cases.
\begin{remark}
HILO-MPC can also optimize the sampling intervals. This will add to Problem \eqref{eq:disc-mpc} the vector $\begin{bmatrix}\Delta t_k&...&\Delta t_{N}\end{bmatrix}$ as optimization variable. This allows the solution of minimum time problems, and optimal sampling time problems.
\end{remark}
The stage cost $l(\cdot)$ and arrival cost $e(\cdot)$ depend on the type of MPC implemented. In the next sections we will go through the different MPC problems HILO-MPC can solve. For simplicity of explanation, we show only quadratic cost functions. Note however that HILO-MPC can use any type of continuous cost functions.
\paragraph{MPC for set point tracking}
In set point tracking MPC the variables $x$, $u$ and $y$ need to track a given reference.
\begin{align*}
l(x(t),u(t),y(t)) &= \Vert x(t) - x_r\Vert^2_Q + \Vert u(t) - u_r\Vert^2_R + \Vert y(t) - y_r\Vert^2_P, \\
e(x(t_0+T),y(t_0+T)) &= \Vert x(t_0+T) - x_r\Vert^2_{Q_{\text{T}}} + \Vert y(t_0+T) - y_r\Vert^2_{P_{\text{T}}},
\end{align*}
where $x_r$, $u_r$ and $y_r$ are fixed references and $Q \in \mathbb{R}^{n_x \times n_x}$, $R \in \mathbb{R}^{n_u \times n_u}$, $P \in \mathbb{R}^{n_y \times n_y}$, $P_{\text{T}} \in \mathbb{R}^{n_y \times n_y}$ and $ Q_{\text{T}} \in \mathbb{R}^{n_x \times n_x}$ are weighting matrices.
For example, once we have defined the bike model like in section \ref{sec:modeling_module}. A possible set point tracking MPC that tracks a reference speed of $v_{\text{r}} = 2 \,\ \mathrm{m}/\mathrm{s}$, with a prediction horizon of 20 steps, can be defined as follows
\begin{minted}[bgcolor=bg,breaklines]{python}
from hilo_mpc import NMPC
nmpc = NMPC(model)
nmpc.horizon = 20
nmpc.quad_stage_cost.add_states(names=['v'], ref=2, weights=10)
nmpc.quad_term_cost.add_states(names=['v'], ref=2, weights=10)
nmpc.setup()
\end{minted}
After the MPC has been setup, it can be used in a control loop as follows
\begin{minted}[bgcolor=bg,breaklines]{python}
# Choose number of steps
n_steps = 100
# Begin simulation loop
for step in range(n_steps):
# find optimal input
u = nmpc.optimize(x0)
# Simulate the plant
model.simulate(u=u)
# Get the current states
x0 = sol['xf']
\end{minted}
As for the the other HILO-MPC objects, the solution of the MPC (e.g., input and state sequences) are stored in the \mintinline{python}{mpc.solution()} object.
\paragraph{MPC for trajectory-tracking}
For trajectory tracking problems the variables need to track a time-varying reference. For quadratic cost function, this looks like the following
\begin{align*}
\label{eq:mpc-traj-tracking}
l(x(t),u(t),y(t)) &= \Vert x(t) - x_r(t)\Vert^2_Q + \Vert u(t) - u_r(t)\Vert^2_R + \Vert y(t) - y_r(t_i)\Vert^2_P, \\
e(x(t_0+T),y(t_0+T)) &= \Vert x(t_0+T) - x_r(t_0+T) \Vert^2_{Q_{\text{T}}} + \Vert y(t_0+T) - y_r(t_0+T) \Vert^2_{P_{\text{T}}},
\end{align*}
where $x_r : \mathbb{R} \rightarrow \mathbb{R}^{n_x}$, $u_r: \mathbb{R} \rightarrow \mathbb{R}^{n_u}$ and $y_r: \mathbb{R} \rightarrow \mathbb{R}^{n_y}$ are time-varying references. For example, a trajectory tracking problem for the racing car problem can be defined as follows
\begin{minted}[bgcolor=bg,breaklines]{python}
from hilo_mpc import NMPC
import casadi as ca
nmpc = NMPC(model)
nmpc.horizon = 20
t = nmpc.create_time_variable()
traj_x = 30 - 14 * ca.cos(t)
traj_y = 30 - 16 * ca.sin(t)
nmpc.quad_stage_cost.add_states(names=['px', 'py'], ref=[traj_x, traj_y], weights=[10, 10], trajectory_tracking=True)
nmpc.quad_term_cost.add_states(names=['px', 'py'], ref=[traj_x, traj_y], weights=[10, 10], trajectory_tracking=True)
nmpc.setup()
\end{minted}
\paragraph{MPC for path following}
While in the MPC for trajectory tracking both the \emph{value of the reference and time are fixed simultaneously}, the MPC for path following has more degrees of freedom by choosing \emph{when} to be on the path.\cite{Matschek2019,faulwasser2016} In this case the model is augmented with a \emph{virtual path state} as follows
\begin{mini!}
{u(\cdot),u_{\text{pf}}(\cdot)}{\int_{t_0}^{t_0+T} l(t,x(t),z(t),u(t),p) \mathrm{d}t + e(t_0+T,x(t_0+T),z(t_0+T),p),}{}{}
\addConstraint{\dot{x}(t)}{=f(t, x(t),z(t),u(t),p),\quad}{x(t_0)=\hat{x}(t_0)}
\addConstraint{0}{=q(t, x(t),z(t),u(t),p)}{}
\addConstraint{\dot{\theta}}{=u_{\text{pf}},\quad}{\theta(t_0)=0}
\addConstraint{0}{\geq g(t,x(t),z(t),u(t),p, \epsilon)}{}
\addConstraint{0}{\geq g_T(t_0+T,x(t_0+T),z(t_0+T),p)}{}
\addConstraint{y(t)}{=h(t, x(t), z(t), u(t), p)}{}
\addConstraint{}{\text{for} \,\ t \in [t_0, t_0+T] \subset \mathbb{R},}{}
\end{mini!}
where $\theta \in \mathbb{R}^{n_\theta}$ is a virtual path state vector and $u_{\text{pf}} \in \mathbb{R}^{n_{u,\theta}}$ is the virtual input that the controller can choose. Hence, the objective function looks like
\begin{align*}
l(x(t),u(t),y(t)) &= \Vert x(t) - x_r(\theta(t))\Vert^2_Q + \Vert u(t) - u_r(\theta(t))\Vert^2_R + \Vert y(t) - y_r(\theta(t))\Vert^2_P, \\
e(x(t_0+T),y(t_0+T)) &= \Vert x(t_0+T) - x_r(\theta(t_0+T)) \Vert^2_{Q_{\text{T}}} + \Vert y(t_0+T) - y_r(\theta(t_0+T)) \Vert^2_{P_{\text{T}}}.
\end{align*}
To force the controller to move into only one direction along the path usually a lower bound on $u_{\text{pf}}$ is added, i.e., $u_{\text{pf}}>u_{\text{pf,min}}$ with $u_{\text{pf,min}} \in \mathbb{R}^{n_{u,\theta}}_{+}$.
HILO-MPC also allows to track a constant $u_{\text{pf}}$, so that the MPC tries to maintain a constant \emph{speed} of the virtual state
\begin{align*}
l(x(t),u(t),y(t)) &= \Vert x(t) - x_r(\theta(t))\Vert^2_Q + \Vert u(t_i) - u_r(\theta(t))\Vert^2_R + \Vert y(t) - y_r(\theta)\Vert^2_P + \Vert u_{\text{pf}} - u_{\text{pf,ref}}\Vert_{R_{\text{pf}}}^2, \\
e(x(t_0+T),y(t_0+T)) &= \Vert x(t_0+T) - x_r(\theta(t_0+T))\Vert^2_Q + \Vert y(t_0+T) - y_r(\theta(t_0+T))\Vert^2_P.
\end{align*}
Contrary to the other available toolboxes, path following MPC problems are automatically generated. The user just needs to activate the path following mode for the desired variables. Here, we show how a path following problem can be defined using HILO-MPC
\begin{minted}[bgcolor=bg,breaklines]{python}
from hilo_mpc import NMPC
import casadi as ca
nmpc = NMPC(model)
nmpc.horizon = 20
theta = nmpc.create_path_variable()
traj_x = 30 - 14 * ca.cos(theta)
traj_y = 30 - 16 * ca.sin(theta)
nmpc.quad_stage_cost.add_states(names=['px', 'py'], ref=[traj_x, traj_y], weights=[10, 10], path_following=True)
nmpc.quad_term_cost.add_states(names=['px', 'py'], ref=[traj_x, traj_y], weights=[10, 10], path_following=True)
nmpc.setup()
\end{minted}
Note that HILO-MPC allows to mix the previous problem formulations together with minimum effort, e.g., the user can track some constant references for some of the variables, while tracking a path with other variables.
\paragraph{Soft constraints}
Soft constraints can be easily implemented. When soft constraints are selected, HILO-MPC automatically adds the slack variables $\epsilon_{\text{p}} \in \mathbb{R}^{n_g}$ and $\epsilon_{\text{T}} \in \mathbb{R}^{n_{g,\text{T}}} $ to the path and terminal constraints, respectively, as follows
\begin{mini!}
{u(\cdot), \epsilon_{\text{T}}, \epsilon_{\text{p}}}{\int_{t_0}^{t_0+T} l(\cdot) \mathrm{d}t + e(\cdot) + \Vert \epsilon_{\text{s}} \Vert^2_{E_{\text{p}}} + \Vert \epsilon_{\text{T}} \Vert^2_{E_{\text{T}}},}{\label{eq:problem-mpc-soft}}{}
\addConstraint{\dot{x}(t)}{=f(t,x(t),z(t),u(t), p),\quad}{x(0)=\hat{x}_0}
\addConstraint{0}{=q(t, x(t),z(t),u(t), p)}{}
\addConstraint{\epsilon_{\text{p}}}{\geq g(t,x(t),z(t),u(t), p)}{}
\addConstraint{\epsilon_{\text{T}}}{\geq g_T(t_0+T,x(t_0+T),z(t_0+T),p)}{}
\addConstraint{y(t)}{=h(t, x(t),z(t), u(t),p)}{}
\addConstraint{0}{\leq\epsilon_{\text{p}} \leq \epsilon_{\text{p,max}}, \quad 0 \leq \epsilon_{\text{T}} \leq \epsilon_{\text{T, max}}}{}
\addConstraint{}{\text{for} \,\ t \in [t_0, t_0+T] \subset \mathbb{R},}{}
\end{mini!}
where and $E_{\text{p}}\in \mathbb{R}^{n_g \times n_{g}}$ and $E_{\text{T}}\in \mathbb{R}^{n_{g,\text{T}} \times n_{g,\text{T}}}$ are weighting matrices that limit the increase of the the slack variables and can be chosen by the user, and $\epsilon_{\text{p,max}}$ and $\epsilon_{\text{T,max}}$ are the maximum constraint violations of path and terminal constraints, respectively.
\subsection{Observer Module}
The observer model contains the moving horizon estimator, Kalman filters and particle filter.
\subsubsection{Moving Horizon Estimation}
The moving horizon estimator (MHE) is an observer based on the solution of an optimization problem similar to the MPC problem.\cite{rawlings2017model} We consider a sample data problem with equidistant sampling times as it is more common in practice. At these sampling times measurements are taken. Since measurements are discontinuous in MHE, the objective function is usually in discrete form. For simplicity we indicate with $(\cdot)_{i|k}$ the variable at time $t_i$, i.e., for the MHE problem solved at time $t_k$ we have
\begin{mini!}
{x_{k-N|k},p_k, w_{(\cdot)|k},z_{(\cdot)|k}}{\left\Vert \begin{bmatrix} x_{k-\bar{N}|k} - \hat{x}_{k-N|k} \\ p_{k} - \hat{p}_{k} \end{bmatrix} \right\Vert^2_{P_k} + \sum^k_{i=k-N} \Vert \hat{y}_i - y_{i|k} \Vert^2_R + \Vert w_{i|k}\Vert^2_W,}{\label{eq:problem-mhe}}{}
\addConstraint{x_{i+1|k}}{=x_{i|k} + \int_{t_i}^{t_i + \Delta t} \left( f(t,x(t),z(t),\hat{u}(t), p_k) + w(t) \right) \mathrm{d}t\label{subeq:mhe-model1}}{}
\addConstraint{y_{i|k}}{=h(t_i,x_{i|k},z_{i|k},\hat{u}_i,p_k) + v_{i|k}}{}
\addConstraint{0}{\geq g(t_i,x_{i|k},u_i)}{}
\addConstraint{}{\text{for} \,\ i \in [k-\bar{N}, k], \,\ k,\bar{N} \in \mathbb{N},}{}
\end{mini!}
where $\bar{N}$ is the horizon length, $\hat{y}_i$ the output measurements and $\hat{u}_i$ the input measurements at time $t_i$. The first term of the objective function is the \emph{arrival cost}, while the second and third weight the measurements and state noise, respectively.\cite{rawlings2017model,allgower1999nonlinear} The matrices $R \in \mathbb{R}^{n_y \times n_y}$, $W \in \mathbb{R}^{n_x \times n_x}$ and $P+k \in \mathbb{R}^{(n_x+n_p) \times (n_x + n_p)}$ are the weighting matrices for the outputs, state noise and arrival cost. The optimization variables are: the state $x_{k-\bar{N}|k}$, i.e., the state at the beginning of the horizon, the state noise $w_{(\cdot)|k} = \{ w_{i|k}, \,\ \forall i \in [k-\bar{N},k] \}$, the algebraic states (for DAE systems) $z_{(\cdot)|k} = \{ z_{i|k}, \,\ \forall i \in [k-\bar{N},k] \}$ and the system parameters $p_k$. Note that the parameters are considered constant in the horizon, but can be updated every time the optimization is run, to adapt to the new measurements.
Also in this case we use direct approaches to solve \eqref{eq:problem-mhe} as for the MPC case. For the bike model, an MHE model with $R=W=P_{k}=10 I$ (where $I$ is the identity matrix of appropriate dimensions) can be easily defined as follows
\begin{minted}[bgcolor=bg,breaklines]{python}
from hilo_mpc import MHE
mhe = MHE(model)
mhe.horizon = 20
mhe.quad_arrival_cost.add_states(weights=[10,10,10,10], guess=x0_est)
mhe.quad_stage_cost.add_measurements(weights=[10,10,10,10])
mhe.quad_stage_cost.add_state_noise(weights=[10,10,10,10]])
mhe.setup()
\end{minted}
After the MHE has been set up, it can be deployed for estimation, for example, as follows
\begin{minted}[bgcolor=bg,breaklines]{python}
# Set number of steps
n_steps = 100
# Simulation/estimation loop
for step in range(n_steps):
# Simulate plant with input
model.simulate(u=u)
# Get measurements
y_meas = model.solution['y'][:, -2]
# Pass the measurements to the MHE
mhe.add_measurements(y_meas, u_meas=u)
# Estimate states
x0 = mhe.estimate()
\end{minted}
As with the other HILO-MPC objects, the solution of the MHE is stored in the \mintinline{python}{mhe.solution()} object.
\subsubsection{Kalman Filters}
The Kalman filter (KF) is an algorithm that allows for estimation of observable states via available measurement data. Generally it consists of two steps: the prediction step and the update step. In the prediction step the estimated states from the previous iteration are propagated through the model dynamics to obtain preliminary values for the states at the current time step, the so-called \emph{a priori} estimates. These \emph{a priori} estimates are then updated in the update step using the measurement data to obtain \emph{a posteriori} estimates. The original formulation of the KF was developed for linear discrete-time systems\cite{kalman1960}
\begin{align*}
x_k&=A_kx_{k-1}+B_ku_k+w_k,\\
y_k&=C_kx_k+v_k,
\end{align*}
where $w_k$ is the process noise and $v_k$ is the measurement noise. The process noise $w_k\sim\mathcal{N}(0,Q_k)$ and the measurement noise $v_k\sim\mathcal{N}(0,R_k)$ are assumed to be zero mean normal distributions with the covariance matrices $Q_k$ and $R_k$, respectively. Accordingly, the prediction step and update step of the KF are as follows
\begin{align}
\hat{x}_{k|k-1}&=A_k\hat{x}_{k-1}+B_ku_k,\\
P_{k|k-1}&=A_kP_{k-1}A_k^\Tr+Q_k,\label{eq:kf_prop}\\
\hat{x}_k&=\hat{x}_{k|k-1}+K(y_k-\hat{y}_k),\\
P_k&=P_{k|k-1}-KP_{y_ky_k}K^\Tr,
\end{align}
with
\begin{align*}
K&=P_{x_ky_k}P_{y_ky_k}^{-1},\\
\hat{y}_k&=C_k\hat{x}_{k|k-1},\\
P_{x_ky_k}&=P_{k|k-1}C_k^\Tr,\\
P_{y_ky_k}&=C_kP_{k|k-1}C_k^\Tr+R_k,
\end{align*}
where $P_k$ is the error covariance matrix, $P_{x_ky_k}$ is the state-measurement cross-covariance matrix, $P_{y_ky_k}$ is the measurement covariance matrix, $K$ is called the Kalman gain and $\hat{y}_k$ is the predicted output. The notation $\hat{x}_{k|k-1}$ represents the estimated state $\hat{x}$ at time step $k$ given observations up to and including time step $k-1$. The covariance matrix $P_k$ is a measure of the estimation accuracy of the estimated states, i.e., the lower the value the closer the estimated state is to the actual one assuming normally distributed noise.
There are also extensions to the original KF formulation for dealing with nonlinear models. The two most known extensions are the extended Kalman filter (EKF),\cite{Jazwinski1970} and the unscented Kalman filter (UKF).\cite{Wan2000} The EKF differs from the original formulation in that the propagation of the error covariance matrix and the consecutive update step are not as straightforward. The \emph{a priori} estimates of the states can be easily obtained by propagating through the nonlinear model dynamics. The propagation of the error covariance matrix is approximated by calculating the Jacobi matrix
\begin{equation*}
A_k=\left.\dfrac{\partial f}{\partial x}\right|_{\hat{x}_{k-1},u_k},
\end{equation*}
and using Equation~\eqref{eq:kf_prop}. The output matrix $C_k$ is approximated analogously for the update step. A drawback is that the approximation via Jacobi matrices can produce highly unstable Kalman filters.\cite{Wan2000} Another nonlinear extension is the UKF. It approximates the nonlinear problem statistically by a linear one. The UKF makes use of the unscented transform to generate a set of samples, the so-called sigma points, that represent the initial probability distribution. These sigma points are then propagated through the nonlinear system dynamics and afterwards used to approximate the a priori estimates as well as the predicted output and the various covariance matrices. The update step is the same as for the KF and EKF.
The setup of a Kalman filter in HILO-MPC can be easily set up as
\begin{minted}[bgcolor=bg,breaklines]{python}
from hilo_mpc import UKF
ukf = UKF(model)
ukf.setup()
\end{minted}
\subsubsection{Particle Filter}
The particle filter (PF) is another algorithm that can be used for estimation.\cite{Simon2006} It works similarly to the UKF in that a set of samples, the ``particles'', is propagated through the nonlinear system dynamics. This set of samples is drawn from a probability density function that is assumed to be known. Next, the relative likelihood of each propagated particle is determined
\begin{equation*}
q_i^*=\dfrac{q_i}{\sum_{i=1}^{n_p}q_i},
\end{equation*}
with
\begin{equation*}
q_i\sim\mathrm{pdf}(y_k|\hat{y}_k),
\end{equation*}
where $n_p$ is the number of particles in the set. This relative likelihood acts as a measure of how good a particle matches the measurement $y_k$ and is used in the resampling step to generate the set of \emph{a posteriori} particles. This resampling step is analogous to the update step of the Kalman filters. The resampling can be achieved by using one of the several available sampling strategies, like e.g., survival of the fittest\cite{Kanazawa1995} or regularization.\cite{Musso2001} Similarly to the UKF, the \emph{a posteriori} particles can now be used to determine the estimated states via computation of the mean of the distribution. For highly nonlinear system dynamics or non-Gaussian probability density functions the PF usually outperforms the Kalman filter algorithms.\cite{Simon2006}
The setup of a PF in HILO-MPC is similar to that of the Kalman filters
\begin{minted}[bgcolor=bg,breaklines]{python}
from hilo_mpc import PF
pf = PF(model)
pf.setup()
\end{minted}
\subsection{Embedded Module}
\label{sec:emb-mod}
\subsubsection{\ensuremath{\mu}\emph{AO-MPC}}
As of now the embedded module consists of a code generation
tool for linear time-invariant discrete-time model
predictive control, tailored for small embedded system,
like microcontrollers.\cite{zometa2013}
We consider the discrete-time linear time-invariant system
\begin{equation}
\label{eqn:system}
x^+ = Ax + Bu
\end{equation}
subject to input and state constraints
\begin{equation*}
\underline{u} \leq u \leq \overline{u}, \ \
\underline{x} \leq x \leq \overline{x},
\end{equation*}
where $x$ and $u$ denote the state and input at the current sampling time
and $x^+$ denotes the state at the next sampling time.
The linear MPC problem formulation considered is
\begin{mini}
{\mathbf{u}}{\frac{1}{2} \sum\limits_{j=0}^{N-1} (\|x_j\|^2_Q + \|u_j\|^2_R)
+ \frac{1}{2} \|x_N\|^2_P,}{\label{eqn:mpcproblem}}{}
\addConstraint{x_{j+1}}{= A x_j + B u_j,\qquad}{j = 0,..., N-1}
\addConstraint{\underline{u}}{\leq u_j \leq \overline{u},}{j = 0,..., N-1}
\addConstraint{\underline{x}}{\leq x_j \leq \overline{x},}{j = 0,..., N-1}
\addConstraint{x_{0}}{=x,}{}
\end{mini}
where the integer $ N \geq 2$ is the prediction horizon, the matrices $Q \inRpow{n
\times n}$, $R \inRpow{m \times m}$ and $P \inRpow{n \times n}$ are the state,
input, and terminal weighting matrix, respectively. The the input sequence
$\vb{u} = \begin{bmatrix}u_{0}&...&u_{N-1}\end{bmatrix}$ is the optimization variable.
To solve problem \eqref{eqn:mpcproblem} efficiently on microcontrollers,
HILO-MPC relies on \ensuremath{\mu}\emph{AO-MPC},\cite{zometa2013} a code generation tool
for a tailored solver based
on the augmented Lagrangian method and Nesterov's fast gradient method.\cite{zometa2012, koegel2011}
By design, the algorithm's implementation is simple
and relies only on matrix-vector operations (i.e., no divisions),
it can easily be warm started and computes good enough MPC inputs
in very few iterations of the algorithm.
Furthermore, for problems with good numerical properties, the computations
can be made more efficient by using
fixed-point arithmetic (instead of
floating point).
The code generation reformulates the MPC optimization problem (\ref{eqn:mpcproblem})
as the following condensed parametric QP $\mathcal{P}(x)$
\begin{mini}
{\vb{u} \in \mathcal{U}}{\frac{1}{2} \vb{u}^\Tr H \vb{u} + \vb{u}^\Tr g(x),}{\label{eqn:qp}}{}
\addConstraint{\underline{\vb{z}}(x)}{\leq E \vb{u} \leq \overline{\vb{z}}(x),}{}
\end{mini}
where the constant Hessian matrix $H^\Tr\!=\!H > 0 \inRpow{Nm \times Nm}$ and the gradient vector $g(x)$
depend on the system (\ref{eqn:system}) and the objective function
in (\ref{eqn:mpcproblem}). Additionally, $g(x)$ depends on the current state $x$ (i.e., the parameter).
The set $\mathcal{U} = \{ \vb{u} \; | \; \underline{u} \leq u_j \leq \overline{u}, \; j\!=\!0,...,N-1\}$ defines the
input constraints.
The constant matrix $E$ and the constraint vectors $\underline{\vb{z}}(x)$ and
$\overline{\vb{z}}(x)$
depend on the
constraints in (\ref{eqn:mpcproblem}).
The generated code is highly portable C-code,
and it is split into the data for the QP $\mathcal{P}(x)$ (e.g., the constant matrix $H$
and the parametric vector $g(x)$)
and the implementation of the optimization
algorithm to solve the QP.\cite{zometa2012, koegel2011}
\begin{figure*}
\centering
\input{figures/codegen}
\caption{Scheme of HILO-MPC embedded module}
\label{fig:hilo-muaompc}
\end{figure*}
Figure~\ref{fig:hilo-muaompc} depicts an overview of the embedded module.
As in other HILO-MPC modules, the \mintinline{python}{LMPC} class is initialized
using a \mintinline{python}{Model} object. In this case, a linear time-invariant discrete-time
model is used. Calling the \mintinline{python}{LMPC.setup()} method, specifying the solver
\mintinline{python}{"muaompc"}, will trigger the code generation procedure.
The code generation automatically creates C-code which can be compiled
to run on an embedded target. Additionally, a Python interface to the
generated C-code is also created. This Python interface is used by the \mintinline{python}{LMPC.optimize()} method to obtain the input that minimizes \ref{eqn:mpcproblem}.
The same procedure is exemplified in the following Python code
\begin{minted}[bgcolor=bg]{python}
from hilo_mpc import LMPC, Model
# We assume that all variables (e.g. dt, matrices A, B, Q, R, etc.)
# have been properly defined elsewhere
# The system matrices are in discrete-time form
model.A = A
model.B = B
# Sampling period is dt
model.setup(dt=dt)
# The linear MPC class is used with the discrete model
mpc = LMPC(model)
# and is configured with the weighting matrices
mpc.Q = Q
mpc.R = R
mpc.P = Q
# and the constraints in states and inputs
mpc.set_box_constraints(x_lb=x_lb, x_ub=x_ub,
u_lb=u_lb, u_ub=u_ub)
# for a given horizon length (in steps)
mpc.horizon = N
# We configure muaompc solver
mpc.set_solver_opts(dict(in_iter=1, ex_iter=5, warm_start=True))
# Using the solver muaompc triggers the code generation
mpc.setup(nlp_solver='muaompc')
# from this point on, the generated MPC code can be used
# directly on an embedded target or via the Python interface.
# HILO-MPC uses the Python interface for simulation
model.set_initial_conditions(x0=x0)
for i in range(sim_steps):
# the generated C-code is called using x as parameter
u = mpc.optimize(x0=x)
model.simulate(u=u)
x = model.solution['xf']
\end{minted}
\section{Examples}
\label{sec:examples}
In the next section, we present some examples that use HILO-MPC. These examples can also be found in the examples repository, as Python files or Jupyter notebooks. For these reasons, here we do not go into the implementation details but give just a high-level description of the problems and present the results. The reader is invited to read the documentation and the examples codes for details.
\subsection{Learning the Dynamics - Race Car}
\label{sec:example-race-car}
We use HILO-MPC for a realistic example of an autonomous racing minicar. The goal is to follow a complex track using a path-following formulation in presence of disturbances. We use a bike model with nonlinear tire models and drive train model (see Figure~\ref{fig:bike}) that has been identified and validated in experiments.\cite{Liniger2015} To this model we add a component describing the effect of lateral drag forces due to, for example, a strong wind. The model is represented by the following system of nonlinear differential equations
\begin{align}
\dot{p}_x &= v_x \cos (\psi) - v_y \sin(\psi), \\
\dot{p}_y &= v_x \sin (\psi) + v_y \cos(\psi), \\
\dot{\psi} &= w, \\
\dot{v}_x &= \frac{1}{m}\left( F_{r,x}-F_{a,x}- (F_{f,y}-F_{a,y}) \sin(\delta) + m v_y \omega \right), \\
\dot{v}_y &= \frac{1}{m}\left( F_{r,y}-F_{a,y}- (F_{f,y}-F_{a,y}) \cos(\delta) - m v_x \omega \right), \\
\dot{\omega} &= \frac{1}{I_z} \left( F_{f,y}l_f \cos(\delta) - F_{r,y}l_r\right),
\end{align}
where $p_x$ and $p_y$ are the coordinates of the center of gravity (CoG), $v_x$ and $v_y$ are longitudinal and later velocities of the CoG. The orientation is denoted by $\psi$ and the yaw rate by $\omega$. The control inputs are the motor duty cycle $d$ and steering angle $\delta$. The two parameters $l_f$ and $l_r$ are the distances from the CoG to the front axle and rear axle respectively, $m$ is the mass of the car and $I_z$ is the inertia. The path to follow is the center of a racing track which has been interpolated using splines.
The tire forces are modeled with a simplified Pacejka Tire Model\cite{bakker1987tyre}
\begin{subequations}
\begin{align}
F_{f,y} &= D_f \sin(C_f \arctan(B_f \alpha_f)), \\
F_{r,y} &= D_r \sin(C_r \arctan(B_r \alpha_r)), \\
F_{r,x} &= (C_{m1}-C_{m2}v_x)d - C_r - C_d v^2_{x}, \\
\alpha_f &= - \arctan\left( \frac{w l_f + v_y}{v_x} \right) + \delta,\\
\alpha_r &= \arctan\left( \frac{w l_r - v_y}{v_x} \right),
\end{align}
\end{subequations}
where $D_f,D_r,B_f,B_r, C_{m1},C_{m2},C_r$ and $C_d$ are parameters. The longitudinal and lateral drag forces are defined respectively as
\begin{align*}
F_{a,x} = 0.5 \,\ c_w \rho A v_x, \\
F_{a,y} = 0.5 \,\ c_w \rho A v_{y,\text{wind}},
\end{align*}
where $c_w$ is the drag coefficient, $\rho$ is the air density, $A$ the effective flow surface and $v_{y,wind}$ is the lateral wind velocity.
The model used by the MPC does not have the drag effect. The goal is learning this effect from data using a Neural Network, and then augment the first principle model with a machine learning component that models the drag effect. After discretization, the hybrid model can be written as
\begin{equation}
x_{k+1} = f(x_k,u_k) + B^T m(x_k,u_k),
\end{equation}
where $m(x_k,u_k)$ is an NN model and
\begin{equation}
B = \begin{bmatrix}
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{bmatrix}.
\end{equation}
The features of the NN are the $v_{y,\text{wind}}$ and $v_x$, the labels are correction terms on $\phi,v_x,v_y$ and $\omega$. To show the effectiveness of the learning, we compare the results of MPC using the perfect model (i.e. with known drag force effects), the hybrid model (using the NN) and the model without drag forces. Furthermore, the measurements of position, velocity and directions are effected by Gaussian noise, and estimated using an Unscented Kalman Filter.
Figure \ref{fig:race_car} shows the results of the simulation. While the hybrid model has similar results as the perfectly-known model, the model without drag exits the race area after the fifth curve. The complete code can be found in the HILO-MPC repository.
\begin{figure*}
\captionsetup[subfigure]{justification=centering,font=large}
\centering
\begin{subfigure}[b]{.49\linewidth}
\input{figures/bike}
\caption{Model used for the racing car.}
\label{fig:bike}
\end{subfigure}
\begin{subfigure}[b]{.49\linewidth}
\input{figures/racing}
\caption{Results of the path-following problem}
\label{fig:race_car}
\end{subfigure}
\caption{Example of a racing car driving on a racetrack following a given path.}
\end{figure*}
\subsection{Learning a Reference - Cooperative Robots}
In this example, inspired by Matschek~et~al.,\cite{Matschek2021a} a follower robot has to track the position of a leader robot. The leader moves in a periodic but unknown trajectory. The objective is learning the trajectory of the leader with GPs and pass the learned trajectory to the follower. Hence, in this case the machine learning model enters the reference function (see Figure~\ref{fig:mobile-robots}). The nonlinear dynamics of the robots is described with the following ODE
\begin{subequations}
\begin{align}
\dot{x}_1 &= u_1 \sin(x_3), \\
\dot{x}_2 &= u_1 \cos(x_3), \\
\dot{x}_3 &= u_2 ,
\end{align}
\end{subequations}
where $x_1$ and $x_2$ are the horizontal and vertical position of the root and $x_3$ its heading angle, $u_1$ is the speed and $u_2$ the turning rate.
The problem we want to solve is of the form \eqref{eq:problem-mpc} with objective function \eqref{eq:mpc-traj-tracking} where $x_r(\cdot)$ is the mean function of a GP trained on the data set collected from the position of the leader.
The trajectory generated by the leader, results from applying the following time-varying forces
\begin{subequations}
\begin{align}
u_1(t) &= 2 + 0.1 \sin(t), \\
u_2(t) &= 0.5 + \sin(t).
\end{align}
\end{subequations}
Figures \ref{fig:mobile-robots-results} show the results of the reference learning and closed-loop simulation.
\begin{figure*}
\centering
\input{figures/mobile_robots}
\caption{Cooperative robot example. The follower needs to track the trajectory of the leader. The trajectory is learned from data using a GP regressor.}
\label{fig:mobile-robots}
\end{figure*}
\begin{figure*}
\centering
\input{figures/tot_mobile_robots}
\caption{Left column: learned GP. Right column: states of the follower robot following the learned references.}
\label{fig:mobile-robots-results}
\end{figure*}
\subsection{Learning the Controller - Spring Damper System}
Solving a MPC requires the solution of a (nonlinear) optimization problem online. This is possible only for applications where the available computational power and energy are sufficient to guarantee a solution within the sampling times. For embedded applications, with low computational power or that can use only a limited amount of energy (for example battery powered systems) this is often not possible. Hence, methods that provide at least a close-to-optimal solution without solving the optimization approach at every time step are necessary. This can be done using \emph{explicit} MPC approaches. Some of these approaches are based on learning the solution of an MPC offline, i.e., the map $x \mapsto \rho_{\theta}(x)$ that approximates the implicit MPC control law, and then using the learned controller online (see Figure~\ref{fig:ml-controller}).\cite{Parisini1995,Karg2020,Maddalena2020,Cao2020,Chen2018,Cseko2015,Ponkumar2018} In this way, the control action can be found with a simple and fast function evaluation.
In this example, we want to control a mass-spring-damper system using a learned controller (Figure \ref{fig:ml-controller}). The models is
\begin{subequations}
\begin{align}
& \dot{x}_1 = x_2, \\
& \dot{x}_2 = \frac{1}{m} (u - k x_1 - d x_2) ,
\end{align}
\end{subequations}
where $x_1$ is the vertical position, $x_2$ the vertical velocity, $u$ the vertical force and $k,d$ the system parameters. The equilibrium point is $x=(0,0), u=0$. The objective is to maintain the reference $x_{\text{ref}}=[1,0]$ using a learned MPC. To do so we use the results of just one closed-loop MPC simulation starting from the initial conditions $x(0)=(12,0)$. In total 667 data points are collected. We use the data collected to train a NN with three fully-connected layers, with 10 neurons each. The features of the NN are $x_1$ and $x_2$, the labels is the input $u$. We test the learned controller starting from a different initial condition $x(0)=(10,0)$. In Figure~\ref{fig:ml-controller-results} the simulation results are shown. The learned controller is able to bring the system to the reference as the original controller.
\begin{figure*}
\centering
\input{figures/machine_learning_controller}
\caption{Scheme of a machine learning controller.}
\label{fig:ml-controller}
\end{figure*}
\begin{figure*}
\centering
\input{figures/learned_mpc}
\caption{Results of the learned MPC.}
\label{fig:ml-controller-results}
\end{figure*}
\subsection{Embedded MPC - HIL Simulation using an Arduino}
Here we present a hardware-in-the-loop (HIL) simulation
based on the code generated by HILO-MPC embedded module (see
Subsection~\ref{sec:emb-mod}).
\begin{figure*}
\centering
\input{figures/hilsim}
\caption{Scheme of hardware-in-the-loop simulation}
\label{fig:hil-sim}
\end{figure*}
Figure~\ref{fig:hil-sim} depicts the HIL simulation setup.
The generated C-code is cross compiled to run on an
microcontroller unit (MCU, an Arduino in this case).
The system dynamics are simulated
on a laptop computer using the nominal linear model $x^+=Ax+Bu$.
For any given input $u$, and using the current state $x$,
the simulation computes the state at the next sampling period $x^+$.
Using a serial interface (i.e. USB to UART), the state $x^+$
is transferred to the Arduino, which uses it as parameter $x$
to solve the MPC problem, i.e. compute a new sequence $\vb{u}$
by solving the parametric QP $\mathcal{P}(x)$.
Only the first part $u_0$ of this sequence is transferred to the computer via the serial interface, and thus closing the loop.
The system to be controlled is a
linearized version of the race car in Subsection~\ref{sec:example-race-car}.
We assume the car is traveling at constant speed,
and only consider the lateral dynamics of the car.
The input is the steering angle $u=\delta$.
The state is defined as $x = [p_y \ \ \psi \ \ \omega]^T$,
where $p_y$ is the lateral position of the center of mass of the car,
and $\psi$ is the orientation
of the car with respect to the $X$ axis and $\omega = \dot \psi$
(refer to Figure~\ref{fig:bike}).
In this case, we constraint the input to be $-0.1 \leq u \leq 0.1$,
and the state $x_1 = \psi$ must satisfy $-0.1 \leq x_1 \leq 0.1$.
The objective is to bring the state to the origin, i.e. $x = \vb{0}$,
while satisfying input and state constraints.
For our HIL simulation, we used an Arduino Nano 33 BLE.\cite{arduinonano}
The microcontroller core is a 32-bit Cortex-M4F (with single precision floating-point unit)
running at 64 MHz.
We use a discretization time of $15 ms$, and a horizon length of $20$ steps.
In our experiments, it takes the Arduino around 10 ms to find (an approximate) solution
to the MPC problem on each iteration. The compiled C-code takes about
$20 kB$ of flash memory.
Figure~\ref{fig:hil-sim-plot}
shows the compared trajectories of states and input
for the HIL simulation (approximate solution on Arduino using single precision float)
and the exact solution computed by in the laptop using HILO-MPC
interface to CPLEX.
In the figures, the difference between exact and approximate solution
are not significant, except where the state $x_1$ is close
to its constraint (around $0.1$ seconds).
Note that in both cases, neither the input nor state constraints are violated.
The main differences are due to truncation due to the early
termination of the embedded algorithm, and the use of
single precision float.
\begin{figure*}
\centering
\input{figures/plot_embmpc}
\caption{Comparison of HIL simulation and exact simulation}
\label{fig:hil-sim-plot}
\end{figure*}
\section{Conclusions and Outlook}
\label{sec:conclusions}
We introduced HILO-MPC, a toolbox for fast development of predictive control and estimation problems that facilitates the use of machine learning models trained using PyTorch and TensorFlow for simulations and embedded applications. We showed how HILO-MPC can be used in four examples, the code of the presented applications is openly available in the HILO-MPC repository.
The toolbox can solve many problems, such as model predictive control, moving horizon estimation, Kalman filters, and particle filters. Its flexibility and simplicity can make it a valuable tool for research and teaching.
Future versions will allow using other machine learning models such as recurrent neural networks and reinforcement learning. Other MPC formulations, such as tube-based MPC, stochastic MPC, \citep{Mayne2005} and multi-mode MPC \citep{bethge2018multi, morabito2019multi} will be available. We are currently working on interfacing the toolbox with an embedded nonlinear MPC software using the Al'brekht's method.\cite{Kallies2020a,Kallies2020b,Kallies2021}
\section*{Acknowledgements}
We would like to thank Michael Maiworm, Tim Zieger, Johanna Bethge, Maik Pfefferkorn, Rudolph Kok, Sebasti{\'a}n Espinel-R{\'i}os and Janine Matschek for providing feedback and examples for HILO-MPC. Special thanks go to Lena Kranert for the cricket.
|
1,116,691,499,255 | arxiv | \section{Introduction}
In this investigation we consider
solution of general {unconstrained} optimization problems using the
steepest-descent method and focus on {modifying the definition of the
gradient} such that in certain circumstances {this approach} will
achieve a quadratic convergence, characteristic of Newton's method.
This is accomplished by judiciously exploiting the freedom inherent in
the choice of different equivalent norms defining the gradient through
the Riesz representation theorem. This freedom can be used to adjust
the definition of the inner product, such that the resulting gradient
will, in a suitable sense, best resemble the corresponding Newton
step. While such ideas can be pursued in both finite-dimensional and
infinite-dimensional settings, the formulation is arguably more
interesting mathematically and more useful in practice in the latter
case. We will thus consider unconstrained optimization problems of the
general form
\begin{equation}
\label{e:min{e(u)}}
e({\widehat{u}}) = \min\{e(u),\; u \in H \},
\end{equation}
where $e \; : \; H \mapsto \mathbb R$ is the objective functional and $H$ is
a suitable function space with Hilbert structure.
Applications which have this form include, for example,
minimization of various energy functionals in physics and optimization
formulations of inverse problems, where evaluation of the objective
functional $e(u)$ may involve solution of a complicated (time-)dependent
partial differential equation (PDE). In such applications $H$ is typically
taken as a Sobolev space $H^p(\Omega)$, $p \in \mathbb N$, where $\Omega
\subset \mathbb R^d$ is the spatial domain assumed to be sufficiently smooth (Lipschitz)
and $d \ge 1$ is the dimension \cite{af05}. Therefore, to fix
attention, here we will assume $H := H^1_0(\Omega)$ with the inner
product and norm in $H$ defined as
\begin{equation}
\label{e:(u,v)_Ht}
(u,v)_H=\int_{\Omega}((\nabla u\cdot\nabla v) + uv)\, dx,
\qquad
\|u\|_H=(u,v)_H^{1/2}.
\end{equation}
\begin{algorithm}
\begin{algorithmic}[1]
\STATE set $u = u_0$
\REPEAT
\STATE compute $e'(u;\cdot)$
\STATE determine $g\in H$ as the solution of $(g,v)_H=e'(u;v)$, $\forall v\in H$
\STATE set $\widetilde{u} = u - \kappa g$
\STATE set $u = \widetilde{u}$
\UNTIL{ \ $|e(u)|<$ \texttt{err}}
\end{algorithmic}
\caption{
(Sobolev) Gradient method \newline
\textbf{Input:} \newline
\hspace*{0.5cm} $u_0 \in H$ --- initial guess \newline
\hspace*{0.5cm} $\kappa > 0$ --- step size (sufficiently small) \newline
\hspace*{0.5cm} $\texttt{err} > 0$ --- tolerance \newline
\textbf{Output:} \newline
\hspace*{0.5cm} $\widetilde{u}$ --- approximation to the solution ${\widehat{u}}$ of problem \eqref{e:min{e(u)}}}
\label{alg:gradient}
\end{algorithm}
\begin{algorithm}
\begin{algorithmic}[1]
\STATE set $u = u_0$
\REPEAT
\STATE {compute $e'(u;\cdot)$}
\STATE {compute $e''(u;\cdot,\cdot)$}
\STATE determine $h\in H$ as the solution of $e''(u;h,v)=e'(u;v)$, $\forall v\in H$
\STATE set $\widetilde{u} = u - h$
\STATE set $u = \widetilde{u}$
\UNTIL{ \ $|e(u)|<$ \texttt{err}}
\end{algorithmic}
\caption{
Newton's method \newline
\textbf{Input:} \newline
\hspace*{0.5cm} $u_0 \in H$ --- initial guess \newline
\hspace*{0.5cm} $\texttt{err} > 0$ --- tolerance \newline
\textbf{Output:} \newline
\hspace*{0.5cm} $\widetilde{u}$ --- approximation to the solution ${\widehat{u}}$ of problem \eqref{e:min{e(u)}}}
\label{alg:newton}
\end{algorithm}
The two most elementary approaches to solve problem
\eqref{e:min{e(u)}} are the gradient and Newton's method which, for
the sake of completeness, are defined in Algorithms \ref{alg:gradient}
and \ref{alg:newton}, respectively. The former approach is sometimes
also referred to as the ``Sobolev gradient'' method \cite{n10}. While
gradient approaches often involve an adaptive step size selection
\cite{nw00}, for simplicity of analysis in Algorithm
\ref{alg:gradient} we consider a fixed step size $\kappa =
\text{Const}$. Likewise, in order to keep the analysis tractable, we
do not consider common modifications of the gradient approach such as
the conjugate-gradient method. As regards convergence of the gradient
and Newton's method, we have the following classical results, see for
example \cite{ciarlet-1}.
\begin{theorem}
\label{th:gradient}
Let ${\widehat{u}}\in H$ be {a} solution of (\ref{e:min{e(u)}}). Assume
that $e$ is twice differentiable near ${\widehat{u}}$ and there exist
$\delta_0>0$ and $\alpha_0>0$ such that $e''(u;v,v)\geq
\alpha_0\|v\|^2_H$ for all $\|u-{\widehat{u}}\|_H\leq \delta_0$ and $v\in H$. Then,
for all $u_0\in B_H({\widehat{u}},\delta_0)$ the gradient {method given by}
Algorithm \ref{alg:gradient} converges linearly to ${\widehat{u}}$ in $H$, i.e.,
\begin{eqnarray*}
\|\widetilde{u}-{\widehat{u}}\|_H
&\leq&
\epsilon\|u-{\widehat{u}}\|_H,
\end{eqnarray*}
with $\epsilon\in(0,1)$ depending only on $e$ and $\kappa$.
\end{theorem}
\begin{theorem}\label{th:newton}
Let ${\widehat{u}}\in H$ be {a} solution of (\ref{e:min{e(u)}}). Assume
that $e$ is three times differentiable near ${\widehat{u}}$ and that the map
$v\in H\mapsto e''({\widehat{u}};v,\cdot)\in H'$ is invertible. Then, there
exists $\delta_0>0$ such that for all $u_0\in B_H({\widehat{u}},\delta_0)$
Newton's {method given by} Algorithm \ref{alg:newton} converges
quadratically to ${\widehat{u}}$ in $H$, i.e.,
\begin{eqnarray*}
\|\widetilde{u}-{\widehat{u}}\|_H
&\leq& C \|u-{\widehat{u}}\|^2_H ,
\end{eqnarray*}
with $C>0$ depending only on $e$.
\end{theorem}
We emphasize that in Algorithm \ref{alg:gradient} the gradient $g$
must be computed with respect to the topology of the space $H$ in
which the solution {to \eqref{e:min{e(u)}}} is sought \cite{n10},
an aspect of the problem often neglected in numerical investigations.
A metric equivalent to $\| \cdot \|_H$ (in the precise sense of norm
equivalence) can be obtained by redefining the inner product in
\eqref{e:(u,v)_Ht} more generally as follows
\begin{equation}
\label{e:(u,v)_lam0}
(u,v)_{\lambda_0}
=
\int_{\Omega}(\lambda_0 (\nabla u\cdot\nabla v) + uv)\, dx,
\end{equation}
where $\lambda_0\in(0,\infty)$ is a fixed constant. While as compared
to \eqref{e:(u,v)_Ht} definition \eqref{e:(u,v)_lam0} does not change
the analytic structure of the optimization problem
\eqref{e:min{e(u)}}, there is abundant computational evidence obtained
in the solution of complicated optimization problems
\cite{pbh02,p07b,ap15a} that convergence of gradient Algorithm
\ref{alg:gradient} may be significantly accelerated by replacing the
inner product from \eqref{e:(u,v)_Ht} with the one introduced in
\eqref{e:(u,v)_lam0} for some judicious {choices} of the
parameter $\lambda_0$. Likewise, a similar acceleration was also
observed when the inner product in \eqref{e:(u,v)_Ht} was replaced
with another equivalent definition motivated by the structure of the
minimization problem and different from \eqref{e:(u,v)_lam0},
cf.~\cite{rssl09,ms10,kd12}. In the absence of an understanding of the
mechanism responsible for this acceleration, the parameter
$\lambda_0$, or other quantities parameterizing the equivalent inner
product, were chosen empirically by trial and error, which is
unsatisfactory.
In the present investigation we will consider a more general form of
the inner product \eqref{e:(u,v)_lam0} in which the constant
$\lambda_0$ is replaced with a space-dependent weight $\lambda =
\lambda(x)$. Our goal is to develop a rational and systematic approach
allowing one to accelerate the convergence of gradient iterations in
Algorithm \ref{alg:gradient} in comparison to the standard case by
adaptively adjusting the weight $\lambda(x)$. This will result in a
reduction of {the total number of iterations needed to solve
problem \eqref{e:min{e(u)}} to a given accuracy,} but each iteration
will be more costly.
Modifications of the inner product with respect to which the gradient
is defined may also be interpreted as gradient preconditioning and
this perspective is pursued in the monograph \cite{fk02} focused on
related problems arising in the solution of nonlinear elliptic
equations. The relationship between the gradient and Newton's methods
was explored in \cite{kn07} where a variable inner product was
considered. In contrast to \revt{the present} approach \revt{in which
the} inner-product \revt{weights are} sought by matching the
projections of \revt{the} gradient and Newton's steps \revt{onto} a
certain subspace, \revt{in \cite{kn07} optimal inner products were
found by maximizing the descent achieved at a given iteration with
respect to the structure of the corresponding preconditioning
operator.}
The structure of the paper is as follows. In the next section we
define the new approach in a general form, whereas in Section
\ref{sec:errors} we prove its convergence properties. Then, in Section
\ref{sec:optimal} we describe the numerical approach implementing the
general method introduced in Section \ref{sec:newgrad} {in two
variants} and analyze its computational complexity. Our model
problem and computational results are presented in Section
\ref{sec:results}, whereas discussion and conclusions are deferred to
Section \ref{sec:final}.
\section{A new gradient method based on an optimal inner product}
\label{sec:newgrad}
In this section we introduce our modified version of Algorithm
\ref{alg:gradient} for the solution of the minimization problem
\eqref{e:min{e(u)}}. We begin by making the following assumptions on
the energy $e(u)$:
\begin{eqnarray}
e&&\mbox{\it is $C^2$ in $H$},
\\
|e''(u;v,w)|
&\leq&
M\|u\|_H \|w\|_H,\quad u,v,w \in H,
\label{e:e->C0}\\
e''(u;v,v)
&\geq&
m\|v\|_H^2,
\label{e:e->convex}
\end{eqnarray}
with certain $M,m>0$.
Let us point out that at each step of both the gradient and Newton's
methods the descent direction is defined by the solution of the
equation
\begin{equation}\label{e:b=e'->1}
b(z,v)=e'(u;v),\quad \forall v\in H,
\end{equation}
where $b(u,v)$, a symmetric bilinear continuous elliptic form, and
$z\in H$ are specific to each method, namely,
\begin{itemize}
\item $b(u,v)=(u,v)_H$ and $z=g$ in the case of {the} gradient method, and
\item $b(u,v)=e''(u;u,v)$ and $z=h$ in the case of Newton's method.
\end{itemize}
Moreover, we note that the solution $z$ of \eqref{e:b=e'->1} is also
the solution of the minimization problem
\begin{equation}
\label{e:min{b-l}}
\min\left\{\frac{1}{2}b(v,v)-e'(u;v),\;\, v\in H\right\}.
\end{equation}
We emphasize that, in fact, Newton's method may be also viewed as a
``gradient'' method with a particular choice of the inner product at
each iteration, namely, the one induced by $e''(u)$. Therefore, the
idea for improving the classical gradient method is to make the
gradient step $g$ ``close'' to the Newton step $h$ by suitably
adapting the inner product in $H$.
We thus propose the following modification of the gradient method from
Algorithm \ref{alg:gradient}. We want to consider the gradient $g$
{defined} with respect to {an} inner product in $H$
depending on a function parameter $\lambda$. Typically, $0<\lambda\in
C^0(\overline{\Omega})$, {however, to make our method more
attractive from the computational point of view} we will consider
$\lambda$ with a finite range. Namely, let
$\{\Omega_{i},\, i=1,\ldots,N\}$, be a partition of $\Omega$ into open Lipschitz sets,
$\Lambda=\{\lambda:\Omega\mapsto\mathbb R,\,\; \lambda(\Omega_i)=\lambda_i
\in\mathbb R\} = \operatorname{span} \{\ell^i,\, i=1,\dots,N\}\subset L^\infty(\Omega)$,
where $\ell^i\in\Lambda$, $\ell^i=\delta_{i,j}$ in $\Omega_j$,
$i,j=1,\ldots,N$ with $\delta_{i,j}$ the Kronecker symbol,
$\Lambda^+=\Lambda\cap\{0<\lambda<\infty\}$. Sometimes {without
the risk of confusion we will write
$\lambda=[\lambda_1,\ldots,\lambda_N]\in\mathbb R^N$ for
$\lambda\in\Lambda$}, meaning $\lambda(\Omega_i)=\lambda_i$, for all
$i=1,\ldots, N$. Then, for $\lambda\in \Lambda^+$, we define the
following inner product and norm in $H$
\begin{equation}\label{e:(u,v)_lambda}
(v,w)_\lambda
=
\int_{\Omega}
\lambda (\nabla v\cdot \nabla w) + v w \, dx,
\quad
\|v\|_\lambda=(v,v)_\lambda^{1/2},
\quad
\forall v,w\in H.
\end{equation}
Clearly, $(\cdot,\cdot)_\lambda$ and $(\cdot,\cdot)_H=(\cdot,\cdot)_1$ are
equivalent in $H$ and therefore we can use $(\cdot,\cdot)_\lambda$
instead of $(\cdot,\cdot)_H$ for the gradient method.
The idea is to use the inner product $(\cdot,\cdot)_\lambda$ in the
gradient method, with $\lambda$ judiciously chosen. More
specifically, for $\lambda\in\Lambda^+$, let $g=g(\lambda)$ be the
solution of
\begin{equation}\label{e:g(lambda)}
(g,v)_\lambda
=
\int_{\Omega}
\lambda (\nabla {g}\cdot \nabla v) + {g} v \, dx
=
e'(u;v),\quad\forall v\in H.
\end{equation}
\begin{remark}
\label{r:Riesz}
In the following we will, \revt{in particular}, refer to the
gradient \revtt{$g_1:=g(1)$} which corresponds to \revt{the usual inner
product \eqref{e:(u,v)_Ht} and is also obtained by setting
$\lambda=1$ in \eqref{e:g(lambda)}}, and to \revt{the gradient}
\revtt{$g_0:=g(0)$} which corresponds to $\lambda=0$ \revt{in
\eqref{e:g(lambda)}}. Usually, \revt{$g_1$ and $g_0$ are referred
to as, respectively, the $H^1$ and $L^2$ Riesz representations of
$e'(u)$.}
In general $g_0\notin H^1_0(\Omega)$, but we have
\begin{eqnarray}
-\Delta g_1+g_1
&=&
g_0
\;\, \mbox{\it in ${\cal D'}(\Omega)$},
\label{e:g_1,g_0}\\
-\nabla\cdot(\lambda\nabla g(\lambda))+g(\lambda)
&=&g_0\;\, \mbox{\it in ${\cal D'}(\Omega)$}.
\label{e:g(l),g_0}
\end{eqnarray}
\end{remark}
Note that we will use the symbol $g$ to denote the operator
$\lambda\in\Lambda\mapsto g=g(\lambda)\in H$, or to denote an element
of $H$ --- the meaning will always be clear from the context.
Now assume we are at a certain iteration of the gradient method with $u$
known, which we seek to update to a new value $\widetilde{u}$, cf.~step
5 in Algorithm \ref{alg:gradient}. For this, first we look for a
certain ${\widehat{\lambda}} \in \Lambda^+$, defined by\footnote{All along this
paper, the symbol ``$\widehat{\phantom{x}}$'' will be used to denote
the solution of a minimization problem, whereas the symbol
``$\widetilde{\phantom{x}}$'' will be used to represent an updated value
of a variable.}
\begin{equation}\label{e:min(lambda)->1}
j({\widehat{\lambda}}) := \min\left\{
j(\lambda):=f\circ g(\lambda),\;\, \lambda\in \Lambda^+
\right\}, \quad \text{\it where}
\quad
f(g):=\frac{\kappa}{2}e''(u;g,g)-e'(u;g).
\end{equation}
The reason for introducing the step size $\kappa$ in this equality
will be clear from Remark \ref{r:name} and also later during the error
analysis in Section \ref{sec:errors}. Note that, if problem
\eqref{e:min(lambda)->1} has a solution ${\widehat{\lambda}} \in \Lambda^+$, then
we will show (see Proposition \ref{p:g'->1}) that ${\widehat{\lambda}}$ solves
\begin{equation}\label{e:e''(lambda)=e'}
e''(u;\kappa g({\widehat{\lambda}}),g'({\widehat{\lambda}};\ell))=e'(u;g'({\widehat{\lambda}};\ell)),\quad\forall \ell\in \Lambda,
\end{equation}
where $g'({\widehat{\lambda}};\ell)$ denotes the derivative of $g$ at ${\widehat{\lambda}}$
in the direction $\ell$. Then, the modified gradient approach will
consist of Algorithm \ref{alg:gradient} with step 4 amended as follows
\begin{eqnarray}
&4.&\mbox{determine $g=g({\widehat{\lambda}})$, where ${\widehat{\lambda}}\in\Lambda^+$ is such that $g({\widehat{\lambda}})\in H$ solves (\ref{e:e''(lambda)=e'})}.
\label{alg:gradient+2}
\end{eqnarray}
\begin{remark}\label{r:name}
Clearly, our approach is equivalent to the gradient method, but with
the classical inner product $(\cdot,\cdot)_H$ replaced with
$(\cdot,\cdot)_\lambda$.
From equation \eqref{e:e''(lambda)=e'} it follows that
$e''(u;h-\kappa g,g'({\widehat{\lambda}};\ell))=0$, where $h$ is
Newton's step. This means that $P_u(h-\kappa g)=0$,
where $P_u:H\mapsto T_u$ is the projection from $H$ to $T_u$, in
which $T_u= \operatorname{span}\{g'({\widehat{\lambda}};\ell),\; \ell\in\Lambda\}$ is the
tangent space to the manifold $\{g(\lambda),\,
\lambda\in\Lambda^+\}\subset H$ at $g({\widehat{\lambda}})$, determined with
respect to the inner product $e''(u;\cdot,\cdot)$.
If $T_u=H$, then $\kappa g({\widehat{\lambda}})=h$ and our method reduces to
Newton's method. However, here we have $dim(\Lambda)=N$, so that
{in general} $T_u\neq H$ and $\kappa g$ will be close to $h$ in
the sense that $P_u(h-\kappa g)=0$. This relation will be the key
ingredient to prove {in the demonstration} that our gradient
method, in addition to the linear convergence of a standard gradient
method, has also a quadratic convergence in a certain sense
depending on $T_u$ and the projection $P_u$. This will be explained
in the next sections.
\end{remark}
Our method critically depends on the choice of $\lambda$ and the
following proposition offers a first glimpse of what may happen with
the solution of problem \eqref{e:min(lambda)->1}.
\begin{proposition}\label{p:lambda^k->}
Let $(\lambda^k)$ be a minimizing sequence of $j$ in $\Lambda^+$ and
$(g(\lambda^k))$ be the \revt{corresponding} sequence of gradients
\revt{defined in} \eqref{e:g(lambda)}. Then, up to a subsequence,
$(g(\lambda^k))$ converges weakly in $H^1_0(\Omega)$ and strongly
in $L^2(\Omega)$ to \revt{an element} $g\in H$, while for the
sequence $(\lambda^k)$ one of the following cases may occur.\\
(i)
There
exist ${\widehat{\lambda}}=[{\widehat{\lambda}}_1,\ldots,{\widehat{\lambda}}_N]\in\Lambda^+$ and a subsequence of $(\lambda^k)$,
still denoted $(\lambda^k)$, such that $\lim_{k\to\infty}\lambda^k_i={\widehat{\lambda}}_i$ for all
$i=1,\dots,N$.
In this case $g=g({\widehat{\lambda}})$, i.e.
\begin{equation}\label{e:g(hlambda),1}
\int_\Omega {\widehat{\lambda}}(\nabla g({\widehat{\lambda}})\cdot\nabla v)+g({\widehat{\lambda}})v \, dx = e'(u;v),\;\, \forall v\in H,
\end{equation}
and ${\widehat{\lambda}}$ solves \eqref{e:min(lambda)->1}.
\\
(ii)
There exist ${\widehat{\lambda}}=[{\widehat{\lambda}}_1,\ldots,{\widehat{\lambda}}_N]\in\partial\Lambda^+$,
$I_0\subset I$, $I_\infty\subset I$, with
${\widehat{\lambda}}_i=0$ for all $i\in I_0$,
${\widehat{\lambda}}_i=+\infty$ for all $i\in I_\infty$,
$0<{\widehat{\lambda}}_i<+\infty$ for all $i\in I\backslash(I_0\cup I_\infty)$,
and a subsequence of $(\lambda^k)$, still denoted $(\lambda^k)$, such that
$\lim_{k\to\infty}\lambda^k_i={\widehat{\lambda}}_i$, for all $i=1,\dots,N$.
In this case $g\in H^1_0(\Omega;\Omega_\infty^\text{const})$ solves
\begin{eqnarray}
\hspace*{-8mm}
\int_{\Omega\backslash \Omega_0}
{\widehat{\lambda}} (\nabla g\cdot\nabla v) + gv \,dx
+
\int_{\Omega_0} gv \, dx
&=& e'(u;v),
\quad
\forall v\in H^1_0(\Omega;\Omega_\infty^\text{const}),
\label{e:g-in-G-G0-Ginf}
\end{eqnarray}
where
\begin{eqnarray}
\hspace*{-6mm}
\Omega_0&=&\cup\{\Omega_i,\; i\in I_0\},
\\
\hspace*{-6mm}
\Omega_\infty&=&\cup\{\Omega_i,\; i\in I_\infty\},
\\
\hspace*{-6mm}
H^1_0(\Omega;\Omega_\infty^\text{const})
&=&
\{
v\in H^1_0(\Omega),
\;\,
v=C_i\in\mathbb R\;\, in\;\, \Omega_i,\;\, \forall i\in I_\infty\}.
\label{e:H1_0(G;G0,Gi)}
\end{eqnarray}
Furthermore, if we define $g({\widehat{\lambda}})=g$, with $g$ given by \eqref{e:g-in-G-G0-Ginf}, we have
\begin{equation}
j({\widehat{\lambda}})\leq\liminf_{k\to\infty}j(\lambda^k).
\label{e:j(hl),=liminf}
\end{equation}
\end{proposition}
{\bf Proof}.
Let $(\lambda^k)$ be a minimizing sequence of $j$ in $\Lambda^+$ and
$\lambda^k=[\lambda^k_1,\ldots,\lambda^k_N]$.
Note that $g(\lambda^k)$ is well defined by
\begin{equation}\label{e:g(lambda^k)}
\int_{\Omega}
\lambda^k(\nabla g(\lambda^k)\cdot\nabla v) + g(\lambda^k)v \, dx
=
e'(u;v),\quad\forall v\in H.
\end{equation}
Note also that from the ellipticity of $f$ in $H$,
cf.~\eqref{e:e->C0}, \eqref{e:e->convex} and \eqref{e:min(lambda)->1},
it follows that the sequence $g(\lambda^k)$ is bounded in
$H^1(\Omega)$. Therefore, up to a subsequence, we may assume that
$g(\lambda^k)$ converges weakly in $H$ and strongly in $L^2(\Omega)$
to a certain $g\in H$.
As ${\rm dim}(\Lambda)=N$, there exist ${\widehat{\lambda}}=[{\widehat{\lambda}}_1,\ldots,{\widehat{\lambda}}_\revt{N}]$, with
${\widehat{\lambda}}_i\in[0,+\infty]$, and a subsequence of $(\lambda^k)$, still denoted $(\lambda^k)$,
such that $\lim_{k\to\infty}\lambda^k_i={\widehat{\lambda}}_i$ for all $i\in I$.
Two cases may occur.\\
1)
${\widehat{\lambda}}\in\Lambda^+$, i.e., ${\widehat{\lambda}}_i\in(0,\infty)$ for all $i\in I$.
From (\ref{e:g(lambda^k)}) we obtain
\begin{eqnarray*}
\int_{\Omega}
{\widehat{\lambda}} (\nabla g\cdot\nabla v) + gv \,dx
&=&
\int_{\Omega}({\widehat{\lambda}}-\lambda^k) (\nabla g\cdot\nabla v) \,dx
\\
&+&
\int_{\Omega}\lambda^k (\nabla(g-g(\lambda^k))\cdot\nabla v) + (g-g(\lambda^k))v \,dx
\\
&+&
\int_{\Omega}\lambda^k (\nabla g(\lambda^k)\cdot \nabla v) + g(\lambda^k)v \,dx
\\
&=&
\int_{\Omega}({\widehat{\lambda}}-\lambda^k) (\nabla g\cdot\nabla v) \,dx
\\
&+&
\int_{\Omega}\lambda^k (\nabla(g-g(\lambda^k))\cdot\nabla v) + (g-g(\lambda^k))v \,dx
\\
&+&
e'(u;v),\quad\forall v\in H.
\end{eqnarray*}
Then, letting $k$ go to infinity gives
\begin{eqnarray*}
\int_{\Omega}
{\widehat{\lambda}} (\nabla g\cdot \nabla v) + gv \,dx
&=&
e'(u;v),\quad\forall v\in H,
\end{eqnarray*}
which proves that $g=g({\widehat{\lambda}})$. Note that it is easy to show that
the subsequence $(g(\lambda^k))$ converges to $g$ strongly in $H$, and therefore
${\widehat{\lambda}}$ is the solution of (\ref{e:min(lambda)->1}) because $j$ is continuous
in $H$.
\\
2)
${\widehat{\lambda}}\in\partial\Lambda^+$.
\revt{Then,} there exist $I_0\subset I$ \revt{and} $I_\infty\subset I$
such that
${\widehat{\lambda}}_i=0$ for $i\in I_0$,
${\widehat{\lambda}}_i=+\infty$ for $i\in I_\infty$,
${\widehat{\lambda}}_i\in(0,+\infty)$ for $i\in I\backslash(I_0\cup I_\infty)$,
and a subsequence of $(\lambda^k)$, still denoted by $(\lambda^k)$, such that
$\lim_{k\to\infty}\lambda^k_i={\widehat{\lambda}}_i$ for all $i\in I$.
From (\ref{e:g(lambda^k)}), for each $g(\lambda^k)$ and $v\in H^1_0(\Omega;\Omega_\infty^{const})$
we have
\begin{equation*}
\int_{\Omega\backslash \Omega_0}
\lambda^k (\nabla g(\lambda^k)\cdot \nabla v) + gv \, dx
+
\int_{\Omega_0}
\lambda^k (\nabla g(\lambda^k)\cdot \nabla v) +gv \, dx
= e'(u;v).
\end{equation*}
Then, letting $k$ go to infinity gives \eqref{e:g-in-G-G0-Ginf},
because $\nabla v = 0$ in $\Omega_\infty$.
To show that $g$ is constant on each $\Omega_i$, $i\in I_\infty$, we take
$v=g(\lambda^k)$ in \eqref{e:g(lambda^k)}, so that we obtain
\begin{eqnarray*}
\int_{\Omega_\infty}
\lambda^k |\nabla g(\lambda^k)|^2 + |g|^2 \, dx
&=&
e'(u;g(\lambda^k))
- \int_{\Omega\backslash\Omega_\infty} \lambda^k |\nabla g(\lambda^k)|^2 + |g|^2 \, dx
\\
&\leq&
C,
\end{eqnarray*}
because $(g(\lambda^k))$ is bounded in $H$ and $(\lambda^k)$ is
bounded in $L^\infty(\Omega\backslash\Omega_\infty)$. It follows that
$\lim_{k\to\infty}|\nabla g(\lambda^k)|=0$ and
$\lim_{k\to\infty}g(\lambda^k)=g$ in $L^2(\Omega_\infty)$. Hence,
$g=C_i$ in $\Omega_i$, $C_i\in\mathbb R$. \revt{Thus, $g\in
H^1_0(\Omega;\Omega_\infty^\text{const})$ solves
\eqref{e:g-in-G-G0-Ginf}}.
Finally, \eqref{e:j(hl),=liminf} follows from the fact that $f$ is
convex and strongly continuous in $H$, so $f$ is weakly lower
semi-continuous, see \cite{brezis-1}. \hfill$\Box$
\begin{remark}\label{r:I0,Iinf}
While analyzing case (ii) we will use the following notation. For
$\lambda=[\lambda_1,\ldots,\lambda_N]\in\partial\Lambda^+$ we write
\begin{equation}
\left\{
\begin{array}{rcl}
I_{0,\lambda}&=&\cup\{i\in I,\;\, \lambda_i=0\},\\
\Omega_{0,\lambda}&=&\cup\{\Omega_i,\;\; i\in I_{0,\lambda}\},
\end{array}
\right.
\quad
\left\{
\begin{array}{rcl}
I_{\infty,\lambda}&=&\cup\{i\in I,\;\, \lambda_i=+\infty\},\\
\Omega_{\infty,\lambda}&=&\cup\{\Omega_i,\;\; i\in I_{\infty,\lambda}\}.
\end{array}
\right.
\end{equation}
For ${\widehat{\lambda}}\in\partial\Lambda^+$ instead we write
\begin{equation}
\left\{
\begin{array}{rcl}
I_0&=&\cup\{i\in I,\;\, {\widehat{\lambda}}_i=0\},\\
\Omega_0&=&\cup\{\Omega_i,\;\; i\in I_0\},
\end{array}
\right.
\quad
\left\{
\begin{array}{rcl}
I_\infty&=&\cup\{i\in I,\;\, {\widehat{\lambda}}_i=+\infty\},\\
\Omega_\infty&=&\cup\{\Omega_i,\;\; i\in I_\infty\}.
\end{array}
\right.
\end{equation}
\end{remark}
\begin{remark}\label{r:hlambda,H10}
In the case when $dim(\Lambda)=1$, i.e., $\Lambda=\mathbb R$, and the space $H$
is equipped with the inner product
\begin{equation}
(u,v)_H=\int_\Omega \nabla u \cdot\nabla v \, dx,
\label{eq:ipH10}
\end{equation}
the optimal weight ${\widehat{\lambda}}$ is given explicitly. Indeed, if
$\displaystyle (u,v)_\lambda=\int_\Omega \lambda(\nabla u \cdot\nabla
v) \, dx$, then $g(\lambda)$ is defined by
\begin{equation*}
\int_\Omega\lambda(\nabla g(\lambda)\cdot\nabla v) \, dx = e'(u;v).
\end{equation*}
This implies $g(\lambda)=\frac{1}{\lambda}g_1$ and then
\begin{equation*}
j(\lambda)=\frac{\kappa}{2}\frac{1}{\lambda^2}e''(u;g_1,g_1)-\frac{1}{\lambda}e'(u;g_1).
\end{equation*}
It follows that the solution ${\widehat{\lambda}}$ of \eqref{e:min(lambda)->1} is given by
\begin{equation}
\label{e:hlambda,N=1}
{\widehat{\lambda}}=\kappa\frac{e''(u;g_1,g_1)}{e'(u;g_1)}.
\end{equation}
Note that ${\widehat{\lambda}}>0$ because $e'(u;g_1)>0$ and
$e''(u;g_1,g_1)>0$. Thus, in the case when $dim(\Lambda)=1$ and the
space $H$ is endowed with inner product \eqref{eq:ipH10}, the proposed
approach will consist of Algorithm \ref{alg:gradient} with step 4
amended as
\begin{eqnarray}
&4.&
\mbox{determine $g=g({\widehat{\lambda}})$, where ${\widehat{\lambda}}\in\Lambda^+$ is given by \eqref{e:hlambda,N=1}}.
\label{alg:gradient+2,N=1}
\end{eqnarray}
We remark that, interestingly, since ${\widehat{\lambda}}$ is proportional to the
step size $\kappa$ and $g({\widehat{\lambda}})$ is proportional to $1/{\widehat{\lambda}}$,
in the present case the iterations produced by Algorithm
\ref{alg:gradient} will not depend on $\kappa$.
The optimal ${\widehat{\lambda}}$ given in \eqref{e:hlambda,N=1} plays a similar
role to the parameter $\alpha$ used in the Barzilai-Borwein version of
the gradient method for minimization in $\mathbb \mathbb R^n$
\cite{barzilai-1}. However, here the idea behind the choice of
${\widehat{\lambda}}$ given by \eqref{e:min(lambda)->1} or \eqref{e:hlambda,N=1}
is to approximate Newton's step. On the other hand, in
\cite{barzilai-1} the optimal $\alpha$ is chosen such that the resulting gradient is a two-point
approximation to the secant direction used in the quasi-Newton methods.
\end{remark}
\section{Error analysis}
\label{sec:errors}
In the following, we first present the analysis of case (i) of
Proposition \ref{p:lambda^k->}.
\subsection{Error analysis: case ${\widehat{\lambda}}\in\Lambda^+$}
\label{s:analysis->1}
\noindent
The following proposition gives the differentiability of the map $g$.
\begin{proposition}
\label{p:g'->1}
Let ${\widehat{\lambda}}\in\Lambda^+$ be a solution of \eqref{e:min(lambda)->1}.
Then $g\in C^1(\Lambda,H)$ and $j\in C^1(\Lambda)$ near ${\widehat{\lambda}}$.
Furthermore, for all $v\in H$ and $\ell\in\Lambda$ we have
\begin{eqnarray}
\int_{\Omega}
{\widehat{\lambda}} (\nabla g'({\widehat{\lambda}};\ell)\cdot\nabla v) + g'({\widehat{\lambda}};\ell)v \, dx
&=&
- \int_{\Omega} \ell (\nabla g({\widehat{\lambda}})\cdot \nabla v) \, dx,
\label{e:g'->1}\\
e''(u;\kappa g({\widehat{\lambda}}),g'({\widehat{\lambda}};\ell))&=&e'(u;g'({\widehat{\lambda}};\ell)),
\label{e:e''=e'->1}
\end{eqnarray}
where $g'({\widehat{\lambda}};\ell)$ is the derivative of $g$ at ${\widehat{\lambda}}$ in the direction $\ell$.
\end{proposition}
{\bf Proof}.
To prove the differentiability of $g$ we consider the map
\begin{eqnarray*}
F:\Lambda\times H&\mapsto&H'
\\
(\lambda,g)&\to& F(\lambda,g),
\quad
F(\lambda,g)(v)
=
\int_{\Omega}
\lambda (\nabla g\cdot\nabla v) + gv \, dx
-
e'(u;v),\;\, v\in H.
\end{eqnarray*}
Note that $F$ is $C^1$ and
\begin{equation*}
\partial_g F({\widehat{\lambda}},g)(z)
=
\int_{\Omega}
{\widehat{\lambda}} (\nabla z\cdot\nabla v) + zv \, dx,\; z\in H.
\end{equation*}
It follows from the Lax-Milgram lemma that $\partial_g F({\widehat{\lambda}},g)$
defines an isomorphism from $H$ to $H'$. Then, the differentiability
of $g$ is easily deduced by using the implicit function theorem and
the fact that the \revt{equation} $F(\lambda,g)=0$ has a unique
solution $g\in H$ for any given $\lambda\in\Lambda^+$. In addition, it
follows that $\lambda\in\Lambda\mapsto j(\lambda)\in\mathbb R$ is also $C^1$
near ${\widehat{\lambda}}$, because $g\in C^1(\Lambda;H)$ and $f$ is continuous
in $H$.
Equalities \eqref{e:g'->1} and \eqref{e:e''=e'->1} are obtained after
straightforward computations.
\hfill$\Box$
\begin{corollary}
\label{c:T->1}
Let ${\widehat{\lambda}}\in\Lambda^+$ be a solution of \eqref{e:min(lambda)->1},
$T_u=\operatorname{span}\{g'({\widehat{\lambda}};\ell),\;\ell\in\Lambda\}$ and $P_u:H\mapsto T_u$
be the projection operator with respect to the inner product
$e''(u;\cdot,\cdot)$, i.e.,
\begin{equation}
e''(u;w-T_u w,v) = 0,\quad \forall w\in H,\;\, v\in T_u.
\end{equation}
Then $P_uh = P_u(\kappa g)$ and $d\leq\dim(T_u)\leq {N}$, where
$d=\rank\{i,\; g({\widehat{\lambda}})\neq g_0 \ \textrm{in} \ {\cal D}'(\Omega_i)
\}$ (we recall that $g_0$ is the $L^2$ \revt{representation of
$e'(u)$, cf.~Remark \ref{r:Riesz}}).
\end{corollary}
{\bf Proof}.
From \eqref{e:e''=e'->1} and
$e''(u;h,v)=e'(u;v)$ for all $v\in H$, it follows $P_u h = P_u(\kappa
g)$.
Clearly $\dim(T_u)\leq N$.
Now we show that $d\leq \dim(T_u)$. For simplicity and without loss
of generality we assume that $g({\widehat{\lambda}})\neq g_0$ in ${\cal D}'(\Omega_i)$ for all $i=1,\ldots,d$.
It is enough to show that $\{g'({\widehat{\lambda}};\ell^i),\; i=1,\ldots,d\}$
are linearly independent. Let
$\sum_{i=1,d}\alpha_i g'({\widehat{\lambda}};\ell^i)=0$, $\alpha_i\in\mathbb R$. From \eqref{e:g'->1} we
obtain
\begin{eqnarray*}
0
&=&
\int_{\Omega}{\widehat{\lambda}}
\nabla \left(\sum_{i=1,d}\alpha_i g'({\widehat{\lambda}};\ell^i)\right)\cdot\nabla v \, dx
+
\left(\sum_{i=1,d}\alpha_i g'({\widehat{\lambda}};\ell^i)\right)v \, dx
\\
&=&
-
\int_{\Omega}
\left(\sum_{i=1,d}\alpha_i\ell^i\right)\nabla g({\widehat{\lambda}})\cdot\nabla v \, dx.
\end{eqnarray*}
Then, taking $v\in{\cal D}(\Omega_i)$ gives
\[
0
=
\sum_{j=1}^d \alpha_j \int_{\Omega} \ell^j (\nabla g({\widehat{\lambda}})\cdot\nabla v) \, dx
=
\alpha_i \int_{\Omega_i} \nabla g({\widehat{\lambda}})\cdot \nabla v \, dx
=
\alpha_i \int_{\Omega_i} \Delta g({\widehat{\lambda}}) v\, dx.
\]
Hence $\Delta g({\widehat{\lambda}})=0$ in ${\cal D}'(\Omega_i)$. Since
$-\nabla\cdot({\widehat{\lambda}}\nabla g({\widehat{\lambda}})) +g({\widehat{\lambda}})=g_0$ in ${\cal
D}'(\Omega)$ and since ${\widehat{\lambda}}$ is constant in each $\Omega_i$, it
follows \revt{that} $\alpha_i(g({\widehat{\lambda}})-g_0)=0$ in ${\cal
D}'(\Omega_i)$, hence $\alpha_i = 0$. \hfill$\Box$
\begin{remark}
The \revt{estimate of} the dimension of $T_u$ is optimal. In fact,
we can prove that $d={\rm dim}\{i\in I,\;
g'({\widehat{\lambda}};\ell^i)\neq0\}$. Indeed, if $\Delta g({\widehat{\lambda}})=0$ in
$\Omega_i$, we can \revt{show that} $\partial_\nu g({\widehat{\lambda}})=0$ on
$\partial\Omega_i$, \revt{where $\nu$ is the direction of the normal
vector on $\partial\Omega_i$,} and then from (\ref{e:g'->1}) we
get $g'({\widehat{\lambda}};\ell^i)=0$.
\end{remark}
\noindent
Now we are able to prove the error estimates for our method.
\begin{theorem}
\label{th:P+newton->1}
Assume $e(u)$ satisfies the assumptions of Theorems \ref{th:gradient}
and \ref{th:newton} near the solution ${\widehat{u}}$ of (\ref{e:min{e(u)}}).
Let $u$ be close to ${\widehat{u}}$ and $\widetilde{u}$ be given by Step 5 of
Algorithm \ref{alg:gradient} with $g=g({\widehat{\lambda}})$ and
${\widehat{\lambda}}\in\Lambda^+$ a solution of (\ref{e:min(lambda)->1}). Then we
have
\begin{eqnarray}
\|\widetilde{u}-{\widehat{u}}\|_{\widehat{\lambda}}
&\leq&
\epsilon \|u-{\widehat{u}}\|_{\widehat{\lambda}},
\label{e:|tu-u|->1}\\
\|P_u(\widetilde{u}-{\widehat{u}})\|_H
&\leq&
C\|u-{\widehat{u}}\|_H^2,
\label{e:|P(tu-u)|->1}
\end{eqnarray}
with $\epsilon\in(0,1)$ depending only on $\kappa$ and $e$ and $C>0$
depending only on $e$.
\end{theorem}
{\bf Proof}. Estimate \eqref{e:|tu-u|->1} follows from
Theorem \ref{th:gradient}, where the norm is changed to
$\|\cdot\|_{\widehat{\lambda}}$, because the gradient is now computed with respect to
the inner product $(\cdot,\cdot)_{\widehat{\lambda}}$.
For \eqref{e:|P(tu-u)|->1},
we note that $P_u$ is a linear continuous operator with
$\|P_u\|_H\leq M$. Then
\begin{eqnarray*}
P_u(\widetilde{u}-{\widehat{u}})
&=&
P_u(u-\kappa g({\widehat{\lambda}}) - {\widehat{u}})
\\
&=&
P_u(u-{\widehat{u}}) - P_u(\kappa g({\widehat{\lambda}})) \qquad(use\;\ Corollary\;\,\ref{c:T->1})
\\
&=&
P_u(u-{\widehat{u}}) - P_u(h)
\\&=&
P_u(u-h - {\widehat{u}}).
\end{eqnarray*}
Therefore
\begin{eqnarray*}
\|P_u(\widetilde{u}-{\widehat{u}})\|_H
& = &
\|P_u(u-h-{\widehat{u}})\|_H
\\
&\leq&
M\|u-h-{\widehat{u}}\|_H \qquad(use\; Theorem\; \ref{th:newton})
\\
&\leq&
C\|u-{\widehat{u}}\|_H^2.
\end{eqnarray*}
\begin{remark}
\label{r:error->1}
Theorem \ref{th:P+newton->1} states that $\|P_u(\widetilde{u}-{\widehat{u}})\|_H$,
the error of our method at a given step projected onto the tangent plane $T_u$,
decreases at least quadratically in terms of $\|u-{\widehat{u}}\|_H$.
\end{remark}
\subsection{Error analysis: case ${\widehat{\lambda}}\in\partial\Lambda^+$}
\label{s:analysis->2}
In case (ii) of Proposition \ref{p:lambda^k->} we are led to consider
$g({\widehat{\lambda}})$ associated to ${\widehat{\lambda}}\in\partial\Lambda^+$ with
${\widehat{\lambda}}_i=0$ for $i\in I_0$ and ${\widehat{\lambda}}_i=+\infty$ for $i\in
I_\infty$, which solves \eqref{e:g(lambda)}. To obtain error
estimates similar to the ones given by Theorem \ref{th:P+newton->1},
we would like to have differentiability results similar to the ones
given by Proposition \ref{p:g'->1}, which means that we \revt{would}
have to compare $g({\widehat{\lambda}})$ with $g(\lambda)$,
$\lambda\in\partial\Lambda^+$. \revt{However}, in general, for
$\lambda\in\partial\Lambda^+$ equation \eqref{e:g(lambda)} does not
provide \revt{an} estimate {in $H$} for $g(\lambda)$ and therefore
{the analysis from the previous section cannot be applied directly}.
\revt{On the other hand,} equation \eqref{e:g(lambda)} with
${\widehat{\lambda}}\in\partial\Lambda^+$ implies extra regularity for $e'(u)$,
in particular in $\Omega_0$. Assuming that $e''(u)$ {possesses} the
same kind of regularity, which comes naturally from the problem, we
will prove an error estimate for case (ii) of Proposition
\ref{p:lambda^k->} similar to the one already given in Theorem
\ref{th:P+newton->1}, but in a weaker norm.
\begin{proposition}\label{p:g=g_0}
Let
$\Omega_{0,{H^1}}$ be the largest union of $\overline{\Omega}_i$ such that $g_0\in H^1(\Omega_{0,H^1})$
and $I_{0,H^1}=\{i\in I,\; \Omega_i\subset\Omega_{0,H^1}\}$.
\\
(i)
If $I_{0,H^1}=\emptyset$, then case (ii) of Proposition \ref{p:lambda^k->} does not happen.
\\
(ii)
If $I_{0,H^1}\neq\emptyset$, then $I_0\subset I_{0,H^1}$, $\Omega_0\subset \Omega_{0,H^1}$ and
\begin{alignat}{2}
g({\widehat{\lambda}})&=g_0 & &\; \text{\it in} \ H^1(\Omega_0).
\label{e:g-in-G0}
\end{alignat}
(iii) Furthermore,
$e'(u;\cdot)$ is continuous in
$H^1_0(\Omega\backslash\Omega_0;\Omega_\infty^\text{const})\cap H^1(\Omega_0)$, where
\begin{eqnarray}
H^1_0(\Omega\backslash\Omega_0;\Omega_\infty^\text{const})\cap H^1(\Omega_0)
&=&
\{
v\in H^1(\Omega\backslash\Omega_0)\cap H^1(\Omega_0),\;\;
\mbox{\it $v=0$\;\, on\;\, $\partial\Omega$\;\, and} \nonumber\\
&&
\hspace*{2mm}
v=C_i\;\,\mbox{\it in}\; \Omega_\infty^i,\;\, i\in I_\infty\}.
\label{e:N1_0(G-G0;Ginf)}
\end{eqnarray}
\end{proposition}
{\bf Proof}. Indeed, from \eqref{e:g-in-G-G0-Ginf}
and \eqref{e:g(l),g_0}, we get $g({\widehat{\lambda}})=g_0$ in ${\cal
D}'(\Omega_0)$. As $g({\widehat{\lambda}})\in H^1_0(\Omega)$, the claims (i) and
(ii) follow.
The form of the inner product $(g,v)_{{\widehat{\lambda}}}$ and the fact that
${\widehat{\lambda}}=0$ in $\Omega_0$ \revt{imply} (iii).
\hfill$\Box$\\
\noindent
Motivated by Proposition \ref{p:lambda^k->}, we are led to the following definition.
\begin{definition}
Let $\lambda=[\lambda_1,\ldots,\lambda_N]\in\partial\Lambda^+$ with
$I_{0,\lambda}\subset I_{0,H^1}$.
We define $g(\lambda)\in
H^1_0(\Omega\backslash\Omega_{0,\lambda};\Omega_{\infty,\lambda}^\text{const})\cap H^1(\Omega_{0,\lambda})$ by
\begin{eqnarray}
\int_{\Omega\backslash\Omega_{0,\lambda}}
\lambda (\nabla g(\lambda)\cdot\nabla v) + g(\lambda)v \, dx
&+&
\int_{\Omega_{0,\lambda}}g(\lambda)v \, dx
=
e'(u;v),\nonumber\\
&&
\forall v\in H^1_0(\Omega\backslash\Omega_{0,\lambda};\Omega_{\infty,\lambda}^\text{const})\cap H^1(\Omega_{0,\lambda}).
\label{e:g''}
\end{eqnarray}
\end{definition}
\begin{proposition}
\label{p:g->2;wd}
Let $\lambda\in\partial\Lambda^+$ with
with $I_{0,\lambda}\subset I_{0,H^1}$.
Then \eqref{e:g''} has a unique solution
$g(\lambda)\in
H^1_0(\Omega\backslash\Omega_{0,\lambda};\Omega_{\infty,\lambda}^\text{const})\cap H^1(\Omega_{0,\lambda})$
and
$g(\lambda)=g_0$ in $H^1(\Omega_{0,\lambda})$.
\end{proposition}
{\bf Proof}.
The existence and uniqueness of
$g(\lambda)$ follows from the Lax-Milgram lemma applied in the space
$H^1_0(\Omega\backslash\Omega_{0,\lambda};\Omega_{\infty,\lambda}^\text{const})\cap L^2(\Omega_{0,\lambda})$
equipped with the inner product
\begin{equation*}
(g,v)_\lambda
=
\int_{\Omega}
\lambda (\nabla g\cdot\nabla v) + gv \, dx
=
\int_{\Omega\backslash\Omega_{0,\lambda}}
\lambda (\nabla u\cdot\nabla v) + gv \, dx
+
\int_{\Omega_{0,\lambda}}gv \, dx.
\end{equation*}
Reasoning as in \revt{case (ii) of} Proposition \ref{p:lambda^k->}, we
see that $g(\lambda)$ is constant in $\Omega_{\infty,\lambda}^i$, for
all $i\in I_{0,\lambda}$. Finally, taking $v\in{\cal D}(\Omega_0)$ we
\revt{obtain} $\int_{\Omega_0}(g(\lambda)-g_0)v \,dx=0$, which implies
\revt{that} $g(\lambda)=g_0$ in ${\cal D}(\Omega_{0,\lambda})$ and, as
$g_0\in H^1(\Omega_{0,H^1})$, completes the proof. \hfill$\Box$
\\
Returning to the minimization problem \eqref{e:min(lambda)->1} and in
view of \revt{case (ii) of} Proposition \ref{p:lambda^k->}, we are led to consider
the problem
\begin{equation}
\label{e:min(lambda)->2}
\text{find ${\widehat{\lambda}}\in\partial\Lambda^+$ such that }\;
j({\widehat{\lambda}}) := \min\{j(\lambda)=(f\circ g)(\lambda),\;\, \lambda\in\partial\Lambda^+\}
\end{equation}
and \revt{from it eventually obtain} a necessary condition
\revt{analogous to} \eqref{e:e''=e'->1}, which was a key element in
proving estimate \eqref{e:|P(tu-u)|->1}.
We would repeat the \revtt{analysis} already applied to problem
\eqref{e:min(lambda)->1}, \revtt{as} in Section \ref{s:analysis->1}.
However,
since $g(\lambda)$ \revt{now} defined via \eqref{e:g''} does not in
general belong to $H^1_0(\Omega)$, $j(\lambda)$ may not be well
defined.
It appears that there are no general conditions on the data which
would ensure that $g(\lambda)\in H^1_0(\Omega)$ when
$\lambda\in\partial\Lambda^+$. We will thus proceed with the analysis
of this case under the following stronger assumptions on $e'$ and
$e''$, which are motivated by the continuity of $e'(u;\cdot)$ in
$H^1(\Omega\backslash\Omega_0)\cap H^1(\Omega_0)$, see Proposition
\ref{p:g=g_0}.
Let us \revt{introduce} the following definitions
\begin{eqnarray}
{\cal H}
&=&
\{u\in H^1(\Omega\backslash\Omega_{0,H^1}),\; v\in H^1(\Omega_i),\;\, i\in I_{0,H^1},\;\,
u=0\;\; on\;\,\partial\Omega\},
\\
(u,v)_{{\cal H}}
&=&
\int_{\Omega\backslash\Omega_{0,H^1}}(\nabla u\cdot\nabla v) + uv \, dx
+
\sum_{i\in I_{0,H^1}}\int_{\Omega_i}(\nabla u\cdot\nabla v) + uv \, dx,
\\
\|v\|_{{\cal H}}^2
&=&
(v,v)_{{\cal H}}.
\end{eqnarray}
The set ${\cal H}$ equipped with the inner product
$(v,v)_{{\cal H}}$ is a Hilbert space.
In the reminder of this section we will assume
\begin{equation}\label{e:assumption(H10)}
\left\{
\begin{array}{l}
\mbox{\it
$e$, $e'$ and $e''$ satisfy all the conditions of Theorems \ref{th:gradient} and \ref{th:newton} with}
\\
\mbox{\it
$H$ replaced by ${\cal H}$}.
\end{array}
\right.
\end{equation}
Moreover, we will assume
\begin{eqnarray}
|e''(u;v,w)|&\leq&M\|v\|_{{\cal H}}\|w\|_{{\cal H}},
\label{e:e->C0,'}\\
e''(u;v,v)
&\geq&
m\|v\|_{{\cal H}}^2,
\label{e:e->convex,'}
\end{eqnarray}
with certain $0<m<M<\infty$.
\begin{proposition}
\label{p:min(lambda)->2}
Assume $e''$ satisfies \eqref{e:assumption(H10)}--\eqref{e:e->convex,'}.
For $\lambda\in\partial\Lambda^+$ with $I_{0,\lambda}\subset I_{0,H^1}$ let
$g(\lambda)\in
H^1_0(\Omega\backslash\Omega_{0,\lambda};\Omega_{\infty,\lambda}^\text{const})\cap H^1(\Omega_{0,\lambda})$
be defined by (\ref{e:g''}).
Then the problem \eqref{e:min(lambda)->2} has a solution ${\widehat{\lambda}}\in\partial\Lambda^+$.
\end{proposition}
{\bf Proof}.
Let $(\lambda^k)$ be a sequence in
$\partial\Lambda^+$ minimizing $j$ in $\partial\Lambda^+$. As
$\dim(\Lambda)=N$, without loss of generality, we may assume that
there exist $I_0\subset I$, $I_\infty\subset I$ such that
$I_{0,\lambda^k}=I_0$, $I_{\infty,\lambda^k}=I_\infty$ for all $k$.
It follows {that}
$\Omega_{0,\lambda^k}=\Omega_0$, $\Omega_{\infty,\lambda^k}=\Omega_\infty$.
Since $f$ is elliptic in ${\cal H}$ and $g(\lambda^k)\in
H^1(\Omega\backslash\Omega_0;\Omega_\infty^{\rm const})\cap
H^1(\Omega_0)$, for all $k$, necessarily $(g(\lambda^k))$ is bounded
in $H^1(\Omega\backslash\Omega_0)\cap H^1(\Omega_0)$. Therefore,
without loss of generality, we may assume that $(g(\lambda^k))$
converge weakly in $H^1(\Omega\backslash\Omega_0)\cap H^1(\Omega_0)$
and strongly in $L^2(\Omega)$ to a certain $g\in
H^1(\Omega\backslash\Omega_0)\cap H^1(\Omega_0)$. As $f$ is convex,
it follows that $f(g)\leq\liminf_{k\to\infty}j(\lambda^k)$, \revt{see
\cite{brezis-1}}.
To conclude that
\eqref{e:min(lambda)->2} has a solution, it is enough to show that
$g=g({\widehat{\lambda}})$ for a certain ${\widehat{\lambda}}\in\partial\Lambda^+$.
For the sequence $\lambda^k$ two cases may occur.\\
(i)
There exist ${\widehat{\lambda}}=[{\widehat{\lambda}}_1,\ldots,{\widehat{\lambda}}_N]$ with
${\widehat{\lambda}}_i=0$ for $i\in I_0$,
${\widehat{\lambda}}_i=+\infty$ for $i\in I_\infty$,
${\widehat{\lambda}}_i\in(0,+\infty)$ for $i\in I\backslash(I_0\cup I_\infty)$,
and a subsequence of $(\lambda^k)$, still denoted $(\lambda^k)$,
such that
$\lim_{k\to\infty}\lambda^k_i={\widehat{\lambda}}_i$ for all $i\in I$.
Note that $g_k=g(\lambda^k)\in
H^1(\Omega\backslash\Omega_0;\Omega_\infty^{\rm const})\cap H^1(\Omega_0)$ satisfies
\eqref{e:g''}, i.e.
\begin{eqnarray}
\int_{\Omega\backslash\Omega_0}
\lambda^k (\nabla g_k\cdot\nabla v) + g_kv \, dx
+
\int_{\Omega_0} g_kv \, dx
&=& e'(u;v),
\label{e:g'',k}
\end{eqnarray}
for all $v\in H^1(\Omega\backslash\Omega_0;\Omega_\infty^{\rm const})\cap H^1(\Omega_0)$.
Passing {to the} limit in
\eqref{e:g'',k}, we find that $g\in H^1_0(\Omega\backslash\Omega_0;\Omega_\infty^\text{const})\cap H^1(\Omega_0)$
solves \eqref{e:g''} so $g=g({\widehat{\lambda}})$.
\\
(ii)
There exist ${\widehat{\lambda}}=[{\widehat{\lambda}}_1,\ldots,{\widehat{\lambda}}_N]$, $i_0\subset I$, $i_\infty\subset I$ with
${\widehat{\lambda}}_i=0$ for $i\in I_0\cup i_0$,
${\widehat{\lambda}}_i=+\infty$ for $i\in I_\infty\cup i_\infty$,
${\widehat{\lambda}}_i\in(0,+\infty)$ for $i\in I\backslash((I_0\cup i_0)\cup (I_\infty\cup i_\infty))$,
and a subsequence of $(\lambda^k)$, still denoted $(\lambda^k)$,
such that
$\lim_{k\to\infty}\lambda^k_i={\widehat{\lambda}}_i$ for all $i\in I$.
We take $v\in H^1_0(\Omega\backslash\Omega_0;(\Omega_\infty\cup\omega_\infty)^\text{const})\cap H^1(\Omega_0)$
in \eqref{e:g'',k}, where
$\omega_0=\cup\{\Omega_i,\; i\in i_0\}$,
$\omega_\infty=\cup\{\Omega_i,\; i\in i_\infty\}$,
and we obtain
\begin{eqnarray*}
\int_{\Omega\backslash(\Omega_0\cup \omega_0)}
\lambda^k (\nabla g_k\cdot\nabla v) + g_kv \, dx
&+&
\int_{\Omega_0}
g_kv \, dx
+
\int_{\omega_0}
\lambda^k (\nabla g_k\cdot\nabla v) + g_kv \, dx
\nonumber\\
&=&
e'(u;v).
\label{e:g'',k'}
\end{eqnarray*}
Letting $k\to\infty$ gives
\begin{eqnarray*}
\int_{\Omega\backslash(\Omega_0\cup \omega_0)}
\hspace*{-5mm}
{\widehat{\lambda}} (\nabla g\cdot\nabla v) + gv \, dx
+
\int_{\Omega_0\cup\omega_0}
\hspace*{-3mm}
gv \, dx
&=&
e'(u;v),
\end{eqnarray*}
for all $v\in
H^1_0(\Omega\backslash\Omega_0;(\Omega_\infty\cup\omega_\infty)^\text{const})\cap
H^1(\Omega_0)$. Reasoning as in Proposition \ref{p:lambda^k->}, we
find that $g=g({\widehat{\lambda}})\in
H^1_0(\Omega\backslash\Omega_0;(\Omega_\infty\cup\omega_\infty)^\text{const})\cap
H^1(\Omega_0)$ solves \eqref{e:g''} with $\lambda={\widehat{\lambda}}$ and
$\Omega_0\cup\omega_0$ (respectively,
$\Omega_\infty\cup\omega_\infty$) instead of $\Omega_0$ (respectively,
$\Omega_\infty$). \hfill$\Box$
\begin{remark}\label{r:sigma(lambda)}
For ${\widehat{\lambda}}\in\partial\Lambda^+$, in order to control the
variations of ${\widehat{\lambda}}$ in the set
$\Omega\backslash(\Omega_0\cup\Omega_\infty)$ we consider ${\mathbb
1}_{0,\infty}\in\partial\Lambda^+$ defined by
\[
{\mathbb 1}_{0,\infty}(x)
=
\left\{
\begin{array}{ll}
0,&x\in \Omega_0\cup\Omega_\infty,\\
1,&x\in\Omega\backslash(\Omega_0\cup\Omega_\infty).
\end{array}
\right.
\]
Then we perturb ${\widehat{\lambda}}$ with the elements of
$\revt{\widehat{\Lambda}}:={\mathbb 1}_{0,\infty}\cdot \Lambda
=
\{{\mathbb 1}_{0,\infty}\cdot\lambda,\; \lambda\in\Lambda\}$.
\end{remark}
\begin{proposition}\label{p:g'->2}
Let ${\widehat{\lambda}}\in\partial\Lambda^+$ be the solution of \eqref{e:min(lambda)->2} as given by Proposition
\ref{p:min(lambda)->2}.
The map
$\lambda\in\hat{\Lambda}
\mapsto
g(\lambda)\in H^1_0(\Omega\backslash\Omega_0;\Omega_\infty^{\rm const})\cap H^1(\Omega_0)$
is $C^1$ near ${\widehat{\lambda}}$.
Furthermore, $g'({\widehat{\lambda}};\ell)\in H^1_0(\Omega\backslash\Omega_0;\Omega_\infty^{\rm const})\cap H^1(\Omega_0)$,
where $g'({\widehat{\lambda}};\ell)$ is the derivative of $g({\widehat{\lambda}})$ at ${\widehat{\lambda}}$ in \revt{the}
direction $\ell\in\hat{\Lambda}$, and satisfies
\begin{eqnarray}
\int_{\Omega\backslash\Omega_0}
{\widehat{\lambda}}(\nabla g'({\widehat{\lambda}};\ell)\cdot\nabla v)
&+&
g'({\widehat{\lambda}};\ell) v \, dx
+
\int_{\Omega_0}
g'({\widehat{\lambda}};\ell) v \, dx
=
-\int_{\Omega\backslash\Omega_0}
\ell (\nabla g({\widehat{\lambda}})\cdot\nabla v) \, dx,
\nonumber\\
&&
\forall
v\in H^1_0(\Omega\backslash\Omega_0;\Omega_\infty^{\rm const})\cap H^1(\Omega_0).
\label{e:Dg''}
\end{eqnarray}
In particular, $g'({\widehat{\lambda}};\ell)=0$ in $\Omega_0$ and for every $\ell\in \revt{\widehat{\Lambda}}$ we have
\begin{equation}\label{e:e''=e'->2}
e''(u;\kappa g({\widehat{\lambda}}),g'({\widehat{\lambda}};\ell))=e'(u;g'({\widehat{\lambda}};\ell)).
\end{equation}
\end{proposition}
{\bf Proof}. The differentiability of $g$ is
deduced from the implicit mapping theorem as follows. Consider the map
$F$
\begin{equation*}
\begin{array}{lcll}
F:&\hat{\Lambda}\times H^1_0(\Omega\backslash\Omega_0;\Omega_\infty^\text{const})\cap L^2(\Omega_0)
&\mapsto &
(H^1_0(\Omega\backslash\Omega_0;\Omega_\infty^\text{const})\cap L^2(\Omega_0))'
\\
&(\lambda,g)&\to&
F(\lambda,g),
\end{array}
\end{equation*}
with
\begin{equation*}
F(\lambda,g)
=
\int_{\Omega\backslash\Omega_0}
\lambda(\nabla g(\lambda)\cdot\nabla v)
+
g(\lambda) v \, dx
+
\int_{\Omega_0}
g(\lambda) v \, dx
-
e'(u;v).
\end{equation*}
Clearly, $F$ is $C^1$ near $({\widehat{\lambda}},g({\widehat{\lambda}}))$. Furthermore, we have
\begin{eqnarray*}
\partial_g F({\widehat{\lambda}},g({\widehat{\lambda}}))(z)
&=&
\int_{\Omega\backslash\Omega_0}
{\widehat{\lambda}}(\nabla z\cdot\nabla v)
+
z v \, dx
+
\int_{\Omega_0}
zv \, dx,
\end{eqnarray*}
which defines an isomorphism from
$H^1_0(\Omega\backslash\Omega_0;\Omega_\infty^\text{const})\cap
L^2(\Omega_0)$ to its dual. Then, the implicit mapping theorem and
the fact that $F(\lambda,g)=0$ has a unique solution $g\in
H^1(\Omega\backslash\Omega_0;\Omega_\infty^\text{const})\cap
H^1(\Omega_0)$ for any $\lambda\in \revt{\widehat{\Lambda}}$ gives the
differentiability of the map $g$. Note that, a priori, \revt{the}
implicit mapping theorem \revt{ensures} the differentiability of the
map $g$ in $L^2(\Omega_0)$. \revt{Then,} as $g(\lambda)=g_0\in
H^1(\Omega_0)$, see Proposition \ref{p:g->2;wd}, the differentiability
of the map $g$ in $H^1(\Omega_0)$ \revt{follows as well}.
Next, by direct computations we can easily show \eqref{e:Dg''}.
Furthermore, $g'({\widehat{\lambda}};\ell)=0$ in $\Omega_0$ because
$g(\lambda)=g_0$ in $H^1(\Omega_0)$.
In regard to \eqref{e:e''=e'->2}, we recall that $e'(u)$ and $e''(u)$
are, respectively, linear and bilinear, and continuous in ${\cal H}$,
which together with the identity $g(\lambda)=g_0$ in $\Omega_0$ and
the differentiability of $g(\lambda)$ \revt{implies} the
differentiability of $\lambda\mapsto e'(u;g(\lambda))$ and
$\lambda\mapsto e''(u;g(\lambda),g(\lambda))$. Then,
(\ref{e:e''=e'->2}) follows from straightforward computations.
\hfill$\Box$
\\
The error estimates are obtained in an analogous way to the
corresponding results in Section \ref{s:analysis->1}. First, we have a
result similar to Corollary \ref{c:T->1}.
\begin{corollary}\label{c:T->2}
Assume $e''(u)$ satisfies conditions \eqref{e:assumption(H10)}--\eqref{e:e->convex,'}.
Let ${\widehat{\lambda}}\in\partial\Lambda^+$ be a solution of
\eqref{e:min(lambda)->2}, $g=g({\widehat{\lambda}})$, $T_u=\operatorname{span}\{g'({\widehat{\lambda}};\ell),\;\ell\in\hat{\Lambda}\}$ and
$P_u:{\cal H}\mapsto T_u$ be
the projection operator with respect to the inner product
$e''(u;\cdot,\cdot)$, i.e.,
\begin{equation}
e''(u;w-P_uw,v) = 0,\quad \forall w\in {\cal H},\;\, \forall v\in T_u.
\end{equation}
Then, $P_u h = P_u(\kappa g)$ and $d\leq \dim(T_u)\leq
N-|I_0|-|I_\infty|$, where $d=\rank\{i,\; g\neq g_0\; \mbox{in}\;
{\cal D}'(\Omega_i)\}$.
\end{corollary}
{\bf Proof}.
From \eqref{e:e''=e'->2} and the relation $e''(u;h,v)=e'(u;v)$ for all $v\in H$,
it follows that $P_u h = P_u(\kappa g)$.
Clearly $\dim(T_u)\leq N-|I_0|-|I_\infty|$.
Now we show that $\dim(T_u)\geq d$. We assume that $g({\widehat{\lambda}})\neq g_0$ in ${\cal D}'(\Omega_i)$,
for all $i=1,\ldots,d$. Let $\ell^i\in\hat{\Lambda}$,
$\ell^i(\Omega_j)=\delta_{ij}$.
It is enough to demonstrate that $\{g'({\widehat{\lambda}};\ell^i),\; i=1,\ldots,d\}$
are linearly independent. Let $\sum_{i=1,d}\alpha_i
g'({\widehat{\lambda}};\ell_i)=0$, $\alpha_i\in\mathbb R$. Then,
\begin{eqnarray*}
0
&=&
\int_{\Omega\backslash\Omega_0}
{\widehat{\lambda}}
\nabla \left(\sum_{i=1,d}\alpha g'({\widehat{\lambda}};\ell^i)\right)\cdot\nabla v
+
\left(\sum_{i=1,d}\alpha_i \nabla g'({\widehat{\lambda}};\ell^i)\right)v \, dx
\\
&+&
\int_{\Omega_0}
\left(\sum_{i=1,d}\alpha_i \nabla g'({\widehat{\lambda}};\ell^i)\right)v \, dx
\\
&=&
\int_{\Omega\backslash\Omega_0}
\left(\sum_{i=1,d}\alpha_i \ell^i\right) (\nabla g({\widehat{\lambda}})\cdot\nabla v) \, dx.
\end{eqnarray*}
In the equality above we take $v\in{\cal D}(\Omega_i)$. Then
\begin{eqnarray*}
0
&=&
\int_{\Omega\backslash\Omega_0}
\alpha_i(\nabla g({\widehat{\lambda}})\cdot \nabla v) \, dx
=
- \alpha_i\int_{\Omega}\Delta g({\widehat{\lambda}}) v \, dx
=
- \alpha_i\int_{\Omega}(g({\widehat{\lambda}})-g_0)v \, dx,
\end{eqnarray*}
because $-{\widehat{\lambda}}\Delta g({\widehat{\lambda}})+g({\widehat{\lambda}})=g_0$ in $\Omega_i$,
which implies $\alpha_i=0$ and proves the claim.
\hfill$\Box$
\\
\noindent
Finally, we are able to prove the error estimate for the case
${\widehat{\lambda}}\in\partial\Lambda^+$.
\begin{theorem}
\label{th:P+newton->2}
Assume $e$ satisfies the conditions of Theorems \ref{th:gradient},
\ref{th:newton} with ${\cal H}$ instead of $H$, and $e''(u)$ satisfies
conditions \eqref{e:assumption(H10)}--\eqref{e:e->convex,'}. Let
${\widehat{\lambda}}\in\partial\Lambda^+$ be a solution of
\eqref{e:min(lambda)->2}, as given by Proposition
\ref{p:min(lambda)->2}. If $\widetilde{u}$ and $u$ are given as in
Algorithm \ref{alg:gradient} with $g=g({\widehat{\lambda}})$, then
\begin{eqnarray}
\|\widetilde{u}-{\widehat{u}})\|_{\widehat{\lambda}}
&\leq&
\epsilon \|u-{\widehat{u}}\|_{\widehat{\lambda}},
\label{e:|tu-u|->2}\\
\|P_u(\widetilde{u}-{\widehat{u}})\|_{{\cal H}}
&\leq&
C\|u-{\widehat{u}}\|_{{\cal H}}^2,
\label{e:|P(tu-u)|->2}
\end{eqnarray}
with
$\epsilon\in(0,1)$ depending only on $\kappa$ and $e$ and $C>0$ depending only on $e$.
\end{theorem}
{\bf Proof}. The proof is analogous to the proof of
Theorem \ref{th:P+newton->1}.
\hfill$\Box$
\begin{remark}
\label{r:error->2}
The estimates in Theorem \ref{th:P+newton->2} are similar to the ones
in Theorem \ref{th:P+newton->1}. However, in general, the quadratic
convergence established in Theorem \ref{th:P+newton->2} is slower than
the one provided by Theorem \ref{th:P+newton->1}, because in Theorem
\ref{th:P+newton->2} the dimension of the space $T_u$ is in general
smaller than the dimension of the space $T_u$ in Theorem
\ref{th:P+newton->1}. Finally, estimate \eqref{e:|P(tu-u)|->2} {might
be of no} interest because we may have
$\Omega\backslash(\Omega_0\cup\Omega_\infty)=\emptyset$ and so
$T_u=\{0\}$. In \revt{such} case $g=g_0$ \revt{would be} the $L^2$
gradient \revt{defined in the entire domain} $\Omega$ and the
question of its impact on the \revt{performance} of the gradient
method is open.
\revt{Lastly}, Theorem \ref{th:P+newton->2} \revt{provides an estimate
applicable at a single step of the gradient approach, cf.~Algorithm
\ref{alg:gradient},} where a certain $u$ is given and the regularity
of $e$ is \revt{determined} in terms of the set where the $L^2$
gradient $g_0=g_0(u)$ is $H^1$. In order to \revt{be able to apply}
Theorem \ref{th:P+newton->2} at each step, \revt{one} should rather
consider \revt{iterations in the space} ${\cal H}=\{u\in
H^1(\Omega_i),\; i\in I,\; u=0\; on\; \partial\Omega\}$ and impose the
same assumptions as in Theorem \ref{th:P+newton->2}.
\end{remark}
\section{Determination of optimal weights ${\widehat{\lambda}}$ and the corresponding gradients}
\label{sec:optimal}
In this section we describe the computational approach which can be
used to determine the optimal form of the inner product
\eqref{e:(u,v)_lambda}, encoded in its weight $\lambda$, and the
corresponding gradient $g(\lambda)$ at each iteration, cf.~modified
step 4 of Algorithm \ref{alg:gradient} given by
\eqref{alg:gradient+2}. We will focus on the case when ${\widehat{\lambda}} \in
\Lambda^+$, cf.~Section \ref{s:analysis->1}, and in order to ensure
non-negativity of the weight, in our approach we will use the
representation $\lambda(x) = \eta^2(x)$, $\forall x \in \Omega$, where
$\eta \; : \; \Omega \mapsto \mathbb R$ is a function {defined} below.
For consistency with the notation introduced in the previous sections
and without risk of confusion, hereafter we will use both $\lambda$
and $\eta$. Relation \eqref{e:g(lambda)} can then be expressed in the
strong form as
\begin{equation}
\left\{
\begin{alignedat}{2}
g -\nabla\cdot(\eta^2 \nabla g) &= g_0(u) & \quad & \text{\it in} \ \Omega, \\
g & = 0 & \quad & \text{\it on} \ \partial\Omega,
\end{alignedat}
\right.
\label{eq:g(eta)}
\end{equation}
where here $g=g(\eta)$, whereas the minimization problem \eqref{e:min(lambda)->1} becomes
\begin{equation}
\label{e:min(eta)->1}
j({\widehat{\eta}}) := \min\left\{
j(\eta):=f\circ g(\eta),\;\, \eta^2 \in \Lambda^+
\right\}.
\end{equation}
We will assume that the function $\eta(x)$ is represented with the ansatz
\begin{equation}
\eta(x) = \sum_{i=1}^N \eta_i \, \ell^i(x),
\label{eq:etaN}
\end{equation}
where $\{ \ell^i\}_{i=1}^N$ is a set of suitable basis functions.
Since in a fixed basis the function $\eta(x)$ is determined by the real
coefficients $\{ \eta_i\}_{i=1}^N$, we will also use the notation $\eta = [
\eta_1,\dots,\eta_N ]$. We thus obtain a finite-dimensional
minimization problem
\begin{equation}
\min\{ j(\eta),\;\, \eta=[ \eta_1,\dots,\eta_N ] \in \mathbb R^N \}.
\label{eq:minj(eta)}
\end{equation}
Its minimizers ${\widehat{\eta}} = [ {\widehat{\eta}}_1,\dots,{\widehat{\eta}}_N ]$ satisfy the
following optimality conditions, which can be viewed as a discrete form
of \eqref{e:e''(lambda)=e'},
\begin{equation}
\left[ F_i({\widehat{\eta}}) \right]
:
=
\left[\Dpartial{j}{\eta_i}({\widehat{\eta}})\right]
=
[e''(u;\kappa g({\widehat{\eta}}),g'_i({\widehat{\eta}}))-e'(u;g'_i({\widehat{\eta}}))]
=
[0], \qquad i=1,\dots,N,
\label{eq:alpha->F=0}
\end{equation}
where $F=[F_1,\ldots,F_N] \; : \; \mathbb R^N \rightarrow \mathbb R^N$ and
$g'_i =g'_i(\eta)=
\left[\Dpartial{g}{\eta_i}({\widehat{\eta}})\right]$ satisfy the equations
\begin{equation}
\left\{
\begin{alignedat}{2}
g'_i - \nabla\cdot\left({\widehat{\eta}}^2 \, \nabla g'_i \right)
&= 2 \nabla\cdot\left( {\widehat{\eta}} \, \ell_i \nabla g\right) & \qquad & \mbox{\it in } \Omega, \\
g'_i &= 0 & & \mbox{\it on }\partial\Omega.
\end{alignedat}
\right.
\label{eq:g'i}
\end{equation}
The optimal weight ${\widehat{\eta}}$ can be found either by
directly minimizing $j(\eta)$, cf.~\eqref{eq:minj(eta)}, using a
version of the gradient-descent method, or by solving the optimality
conditions \eqref{eq:alpha->F=0} with a version of Newton's method.
The two approaches are described below.
\subsection{Optimal weights via gradient minimization}
\label{sec:etagrad}
While in practice one may prefer to use a more efficient minimization
approach, such as, e.g., the nonlinear conjugate-gradients method
\cite{nw00}, for simplicity of presentation here we focus on the
gradient steepest-descent method. The step size $\tau_k$ along the
gradient can be determined by solving a line-minimization problem,
which can be done efficiently using for example Brent's method
\cite{nw00}. Step 4 of Algorithm \ref{alg:gradient},
cf.~\eqref{alg:gradient+2}, is then realized by the operations
summarized as Algorithm \ref{alg:etagrad}. In actual computations it
may also be beneficial to prevent any of the values $\eta_i$ from
becoming too close to zero, which is achieved easily by imposing a
suitable bound on the step size $\tau_n$.
Having in mind the complexity analysis presented in Section
\ref{sec:complexity}, the termination condition for the main loop in
Algorithm \ref{alg:etagrad} is expressed in terms of the maximum
number $N_g$ of iterations, although in practice it will be more
convenient to base this condition on the relative decrease of
$j(\eta)$.
\begin{algorithm}[h!]
\begin{algorithmic}[1]
\STATE evaluate adjoint states $z$ and $z(g)$ (if $e(u)$ depends on
a PDE equation)
\STATE set $k = 1$
\REPEAT
\STATE evaluate $g(\eta)$ by solving \eqref{eq:g(eta)}
\STATE evaluate $g'_i(\eta)$, $1=1,\dots,N$, by solving problems \eqref{eq:g'i}
\STATE evaluate $e'(u; g'_i(\eta))$, $i=1,\ldots,N$
\STATE evaluate $e''(u; \kappa g(\eta), g'_i(\eta))$, $i=1,\ldots,N$
\STATE evaluate $F(\eta)$, cf.~\eqref{eq:alpha->F=0}
\STATE perform line-minimization to determine optimal step size \newline
\hspace*{1.0cm} $\tau_k = \argmin_{\tau>0} j(\eta - \tau F(\eta))$ \qquad (Brent's method \cite{nw00})
\\
\STATE set ${\widehat{\eta}} = \eta - \tau_k F(\eta)$
\STATE set $\eta = {\widehat{\eta}}$
\STATE set $k = k + 1$
\UNTIL{ \ $k = N_g$}
\STATE obtain $g({\widehat{\eta}})$ by solving \eqref{eq:g(eta)} with $\eta = {\widehat{\eta}}$
\end{algorithmic}
\caption{Determination of optimal weight ${\widehat{\lambda}}$ via gradient minimization \newline
\textbf{Input:} \newline
\hspace*{0.5cm} $N$ --- dimension of the space in which optimal weights are sought \newline
\hspace*{0.5cm} $u \in H$ --- current approximation of minimizer ${\widehat{u}}$ \newline
\hspace*{0.5cm} $\kappa > 0$ --- step size in the outer loop (Algorithm \ref{alg:gradient}) \newline
\hspace*{0.5cm} $\{ \ell^i\}_{i=1}^N$ --- basis function for ansatz \eqref{eq:etaN} \newline
\hspace*{0.5cm} $N_g$ --- maximum number of gradient iterations \newline
\hspace*{0.5cm} $\eta$ --- initial guess for the weight \newline
\textbf{Output:} \newline
\hspace*{0.5cm} ${\widehat{\eta}}$ --- optimal weight \newline
\hspace*{0.5cm} $g({\widehat{\eta}})$ --- corresponding optimal gradient
}
\label{alg:etagrad}
\end{algorithm}
\subsection{Optimal weights via Newton's method}
\label{sec:etanewton}
In addition to the gradient of $j(\eta)$ already given in
\eqref{eq:alpha->F=0}--\eqref{eq:g'i}, the key additional step
required for Newton's method is the evaluation of the Hessian of
$j(\eta)$, i.e.,
\begin{align}
\left[ \partial_jF_i(\eta) \right]
& = \left[\Dpartialmix{j}{\eta_i}{\eta_j}(\eta)\right] \qquad\qquad \qquad\qquad
(i,j=1,\dots,N) \nonumber \\
& = \kappa
(e''(u;g'_j(\eta),g'_i(\eta)) + e''(u;g(\eta),g''_{ij}(\eta)))
-
e'(u;g''_{ij}(\eta)),
\label{eq:DF}
\end{align}
where $g(\eta)$ is given by \eqref{eq:g(eta)}, $g'_i(\eta)$ is given by \eqref{eq:g'i} and
$g''_{ij}=g''_{ij}(\eta) = \left[\Dpartialmix{g}{\eta_i}{\eta_j}({\widehat{\eta}})\right]$ satisfy the equations
\begin{equation}
\begin{alignedat}{2}
g''_{ij}(\eta)-\nabla\cdot(\eta^2\nabla g''_{ij}(\eta))
&= 2\left( \nabla\cdot(\ell_j \eta\nabla g'_i(\eta)) + \nabla\cdot(\ell_i \eta\nabla g'_j(\eta))\right) \\
&+ 2 \nabla\cdot(\ell_i \ell_j\nabla g(\eta)) &
\quad & \text{\it in} \ \Omega,\\
g''_{ij}(\eta)
& = 0 &
\quad & \text{\it on} \ \partial\Omega.
\end{alignedat}
\label{eq:g''}
\end{equation}
For brevity, Newton's approach is stated in Algorithm
\ref{alg:etanewton} in its simplest form and in practice one would
typically use its damped (globalized) version in which the step along
Newton's direction $-\left[ DF(\eta) \right]^{-1}\cdot F(\eta)$ may be
reduced to ensure the residual $\| F(\eta)\|_2$ of equation
\eqref{eq:alpha->F=0} decreases between iterations \cite{k03}. A
similar step-size limitation may also be imposed in order to prevent
any of the values $\eta_i$ from becoming too close to zero. In
addition, in practice, a termination criterion based on the residual
$\| F(\eta)\|_2$ will be more useful. The criterion involving the
total number of iterations $N_n$ is used in Algorithm
\ref{alg:etanewton} only to simplify the complexity analysis which is
presented next.
\begin{algorithm}[h!]
\begin{algorithmic}[1]
\STATE evaluate adjoint states $z$ and $z(g'_i)$, $i=1,\ldots,N$ (if $e(u)$ depends on
a PDE equation)
\STATE set $k = 1$
\REPEAT
\STATE evaluate $g(\eta)$ by solving \eqref{eq:g(eta)}
\STATE evaluate $g'_i(\eta)$, $i=1,\dots,N$, by solving \eqref{eq:g'i}
\STATE evaluate $g''_{ij}(\eta)$, $i,j=1,\dots,N$, by solving \eqref{eq:g''}
\STATE evaluate $e'(u; g'_i(\eta))$, $i=1,\ldots,N$
\STATE evaluate $e'(u; g'_{ij}(\eta))$, $i,j=1,\ldots,N$
\STATE evaluate $e''(u; g'_i(\eta),g'_j(\eta))$, $i,j=1,\ldots,N$
\STATE evaluate $e''(u; g(\eta),g'_{ij}(\eta))$, $i,j=1,\ldots,N$
\STATE evaluate the function $F(\eta)$, cf.~\eqref{eq:alpha->F=0}
\STATE evaluate the Hessian $DF(\eta)$, cf.~\eqref{eq:DF}
\STATE set ${\widehat{\eta}} = \eta - \left[ DF(\eta) \right]^{-1}\cdot F(\eta)$
\STATE set $\eta = {\widehat{\eta}}$
\STATE set $k = k + 1$
\UNTIL{ \ $k = N_n$}
\STATE obtain $g({\widehat{\eta}})$ by solving \eqref{eq:g(eta)} with $\eta = {\widehat{\eta}}$
\end{algorithmic}
\caption{Determination of optimal weight ${\widehat{\lambda}}$ using Newton's method \newline
\textbf{Input:} \newline
\hspace*{0.5cm} $N$ --- dimension of the space in which optimal weights are sought \newline
\hspace*{0.5cm} $u \in H$ --- current approximation of minimizer ${\widehat{u}}$ \newline
\hspace*{0.5cm} $\kappa > 0$ --- step size in the outer loop (Algorithm \ref{alg:gradient}) \newline
\hspace*{0.5cm} $\{ \ell^i\}_{i=1}^N$ --- basis function for ansatz \eqref{eq:etaN} \newline
\hspace*{0.5cm} $N_n$ --- maximum number of Newton iterations \newline
\hspace*{0.5cm} $\eta$ --- initial guess for the weight \newline
\textbf{Output:} \newline
\hspace*{0.5cm} ${\widehat{\eta}}$ --- optimal weight \newline
\hspace*{0.5cm} $g({\widehat{\eta}})$ --- corresponding optimal gradient
}
\label{alg:etanewton}
\end{algorithm}
\FloatBarrier
\subsection{Complexity analysis}
\label{sec:complexity}
In this section we estimate the computational cost of a single
iteration of Algorithms \ref{alg:etagrad} and \ref{alg:etanewton} in
which the optimal weight ${\widehat{\lambda}}$ is computed using gradient
minimization and Newton's method, respectively, as described in
Sections \ref{sec:etagrad} and \ref{sec:etanewton}. This cost will be
expressed in terms of:
(i)
the number $N$ of the degrees of freedom
characterizing the dimension of the weight space $\Lambda$,
cf.~\eqref{eq:etaN};
(ii)
the number $M$ {determining the cost of} the numerical solution of
the elliptic boundary-value problems \eqref{eq:g(eta)}, \eqref{eq:g'i},
\eqref{eq:g''}; this latter quantity can be viewed as the number of
computational elements used to discretize the domain $\Omega$ (such as
finite elements/volumes, grid points or spectral basis functions);
(iii)
the number $K$ which is the typical number of line-search
iterations (line 9 in Algorithm \ref{alg:etagrad}). In the following we
will assume that the constants $C_1, C_2, \dots$ are all positive and
$\mathcal O(1)$.
Both algorithms require first the evaluation of $e'(u;\ell_i)$, $i=1,\ldots,N$.
{In general, the linear form can be expressed as}
\begin{equation}
e'(u;v) = \int_{\Omega} z v \, dz
\label{eq:z}
\end{equation}
{and, assuming that $v$ is already available, the cost of its
approximation is determined by the cost of evaluating $z$ on
$\Omega$ and the cost of the quadrature which is typically
$\mathcal O(M)$.} If $z$ is a function given explicitly in terms of $u$,
then {it can be evaluated on $\Omega$ in terms of $\mathcal O(M)$ operations.
However,} in general, {when the energy depends on $u$ through
some PDE,} which is the case of interest {here,} $z$ will be
given in terms of the solution of a suitably-defined adjoint PDE
problem. {Then,} for example, if the governing system is an
elliptic PDE problem in dimension $(d+1)$ with $u$ acting as the
boundary condition, {the numerical solution of the PDE will
require discretization with $\mathcal O(M^q)$, $q=\frac{d+1}{d}$, degrees
of freedom} and, assuming direct solution of the resulting
algebraic problems, the cost of evaluating $z$ on $\Omega$ will be
$\mathcal O(M^{3q})$. {Thus,} for simplicity, we will restrict our
attention to problems in which the cost of approximating $z$ on $M$
points/elements discretizing the domain $\Omega$ will be $C_1(M^{3q}+M)$,
$q\in\mathbb N$ ($q=0$ represents the case when the dependence of
$e$ on $u$ does not involve a PDE).
A similar argument applies to the evaluation of the second derivative
$e''(u;v,w) = \int_{\Omega} z(v) w \, dx$, except that now $z = z(v)$.
As the {operator defining the} adjoint PDE is the same for
{both} $z$ and $z(v)$, {to determine} $z(v)$ we {only
need} to perform a {back-}substitution at a computation cost
$C_2 M^{2q}$, {as explained below}.
We note that the cost of evaluating the gradient $g$ corresponding to
a certain $\lambda$ (or equivalently $\eta$) and its derivatives
$g'_i$, $g''_{ij}$, see \eqref{eq:g(eta)}, \eqref{eq:g'i},
\eqref{eq:g''}, will primarily depend on $M$. In general, solution of
each problem of this type requires $\mathcal O(M^3)$ operations. However,
when several such problems need to be solved with the same
differential operator, then it is more efficient to perform an LU-type
matrix factorization, at the cost $C_3 M^3$, followed by solution of
individual problems via back-substitution, each at the cost $C_4 M^2$.
With these estimates in place and assuming $K \ll M$ and $N_g, N_n
\ll M$, we are now in the position to characterize the complexity of
Algorithms \ref{alg:etagrad} and \ref{alg:etanewton}. The cost of a
single iteration of the gradient-minimization approach in Algorithm
\ref{alg:etagrad} will be dominated by:
\begin{itemize}
\item[g.1)] one evaluation of $z$ at the computational cost $C_1
(M^{3q}+M)$,
\item[g.2)] one evaluation of $z(g)$ at the computational cost $C_2
M^{2q}$,
\item[g.3)] the following computations {repeated $N_g$ times:}
\begin{itemize}
\item[i.1)] $N+K$ elliptic solves (with factorization) for $g$, $g'_i$
and $g(\eta-\tau F(\eta))$ at the cost $C_3 M^3 + C_4 (N+K) M^2$,
\item[i.2)] $N+K$ evaluations of $e'(u;v)$ ($v=g$, $v=g'_i$,
$v=g(\eta-\tau F(\eta))$), and $N+K$ evaluations of $e''(u;g,v)$
($v=g$, $v=g'_i$) at the cost at the cost $C_5 (N+K) M$.
\end{itemize}
\end{itemize}
Thus, finding ${\widehat{\lambda}}$ and $g({\widehat{\lambda}})$ with Algorithm
\ref{alg:etagrad} will require
\begin{equation}
{\cal C}_g
=
\mathcal O(1)
\left(
M^{3q}
+
(M^3 + (N+K)M^2)N_g
\right)
\approx
\mathcal O(1)
\left(
M^{3q}
+
M^3N_g
\right)
\
\mbox{\it flops}.
\label{eq:cost(gradient)}
\end{equation}
The cost of a single iteration of Newton's approach in Algorithm
\ref{alg:etanewton} will be dominated by:
\begin{itemize}
\item[n.1)]
one evaluation of $z$ at the computational cost $C_1 M^{3q}$,
\item[n.2)]
$N$ evaluation of $z(g'_i)$ at the computational cost
$C_2 N M^{2q}$,
\item[n.3)] the following computations {repeated $N_n$ times:}
\begin{itemize}
\item[i.1)] $\frac{1}{2}N^2$ elliptic solves (with factorization) for
$g''_{ij}$ (noting that $g''_{ij} = g''_{ji}$) at the total cost
proportional to $C_3 M^3 + C_4 N^2 M^2$,
\item[i.2)] $\frac{1}{2}N^2$ evaluations of $e'(u;v)$ (with $v=g''_{ij}$,
$1\leq i\leq j\leq N$), and
$\frac{1}{2}N^2$ evaluations of $e''(u;v,w)$ (with $(v,w)=(g,g''_{ij})$,
$(v,w)=(g'_i,g'_j)$, $i,j=1,\dots,N$),
at the cost $C_6 N^2 M$,
\item[i.3)] one evaluation of $[\partial_j F_i(\eta)]^{-1}\cdot
[F_i(\eta)]$ at the cost $C_7 N^3$.
\end{itemize}
\end{itemize}
Thus, the cost for computing ${\widehat{\lambda}}$ and $g({\widehat{\lambda}})$ with
Algorithm \ref{alg:etanewton} would require
\begin{eqnarray}
{\cal C}_{n}
&=&
\mathcal O(1)
\left(
M^{3q}+ N M^{2q}
+
(M^3 + N^2 M^2 +N^3)N_n
\right)
\nonumber\\
&\approx&
\mathcal O(1)
\left(
M^{3q}+ N M^{2q} +
(M^3 + N^2 M^2)N_n
\right)
\quad
\mbox{\it flops}.
\label{eq:cost(newton)}
\end{eqnarray}
Note that the cost of an iteration of a simple gradient algorithm is
\begin{equation}
{\cal C}_{sg}
=
\mathcal O(1)
\left(
M^{3q}+ M^3
\right)
\quad
\mbox{\it flops},
\label{eq:cost(g)}
\end{equation}
Then we {obtain}
\begin{alignat}{2}
\lim_{N/M\to0}
\frac{{\cal C}_g}{{\cal C}_{sg}}
& =
{\cal O}(1)
\left(1+M^{3(1-q)}N_g\right),
& \quad
\lim_{N/M\to1}
\frac{{\cal C}_g}{{\cal C}_{sg}}
& =
{\cal O}(1)
\left(1+M^{3(1-q)}N_g\right),
\label{eq:Cg/Csg}
\\
\lim_{N/M\to0}
\frac{{\cal C}_{n}}{{\cal C}_g}
& =
{\cal O}(1)
\frac{1+M^{3(1-q)}N_n}{1+M^{3(1-q)}N_g},
& \quad
\lim_{N/M\to1}
\frac{{\cal C}_{n}}{{\cal C}_g}
& =
{\cal O}(1)
\frac{1+MM^{3(1-q)}N_n}{1+M^{3(1-q)}N_g}.
\label{eq:Cn/Cg}
\end{alignat}
Equations \eqref{eq:Cg/Csg} show that the ratio of the cost of our
method using Algorithm \ref{alg:etagrad} {and} the cost of the simple
gradient method is of the same order ${\cal
O}(1)\left(1+M^{3(1-q)}N_g\right)$, regardless {of} $N$.
Furthermore, the methods tend to {have a comparable cost when $q\geq1$
and $M$ is} large. In view of \eqref{eq:Cn/Cg}, it follows that the
same conclusion {also} holds when comparing our method using Algorithm
\ref{alg:etagrad} and Algorithm \ref{alg:etanewton} for $N \ll M$.
{However,} when $N\approx M$, equations \eqref{eq:Cn/Cg} {indicate}
that the cost of our method with Algorithm \ref{alg:etanewton} becomes
substantially higher {(by a factor of $M$) as} compared to the cost
when Algorithm \ref{alg:etagrad} {is used}. These comments suggest
that it {may be more cost efficient} to use Algorithm
\ref{alg:etagrad} with {large $N$} (under the assumption $K \ll M$),
or Algorithm \ref{alg:etanewton} with $N \ll M$. {In either case,}
the cost will depend also on $N_g$ {and} $N_n$, i.e., on how fast
Algorithms \ref{alg:etagrad} {and \ref{alg:etanewton} can converge to
${\widehat{\eta}}$}. In conclusion, the relative efficiency of {original
Algorithm \ref{alg:gradient} versus its versions using Algorithms
\ref{alg:etagrad} or \ref{alg:etanewton} to find the optimal
gradients} will depend on the extend to which the increased
per-iteration cost {in the latter cases} can be offset by the reduced
number of iterations. This trade-off is illustrated based on a simple
model in the next section.
\section{A model problem and computational results}
\label{sec:results}
In order to illustrate the approach developed in this study, in the
present section we consider the following model problem defined on the
domain $\Omega = (-1,1)$
\begin{equation}
e({\widehat{u}})
=
\inf\left\{e(u)
:=
\int_\Omega
\left( 1+ a \, u^2 + a \, \left( \frac{du}{dx} \right)^2 \right)^{1/2} \, dx,
\quad u \in H^1_0(\Omega) \right\},
\label{eq:E}
\end{equation}
where $a = a(x) = 1 - x^2 / 2$. Clearly, the solution is ${\widehat{u}}=0$ and
$e({\widehat{u}}) = 2$. Energy \eqref{eq:E} gives rise to the following
expressions for its first and second derivative
\begin{align*}
e'(u;v)
& = \int_{-1}^1 \left\{ \frac{ a u }
{\left[ 1+ a \, u^2 + a \, \left( \frac{du}{dx} \right)^2 \right]^{1/2}} -
\frac{d}{dx} \left(\frac{ a \frac{du}{dx} }
{\left[ 1+ a \, u^2 + a \, \left( \frac{du}{dx} \right)^2 \right]^{1/2}}\right)\right\} v \, dx, \\
e''(u;v,w) & = \int_{-1}^1 \left\{ \frac{ a v w + a \frac{dv}{dx} \frac{dw}{dx} }
{\left[ 1+ a \, u^2 + a \, \left( \frac{du}{dx} \right)^2 \right]^{1/2}} -
\frac{ \left(a u v + a \frac{du}{dx} \frac{dv}{dx}\right)
\left(a u w + a \frac{du}{dx} \frac{dw}{dx}\right)}
{\left[ 1+ a \, u^2 + a \, \left( \frac{du}{dx} \right)^2 \right]^{3/2}} \right\} \, dx.
\end{align*}
To solve problem \eqref{eq:E} we will use the initial guess $u_0(x) =
(1-x^2) \cos(6x) e^x$ chosen such that $u_0\in H^1_0(\Omega)$ and it
has a large $H^1$ norm ensuring that $u_0$ is a ``significant
distance'' away from the solution ${\widehat{u}}$.
In order to mimic the setting with a refined discretization of the
domain $\Omega$, i.e., the case when $M \rightarrow \infty$, in our
computations all functions defined on $\Omega$ (i.e, $u$,
$\widetilde{u}$, $\lambda$, $g_0(u)$, $g(\lambda)$, $g'_i(\lambda)$
and $g''_{ij}(\lambda)$) will be approximated using {\tt Chebfun}
\cite{chebfun}. In this approach all the functions involved are
represented in terms of Chebyshev-series expansions truncated
adaptively to ensure that the truncation errors do not exceed a
prescribed bound (typically related to the machine precision). {\tt
Chebfun} also makes it possible to solve elliptic boundary-value
problems such as \eqref{eq:g(eta)},\eqref{eq:g'i} and \eqref{eq:g''}
with comparable accuracy. By minimizing the errors related to the
discretization in space, this approach allows us to focus on the
effect of the main parameter in the problem, namely, the dimension $N$
of the space $\Lambda$ in which the optimal weights are constructed,
cf.~\eqref{eq:etaN}. In terms of the basis $\{ \ell^i \}_{i=1}^N$ we
take the standard piecewise-linear ``hat'' functions which, unless
stated otherwise, are constructed based on an equispaced grid. With
such data and choice of the discretization parameters, minimization
problem \eqref{eq:E} is already rich enough to reveal the effect of
the parameter $N$ on convergence and the differences between different
approaches.
We now move on to present computational results obtained solving
problem \eqref{eq:E} using the following approaches:
\begin{itemize}
\item[(a)] steepest-descent method from Algorithm \ref{alg:gradient}
with Sobolev gradients $g(\lambda_0)$ defined through the inner
product \eqref{e:(u,v)_lam0} with {\em constant} weight $\lambda_0 =
10$ (this value of $\lambda_0$ was found by trial and error to
produce fastest convergence),
\item[(b)] steepest descent method from Algorithm \ref{alg:gradient}
with optimal Sobolev gradients $g({\widehat{\lambda}})$ determined using
Algorithm \ref{alg:etagrad} for different values $N$; at every iteration
Algorithm \ref{alg:etagrad} is restarted
with the same initial guess $\lambda(x) = \lambda_0$,
\item[(c)] Newton's method from Algorithm \ref{alg:newton},
\item[(d)] steepest descent method from Algorithm \ref{alg:gradient}
with optimal Sobolev gradients $g({\widehat{\lambda}})$ determined using a
simplified version of Algorithm \ref{alg:etanewton} for different
values $N$ (see below for details); at every iteration simplified
Algorithm \ref{alg:etanewton} is restarted with the same initial
guess $\lambda(x) = \lambda_0$.
\end{itemize}
Approaches (a), (b) and (d) use the same fixed step size $\kappa =
50$. Approximations of the exact solution $\widehat{u}$ obtained at
the $n$th iteration will be denoted $u_n$. In order to prevent the
optimal weights ${\widehat{\lambda}}(x)$ from becoming too close to zero for
certain $x$, which would complicate the numerical solution of problems
\eqref{eq:g(eta)}, \eqref{eq:g'i} and \eqref{eq:g''}, the line-search
in Algorithm \ref{alg:etagrad} and the length of Newton's step in
Algorithm \ref{alg:etanewton} are restricted such that $\min_{x \in
[-1,1]} {\widehat{\lambda}}(x) > \epsilon_{\tau} \lambda_0$, where we used
$\epsilon_{\tau} = 10^{-2}$. In addition, since this will make it
possible to objectively compare cases with different values of $N$,
here we modify the termination condition in Algorithm
\ref{alg:etagrad}, cf.~line 13, by replacing it with one given in
terms of a minimum relative decrease of $j(\eta)$, i.e., $|j({\widehat{\eta}}) -
j(\eta)| / j(\eta) \le \epsilon_{\lambda}$, where $\epsilon_{\lambda}$
is a prescribed tolerance.
We now examine the effect of different parameters on the results
obtained with each of the approaches (a)--(d) defined above.
\subsection{Analysis of the effect of the tolerance $\epsilon_\lambda$}
The decrease of the (shifted) energy $e(u_n)-e(\hat{u})$ and of the
$H^1$ approximation error $\|u_n - {\widehat{u}}\|_{H^1}$ are shown for
approaches (a), (b) and (c) in Figures \ref{fig:eps}a and
\ref{fig:eps}b, respectively, where in case (b) we used a single value
$N=50$ and three different tolerances $\epsilon_{\lambda} = 10^{-1},
10^{-2}, 10^{-3}$. In Figure \ref{fig:eps}a we see that minimization
with optimal gradients $g({\widehat{\lambda}})$ produces a significantly faster
decrease of energy $e(u_n)$ than optimization with ``standard''
Sobolev gradients $g(\lambda_0)$ and analogous trends are also evident
in the decrease of the approximation error $\|u_n - {\widehat{u}}\|_{H^1}$,
cf.~Figure \ref{fig:eps}b. We add that in order to solve the
minimization problem to the same level of accuracy the method based on
the ``standard'' Sobolev gradients $g(\lambda_0)$ requires as many as
42 iterations (for clarity, these later stages are not shown in the
figures).
In addition, in Figures \ref{fig:eps}a and \ref{fig:eps}b
we also observe that convergence of the proposed method systematically
accelerates as the tolerance $\epsilon_{\lambda}$ is refined, i.e., as
the optimal weights ${\widehat{\lambda}}$ are approximated more accurately.
However, we remark that reducing $\epsilon_{\lambda}$ below $10^{-3}$
did not produce further improvement of convergence. Hence, hereafter
we will set $\epsilon_{\lambda} = 10^{-3}$.
\begin{figure}[!h]
\centering
\mbox{
\subfloat[]{\includegraphics[width=0.475\textwidth]{./Figs/Emin_eps_v2.eps}}
\quad
\subfloat[]{\includegraphics[width=0.475\textwidth]{./Figs/Errmin_eps_v2.eps}}
}
\caption{(a) (Shifted) energy $e(u_n)-e({\widehat{u}})$ and (b) the $H^1$
approximation error $\|u_n - {\widehat{u}} \|_{H^1}$ as functions of the
iteration count $n$ where the optimal gradients $g({\widehat{\lambda}})$ are
obtained {using} Algorithm \ref{alg:etagrad} with $N=50$ and
different tolerances $\epsilon_{\lambda} = 10^{-1}, 10^{-2},
10^{-3}$. For comparison, the results obtained using standard
Sobolev gradients $g(\lambda_0)$ with a constant weight $\lambda_0 =
10$ and with Newton's method, cf.~Algorithm \ref{alg:newton}, are
also presented.}
\label{fig:eps}
\end{figure}
\subsection{Analysis of the effect of the dimension $N$ of the
approximation space $\Lambda$}
{The results concerning} the effect of $N$ on the performance of
approach (b) are compared with the data for approaches (a) and (c) in
Figures \ref{fig:min}a and \ref{fig:min}b for the (shifted) energy
$e(u_n)-e(\widehat{u})$ and the $H^1$ approximation error $\|u_n -
{\widehat{u}}\|_{H^1}$, respectively. We observe that, when optimal gradients
$g({\widehat{\lambda}})$ are used, both $e(u_n)-e({\widehat{u}})$ and $\|u_n - {\widehat{u}}\|_{H^1}$
initially reveal a quadratic convergence, similar to the behavior of
these quantities in Newton's method, followed at later iterations by a
linear convergence, typical of the standard gradient method. In the
light of Theorem \ref{th:P+newton->1}, cf.~estimate
\eqref{e:|P(tu-u)|->1}, this observation can be explained by the fact
that at early iterations dominant components of the error $(u_n-{\widehat{u}})$
are contained in the subspaces $T_{u_n}$ where the optimal gradients
$g({\widehat{\lambda}})$ are consistent with Newton's steps $h$, cf.~Remark
\ref{r:name}. Then, once these error components are eliminated, at
later iterations the error $(u_n - {\widehat{u}})$ is dominated by components in
directions orthogonal to $T_{u_n}$ where the optimal gradients
$g({\widehat{\lambda}})$ do not well reproduce the Newton steps $h$. In Figures
\ref{fig:min}a and \ref{fig:min}b we also see that the convergence
improves as the dimension $N$ is increased until it saturates for $N$
large enough (here $N \gtrapprox 25)$. This could be explained by the
conjecture that increasing $N$ above a certain limit (approximately
$25$ in this case) does not increase the ``effective'' dimension of
$T_{u_n}$ in $H$ anymore (such a possibility is allowed by the error
analysis presented in Section \ref{s:analysis->1}).
In this context it
is also interesting to investigate the evolution of the spatial
structure of the optimal weights ${\widehat{\lambda}}(x)$ and these results are
shown for different values of $N$ at an early ($n=2$) and a later
($n=8$) iteration in Figures \ref{fig:lam}a and \ref{fig:lam}b,
respectively. In the first case ($n=2$ corresponding to the quadratic
convergence) we see that the optimal weights ${\widehat{\lambda}}(x)$ converge to
a well-defined profile as $N$ increases, which features a number of
distinct ``spikes''. On the other hand, at later iterations ($n=8$
corresponding to the linear regime) the convergence of the optimal
weights ${\widehat{\lambda}}(x)$ with $N$ is less evident and the resulting
profiles tend to be more uniform.
\begin{figure}[!h]
\centering
\mbox{
\subfloat[]{\includegraphics[width=0.475\textwidth]{./Figs/Emin_v2.eps}}
\quad
\subfloat[]{\includegraphics[width=0.475\textwidth]{./Figs/Errmin_v2.eps}}
}
\caption{(a) (Shifted) energy $e(u_n)-e({\widehat{u}})$ and (b) the $H^1$
approximation error $\|u_n - {\widehat{u}} \|_{H^1}$ as functions of the
iteration count $n$ for different dimensions $N$ where the optimal
gradients $g({\widehat{\lambda}})$ are obtained with Algorithm
\ref{alg:etagrad}. For comparison, the results obtained using
standard Sobolev gradients $g(\lambda_0)$ with a constant weight
$\lambda_0 = 10$ and with Newton's method, cf.~Algorithm
\ref{alg:newton}, are also presented.}
\label{fig:min}
\centering
\mbox{
\subfloat[]{\includegraphics[width=0.475\textwidth]{./Figs/lam_it2_v2.eps}}
\quad
\subfloat[]{\includegraphics[width=0.475\textwidth]{./Figs/lam_it8_v2.eps}}
}
\caption{Optimal weights ${\widehat{\lambda}}(x)$ as functions of $x$ obtained
for different $N$: {(a) an early ($n=2$) iteration and (b) a
late ($n=8$)} iteration of Algorithm \ref{alg:etagrad}, cf.~Figure
\ref{fig:min}.}
\label{fig:lam}
\end{figure}
\noindent
We want to {highlight} the case when $N=1$ and space $H$ is
endowed with the inner product redefined as in \eqref{eq:ipH10}. As
shown in Remark \ref{r:hlambda,H10}, in such circumstances the optimal
$\lambda$ can be found analytically, cf.~relation
\eqref{e:hlambda,N=1}, at essentially no cost and the iterations
produced by Algorithm \ref{alg:gradient} do not depend on the step
size $\kappa$. The results obtained with this approach and using the
optimal gradients $g({\widehat{\lambda}})$ defined in terms of the inner product
\eqref{e:(u,v)_lambda} are compared in Figures \ref{fig:N1}a and
\ref{fig:N1}b for the (shifted) energy $e(u_n)-e(\widehat{u})$ and the
$H^1$ approximation error $\|u_n - {\widehat{u}}\|_{H^1}$, respectively. As is
evident from these figures, the performance of the approaches
corresponding to the two definitions of the inner product,
\eqref{e:(u,v)_lambda} and \eqref{eq:ipH10}, is comparable and
{in both cases} much better than when a fixed weight $\lambda_0$
is used. We stress that in the case corresponding to the inner product
\eqref{eq:ipH10} determination of the optimal $\lambda$ does not
require an iterative solution.
\begin{figure}[!h]
\centering
\mbox{
\subfloat[]{\includegraphics[width=0.475\textwidth]{./Figs/EminN1_v2.eps}}
\quad
\subfloat[]{\includegraphics[width=0.475\textwidth]{./Figs/ErrminN1_v2.eps}}
}
\caption{(a) (Shifted) energy $e(u_n)-e({\widehat{u}})$ and (b) the $H^1$
approximation error $\|u_n - {\widehat{u}} \|_{H^1}$ as functions of the
iteration count $n$ for the case when $N=1$ and the optimal
gradients $g({\widehat{\lambda}})$ are obtained using the inner product
definitions \eqref{e:(u,v)_lambda} combined with Algorithm
\ref{alg:etagrad} and \eqref{eq:ipH10} combined with {the}
explicit relation \eqref{e:hlambda,N=1}. For comparison, the results
obtained using standard Sobolev gradients $g(\lambda_0)$ with a
constant weight $\lambda_0 = 10$ and with Newton's method,
cf.~Algorithm \ref{alg:newton}, are also presented. The step size
used in these calculations is $\kappa = 25$.}
\label{fig:N1}
\end{figure}
\subsection{Analysis of the robustness of approach (b) with respect to
{variations of the basis functions defining $\eta$}}
This {analysis is performed} by constructing basis functions $\{
\ell^i \}_{i=1}^N$ based on grid points distributed randomly with an
uniform probability distribution over the interval $(-1,1)$, except
for the leftmost and the rightmost grid points which are always at $x
= \pm 1$. The results obtained in several realizations with $N=5$ are
compared to the reference case of basis functions constructed based on
equispaced grid points as well as with the results obtained with
approaches (a) and (c) in Figures \ref{fig:rand}a and \ref{fig:rand}b
for the (shifted) energy $e(u_n)-e(\widehat{u})$ and the $H^1$
approximation error $\|u_n - {\widehat{u}}\|_{H^1}$, respectively. One can see
in these figures that, expect for one {realization} corresponding
to a very special distribution of the grid points, the convergence is
little affected by the choice of the basis $\{ \ell^i \}_{i=1}^N$.
\begin{figure}[H]
\centering
\mbox{
\subfloat[]{\includegraphics[width=0.475\textwidth]{./Figs/Eminrand_v2.eps}}
\quad
\subfloat[]{\includegraphics[width=0.475\textwidth]{./Figs/Errrand_v2.eps}}
}
\caption{(a) (Shifted) energy $e(u_n)-e({\widehat{u}})$ and (b) the $H^1$
approximation error $\|u_n - {\widehat{u}} \|_{H^1}$ as functions of the
iteration count $n$ for the case when $N=5$ and the optimal
gradients $g({\widehat{\lambda}})$ are obtained with Algorithm
\ref{alg:etagrad} using uniform and random distributions of grid
points defining the basis functions $\{ \ell^i \}_{i=1}^N$. For
comparison, the results obtained using standard Sobolev gradients
$g(\lambda_0)$ with a constant weight $\lambda_0 = 10$ and with
Newton's method, cf.~Algorithm \ref{alg:newton}, are also
presented.}
\label{fig:rand}
\end{figure}
\subsection{{Analysis of the performance of a simplified version of Algorithm \ref{alg:etanewton}}}
Finally, we consider approach (d) where the optimal weights
${\widehat{\lambda}}(x)$ and the corresponding optimal gradients $g({\widehat{\lambda}})$
are determined with Algorithm \ref{alg:etanewton} simplified as
follows. The complexity analysis presented in Section
\ref{sec:complexity} shows that Algorithm \ref{alg:etanewton} may be
quite costly from the computational point of view when $N \gg 1$. To
alleviate this difficulty, we consider its simplified version where
only one iteration ($N_n = 1$) is performed on system
\eqref{eq:alpha->F=0} in which the ``test'' functions $g'_i$,
$i=1,\dots,N$, are assumed not to depend on $\lambda$ (or $\eta$). In
other words, since instead of $g'_i(\eta)$, $i=1,\dots,N$, the
functions $g'_i(\eta_0)$ are used to obtain expressions for
$[F(\eta)]_i$, $i=1,\dots,N$, in \eqref{eq:alpha->F=0}, the second
derivatives $g''_{ij}$ are eliminated from the Hessian
$[DF(\eta)]_{ij}$, $i,j=1,\dots,N$ in \eqref{eq:DF}, which very
significantly reduces the computational cost. The results obtained
with this simplified approach are shown in Figures \ref{fig:lin}a and
\ref{fig:lin}b, respectively, for the decrease of the (shifted) energy
$e(u_n)-e({\widehat{u}})$ and for the decrease of the $H^1$ approximation error
$\|u_n - {\widehat{u}}\|_{H^1}$. In these figures we observe general trends
qualitatively similar to those evident in Figures \ref{fig:min}a and
\ref{fig:min}b, except that the convergence is slower and the
transition from the quadratic to linear convergence tends to occur at
earlier iterations.
\begin{figure}[!h]
\centering
\mbox{
\subfloat[]{\includegraphics[width=0.475\textwidth]{./Figs/Elin_v1.eps}}
\quad
\subfloat[]{\includegraphics[width=0.475\textwidth]{./Figs/Errlin_v1.eps}}
}
\caption{(a) (Shifted) energy $e(u_n)-e({\widehat{u}})$ and (b) the $H^1$
approximation error $\|u_n - {\widehat{u}} \|_{H^1}$ as functions of the
iteration count $n$ for different dimensions $N$ where the optimal
gradients $g({\widehat{\lambda}})$ are obtained with the simplified version of
Algorithm \ref{alg:etanewton} described above. For comparison, the
results obtained using standard Sobolev gradients $g(\lambda_0)$
with a constant weight $\lambda_0 = 10$ and with Newton's method,
cf.~Algorithm \ref{alg:newton}, are also presented.}
\label{fig:lin}
\end{figure}
\section{Conclusions}
\label{sec:final}
We have developed a gradient-type numerical approach for
{unconstrained} optimization problems in infinite-dimensional
Hilbert spaces.
Our method consists in
finding an optimal inner product among a family of equivalent inner
products parameterized by a space-dependent weight $\lambda$ function.
The optimal weight ${\widehat{\lambda}}$ solves a nonlinear optimization problem in a
finite dimensional subspace.
Rigorous analysis demonstrates that, in addition to the linear
convergence characterizing the standard gradient method, the proposed
approach also attains quadratic convergence in the sense that the
projection error in a finite-dimensional subspace generated in the
process decreases quadratically. Or, equivalently, in this finite
dimensional subspace, the optimal gradients and Newton's steps are
equivalent. The dimension of these subspaces is determined by the
number $N$ of discrete degrees of freedom parameterizing the inner
products through the weight $\lambda$.
This analysis is confirmed by numerical experiments,
performed based on a simple optimization problem in a setting
mimicking high spatial resolution. More specifically, at early
iterations both the minimized energy and the error with respect to the
exact solution exhibit quadratic convergence followed by the linear
convergence at later iterations. The behavior of the proposed method
also reveals expected trends with variations of the numerical
parameters, namely, the dimension of the space in which the optimal
weights ${\widehat{\lambda}}$ are constructed, properties of the basis in this
space and the accuracy with which the inner optimization problems are
solved. In all cases the convergence of the proposed approach is much
faster than obtained using Sobolev gradients with fixed weights. For
the ease of analysis and comparisons we focused on a gradient-descent
method with a fixed step size $\kappa$, but it can be expected that a
similar behavior will also occur when the step size is determined
adaptively through suitable line minimization.
The complexity analysis performed in Section \ref{sec:complexity}
indicates that the per-iteration cost of the proposed approach
and of the standard Sobolev gradient method have the same order if,
for example, the energy depends on a elliptic PDE. When the dimension
of weight space is $N = 1$ and the inner product does not have the
$L^2$ term, cf.~\eqref{eq:ipH10}, then the optimal weight is given
explicitly, eliminating the need for its numerical determination. In
this particular case the proposed approach has some similarity to the
Barzilai-Borwein algorithm \cite{barzilai-1} and produces iterates
which do not depend on the step size in the gradient method. The
computational cost of the proposed approach is also significantly
reduced when Algorithm \ref{alg:etanewton} is used in a simplified
form, as described in Section \ref{sec:results}. {We thus
conclude that the gradient-descent method from Algorithm
\ref{alg:gradient} combined with Algorithms \ref{alg:etagrad} and
\ref{alg:etanewton} used to find optimal gradients} are promising
approaches suitable for large-scale optimization problems and
applications to some such problems will be investigated in the near
future.
Our approach based on optimal gradients differs from the family of
quasi-Newton methods in that instead of approximating the full Hessian
using gradients from earlier iterations, see for example \cite{nw00},
it relies on computing the action of the exact \revt{Hessian and
gradients,} but only on a few judiciously selected directions, and
then matching them by appropriately choosing the inner product.
Consequently, the resulting algebraic problem is of a much smaller
dimension thereby avoiding complications related to poor conditioning
and computational cost.
Finally, we believe that the analysis and results presented here
explain the acceleration of gradient minimization reported for a range
of different problems in \cite{pbh02,p07b,rssl09,r10,ms10,dk10} when
Sobolev gradients with suitable (constant) weights were used.
Moreover, our work also provides a rational and constructive answer to
the open problem of finding an optimal form of the inner product
defining the Sobolev gradients.
\bibliographystyle{plain}
|
1,116,691,499,256 | arxiv | \section{Introduction}\label{sec:1}
The use of combinatorial methods in knot theory, as is well known, has led to connections between graph and link polynomials, the most famous of which is a relation between the Jones polynomial and the Tutte polynomial. This appeared early on in the development of quantum topology (see \cite{kauf,this,jones}). Jaeger also found a connection between the Tutte and HOMFLY polynomials (\cite{Ja}). The relation was extended to a larger class of links by Traldi in \cite{Tr}.
Here we are interested in the extension of these relationships to polynomials of embedded graphs and
invariants of links in $\Sigma \times I$, the product of a surface $\Sigma$ and the unit interval $I = [0,1]$. We refer the reader to \cite{CRR,hp,ik,Li,tu} for various approaches to the generalization of invariants of links in the $3$-sphere $S^3$ to links in $\Sigma \times I$ and to \cite{aflt,CP2,Da,dk,m} for recent related results.
In the recent preprint \cite{CP} (subsequently revised to become paper \cite{CP2}) Chmutov and Pak generalized the connection between the Jones and Tutte polynomials by relating the Bollob\'{a}s-Riordan polynomial of a ribbon graph $F$, which generalizes the Tutte polynomial to embedded graphs, to the Jones polynomial of a link in $F\times I$.
Rather than using graph theory to develop knot theory, as would be more usual, in this note we are interested in doing things the other way round: we use knot theory to advance our knowledge of graph polynomials. Our approach is to relate the Bollob\'{a}s-Riordan polynomial of an embedded graph $G \subset \Sigma$ to the HOMFLY polynomial of a link in $\Sigma \times I$ and then to use the topology of knots to deduce results for the graph polynomial. This relation between polynomials generalizes the results of Jaeger and Traldi mentioned above and also answers a question of Chmutov and Pak posed in \cite{CP}.
We will show that the HOMFLY and the Bollob\'{a}s-Riordan polynomials are equivalent along the surface $\{ (x,y,z)|xyz^2=1 \}$. This observation allows us to answer graph theoretical questions with knot theory. We use knot theory to prove a duality relation for the Bollob\'{a}s-Riordan polynomial (see also \cite{EMS}). Finding such a duality relation was a problem posed by Bollob\'{a}s and Riordan in \cite{BR}.
We also give an application to knot theory by showing that the genus of the smallest surface containing a link projection can be recovered from its HOMFLY polynomial.
We then go on to consider the Jones polynomial as a specialization of the HOMFLY, and the relation of this with Chmutov and Pak's result in \cite{CP}. From this we find a relation between the Bollob\'{a}s-Riordan polynomial of an embedded graph and its tensor product with the $(2^p+1)$-cycle $C_{2^p+1}$ (see also \cite{JVW,Wo}).
In section~\ref{sec:2} we recall how embedded graphs give rise to ribbon graphs and we define the Bollob\'{a}s-Riordan polynomial. We then discuss how to construct oriented links in $\Sigma \times I$ from an embedded graph $G \subset \Sigma$.
In section~\ref{sec:3}, after defining the HOMFLY polynomial of a link in $\Sigma \times I$, we prove our first result which relates it to the Bollob\'{a}s-Riordan polynomial.
This is generalized in two directions in section~\ref{sec:4} by considering weighted graphs and a multivariate Bollob\'{a}s-Riordan polynomial. We then go on to discuss the question of equivalence between the two polynomials and prove a duality relation for the Bollob\'{a}s-Riordan polynomial.
The final section concerns the specialization to the Jones polynomial. We relate our earlier formulae for the HOMFLY polynomial to Chmutov and Pak's results on the Jones polynomial and prove a formula for the tensor product of a graph and its Bollob\'{a}s-Riordan polynomial.
\section{Links and embedded graphs}\label{sec:2}
\subsection{Ribbon graphs}
A {\em ribbon graph} (or {\em fatgraph}) $F$ is a graph together with a fixed cyclic order of the incident half-edges at each vertex. Ribbon graphs can be regarded as orientable surfaces with boundary by fattening each vertex into a disk, $D^2$, and fattening each edge into an `untwisted ribbon'.
Notice that if $G \subset \Sigma$ is a graph embedded in an orientable surface $\Sigma$, then a ribbon graph $F$ arises naturally as a neighbourhood of $G$ (we retain information on the position of the vertices). This is indicated in figure~\ref{fig:links}. In such a situation we will say that $F$ is the ribbon graph {\em associated} with the embedded graph $G$. We will generally denote a ribbon graph by $F$ and an embedded graph by $G$ and move freely between the two concepts.
\begin{figure}
\[ \epsfig{file=figs/fig1, height=1.5cm} \]
\caption{}
\label{fig:jaeger}
\end{figure}
\begin{figure} \begin{center}
\epsfig{file=figs/fig2a, height=2cm}
\hspace{1.5cm}
\epsfig{file=figs/fig2b, height=2cm}
\hspace{1.5cm}
\epsfig{file=figs/fig2c, height=2cm} \end{center}
\caption{}
\label{fig:links}
\end{figure}
We are interested in the Bollob\'{a}s-Riordan polynomial of ribbon graphs (\cite{BR1,BR}) which is the natural generalization of the Tutte polynomial to embedded graphs.
We need some notation. Let $G=(V,E)$ be a graph then we set $v(G)= |V|$,
$e(G)=|E|$, $r(G)= |V|- k(G)$ and $n(G)= |E|- r(G)$ where $k(G)$ denotes the number of connected components
of $G$. We use a similar notation for ribbon graphs. In addition, if $F$ is a ribbon graph regarded as a surface, then we set $p(F)=|\partial(F)|$, the number of its boundary components. A {\em state} of a (ribbon) graph $F$ is a spanning sub(ribbon)~graph and we denote the set of states by ${\mathcal{S}}(F)$. The {\em Bollob\'{a}s-Riordan polynomial} of a ribbon graph $F$ can then be defined as the sum over states:
\begin{equation}\label{eq:BRpoly}
R(F;\alpha , \beta, \gamma) = \sum_{H\in {\mathcal{S}}(F)}\alpha^{r(F)-r(H)}\beta^{n(H)}\gamma^{k(H)-p(H)+n(H)}.
\end{equation}
By the \mbox{Bollob\'{a}s-Riordan } polynomial of an embedded graph we mean the Bollob\'{a}s-Riordan polynomial of the associated ribbon graph.
Notice that
$R(F; \alpha-1, \beta-1 , 1)= T(F; \alpha , \beta)$ the Tutte polynomial,
and, since the exponent of $\gamma$ is exactly twice the genus of the ribbon graph regarded as a surface (see \cite{BR,LM}),
$R(G; \alpha-1, \beta-1 , \gamma)= T(G; \alpha , \beta)$ whenever $G \subset \mathbb{R}^2$.
We will also make use of the following rearrangement of the Bollob\'{a}s-Riordan polynomial which is obtained by expanding the rank and nullity
\begin{equation}\label{eq:br}
R(F;\alpha , \beta, \gamma)= \alpha^{-k(F)} (\beta \gamma)^{-v(F)} \sum_{H\in {\mathcal{S}}(F)} (\alpha\beta\gamma^2)^{k(H)} (\beta\gamma)^{e(H)} (\gamma)^{-p(H)}.
\end{equation}
\subsection{Links in $\Sigma \times I$}
Having discussed embedded graphs, we move onto our second main object.
A {\em link} is an embedding of a finite number of copies of the unit circle $S^1$ into a 3-manifold $M$. Knot theory is mostly concerned with the special case $M=S^3$. Here, however, we are concerned with the more general case of links in $\Sigma \times I$, where $\Sigma$ is an orientable surface and $I=[0,1]$ is the unit interval. Note that knot theory in the $3$-ball $D^2 \times I$ is equivalent to knot theory in the $3$-sphere $S^3$.
Given a link $L \subset \Sigma \times I$ there is a generic projection on to the surface $\Sigma$ by projection onto the first variable. It is obvious how to construct a link in $\Sigma \times I$ from its projection.
If we are given an embedded graph $G \subset \Sigma$ we can associate an oriented link projection by replacing each edge with the oriented tangle of figure~\ref{fig:jaeger} and connecting these tangles according to the cyclic order at the vertices. An example is shown in figure~\ref{fig:links}. We denote a link in $\Sigma \times I$ constructed in this way by ${\mathcal{L}}(G)$.
We may assume that the link lies in a sufficiently small neighbourhood of $G$ so that we obtain a projection ${\mathcal{L}}(F)$ on the associated ribbon graph $F$. This gives rise to a link in $F\times I$. Again this is indicated in figure~\ref{fig:links}.
\section{The HOMFLY in $F \times I$}\label{sec:3}
\subsection{The HOMFLY polynomial}
Let $\Sigma$ be an orientable surface (possibly with boundary) and $L\subset \Sigma \times I$ be a link.
The HOMFLY polynomial $P(L)$ is a link invariant which satisfies the {\em HOMFLY skein relation}
\[ xP(L_+) -x^{-1}P(L_-) = yP(L_0) ,\]
where $L_+$, $L_-$ and $L_0$ are links which differ only in the locality of a single crossing as shown in figure~\ref{fig:skein}.
\begin{figure}
\[ \begin{array}{ccccc}
\epsfig{file=figs/fig3a, height=1.5cm} & & \epsfig{file=figs/fig3b, height=1.5cm} & & \epsfig{file=figs/fig3c, height=1.5cm} \\
L_+ & & L_- & & L_0
\end{array}\]
\caption{}
\label{fig:skein}
\end{figure}
We will also require that the invariant takes the value $1$ on the unknot, {\em i.e.} $ P(\mathcal{O})=1$.
If we were only considering links in $S^3$ or $D^2 \times I$ then this would be enough to uniquely determine a link invariant $P(L)\in {\mathbb{Z}} [x^{\pm 1}, y^{\pm 1}]$.
However, for a general orientable surface $\Sigma$ this is not enough. To describe a basis for the HOMFLY skein module we need to introduce the notion of a descending link.
First notice that there is a natural {\em product} of links in $\Sigma\times I$ given by reparameterizing the two copies of $\Sigma\times I$ and stacking them:
\[
(\Sigma \times I) \times (\Sigma \times I) \cong (\Sigma \times [1/2,1]) \times (\Sigma \times [0,1/2])
\rightarrow (\Sigma \times I) .
\]
We denote the projections from $\Sigma \times I$ to $\Sigma $ and $I$ by $p_\Sigma $ and $p_I$ respectively. The value $p_I(x)$ is called the {\em height} of $x$. We can now make our definition.
\begin{definition}
(1) A knot $K \subset \Sigma \times I$ is {\em descending} if
it is isotopic to a knot $K^{\prime} \subset \Sigma \times I$ with the property that
there is a choice of basepoint $a$ on $K^{\prime}$ such that if we travel along $K^{\prime}$ in the direction of the orientation from the basepoint the height of $K^{\prime}$ decreases until we reach a point $a^{\prime}$ with $p_\Sigma (a)=p_\Sigma (a^{\prime})$ from which $K^{\prime}$ leads back to $a$ by increasing the height and keeping the projection onto $F$ constant.
(2) A link $L \subset \Sigma \times I$ is said to be {\em descending} if it is isotopic to a product of descending knots.
\end{definition}
Clearly each (isotopy class of a) descending knot uniquely determines a conjugacy class of the fundamental group $\pi_1(\Sigma )$.
Moreover there is a bijection between the conjugacy classes in $\pi_1(\Sigma )$ and isotopy classes of descending knots. In other words a conjugacy class determines a descending knot.
In \cite{Li}, Lieberum gives a procedure for choosing a canonical element of the conjugacy classes of $\pi_1(\Sigma )$. We will denote this set of choices by $S(\Sigma ) = \{t_w\}$. We do not need to worry about the exact choices here.
This means that each descending knot $K$ determines some $t_K \in S(\Sigma)$ and each descending link determines a monomial $t_L$ in commuting indeterminates $\{t_w\}$ (we have $t_{\mathcal{O}}=1$).
For a descending link $L$ we set
\begin{equation}\label{eq:basis}
P(L) = t_L \left( \frac{x-x^{-1}}{y} \right)^{k(L)-1},
\end{equation}
where $k(L)$ is the number of components of the link $L$.
The {\em HOMFLY} (or {\em HOMFLYPT}) polynomial is then defined by the following theorem.
\begin{theorem}[\cite{Li}]
There exists a unique invariant $P(L) \in {\mathbb{Z}} [x^{\pm 1}, y^{\pm 1}]\otimes {\mathbb{Z}} [\{t_w\}]$ of links $L \subset \Sigma \times I$ that satisfies the HOMFLY skein relation and equation~\ref{eq:basis}.
\end{theorem}
If we set each $t_w=1$ then we obtain a polynomial in ${\mathbb{Z}} [x^{\pm 1}, y^{\pm 1}]$, which we denote by $P(L; x,y )$.
We consider this $2$-variable polynomial first.
\subsection{The relation to the Bollob\'{a}s-Riordan polynomial}
In our first result we relate the Bollob\'{a}s-Riordan polynomial of an embedded graph $G \subset \Sigma$ to the HOMFLY polynomial of the associated link ${\mathcal{L}}(G) \subset \Sigma \times I$. In order to do this we need to set each of the variables $t_w$ equal to $1$ and consider the invariant $P(L; x,y )$. In the following section we will discuss the full HOMFLY invariant $P(L) \in {\mathbb{Z}} [x^{\pm 1}, y^{\pm 1}]\otimes {\mathbb{Z}} [\{t_w\}]$ and relate it to a multivariate Bollob\'{a}s-Riordan polynomial. The relation proved in this section will follow from the more general result, however we feel that it is more clear if we give a separate proof.
As promised, the following theorem gives a relationship between the HOMFLY and Bollob\'{a}s-Riordan polynomials.
\begin{theorem}\label{th:homfly}
Let $\Sigma$ be an orientable surface, possibly with boundary, and $G \subset \Sigma$ be an embedded graph. Then
\begin{multline}\label{eq:homfly}
P({\mathcal{L}}(G); x,y ) = \left( \frac{1}{xy} \right)^{v(G)-1}
\left( \frac{y}{x}\right)^{e(G)}
\left( x^{2}-1 \right)^{k(G)-1}
\\ R\left(G; x^{2}-1 , \; \frac{x-x^{-1}}{xy^2},\; \frac{y}{x - x^{-1}}\right).
\end{multline}
\end{theorem}
\begin{proof}
An application of the HOMFLY skein and isotopy of the link gives the equation
\begin{equation}\label{eq:skstate}
P\left( \raisebox{-5mm}{\epsfig{file=figs/st0 ,height=1cm}} \right) =
\frac{1}{x^2} \; P\left( \raisebox{-5mm}{\epsfig{file=figs/st1 ,height=1cm}} \right)
+\frac{y}{x} \; P\left( \raisebox{-5mm}{\epsfig{file=figs/st2 ,height=1cm}} \right).
\end{equation}
We will call a link obtained by eliminating the crossings in this way a {\em resolution}.
There is a clear correspondence between resolutions of the link ${\mathcal{L}} (G)$ and the states of the associated ribbon graph $F$ given by including an edge in the state whenever the corresponding link is resolved as
\raisebox{-2mm}{\epsfig{file=figs/st1 , height=0.6cm}}
and excluding an edge otherwise.
The links corresponding to a given state have no crossings and are therefore descending. It is then clear that
\begin{equation*}\begin{split}
P({\mathcal{L}}(G); x,y ) & =
\sum_{H \in {\mathcal{S}} (G)} \left( \frac{1}{x^2} \right)^{e(H)}
\left( \frac{y}{x} \right)^{e(G)-e(H)}
\left( \frac{x-x^{-1}}{y} \right)^{p(H)-1}\\
& =\left( \frac{y}{x-x^{-1}} \right) \left( \frac{y}{x} \right)^{e(G)}
\sum_{H \in {\mathcal{S}} (F)} \left( \frac{1}{xy} \right)^{e(H)}\left( \frac{x-x^{-1}}{y} \right)^{p(H)}.
\end{split}
\end{equation*}
Now setting $\alpha = x^2-1$, $\beta=(x-x^{-1})/xy^2$ and $\gamma=y/(x-x^{-1})$, so that $\alpha\beta\gamma^2 = 1$ and $\beta\gamma=1/xy$, we see that by applying (\ref{eq:br}), the above can be written as
\begin{equation*}\begin{split}
\left(\frac{y}{x-x^{-1}}\right)
\left( \frac{y}{x} \right)^{e(G)}
(x^2-1)^{k(G)}
\left( \frac{1}{xy} \right)^{v(G)}
\; R\left(F; x^{2}-1 , \; \frac{x-x^{-1}}{xy^2},\; \frac{y}{x - x^{-1}}\right)
\\
=\left( \frac{1}{xy} \right)^{v(G)-1}
\left( \frac{y}{x}\right)^{e(G)}
\left( x^2 -1 \right)^{k(G)-1}
\; R\left(F; x^{2}-1 , \; \frac{x-x^{-1}}{xy^2},\; \frac{y}{x - x^{-1}}\right)
\end{split}
\end{equation*}
as required.
\end{proof}
Notice that when $\Sigma$ is a disc then, using the relation between the Bollob\'{a}s-Riordan and Tutte polynomials, we recover Jaeger's expression for the HOMFLY from \cite{Ja}.
\section{The full polynomial}\label{sec:4}
In order to find a formula for the full HOMFLY invariant, rather than just its specialization at $t_w=1$, we need to consider a ``Sokalization'' ({\em c.f.} \cite{So}) of the Bollob\'{a}s-Riordan polynomial.
The Bollob\'{a}s-Riordan polynomial can be expressed as
\begin{equation*}\begin{split}
R(F;\alpha,\beta,\gamma)
& = \alpha^{-k(F)} (\beta \gamma)^{-v(F)} \sum_{H \in {\mathcal{S}} (F)} (\alpha\beta\gamma^2)^{k(H)} (\beta\gamma)^{e(H)} (\gamma^{-1})^{p(H)} \\
& = ((ac)^{-1}b)^{k(F)} b^{-v(F)} \sum_{H \in {\mathcal{S}} (F)} a^{k(H)} b^{e(H)} c^{p(H)},
\end{split}
\end{equation*}
where $a=\alpha\beta\gamma^2$, $b=\beta\gamma$ and $c=\gamma^{-1}$. Therefore the Bollob\'{a}s-Riordan polynomial can be equivalently formulated as a polynomial
\[
B(F; a,b,c) = \sum_{H \in {\mathcal{S}} (F)} a^{k(H)} b^{e(H)} c^{p(H)}.
\]
A {\em weighted ribbon graph} is a ribbon graphs equipped with a map from its edge set $E$ to a set of {\em weights} $\{b_e \}_{e \in E(F)}$. Notice that we can (and will) regard the {\em weights} $b_e$ as formal commuting variables. We can regard an unweighted ribbon graph as a weighted graph all of whose weights are equal. We will usually denote this single weight by $b$.
For a weighted ribbon graph $F$ we define the {\em weighted Bollob\'{a}s-Riordan polynomial} as
\[
B(F; a,\textbf{b} ,c) = \sum_{H \in {\mathcal{S}} (F)} a^{k(H)} \left(\prod_{e\in E(H)} b_e \right) c^{p(H)},
\]
where $\textbf{b} = \{b_e \}_{e \in E}$ is the set of edge weights. We will sometimes exclude the variables from the notation and just write $B(F)$. As before, for an embedded graph $G$, $B(G):=B(F)$ where $F$ is the associated ribbon graph.
Notice that for an unweighted ribbon graph this polynomial is equivalent to the Bollob\'{a}s-Riordan polynomial.
We now state the general form of Theorem~\ref{th:homfly}.
\begin{theorem}\label{th:fullhomfly}
Let $\Sigma$ be an orientable surface, possibly with boundary, and $G \subset \Sigma$ be an embedded graph
with edge weights $\textbf{b} = \{b_e \}_{e \in E(F)} $. Then
\begin{multline}\label{eq:full}
P({\mathcal{L}}(G); x,y, \{ t_w \} ) =\\ f \left( \
\begin{array}{c}
\left( y/(x-x^{-1}) \right)
\left( y/x\right)^{e(G)}
\; B\left(G; 1 , \; \left(b_e/xy\right)_{e\in E(G)},\; (x - x^{-1})/y\right)
\end{array}
\right)
\end{multline}
where $f$ is a function from polynomials in $\textbf{b}$ to polynomials in the conjugacy classes $S(\Sigma) =\{t_w\}$ of $\pi_1 (\Sigma)$.
\end{theorem}
\begin{proof}
As in the proof of Theorem~\ref{th:homfly}, there is a correspondence between the states of the ribbon graph and the resolutions of the link shown in Equation~\ref{eq:skstate}. So it is enough to show that each corresponding term on the left and right hand sides of (\ref{eq:full}) is assigned the same value.
Let $F$ be the ribbon graph associated with $G$. Observe that a state $H$ of the ribbon graph $F$ is assigned the monomial in $\textbf{b}$
\[
\left( \frac{y}{x-x^{-1}} \right)
\left( \frac{y}{x}\right)^{e(F)}
\left(\prod_{e\in E(H)} \frac{b_e}{xy} \right)
\left(\frac{x - x^{-1}}{y} \right)^{p(H)} \]
by the right hand side. This is equal to
\begin{equation}\label{eq:fullpf1}
\left( \frac{y}{x}\right)^{e(F)-e(H)}
\left( \frac{1}{x^2} \right)^{e(H)}
\left(\frac{x - x^{-1}}{y} \right)^{p(H)-1}
\left(\prod_{e\in E(H)} b_e \right).
\end{equation}
Now $H$ is a state of $F$ so we can regard it as a subsurface of $F$. The boundary $\partial H$ determines a descending link $L_H \subset F\times I$.
Recalling the correspondence between states of $F$ and resolutions of ${\mathcal{L}} (F)$ in the proof of Theorem~\ref{th:homfly}, it is clear that $L_H$ is a resolution of ${\mathcal{L}} (F)$. By (\ref{eq:skstate}) and (\ref{eq:basis}) this resolution contributes the expression
\[
\left( y/x\right)^{e(F)-e(H)}
\left( 1/x^2 \right)^{e(H)}
\left((x - x^{-1})/y \right)^{p(H)-1}
t_{L_H}.
\]
Comparing this to (\ref{eq:fullpf1}) we see that all that remains is to describe the map $f: \prod_{e\in E(H)} b_e \mapsto t_{L_H}$ which induces the map of the theorem.
Now the monomial $\prod_{e\in E(H)} b_e$ tells us which edges are in the state $H$ and therefore uniquely determines the subsurface $H$
of $F$ and the link diagram $L_H$ on the surface $\Sigma$.
We simply define $f$ to be the map which assigns the appropriate conjugacy class of $\pi_1 (\Sigma)$ and if a monomial contains an element $b_e$ more than once then send it to zero.
\end{proof}
\begin{remark}
It is a simple exercise to give an explicit construction of the map $f$ into the representatives of the conjugacy classes described in \cite{Li}. All that is required is the observation that any ribbon graph $F$ is homeomorphic to a decomposed surface of \cite{Li} then simply use this homeomorphism to pull back the link $L_H$ to the set of canonical conjugacy generators of the decomposed surface.
\end{remark}
\subsection{On Traldi's extension}
So far we have discussed the construction of links using Jaeger's idea of replacing an edge of the ribbon graph as in figure~\ref{fig:jaeger}. However in \cite{Tr} Traldi extended this idea by replacing the edges of a weighted graph by various tangles according to the weight. We use this idea to extend Theorem~\ref{th:fullhomfly}.
Let $F$ be a ribbon graph with edge weights in the set
$\{ b_e \}_{e\in E(F)}\times \{ w_1, \ldots , w_4 \}$.
Construct a link ${\mathcal{L}} (F)$ by associating a tangle to each edge according to the weight $w_i$ as indicated in figure~\ref{fig:traldi}.
\begin{figure}
\[\begin{array}{ccccccc}
\epsfig{file=figs/fig4a, height=1.2cm} &&
\epsfig{file=figs/fig4b, height=1.2cm} &&
\epsfig{file=figs/fig4c, height=1.2cm} &&
\epsfig{file=figs/fig4d, height=1.2cm}
\\
w_1 & &w_2 & &w_3 & &w_4
\end{array}\]
\caption{}
\label{fig:traldi}
\end{figure}
The following result then holds.
\begin{theorem}
Let $\Sigma$ be an orientable surface, possibly with boundary, and $G \subset \Sigma$ be an embedded graph
with edge weights
$\{ b_e \}_{e\in E(F)}\times \{ w_1, \ldots , w_4 \}$. Then
\begin{multline}\label{eq:traldi}
P({\mathcal{L}}(G)) = f \left(
\begin{array}{c}
\left( y/(x-x^{-1}) \right)
\left( y/x \right)^{w_1(G)}
\left( -yx \right)^{w_2(G)}
\left( 1/x^2 \right)^{w_3(G)}
\left( x^2 \right)^{w_4(G)}
\end{array}
\right. \\
\left.
\begin{array}{c}
B\left(G; 1 , \; \left( b_e w_e \right)_{e\in E(G)},\; (x - x^{-1})/y\right)
\end{array}
\right)
\end{multline}
where $w_1 = 1/xy$, $w_2=-x/y$, $w_3=xy$ and $w_4=-y/x$;
$f$ is a function from polynomials in $\textbf{b}$ to polynomials in the conjugacy classes $S(\Sigma) =\{t_w\}$ of $\pi_1 (\Sigma)$ and $w_i(G)$ denotes the number of edges of $G$ with the edge weight of the form $(\cdot , w_i)$.
\end{theorem}
The proof of this result is a straight forward generalization of the proof of Theorem~\ref{th:fullhomfly} and is therefore excluded.
Notice that Traldi's Theorem~5 in \cite{Tr} and also Theorem~\ref{th:fullhomfly} (by setting all $w_e = w_1$) may be recovered from this.
\subsection{Determination of the Bollob\'{a}s-Riordan polynomial and a duality relation}
For planar graphs, Jaeger showed that the Tutte polynomial of a graph $G$ and the HOMFLY polynomial of the associated link ${\mathcal{L}} (G)$ are equivalent on domain $\{ (u,v)\in \mathbb{R}^2 | u \neq 0 \}$. Since the Bollob\'{a}s-Riordan polynomial is a generalization of the Tutte polynomial for non-planar embedded graphs, it is natural to ask if the HOMFLY polynomial of ${\mathcal{L}} (F)$ determines the Bollob\'{a}s-Riordan polynomial of $F$. We will see that this isn't quite the case.
To answer this question we need to find non-zero values of $x$ and $y$ such that
$\alpha = x^2-1$, $\beta = (x-x^{-1})/xy^2$ and $\gamma = y/(x-x^{-1})$. We see that this is possible for the choices $x= \sqrt{\alpha+1} \neq 1$ and
$y= \sqrt{\alpha/(\beta(\alpha +1))}$. Notice that $\alpha \beta \gamma^2 =1$.
The proof of Proposition~\ref{pr:det} then follows immediately.
\begin{proposition}\label{pr:det} For any ribbon graph $F$ and $\alpha \neq 0,1$, $\beta \neq 0$
\begin{multline}\label{pr:determination}
R\left( F; \alpha, \beta , 1/\sqrt{\alpha \beta} \right) =
\left( \sqrt{\alpha}/ \sqrt{\beta} \right)^{v(F)-e(F)-1}
(\alpha+1)^{e(F)} \alpha^{1-k(F)} \\
P\left({\mathcal{L}}(F); \sqrt{\alpha-1} , \sqrt{\frac{\alpha}{\beta (\alpha+1)}} \right).
\end{multline}
\end{proposition}
We observe the following intriguing application of the above to knot theory.
We say that a link $L \subset \Sigma \times I$ is {\em essential} in $\Sigma$ if there is no embedded surface $\Sigma^{\prime} \subset \Sigma$, whose genus is smaller than that of $\Sigma$, such that $L \subset \Sigma^{\prime}\times I$.
Also we say that a link $L \subset \Sigma \times I$ is {\em split} if it is isotopic to a link $L^{\prime}$ with the property that $ L^{\prime} \cap (\Sigma \times [0,1/2]) \neq \emptyset$, $L^{\prime} \cap (\Sigma \times [1/2,1])\neq \emptyset$ and $L^{\prime} \cap (\Sigma \times \{ 1/2 \}) = \emptyset$.
\begin{corollary}
Let $L \subset \Sigma \times I$ be a non-split alternating link essential in $\Sigma$. If $L$ has a projection of the form ${\mathcal{L}}(G)$ for some graph $G \subset \Sigma$, then the genus of $\Sigma$ can be recovered from the HOMFLY polynomial of $L$.
\end{corollary}
\begin{proof}
Since $L$ has a projection of the form ${\mathcal{L}}(G)$, $G \subset \Sigma$, and the HOMFLY polynomial is equivalent to the Bollob\'{a}s-Riordan polynomial on the surface $\{ (x,y,z)|xyz^2=1 \}$, we see it is enough to show that the genus of $\Sigma$ can be recovered from $R\left( G; \alpha , \beta, 1/\sqrt{\alpha\beta} \right)$.
Now by \cite{LM},
\[ R\left( G; \alpha , \beta, 1/\sqrt{\alpha\beta} \right)
= \sum_{H\in {\mathcal{S}}(F)}\alpha^{r(F)-r(H)}\beta^{n(H)} (\alpha \beta)^{-g(H)}.
\]
By setting $\beta=1$ and observing that $G$ must be connected the above sum becomes \\ $\sum_{H\in {\mathcal{S}}(G)} \alpha^{k(H)-1 -g(H)}$. Therefore the lowest degree of $\alpha$ is equal to $-g(\Sigma)$ (since $L$ is essential in $\Sigma$), and we are done.
\end{proof}
This corollary motivates the following conjecture.
\begin{conjecture}
Let $L \subset \Sigma \times I$ be a link. Then the genus of an essential surface for $L$ can be recovered from its HOMFLY polynomial $P(L)$.
\end{conjecture}
\begin{remark}
We consider the 2-variable HOMFLY skein. However there is also a 3-variable version of the skein relation. It seems reasonable to conjecture that the 3-variable HOMFLY polynomial has the basis
$ t_L \left( -(x+y)/z \right)^{k(L)-1}$ on descending links. In which case one could make similar arguments as above and relate the 3-variable HOMFLY and Bollob\'{a}s-Riordan polynomials. However, with regards to the proposition above, the additional variable in the HOMFLY does not provide any further information about the Bollob\'{a}s-Riordan polynomial. In fact all that happens is that we would introduce a redundant third variable into the right hand side of equation~\ref{pr:determination}. This is similar to Proposition~2 of \cite{Ja}.
\end{remark}
\medskip
Given a graph $G$ embedded in a 2-manifold without boundary, one can form a {\em dual} embedded graph $G^*$ in the usual way. From this we can obtain the dual $F^*$ of a ribbon graph $F$. In the remainder of this section we study the relation between the Bollob\'{a}s-Riordan polynomial of a ribbon graph and its dual.
As far as the author is aware, there is no known duality relation for the full 3-variable Bollob\'{a}s-Riordan polynomial. In \cite{BR}, Bollob\'{a}s and Riordan prove a 1-variable relation
$R(F; \alpha, \alpha ,\alpha^{-1}) = R(F^*; \alpha, \alpha ,\alpha^{-1})$ leaving it as an open problem to find a multi-variable relation. Ellis-Monaghan and Sarmiento in \cite{EMS} extended this to the 2-variable relation:
\begin{theorem}[\cite{EMS}] \label{th:duality}
Let $F$ be a connected ribbon graph and $F^*$ its dual. Then
\[
R(F; \alpha, \beta , 1/\sqrt{\alpha \beta}) = \left( \beta /\alpha\right)^{g(F)} R(F^*; \beta, \alpha , 1/\sqrt{\alpha \beta}),\]
where $g(F)$ is the genus of the ribbon graph regarded as a surface.
\end{theorem}
One notices immediately that the specialization of the Bollob\'{a}s-Riordan polynomial in this relation is exactly that which is determined by the HOMFLY polynomial in (\ref{pr:determination}). We will provide a new proof for the duality relation above using knot theory and we will see that the duality relation holds for the specialization $R(F; \alpha, \beta , 1/\sqrt{\alpha \beta})$ precisely because it is determined by the HOMFLY polynomial.
\begin{proof}[Proof of Theorem~\ref{th:duality}.]
Let ${\mathcal{L}}_1 (F)$ be the link associated the ribbon graph $F$ by associating tangles $w_1$ of figure~\ref{fig:traldi} to edges of $F$ and let ${\mathcal{L}}_3 (F^*)$ be the link associated the ribbon graph $F^*$ by associating tangles $w_3$ of figure ~\ref{fig:traldi} to edges and reversing the orientation of all components of the link. Clearly these two links are isotopic and therefore
$P({\mathcal{L}}_1 (F))=P({\mathcal{L}}_3 (F^*))$.
Now by (\ref{eq:homfly}) we have
\begin{equation*}
P({\mathcal{L}}_1(F); x,y ) = \left( xy \right)^{v(G)-1}
\left( 1/x^2\right)^{e(G)}
\; R\left(F; x^{2}-1 , \; \frac{x-x^{-1}}{xy^2},\; \frac{y}{x - x^{-1}}\right),
\end{equation*}
and since the reversal of the orientation of a link does not change its HOMFLY polynomial, equation~\ref{eq:traldi} gives
\begin{equation*}
P({\mathcal{L}}_3(F^*); x,y ) = \left( \frac{1}{xy} \right)^{v(G)-1}
\left( \frac{y}{x}\right)^{e(G)}
\; R\left(F^*; \; \frac{x-x^{-1}}{xy^2}, x^{2}-1 , \; \frac{y}{x - x^{-1}}\right).
\end{equation*}
Then, by the isotopy of the link, we have
\begin{multline*}
R\left(F; x^{2}-1 , \; \frac{x-x^{-1}}{xy^2},\; \frac{y}{x - x^{-1}}\right) = \\
\left( xy \right)^{v(F)+v(F^*)-e(F)-2}
R\left(F^*; \; \frac{x-x^{-1}}{xy^2}, x^{2}-1 , \; \frac{y}{x - x^{-1}}\right).
\end{multline*}
Notice that since $v(F^*)$ equals the number of faces of $F^*$ (embedded in a non-punctured surface), we have $ v(F)+v(F^*)-e(F)-2 = \chi (F) -2 =-2g(F)$.
Finally the substitutions
$x= \sqrt{\alpha+1} \neq 1$ and $y= \sqrt{\alpha/(\beta(\alpha +1))}$ give the relation
\[
R(F; \alpha, \beta , 1/\sqrt{\alpha \beta}) = \left( \beta /\alpha\right)^{g(F)} R(F^*; \beta, \alpha , 1/\sqrt{\alpha \beta})
\]
as required.
\end{proof}
\section{The Jones polynomial}\label{sec:5}
Given a digraph $G$, a graph $H$ and a distinguished oriented edge of $H$. The {\em tensor product} $G \otimes H$ is defined to be the graph obtained by identifying each edge of $G$ with the distinguished edge of a copy of $H$ and then deleting each of the edges of $G$ ({\em i.e.} we take the 2-sum with $H$ at every edge of $G$). In general the graph obtained depends upon the various choices made. However this is not always the case. One example for which the tensor product is well defined is when $H$ is the $p$-cycle $C_p$. In this case notice also that the tensor product is independent of the orientation of the edges of $G$ and so the tensor product makes sense for (embedded) graphs $G$.
Tensor products and their effect on the Tutte polynomial have been considered previously in \cite{JVW} and \cite{Wo}.
In this section we consider the connection between the tensor product $G\otimes C_3$, Theorem~\ref{th:homfly} and a result of Chmutov and Pak which relates the Jones polynomial of a link in $F\times I$ to the Bollob\'{a}s-Riordan polynomial. This generalizes results of Huggett which appeared in \cite{Hu}.
So far we have constructed oriented links from embedded graphs by replacing each edge with a tangle as in figures~\ref{fig:jaeger}~and~\ref{fig:traldi}. However an unoriented link can be associated to an embedded graph by replacing each edge with the tangle indicated in figure~\ref{fig:medial}. This is known as the {\em medial link} and we denote the medial link associated with a graph $G$ by $L(G)$.
\begin{figure} \begin{center}
\epsfig{file=figs/fig5, height=1.2cm} \end{center}
\caption{}
\label{fig:medial}
\end{figure}
The Jones polynomial $J(L)$ of a link $L$ is defined by the skein
$t^{-1} J(L_+) - t J(L_-) = (t^{1/2} - t^{-1/2})J(L_0)$ and $J(\mathcal{O})=1$.
There is a well known formula relating the Tutte polynomial $T(G;-t, -t^{-1})$ of a planar graph $G$ to the Jones polynomial of its medial link. This result was recently generalized by Chmutov and Pak who related the Bollob\'{a}s-Riordan polynomial of a ribbon graph $F$ and the Jones polynomial of its medial link $L(F) \subset F\times I$:
\begin{theorem}[\cite{CP}]
Let $F$ be a ribbon graph and $L(F) \subset F\times I$ be its medial link then
\begin{multline}\label{eq:cp}
J(L(F); t)= (-1)^{\omega}t^{(3\omega - r(F) +n(F))/4} (-t^{1/2} - t^{-1/2})^{k(F)-1} \\
R\left(F; -t-1 , -t^{-1}-1, 1/(-t^{1/2} - t^{-1/2})\right),
\end{multline}
where $\omega$ is the writhe of the link (the writhe is the number of $L_+$ crossings minus the number of $L_-$ crossings, where $L_{\pm}$ are as in figure~\ref{fig:skein}.
\end{theorem}
Chmutov and Pak proved this result by considering the Kauffman bracket construction of the Jones polynomial.
Of course the Jones polynomial is the specialization of the HOMFLY polynomial at $x=t^{-1}$ and $y=t^{1/2}-t^{-1/2}$ and so by (\ref{eq:homfly}) we also have
\begin{multline}\label{eq:joho}
J(\mathcal{L}(F);t) =
(t^{1/2}-t^{-1/2})^{(e(F)-v(F)+1 )} t^{(e(F)+v(F)-1)}
(t^{-2}-1)^{k(F)-1} \\
\; R\left( F; \, t^{-2}-1 , \, \frac{1-t^{2}}{(t^{1/2}- t^{-1/2})^2} , \,
\frac{1}{-t^{1/2}-t^{-1/2}} \right).
\end{multline}
Now tensors return to the story. It is clear that, forgetting the orientations, the links $L(F\otimes C_3)$ and the mirror image of ${\mathcal{L}}(F)$ (obtained by reversing all of the crossings) are isotopic. Therefore $J(L(F\otimes C_3); t^{-1}) = J(\mathcal{L}(F);t)$ (since taking the mirror image of a link changes the Jones polynomial by the substitution $t \mapsto t^{-1}$), where the orientation on $L(F\otimes C_3)$ is induced by that of ${\mathcal{L}}(F)$. We use this observation to prove the following formula for the tensor product of a ribbon graph with $C_3$.
\begin{theorem}\label{th:tens}
Let $F$ be a ribbon graph; $\alpha \neq -2, 0, 1$; $\beta = \alpha (1-\alpha)$ and $\gamma = 1/ \sqrt{\alpha\beta}$. Then
\[R(F\otimes C_3;\alpha, \beta, \gamma) = (\alpha +2)^{n(F)} R(F; \alpha (\alpha +2) ,\beta/(\alpha +2) , \gamma). \]
\end{theorem}
\begin{proof}
Consider the links ${\mathcal{L}} (F)$ and $L=L(F\otimes C_3)$ described above. We will use the relation between their Jones polynomials to prove the result. For brevity we set $A:=F \otimes C_3$.
By (\ref{eq:cp}) we have
\begin{multline*}
J(L(A); t) =
(-1)^{\omega}t^{(3\omega(L) - r(A) +n(A))/4} (-t^{1/2} - t^{-1/2})^{k(A)-1} \\
R\left(A; -t-1 , -t^{-1}-1, 1/(-t^{1/2} - t^{-1/2})\right).
\end{multline*}
Clearly $\omega(L) = -e(A)=-2e(F)$, $e(A)=2e(F)$ and $v(A)=v(F)+e(F)$ and the above becomes
\begin{equation}\label{eq:proof1}
t^{(-3e(F) -v(F) +k(F))/2} (-t^{1/2} - t^{-1/2})^{k(G)-1}
R\left(A; -t-1 , -t^{-1}-1, 1/(-t^{1/2} - t^{-1/2})\right).
\end{equation}
On the other hand equation~\ref{eq:joho}, coming from the HOMFLY polynomial gives
\begin{multline*}
J({\mathcal{L}}(A); t^{-1}) = (t^{2}+1)^{k(F)-1}
t^{(-e(F) -v(F) +1)/2} (-t^{1/2}+t^{-1/2})^{e(F)-v(F)+1} \\
R\left( F; \, t^{2}-1 , \, \frac{1-t^{-2}}{(t^{-1/2}- t^{1/2})^2} -1 , \,
\frac{1}{-t^{1/2}-t^{-1/2}} \right),
\end{multline*}
which can be written as
\begin{multline}\label{eq:proof2}
(t^{2}+1)^{k(F)-1}
t^{(-e(F) -v(F) +1)/2} (-t^{1/2}+t^{-1/2})^{e(F)-v(F)+1} \\
R\left( F; \, t^{2}-1 , \, \frac{1-t^{-2}}{(t^{-1/2}- t^{1/2})^2} -1 , \,
\frac{1}{-t^{1/2}-t^{-1/2}} \right).
\end{multline}
As observed above, we have $J(L(F\otimes C_3); t) = J(\mathcal{L}(F);t^{-1})$ and therefore, using (\ref{eq:proof1}) and (\ref{eq:proof2}), we have
\begin{multline*}
R\left(A; -t-1 , -t^{-1}-1, 1/(-t^{1/2} - t^{-1/2})\right) = \\
(-t+1)^{n(F)}
R\left( F; \, t^{2}-1 , \, \frac{1-t^{-2}}{(t^{-1/2}- t^{1/2})^2} -1 , \,
\frac{1}{-t^{1/2}-t^{-1/2}} \right).
\end{multline*}
Finally, substituting $t = 1-\alpha$ gives
\[R(A;\alpha, \beta, \gamma) = (\alpha +2)^{n(F)} R(F; \alpha (\alpha +2) ,\beta/(\alpha +2) , \gamma) , \]
where $\beta$ and $\gamma$ are as in the statement of the lemma.
\end{proof}
Induction gives the following corollary.
\begin{corollary}
Let $F$ be a ribbon graph; $\alpha \neq -2, 0, 1$; $\beta = \alpha (1-\alpha)$ and $\gamma = 1/ \sqrt{\alpha\beta}$. Then
\[R(F\otimes C_{2^{p}+1};\alpha, \beta, \gamma) = \left(\sum_{i=0}^{2^p} (\alpha +1)^i \right)^{n(F)}
R\left(F; \; (\alpha +1)^{2^p}-1 , \; \frac{\beta}{\sum_{i=0}^{2^p} (\alpha +1)^i } , \; \gamma \right). \]
\end{corollary}
\begin{remark}
In fact, it is not difficult to prove that for any $\alpha$, $\beta$ and $\gamma$,
$R(F\otimes C_3;\alpha, \beta, \gamma) = (\alpha +2)^{n(F)} R(F; \alpha (\alpha +2) ,\beta/(\alpha +2) , \gamma)$. Taking this as our starting point, we see that formula~\ref{eq:cp}, which is proven by considering the Kauffman bracket, and formula~\ref{eq:joho} are related through the notion of the tensor product of an embedded graph. This generalizes the main result of \cite{Hu}.
\end{remark}
|
1,116,691,499,257 | arxiv | \section{Introduction}
By $\mathbb{D}$ be the open unit disc in the complex plane $\mathbb{C}$, $H(\mathbb{D})$ is denoted the space of all analytic functions on $\mathbb{D}$.
The classic {Zygmund space} $\mathcal{Z}$ consists of all functions $f \in H(\mathbb{D})$ which are continuous on the closed unit ball
$ \overline{\mathbb{D}}$ and
$$\sup \frac{|f(e^{i(\theta +h)}) + f(e^{i(\theta -h)}) - 2 f (e^{i \theta})|}{h} < \infty,$$
where the supremum is taken over all $\theta \in \mathbb{R}$ and $h> 0$.
By \cite[Theorem 5.3]{peter}, an analytic function $f$ belongs to $\mathcal{Z}$ if and only if $\sup_{z\in \mathbb{D}} (1-|z|^2) |f''(z)| < \infty$.
Motivated by this, for each $\alpha>0$, the {Zygmund type space} $\mathcal{Z}_{\alpha}$ is defined to be the space of all functions $f \in H(\mathbb{D})$ for which
$$\|f\|_{s\mathcal{Z}_{\alpha}} = \sup_{z\in \mathbb{D}} (1-|z|^2)^{\alpha} |f''(z)| < \infty.$$
The space $\mathcal{Z}_{\alpha}$ is a Banach space equipped with the norm
$$\|f\|_{\mathcal{Z}_{\alpha}} = |f(0)| + |f'(0)| + \|f\|_{s\mathcal{Z}_{\alpha}},$$
for each $f\in \mathcal{Z}_{\alpha}$.
Let $Aut(\mathbb{D})$ be the group of all conformal automorphisms of $\mathbb{D}$ which is also called M\"{o}bius group. It is well-known that
each element of $Aut(\mathbb{D})$ is of the form
$$ e^{i \theta} \sigma_a (z)= e^{i \theta} \frac{a-z}{1- \overline{a}z}, \ \ \ \ a,z \in \mathbb{D}, \ \ \ \ \theta \in \mathbb{R}. $$
Let $X$ be a linear space of analytic functions on $\mathbb{D}$, which is complete. $X$ is called M\"{o}bius invariant if for each function $f$ in $X$ and each element $\psi$ in $Aut(\mathbb{D})$, the composition function $f \circ \psi$ also lies in $X$ and satisfies that
$||f \circ \psi||_X = ||f||_{X}$. For example, the space $H^{\infty}$ of all bounded analytic functions is M\"{o}bius invariant.
Also the Besov space $B_p(1 < p < \infty)$, is M\"{o}bius invariant which is the space of all $f \in H(\mathbb{D})$ such that
\begin{equation}\label{e10}
\int_{\mathbb{D}} |f'(z)|^p (1-|z|^2)^{p-2} dA(z) < \infty.
\end{equation}
If $p= 2$, we have the well-known Dirichlet space. For $p=\infty$, $B_{\infty} = \mathcal{B}$ the classic Bloch space. The space
$B_1$ which is called the minimal M\"{o}bius invariant is defined separately. The function $f \in H(\mathbb{D})$ belongs to
$B_1$ if and only if it has representation as
$$ f(z) = \sum_{k=1}^{\infty} c_k \sigma_{a_k} \quad \mbox{where} \ \ \ a_k \in \mathbb{D} \ \ \mbox{and }\ \ \sum_{k=1}^{\infty} |c_k| < \infty. $$
The norm on $B_1$ defines as infimum of $\sum_{k=1}^{\infty} |c_k| $ for which the above statement holds. $B_1$ is contained in any
M\"{o}bius invariant space and it has been proved that is the set of all analytic functions $f$ on $\mathbb{D}$ such that
$f''$ lies in $L^1 (\mathbb{D}, dA)$. Also there exist positive constants $C_1$ and $C_2$ such that
\begin{equation}\label{e11}
C_1 ||f||_{B_1} \leq |f(0)| + |f'(0)| + \int_{\mathbb{D}} |f''(z)| dA(z) \leq C_2 ||f||_{B_1}.
\end{equation}
Let $u, v, \varphi \in H(\mathbb{D})$ and $\varphi: \mathbb{D} \rightarrow \mathbb{D}$. The Stevi\'c-Sharma type operator is defined as follows
\begin{align*}
T_{u, v, \varphi}f(z) = u(z) f{(\varphi(z))}+ v(z) f'(\varphi(z)) ,\quad f\in H(\mathbb{D} ), \quad z\in \mathbb{D}.
\end{align*}
Indeed $T_{u, v, \varphi}= uC_\varphi+vC_\varphi D$ where $D$ is the differentiation operator and $C_\varphi$ is composition operator.
More information about this operator can be found in \cite{ly, s1, s2}.
The generalized Stevi\'c-Sharma type operator $T^m_{u,v,\varphi}$ is defined by the second auther of this paper and et al. in \cite{eym} as follows
\begin{align*}
T^m_{u, v, \varphi}f(z)
= (u C_\varphi f)(z) + (D^m_{\varphi,v} f)(z) = u(z) f(\varphi(z)) + v(z) f^{(m)}(\varphi(z)),
\end{align*}
where $m\in\mathbb{N}$ and $D^m_{\varphi,u}$ is
the generalized weighted composition operator. When $v=0$, then $T^m_{u, 0, \varphi} = u C_\varphi $ is the well-known weighted composition operator. If $u = 0,$ then $T^m_{0, v, \varphi} = D^m_{\varphi,v}$ and for $m=1$, $T^m_{u, v, \varphi} = T_{u, v, \varphi}$ is Stevi\'c-Sharma type operator.
$T^m_{u, v, \varphi}$ also include other operators as well as product type operators which are studied in several papers in recent years.
The results of the papers can be stated to many operators and obtained the results of the papers published before.
For Banach spaces X and Y and a continuous linear operator $T: X\rightarrow Y$, the essential norm is the distance of $T$ from the space of all compact operators, that is
\[
\|T\|_{e} = \inf\{ \| T-K\| : K: X\rightarrow Y\hspace{0.1cm} \text{is compact}\}.
\]
$T$ is compact if and only if $\|T\|_e =0$.
In this paper, we study the operator-theoretic properties in minimal M\"{o}bius invariant space.
In section 2, we first bring some lemmas on the space $B_1$ and then obtain some characterizations for boundedness of operator $T^m_{u,v,\varphi}: B_1\rightarrow \mathcal{Z}_{\alpha}$. In section 3, some estimations for the essential norm of these operators are given. As a result, some new criteria for the compactness of $T^m_{u,v,\varphi}$ are presented.
By $A\succeq B$ we mean there exists a constant $C$ such that $A\geq CB$ and $A\approx B$ means that $A\succeq B\succeq A$.
\section{Boundedness}
In this section, we give some necessary and sufficient conditions for the generalized Stevi\'c-Sharma type operators to be bounded.
Firstly, we state some lemmas which are needed for proving the main results.
According to the definition of the norm in minimal M\"{o}bius invariant space, for each $f \in B_1$, $||f||_{\infty} \leq ||f||_{B_1}$. Thus, from
\cite[Proposition 5.1.2]{far11} and \cite[Proposition 8]{far12} we have the following lemma.
\begin{lemma} \label{l10}
Let $n \in \mathbb{N}$. Then there exists a positive constant $C$ such that for each $f \in B_1$
$$ (1-|z|^2)^n |f^{(n)} (z)| \leq C ||f||_{B_1}. $$
\end{lemma}
As a similar proof in Lemma 2.5 of \cite{{xea1}} we get the following lemma.
\begin{lemma} \label{l11}
Let
$$ f_{j,a}(z)= \left ( \frac{1-|a|^2}{1-\overline{a}z} \right)^j, \ \ \ \ j \in \mathbb{N}, \ a \in \mathbb{D}. $$
Then
$$ f_{j,a} \in B_1, \ \ \ \ \sup_{a \in \mathbb{D}} ||f_{j,a}||_{B_1} < \infty. $$
Moreover, $\{ f_{j,a} \} \rightarrow 0$ uniformly on compact subsets of $\mathbb{D}$ as $|a| \rightarrow 1$.
\end{lemma}
The proof of the following lemmas are similar to the proof of Lemmas 2.6 and 2.5 \cite{eym}, so they are omitted.
\begin{lemma} \label{l12}
For any $m \in \mathbb{N}-\{1,2\}$, $0 \not = a \in \mathbb{D}$ and
$i,k \in \{ 0,1, 2, m, m+1, m+2 \}$, there exits a function $g_{i,a} \in B_1$ such that
$$ g_{i,a}^{(k)} (a) = \frac{\delta_{ik} \overline{a}^{k}}{(1-|a|^2)^k}, $$
where $\delta_{ik}$ is Kronecker delta. For each $i\in\{0,1,2\}$ and $i\in\{m,m+1,m+2\}$ respectively
\[
g_{i,a}(z) = \sum_{j=1}^{3} c_j^i f_{j,a} (z), \qquad\quad g_{i,a}(z) = \sum_{j=m+1}^{m+3} c_j^i f_{j,a} (z)
\]
where $c_j^i$ is independent of $a$.
\end{lemma}
\begin{lemma} \label{m123}
Let $m=1$ or $2$, $0 \not = a \in \mathbb{D}$ and
$i,k \in \{ 0,1, \cdots, m+2 \}$, there exists a function $g_{i,a} \in B_1$ such that
$$ g_{i,a}^{(k)} (a) = \frac{\delta_{ik} \overline{a}^{k}}{(1-|a|^2)^k}. $$
\end{lemma}
Let $f \in B_1$. Then
\begin{align*}
|| T^m_{u, v, \varphi} f ||_{\mathcal{Z}_{\alpha}} = |T^m_{u, v, \varphi} f (0)| + |(T^m_{u, v, \varphi} f)' (0)| + \sup_{z \in \mathbb{D}}(1-|z|^2)^{\alpha} | (T^m_{u, v, \varphi} f)'' (z) |.
\end{align*}
We compute the above sentences separately. We have
\begin{align}\label{e12}
& (T^m_{u, v, \varphi} f)' (0) =\\
&u'(0) f(\varphi(0)) + u(0) \varphi'(0) f'(\varphi(0)) + v'(0) f^{(m)} (\varphi(0)) + v (0) \varphi'(0) f^{(m+1)}(\varphi(0)).\nonumber
\end{align}
And
\begin{align*}
(T^m_{u, v, \varphi} f)'' (z) = & \sum_{i=0}^2 \big(I_i(z) f^{(i)}(\varphi(z))+ I_{i+m}(z) f^{(i+m)}(\varphi(z))\big),
\end{align*}
where,
$$ I_0 = u'' \ \ \ \ I_1 = 2 u'\varphi' + u \varphi'', \ \ \ \ I_2 =u \varphi'^2 $$
$$ I_m = v'' \ \ \ \ I_{m+1} = 2 v'\varphi' + v \varphi'' \ \ \ \ I_{m+2} = v \varphi'^2 $$
\begin{theorem}\label{th10}
Let $\alpha>0$, $u, v, \varphi \in H(\mathbb{D})$, $\varphi: \mathbb{D} \rightarrow \mathbb{D}$ and $m > 2$ be an integer. Then the following conditions are equivalent:
\begin{enumerate}
\item [(i)]
The operator $T^m_{u, v, \varphi} : B_1 \rightarrow \mathcal{Z}_{\alpha} $ is bounded.
\item [(ii)] For each $ j \in \{0,1,2,m,m+1, m+2 \}=\mathfrak{Q}$
\[
\max\{\sup_{a \in \mathbb{D}} ||T_{u,v,\varphi}^m f_{j,a}||_{\mathcal{Z}_{\alpha}},\ \ \sup_{z \in \mathbb{D}} (1-|z|^2)^{\alpha} |I_j (z)| \}< \infty.
\]
\item [(iii)]
$$ \sup_{z \in \mathbb{D}} \frac{(1-|z|^2)^{\alpha} |I_k (z)|}{(1-|\varphi(z)|^2)^{k}} < \infty, \ \ \ \ k \in\mathfrak{Q}. $$
\end{enumerate}
\end{theorem}
\begin{proof}
$(iii)\Rightarrow (i)$ Suppose that $f \in B_1$. By using Lemma \ref{l10}
\begin{align*}
(1-|z|^2)^{\alpha} |(T^m_{u, v, \varphi} f)'' (z)| = & (1-|z|^2)^{\alpha} |\sum_{k \in\mathfrak{Q}} I_k (z) f^{(k)}(\varphi(z)) | \\
\leq &\sum_{k \in\mathfrak{Q}} |I_k (z)| (1-|z|^2)^{\alpha} | f^{(k)}(\varphi(z))| \\
\leq & C \sum_{k \in\mathfrak{Q}} \frac{(1-|z|^2)^{\alpha}|I_k (z) |}{(1-|\varphi(z)|^2)^{k}} ||f||_{B_1}.
\end{align*}
Also using the fact that $||f||_{\infty} \leq ||f||_{B_1}$ and Lemma \ref{l10}, we have
\begin{align*}
|T^m_{u, v, \varphi} f (0)| \leq & |u(0) f(\varphi(0))| + |v(0) f^{(m)} (\varphi(0))| \\
\leq & |u(0)| ||f||_{B_1} + C \frac{|v(0)|}{(1-|\varphi(0)|^2)^{m}} ||f||_{B_1}
\end{align*}
and
\begin{align*}
|(T^m_{u,v,\varphi}f)' (0)|
\leq C\left (|u'(0)|+\frac{|u(0) \varphi'(0)|}{1-|\varphi(0)|^2} + \frac{|v'(0)|}{(1-|\varphi(0)|^2)^{m}} + \frac{|v (0)\varphi'(0)|}{(1-|\varphi(0)|^2)^{m+1}} \right) ||f||_{B_1}.
\end{align*}
Therefore $T^m_{u, v, \varphi} : B_1 \rightarrow \mathcal{Z}_{\alpha} $ is bounded.
$(i)\Rightarrow (ii)$ Suppose that $T^m_{u, v, \varphi} : B_1 \rightarrow \mathcal{Z}_{\alpha} $ be a bounded operator. Lemma \ref{l11} implies that $||f_{j,a}||_{B_1} < \infty$. So
$$ ||T_{u,v,\varphi}^m f_{j,a}||_{\mathcal{Z}_{\alpha}} \leq
|| T_{u,v,\varphi}^m || ||f_{j,a}||_{B_1} < \infty. $$
Then
$$ \sup_{a \in \mathbb{D}} ||T_{u,v,\varphi}^m f_{j,a}||_{\mathcal{Z}_{\alpha}} \leq || T_{u,v,\varphi}^m || \sup_{a \in \mathbb{D}, j\in\mathfrak{Q}} ||f_{j,a}||_{B_1} < \infty. $$
Define $f_0 (z) =1 \in B_1$. Boundedness of the operator implies that
\begin{align*}
\sup_{z \in \mathbb{D}} (1-|z|^2)^{\alpha} |I_0 (z)| = \sup_{z \in \mathbb{D}} (1-|z|^2)^{\alpha} |u''(z)| \leq ||T_{u,v,\varphi}^m f_0||_{\mathcal{Z}_{\alpha}} \leq || T_{u,v,\varphi}^m || ||f_0||_{B_1} < \infty.
\end{align*}
Take $f_1 (z) = z \in B_1$. Then we have
\begin{align*}
\sup_{z \in \mathbb{D}} (1-|z|^2)^{\alpha} |u''(z) \varphi(z) + 2 u'(z) \varphi'(z) + u (z) \varphi''(z)| \leq ||T_{u,v,\varphi}^m f_1||_{\mathcal{Z}_{\alpha}} \leq || T_{u,v,\varphi}^m || ||f_1||_{B_1} < \infty.
\end{align*}
Using the previous equations, we can get that
\begin{align*}\label{e15}
\sup_{z \in \mathbb{D}} (1-|z|^2)^{\alpha} |I_1 (z)|< \infty.
\end{align*}
Similarly by employing the functions $f_2 (z) = z^2$, $f_m (z) = z^m$, $f_{m+1} (z) = z^{m+1}$ and $f_{m+2} (z) = z^{m+2}$ for operator $ T_{u,v,\varphi}^m$ we get the other part of $(ii)$.
$(ii)\Rightarrow (iii)$ For any $i \in\mathfrak{Q}$ and $a \in \mathbb{D}$, by applying Lemma \ref{l12} we have
\begin{align*}
\frac{(1-|z|^2)^{\alpha} |I_i (a)| |\varphi(a)|^i}{(1-|\varphi(a)|^2)^i} \leq & (1-|z|^2)^{\alpha} |(T_{u,v,\varphi}^m g_{i, \varphi(a)})''(a)| \\
\leq & \sup_{a \in \mathbb{D}} ||T_{u,v,\varphi}^m g_{i, \varphi(a)}||_{\mathcal{Z}_{\alpha}} \\
\leq &\max\{\sum_{j=1}^{3} c_j^i \sup_{a \in \mathbb{D}} ||T_{u,v,\varphi}^m f_{j,a}||_{\mathcal{Z}_{\alpha}}, \ \
\sum_{j=m+1}^{m+3} c_j^i \sup_{a \in \mathbb{D}} ||T_{u,v,\varphi}^m f_{j,a}||_{\mathcal{Z}_{\alpha}}\}\\ <& \infty.
\end{align*}
So, for any $i \in\mathfrak{Q}$
\begin{equation}\label{e16}
\sup_{|\varphi(a)| > 1/3} \frac{(1-|z|^2)^{\alpha} |I_i (a)| }{(1-|\varphi(a)|^2)^i} < \infty.
\end{equation}
On the other hand
\begin{equation}\label{e17}
\sup_{|\varphi(a)| \leq 1/3} \frac{(1-|z|^2)^{\alpha} |I_i (a)| }{(1-|\varphi(a)|^2)^i} \leq C
\sup_{a \in \mathbb{D}} (1-|z|^2)^{\alpha} |I_i (a)| < \infty.
\end{equation}
From the last inequalities, we get the desired result.
\end{proof}
In the special case $m\leq2$, by using Lemma \ref{m123}, we have the following theorems which is stated without proof.
\begin{theorem}\label{th11}
Let $\alpha>0$, $u, v, \varphi \in H(\mathbb{D})$ and $\varphi: \mathbb{D} \rightarrow \mathbb{D}$. Then the following conditions are equivalent:
\begin{enumerate}
\item[(i)] The operator $T^2_{u, v, \varphi} : B_1 \rightarrow \mathcal{Z}_{\alpha} $ is bounded.
\item[(ii)] For $j\in\{1,...,5\}$, $ \sup_{a \in \mathbb{D}} ||T_{u,v,\varphi}^m f_{j,a}||_{\mathcal{Z}_{\alpha}} < \infty$ and
\begin{align*}
&\sup_{z \in \mathbb{D}} (1-|z|^2)^{\alpha}\left |u''(z)| +|( 2 u'\varphi' + u \varphi'')(z)|+|(u \varphi'^2 +v'')(z)|\right)< \infty,\\
&\sup_{z \in \mathbb{D}} (1-|z|^2)^{\alpha}\left |(2 v'\varphi' + v \varphi'')(z)|+|(v \varphi'^2)(z)|\right)<\infty.
\end{align*}
\item[(iii)]
\begin{align*}
&\sup_{z \in \mathbb{D}} (1-|z|^2)^{\alpha}\left( |u''(z)| + \frac{|( 2 u'\varphi' + u \varphi'')(z)|}{(1-|\varphi(z)|^2)}+\frac{|(u \varphi'^2 +v'')(z)|}{(1-|\varphi(z)|^2)^2}\right)< \infty,\\
&\sup_{z \in \mathbb{D}} (1-|z|^2)^{\alpha}\left( \frac{|(2 v'\varphi' + v \varphi'')(z)|}{(1-|\varphi(z)|^2)^3}+\frac{|(v \varphi'^2)(z)|}{(1-|\varphi(z)|^2)^4}\right)<\infty.
\end{align*}
\end{enumerate}
\end{theorem}
\begin{theorem}\label{th11z}
Let $\alpha>0$, $u, v, \varphi \in H(\mathbb{D})$ and $\varphi: \mathbb{D} \rightarrow \mathbb{D}$. Then the following conditions are equivalent:
\begin{enumerate}
\item[(i)] The operator $T_{u, v, \varphi} : B_1 \rightarrow \mathcal{Z}_{\alpha} $ is bounded.
\item[(ii)] For $j\in\{1,...,4\}$, $ \sup_{a \in \mathbb{D}} ||T_{u,v,\varphi}^m f_{j,a}||_{\mathcal{Z}_{\alpha}} < \infty$ and
\begin{align*}
&\sup_{z \in \mathbb{D}} (1-|z|^2)^{\alpha}\left |u''(z)| +|( 2 u'\varphi' + u \varphi'' +v'')(z)|\right)< \infty,\\
&\sup_{z \in \mathbb{D}} (1-|z|^2)^{\alpha}\left |(u \varphi'^2 + 2 v'\varphi' + v \varphi'')(z)|+|(v \varphi'^2)(z)|\right)<\infty.
\end{align*}
\item[(iii)]
\begin{align*}
&\sup_{z \in \mathbb{D}} (1-|z|^2)^{\alpha}\left( |u''(z)| + \frac{|( 2 u'\varphi' + u \varphi'' +v'')(z)|}{(1-|\varphi(z)|^2)}\right)< \infty,\\
&\sup_{z \in \mathbb{D}} (1-|z|^2)^{\alpha}\left( \frac{|(u \varphi'^2+2 v'\varphi' + v \varphi'')(z)|}{(1-|\varphi(z)|^2)^2}+\frac{|(v \varphi'^2)(z)|}{(1-|\varphi(z)|^2)^3}\right)<\infty.
\end{align*}
\end{enumerate}
\end{theorem}
\section{Essential Norm}
In this section, some estimations for the essential norm of the operator $T^m_{u,v,\varphi}$ from minimal M\"{o}bius invariant spaces into Zygmund type spaces are given.
\begin{theorem}\label{th15}
Let $u, v, \varphi \in H(\mathbb{D})$, $\varphi: \mathbb{D} \rightarrow \mathbb{D}$ and $2<m\in\mathbb{N}$. Let the operator $T^m_{u, v, \varphi} : B_1 \rightarrow \mathcal{Z}_{\alpha} $ be bounded then
$$ ||T^m_{u, v, \varphi}||_e \approx \max \{ E_i \}_{i=1}^6 \approx \max \{ F_k \}_{k \in \{0,1,2,m,m+1, m+2 \}}$$
where,
$$ E_i = \limsup_{|a| \rightarrow 1} ||T_{u,v,\varphi}^m f_{i,a}||_{\mathcal{Z}_{\alpha}} \ \mbox{and}\ \ F_k = \limsup_{|\varphi(z)| \rightarrow 1} \frac{(1-|z|^2)^{\alpha} |I_k (z)|}{(1-|\varphi(z)|^2)^{k}}.$$
\end{theorem}
\begin{proof}
First we prove the lower estimates. Suppose that $K : B_1 \rightarrow \mathcal{Z}_{\alpha} $ be an arbitrary compact operator. Since $\{ f_{i,a} \} $ is a bounded sequence in $B_1$ and converges to $0$ uniformly on compact subsets of $\mathbb{D}$ as $|a| \rightarrow 1$, we have
$\limsup_{|a| \rightarrow 1} ||K f_{i,a}||_{\mathcal{Z}_{\alpha}} =0.$
So
\begin{align*}
\|T^m_{u, v, \varphi}-K\|_{B_1 \rightarrow \mathcal{Z}_{\alpha}} &\succeq
\limsup_{|a|\rightarrow 1}\|(T^m_{u, v, \varphi}-K)f_{i,a}\|_{\mathcal{Z}_{\alpha}}=E_i.
\end{align*}
Then
\begin{align*}
\|T^m_{u, v, \varphi}\|_{e}= \inf _{K}\|T^m_{u, v, \varphi}-K\|_{B_1\rightarrow \mathcal{Z}_{\alpha}} \succeq
\max\{E_i\}_{i=1}^{6}.
\end{align*}
For the other part, let $\{z_j\}_{j\in\mathbb{N}}$ be a sequence in $\mathbb{D}$ such that $|\varphi(z_j)|\rightarrow 1$ as $j\rightarrow \infty$. Since $T^m_{u, v, \varphi} : B_1 \rightarrow \mathcal{Z}_{\alpha}$ is bounded, using Lemmas $\ref{l11}$ and $\ref{l12}$ for any compact operator $K: B_1 \rightarrow \mathcal{Z}_{\alpha}$ and $i \in \{0,1,2,m,m+1, m+2 \}$, we obtain
\begin{align*}
\| T^m_{u, v, \varphi}-K\|_{B_1 \rightarrow \mathcal{Z}_{\alpha}} &\succeq
\limsup_{j\rightarrow \infty}\| T^m_{u, v, \varphi}(g_{i, \varphi(z_j)})\|_{\mathcal{Z}_{\alpha}}-
\limsup_{j\rightarrow \infty}\| K(g_{i, \varphi(z_j)})\|_{\mathcal{Z}_{\alpha}}\nonumber\\
&\succeq \limsup_{j\rightarrow \infty}\frac{(1-|z_j|^2)^{\alpha} |\varphi(z_j)|{i} |I_i (z_j)|}{ (1- | \varphi(z_j)|^{2})^{i}}=F_i.
\end{align*}
Therefore,
\begin{align*}
\|T^m_{u, v, \varphi}\|_{e}= \inf _{K}\|T^m_{u, v, \varphi}-K\|_{B_1\rightarrow \mathcal{Z}_{\alpha}} \succeq
\max\{F_i\}.
\end{align*}
Now we prove the upper estimates. Consider the operators $K_r$ on $B_1$, $K_r f(z) = f_r(z) = f(rz)$, where $0 < r <1$. $K_r$ is a compact operator and $||K_r|| \leq 1$. Let $\{r_j\}\subset (0,1)$ be a sequence such that $r_j\rightarrow 1$ as $j\rightarrow \infty$.
For any positive integer $j$, the operator $T^m_{u,v,\varphi}K_{r_j} : B_1\rightarrow \mathcal{Z}_{\alpha}$ is compact. Thus
\begin{align}\label{e20}
\|T^m_{u,v,\varphi}\|_{e} \leq \limsup_{j\rightarrow \infty}\| T^m_{u,v,\varphi} - T^m_{u,v,\varphi}K_{r_j}\|.
\end{align}
So it will be sufficient to prove that
\begin{align*}
\limsup_{j\rightarrow \infty}\| T^m_{u,v,\varphi} - T^m_{u,v,\varphi}K_{r_j}\|\preceq \min\{ \max\{E_i\}, \max\{F_i\} \}.
\end{align*}
For any $f\in B_1$ such that $\|f\|_{B_1}\leq 1$,
\begin{align*}
||( T^m_{u, v, \varphi} -T^m_{u, v, \varphi} K_{r_j}) f ||_{\mathcal{Z}_{\alpha}} = A_1 + A_2 + A_3
\end{align*}
where
$$ A_1 =: \left|T^m_{u, v, \varphi} f (0) - T^m_{u, v, \varphi} f_{r_j}(0)\right| = \left|u(0) (f-f_{r_j})(\varphi(0)) + v(0) (f-f_{r_j})^{(m)}(\varphi(0))\right|$$
\begin{align*}
A_2=: & \left|(T^m_{u, v, \varphi} f - T^m_{u, v, \varphi}f_{r_j})' (0)\right|
=| u'(0)(f-f_{r_j})(\varphi(0)) + u(0) \varphi'(0) (f-f_{r_j})'(\varphi(0)) \\&+ v'(0) (f-f_{r_j})^{(m)}(\varphi(0)) + v(0) \varphi'(0) (f-f_{r_j})^{(m+1)}(\varphi(0)) |
\end{align*}
\begin{align*}
A_3 =: & \sup_{z \in \mathbb{D}}(1-|z|^2)^{\alpha} \left|(T^m_{u, v, \varphi} f - T^m_{u, v, \varphi} f_{r_j})'' (z)\right| \\
= & \sup_{z \in \mathbb{D}} (1-|z|^2)^{\alpha} \left|\sum_{k \in \{0,1,2,m,m+1, m+2 \}} I_k (z) (f-f_{r_j})^{(k)}(\varphi(z))\right| \\
\leq & \sup_{|\varphi(z)| \leq r_N} (1-|z|^2)^{\alpha} \sum_{k \in \{0,1,2,m,m+1, m+2 \}} \left|I_k (z) (f-f_{r_j})^{(k)}(\varphi(z)) \right| \\
& + \sup_{|\varphi(z)| > r_N} (1-|z|^2)^{\alpha} \sum_{k \in \{0,1,2,m,m+1, m+2 \}} \left|I_k (z) (f-f_{r_j})^{(k)}(\varphi(z)) \right| \\
=: & A_4 + A_5
\end{align*}
Since $(f-f_{r_j})^{(i)} \rightarrow 0$ uniformly on compact subsets of ${\mathbb{D}}$ as $j\rightarrow \infty$, for any nonnegative integer $i$, then using Theorem \ref{th10}, we get
$$ \limsup_{j \rightarrow \infty} A_1 =\limsup_{j \rightarrow \infty} A_2 =\limsup_{j \rightarrow \infty} A_4 =0. $$
About $A_5$, we get
\begin{align*}
A_5 \leq & \sum_{k \in \{0,1,2,m,m+1, m+2 \}} \sup_{|\varphi(z)| > r_N} (1-|z|^2)^{\alpha} |I_k (z)| | f^{(k)}(\varphi(z)) | \\
& + \sum_{k \in \{0,1,2,m,m+1, m+2 \}} \sup_{|\varphi(z)| > r_N} (1-|z|^2)^{\alpha} |I_k (z)| | r_j^k f^{(k)}(r_j \varphi(z)) | \\
= & \sum_{k \in \{0,1,2,m,m+1, m+2 \}} A_{k,6} + \sum_{k \in \{0,1,2,m,m+1, m+2 \}} A_{k,7}.
\end{align*}
For $A_{k,6}$, using Lemmas \ref{l10}, \ref{l11} and \ref{l12}, we obtain
\begin{align*}
A_{k,6} = & \sup_{|\varphi(z)| > r_N} \frac{(1-|\varphi(z)|^2)^k f^{(k)}(\varphi(z))}{|\varphi(z)|^k} \frac{(1-|z|^2)^{\alpha}|I_K(z)| |\varphi(z)|^k}{1-|\varphi(z)|^2)^k} \\
\leq & ||f||_{B_1} \sup_{|\varphi(z)| > r_N} ||( T^m_{u, v, \varphi} g_{k, \varphi(z)}||_{\mathcal{Z}_{\alpha}} \\
\preceq & \sum_{j=1}^6 c_j^k \sup_{|a| > r_N} ||T^m_{u, v, \varphi} f_{j,a}||_{\mathcal{Z}_{\alpha}}
\end{align*}
where $k \in \{0,1,2,m,m+1, m+2 \}$.
As $N \rightarrow \infty$,
$$ \limsup_{j \rightarrow \infty} A_{k,6} \preceq \sum_{j=1}^6 \limsup_{|a| \rightarrow 1} || T^m_{u, v, \varphi} f_{j,a}||_{\mathcal{Z}_{\alpha}} \preceq \max \{ E_j \}_{j=1}^6. $$
Also for $A_{k,6}$ we can write
\begin{align*}
A_{k,6} = & \sup_{|\varphi(z)| > r_N} (1-|\varphi(z)|^2)^k f^{(k)}(\varphi(z)) \frac{(1-|z|^2)^{\alpha}|I_K(z)|}{1-|\varphi(z)|^2)^k} \\
\preceq & ||f||_{B_1} \sup_{|\varphi(z)| > r_N} \frac{(1-|z|^2)^{\alpha}|I_K(z)|}{(1-|\varphi(z)|^2)^k}
\end{align*}
which can be deduced that
$$\limsup_{j \rightarrow \infty} A_{k,6} \preceq \limsup_{|\varphi(z)| \rightarrow 1} \frac{(1-|z|^2)^{\alpha}|I_k(z)|}{(1-|\varphi(z)|^2)^k} \leq
\max \{ F_k \}_1^6. $$
A similar argument can de done for $A_{k,7}$. Thus we prove that
$$ \sup_{||f||_{B_1} \leq 1} ||( T^m_{u, v, \varphi} -T^m_{u, v, \varphi} K_{r_j}) f ||_{\mathcal{Z}_{\alpha}} \preceq \max \{ E_j \}_{1}^6 $$
and
$$ \sup_{||f||_{B_1} \leq 1} ||( T^m_{u, v, \varphi} -T^m_{u, v, \varphi} K_{r_j}) f ||_{\mathcal{Z}_{\alpha}} \preceq \max \{ F_k \}_1^6. $$
Finally we have
\begin{align*}
\limsup_{j\rightarrow \infty}\| T^m_{u,v,\varphi} - T^m_{u,v,\varphi}K_{r_j}\|\preceq \min\{ \max\{E_i\}_1^6, \max\{F_k\}_1^6 \}.
\end{align*}
\end{proof}
In the case $m\leq 2$, the similar result can be stated using Theorems \ref{th11} and \ref{th11z}.
\begin{theorem}\label{th112}
Let $\alpha>0$, $u, v, \varphi \in H(\mathbb{D})$ and $\varphi: \mathbb{D} \rightarrow \mathbb{D}$ and the operator $T^2_{u, v, \varphi} : B_1 \rightarrow \mathcal{Z}_{\alpha} $ be bounded. Then
\begin{align*}
&||T^2_{u, v, \varphi}||_e \approx \max \{\limsup_{|a|\rightarrow 1} ||T_{u,v,\varphi}^2 f_{j,a}||_{\mathcal{Z}_{\alpha}} \}_{1}^5 \approx\\
&\limsup_{|\varphi(z)|\rightarrow 1} (1-|z|^2)^{\alpha}\left( |u''(z)| + \frac{|( 2 u'\varphi' + u \varphi'')(z)|}{(1-|\varphi(z)|^2)}+\frac{|(u \varphi'^2 +v'')(z)|}{(1-|\varphi(z)|^2)^2}\right)+\\
&\limsup_{|\varphi(z)|\rightarrow 1} (1-|z|^2)^{\alpha}\left( \frac{|(2 v'\varphi' + v \varphi'')(z)|}{(1-|\varphi(z)|^2)^3}+\frac{|(v \varphi'^2)(z)|}{(1-|\varphi(z)|^2)^4}\right).
\end{align*}
\end{theorem}
\begin{theorem}\label{th112z}
Let $\alpha>0$, $u, v, \varphi \in H(\mathbb{D})$ and $\varphi: \mathbb{D} \rightarrow \mathbb{D}$ and the operator $T_{u, v, \varphi} : B_1 \rightarrow \mathcal{Z}_{\alpha} $ be bounded. Then
\begin{align*}
&||T_{u, v, \varphi}||_e \approx \max \{\limsup_{|a|\rightarrow 1} ||T_{u,v,\varphi} f_{j,a}||_{\mathcal{Z}_{\alpha}} \}_{1}^4 \approx\\
&\limsup_{|\varphi(z)|\rightarrow 1} \left( |u''(z)| + \frac{|( 2 u'\varphi' + u \varphi'' +v'')(z)|}{(1-|\varphi(z)|^2)}\right)+\\
&\limsup_{|\varphi(z)|\rightarrow 1} (1-|z|^2)^{\alpha}\left( \frac{|(u \varphi'^2+2 v'\varphi' + v \varphi'')(z)|}{(1-|\varphi(z)|^2)^2}+\frac{|(v \varphi'^2)(z)|}{(1-|\varphi(z)|^2)^3}\right).
\end{align*}
\end{theorem}
By using Theorem \ref{th15}, we have the following Corollary.
\begin{corollary}\label{th16}
Let $u, v, \varphi \in H(\mathbb{D})$, $\varphi: \mathbb{D} \rightarrow \mathbb{D}$ and $2<m\in\mathbb{N}$. Let operator $T^m_{u, v, \varphi} : B_1 \rightarrow \mathcal{Z}_{\alpha} $ be bounded. Then the following conditions are equivalent:
\begin{enumerate}
\item [(i)]
The operator $T^m_{u, v, \varphi} : B_1 \rightarrow \mathcal{Z}_{\alpha} $ is compact.
\item [(ii)]
$$ \limsup_{|a| \rightarrow 1} ||T_{u,v,\varphi}^m f_{i,a}||_{\mathcal{Z}_{\alpha}} =0, \ \ \ \ i = 1, 2,\cdots, 6 $$
\item [(iii)]
$$ \limsup_{|\varphi(z)| \rightarrow 1} \frac{(1-|z|^2)^{\alpha} |I_k (z)|}{(1-|\varphi(z)|^2)^{k}} =0, \ \ \ \ k \in \{0,1,2,m,m+1, m+2 \}. $$
\end{enumerate}
\end{corollary}
From Theorems \ref{th112} and \ref{th112z}, we obtian similar results for compactness of operator $T^2_{u, v, \varphi} : B_1 \rightarrow \mathcal{Z}_{\alpha}$
and $T_{u, v, \varphi} : B_1 \rightarrow \mathcal{Z}_{\alpha}$ respectively.
\begin{rem}
By taking $u=0(v=0)$, we can get the results of the paper for generalized (weighted) composition operators.
\end{rem}
|
1,116,691,499,258 | arxiv | \section{Introduction}
\label{intro}
Understanding the effect of hydrogen on dislocations \citep{Robertson2001} is essential to understanding how hydrogen reduces ductility in metals. Using in situ TEM testing, hydrogen has been observed to enhance dislocation mobility in a variety of materials \citep{Tabata1983,Robertson1986,Shin1988}. \cite{Ferreira1998} found that hydrogen decreases the equilibrium dislocation spacing in a pileup in 310 stainless steel and Al. The same authors then observed hydrogen suppresses dislocation cross slip thereby increasing slip planarity \citep{Ferreira1999}. Motivated by these experimental observations, \cite{Birnbaum1994} used analytic solutions to calculate the elastic stress field generated by a hydrogen concentration field that is in equilibrium with an edge dislocation (assuming plane strain). The elastic interaction between two dislocations or between a dislocation and a centre of dilation (defect) is reduced in the presence of hydrogen, leading to reduced spacing and enhanced mobility. This mechanism is able to partially account for the experimental observation of hydrogen enhanced plasticity, and is referred to as the hydrogen elastic shielding mechanism. Based on this mechanism, the effects of hydrogen on dislocation nucleation (Frank-Read source operation) and expansion and cross slip have been discussed \citep{Delafosse2008}. All of these studies focused on dislocation behaviour limited to a single slip plane, and considered only the long range elastic stress, which is important for dislocation glide, but neglected the near-core contribution of hydrogen. Recently, it was revealed that hydrogen weakens dislocation junctions in fcc materials which will likely play an important role in work hardening \citep{Yu2019b}.
A sessile dislocation junction forms when a glissile dislocation cuts through a forest dislocation that does not lie on the same slip plane. This mechanism contributes to the formation of complex, tangled dislocation arrays during crystal deformation \citep{Grilli2018}, and is a key source of strain hardening. Given their importance, dislocation junctions have been extensively studied \citep{Bulatov2006}. TEM observations of dislocation structures of hydrogen-charged specimens in bcc Fe show that a homogeneous dislocation forest in a hydrogen-free sample transforms into cell walls that partition dislocation-free regions when it is charged. The cell walls can be regarded as dense dislocation tangles. TEM images reveal that the area of the dislocation free zones and the density in the dislocation tangles increase with hydrogen concentration in the interval 0-25 appm \citep{Wang2013}. The physics underlying dislocation reorganization due to hydrogen is not yet well established, but investigations into the effect of hydrogen on dislocation junctions provide a useful starting point for understanding this phenomenon.
It is instructive to observe a single dislocation junction and study its formation and destruction. This will help predict and understand the mergent properties of dislocation tangles. Due to the difficulty in pinpointing and following the development of a single junction, only a handful of experimental observations have been reported in the literature \citep{Wu2009,Ma2010,Lee2013,Pardoen2015}. This research investigated the formation and destruction of Lomer junctions using TEM, in the absence of hydrogen. Experimental investigation becomes even more challenging when hydrogen is present. Conditions apart from the hydrogen concentration ought to be equivalent between the charged and uncharged experiments, including the initial configuration of the junction, (strictly speaking) its surrounding dislocation structure and the loading history. Due to the statistical nature of dislocation activity, exact comparison between hydrogen charged and unchanged specimens is out of the question. Even for in situ tests, it is extremely difficult, if not impossible, to keep all the control variables unchanged. For instance, hydrogen charging will inevitably alter the dislocation structure, and the loading history will be distinct for different tests. No experimental observations on the effect of hydrogen on junctions have been reported for individual dislocation junctions, and it is questionable whether this will be achieved in the near future.
In the absence of suitable experiments, modelling can provide detailed and important information about the development of junctions and how they influence cell formation. Molecular dynamics (MD) \citep{Bulatov1998} and 3D discrete dislocation dynamics (DDD) \citep{Cai2006book} are powerful tools for simulating dislocation junctions. \cite{Rodney1999} employed the quasicontinuum (QC) method, which combines MD and the finite element method (FEM), to study the formation and destruction of Lomer-Cottrell junctions. \cite{Bulatov1998,Yamakov2003} simulated Lomer-Cottrell junctions using molecular dynamics simulation. Molecular statics simulation can also be used to simulate dislocation locks \citep{Parthasarathy1996,Madec2003}. Atomistic methods are limited to small temporal and spatial scales, compromising their ability to capture the long-range character of dislocation stress fields \citep{Martinez2008}; in general, the results obtained via atomistic methods cannot be directly implemented in large scale (e.g. crystal plasticity) models. DDD is an ideal tool for bridging the gap in space and time scales between atomistic and continuum models. DDD has a dislocation line segment as its basic element and uses analytic solutions for the elastic fields so is able to simulate the collective behaviour of a large array of dislocations. To date, DDD has been used to study problems ranging from individual dislocation junctions \citep{Madec2002}, to large scale plasticity arising from the motion of a large number of dislocations \citep{Zbib1998}. \cite{Shenoy2000} performed DDD simulations of a Lomer-Cottrell junction, reproducing the atomistic results obtained by \cite{Rodney1999}. \cite{Madec2002} performed systematic DDD simulations of Lomer-Cottrell junctions covering all possible initial configurations, and further evaluated the effect of forest hardening on plasticity using large scale DDD simulations. \cite{Madec2003} utilised the DDD approach in combination with molecular statics simulations to study the collinear interaction of dislocations. \cite{Bulatov2006} probed the properties of multi-junctions using DDD and \cite{Lee2011} performed DDD simulations of Lomer junctions in a free-standing thin film. \cite{Wu2016} studied binary junctions in hexagonal close-packed crystals, combining the line tension model and DD simulations.
Despite the large literature on modelling the effects of hydrogen on mechanical properties, across a wide range of scales, very few have investigated the effect of hydrogen on dislocation junctions. \cite{Hoagland1998} studied the effects of hydrogen on a Lomer-Cottrell lock using Monte Carlo (MC) calculations. \cite{Chen2008} investigated the effects of a general solute on a Lomer junction, considering different solute properties, using a kinetic Monte Carlo (kMC) coupled DDD approach. Hydrogen redistribution during the destruction of the dislocation junction was not considered in these works. Recently, \cite{Yu2019b} utilised a hydrogen informed discrete dislocation dynamics (DDD) approach to investigate the influence of hydrogen on Lomer junctions; hydrogen redistribution was taken into account and various initial configurations \citep{Madec2002} were examined. This could provide reference for modelling hydrogen with crystal plasticity approach considering dislocation interactions \citep{Gustavo2018}. \cite{Zhao2018} performed MD simulations of a hydrogen charged nanoindentation test and discussed the effect of hydrogen on dislocation entanglement.
Recently, \cite{Gu2018} proposed a framework for incorporating hydrogen into the non-singular DDD formulation \citep{Cai2006}. Hydrogen is treated as an isotropic point dilatation in an infinite elastic medium, based on which the hydrogen elastic stress field and its influence on dislocation motion are evaluated. This formulation accounts only for the linear elastic contribution of hydrogen residing outside the dislocation core region, which is consistent with the hydrogen elastic shielding mechanism. Similar formulations were also proposed in 2D by \cite{Cai2014,Song2018}. However, this is only a partial contribution of hydrogen. Neglecting the core contribution means an unrealistically high bulk concentration must be assigned in order to produce a noticeable hydrogen influence.
As described by the hydrogen elastic shielding theory, hydrogen forms atmospheres around dislocations, reducing the long-range elastic dislocation-dislocation interactions \citep{Birnbaum1994,Gu2018}. At high hydrogen concentrations, the hydrogen shielding effect increases, which could affect the formation of dislocation structures. Although the shielding effect remains commonly quoted in the current literature as a viable mechanisms for the reduction of elastic interactions between dislocations, it is not adequate to explain the experimental observations in Fe. Theoretical studies using continuum models show that the hydrogen shielding effect is relatively short range and unlikely unless hydrogen concentrations are extremely high ($> 10^{4}$ appm) \citep{Birnbaum1994}. Experiments using in-situ TEM on samples deformed in tension \citep{Ferreira1998} also show that hydrogen has a significant effect on dislocation spacing at distances greater than 20b, indicating that hydrogen shielding of the dislocation-dislocation interactions cannot, by itself, account for the observations. Atomistic simulations of the effects of hydrogen on $1/2[111]$ edge dislocation pile-ups in bcc-Fe also indicate that the shielding mechanism is not operative at hydrogen concentrations lower than $10^{5}$ appm \citep{Song2014}. Thus, other mechanisms, such as the effect of hydrogen on the dislocation core must be examined in more detail.
To account for the hydrogen core contribution, \cite{Yu2019a} implemented a hydrogen dependent dislocation mobility law where the velocity of a dislocation is increased in the presence of hydrogen. The increased mobility is attributed to hydrogen promoted kink-pair nucleation in the near core region, which was revealed using kMC simulations \citep{Katzarov2017}. Therefore, a hydrogen dependent dislocation mobility law, to some extent, accounted for the effect of hydrogen residing near the dislocation core, and allowed the effect of hydrogen on a microcantilever to be simulated at a realistic concentration. Hydrogen residing in the core will not only affect kink-pair formation and migration but also change the dislocation core energy profile which is critical for dislocation line tension and near-field interactions.
\cite{Cai2018} suggested introducing a solute-solute interaction energy term to the hydrogen stress formulation. In this way, an additional free energy contribution from the near core regime is incorporated, and this can be approximated as a change in the core energy due to hydrogen. With this method, \cite{Cai2018} were able to observe the effects of hydrogen on a dislocation loop and a Frank-Read (F-R) source at realistic bulk concentrations. In particular, a large decrease in the activation stress of the F-R source was observed, indicating the importance of the hydrogen core contribution in the activation of dislocation sources.
The force acting on a dislocation can be partitioned into two parts \citep{Arsenlis2007}: the elastic force and the core force, due to the dislocation core energy $E^{c}$. Hydrogen modifies this core energy and so contributes an additional hydrogen core force. The hydrogen core contribution is key in hydrogen-junction interactions, where the dislocation self force and short-range interactions are crucial. Hydrogen elastic shielding and hydrogen dependent dislocation mobility contribute marginally to dislocation junction formation which is regarded as a quasi-static process. As a demonstration, we performed DDD simulations of junction formation at low concentrations ($0-60$appm) considering these contributions and observed practically no change in junction length in the presence of hydrogen; see \autoref{subsec:application} for further details. To capture the effect of hydrogen on dislocation junctions, the hydrogen core force needs to be incorporated. The approach proposed by \cite{Cai2018} is a viable one, but as they noted, the near-core contribution was based on the stress field derived from linear elasticity theory, and the nonlinear interatomic interactions at the dislocation core were neglected. These nonlinear effects should be incorporated, which can be achieved by calibrating the dislocation core energy in the presence of hydrogen with atomistic simulations.
In this work we assume the hydrogen core force arises as a result of hydrogen lowering the dislocation core energy, which is calibrated based on first principles (DFT) calculations for bcc iron. \textcolor{black}{We will show that the effects of hydrogen on dislocation junctions are dominated by the hydrogen core force. This highlights the critical role of dislocation core energy in near-field dislocation interactions. This work, in a sense, depicts a more general picture of how dislocation reactions are influenced by a change in dislocation core energy, using hydrogen as a medium to trigger the change.}
This paper is organised as follows: \autoref{sec:DFT} presents the details of the calibration of the hydrogen core energy using DFT. \autoref{sec:linetension} discusses the effects of the hydrogen modified core energy on dislocation junction formation in bcc iron, using a line tension model. \autoref{sec:DDD} implements a hydrogen core force in the DDD framework and employs it to study the influence of hydrogen on dislocation junction properties. A summary is presented in \autoref{sec:sum}.
\section{\textcolor{black}{Atomic calibration}}
\label{sec:DFT}
The line energy, $E$, of a dislocation can be partitioned in two parts \citep{Arsenlis2007},
\begin{equation} \label{eq:DFT1}
E=E^{e}+E^{c},
\end{equation}
the elastic energy $E^{e}$ and the core energy $E^{c}$. $E^{e}$ can be calculated with linear elasticity theory whereas $E^{c}$ is the energy contained within the dislocation core, \textcolor{black}{typically} within $5b$ from the dislocation line.
In an isotropic crystal the elastic energy per unit length of a straight dislocation line is
\begin{equation} \label{eq:DFT2}
E^{e}=\frac{\mu b^{2}}{4\pi(1-\nu)}\ln\left( \frac{R}{r_{c}} \right) (1-\nu \cos^{2}\theta),
\end{equation}
$\mu$ is the shear modulus, $\nu$ is Poisson's ratio, $R$ and $r_c$ are the outer and inner cutoff radii,
and $\theta$ is the character angle between the dislocation's Burgers vector and line direction. Material parameters for iron are used in this work: $a=2.856$ \r{A} is the lattice parameter, $\mu=82$ GPa and $\nu=0.29$. Core energies in bcc Fe were determined by \cite{Clouet2009,Clouet2009b,Zhang2011} who obtained values of $E^c_{s}=0.219$ eV/\AA{} for a $1/2[111]$ screw, $E^c_{e}=0.286$ eV/\AA{} for a $1/2[111](110)$ edge, and $E^{c}_{e}=0.62$ eV/\AA{} for a $[100]$ edge dislocation, calculated with a core radius $r_{c}$, of 3 \AA{}, 2.45 \AA{} and 5.16 \AA{},
respectively.
We will notionally divide the hydrogen atoms into those which are trapped at distances greater than the core radius and usually referred to as forming an "atmosphere", and those which are occupying deep traps within the dislocation core.
Due to the fast hydrogen diffusion compared to the low mobility of intersecting dislocations, it is reasonable to assume that
dislocations are in equilibrium with hydrogen \citep{Gu2018,Yu2019a}.
In the case of the hydrogen in the atmosphere, these are occupying sites that we will assume to be equivalent to sites in an undistorted tetrahedral interstice and since these sites are not regular tetrahedra the associated strain is not purely dilatational.
According to calculations using the density functional theory \citep{Wang2013b}, the strain field of a single hydrogen atom in
bcc Fe is
\begin{equation}
\bm{\varepsilon}^H = \begin{bmatrix}
\varepsilon^H_{11} & 0 & 0 \\
0 & \varepsilon^H_{22} & 0 \\
0 & 0 & \varepsilon^H_{33}
\end{bmatrix}
= \begin{bmatrix}
0.014 & 0 & 0 \\
0 & 0.041 & 0 \\
0 & 0 & 0.041
\end{bmatrix}.
\end{equation}
The interaction energy per unit dislocation length, between a hydrogen in the atmosphere and a straight
dislocation can then be determined following the continuum approach developed by \cite{Cochardt1955}.
The interaction energy is
\begin{equation} \label{eq:DFT5}
E^{b}(r)=a^{3}\sum_{ij}\sigma_{ij}(r) \varepsilon^{H}_{ij}
\end{equation}
where $a$ is the lattice constant, $\sigma_{ij}$ is the stress field generated by the
straight dislocation
expressed in the same coordinate system as $\bm{\varepsilon}^{H}$.
\textcolor{black}{The total interaction energy per unit dislocation length between the hydrogen atmosphere and a straight
dislocation is
\begin{equation}\label{eq:continuum}
E^{H}(\chi_{0}) = \int_{0}^{2\pi}\int_{r_{H}}^{R} C_{max} \chi(\varphi, r) E^{b}(\varphi, r) r d\varphi dr
\end{equation}
where $r_{H}$ and $R_{H}$ are the inner and outer cut-off radii of the hydrogen atmosphere centred at the
dislocation. $C_{max}$ is the maximum solute concentration for the solid (in solutes per unit volume), and $\chi\equiv C/C_{max}$
is the fraction of available lattice sites that are occupied by hydrogen \citep{Cai2018}.}
Turning now to the hydrogen atoms trapped within the core,
their binding energies, $E^b$, can be calculated using atomistic simulations
\citep{Kimizuka2011}. These show that $1/2[111](110)$ edge dislocations
generate seven inequivalent hydrogen trap sites in the plane perpendicular to the dislocation line sense, with binding energy higher
than 100 meV (one with binding energy $E^{b}_{1}=0.34~\textrm{eV}$, two with binding energies $E^{b}_{2}=0.40~\textrm{eV}$,
$E^{b}_{3}=0.27~\textrm{eV}$, $E^{b}_{4}=0.12~\textrm{eV}$).
The contribution to total hydrogen--dislocation interaction energy per periodic distance along the
dislocation line from the deep near-core hydrogen binding sites is
\begin{equation} \label{eq:atomic}
G^{H}(\chi_{0}) = \sum_{i}\chi^{b}_{i}(\chi_{0}) E^{b}_{i}
\end{equation}
where the sum is over the binding sites $i$ \citep{Kimizuka2011}.
The probability that a trap site is occupied by hydrogen is calculated from the McLean isotherm,
\begin{equation} \label{eq:DFT4}
\chi^{b}_{i} = \frac{\frac{1}{6}\chi_{0}\exp \left( E^{b}_{i}/k_{B}T \right)}{1+\frac{1}{6}\chi_{0} \exp\left( E^{b}_{i}/k_{B}T \right)}.
\end{equation}
where the factor $1/6$ accounts for there being six tetrahedral sites per iron atom in good crystal. $\chi_{0}$ is the nominal number of hydrogen atoms per iron atom: the concentration of hydrogen in atomic parts per million (appm) is thereby $c_0=10^{6}\chi_0$.
The change in energy per unit length, $E^{H}(\chi_{0})$, is then
\begin{equation}
E^H = \frac{G^H(\chi_{0}) }{L}, \quad
L=\left\{\begin{array}{cc}
\sqrt{6}a & \textrm{edge}\\
\sqrt{3}a/2 & \textrm{screw}
\end{array}\right.
\end{equation}
where L is the periodic spacing along the dislocation line
The hydrogen concentration that we use in this work will range between zero and 60~appm; typical amounts found in industrial contexts are 1--10~appm, whereas specimens charged in the laboratory may reach 100~appm.
In this range, the contribution to the total hydrogen--dislocation interaction energy from the near-core deep binding sites is dominant. For instance, for a $1/2[111](110)$ edge dislocation with $c_{0}=30$~appm the contribution to the total binding energy per unit dislocation length due to "core" hydrogen calculated from \autoref{eq:atomic} is 0.153~eV/\AA\,
The contributions from the hydrogen atmosphere
calculated from \autoref{eq:continuum} with inner cut-off radius $r_{H}=a$ and outer cut-off radii $R_{H}=100a$ and $R_{H}=1000a$ are correspondingly 0.0018 eV/\AA\,and 0.0022 eV/\AA.
In the case of the $1/2[111]$ screw dislocation, we use the deep near-core hydrogen binding sites calculated by \cite{Itakura2013}. Because the strain field is purely deviatoric in linear elasticity the change in energy due to the hydrogen atmosphere is yet smaller than for the edge dislocation. When it comes to the hydrogen binding to $[100]$ dislocations we consider only the atmospheric hydrogen and neglect core binding.
The interaction energies, $E^{H}$, as functions of nominal hydrogen concentration, $c_0$ are shown in \autoref{fig:Henergy}.
\begin{figure}[!htb]
\centering
\includegraphics[width=0.7\linewidth]{bindingE}
\caption{\textcolor{black}{The calibrated hydrogen core energy for $1/2[111]$ and $[100]$ dislocations. The hydrogen core energy is negative and so will reduce the total core energy and total core force. We refer to this as \emph{hydrogen core force shielding}, as it is analogous to hydrogen elastic shielding \citep{Birnbaum1994}.} }
\label{fig:Henergy}
\end{figure}
The hydrogen core energy is negative and so will reduce the total core energy and total core force. This trend is consistent with the defactant theory proposed by \cite{Kirchheim2007,Kirchheim2010}. It is noted that the hydrogen binding energy for a $1/2[111]$ screw dislocation is non-zero. This is because hydrogen has a tetragonal misfit strain which couples with the shear field of the 1/2[111] screw dislocation to produce non-zero interaction energy.
In contrast, the hydrogen binding energy for a $[100]$ screw dislocation is zero, because the misfit strain of hydrogen does not couple with the elastic field in this case. When two dislocation segments attract and intersect to form a junction, the dislocation lines rotate, resulting in a change in the angles between the Burgers vectors and line directions (the character angle, $\theta$). The change in the character angle leads to rearrangement of hydrogen atoms around the dislocation which changes the interaction energy. The continuum method (\autoref{eq:continuum}) can be used to calculate the interaction energy between the hydrogen atmosphere and a mixed straight dislocation. However, the comparison of $E^{H}$ calculated by the atomistic (\autoref{eq:atomic}) and continuum methods shows that $r_{H}$ for $1/2[111]$ screw and edge dislocations are different, which indicates that the inner cut-off radius depends on the character angle. Therefore, accurate determination of the interaction energy between hydrogen and a mixed dislocation is not feasible with the continuum approach without using reliable data for the inner cut-off radius, $r_{H}$, as a function of the character angle. In this study, we approximate the interaction energy $E(\theta)$ between a hydrogen atmosphere and a mixed $1/2[111]$ dislocation segment with a character angle $\theta$ as an interpolation between the interaction energies $E_H^{s}$ and $E_H^{e}$ for the screw and edge segments calculated atomistically. Inspired by the observation in \cite{Cai2018} that $E_H$ seems to have a roughly sinusoidal dependence on $\theta$, the following interpolation function is used
\begin{equation}
\label{eq:HcoreEinter0}
E^{H} = E^{H}_e\sin^2\theta + E^{H}_s\cos^2\theta,
\end{equation}
Note that this interpolation is not unique, any number of schemes which enable a smooth transition from $E^H(0)=E^H_s$ to $E^H(\pi/2)=E^H_e$ could be used. We tested other interpolation schemes, including a linear interpolation, in the line tension model, and found that the influence of the interpolation scheme on dislocation junction formation is limited.
\textcolor{black}{Finally, it is acknowledged that the atomic calibration presented in this section is based on certain simplifications. For instance, the continuum approach is used for the $[100]$ case and a simple interpolation is applied to mixed dislocations. The key features of the hydrogen binding energy profile are captured: the binding energy is small but non-zero for a $1/2[111]$ screw and zero for a $[100]$ screw; the magnitude of the binding energy is higher for edge than screw dislocations. The main goal of this work is to present the first discrete dislocation dynamics framework utilizing atomistic-level hydrogen energetics for the dislocation core as input and investigate any emergent influence on plasticity. The outcomes of this study are more qualitative and conceptual than quantitative.}
\section{Line tension model}
\label{sec:linetension}
To investigate the role of hydrogen trapped in the dislocation core on the formation of dislocation junctions, we investigate the intersection between straight dislocations in Fe moving on different glide planes. Our aim here is to evaluate hydrogen core effects on dislocation reactions, and ignore the external stress and elastic interactions between dislocations. In bcc materials, both binary and triple junctions are important in plasticity \citep{Bulatov2006}. Binary junctions result from the interaction of two dislocations on different slip systems, and can influence the yield stress and early stages of work hardening, whereas triple junctions result from the interaction of three dislocations, and are important during the late stages of work hardening. Triple junctions are strong anchors for dislocation entanglements and are potentially important in the formation of dislocation cell structures \citep{Bulatov2006}.
The binary junction geometry is illustrated in \autoref{fig:biillustration}. Consider two straight dislocation line segments, with Burgers vectors $b_{1}$ and $b_{2}$. Both segments are bound by a pair of pinning points.
\begin{figure}[ht]
\centering\includegraphics[width=0.8\linewidth]{biillustration}
\caption{Illustration of the binary junction geometry: (a) the initial configuration and (b) the junction formed after relaxation. The initial dislocations and the junction are of pure edge type.}
\label{fig:biillustration}
\end{figure}
The dislocation segments are in the slip planes with normals $\bm{n}_{1}$ and $\bm{n}_{2}$ and intersect at their mid-point; which is on the line of intersection of the two slip planes, $\bm{L}=\bm{n}_2\times\bm{n}_1=[001]$. The dislocations are at angles $\phi_1 $ and $\phi_2 $ to $\bm{L}$. A junction (a straight dislocation segment along $\bm{L}$) with a Burgers vector $\bm{b}_{3}=\bm{b}_{1}+\bm{b}_{2}$ will form if it reduces the energy of the system. If $E_{1}$, $E_{2}$ and $E_{3}$ are respectively the line energies of the reacting dislocations and junction, a junction will form if the following criterion is fulfilled \citep{Baird1965}
\begin{equation} \label{eq:linetension1}
E_1\cos\phi_1+E_2\cos\phi_2 > E_{3}
\end{equation}
The energy of the configuration is a minimum when the condition
\begin{equation} \label{eq:linetension2}
E_1\cos\psi_1+E_2\cos\psi_2 - E_{3} = 0
\end{equation}
is satisfied with equilibrium angles $\psi_1 $ and $\psi_2 $, as illustrated in \autoref{fig:biillustration}(b).
In order to evaluate the effect of hydrogen on junction formation we consider the interaction between two $1/2\langle111\rangle$ edge dislocations, namely, $1/2[1\bar{1}1](110)$ and $1/2[\bar{1}\bar{1}\bar{1}](1\bar{1}0)$, as illustrated in \autoref{fig:biillustration}.
The dislocations are of length $l_1=l_2=200b$, and initially intersect at their mid point. The dislocations attract each other and their intersection results in an $[0\bar{1}0](\bar{1}00)$ edge junction:
\begin{equation} \label{eq:linetension4}
1/2[1\bar{1}1](110)+1/2[\bar{1}\bar{1}\bar{1}](1\bar{1}0)\rightarrow [0\bar{1}0](\bar{1}00).
\end{equation}
After including the dislocation core energy change due to hydrogen $E^{H}$ (shown in \autoref{fig:Henergy}) into \autoref{eq:DFT1}, the energy per unit dislocation length in the presence of hydrogen is
\begin{equation}
\label{eq:linetension5}
E=E^{e}+E^{c}+E^{H}.
\end{equation}
To evaluate the effect of hydrogen on the dislocation reaction, at different bulk hydrogen concentrations, we determine the junction length which minimises the energy of the configuration; by substituting the energies determined in \autoref{sec:DFT} into \autoref{eq:linetension2}. The calculations show an increase in binary junction length but almost no change in the triple junction length due to hydrogen. Since a $\langle100\rangle$ edge dislocation generates deeper binding sites (compared to a $1/2\langle111\rangle$ edge segment), the junction generated by the reaction traps more hydrogen atoms, reducing the energy of the system, and increasing the equilibrium binary junction length, as shown in \autoref{fig:LTresults}.
\begin{figure}[!ht]
\centering\includegraphics[width=0.7\linewidth]{LTlength}
\caption{Variation of junction length with hydrogen concentration calculated via the LT model.}
\label{fig:LTresults}
\end{figure}
In the triple junction case we adopt the same configuration as in \cite{Bulatov2006}. Three $\langle111\rangle\{1\bar{1}0\}$ edge dislocations
intersect at their midpoint and a $1/2[111]$ screw type triple junction is formed,
\begin{equation} \label{eq:triple}
1/2[\bar{1}11](01\bar{1})+1/2[1\bar{1}1](10\bar{1})+1/2[11\bar{1}](1\bar{1}0)\rightarrow 1/2[111].
\end{equation}
In this case, the screw dislocation does not generate additional binding sites and hydrogen atoms released during the reduction of the lengths of the reacting edge dislocations increases the energy of the configuration, resulting in a very slight reduction in the screw junction length with hydrogen, as shown in \autoref{fig:LTresults}. The effect of hydrogen in this case is a second order effect since the triple junction is screw type, its core energy is independent of hydrogen. It is only the edge components of the reacting dislocations (rotated $\langle111\rangle$ screw dislocations) which have a reduced core energy. This increases the equilibrium length of the reacting dislocations and therefore reduces the equilibrium length of the triple junction.
The line tension model is a static approach for evaluating hydrogen core effects on dislocation reactions. It is applicable only for straight dislocation segments, while the external stress and dislocation interactions are ignored. In reality, under an applied stress, the dislocation segments will bow out. To study the effects of hydrogen on the intersection of curved dislocations requires including both the long-range elastic interaction due to hydrogen and the short range interaction between dislocations and hydrogen at the core. Discrete dislocation dynamics (DDD) simulations on the effects of hydrogen on dislocation junctions are performed subsequently for this purpose. Furthermore, DDD can be used to determine the junction strength, which is not possible with the line tension model.
\section{DDD simulation with hydrogen core force}
\label{sec:DDD}
Hydrogen influences dislocation plasticity by exerting elastic shielding, core force shielding and modifying the dislocation mobility. Long-range hydrogen elastic shielding and a hydrogen dependent mobility law were implemented by \cite{Yu2019a}. However, the model was not able to capture the effect of hydrogen on dislocation junction formation, due to the short-range and quasi static nature of the process. To some extent, this was implied in our recent work on the influence of hydrogen on Lomer junctions \citep{Yu2019b}, where hydrogen effects were observed with an unrealistically high bulk concentration due to the omission of hydrogen core effects. DDD simulations of binary and triple junctions are performed here, considering only hydrogen elastic shielding and a hydrogen dependent mobility. The initial dislocation configurations are the same as for the line tension model in \autoref{sec:linetension}.
\begin{figure}[!ht]
\centering\includegraphics[width=0.9\linewidth]{HDDDlengths}
\caption{(a) Variation of junction length with hydrogen concentration calculated via DDD without considering hydrogen core force shielding and (b) with hydrogen core force shielding.}
\label{fig:DDDlengths}
\end{figure}
As shown in \autoref{fig:DDDlengths}(a), the effect of hydrogen on junction length is not captured at these low concentrations. Comparison to the line tension model results (\autoref{fig:LTresults}) indicates that hydrogen core force shielding is dominant in the influence of hydrogen on junction formation.
\subsection{Formulation}
\label{subsec:formulation}
In the DDD simulation, dislocations are represented by discrete straight segments. The dislocation nodal velocity $\bm{V}_k$ is determined by balancing the drag force with the nodal driving force $\bm{F}_k$ on node $k$ at position $\bm{X}_k$. The nodal force is evaluated based on the non-singular continuum theory of dislocations developed by \cite{Cai2006}. In general, it consists of five parts in the presence of hydrogen
\begin{equation}
\label{eq:Fk}
\begin{aligned}
\bm{F}_{k} & = \sum_{l}\sum_{i,j}\tilde{\bm{f}}^{ij}_{kl}(\bm{X}_k) + \sum_{l}\sum_{i,j}\bar{\bm{f}}^{ij}_{kl}(\bm{X}_k) + \sum_{l}\bm{f}^c_{kl}(\bm{X}_k)+\sum_{l}\bm{f}^{c,H}_{kl}(\bm{X}_k)+\sum_{l}\hat{\bm{f}}_{kl}(\bm{X}_k)\\
& = \tilde{\bm{F}}_k+\bar{\bm{F}}_k+\bm{F}^c_k+\bm{F}^{c,H}_k+\hat{\bm{F}}_k,
\end{aligned}
\end{equation}
where $\tilde{\bm{f}}^{ij}_{kl}(\bm{X}_k)$ is the interaction force at node $k$, due to segment $i\rightarrow j$ integrated along segment $k\rightarrow l$; this is summed over all segments $i\rightarrow j$ inside the domain, including the self force due to segment $k\rightarrow l$, this is then summed over all nodes $l$ which are connected to node $k$. Similarly, $\bar{\bm{f}}^{ij}_{kl}(\bm{X}_k)$ is the hydrogen elastic shielding force evaluated at node $k$, implemented following the formulation proposed by \cite{Gu2018}. $\bm{f}^c_{kl}(\bm{X}_k)$ is the dislocation core force and $\bm{f}^{c,H}_{kl}(\bm{X}_k)$ is the hydrogen core shielding force (that is missing in previous simulations). $\hat{\bm{f}}_{kl}(\bm{X}_k)$ is the corrective elastic force for finite boundary value problem, which is evaluated with the finite element method using the superposition principle \citep{Needleman1995}.
In the presence of hydrogen, the core energy is
\begin{equation}
\label{eq:coreE}
E^{c}_\textrm{eff}(\theta) = E^{c}(\theta) + E^H(\theta)= \sin^2(\theta) (E^{c}_e+E^{H}_e) + \cos^2(\theta) (E^{c}_s+E^{H}_s),
\end{equation}
\textcolor{black}{where $E^{c}_e$ and $E^{c}_s$ are the edge and screw core energies in the absence of hydrogen and $E^H_e$ and $E^H_s$ are the hydrogen core binding energies presented in \autoref{fig:Henergy}.} The core force \citep{Cai2018} in the presence of hydrogen is then
\begin{equation}
\label{eq:coreF}
\bm{f}^c_{kl}(\bm{X}_k)+\bm{f}^{c,H}_{kl}(\bm{X}_k) = -E^{c}_\textrm{eff}(\theta)\bm{l}_{kl} + \frac{dE^{c}_\textrm{eff}(\theta)}{d\theta}\left(\frac{\bm{b}^e_{kl}}{ \lvert\bm{b}^e_{kl}\rvert} \right),
\end{equation}
where $\theta$ is the angle between $\bm{l}_{kl}$ the line direction and Burgers vector of segment $k\rightarrow l$ with length $L_{kl}$. $\bm{b}^e_{kl}$ is the edge component of the Burgers vector. The first term on the right hand side acts along the line direction to shrink the segment, while the second term is a moment tending to rotate the segment to the orientation with lowest core energy \citep{Arsenlis2007}; in the absence of hydrogen this is the screw orientation. \textcolor{black}{As discussed in \autoref{sec:DFT}, different dislocation core radii and hydrogen free core energies were used for the $1/2[111]$ and $[100]$ dislocations in the atomistic calibration. To be consistent, we detected the dislocation type during the DDD simulation and assigned the core radii and hydrogen free core energies accordingly. For instance, for $1/2[111]$ dislocations, a core radius of $3\AA$ was used, and a core radius of $5.16\AA$ was used for $[100]$ dislocations. The hydrogen free core energies for $1/2[111]$ edge and screw dislocations are respectively $E^{c}_e=0.286$ eV/\AA{} and $E^{c}_s=0.219$ eV/\AA{} \citep{Clouet2009,Clouet2009b}; for $[100]$ edge dislocation the value is $E^{c}_e=0.62$ eV/\AA{} \citep{Zhang2011}. To the best of our knowledge, the dislocation core energy for $[100]$ screw dislocation in iron has not been reported in the literature, therefore, the core energy for this case is approximated following the relation $E^{c}_s=(1-\nu)E^{c}_e=0.44$ eV/\AA{}. It is noted that the atomistically determined core energies in the $1/2[111]$ cases also follow this relation ($0.219/0.286=0.766\approx 1-\nu$). The validity of the treatment of dislocation core radii and energies is verified in \autoref{fig:DDDlengths}(b), where good agreement is achieved in predicting the hydrogen free junction lengths using two different mthods; the line tension model and DDD.}
A linear dislocation mobility law \citep{Arsenlis2007,Yu2019a} is used. For each segment $kl$, a drag tensor $\bm{B}_{kl}$ is determined according to the segment character, the nodal velocity $\bm{V}_k$ at node $k$ is then obtained using
\begin{equation}
\label{eq:linearmob}
\left[\frac{1}{2}\sum_{l}L_{kl}\bm{B}_{kl}\right]^{-1}\bm{F}_{k} = \bm{V}_k,
\end{equation}
where the sum is over all nodes $l$ connected to node $k$, and $\bm{F}_k$ is the nodal force determined with \autoref{eq:Fk}. The mobility of a dislocation segment is anisotropic with respect to glide and climb and line directions,
\begin{equation}
\label{eq:dragedge}
\bm{B}_{kl}(\bm{l}_{kl}) = B_{g}(\bm{m}_{kl}\otimes\bm{m}_{kl})+B_{c}(\bm{n}_{kl}\otimes\bm{n}_{kl})+B_{l}(\bm{l}_{kl}\otimes\bm{l}_{kl}),\quad \mathrm{when}\ \bm{l}_{kl}\perp\bm{b}_{kl},
\end{equation}
where $B_{g}$ and $B_{c}$ are the drag coefficients for glide and climb, respectively. The unit vectors are the plane normal $\bm{n}_{kl}$ and glide direction $\bm{m}_{kl}$.
The dislocation mobility is inversely proportional to the drag coefficient. In bcc materials, the mobility of a pure edge segment should be greater than a pure screw segment, which is accounted for by assigning $B_{eg} < B_s$. In the absence of hydrogen, the edge and screw drag coefficients are $B_{eg}=5\times10^{-4}~\textrm{Pa s}$ and $B_{s}=1\times10^{-2}~\textrm{Pa s}$ as used by \citet{Wang2011}.
The glide drag coefficient for a mixed segment is determined by interpolation between the screw and edge values using
\begin{equation}
\label{eq:Bkl}
B_g = \left[B^{-2}_{eg}\sin^2(\theta)+B^{-2}_{s}\cos^2(\theta)\right]^{-1/2}
\end{equation}
the drag coefficient for climb is set sufficiently high, $B_c=1\times10^{6}B_{eg}$ so that only glide occurs and the line drag coefficient is sufficiently low, $B_l=1\times10^{-4}B_{eg}$, to allow nodes to move freely along the dislocation line.
Cross slip of screw segments is considered in a very simplified manner. For a screw segment, all possible cross slip planes are checked and the plane with the maximum resolved shear stress is used as the cross slip plane. The motion of the screw segment is then limited to this plane. For example, in the triple junction case, a $1/2[111]$ screw junction is formed, whose possible cross slip planes are the $(\bar{1}10)$, $(\bar{1}01)$ and $(0\bar{1}1)$ planes. Under uniaxial tension applied along $[100]$, the screw junction will cross slip on the $(\bar{1}10)$ plane.
\subsection{DDD modeling of dislocation junctions}
\label{subsec:application}
\subsubsection{Binary junction}
\label{subsubsec:binary}
The hydrogen elastic shielding and hydrogen dependent mobility law were found to be unimportant over the range of concentrations considered here. Therefore, only hydrogen core force shielding is implemented unless stated otherwise.
Using the binary junction model in \autoref{sec:linetension}, the effect of hydrogen on junction length is studied with DDD. The results are plotted against the line tension model predictions in \autoref{fig:DDDlengths}(b). In contrast to \autoref{fig:DDDlengths}(a), a significant hydrogen effect is captured, indicating the dominant role of hydrogen core force shielding in dislocation junctions. Good agreement is achieved between the DDD and line tension model. Some discrepancy is expected, as DDD accounts for the elastic interactions between segments, which is neglected in the line tension calculations.
A binary junction can be destroyed under an applied stress. The non-junction segments can bow out, unzipping the junction, and restoring the initial configuration.
After a sufficient relaxation time, we apply a uniaxial tensile stress along $[100]$.
As the magnitude of the stress gradually increases, the junction will be unzipped reducing the junction length. We take the stress $\sigma_c$ at which the junction length first becomes zero as the strength of the binary junction. We simulate a hydrogen concentration of $c_0=10.0$appm. The initial dislocation length is varied from $l_0=200b$ to $l_0=800b$. The junction length and strength are shown in \autoref{fig:DDDbistrength}.
\begin{figure}[!ht]
\centering\includegraphics[width=0.9\linewidth]{binarystrength}
\caption{The effect of hydrogen on binary junction (a) length and (b) strength.}
\label{fig:DDDbistrength}
\end{figure}
Increasing the initial dislocation length, increases the junction length (to obtain the equilibrium angles $\Psi_1=\Psi_2\approx 53.5^\circ$ without hydrogen and $\Psi_1=\Psi_2\approx 56.5^\circ$ with hydrogen) and reduces the junction strength. Consistent with the observation in \autoref{fig:DDDlengths}, hydrogen increases the junction length, independently of the initial dislocation length. Hydrogen increases the strength of the junction for mixed initial segments and with initial edge segments the effect is negligible. The effect of hydrogen is dependent on the initial orientation of the intersecting dislocations, even if the line direction of the junction is fixed \citep{Yu2019b}.
To be systematic, we rotated the initial dislocations inside their slip plane, changing their angles ($\phi_1$,$\phi_2$) with respected to the line direction of the junction ($[001]$); we tested all the cases with $\phi_1=\phi_2$ and found that hydrogen increases the strength of the $[001]$ junctions. A typical example with initially mixed segments at $\phi_1=\phi_2=20^\circ$ is shown in \autoref{fig:DDDbistrength}.
It might be assumed that hydrogen increases the junction strength by increasing the junction length, thus shortening the initial segments; making it more difficult for them to bow out and unzip the junction. However, in \autoref{fig:DDDbistrength}(a), a large difference in junction length is observed between the two initial configurations in the absence of hydrogen; in \autoref{fig:DDDbistrength}(b), however, the corresponding junction strengths are identical. This indicates the increase in junction strength due to hydrogen is not attributed to the hydrogen increased junction length but to some other mechanism.
The driving force for dislocation motion is partitioned into the hydrogen core force shielding term, $\bm{F}^{c,H}$, and the hydrogen free forces, $\tilde{\bm{F}}_k + \hat{\bm{F}}_k + \bm{F}^{c}$, as shown in \autoref{fig:biforce}.
\begin{figure}[!ht]
\centering\includegraphics[width=0.9\linewidth]{biforce}
\caption{The nodal force partitioned into hydrogen, $\bm{F}^{c,H}$ (blue arrows), and non-hydrogen parts $\tilde{\bm{F}}_k + \hat{\bm{F}}_k + \bm{F}^{c}$ (green arrows). (a) shows the formation (zipping) of a $[001]$ junction. (b) shows the destruction (unzipping) of the junction. The end points of the junction are marked with red circles; note ${\bm{F}}^{c,H}$ has been scaled $5\times$ for clarity.}
\label{fig:biforce}
\end{figure}
During formation, hydrogen lowers the core energy of the $\langle100\rangle$ junction. The hydrogen core forces act to stretch the junction, (reducing the total core force) and increasing the junction length. The destruction of the junction is realised via a procedure similar to the operation of a F-R source. A $1/2[111]$ arm which is pinned at one end and constrained to move along the line of the junction at the other end is activated and bows out, which drives the mobile node along the line, unzipping the junction. The hydrogen core force tends to impede the expansion of this F-R source making the destruction of the junction more difficult. Finally, it should be noted that the opposite was observed in fcc materials where hydrogen elastic shielding weakens Lomer junctions \citep{Yu2019b}.
\subsubsection{Triple junction}
\label{subsubsec:triple}
Triple junctions are stronger than binary junctions. \cite{Bulatov2006} observed more pronounced strain hardening in the presence of triple junctions and attributed the phenomenon to the propensity of these junctions to form additional dislocation sources. Triple junctions occur at a late stage of loading, serving as strong anchors which form complex dislocation networks which impede dislocation motion. Unlike the $\langle001\rangle$ binary junctions which are sessile, the $1/2\langle111\rangle$ triple junctions are glissile and considerably longer than the other $1/2\langle111\rangle$ arms. As a result, the junction (rather than the non-junction segments), will bow out under an external stress, meaning rather than unzipping, the junction operates like a F-R source.
Therefore, the strength of a triple junction is defined here as the stress required to nucleate a loop. After the pure screw triple junction is formed, out-of-plane pure shear is applied to one of its possible cross slip planes, the $(1\bar{1}0)$ plane,
and the motion of the $1/2[111]$ screw dislocation is limited to this plane.
When the applied stress is sufficiently high, the triple junction will bow out and generate dislocation loops, as shown in \autoref{fig:tripleFRprocess}.
\begin{figure}[!ht]
\centering\includegraphics[width=0.9\linewidth]{tripleFRprocess}
\caption{A screw triple junction operating as a F-R source. (a) shows the initial triple junction before bow-out; (b) shows the interaction between the $1/2[111]$ (black) dislocation loop and the $1/2[11\bar{1}]$ (green) arm on the left, forming a $[00\bar{1}]$ sessile dislocation (red): $1/2[11\bar{1}](1\bar{1}0)+1/2[\bar{1}\bar{1}\bar{1}](1\bar{1}0)\rightarrow [00\bar{1}](1\bar{1}0)$; (c) the $1/2[111]$ loop has bowed back restoring the $1/2[11\bar{1}]$ arm and the process then occurs on the right. (d) Shows the moment just before the emission of a full loop, restoring the triple junction as in (a) and the process repeats.}
\label{fig:tripleFRprocess}
\end{figure}
This procedure is similar to the F-R operation but has some unique features which haven't been discussed before. On each of the possible cross slip planes (i.e. the possible F-R operation planes), there lie a pair of $1/2\langle111\rangle$ arms formed during the relaxation process. The bowing out loop will inevitably interact with one of these pairs during the operation. In the case selected here, it is the $1/2[11\bar{1}]$ (green) pair. As shown in \autoref{fig:tripleFRprocess}(b), the junction encounters the arm on the left as soon as it bows out, which yields a sessile $[00\bar{1}]$ segment (red): $1/2[11\bar{1}](1\bar{1}0)+1/2[\bar{1}\bar{1}\bar{1}](1\bar{1}0)\rightarrow [00\bar{1}](1\bar{1}0)$. Note that the line direction of the junction is opposite to the $1/2[11\bar{1}]$ arm when they first interact. This impedes the expansion of the loop on the left and leads to the unsymmetric development of the loop as shown in \autoref{fig:tripleFRprocess}(b). As expansion continues, the loop will eventually bypass the pinning point, bow sharply back and wrap around the now sessile arm (red) restoring it back to it's initial state (green): $[00\bar{1}](1\bar{1}0)+1/2[111](1\bar{1}0)\rightarrow 1/2[11\bar{1}](1\bar{1}0)$. The same process, where a tempory sessile $[00\bar{1}]$ segment is formed, then occurs on the other pinning point; as shown on the right of \autoref{fig:tripleFRprocess}(c).
During the activation of a F-R source and similarly of the triple junction, there exists a certain geometric configuration beyond which the dislocation line becomes unstable and keeps growing and emits loops. During the simulation, the magnitude of the applied stress is gradually increased until the triple junction starts to operate and reaches the critical geometry \citep{Cai2018}, and thus the critical activation stress $\tau_c$, i.e. junction strength, is determined. We have verified that the predicted activation stresses obtained for pure screw F-R sources in the absence of hydrogen agree with the $\frac{3}{2}\mu b/L$ relation expected by theoretical calculation \citep{anderson2017}.
The formation length and destruction strength of the triple junctions of varying length with and without hydrogen ($c_0=10.0$appm) are shown in \autoref{fig:triplestrength}.
\begin{figure}[!ht]
\centering\includegraphics[width=0.9\linewidth]{triplestrength}
\caption{Variation of (a) triple junction length and (b) strength with initial dislocation length, with and without hydrogen.}
\label{fig:triplestrength}
\end{figure}
Consistent with the results in \autoref{fig:DDDlengths}(b), the hydrogen core force slightly reduces the length and strength of the triple junction. However the effect is weak as it is a second order effect.
The nodal forces are shown in \autoref{fig:tripleforce}.
\begin{figure}[!ht]
\centering\includegraphics[width=0.9\linewidth]{tripleforce}
\caption{The nodal force partitioned into hydrogen core force, ${\bm{F}}^{c,H}$ (blue arrows), and non-hydrogen parts $\tilde{\bm{F}}_k + \hat{\bm{F}}_k$ (green arrows). (a) shows the formation of the triple junction. (b) shows the bow-out of the junction. The ending points of the junction in (a) are marked with red circles; note ${\bm{F}}^{c,H}$ has been scaled $20\times$ with respect to $\tilde{\bm{F}}_k + \hat{\bm{F}}_k$ for clarity.}
\label{fig:tripleforce}
\end{figure}
In \autoref{fig:tripleforce}(a), the hydrogen core force tends to shrink the triple junction. During the activation of the triple junction, as shown in \autoref{fig:tripleforce}(b), the hydrogen core force tends to enhance bowing out of the mixed dislocation segments while impeding the edge segments, which facilitates the bow-out and nucleation of a loop. This accounts for the slight reduction in stress with hydrogen in \autoref{fig:triplestrength}(b). A hydrogen dependent mobility law \citep{Yu2019a} was not incorporated in the current simulations. A significant enhancement in the operation of the triple junction/F-R source is expected if hydrogen enhanced screw mobility is implemented.
It is hard to anticipate the consequence of hydrogen on the mechanical response by analysis of individual junctions. For this purpose, hydrogen-junction interactions should be discussed in the context of a complex dislocation network formed in a volume of material under an applied load.
\subsection{Microcantilever simulation with hydrogen core force shielding}
\label{subsec:microcantilever}
To examine if hydrogen core force shielding could effect mechanical properties we performed DDD simulations of a microcantilever bend test. The beam is aligned along the $\langle 100 \rangle$ crystallographic axes and has dimensions of $12\times3\times3~\mu\textrm{m}$ with $\bm{u}(0,y,z)=[0,0,0]$, $\bm{u}(L_x,y,L_z) = [0,0,U]$ with an applied displacement of $U=-0.08~\mu\textrm{m}$, and the remaining surfaces are traction free. Fully integrated linear brick finite elements, $0.24\mu \textrm{m}$ in size, were used to solve for the corrective elastic fields at every time increment \citep{Yu2019a}. The initial dislocation structure consisted of $40$ square prismatic loops with a length of $0.39\mu \textrm{m}$ randomly generated from the three slip systems of interest in \autoref{eq:triple}, $1/2[\bar{1}11](01\bar{1})$, $1/2[1\bar{1}1](10\bar{1})$ and $1/2[11\bar{1}](1\bar{1}0)$, positioned randomly in the high stress region near the fixed end $x<4~\mu\textrm{m}$. This enables the formation of binary junctions as well as triple and higher order junctions referred to here as multi junctions. A hydrogen concentration of $10$ appm was applied when computing the core force, apart from this the simulations with and without hydrogen were identical.
\begin{figure}[!ht]
\centering\includegraphics[width=0.95\linewidth]{load-disp}
\caption{(a) The load-displacement curve; (b) the evolution of the total dislocation density.}
\label{fig:load-disp}
\end{figure}
The evolution of the total dislocation density, together with the load-displacement curves are plotted in \autoref{fig:load-disp}. During the early loading stage ($0.03<U<0.07~\mu\textrm{m}$), the dislocation density is slightly higher in the presence of hydrogen, due to hydrogen promoted dislocation generation from the prismatic loops, due to a similar mechanism as illustrated in \autoref{fig:tripleforce}(b). Consequently, yielding occurs earlier with hydrogen.
\begin{figure}[!ht]
\centering\includegraphics[width=0.95\linewidth]{junctiondensity}
\caption{(a) The evolution of binary dislocation junction density; (b) the evolution of multiple dislocation junction density.}
\label{fig:junctiondensity}
\end{figure}
The evolution of the binary and multi junction densities with and without hydrogen are shown in \autoref{fig:junctiondensity}. During the entire loading history, binary junctions are the dominant junction type. The density of binary junctions is significantly higher in the presence of hydrogen, as shown in \autoref{fig:junctiondensity}(a). This is consistent with hydrogen increasing the length and strength of binary junctions as discussed earlier. In \autoref{fig:junctiondensity}(b), hydrogen also increases the proportion of multi junctions, which usually form when a mobile dislocation interacts with a binary junction to form a triple junction.
The increased junction density with hydrogen eventually reduces the mobile density production rate leading to subsequent hardening, as evident in \autoref{fig:load-disp}(a) for $U>0.07 ~\mu\textrm{m}$. This corresponds to the mobile density with hydrogen dropping below that without hydrogen. A snap shot (at $U=-0.0524\mu m$) of the dislocation structure in the initial (hydrogen softening) stage is shown in \autoref{fig:dislocationstruct}.
\begin{figure}[!ht]
\centering\includegraphics[width=0.9\linewidth]{dislocationstruct}
\caption{Dislocation structure in the initial loading stage $U=-0.0524~\mu\textrm{m}$: (a) without hydrogen and (b) with hydrogen. The dislocations belonging to the three initial slip systems are $1/2[\bar{1}11](01\bar{1})$ (black), $1/2[1\bar{1}1](10\bar{1})$ (blue) and $1/2[11\bar{1}](1\bar{1}0)$ (green) with binary junctions (red) and multiple dislocation junctions (magenta).}
\label{fig:dislocationstruct}
\end{figure}
The $1/2[1\bar{1}1](10\bar{1})$ and $1/2[11\bar{1}](1\bar{1}0)$ slip systems have the highest Schmid factor and so are the most active and make up the majority of the mobile density. The higher binary junction density (red) with hydrogen is also visible in \autoref{fig:dislocationstruct}(b).
The dislocation structure at the end of the simulation ($U=-0.09\mu m$) is shown in \autoref{fig:dislocationstruct2}
\begin{figure}[!ht]
\centering\includegraphics[width=0.9\linewidth]{dislocationstruct2}
\caption{Dislocation structure at $U=-0.09\mu m$: (a) without hydrogen and (b) with hydrogen. With the same colour coding as \autoref{fig:dislocationstruct}.}
\label{fig:dislocationstruct2}
\end{figure}
Comparing \autoref{fig:dislocationstruct2}(a) with \autoref{fig:dislocationstruct}(a) shows an increase in the mobile density (green and blue) in the absence of hydrogen. Comparing \autoref{fig:dislocationstruct2}(b) with \autoref{fig:dislocationstruct2}(a) shows the higher junction density and lower mobile density with hydrogen.
Hydrogen significantly increases the density of junctions, leading to more complicated dislocation tangles, which will inevitably impede the motion and generation of dislocations as loading increases. Therefore, the overall influence of hydrogen core force shielding is expected to be a slight softening effect initially followed by a hardening effect. Recently, \cite{Wang2019} observed that hydrogen decreases the yield stress and increases strain hardening in an fcc TWIP steel and hydrogen hardening was observed in bcc iron, both experimentally \citep{Depover2016} and numerically \citep{Xie2011}.
\section{Summary}
\label{sec:sum}
The hydrogen core force was calibrated using atomistic modelling and shows that hydrogen reduces or shields the core force of edge dislocations. This effect plays a dominant role in the formation of dislocation junctions even at low concentrations, where the contribution from hydrogen elastic shielding is negligible.
Hydrogen was found to increase the length and strength of $\langle100\rangle$ binary edge junctions whereas the influence on a $1/2\langle111\rangle$ screw triple junctions was negligible. \textcolor{black}{However the increased binary junction strength leads to a significant increase in the number of triple junctions forming in microscale plasticity simulations}. The triple junction can operate like a F-R source, hydrogen facilitates this by exerting a core force promoting the bowing-out of the junction and enhancing the expansion of the nucleated loops. By strengthening binary junctions, hydrogen tends to limit the motion and generation of dislocations; while it tends to accelerate dislocation generation from the triple junctions.
Hydrogen was found to increase the total dislocation density and cause plastic softening at an early stage in a microcantilever. Hydrogen was also observed to increase the density of binary and multi junctions. Junctions act as obstacles to dislocation motion. Consequently, dislocation generation and motion were suppressed by hydrogen as the load increased, enhancing strain hardening. \textcolor{black}{The hydrogen core force shielding effect analysed here appears to share many similarities with the hydrogen elastic shielding effect but occurs at much lower concentrations as typically found in bcc materials.}
\section*{Acknowledgements}
This work was supported by the Engineering and Physical Sciences Research Council (EPSRC) under Programme grant EP/L014742/1 and Fellowship grant EP/N007239/1; IHK acknowledges support from Bulgaria National Science Fund (BNSF) under Programme grant KP-06-H27/19; HY acknowledges Bruce Bromage for useful discussions.
\section*{References}
\bibliographystyle{model2-names}
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1,116,691,499,259 | arxiv | \subsection{\@startsection
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\begin{center}
{\sc\large Elena Angelini}
\end{center}
\vspace {1.5cm}
\centerline{\large{\textbf{On complex and real identifiability of tensors}}}
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\footnotetext{
This research was partially supported by the Italian GNSAGA-INDAM and by the Italian PRIN2015 - Geometry of Algebraic Varieties (B16J15002000005)}
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\small{ \noindent \textbf{Abstract.} We report about the state of the art on complex and real generic identifiability of tensors, we describe some of our recent results obtained in \cite{ABC} and we present perspectives on the subject.
\medskip
\noindent \textbf{Keywords.} Tensor decomposition, complex identifiability, real identifiability, elliptic curves.
\medskip
\noindent \textbf{Mathematics~Subject~Classification~(2010):}
14N05, 15A69, 14P05.
}
\end{minipage}
\end{center}
\bigskip
\section{Introduction}
Identifiability is a very special property for tensors. This interesting topic has been extensively investigated starting from the $XIX^{th}$ century, related to the Waring problem \cite{Hi,K,R,S,Sy,We}, however the theory is far from being complete. The notion of identifiability has many important applications, that go beyond Algebraic Geometry: for example, in engineering, with the Blind Source Separation problem \cite{AGHKT}, in chemistry, when one deals with mixtures of fluorophores \cite{AD}, or in statistics \cite{AMR}.
The paper is organized as follows. In section $ 2 $ we introduce preliminary material. Section $ 3 $ is devoted to the complex case, with particular emphasis on symmetric, partially-symmetric and skew-symmetric tensors, which are three significant special cases. In section $ 4 $ we focus on the real case, describing some of our recent results obtained in \cite{ABC}. Finally in section $ 5 $ we present open problems and perspectives on the subject.
The main purpose of this note is to report about the state of the art on generic identifiability of tensors, stressing, in a clear and simple way, analogies and differences between the complex and real case.
\section{Preliminaries}
In this section we recall basic definitions about tensor decomposition, rank and identifiability, mainly referring to the introductory handbook \cite{Lan}. Moreover we describe some properties of elliptic normal curves, that play a special role in the discovery of unidentifiable cases.
\subsection{Tensor decomposition and rank}
Let $\mathbb{F}$ be either the complex field $ \mathbb{C} $ or the real field $ \mathbb{R} $, let $ n_{1}, \ldots, n_{d} \in \mathbb{N} $ such that $ n_{1} \leq \ldots \leq n_{d} $ and let $ \mathbb{F}^{n_{1}} \otimes \ldots \otimes \mathbb{F}^{n_{d}} $ be the space of tensors of type $ n_{1}, \ldots, n_{d} $ over $\mathbb{F}$. Let $ T \in\mathbb{F}^{n_{1}} \otimes \ldots \otimes \mathbb{F}^{n_{d}} $.
\begin{definition}
A \emph{tensor decomposition} of $ T $ is given by vectors $ {\bold{v}}_{i}^{j} \in\mathbb{F}^{n_{j}} $, with $ i \in\{1, \ldots, k\} $ and $ j \in \{1, \ldots, d\} $, such that
\begin{equation}\label{eq:tensordec}
T = \displaystyle{\sum_{i=1}^{k} {\bold{v}}_{i}^{1} \otimes \ldots \otimes {\bold{v}}_{i}^{d}}.
\end{equation}
\end{definition}
\begin{definition}
The minimal $ k $ appearing in (\ref{eq:tensordec}) is the \emph{rank} of $ T $ \emph{over} $ \mathbb{F} $.
\end{definition}
\begin{remark}
Any $ {\bold{v}}_{i}^{1} \otimes \ldots \otimes {\bold{v}}_{i}^{d} $ has rank $ 1 $ over $ \mathbb{F} $.
\end{remark}
\begin{definition}
A \emph{typical rank} over $ \mathbb{F} $ for tensors of type $ n_{1}, \ldots, n_{d} $ is any $ k $ such that the set of tensors having rank $ k $ has positive Euclidean measure.
\end{definition}
According to section 5.2.1 of \cite{Lan}, we introduce the following:
\begin{definition}
There exists a unique typical rank for tensors of type $ n_{1},\dots, n_{d} $ over $ \mathbb{C} $, which we call the \emph{generic rank} for that space of tensors.
\end{definition}
\begin{remark}
The \emph{expected generic rank} for $ \mathbb{C}^{n_{1}} \otimes \ldots \otimes \mathbb{C}^{n_{d}} $ is
$$ k_{g} = \left\lceil \frac{\prod_{i=1}^{d}n_{i}}{1+\sum_{i=1}^{d}(n_{i}-1)} \right\rceil. $$
\end{remark}
According to \cite{COV}, we give the following:
\begin{definition}
Tensors of type $ n_{1}, \dots , n_{d} $ over $ \mathbb{C} $ and rank $ k < k_{g} $ are of \emph{sub-generic rank}.
\end{definition}
\begin{remark}\label{rem:c&r}
It is well known that it is possible to have more than one typical rank for tensors of type $ n_{1}, \ldots, n_{d} $ over $ \mathbb{R} $. The smallest typical rank over $ \mathbb{R} $ coincides with the generic rank over $ \mathbb{C} $ \cite{BT}. Any rank over $ \mathbb{R} $ between the minimal typical rank and the maximal typical rank is also typical and it is an open problem to determine an expression for the expected maximal typical rank over $ \mathbb{R} $ in the general case. Partial results have been obtained for symmetric tensors \cite{BBO}.
\end{remark}
\subsection{Identifiability}\label{subsec:id}
\begin{definition}
A rank-$ k $ tensor $ T \in \mathbb{F}^{n_{1}} \otimes \ldots \otimes \mathbb{F}^{n_{d}} $ is \emph{identifiable over} $ \mathbb{F} $ if the presentation (\ref{eq:tensordec}) is unique up to a permutation of the summands and scaling of the vectors.
\end{definition}
The above definition extends as follows:
\begin{definition}
The set of tensors of type $ n_{1}, \ldots, n_{d} $ and rank $ k $ over $\mathbb{F}$ is \emph{generically identifiable over} $ \mathbb{F} $ if identifiability over $ \mathbb{F} $ holds on a Zariski dense open subset of the variety of tensors of rank $ \leq k $.
\end{definition}
\subsection{Elliptic normal curves}\label{subsec:enc}
\indent Let $ \mathbb{P}^{n} = {\mathbb{P}}_{\mathbb{C}}^{n} $ be the $ n $-dimensional complex projective space.
\begin{definition}
An \emph{elliptic normal curve} $ \mathcal{C} \subset \mathbb{P}^{n} $ is a smooth curve of genus $ 1 $ and degree $ n+1 $ that is contained in no hyperplane.
\end{definition}
For any such curve $ \mathcal{C} \subset \mathbb{P}^{2k-1} $ of even degree $ n+1 = 2k $, it is known that $ k-1 $ is the minimal dimension of a $ \mathbb{P}^{h-1} $ $ h $-secant to $ \mathcal{C} $ which contains the general point $ P \in \mathbb{P}^{2k-1} $. In particular the following holds:
\begin{proposition}[Chiantini-Ciliberto, 2006, \cite{CC}]\label{prop:CC}
Under the above assumptions, the number of $ k $-secant $ \mathbb{P}^{k-1} $ to $ \mathcal{C} $ passing through $ P $ is $ 2 $.
\end{proposition}
\begin{remark}
As a consequence of Proposition \ref{prop:CC}, we can prove the existence of two decompositions over $ \mathbb{C} $ in some cases. Precisely, up to now, the only known examples are those listed in Table $ 4 $ of section $ 4 $. Indeed, let $ \mathbb{T} $ be the space of tensors object of study and $ \mathbb{T}_{1} \subset \mathbb{P}(\mathbb{T}) $ the corresponding variety of rank-$1$ tensors over $ \mathbb{C} $. We choose a general tensor $ T \in \mathbb{P}(\mathbb{T}) $ of rank $ k $ over $ \mathbb{C} $ and one of its complex decompositions, i.e.\ $T_{1}, \ldots, T_{k} \in \mathbb{T}_{1} $ such that $ T = \sum_{i=1}^{k} T_{i} $. We can show that there exists a unique irreducible elliptic normal curve $ \mathcal{C} $ of even degree $ 2k $ entirely contained in $ \mathbb{T}_{1} $, passing through the summands $ T_{1}, \ldots, T_{k} $ and such that $ T $ belongs to the odd dimensional projective space $ \mathbb{P}^{2k-1} $ spanned by $ \mathcal{C} $. By Proposition \ref{prop:CC}, $ T $ belongs to two $\mathbb{P}^{k-1}$ $k$-secant to $ \mathcal{C} $, which provide the two complex decompositions of $ T $. We notice that, for the cases of sub-generic rank, the elliptic normal curve is the \emph{$ k $-contact locus} of $ \mathbb{T}_{1} $ (see \cite{CC} for more details on this concept). As will be explained in our forthcoming paper \cite{ABC1}, for the last example of Table $ 4 $ it is not possible to define a contact locus, since in this case the $k$-secant variety of $ \mathbb{T}_{1} $ fills the ambient space, so that the curve represents a ``known" subvariety of $ \mathbb{T}_{1} $ bounding the decompositions of $ T $.
\end{remark}
\section{Complex identifiability}
Generic identifiability over $ \mathbb{C} $ has been largely investigated, in particular we refer to \cite{BCV,BC,BCO,CO,COV,COV1,K,S}.
The first interesting result concerns the case $ d = 3 $:
\begin{theorem}[Kruskal, 1977, \cite{K}]\label{thm:K}
The set of tensors of type $ n_{1},n_{2},n_{3} $ and rank $ k $ over $ \mathbb{C} $ is generically identifiable over $ \mathbb{C} $ if
$$ k \leq \frac{1}{2} (\min(n_{1},k) + \min(n_{2},k) + \min(n_{3},k) -2). $$
\end{theorem}
Theorem \ref{thm:K} has been improved for $ n_{1} = n_{2} = n_{3} = n $ odd as follows:
\begin{theorem}[Strassen, 1983, \cite{S}]
The set of tensors of type $ n, n, n $ with $ n $ odd and rank $ k $ over $ \mathbb{C} $ is generically identifiable over $ \mathbb{C} $ if
$$ k \leq \left\lfloor\frac{n^3}{3n-2} \right\rfloor - n. $$
\end{theorem}
Furthermore we have the following:
\begin{theorem}[Bocci-Chiantini-Ottaviani, 2014, \cite{BCO}]
The set of tensors of type $ n_{1},n_{2},n_{3} $ with $ 3 < n_{1} \leq n_{2} \leq n_{3} $ and rank $ k $ over $ \mathbb{C} $ is generically identifiable over $ \mathbb{C} $ if
$$ k \leq \frac{n_{1}n_{2}n_{3}}{n_{1}+n_{2}+n_{3}-2}-n_{3}. $$
\end{theorem}
Admitting a unique minimal decomposition is a quite rare phenomenon for tensors of generic rank over $ \mathbb{C} $. As an example, in the \emph{symmetric} setting, i.e. \ when $ n_{1} = \ldots = n_{d} = n $ and we deal with Sym$^{d}\mathbb{C}^n \subset (\mathbb{C}^n)^{\otimes d}$, the complex identifiable cases of tensors of generic rank are only the one classically known described in Table $ 1 $, as stated in \cite{GM}. We would like to emphasize that the first and third examples are due to Sylvester, while the second one to Hilbert.
\begin{table}[h]
\begin{center}
\begin{tabular} {c |c |c |c }
{\bf Space of tensors} & {\bf Feature} & {\bf Rank} & {\bf Ref.} \\
$(\mathbb{C}^2)^{\otimes 2t+1}$ & symmetric & $t+1$ & \cite{Sy} \\
$(\mathbb{C}^3)^{\otimes 5}$ & symmetric & $7$ & \cite{Hi} \\
$(\mathbb{C}^4)^{\otimes 3}$ & symmetric & $5$ & \cite{Sy} \\
\end{tabular}\caption{Symmetric generically identifiable cases of generic rank over $ \mathbb{C} $}
\end{center}
\end{table}
\noindent In the \emph{partially-symmetric} case, that is when we work with vectors of symmetric tensors in the same variables, searching for simultaneous decompositions, the complete list of identifiable cases of generic rank is not known. The discovered ones, up to now, are collected in Table $ 2 $. Besides of Veronese's result, which goes back to 1880, the second and fourth examples are classical too, respectively, due to Weierstrass and Roberts. For a more detailed discussion on simultaneous decompositions of vectors of symmetric tensors and identifiability, we refer to our papers \cite{A} and \cite{AGMO}.
\begin{table}[h]
\begin{center}
\begin{tabular} {c |c |c |c }
{\bf Space of tensors} & {\bf Feature} & {\bf Rank} & {\bf Ref.} \\
$\bigoplus_{j=1}^{r} (\mathbb{C}^2)^{\otimes d_{j}}, \, d_{1}+1 \geq k $ & part.-symm. & $ \left \lceil {\frac{1}{1+r}}\sum_{j=1}^{r}{1+d_{j} \choose d_{j}} \right \rceil $ & \cite{CR} \\
$((\mathbb{C}^n)^{\otimes 2})^{\oplus 2} $ & part.-symm. & $n+1$ & \cite{We} \\
$((\mathbb{C}^3)^{\otimes 2})^{\oplus 4}$ & part.-symm. & $4$ & Veronese \\
$(\mathbb{C}^3)^{\otimes 2} \oplus (\mathbb{C}^3)^{\otimes 3} $ & part.-symm. & $4$ & \cite{R} \\
$((\mathbb{C}^3)^{\otimes 3})^{\oplus 2} \oplus (\mathbb{C}^3)^{\otimes 4} $ & part.-symm. & $7$ & \cite{AGMO} \\
\end{tabular}\caption{Part.-symm. generically identifiable cases of generic rank over $ \mathbb{C} $}
\end{center}
\end{table}
On the other side, generic identifiability over $ \mathbb{C} $ is expected for tensors of sub-generic rank over $ \mathbb{C} $. In this direction, by means of an algorithm based on the so-called \emph{Hessian criterion}, in \cite{COV} it is proved generic identifiability over $ \mathbb{C} $ for a large number of spaces of tensors in sub-generic rank cases. For the symmetric case (with $ d \geq 3 $), in \cite{COV1} it is proved that there are no exceptions besides the ones appearing in Table $ 3 $.
\begin{table}[h]
\begin{center}
\begin{tabular} {c |c |c |c }
{\bf Space of tensors} & {\bf Feature} & {\bf Rank} & {\bf Ref.} \\
$(\mathbb{C}^3)^{\otimes 6}$ & symmetric & $9$ & \cite{AC,CC} \\
$(\mathbb{C}^4)^{\otimes 4}$ & symmetric & $8$ & \cite{CC,M}\\
$(\mathbb{C}^6)^{\otimes 3}$ & symmetric & $9$ & \cite{COV1,RV}\\
\end{tabular}\caption{Symmetric generically unidentifiable cases of sub-generic rank over $ \mathbb{C} $}
\end{center}
\end{table}
\noindent An account of this topic in the partially-symmetric setting can be found in our forthcoming paper \cite{A}.
Concerning generic identifiability over $ \mathbb{C} $ of \emph{skew-symmetric} cases, i.e.\ when $ n_{1} = \ldots = n_{d} = n $ and it is investigated $ \bigwedge^d \mathbb{C}^n \subset (\mathbb{C}^n)^{\otimes d}$, we refer to \cite{BV} (Theorem 1.1).
Besides the cases described above, generic identifiability over $ \mathbb{C} $ does not hold anytime the projective algebraic variety $ \mathbb{T}_{1} $ of rank-$1$ tensors object of study (Segre, Veronese and Grassmann variety, respectively, for the general, the symmetric and the skew-symmetric case) is \emph{$ k $-defective} (we refer to \cite{Lan} for the basics about secant varieties and the defectivity problem and to \cite{AOP,AH,BDdG} for an account on Segre, Veronese and Grassmann defective varieties), in which case we deal with rank-$k$ tensors admitting infinitely many complex decompositions.
\section{Real identifiability}\label{sec:recres}
Recent interest has been devoted to the real case \cite{COV2,CLQ,DDL}, very useful in applications.
If a set of tensors, of fixed type and rank over $\mathbb{C}$, is generically identifiable over $ \mathbb{C} $, then, necessarily, the same set of tensors seen over $\mathbb{R}$ is generically identifiable over $ \mathbb{R} $. Equivalently, with the above assumption, the unique complex decomposition of the general real tensor of this type and rank over $\mathbb{R}$ is completely real.
At this point a natural question arises: if identifiability over $ \mathbb{C} $ fails, what happens to identifiability over $\mathbb{R}$? For example, one may wonder if there exist real tensors admitting several decompositions in terms of complex rank-$ 1 $ summands but only one of them is a decomposition over $ \mathbb{R} $.
In \cite{ABC} we answered this question in some cases.
The main result is for tensors having two complex decompositions explained through elliptic normal curves:
\begin{theorem}[Angelini-Bocci-Chiantini, 2017, \cite{ABC}]\label{thm:ABC}
Let $ \mathbb{T} $ be any space of tensors in Table $ 4 $ of fixed feature and $ k $ the rank over $ \mathbb{C} $ under investigation.
\begin{table}[h]
\begin{tabular} {c |c |c |c }
{\bf Space of tensors} & {\bf Feature} & {\bf Rank} & {\bf Ref.} \\
$(\mathbb{C}^2)^{\otimes 5}$ & & $5$ (sg) & \cite{BC} \\
$(\mathbb{C}^3)^{\otimes 6}$ & symmetric & $9$ (sg) & \cite{AC,CC} \\
$(\mathbb{C}^4)^{\otimes 3}$ & & $6$ (sg) & \cite{CO} \\
$(\mathbb{C}^4)^{\otimes 4}$ & symmetric & $8$ (sg) & \cite{CC,M}\\
$(\mathbb{C}^6)^{\otimes 3}$ & symmetric & $9$ (sg) & \cite{COV1,RV}\\
$(\mathbb{C}^{10})^{\otimes 3}$ & skew-symmetric & $5$ (sg) & \cite{BV} \\
$((\mathbb{C}^{3})^{\otimes 3})^{\oplus 3} \oplus ((\mathbb{C}^{3})^{\otimes 2})^{\oplus m}, \, m \geq 0 $ & partially-symmetric & $6$ (g) & \cite{ABC1,L} \\
\end{tabular}\caption{Gen. unidentifiable cases of sub-generic and generic rank over $ \mathbb{C} $}
\end{table}
\noindent Then there exist non-trivial Euclidean open subsets $ U_{1}, U_{2}, U_{3} $ of the variety of real tensors in $ \mathbb{T} $ of rank $ \leq k $ over $ \mathbb{C} $, whose elements have two complex decompositions and:
\begin{itemize}
\item[$\bullet$] $ \forall \, T \in U_{1} $, only one decomposition is real;
\item[$\bullet$] $ \forall \, T \in U_{2} $, both decompositions are real;
\item[$\bullet$] $ \forall \, T \in U_{3} $, both decompositions are not real.
\end{itemize}
\end{theorem}
In column $ 2 $ of Table $ 4 $, when not declared, we intend any tensor of the corresponding space. The last row is devoted to the partially-symmetric case of vectors of symmetric tensors with three ternary cubics and an arbitrary number of conics. This example will be extensively treated in our forthcoming paper \cite{ABC1}.
The proof of Theorem \ref{thm:ABC}, the details of which we refer to \cite{ABC}, is based on a study of real elliptic normal curves of even degree $ 2k $ in projective spaces of odd dimension $ 2k-1 $, through techniques of projective geometry. In particular we first describe the case of quartics showing the following:
\begin{proposition}[Angelini-Bocci-Chiantini, 2017, \cite{ABC}]\label{prop:quartics}
Let $ \mathcal{C} \subset \mathbb{P}^{3} $ be an irreducible real elliptic normal quartic. Then there exist $ B_{1}, B_{2}, B_{3}, B_{4} \subset {\mathbb{P}}_{\mathbb{R}}^{3} $ non-trivial open balls entirely composed of points $ P $ such that the two secant lines to $ \mathcal{C} $ through $ P $ intersect $ \mathcal{C} $, respectively, in $ 4 $ real points, $ 2 $ real and $ 2 $ not real conjugate points, $ 4 $ not real points pairwise conjugate (for $ B_{3} $ the conjugate points being on the same line and for $ B_{4} $ otherwise).
\end{proposition}
By induction on $ k $ we extend Proposition \ref{prop:quartics} to curves in higher dimensional spaces getting the following:
\begin{proposition}[Angelini-Bocci-Chiantini, 2017, \cite{ABC}]\label{prop:curves}
Let $ \mathcal{C} \subset \mathbb{P}^{2k-1} $ be an irreducible real elliptic normal curve of degree $ 2k $. Then, for any $ (a_{1}, a_{2})\in \mathbb{N}^2 $ such that $ 0 \leq 2a_{1}+2a_{2} \leq k $, there exists $ B_{a_{1},a_{2}}\subset {\mathbb{P}}_{\mathbb{R}}^{2k-1} $ non-trivial open ball entirely composed of points $ P $ such that each of the two secant spaces $ \Pi_{i}(P) \cong \mathbb{P}^{k-1} $, $i \in \{1,2\}$, to $ \mathcal{C} $ through $ P $ intersects $ \mathcal{C} $ in k-2$a_{i}$ real points and 2$a_{i}$ not real points.
\end{proposition}
We notice that, with the notation introduced in Proposition \ref{prop:curves}, if one of the $ a_{i} $'s equals $ 0 $ and the other one is different from $ 0 $, then we get real points $ P $ admitting a unique real decomposition with respect to $ \mathcal{C} $. If $ a_{1} = a_{2} = 0 $, then the two decompositions are real. Finally, if the $ a_{i} $'s are both different from $ 0 $, then the two decompositions are complex but not real. In particular, the first case will lead to real identifiability.
Proposition \ref{prop:curves} can be generalized to families of irreducible real elliptic normal curves. This step allows us to pass from curves to tensors, obtaining the proof of Theorem \ref{thm:ABC}.
\section{Open problems and perspectives}\label{sec:persoppbs}
We conclude this note by presenting a short list of open problems that have arisen mainly from discussions within the seminars I held on the subject.\\
\indent $1)$ Classifying spaces of tensors, of fixed type and rank, such that the general tensor has two complex decompositions, due to the presence of elliptic normal curves, is, at the moment, an open problem.
$2)$ According to Remark \ref{rem:c&r}, from Theorem \ref{thm:ABC} we immediately get some information about the rank over $ \mathbb{R} $ and identifiability over $ \mathbb{R} $ of tensors in $ U_{1}, U_{2}, U_{3} $. Indeed, in $ U_{1} $ the rank over $ \mathbb{R} $ equals the rank over $ \mathbb{C} $ and real identifiability holds. In $ U_{2} $ the same condition on the ranks holds but identifiability over $ \mathbb{R} $ fails. In $ U_{3} $ the rank over $ \mathbb{R} $ is strictly greater than the rank over $ \mathbb{C} $ and we can't say anything about real identifiability. Therefore we can ``recover" identifiability only in $ U_{1} $. At this point it is reasonable to ask how large the three open sets of Theorem \ref{thm:ABC} are and, till now, also determining an estimate of their measure represents an interesting open problem.
$3)$ It is not known completely how identifiability over $ \mathbb{R} $ behaves when identifiability over $ \mathbb{C} $ fails for reasons other than the presence of elliptic normal curves. In this direction, in \cite{ABC} we showed that, if the general tensor of rank $ k $ over $ \mathbb{C} $ admits infinitely many decompositions, then there are no Euclidean open sets of tensors in which real identifiability holds.
$4)$ Several situations seem to occur in the cases with a finite number, greater than $ 1 $, of complex decompositions. For instance, let us consider tensors with complex sub-generic rank $ k = n_{d}-1 $ in \emph{almost unbalanced} spaces $ \mathbb{C}^{n_{1}} \otimes \ldots \otimes \mathbb{C}^{n_{d}} $ (i.e.\ $ n_{1} \leq \ldots \leq n_{d} $ and $ n_{d}-1 = \prod_{i=1}^{d-1}(n_{i}) - \sum_{i=1}^{d-1}(n_{i}-1) $: it is known, by \cite{BCO}, that they have $ {D}\choose{k} $ complex decompositions, with $ D = \deg(\mathbb{P}^{n_{1}-1} \times \ldots \times \mathbb{P}^{n_{d-1}-1} ) $, and we proved in \cite{ABC} that, whenever $ D - n_{d}+1 $ is odd, then there are no Euclidean open sets where identifiability over $ \mathbb{R} $ holds, while, when $ D - n_{d}+1 $ is even, we believe in the existence of such open sets. In this sense, this is true when $ d = n_{1} = 3, n_{2} = 5, n_{3} = 10 $, see Example 5.2 of \cite{ABC}. Concerning the cases of complex generic rank, we can apply the computer-aided procedure introduced in \cite{AGMO} and based on \emph{homotopy continuation techniques} and \emph{monodromy loops}, to produce examples where real identifiability is recovered at least in Euclidean open sets (see Table $ 5 $ for the cases analyzed with this method in our papers \cite{ABC} and \cite{A}).
\begin{table}[h]
\begin{center}
\begin{tabular} {c |c |c |c |c }
{\bf Space of tensors} & {\bf Feature} & {\bf Rank} & {\bf Dec.} & {\bf Ref.} \\
$(\mathbb{C}^3)^{\otimes 7}$ & symmetric & $12$ (g) & 5 & \cite{RS} \\
$(\mathbb{C}^3)^{\otimes 8}$ & symmetric & $15$ (g) & 16 & \cite{RS} \\
$((\mathbb{C}^{3})^{\otimes 3})^{\oplus 3} \oplus (\mathbb{C}^{3})^{\otimes 2} $ & partially-symmetric & $6$ (g) & 2 & \cite{ABC1} \\
\end{tabular}\caption{Generically unidentifiable cases analyzed through computations}
\end{center}
\end{table}
\vspace{0.5cm} \indent {\it
A\,c\,k\,n\,o\,w\,l\,e\,d\,g\,m\,e\,n\,t\,s.\;} This note, based on joint works with C. Bocci and L. Chiantini, arises partially from the conference ``Real identifiability and complex identifiability" given by the author within the cycle of study seminars on ``Algebraic Geometry and Tensors", held among Bologna, Ferrara, Firenze and Siena in 2016-2017. The author would like to thank all the participants for fruitful and stimulating discussions.
\bigskip
\begin{center}
|
1,116,691,499,260 | arxiv | \section{Introduction}
A common method in the economic literature to determine the price of an asset is the concept of equilibrium pricing~\cite{MasColell1995,Dixon2001}. The price is the result of the available supply and demand. In the context of stock markets supply and demand are often identified with the market expectation of the traders to sell or buy a stock. For example, such a pricing scheme is frequently employed within Ising-type models~\cite{Bornholdt2001,Kaizoji2002,Kim2008}, where the market expectation of each trader is symbolized by a spin on a lattice. There the decision to sell or buy a stock is mapped to spin down and up. In each time step an update of the market expectations is performed by taking the market expectations of the nearest neighbors into account~\cite{Bornholdt2001}. A possible criterion to derive the equilibrium price is to take the average of all market expectations.
In contrast, the standard pricing mechanism at stock exchanges is the double auction order book~\cite{Friedman1993,Johnson2010}. The order book lists all current propositions to sell or buy at a given price. This information is available to all traders. Here, the price is the result of a new incoming order matching a limit order already in the book.
The empirical return time series show a collection of remarkable properties, which are called stylized facts, see Ref.~\cite{Chakraborti2011} for a review. The empirical distribution of returns has heavy tails, \textit{i.e.}, the tails are more pronounced compared to a normal distribution~\cite{Mandelbrot1963,Plerou1999,Mantegna1999,Plerou2004}. In the literature many approaches are discussed to explain this feature~\cite{Clark1973,Arthur1996,Mandelbrot1997,Sornette1998,Lux1999,Challet2000,Farmer2004,Gabaix2003,Farmer2004a}. One point of view claims that large trading volumes are responsible for the heavy tails~\cite{Clark1973,Arthur1996,Mandelbrot1997,Sornette1998,Lux1999,Challet2000}. Another approach holds gaps in the order book structure responsible. If gaps are present between the price levels occupied by limit orders, even small volumes can cause large price shifts~\cite{Farmer2004}. According to this reasoning the order book plays an important role in the emergence of heavy tails.
Agent-based modeling makes it possible to microscopically understand trading mechanisms~\cite{Cohen1983,Kim1989,Frankel1988,Chiarella1992,Beltratti1993,Levy1994,Lux1997,Mike2008,Gu2009} and to study trading strategies~\cite{LeBaron2002,Ehrentreich2007}. Here, we extend the agent-based stock market introduced by Schmitt \textit{et al.}~\cite{Schmitt2012} which implements a double auction order book. The model is capable of reproducing the gap structure described in Ref.~\cite{Farmer2004} and yields heavy-tailed return distributions. The traders employed in the model act randomly, independently of each other and follow no strategy.
We extend this model by designing a trader that acts with respect to a decision model. The traders are indirectly coupled to each other because their decisions influence the price which, in turn, affects their decision to buy or to sell an asset, see Sec. 2. Since the model has two parameters we initially analyze the parameters' influence on the price with an equilibrium pricing. Then we analyze the persistence of this behavior if we put the traders' decision model into an order book setting. As results of our simulations we discuss the return distribution and the order book structure in Sec. 3. We conclude our results in Sec. 4.
\section{Model}
We present a decision model in an equilibrium pricing and discuss how to implement it in an order book driven agent-based model. In reality, traders employ all kinds of strategies. Unfortunately, it is not possible to capture the decision making process of any trader. Furthermore, his strategy could be erratic or could at least contain erratic elements, so that it would be impossible to exactly determine how he would react in a particular case. Also, the more complex a system is, the less it is possible to trace back features in the observables, \textit{e.g.} the price, to individual decisions. Therefore we restrict ourselves to a simple decision model where this is still possible.
In an agent-based model for a financial market the decisions of every trader (agent) and their influence on the price are simulated. Thus, agent-based models have to take into account two different aspects: decision making and price formation. A decision model describes how a trader reacts on events, \textit{e.g.} when the price changes, and price formation is a mechanism describing how decisions manifest themselves in the price. Here we analyze one decision model in combination with two different pricing models:
Our decision model demands that every trader has his own price estimation for a traded asset. Of course this influences his market expectation and, thus, his decision either to buy or sell. Depending on the current asset price his individual price estimation develops in time. He changes his decision to buy or to sell, respectively, depending on the relative deviation between the individual and the asset price; a distribution function determines the probability that he changes his market expectation.
The first pricing model calculates the relative price change between two time steps as the mean value of the market expectations of all traders. This corresponds to an equilibrium pricing, balancing supply and demand. In the second price formation scheme the price is determined in the framework of an order book setting. Here we embed the decision model into an agent-based model with a double auction order book pricing. We will now look deeper into these aspects.
\subsection{Decision making}
By their very nature decision making models are capable of describing a large variety of scenarios, wherever a choice between alternatives has to be made. Contrary to approaches in the literature where the decisions are based on mutual decisions between the agents~\cite{Torok2013}, the decisions in the present setting are only made with the reference to a general trend, which can be understood as a mean field.
Let us consider $i=1,\dots ,N$ agents who order to buy or offer to sell a certain asset. Only a single asset is considered. Its supply, quality, usefulness etc. are not taken into account. Whether an agent orders to buy or offers to sell depends on his market expectation $m_i(t)$ about the current price. If he thinks the asset is overpriced $m_i(t)=-1$ he offers to sell, if he thinks it is more worth than the current price $m_i(t)=1$, he buys. To determine what an agent regards as the appropriate price constitutes the crucial part of the model. For simplicity it is assumed that every agent follows the price evolution from the time instant on when he last changed his market expectation. We call this time ``individual reset time'' $\tau_{i,j}$, where $j$ just enumerates those instants of time. What an agent thought before that instant is forgotten. Between individual reset times the individual market expectation $m_i(t)$ is constant.
At the reset time an agent $i$ changes his market expectation because at this time instant agent he felt the asset was more mispriced than he would tolerate. A simple ansatz for what he regards as the appropriate price at this time instant is
\begin{align}
s_i(\tau_{i,j}) = {S}(\tau_{i,j}) (1+m_i(\tau_{i,j})) \ .
\label{eq:reset}
\end{align}
The function $S(t)$ describes the time development of the asset price. Here, $S(\tau_{i,j})$ is the current price at the individual reset time $\tau_{i,j}$. We distinguish the quantity $s_i$ which is used to assess the benefit of the trader's market expectation with reference to the present price from the quantity $m_i$ which is a binary measure for the trader's market expectation. Eq.~\eqref{eq:reset} is a rather bold ansatz. Its advantage is the absence of free parameters. It means that the asset becomes worthless to the agent $s_i(\tau_{i,j})=0$, if he decides that he wants to sell $m_i(\tau_{i,j})=-1$. If he decides to buy, however, he thinks the appropriate price may be as much as twice as high as the current price $s_i(\tau_{i,j}) = 2 {S}(\tau_{i,j})$.
As long as agent $i$ does not change his market expectation again, the price he thinks appropriate evolves during a discrete time step $\Delta t$ according to
\begin{align}
s_i(t+\Delta t) = (1-\alpha)s_i(t)
+ \alpha {S}(t+\Delta t) \ .
\end{align}
The parameter $\alpha \in [0,1]$ quantifies how readily the agent adjusts his estimation what the appropriate price should be compared to the actual price, taking the trend into account which explains why we use $S(t+\Delta t)$ instead of $S(t)$ on the right-hand side. Trivial cases are $\alpha=0$ (no adjustment) and $\alpha=1$ (current price always regarded as appropriate).
The last ingredient of the model concerns the condition that leads agent $i$ to change his market expectation. It depends on the relative deviation $h_i(t)$ of the current price from the one he regards as appropriate,
\begin{align}
h_i(t) = \frac{s_i(t)-{S}(t)}{{S}(t)} \ .
\end{align}
If $m_i(t) h_i(t)>0$, there is no incentive to change. Either the actual price is lower than what is regarded as appropriate and the agent orders to buy, or the actual price is higher than what is regarded as appropriate and the agent wants to sell. However, if $m_i(t) h_i(t)<0$ the agent must consider to change his market expectation in order to avoid losses. On the other hand, a small mispricing might be a transient fluctuation which the agent may tolerate for a while without changing his market expectation. This means that there is a certain inertia in the variables $m_i(t)$.
In this model, changing ones market expectation happens with a certain probability $w_i(m_i(t)h_i(t))$ which approaches zero for $m_i(t)h_i(t) \rightarrow \infty$ and one for $m_i(t)h_i(t)\rightarrow -\infty$. A possible choice is
\begin{align}
w_i(m_i(t)h_i(t)) = \min(1,-\beta m_i(t)h_i(t))
\label{eq:rate}
\end{align}
if $m_i(t)h_i(t)<0$ and zero otherwise, see Fig.~\ref{fig:dmm:rate}.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.75\textwidth]{modell.pdf}
\end{center}
\caption{Comparison of the changing probability Eq.~\eqref{eq:rate} for several values of $\beta$}
\label{fig:dmm:rate}
\end{figure}
Here, $1/\beta$ has the meaning of tolerance of an agent, ranging from zero to infinity. The changing probability is asymmetric, that is, depending on $\beta$, the trader only changes his market expectation if $m_i(t) h_i(t)$ is less than zero. It is worth mentioning that for $\beta \rightarrow 0$ the trader never changes his market expectation.
Obviously, other market expectation switching rules can be constructed in the same spirit. We discovered that choosing a point symmetric function (around $(0,0.5)$, like a Fermi-Dirac distribution) always leads to an exponential decrease of the price for increasing $t$. Furthermore we found that for $w_i(0)=0.5$ any local imbalance, that is, there are temporarily more traders who want to buy than who want to sell or \textit{vice versa}, vanishes from one time step to the next.
\subsection{Equilibrium pricing}
For a simple price formation the relative price change is only determined by the mean value of the variables $m_i(t)$:
\begin{align}
\frac{{S}(t+\Delta t)-{S}(t)}{{S}(t)} = \frac{1}{N} \sum_{i=1}^N m_i(t).
\label{eq:return}
\end{align}
Furthermore $N$ should be odd, so that this quantity is never exactly zero.
\subsection{Order book pricing}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.75\textwidth]{order.pdf}
\end{center}
\caption{Exemplary order book with bids/asks and their quantities}
\label{fig:pm:order}
\end{figure}
At a stock exchange the clearing office manages an order book which is publicly available. The order book is filled by limit orders so that it contains the information about prices and volumes of all bids and asks. If the best (\textit{i.e.} lowest) ask is at least equal to the best (\textit{i.e.} highest) bid, a trade takes place. The traded price at this moment constitutes the current stock price. A trader decides if he wants to buy or sell a specific amount of stocks and he decides on the price he would agree with. In addition, the trader determines how long this limit order should exist in the order book at most. This is the so-called lifetime of an order. If he does not care about the price he is also able to submit a market order. Such a market order is executed immediately. As market orders remove limit orders, they enlarge the bid-ask spread in the order book and reduce liquidity.
In the exemplary order book in Fig.~\ref{fig:pm:order} there are gaps at the prices $9.93$, $10.05$ to $10.07$ and $10.09$. They can appear if limit orders vanish due to lifetime expiration or if no limit orders have been placed at that price level. If market orders remove enough volume, including multiple price levels with gaps, larger price shifts (\textit{i.e.} larger returns) occur.
\section{Results}
It is our goal to study how different pricing mechanisms, equilibrium or order book pricing, respectively, influence the observables considering our decision model.
\subsection{Equilibrium pricing}
We study our decision model using the equilibrium pricing in Eq.~\eqref{eq:return} where $50$ traders start with $m_i(0)=-1$ and $51$ with $m_i(0)=1$. The simulated price time series behave like a geometric Brownian motion for $\alpha=0.5$ and $\beta=0.5$.
Analyzing the dependencies on $\alpha$ and $\beta$, we find three regimes, see Fig.~\ref{fig:results:B3D}:
\begin{itemize}
\item exponential increase of the price (strong positive drift),
\item fluctuations of the price without a drift and
\item exponential decrease of the price (strong negative drift).
\end{itemize}
In principle, $\alpha$ and $\beta$ determine the drift of the price time series. It turns out that there is a large parameter space ($\ln{\langle S \rangle}\approx 0$) in which the price neither diverges nor decays. This is important to generate sufficiently long time series for better statistics. We find that the returns are normal distributed. Given the equilibrium pricing the price time series of our decision model follows a geometric Brownian motion.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.75\textwidth]{B23D_cont.pdf}
\end{center}
\caption{Natural logarithm of the mean price $\langle S \rangle$ depending on $\alpha$ and $\beta$ as a mean value of $100$ seeds after $10^6$ time steps}
\label{fig:results:B3D}
\end{figure}
\subsection{Order book pricing}
In combination with the equilibrium pricing the price time series does not show any interesting features: Our simple decision model only yields a geometric Brownian motion. We are now going to investigate whether this is still the case in an order book setting.
We extend the agent-based model program by Schmitt \textit{et al.}~\cite{Schmitt2012} which implements a double auction order book. The hitherto used equilibrium pricing in Eq.~\eqref{eq:return} is no longer relevant, but for decision making the relation in Eq.~\eqref{eq:rate} is still used. Our model traders ($200$ of them are used in our simulations) only submit market orders. Their market expectations $m_i(t)$ determine if they submit a sell or a buy market order. In our simulations $400$ \texttt{RandomTrader}s act as liquidity providers; they fill the order book with limit orders whose values are randomly drawn from a normal distribution with a mean value of the current best bid or best ask, respectively. Their order lifetime influences the kurtosis of the return distribution drastically and is chosen in such a way that the return distribution is normal without any of our model traders, see~\cite{Schmitt2012}.
Our program adapts real trading behavior in so far as not every trader acts at every time step (here: one second). During $10^6$ seconds the traders each submit 1500 orders on average. In the decision model the individual price expectation $s_i$ updates every second. As time steps can be skipped in the course of the simulation, every time an agent trades we have to calculate the individual price expectation for the corresponding delay afterwards. Therefore we will see that $\alpha_\mathrm{abm} \sim \alpha / 1000$, that is, his price expectation in our agent-based model develops much more slowly than in our study with equilibrium pricing.
The criterion for changing the market expectation is often fulfilled within the time interval between two trades of an agent because of the previously mentioned missed time steps. Therefore in the agent-based model $\beta$ (inverse tolerance) is of minor importance. The simulations are almost independent of this parameter if it is within $[0.1, 1]$. For the results presented here, we chose $\beta=0.5$.
Figure~\ref{fig:results:abmalpha} shows that there are also regimes where the price decreases, increases and fluctuates without a global drift.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.75\textwidth]{alpha.pdf}
\end{center}
\caption{Mean price $\langle S \rangle$ depending on $\alpha_\mathrm{abm}$ as a mean value of 100 seeds after $10^6$ time steps ($S(0)=100$ as starting value for every simulation)}
\label{fig:results:abmalpha}
\end{figure}
Since the initial price is $S(0)=100$, we analyze $\alpha_\mathrm{abm}=10^{-4}$: The return distribution which we calculated for a return interval of $120\,\mathrm{s}$ is heavy-tailed with a kurtosis of about six, see Fig.~\ref{fig:results:abm4ret}. As already mentioned above, this shape can be modified by varying the lifetime of the limit orders.
Another feature of this decision model is the herding behavior which can be observed on the order book volume, see Fig.~\ref{fig:results:abmobook}. There are time intervals in which much more sell orders are stored in the order book and time intervals in which much more buy orders are stored therein. This behavior is a feature of our decision model and cannot be reproduced by solely using $\mathtt{RandomTrader}$s.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.75\textwidth]{hist_m5.pdf}
\end{center}
\caption{Return distribution for $\alpha_\mathrm{abm}=10^{-4}$ with return interval $120\,\mathrm{s}$ and a normal distribution for comparison}
\label{fig:results:abm4ret}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.75\textwidth]{obook_m5.pdf}
\end{center}
\caption{Typical order book volume with sell (above) and buy orders (below)}
\label{fig:results:abmobook}
\end{figure}
\section{Conclusion}
For the equilibrium pricing we see a large space of parameters for which the model produces sustainable price time series, \textit{i.e.}, the price does not diverge or collapse. Heavy tails are not observed. The time series is reminiscent of a geometric Brownian motion where $\alpha$ and $\beta$ determine its global trend.
Embedding the decision model into an agent-based model with order book pricing causes the scale of the parameters to change. However, the parameter $\alpha$ of the decision model has comparable ramifications for the price time series. More importantly, we observe heavy tails for the return distribution. Thus, we conclude that an order book is necessary for the emergence of heavy tails in the context of our decision model. Combining a decision model with an order book pricing allows us to study the effects of the decision model within a realistic environment. We emphasize that in contrast to Schmitt \textit{et al.}~\cite{Schmitt2012} the lifetime of the \texttt{RandomTrader}s' limit orders is chosen so that near the best prices less gaps exist. At deeper price levels gaps do exist, but typically they play no role because the order volumes of the \texttt{RandomTrader}s do not dig deep enough into the order book.
The herding behavior of our model traders leads to a unidirectional stress to either the available ask or bid volume. By acting in the same direction the model traders are able to remove the densely occupied price levels around the best ask or bid and reach the gaps deeper in the order book. This is a crucial factor in generating heavy tails.
In general, implementing a decision model in the setting of an order book pricing leads to new insights and gives the possibility to study it in a realistic environment. The order book makes it possible to gain the understanding of a microscopic level and to access new features. Put differently, decision making models that are not embedded into an order book environment can yield unrealistic or even misleading results.
\section*{Acknowledgments}
We thank S\'ilvio R. Dahmen and Ana L.C. Bazzan for preliminary studies on the model.
\bibliographystyle{elsarticle-num-names}
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1,116,691,499,261 | arxiv | \section*{Introduction}
Let $K$ be a number field, let $X/K$ be a nice (i.e. smooth, projective, geometrically integral) curve of genus $g$, and let $J_X/K$ be the Jacobian of $X$. The Mordell-Weil Theorem says that $J_X(K)$ is a finitely generated abelian group. The \defi{rank} of $X/K$ is the integer
\begin{equation*}
\rk X/K \coloneqq \dim_{\mathbb{Q}} J_X(K) \otimes_{\mathbb{Z}} \mathbb{Q}.
\end{equation*}
Faltings' Theorem~\cite{faltings}*{Satz~7}, formerly known as the Mordell Conjecture, says that if $g \geqslant 2$, then the set of rational points $X(K)$ of $X$ is finite. Given these finiteness theorems, it is natural to wonder, for a fixed genus $g$ and number field $K$, how large the integers $\#X(K)$ and $\rk X/K$ can be. We define two integers, $R(K,g)$ and $N(K,g)$, if they exist, as follows:
\begin{align}\label{eq:constants}
\begin{split}
R(K,g) & \coloneqq \sup \{ R \mid \rk X/K \geqslant R \textup{ for infinitely many genus $g$ curves } X/K\}\\
N(K,g) & \coloneqq \sup \{ N \mid \#X(K) \geqslant N \textup{ for infinitely many genus $g$ curves } X/K\}.
\end{split}
\end{align}
Above, and for the rest of the article, by ``infinitely many curves,'' we mean `` infinitely many nice curves that are pairwise non-isomorphic over an algebraic closure $\smash{\overline{k}}$.''
For any number field $K$ and positive $g$, whether the constant $R(K,g)$ exists is an open question. On the other hand, it is a theorem of Caporaso, Harris, and Mazur~\cite{CHM}*{Theorem~1.2} that if the Weak Lang Conjecture~\cite{CHM}*{Conjecture~A} is true (which is not currently known), then the integer $N(K,g)$ exists for any number field $K$ and $g\geqslant 2$.
A genus $1$ curve $X/K$ with a rational point is an elliptic curve, and $X \cong J_X$. The existence of $R(\mathbb{Q},1)$ has been the subject of multiple folklore conjectures; the papers \cites{Park-wood,discursus} give heuristics that suggest $R(\mathbb{Q},1) \leqslant 21$. For $g \geqslant 2$, to the best of our knowledge, there exist no such heuristics, nor a folklore conjecture in either direction. \\
Producing an infinite set of genus $g$ curves $X/K$ satisfying $\rk X/K \geqslant R$ and $\#X(K) \geqslant N$ would yield lower bounds $R(K,g) \geqslant R$ and $N(K,g) \geqslant N$. The standard way to do this is by the method of \defi{specialization}; one constructs over the function field $K(\mathcal{B})$ of a $K$-variety $\mathcal{B}$ a curve $X/K(\mathcal{B})$ that has the desired properties (genus $g$, many rational points, large rank), and then one takes specializations of $X$ that preserve these properties.
The use of the specialization method to construct curves of high rank originated in the work of N\'eron~\cites{neronhilbert,neronhilbert2}, who proved a theorem~\cite{neronhilbert}*{Th\'eor\`eme~6} on the injectivity of the specialization map (cf. \ref{emp:specialization}) for Jacobians of curves $X/K(\mathcal{B})$, with $\mathcal{B}$ rational over $K$, and applied it to construct infinitely many elliptic curves of rank at least $10$ (i.e. $R(\mathbb{Q},1)\geqslant 10$), and for $g\geqslant2$, infinitely many genus $g$ curves of rank at least $3g+6$ (see also~\cite{umezu1} for a correction of N\'eron's original claim of $3g+7$).
Mestre~\cites{mestre11,mestre12} gave a way to construct genus $g$ curves $X/K(t_1,\dotsc,t_n)$ with at least $8g+12$ rational points; here, $n$ is an integer that depends on $g$. His construction yields $N(\mathbb{Q},g) \geqslant 8g+12$, and he used it in the case $g=1$ to prove $R(\mathbb{Q},1) \geqslant 12$. Shioda~\cite{shiodasymmetry} generalized Mestre's construction and proved $R(\mathbb{Q},g) \geqslant 4g+7$ and $N(\mathbb{Q},g) \geqslant 8g+16$ for all $g \geqslant 2$; we refer to this as the \emph{Mestre--Shioda construction} (see \Cref{sec:construction}).
In this paper, largely inspired by the ideas in \cite{shiodasymmetry}, we produce a refinement of the Mestre--Shioda construction and use it to improve on the known lower bounds for the constants defined in \eqref{eq:constants}.
Our first theorem is the following.
\begin{thm}[see \ref{emp:proof of 1.2}]\label{thm:1.2}
Let $g$ be a positive integer not divisible by $6$. Then
\begin{align*}
R(\mathbb{Q},g) & \geqslant
\begin{cases}
\, 4g + 8 & \textup{ if } g \geqslant 4 \textup{ and } g \equiv 4 \pmod{6}\\
\, 4g + 11 & \textup{ if } g \geqslant 7 \textup{ and } g \equiv 1,3 \pmod{6}\\
\, 4g + 15 & \textup{ if } g \geqslant 8 \textup{ and } g \equiv 2,5 \pmod{6},
\end{cases}\\
N(\mathbb{Q},g) & \geqslant
\begin{cases}
\, 8g + 17 & \textup{ if } g \equiv 4 \pmod{6}\\
\, 8g + 24 & \textup{ if } g \equiv 1,3 \pmod{6}\\
\, 8g + 32 & \textup{ if } g \equiv 2,5 \pmod{6}.
\end{cases}
\end{align*}
\end{thm}
\begin{emp}\textbf{Current records.}
The following table summarizes the current records for lower bounds on $R(\mathbb{Q},g)$. The records below for $g=2$ and $5$ have not explicitly appeared in print; the former follows by applying \cite{kulesz}*{Th\'eor\`eme~2.2.1} to the elliptic curve $E/\mathbb{Q}$ with $\rk E/\mathbb{Q} \geqslant 28$ found by Elkies~\cite{Elkies2006}, which gives an infinite set of genus $2$ curves given by an equation $y^2 = f(x)$, with $\deg f = 6$. The record for $g=5$ then follows by considering the set of genus $5$ curves given by $y^2 = f(x^2 + bx)$, with $b \in \mathbb{Q}$; we leave the details to the reader.
\end{emp}
\begin{table}[H]\label{tab:records}
\centering
\setlength{\tabcolsep}{5pt}
\adjustbox{max width=\textwidth}{
\caption{Rank records}
\begin{tabular}{|c|c|c|c|c|}
\hline
$g$ & $1$ & $2,5$ & $3$ & $4,\geqslant 6$ \\
\hline
$R(\mathbb{Q},g)\geqslant$ & $19$~\cite{Elkies2006} & $32$ & $26$~\cite{kulesz}*{Thm.~3.2.1} & $4g+7$~\cite{shiodasymmetry}*{Cor.~2}\\
\hline
\end{tabular}}
\end{table}
\noindent The curves of Shioda's family that witness the bound $R(\mathbb{Q},g) \geqslant 4g+7$ also yield the current record $N(\mathbb{Q},g) \geqslant 8g+16$ for all $g \geqslant 3$ with the exception of $g=4$, for which the record is $N(\mathbb{Q},4) \geqslant 126$~\cite{elkiesgenus4}*{Page~9}; the record for $g=2$ is $N(\mathbb{Q},2) \geqslant 150$~\cite{elkiescover}. Thus, \Cref{thm:1.2} gives a new record for $R(\mathbb{Q},g)$ for all $g\geqslant 5$ not divisible by $6$, and a new record for $N(\mathbb{Q},g)$ for all $g\geqslant 3$ not divisible by $6$ except $g=4$.\\
Our next theorem produces a ``better-than-constant'' improvement in the bound for $R(K,g)$ for certain $K$ and $g$.
\begin{thm}[see \ref{emp:proofcyc}]\label{thm:cyc}
Let $g \geqslant 2$ be an integer such that $g+1$ is prime, and let $K$ be a number field that contains the $(g+1)$-th roots of unity. Then $R(K,g) \geqslant 6g$.
\end{thm}
Let $T$ be a finite abelian group and consider, if it exists, the constant:
\begin{equation}\label{eq:rkgt}
R(K,g;T) \coloneqq \sup \{ R \mid J_X(K) \supset \mathbb{Z}^{R} \oplus T \textup{ for infinitely many genus $g$ curves } X/K\}.
\end{equation}
Fix $T= \mathbb{Z}/2\mathbb{Z}$ and $K=\mathbb{Q}$. The current record for $g=1$ (elliptic curves) is $R(\mathbb{Q},1;\mathbb{Z}/2\mathbb{Z})\geqslant 11$, due to Elkies~\cite{elkiesthreelectures}, and for $g\geqslant 2$, the record is $R(\mathbb{Q},g;\mathbb{Z}/2\mathbb{Z}) \geqslant 4g+2$, due to Shioda~\cite{shiodasymmetry}*{Theorem~7} (see also \Cref{rem:work of shioda}). The next theorem improves the record for certain $g$'s.
\begin{thm}[see \ref{proof:2torsion}]\label{thm:2torsion}
Let $g$ be a positive integer divisible by $3$. Then $R(\mathbb{Q},g;\mathbb{Z}/2\mathbb{Z}) \geqslant 4g+6$.
\end{thm}
\begin{rem}
A few remarks are in order.
\begin{enumerate}
\item The definition in \eqref{eq:constants} implies that $R(K,g) \geqslant R(\mathbb{Q},g)$ for any number field $K$. It is perhaps natural to expect that if we fix $g$, then $R(K,g)$ goes to infinity as $[K:\mathbb{Q}]$ goes to infinity. It appears, however, to be a non-trivial task to \emph{exhibit} a $K$ and $g$ for which $R(K,g)$ enjoys a bound better than one of the form $R(K,g)\geqslant 4g+C$, and we know of no results in this direction in the literature. In fact, we know of only one other construction that yields a bound of the form $4g+C$, namely, the authors of \cite{rankbias} prove $R(K,g) \geqslant 4g+2$, conditional on a conjecture of Nagao, using a method distinct from Mestre--Shioda.
\item There exist number fields $K$ and small genera $g$ for which $N(K,g)$ is known to have a significantly better lower bound than what is mentioned above (see~\cite{elkiesgenus4}). In particular, for large $g$ and $K$ containing the primitive $(g+1)$-th roots of unity, the current record is $N(K,g) \geqslant 16(g+1)$, due to Brumer~\cite{capratpts}*{Section~5} (see \Cref{rem:brumer}).
\item Fix a number field $K$ and consider, if it exists, the integer $$\rho_K \coloneqq {\limsup}_{g\to \infty} ({R(K,g)}/{g}).$$ N\'eron's construction in the 1950's gives $\rho_K \geqslant 3$. In the 1990's, this was improved to $\rho_K \geqslant 4$ by Shioda, and this remains the best known lower bound on $\rho_K$ for any number field $K$. It would be very interesting if one could produce a construction that improves this bound for some $K$.
\item The analogues of Faltings' Theorem and the Mordell-Weil Theorem also hold for curves $X/K$, with $K$ the function field of a curve in positive characteristic. It is proved in \cite{ulmerpoints} (resp. \cite{ulmerranks}) that $N(\mathbb{F}_p(t),g)$ (resp. $R(\mathbb{F}_p(t),g)$) does not exist.
\end{enumerate}
\end{rem}
\begin{emp}\textbf{Outline of paper.}
In \Cref{sec:prelims}, we recall some facts about curves and Jacobians. In \Cref{sec:specialization}, we review the specialization method described in the introduction. In \Cref{sec:construction}, we review the Mestre--Shioda construction of curves with many rational points. In \Cref{sec:strategyW}, we outline the strategy behind our refinement of the Mestre--Shioda construction. In \Cref{sec: composite tuples}, we discuss how our strategy is related to a classical problem of number theory, the Prouhet-Tarry-Escott problem. In Sections \ref{sec:rational base} -- \ref{sec:finale}, we prove \Cref{thm:2torsion} and establish the bounds in \Cref{thm:1.2} for the case $g \equiv 2,4,5 \pmod{6}$. In Sections \ref{sec:calz} and \ref{sec:odd positive g}, we establish the bounds in \Cref{thm:1.2} for the case $g \equiv 1,3 \pmod{6}$. In \Cref{sec:kummer}, we prove \Cref{thm:cyc}.
\end{emp}
\begin{emp} \textbf{Acknowledgements.}
This work is part of the author's doctoral dissertation at the University of Georgia. The author is grateful to UGA and its math department for providing a stimulating environment for research. The author is greatly indebted to his advisor, Dino Lorenzini, for carefully reading the manuscript and providing thoughtful feedback.
\end{emp}
\tableofcontents
\section{Preliminaries}\label{sec:prelims}
\begin{emp}\label{emp:conventions} \textbf{Notation and Conventions.}
A scheme $X/K$ is a $K$-variety if $X$ is geometrically integral, separated, and of finite type over $K$; the function field of $X$ is denoted by $K(X)$. A nice curve is a smooth, projective, one-dimensional $K$-variety. We will sometimes regard $K$-points of $X$ as closed points with residue field $K$.
We use bold font as short-hand to denote tuples of indeterminates; for instance, if we say ``$\mbf{t}$ denotes the tuple of indeterminates $t_1,\dotsc,t_n$,'' then $K[\mbf{t}]$ denotes $K[t_1,\dotsc,t_n]$, $K(\mbf{t})$ the function field $K(t_1,\dotsc,t_n)$, and $\At{n}$ the affine space $\spec K[\mbf{t}]$. If we make an assertion about the Galois group $\textup{Gal}(L/F)$ of an extension of fields $L/F$, then it is implied that $L/F$ is Galois.
\end{emp}
\begin{emp}\label{emp:curves and divisors} \textbf{Divisors and Jacobians.}
Let $K$ be a field, and $X/K$ a nice curve of genus $g$. We denote by $\Div(X)$ the group of divisors on $X$, and by ${\pic^0(X)}$ the group of degree $0$ divisors modulo the subgroup of principal divisors. If $L/K$ is a field extension, then we will often use the natural injection $\Div(X) \lhook\joinrel\longrightarrow \Div(X_L)$ to regard divisors on $X$ as divisors on $X_L$. For $D \in \Div(X)$, we denote its class in ${\pic^0(X)}$ by $[D]$. The \defi{Jacobian} of $X$, denoted $J_X/K$, is an abelian variety of dimension $g$ such that $J_X(L) \cong {\pic^0(X_L)}$ for any field extension $L/K$ (not necessarily finite) that verifies the condition $X(L) \neq \emptyset$.
For a $K$-morphism $f:X \longrightarrow Y$ of curves, we denote by $f^*$ (resp. $f_*$) the induced pull-back (resp. pushforward) homomorphism on $\Div$ and $\pic$.
\end{emp}
\begin{emp}\label{emp:hyperelliptic} \textbf{Hyperelliptic curves.}
Assume $\chr K \neq 2$. Let $f(x) \in K[x]$ be a non-constant polynomial. The affine curve $X_0/K$ defined by the equation ${y^2 = f(x)}$ is smooth if and only if $f(x)$ is separable. The \defi{degree} of the equation $y^2 = f(x)$ (or of $X_0$) is the integer $d\coloneqq \deg f$.
Assume now that $X_0$ is smooth. If $g$ denotes the genus of $X_0$, then we have $d=2g+1$ (resp. $2g+2$) if $d$ is odd (resp. even). Let $h(x,z) \in K[x,z]$ denote the binary form $z^{2g+2}f(x/z)$, and let $K[x,y,z]$ denote the graded $K$-algebra in which $\deg x = \deg z = 1$ and $\deg y = g+1$. Then, $X\coloneqq \textup{Proj}\, K[x,y,z]/(y^2 - h(x,z))$ is a nice curve of genus $g$, and $X_0$ is naturally isomorphic to the open subset $\{z \neq 0 \} \subset X$. The natural projection $\pi_X: X \longrightarrow \mathbb{P}^1_K \coloneqq \textup{Proj} \,K[x,z]$ is finite of degree $2$, with branch divisor given by $\{h(x,z) = 0\}$; this is the quotient by the involution $\iota : X \longrightarrow X$ defined by $(x,y) \longmapsto (x,-y)$. If $d = 2g+1$ is odd, then $X$ has a single ramified $K$-point lying over $\infty \coloneqq [1:0] \in \mathbb{P}^1_K$, and if $d = 2g+2$ is even, then $X$ has two (geometric) points lying over $\infty$; in either case, we let $D_{X,\infty} \in \Div(X)$ denote the degree $2$ divisor $\pi^*(\infty)$, and for any degree $k$ divisor $E \in \Div(X)$, we let $\epsilon(E) \coloneqq [2E - kD_{X,\infty}] \in \pic^0(X)$.
Since $\chr K \neq2$, a nice curve $X/K$ is birational to an affine curve $y^2 = f(x)$ if and only if $X$ admits a degree $2$ morphism to $\mathbb{P}^1_K$; for $g \geqslant 2$, such curves are called \defi{hyperelliptic curves}.
\end{emp}
\begin{lem}\label{lem:D infty}
Let $X/K$ (resp. $Y/K$) be the nice curve associated (as in \ref{emp:hyperelliptic} above) to the smooth, affine curve given by $y^2 = g_1(x)$ (resp. $y^2 = g_2(x)$).
Suppose $f: X \longrightarrow Y$ is a finite $K$-morphism of degree $k$, and $h:\mathbb{P}^1_K \longrightarrow \mathbb{P}^1_K$ is finite of degree $k$ and totally ramified at $\infty$, and suppose that $f \circ \pi_Y = h\circ \pi_X$. Then, $f_*D_{X,\infty} = D_{Y,\infty}$ and $f^*D_{Y,\infty} = kD_{X,\infty}.$ Consequently, for points $P \in X(K)$ and $Q \in Y(K)$, we have
\begin{equation*}
f_*(\epsilon(P)) = \epsilon(f(P)) \in \pic(Y), \quad \textup{ and }\quad f^*(\epsilon(Q)) = \epsilon(f^*Q) \in \pic(X).
\end{equation*}
\end{lem}
\begin{proof}
The diagram immediately gives $$f^*D_{Y,\infty} = f^*\pi_Y^*\infty = \pi_X^*h^*\infty = \pi_X^*(k\infty) = kD_{X,\infty}.$$ On the other hand, since $\iota_Y \circ f = f \circ \iota_X$, we have $$(\iota_Y)_*f_*D_{X,\infty} = f_*(\iota_X)_*D_{X,\infty} = f_*D_{X,\infty}.$$ So, $f_*D_{X,\infty} \in \Div(Y)$ is a degree $2$ divisor lying over $\infty \in \mathbb{P}^1_K$, and which is fixed by $\iota_Y$; this implies that $f_*D_{X,\infty} = D_{Y,\infty}$. The reader can now easily verify that the last statement in the lemma also holds.
\end{proof}
\section{The specialization method}\label{sec:specialization}
Now, and for the rest of this article, \emph{$K$ will denote a number field}.
\begin{emp}\textbf{Non-constant curves.}\label{emp:families} Let $\mathcal{K}/K$ be an extension of fields such that $\trdeg_K \mathcal{K} =n >0$ and $K$ is algebraically closed in $\mathcal{K}$. Given a nice curve $X/\mathcal{K}$ of genus $g\geqslant 2$, we can find a $K$-variety $V$ with $K(V) \cong \mathcal{K}$, and a smooth, projective family of curves $\mathscr{X} \longrightarrow V$ with geometrically integral fibers, and generic fiber $X/\mathcal{K}$ (cf.~\cite{Qpoints}*{Theorem~3.2.1 (i)}). This family induces a $K$-morphism $V \longrightarrow M_{g,K}$, where $M_{g,K}$ denotes the coarse moduli scheme of genus $g$ curves over $K$ (cf. ~\cite{mumfordGIT}*{Page 103, Proposition~5.4}); we say that $X/\mathcal{K}$ is \defi{non-constant} if this morphism is non-constant (this definition is independent of the choice of $V$ and family $\mathscr{X}/V$). An elliptic curve $X/\mathcal{K}$ is called non-constant if its $j$-invariant is in $\mathcal{K} \backslash K$.
\end{emp}
\begin{emp}\textbf{The specialization map.}\label{emp:specialization}
With $\mathscr{X}/V$ and $X/K(V)$ as above, for a point $s \in V$, the fiber $\mathscr{X}_s / \kappa(s)$ is called a \defi{specialization} of $X$. The \defi{relative Jacobian} of $\mathscr{X} /V$ is an abelian $V$-scheme $\smash{\mathscr{J}}/V$~\cite{neronmodels}*{Proposition~9.4.4} such that, for any point $s \in V$, the fiber $\smash{\mathscr{J}}_s / \kappa(s)$ is isomorphic to the Jacobian of $\mathscr{X}_s / \kappa(s)$; in particular, the generic fiber of $\smash{\mathscr{J}}/V$ is isomorphic to $J_X/K(V)$.
If $V$ is regular, then we have an isomorphism of groups $J_X(K(V)) \cong \smash{\mathscr{J}}(V)$; indeed,~\cite{movinglemma}*{Proposition~6.2} implies that the natural injective map $\smash{\mathscr{J}}(V) \lhook\joinrel\longrightarrow J_X(K(V))$ is also surjective. For any $P \in V(K)$, we also have a natural map $\smash{\mathscr{J}}(V) \longrightarrow \smash{\mathscr{J}}_P(K)$; pre-composing this with the isomorphism $J_X(K(V)) \cong \smash{\mathscr{J}}(V)$, we get the \defi{specialization map} $\sigma_P: J_X(K(V)) \longrightarrow \smash{\mathscr{J}}_P(K),$ which is a group homomorphism.
The Lang-N\'eron Theorem (see~\cite{lang-neron}*{Theorems~1,~2}), a generalization of the Mordell-Weil Theorem, implies that $J_X(K(V))$ is a finitely generated abelian group; we define as usual the integer $\rk X/K(V) \coloneqq \dim_{\mathbb{Q}} J_X(K(V)) \otimes_{\mathbb{Z}} \mathbb{Q}.$
\end{emp}
\begin{emp} \textbf{Thin sets.}\label{emp:thin sets}
Let $V$ be a $K$-variety. We recall from ~\cite{serremw}*{Page~121, Section~9.5} or~\cite{CTrankjump}*{Section 2} the notion of a \defi{thin} set $T\subset V(K)$. Thin sets are useful for the following reason: if $V(K)$ is \emph{not} thin, then $\{V(K)\backslash T\mid T \subset V(K) \textup{ is thin}\}$ is a collection of Zariski-dense subsets of $V(K)$ that is closed under finite intersections. It is well-known (see \cite{serremw}*{Section 9.6}) that if $X/K$ is a \defi{$K$-rational variety} (i.e. $X$ is $K$-birational to an affine space $\mathbb{A}^n_K$), then $X(K)$ is not thin.
\end{emp}
\begin{thm}[{\cite{CTrankjump}*{Theorem~2.4}, \cite{serremw}*{Section~11.1}}] \label{thm:specialization}
Let $V$ be a smooth $K$-variety, and let $\mathscr{A}/V$ be an abelian scheme with generic fiber $A/K(V)$. The set
\begin{equation*}
\{\, P \in V(K) \mid \sigma_P : A(K(V)) \longrightarrow \mathscr{A}_P(K) \textup{ is not injective } \}
\end{equation*}
is thin.
\end{thm}
Recall the constants defined in \eqref{eq:constants} and \eqref{eq:rkgt}.
\begin{cor}\label{cor:rank cor}
Let $R$ and $N$ be non-negative integers, let $T$ be a finite abelian group, and let $\mathcal{B}/K$ be a $K$-rational variety. Suppose there exists a curve $X/K(\mathcal{B})$ of positive genus such that
\begin{enumerate}[(a)]
\item $X/K(\mathcal{B})$ is non-constant,\label{part:speccor a}
\item $\#X(K(\mathcal{B})) \geqslant N$, \label{part:speccor b}
\item $J_X(K(\mathcal{B}))$ contains a subgroup isomorphism to $\mathbb{Z}^R \oplus T$.\label{part:speccor c}
\end{enumerate}
Then $N(K,g) \geqslant N$ and $R(K,g;T) \geqslant R$ (and therefore also $R(K,g) = R(K,g;\{1\}) \geqslant R$).
\end{cor}
\begin{proof}
Suppose $V \subset \mathcal{B}$ is an open subscheme that is smooth, and which admits a smooth, projective family of curves $\mathscr{X}/V$ with generic fiber $X/K(V)$ (as in \ref{emp:families}). Since $V$ is $K$-rational, for any thin set $S\subset V(K)$, the complement $V(K) \backslash S$ is Zariski-dense in $V$. Combining this with assumption \ref{part:speccor a}, we see that: \emph{if $S \subset V(K)$ is thin, then the set $\{\mathscr{X}_P/K \}_{P \in V(K)\backslash S}$ contains infinitely many nice genus $g$ curves over $K$.}
By assumption \ref{part:speccor b}, there exist distinct points $P_1,\dotsc,P_N \in X(K(V))$, which extend to $N$ sections $V \longrightarrow \mathscr{X}$ of $\mathscr{X}/V$ (cf. \cite{movinglemma}*{Proposition~6.2}); by replacing $V$ with a non-empty open subset (if necessary), we may assume that these sections specialize to $N$ distinct $K$-points on any $K$-fiber of $V$. Then, the curves $\{\mathscr{X}_P/K\}_{P \in V(K)}$ witness the claim $N(K,g) \geqslant N$.
Similarly, applying \Cref{thm:specialization} to the relative Jacobian $\smash{\mathscr{J}}/V$ of $\mathscr{X}/V$, we find that there is a thin set $S \subset V(K)$ such that $\smash{\mathscr{J}}_P(K) \supset \mathbb{Z}^R\oplus T$ for any $P \in V(K) \setminus S$, so the curves $\{\mathscr{X}_P/K \}_{P \in V(K)\backslash S}$ witness the claim $R(K,g;T) \geqslant R$.
\end{proof}
\section{The Mestre--Shioda construction}\label{sec:construction}
In this section, we outline the Mestre--Shioda construction.
\begin{lem}\label{lem:kthroot}
Let $d\geqslant 2$ be an integer. Let $\mbf{m},\mbf{h},$ and $\Bell$ denote the tuples of indeterminates $(m_0,\dotsc,m_{2d-1})$, $(h_0,\dotsc,h_{d-1}),$ and $(l_0,\dotsc,l_{d-1})$ respectively, and define:
\begin{equation}\label{def:polys}
\begin{alignedat}{2}
m(x) & \coloneqq x^{2d} + m_{2d-1}x^{2d-1} + \dotsb + m_0 \quad && \in K[\mbf{m}][x]\\
h(x) & \coloneqq x^{d} + h_{d-1}x^{d-1} + \dotsb + h_0 \quad && \in K[\mbf{h}][x]\\
\ell(x) & \coloneqq l_{d-1}x^{d-1} + \dotsb + l_0 \quad && \in K[\Bell][x].
\end{alignedat}
\end{equation}
The $K$-morphism $\phi_d : \Ah{d}\times \Al{d} \longrightarrow \Am{2d}$, defined on coordinate rings by
\begin{equation*}
m_i \longmapsto i^{\textup{th}} \textup{ coefficient of } (h(x)^2 - \ell(x)), \quad \; i=0,\dotsc,2d-1,
\end{equation*}
is an isomorphism. Thus, given a monic $M(x) \in K[x]$ of degree $2d$, there exists a \textbf{unique} monic $H(x) \in K[x]$ of degree $d$ and \textbf{unique} $L(x) \in K[x]$ of degree at most $d-1$, satisfying the relation
\begin{equation}\label{eq:kthrootidentity}
M(x) = H(x)^2 - L(x).
\end{equation}
\end{lem}
\begin{proof}
To show that $\phi_d$ is an isomorphism, we need to express each $h_i,l_i$ as a polynomial in the variables $m_0,\dotsc,m_{2d-1}$;
for this, we can compare coefficients (in descending order) in the identity $m(x) = h(x)^2 - \ell(x)$ and sequentially ``solve'' for the variables $h_i,l_i$ in terms of $m_i$. Indeed, we have $h_{d-1} = m_{2d-1}/2$. For $j = 2,\dotsc, d$, the coefficient of $x^{2d- j}$ in $h(x)^2$ is of the form $2h_{d-j} + g_j$ for some polynomial $g_j \in K[h_{d-1},h_{d-2},\dotsc , h_{d-j+1}]$, which yields $h_{d-j} = (m_{2d - j} - g_j)/2,$ for $j = 2,\dotsc , d$.
This expresses each coefficient of $h(x)$ as a polynomial in the $m_i$. For $0 \leqslant i \leqslant d-1$, the coefficient of $x^i$ in $h(x)^2 - \ell(x)$ is of the form $q_i - l_i$, where $q_i \in K[h_0,\dotsc , h_{d-1}]$, which yields $l_i = q_i - m_i,$ for $i=0,\dotsc,d-1$.
This expresses each coefficient of $\ell(x)$ as a polynomial in the $m_i$, and concludes the proof that $\phi_d$ is an isomorphism.
For the second part, consider the inverse $\phi_d^{-1} : \Am{2d} \longrightarrow \Ah{d}\times \Al{d}$, and suppose $M(x) \in K[x]$ is monic of degree $2d$. There is a unique point $P \in {\Am{2d}(K)}$ such that $M(x)$ is the specialization of $m(x)$ at $P$. By construction, the specializations $H(x)$ and $L(x)$ of $h(x)$ and $\ell(x)$ respectively at $\phi_d^{-1} (P)$ satisfy the identity \eqref{eq:kthrootidentity}, $L(x)$ is of degree at most $d-1$, and $H(x)$ is monic of degree $d$. The uniqueness is obvious.
\end{proof}
\begin{defn}\label{def: sqroot rem}
In the setting of \Cref{lem:kthroot}, we refer to $H(x)$ (resp. $L(x)$) as the \defi{square-root} \defi{approximation} (resp. \defi{square-root remainder}) of $M(x)$.
\end{defn}
\begin{emp}\textbf{The Mestre--Shioda construction.}\label{emp:mestre} Let $\mbf{u}$ denote the tuple of indeterminates $u_1,\dotsc,u_{2d}$, let $s_j(\mbf{u})$ denote the $j$-th elementary symmetric polynomial in the $u_i$'s, and consider the morphism $F: \Au{2d} \longrightarrow \Am{2d}$, defined on coordinate rings by $m_i \longmapsto (-1)^{2d-j} s_j(\mbf{u})$. The classical \emph{Fundamental theorem of symmetric functions} says that $F$ is finite and surjective, and the extension $K(\mbf{u})/K(\mbf{m})$ is Galois, with group isomorphic to the symmetric group $S_{2d}$.
Using the isomorphism $K[\Bell,\mbf{h}] \cong K[\mbf{m}]$ provided by \Cref{lem:kthroot}, along with the injection $K[\mbf{m}] \lhook\joinrel\longrightarrow K[\mbf{u}]$ corresponding to $F$, we can regard $m(x),\ell(x),h(x)$ as polynomials in $K[\mbf{u}][x]$, which satisfy the identity
\begin{equation}\label{eq:points on curves}
(x-u_1)\dotsb(x-u_{2d}) = m(x) = h(x)^2 - \ell(x) \in K[\mbf{u}][x].
\end{equation}
Now, consider the smooth affine curve $X_0/K(\mbf{u})$ given by the degree $d-1$ equation $y^2 = \ell(x)$. The identity \eqref{eq:points on curves} implies that $X_0$ has the $K(\mbf{u})$-rational points $P_i \coloneqq (u_i,\pm h(u_i))$, for $i=1,\dotsc,2d$.
Shioda showed in \cite{shiodasymmetry}*{Theorems~5, 6} that the divisor classes $[2P_i - D_{X,\infty}]$ (cf. \ref{emp:hyperelliptic}) generate a subgroup of rank $2d-1$ in $J_X(K(\mbf{u}))$, where $X/K(\mbf{u})$ is a smooth, proper model of $X_0$.
\end{emp}
\section{Strategy behind improvement}\label{sec:strategyW}
We outline in this section our strategy to produce curves with \emph{more} rational points relative to their genus that what is obtained from the Mestre--Shioda construction.
Continuing with the notation of \ref{emp:mestre}, for a point $P \in \Au{2d}(K)$, let $\ell_P(x),h_P(x), m_P(x) \in K[x]$ denote the specializations of the polynomials $\ell(x),h(x),m(x) \in K[\mbf{u}][x]$ respectively. Our basic strategy is to find $K$-points of the closed subset $\{l_{d-1} = 0 \} \subset \Au{2d}$, i.e. points $P \in \Au{2d}(K)$ for which $\deg \ell_P \leqslant d - 2$. If $\deg \ell_P$ is small enough, then the genus of the specialized curve $C_P/K$ defined by $y^2 = \ell_P(x)$ will be smaller than that of $X_0/K(\mbf{u})$; we want to find such $K$-points $P$ for which $\#C_P(K) \geqslant 4d$ and $\rk C_P/K \geqslant 2d-1$.
\begin{exmp}
Let $d=6$, so that $\mathbb{A}^{2d}_K = \mathbb{A}^{12}_K$. In~\cite{mestre11}*{Page~142, Section~2}, Mestre described a way to systematically find $\mathbb{Q}$-rational curves inside $\{l_4 \neq 0\} \cap \{l_5 = 0 \} \subset \mathbb{A}^{12}_{\mathbb{Q}}$; he used it to produce a non-isotrivial elliptic curve over $\mathbb{Q}(t)$ of rank at least $12$ over $\mathbb{Q}(t)$. Subsequently, Nagao~\cite{nagao13} used the same method to find one of rank at least $13$ over $\mathbb{Q}(t)$, and Kihara~\cite{kihara14} used it to find one of rank at least $14$ over $\mathbb{Q}(t)$. We refer the reader to~\cite{elkiesthreelectures} for an exposition of these and other constructions of high rank elliptic curves over $\mathbb{Q}(t)$ and $\mathbb{Q}$.
\end{exmp}
The starting point for the results of this paper is \Cref{obs:compiswin} below, which gives a ``natural'' sufficient condition for a point $P \in {\Au{2d}}(K)$ to satisfy $\deg \ell_P(x) \leqslant d-2$. First, we need the following definition.
\begin{defn}\label{def:en composite}
A point $(a_1,\dotsc,a_{en}) \in \mathbb{A}^{en}(K)$ is \defi{$(e,n)$-composite} if there exist polynomials $G(x),M(x) \in K[x]$, of degree $e$ and $n$ respectively, such that $(x-a_1)\dotsb (x-a_{en}) = M(G(x))$.
\end{defn}
\begin{obs}\label{obs:compiswin}
Recall the space $\Au{2d} = \spec K[u_1,\dotsc,u_{2d}]$ from the construction in \ref{emp:mestre}. Suppose $d$ factors as $d = ed'$, with $e\geqslant 2$ and $d'\geqslant 2$. Then, if
$P = (a_1,\dotsc,a_{2d})\in {\Au{2d}(K)}$ is $(e,2d')$-composite, then $\deg \ell_P(x) = d - ej$, for some $j\geqslant 1$. In particular, $\deg \ell_P(x) \leqslant d - e$.
\end{obs}
\begin{proof}
By assumption, there exist polynomials $G(x),M(x) \in K[x]$, of degree $e$ and $2d'$ respectively, such that $m_P(x) \coloneqq (x-a_1)\dotsb (x-a_{2d})= M(G(x))$. If $b$ denotes the leading coefficient of $G(x)$, we can replace $G(x)$ with $G(x)/b$ and $M(x)$ with $M(bx)$, and hence assume without loss of generality that $M(x)$ and $G(x)$ are both monic. Let $H(x), L(x)$ denote the square-root approximation and remainder of $M(x)$, so that $H(x) \in K[x]$ is monic of degree $d'$, $L(x)\in K[x]$ is of degree $d'-j$ for some $j\geqslant 1$, and we have $M(x) = H(x)^2 - L(x)$ as in \eqref{eq:kthrootidentity}. Replacing $x$ with $G(x)$, we get $m_P(x) = M(G(x)) = H(G(x))^2 - L(G(x)).$
Observe now that $H(G(x))$ is monic of degree $d$, and
\begin{equation*}
\deg L(G(x)) = e(d'-j) = d - ej \leqslant d-e.
\end{equation*}
By the uniqueness statement in \Cref{lem:kthroot}, we see that $h_P(x) = H(G(x))$ and $\ell_P(x) = L(G(x))$, so we conclude $\deg \ell_P = d - ej$ for some $j\geqslant 1$, as desired.
\end{proof}
\begin{emp}\label{emp:poa}\textbf{Plan of action.}
Let us now summarize how we will use \Cref{obs:compiswin} going forward. For certain integers $e\geqslant 2$, and for any integer $d\geqslant 2$, we will define a positive dimensional affine $K$-rational variety $V \coloneqq \spec S$ with the following property: There exist elements $t_1,\dotsc,t_{2de} \in S$ and monic polynomials $g(x)$ and $m(x)$ in $S[x]$, of degree $e$ and $2d$ respectively, such that $m(g(x)) = (x - t_1)\dotsb (x-t_{2de}) \in S[x].$ Thus, $V$ \defi{parametrizes $(e,2d)$-composite tuples} in the sense that to any point $P \in V(K)$, corresponding to a homomorphism $P^{\#}: S \longrightarrow K$, we can associate the tuple $(P^{\#}(t_1),\dotsc,P^{\#}(t_{2de})) \in K^{2de}$, which is $(e,2d)$-composite.
Then, we will apply \Cref{lem:kthroot} to obtain polynomials $h(x)$ and $\ell(x)$ in $S[x]$, of degree $d$ and $d-1$ respectively, such that $m(x) = h(x)^2 - \ell(x)$, giving the identity $(x-t_1)\dotsb (x-t_{2de}) = h(g(x))^2 - \ell(g(x))\in S[x].$ We will then consider (variants of) the curve $\wt{X}/K(V)$ defined by $y^2 = \ell(g(x))$, which has the $K(V)$-points $P_i \coloneqq (t_1,\pm h(g(t_i)))$, for $i=1,\dotsc,2de$. We will determine the rank of the subgroup of $J_X(K(V))$ generated by the divisor classes $[2P_i - D_{\wt{X},\infty}]$ (cf. \ref{emp:hyperelliptic}), and finally, we will apply \Cref{cor:rank cor} to get lower bounds on $N(K,g)$ and $R(K,g)$.
\end{emp}
\begin{rem}
Mestre (cf. \cite{elkiesgenus4}*{Page 4}) and Shioda (see \ref{emp:way forward} \ref{part:emp cases 1}) produced improvements of the construction in \ref{emp:mestre} by applying \Cref{obs:compiswin} in the special case when the ``inner polynomial'' $G(x)$ is $x^e$. In fact, we came upon \Cref{obs:compiswin} by noting that there is nothing special about $x^e$, and that any choice of $G(x)$ would yield a similar improvement.
\end{rem}
\section{Composite tuples}\label{sec: composite tuples}
The purpose of this section is to provide context to the reader for \emph{how} we find $K$-rational varieties $V$ that parametrize $(e,n)$-composite tuples (as in \ref{emp:poa}); the results in this section are not needed or used in the subsequent sections.
\begin{emp}\label{emp:calgen
Let $e\geqslant 2$ and $n\geqslant 2$ be integers. For $i=1,\dotsc,n$, let $\mbf{t}_i$ denote the tuple of indeterminates $t_{i1},\dotsc,t_{ie}$. For each $j=1,\dotsc,e$, let $s_j(\mbf{t}_i)$ denote the $j^\textup{th}$ elementary symmetric polynomial in the tuple $\mbf{t}_i$. The \defi{$\mathbb{Q}$-variety of $(e,n)$-composite tuples}, denoted $\mathcal{G}(e,n)$, is the closed subscheme of $\mathbb{A}^{en}_{\mathbb{Q}} = \spec \mathbb{Q}[\mbf{t}_1,\dotsc,\mbf{t}_n]$ defined by the equations:
\begin{equation}\label{eq: calg scheme}
s_j(\mbf{t}_1) = \dotsb = s_j(\mbf{t}_n), \qquad j=1,\dotsc,e-1.
\end{equation}
Note that if we set $g(x) \coloneqq x^e + \sum_{j=1}^{e-1}(-1)^js_j(\mbf{t}_1)x^{e-j}$ and $m(x) \coloneqq \prod_{i=1}^n (x + (-1)^es_e(\mbf{t}_i))$, then we have $\prod_{i,j} (x-t_{ij}) = m(g(x))\in S[x]$, where $S$ denotes the coordinate ring of $\mathcal{G}(e,n)$. It is not hard to show that $\mathcal{G}(e,n)/\mathbb{Q}$ is geometrically integral, i.e. it is indeed a $\mathbb{Q}$-variety.
It is an easy consequence of {\cite{rickards}*{Proposition~1}} that, for a field $K\supset \mathbb{Q}$, a $K$-point $P$ of the ambient affine space $\mathbb{A}^{en}_{\mathbb{Q}}$ is an $(e,n)$-composite tuple if and only if there is a permutation $\sigma$ of the coordinates of $P$ such that $\sigma(P)\in \mathcal{G}(e,n)(K)$. Thus, to carry out the method of \ref{emp:poa} for some $e\geqslant2$ and $d\geqslant2$, we need to: \emph{find a $K$-rational sub-variety $V \subset \mathcal{G}(e,2d)_K$.}
\end{emp}
\begin{rem}\label{emp:PTE}
For a positive integer $j$, let $p_j(\mbf{t}_i)\coloneqq {t_{i1}^j + \dotsb + t_{ie}^j}$. The classical \defi{Prouhet--Tarry--Escott (PTE) problem} of size $e$ and length $n$ asks for integer solutions to the system of equations:
\begin{equation}\label{eq: PTE equations}
p_j(\mbf{t}_1) = \dotsb = p_j(\mbf{t}_n), \qquad j=1,\dotsc,e-1.
\end{equation}
We note here that \emph{the equations in \eqref{eq: PTE equations} define the closed subscheme $\mathcal{G}(e,n) \subset \spec \mathbb{Q}[\mbf{t}_1,\dotsc,\mbf{t}_n]$}; in other words, a tuple $(a_i) \in K^{en}$ is $(e,n)$-composite if and only if the coordinates give a ($K$-rational) solution to the PTE problem of size $e$ and length $n$. The equivalence between the systems of equations \eqref{eq: calg scheme} and \eqref{eq: PTE equations} is an easy application of the \emph{Newton Identities} (cf. \cite{newtonformulas}); we leave the details to the reader.
\end{rem}
\begin{emp}\textbf{Known solutions to the PTE problem.} Most investigations of $\mathcal{G}(e,n)(\mathbb{Q})$ have focused on the case $n=2$, and even in this case, non-trivial points in $\mathcal{G}(e,n)(\mathbb{Q})$ are known only for $e \leqslant 12$; here, by ``\emph{non-trivial}'' we mean that the coordinates of the point are distinct. For our purposes (cf. \ref{emp:poa}), we need $n = 2d \geqslant 4$, so we do not dwell on what is known for $n \leqslant 3$, instead referring the reader to the survey article \cite{PTE-Raghavendran}. We summarize below the cases, with $n\geqslant 4$, in which $\mathcal{G}(e,n)$ is known to have a non-trivial $\mathbb{Q}$-point. First, we define a closed subscheme $\mathcal{G}_0(e,n)\coloneqq \{s_1(\mbf{t}_1) = 0\} \subset \mathcal{G}(e,n)$ that will appear below. \label{emp:way forward}
\begin{enumerate}[(i)]
\item \textbf{The case $(e,n) = (2,n)$.} The reader can easily verify that $\mathcal{G}(2,n) \cong \mathbb{A}^{n+1}_{\mathbb{Q}}$, and $\mathcal{G}_0(2,n) \cong \mathbb{A}^n_{\mathbb{Q}}$. In~\cite{shiodasymmetry}, Shioda carried out the method of \ref{emp:poa} by taking $V = \mathcal{G}_0(2,n)$, and thereby proved $N(\mathbb{Q},g)\geqslant 8g+16$ and $R(\mathbb{Q},g) \geqslant 4g+7$ for $g\geqslant 2$. \label{part:emp cases 1}
\item \textbf{The case $(e,n) = (3,n)$.} Dickson showed in \cite{dickson}*{Theorem~47, Page~52} that $\mathcal{G}(3,2)$ is $\mathbb{Q}$-rational; Gloden showed in \cite{gloden}*{Pp.~74-81} that for arbitrary $n$, the subscheme $\mathcal{B}_n \coloneqq \mathcal{G}_0(3,n)$ (see \ref{variety:cals}) contains infinitely many non-trivial $\mathbb{Q}$-points. We show in \Cref{thm:calg03n rational} that $\mathcal{B}_n$ is a $\mathbb{Q}$-rational variety; in fact, one can show that $\mathcal{G}(3,n)$ is $\mathbb{Q}$-birational to $\mathcal{B}_n \times \mathbb{A}^1_{\mathbb{Q}}$, so $\mathcal{G}(3,n)$ is itself a $\mathbb{Q}$-rational variety. \label{part:emp cases 2}
\item \textbf{The case $(e,n) = (4,n)$.} Chernick showed in \cite{PTE-chernick} that $\mathcal{G}(4,2)$ is $\mathbb{Q}$-rational; Gloden showed in \cite{gloden}*{Pp.~81-83} that for arbitrary $n$, $\mathcal{G}_0(4,n)$ contains infinitely many non-trivial $\mathbb{Q}$-points. In \Cref{prop:calz rational}, we show that these points live on a $\mathbb{Q}$-rational sub-variety $\mathcal{Z}_n \subset \mathcal{G}(4,n)$, and in \Cref{sec:odd positive g}, we carry out the method of \ref{emp:poa} with $V=\mathcal{Z}_{2d}$; this culminates in \Cref{thm:odd g result}.\label{part:emp cases 3}
\item \textbf{The case $(e,n) = (6,n)$.} Gloden noted in~\cite{gloden}*{Page~87} that if $P$ is a $\mathbb{Q}$-point of $\mathcal{B}_n = \mathcal{G}_0(3,n)$, and ``$-P$'' denotes the tuple whose coordinates are the negatives of the coordinates of $P$, then the tuple $(P,-P) \in \mathbb{Q}^{6n}$ is $(6,n)$-composite. In Sections \ref{sec:rational base} -- \ref{sec:finale}, we carry out the method of \ref{emp:poa} with $V=\mathcal{B}_{2d}$; this culminates in Theorems \ref{thm:2mod3}, \ref{thm:1mod3} and \ref{thm:2torsion}. \label{part:emp cases 4}
\item \textbf{The ``Kummer'' case.} There is a closed subscheme $\mathcal{K}(e,n)\subset \mathcal{G}(e,n)$, whose geometric irreducible components are affine spaces defined over the $e$-th cyclotomic field $\mathbb{Q}(\zeta_e)$, such that for any field $K\supset \mathbb{Q}(\zeta_e)$, the $K$-points of $\mathcal{K}(e,n)$ are tuples $(a_i) \in K^{en}$ such that $\prod_i(x-a_i) = M(x^e)$, for some $M(x)\in K[x]$ of degree $n$ (see \Cref{rem:kummer locus}). In \Cref{sec:kummer}, we carry out the method of \ref{emp:poa} by taking $V$ to be one of these affine spaces; this culminates in \Cref{thm:cyc}.
\end{enumerate}
\end{emp}
\begin{rem}
Choudhry~\cite{choPTWellcurve} has constructed elliptic curves of large rank (at least $12$) using solutions to the PTE problem, but his method is distinct from the method of this paper. We note (see Tables \ref{tab: genus and ranks gammas} and \ref{tab: genus and ranks thetas}) that for genus $g=1$, our method cannot do any better than the ``baseline'' method of \ref{emp:mestre}, which produces elliptic curves of rank at least $8$.
\end{rem}
\section{Parametrizing $(6,n)$-composite tuples}\label{sec:rational base}
Sections \ref{sec:rational base} -- \ref{sec:finale} form one block, culminating in Theorems \ref{thm:2mod3}, \ref{thm:1mod3}, and the proof of \Cref{thm:2torsion} in \ref{proof:2torsion}.
In this section, we fix an integer $n\geqslant 2$, an arbitrary number field $K$, and define a $K$-variety $\mathcal{B}_n$ (see \ref{emp:way forward} \ref{part:emp cases 2} and \ref{part:emp cases 4}), which parametrizes $(6,n)$-composite tuples (as in \ref{emp:poa}).
\begin{emp} \label{variety:cals} \textbf{The rational variety $\mathcal{B}_n$. }
Let $\mathcal{B}_n$ denote the closed subscheme of $\mathbb{A}^{3n}_K \coloneqq \spec K[T_{i1},T_{i2},T_{i3}\mid i=1,\dotsc,n]$ defined by the equations:
\begin{align}
\begin{split}\label{eq:calsn equations}
T_{11} + T_{12} + T_{13} = & \dotsb = T_{n1} + T_{n2} + T_{n3} = 0,\\
T_{11}T_{12} + T_{11}T_{13} + T_{12}T_{13} = & \dotsb = T_{n1}T_{n2} + T_{n1}T_{n3} + T_{n2}T_{n3}.
\end{split}
\end{align}
More formally, if we let $P_i$ (resp. $G_i$) denote the polynomial $T_{i1} + T_{i2} + T_{i3}$ (resp. $T_{i1}T_{i2} + T_{i1}T_{i3} + T_{i2}T_{i3}$), then we define:
\begin{equation*}
S \coloneqq \dfrac{K[T_{i1},T_{i2},T_{i3} \mid i=1,\dotsc,n]}{(P_1,\dotsc,P_n,G_1-G_2,\dotsc,G_{n-1}-G_n)},\qquad \quad \mathcal{B}_n \coloneqq \spec S.
\end{equation*}
\end{emp}
\begin{emp}\label{emp:notation cals} \textbf{Notation.} The following notation will recur (until \Cref{sec:finale}).
\begin{enumerate}[(i)]
\item We define the following elements of $S$:\label{part:cals notation i}
\begin{equation}\label{eq:functions}
\begin{alignedat}{2}
b & \coloneqq T_{11}T_{12} + T_{11}T_{13} + T_{12}T_{13} = G_1, && \quad \\
t_i & \coloneqq T_{i1}T_{i2}T_{i3} &&\qquad i=1,\dotsc,n,\\
u_i & \coloneqq t_i^2, && \qquad i=1,\dotsc,n,\\
m_j & \coloneqq (-1)^{n-j}s_{n-j}(u_1,\dotsc,u_n), && \qquad j=0,\dotsc,n-1,\\
U_{ij} & \coloneqq T_{ij}^2, && \qquad i=1,\dotsc,n, \; j=1,2,3.\\
\end{alignedat}
\end{equation}
We let $\mbf{t},\mbf{u},$ and $\mbf{m}$ denote the tuples of elements $(t_1,\dotsc,t_n), (u_1,\dotsc,u_n),$ and $(m_0,\dotsc,m_{n-1})$ respectively. We let $R$ denote the $K$-subalgebra of $S$ generated by the $U_{ij}$'s, and we set $\mathcal{H}_n \coloneqq \spec R$.
\item We define the following polynomials in $S[x]$:
\begin{align}\label{eq:g and ghat}
\begin{split}
g(x) & \coloneqq x(x^2+b), \\
\widehat{g}(x) & \coloneqq x(x+b)^2, \\
m(x) & \coloneqq x^n + m_{n-1}x^{n-1} + \dotsb + m_0.
\end{split}
\end{align}
\end{enumerate}
\end{emp}
The fourth line of \eqref{eq:comp factorization} below implies that $\mathcal{B}_n$ parametrizes $(6,n)$-composite tuples (as in \ref{emp:poa}).
\begin{prop}\label{prop: poly identities}
Continuing with the notation of \ref{emp:notation cals}, we have the following identities in $S[x]$:
\begin{enumerate}[(i)]
\item For $i=1,\dotsc, n$, we have
\begin{align}\label{eq:g - ti}
\begin{split}
g(x) + t_i & = (x + T_{i1})(x + T_{i2})(x + T_{i3})\\
g(x) - t_i & = (x - T_{i1})(x - T_{i2})(x - T_{i3})\\
\widehat{g}(x) - u_i & = (x - U_{i1})(x-U_{i2})(x - U_{i3})\\
g(x)^2 - u_i & = (x - T_{i1})(x - T_{i2})(x - T_{i3})(x + T_{i1})(x + T_{i2})(x + T_{i3})\\
& = \widehat{g}(x^2) - u_i.
\end{split}
\end{align}
\item We have
\begin{align}\label{eq:comp factorization}
\begin{split}
m(x) & = (x-u_1) \dotsb (x-u_n)\\
m(x^2) & = \prod_{i=1}^n (x+t_i)(x-t_i)\\
m(\widehat{g}(x)) & = \prod_{i=1}^n (x-U_{i1})(x-U_{i2})(x-U_{i3})\\
m(g(x)^2) & = \prod_{i=1}^n (x - T_{i1})(x - T_{i2})(x - T_{i3})(x + T_{i1})(x + T_{i2})(x + T_{i3}) \\
& = m(\widehat{g}(x^2)).
\end{split}
\end{align}
\end{enumerate}
\end{prop}
\begin{proof}
The reader can check that these identities follow from the definitions in \ref{emp:notation cals}.
\end{proof}
\section{Normality and rationality of $\mathcal{B}_n$}\label{sec:normality rationality calsn}
As in the previous section, $n\geqslant 2$ is a fixed integer. In this section, we prove that $\mathcal{B}_n$ is a normal, $K$-rational variety; the normality of $\mathcal{B}_n$ is used in the next section, and the rationality in \Cref{sec:finale}.
Recall that an affine $K$-scheme $Y$ is a \defi{complete intersection} over $K$ if it admits a presentation $Y \cong \spec K[x_1,\dotsc,x_m]/(f_1,\dotsc,f_r)$, with $\dim Y = m-r$.
\begin{lem}\label{lem:dim Bn = n+1}
Let $\wt{\mbf{m}}$ denote the tuple of indeterminates $\wt{m}_0,\dotsc,\wt{m}_{n-1}$.
\begin{enumerate}[(a)]
\item The $K$-morphism $f: \mathcal{B}_n \longrightarrow \mathbb{A}^{n+1}_K \coloneqq \spec K[\wt{b},\wt{\mbf{m}}]$, defined on coordinate rings by $\wt{b} \longmapsto b, \wt{m}_i \longmapsto m_i$, is finite and surjective.\label{part:dim (a)}
\item Each of the tuples $(b,\mbf{m}), (b,\mbf{u})$ and $(b,\mbf{t})$ generates a $K$-subalgebra of $S$ that is isomorphic to an $n+1$-dimensional polynomial ring over $K$.\label{part:dim (b)}
\item The $K$-scheme $\mathcal{B}_n$ is an $n+1$-dimensional complete intersection over $K$. \label{part:dim (c)}
\end{enumerate}
\end{lem}
\begin{proof}
We first note that \ref{part:dim (c)} follows from \ref{part:dim (a)}; indeed, \ref{part:dim (a)} implies that $\dim \mathcal{B}_n = n+1$, and it is then clear from the definition in \ref{variety:cals} that $\mathcal{B}_n$ is a complete intersection over $K$.
We now prove \ref{part:dim (a)}. To see that $f$ is surjective, suppose $Q = (b_0,k_0,\dotsc,k_{n-1})$ is a $\overline{k}$-point of the target. Let $\widehat{G}(x) \coloneqq x(x + b_0)^2 \in \overline{k}[x]$, let $M(x) \coloneqq x^n + k_{n-1}x^{n-1} + \dotsb + k_0 \in \overline{k}[x]$, and let $c_1,\dotsc,c_n \in \overline{k}$ denote the roots of $M(x)$ (counted with multiplicity). For each $i=1,\dotsc,n$, let $a_{i1},a_{i2},a_{i3},-a_{i1},-a_{i2},-a_{i3} \in \overline{k}$ denote the roots of $\widehat{G}(x^2) - c_i$ (counted with multiplicity). Then, $P = (a_{ij})$ is a $\overline{k}$-point of the fiber $f^{-1} (Q)$. Since $f$ is dominant and the target is reduced, the homomorphism $K[\wt{b},\wt{\mbf{m}}] \longrightarrow S$ is injective and identifies $K[\wt{b},\wt{\mbf{m}}]$ with the $K$-subalgebra of $S$ generated by $b,\mbf{m}$ (which we denote by $K[b,\mbf{m}]$). The finiteness of $f$ then follows from the last line of \eqref{eq:comp factorization}, which implies that each $T_{ij}$ is the root of the monic polynomial $m(g(x)^2) \in K[b,\mbf{m}][x]$, and hence, that $S$ is a finite $K[b,\mbf{m}]$-module.
For \ref{part:dim (b)}, we need to show that each $n+1$-tuple in the statement consists of elements of $S$ that are algebraically independent over $K$; we already saw above that this is true of the tuple $(b,\mbf{m})$. For the remaining tuples, is suffices to show they each generate a field $F$ of transcendence degree $n+1$ over $K$. For this, we note that $K(b,\mbf{m}) \subset F \subset \mathcal{K}$, and since $[\mathcal{K}:K(b,\mbf{m})]$ is finite (by \ref{part:dim (a)}), we have $\trdeg_K F = \trdeg_K \mathcal{K} = n+1$, so we are done.
\end{proof}
\begin{emp}\textbf{A different presentation of $\mathcal{B}_n$.} \label{emp:cals second pres}
For each $i=1,\dotsc,n$, we have $T_{i3} = -(T_{i1} + T_{i2}) \in S$, and the reader can check that we have an isomorphism $S \cong K[T_{i1},T_{i2} \mid 1\leqslant i \leqslant n]/(F_1,\dotsc,F_{n-1})$, where $F_i \coloneqq (T_{i1}^2 + T_{i1}T_{i2} + T_{i2}^2) - (T_{i+1,1}^2 + T_{i+1,1}T_{i+1,2} + T_{i+1,2}^2)$ for $i=1,\dotsc,n-1$. Note that the element $b\in S$ from \eqref{eq:functions} is identified with $-(T_{11}^2 + T_{11}T_{12} + T_{12}^2)$ under this isomorphism.
For each $i = 1,\dotsc,n$, if we let $N: \mathbb{A}^2_K \longrightarrow \spec K[z_i]$ be the morphism defined on coordinate rings by $z_i \longmapsto (T_{i1}^2 + T_{i1}T_{i2} + T_{i2}^2)$, then $\mathcal{B}_n$ fits into a \emph{cartesian diagram}:
\begin{equation}\label{diag:cals fib}
\begin{tikzcd}
\mathcal{B}_n \arrow[d] \arrow[r, hook] \drar[phantom, "\boxtimes"] & (\mathbb{A}^2_K)^n \arrow[d, "(N)^n"]\\
\mathbb{A}^1_K \arrow[r, hook, swap, "\Delta"] & (\mathbb{A}^1_K)^n,
\end{tikzcd}
\end{equation}
in which the bottom horizontal morphism is the diagonal closed immersion.
\end{emp}
We recall before the next lemma that a $K$-group scheme $Y$ is an \defi{algebraic $K$-torus} if there is an isomorphism of $\overline{k}$-group schemes $\smash{Y_{\overline{k}} \cong \gm{\overline{k}}^r}$, for some $r$.
\begin{lem}\label{lem:calv rational}
The open subscheme $\mathcal{V}_n \coloneqq \{ b \neq 0 \} \subset \mathcal{B}_n$ is an $n+1$-dimensional algebraic $K$-torus that is rational over $K$.
\end{lem}
\begin{proof}
Let $L$ denote the $K$-algebra $K[x]/(x^2 + x + 1)$, and let $\zeta \in L$ denote the class of $x$. The set $\{ 1, -\zeta \}$ is a $K$-basis for $L$, and the bijection $\mathbb{A}^2_K (R) = \mf{R}_{L/K} \mathbb{A}^1_L(R)$ obtained by identifying $(a_{i1},a_{i2}) \in \mathbb{A}^2_K(R)$ with $(a_{i1} - a_{i2} \zeta) \in R \otimes_K L$ is natural in $R$, so we can identify $\spec K[T_{i1},T_{i2}]$ with the \defi{Weil Restriction} $\mf{R}_{L/K} \mathbb{A}^1_L$ (cf. \cite{neronmodels}*{Chapter.~7.6}). An element $a_{i1} - \zeta a_{i2} \in R\otimes_K L$ is invertible if and only if its \emph{norm} $N_{R\otimes L/R}(a_{i1} - \zeta a_{i2}) \coloneqq a_{i1}^2 + a_{i1}a_{i2} + a_{i2}^2$ is invertible in $R$. Thus, the open subscheme $\{T_{i1}^2 + T_{i1}T_{i2} + T_{i2}^2 \neq 0 \} \subset \mf{R}_{L/K} \mathbb{A}^1_L$ is naturally identified with $\mf{R}_{L/K} \gm{L}$, and moreover, the map $N : \mf{R}_{L/K} \mathbb{A}^1_L \longrightarrow \mathbb{A}^1_K$ from \ref{emp:cals second pres} restricts to a group scheme homomorphism $N: \mf{R}_{L/K} \gm{L} = \spec K[T_{i1},T_{i2},(T_{i1}^2 + T_{i1}T_{i2} + T_{i2}^2)^{-1}] \longrightarrow \gm{K} \coloneqq \spec K[z_i,z_i^{-1}]$. By restricting everything in \eqref{diag:cals fib} to $(\gm{K})^n \subset (\mathbb{A}^1_K)^n$, we see that $\mathcal{V}_n$ fits into a \emph{cartesian diagram}:
\begin{equation}\label{diag:calv fib}
\begin{tikzcd}
\mathcal{V}_n \arrow[d] \arrow[r, hook] \drar[phantom, "\boxtimes"] & (\mf{R}_{L/K} \gm{L})^n \arrow[d, "(N)^n"]\\
\gm{K} \arrow[r, hook, swap, "\Delta"] & (\gm{K})^n.
\end{tikzcd}
\end{equation}
Since the right vertical map and the bottom closed map are both group-scheme homomorphisms, $\mathcal{V}_n$ is a subgroup scheme of $(\mf{R}_{L/K} \gm{L})^n$. Note that $L/K$ is \'etale, so $\mf{R}_{L/K} \gm{L}$ is an algebraic $K$-torus.
We claim that there is an isomorphism of algebraic $K$-tori:
\begin{equation*}
\phi: \mathcal{V}_n \xrightarrow{\raisebox{-0.7ex}[0ex][0ex]{$\;\sim\;$}} H \coloneqq \mf{R}_{L/K} \gm{L} \times (\mf{R}^1_{L/K} \gm{L})^{n-1},
\end{equation*}
where $\mf{R}^1_{L/K} \gm{L}$ denotes the \defi{norm-one torus}, i.e. the kernel of the norm map $N : \mf{R}_{L/K} \gm{L} \longrightarrow \gm{K}$. Since $\mf{R}_{L/K} \gm{L}$ is $K$-rational and one-dimensional $K$-tori are $K$-rational, this isomorphism implies that $H$ (and therefore, also $\mathcal{V}_n$) is $K$-rational.
Now, we construct the isomorphism $\phi$. For any commutative group scheme $\mathcal{G}$, let $m : \mathcal{G}^2 \longrightarrow \mathcal{G}$ denote the multiplication, $\textup{inv}: \mathcal{G} \longrightarrow \mathcal{G}$ the inversion, $p_i : \mathcal{G}^n \longrightarrow \mathcal{G}$ the $i^\textup{th}$-projection, and let $p_{ij} \coloneqq (p_i,p_j) : \mathcal{G}^n \longrightarrow \mathcal{G}^2$. We define the homomorphism $\phi : \mathcal{G}^n \longrightarrow \mathcal{G}^n$ to be the composition
\begin{equation*}
\mathcal{G}^n \xlongrightarrow{(\,\textup{inv}\circ p_1 ,\,p_2,\,\dotsc\,,\,p_n)} \mathcal{G}^n \xlongrightarrow{(\, \textup{Id},\, m\circ p_{12},\, \dotsc \,, \,m \circ p_{1n})} \mathcal{G}^n.
\end{equation*}
The reader can easily check that $\phi \circ \phi = \textup{Id}$, so any subgroup scheme of $\mathcal{G}^n$ is mapped isomorphically onto its image. In particular, if $\mathcal{G} = \gm{K}$, then $\phi:\gm{K}^n \longrightarrow \gm{K}^n$ is given by $(z_1,\dotsc,z_n) \longmapsto (z_1^{-1} , z_2z_1^{-1} , \dotsc , z_n z_1^{-1}),$ and visibly, $\phi$ gives an isomorphism from the diagonal $\Delta(\gm{K}) = \{z_1 = \dotsb = z_n\} \subset \gm{K}^n$ onto the subgroup scheme $\gm{K} \times (1,\dotsc,1) \subset \gm{K}^n$ (i.e: the subgroup scheme on which the $i^\textup{th}$-coordinate is $1$ for $i=2,\dotsc,n$). We have a commutative diagram:
\[
\begin{tikzcd}
(\mf{R}_{L/K} \gm{L})^n \arrow[d, "{(N)^n}",swap] \arrow[r,"\phi"] & (\mf{R}_{L/K} \gm{L})^n \arrow[d, "{(N)^n}"]\\
\gm{K}^n \arrow[r,"\phi",swap] & \gm{K}^n.
\end{tikzcd}
\]
By \eqref{diag:calv fib}, the pre-image under $(N)^n$ of the diagonal $\Delta(\gm{K}) \subset \gm{K}^n$ is $\mathcal{V}_n$, and this is mapped isomorphically onto the pre-image of $\gm{K} \times (1,\dotsc,1)$, which by definition is $H = \mf{R}_{L/K} \gm{L} \times (\mf{R}^1_{L/K} \gm{L})^{n}$, so we are done.
\end{proof}
\begin{thm}\label{thm:calg03n rational}
The $K$-scheme $\mathcal{B}_n$ is a normal, $K$-rational variety of dimension $n+1$.
\end{thm}
\begin{proof}
Recall that $S$ denotes the coordinate ring of $\mathcal{B}_n$. First, we note that the closed subset $\{b = 0\} \subset \mathcal{B}_n$ is simply the $n$-fold product of the affine curve given by $x^2 + xy + y^2 = 0$, so it is of dimension $n$. Since $\mathcal{B}_n$ is a complete intersection (cf. \Cref{lem:dim Bn = n+1}), its irreducible components are equi-dimensional, of dimension $n+1$, so $\{b = 0\}$ cannot contain an irreducible component of $\mathcal{B}_n$. Thus, $b \in S$ is not a zero-divisor, the localization map $S \longrightarrow S[b^{-1}]$ is injective, and \Cref{lem:calv rational} implies that $\mathcal{B}_n$ is a $K$-rational variety of dimension $n+1$.
Next, we prove that $\mathcal{B}_n$ is normal; in light of \cite{stacks-project}*{Lemma \href{https://stacks.math.columbia.edu/tag/038O}{038O}}, we may assume that $K$ is algebraically closed, and in particular, that $K$ contains a primitive third root of unity $\zeta_3$. \Cref{lem:dim Bn = n+1} \ref{part:dim (c)} implies (using \cite{stacks-project}*{Lemma \href{https://stacks.math.columbia.edu/tag/00SB}{00SB}} and \cite{stacks-project}*{Lemma \href{https://stacks.math.columbia.edu/tag/045N}{045N}}) that $\mathcal{B}_n$ is Cohen-Macaulay, so it satisfies Serre's condition (S2) (by~\cite{stacks-project}*{Lemma \href{https://stacks.math.columbia.edu/tag/0342}{0342}}). Using~\cite{stacks-project}*{Lemma \href{https://stacks.math.columbia.edu/tag/031S}{031S}}, we are therefore reduced to showing that: \emph{$\mathcal{B}_n$ is regular in codimension $1$}. To see that this is true, we first note that $\mathcal{B}_n$ is $K$-isomorphic to the closed subscheme of $\spec K[z_{i1},z_{i2} \mid i=1,\dotsc,n]$ given by the equations $z_{11}z_{12} = \dotsb = z_{n1}z_{n2},$
with the isomorphism defined on coordinate rings by $z_{ij} \longmapsto T_{i1} - \zeta_3^jT_{i2}$, for $i=1,\dotsc,n$ and $j=1,2$.
For each $i=1,\dotsc,n$, let $D_i$ denote the closed subset $\{z_{i1} = z_{i2} = 0 \} \subset \mathcal{B}_n$; note that $D_i$ is isomorphic to the $(n-1)$-fold product of the affine curve given by $xy=0$, so it is of codimension $(n+1) - (n-1) = 2$ in $\mathcal{B}_n$. We claim that the singular locus of $\mathcal{B}_n$ is contained in the codimension $2$ subset $D \coloneqq D_1 \cup \dotsb \cup D_n$. Indeed, any point in the complement of $D$ is contained in an open subset of the form $U(\epsilon(1),\dotsc,\epsilon(n)) \coloneqq \{ z_{1,\epsilon(1)} \dotsb z_{n,\epsilon(n)} \neq 0 \} \subset \mathcal{B}_n,$
where each $\epsilon(i) \in \{1,2\}$. This open subset is isomorphic to $\spec K[w,v_i,v_i^{-1} \mid i=1,\dotsc,n]$ via the morphism defined on coordinate rings by $w \longmapsto z_{1,3-\epsilon(1)}$ and $v_i \longmapsto z_{i,\epsilon(i)}$ for $i=1,\dotsc,n$; the inverse is defined by $z_{i,\epsilon(i)} \longmapsto v_i$ and $z_{i,3-\epsilon(i)} \longmapsto wv_1v_i^{-1}$, for each $i=1,\dotsc,n$. Thus $U(\epsilon(1),\dotsc , \epsilon(n))$ is smooth, and we are done.
\end{proof}
\section{Some properties of the function field $K(\mathcal{B}_n)$}\label{sec:properties of kk}
As in the previous section, $n\geqslant 2$ is a fixed integer. In this section, we study a Galois action on the function field $\mathcal{K} \coloneqq K(\mathcal{B}_n)$. If $n=2d$ for some integer $d\geqslant 2$, we define a certain subfield $K(b,\Bell) \subset \mathcal{K}$ and show that it is algebraically closed in $\mathcal{K}$.
Propositions \ref{prop: cals galois group} and \ref{prop:alg closed} together allow us to apply the results of the next section to determine (in \Cref{sec:ranks of curves}) the ranks of various curves parametrized by $\mathcal{B}_{2d}$.
\begin{emp}\label{emp:function fields} \textbf{Subfields of $K(\mathcal{B}_n)$.} Let $\mathcal{K}$ denote the function field $K(\mathcal{B}_n)$, and recall the various elements of $S$ defined in \eqref{eq:functions}. By \Cref{lem:dim Bn = n+1}, we have subrings $K[b,\mbf{m}],K[b,\mbf{u}]$ and $K[b,\mbf{t}]$ of $S$. Note moreover that $K[b,\mbf{u}]\subset R$; indeed, the identity $(x - U_{i1})(x-U_{i2})(x-U_{i3}) = x^3 + 2bx^2 + b^2x - u_i$ (cf. \eqref{eq:g - ti}) implies that $b = -(U_{i1} + U_{i2} + U_{i3})/2$ and $u_i = U_{i1}U_{i2}U_{i3}$. Putting it all together, we get the following diagrams of rings and their function fields:
\begin{equation}\label{diag:rings and fields}
\begin{tikzcd}
& S \arrow[dl, dash] \arrow[dr, dash] & & & \mathcal{K} \arrow[dl, dash] \arrow[dr, dash]\\
K[b,\mbf{t}] \arrow[dr, dash] & & R \arrow[dl, dash] & K(b,\mbf{t}) \arrow[dr, dash] & & K(\mathcal{H}_n) \arrow[dl, dash]\\
& K[b,\mbf{u}] \arrow[d, dash] & & & K(b,\mbf{u}) \arrow[d, dash]\\
& K[b,\mbf{m}] & & & K(b,\mbf{m}).
\end{tikzcd}
\end{equation}
Let $S_n$ denote the symmetric group on $n$ letters. Recall that for a finite group $G$, the \defi{wreath product} of $G$ by $S_n$, denoted $G \wr S_n$, is the semi-direct product $G^n \rtimes_q S_n$, where $q:S_n \longrightarrow \aut(G^n)$ is the natural permutation action. We refer the reader to ~\cite{odoni}*{Section~4} for a discussion of wreath products arising as Galois groups of composite polynomials (note that Odoni's notation $G[H]$ corresponds to our notation $H \wr G$).
\end{emp}
\begin{prop}\label{prop: cals galois group}
\begin{enumerate}[(i)]
\item\label{part:cals galois i} The following isomorphisms hold:
\begin{equation}
\begin{alignedat}{4}
\alpha_1:(S_3)^n & \xrightarrow{\raisebox{-0.7ex}[0ex][0ex]{$\;\sim\;$}} \textup{Gal}(\mathcal{K}/K(b,\mbf{t})) && \qquad &&\quad \alpha_2:(S_3)^n && \xrightarrow{\raisebox{-0.7ex}[0ex][0ex]{$\;\sim\;$}} \textup{Gal}(K(\mathcal{H}_n)/K(b,\mbf{u}))\\
(\sigma_1,\dotsc,\sigma_n) & \longmapsto (T_{ij} \mapsto T_{i,\sigma_i(j)}) && \qquad && (\sigma_1,\dotsc,\sigma_n) && \longmapsto (U_{ij} \mapsto U_{i,\sigma_i(j)})\\
\; & \; && \; && \; && \\
\beta_1:(S_2)^n & \xrightarrow{\raisebox{-0.7ex}[0ex][0ex]{$\;\sim\;$}} \textup{Gal}(\mathcal{K}/K(\mathcal{H}_n)) && \qquad &&\quad \beta_2:(S_2)^n && \xrightarrow{\raisebox{-0.7ex}[0ex][0ex]{$\;\sim\;$}} \textup{Gal}(K(b,\mbf{t})/K(b,\mbf{u}))\\
(e_1,\dotsc,e_n) & \longmapsto (T_{ij} \mapsto e_iT_{ij}) && \qquad && (e_1,\dotsc,e_n) && \longmapsto (t_i \mapsto e_it_i),
\end{alignedat}
\end{equation}
where each $e_i \in \{\pm 1\} = S_2$.
\item\label{part:cals galois ii} The maps $\alpha_1$ and $\beta_1$ induce an isomorphism $$(\alpha_1,\beta_1) :(S_3\times S_2)^n \xrightarrow{\raisebox{-0.7ex}[0ex][0ex]{$\;\sim\;$}} \textup{Gal}(\mathcal{K}/K(b,\mbf{u})).$$
\item \label{part:cals galois iii} We have an injection $\gamma : S_n \longrightarrow \textup{Gal}(\mathcal{K}/K(b,\mbf{m}))$ defined by $\tau \longmapsto (T_{ij} \mapsto T_{\tau(i),j})$, which induces homomorphisms:
\begin{equation}
\begin{alignedat}{3}
\gamma_1 : S_n & \lhook\joinrel\longrightarrow \aut_{K(b,\mbf{m})}(K(b,\mbf{t})) && \qquad \gamma_2 : S_n && \xrightarrow{\raisebox{-0.7ex}[0ex][0ex]{$\;\sim\;$}} \textup{Gal}(K(b,\mbf{u})/K(b,\mbf{m}))\\
\tau & \longmapsto (t_i \mapsto t_{\tau(i)}) &&\qquad\qquad \tau && \longmapsto (u_i \mapsto u_{\tau(i)})
\end{alignedat}
\end{equation}
\item \label{part:cals galois iv} The maps $(\alpha_1,\beta_1)$ and $\gamma$ induce an isomorphism
\begin{align}
\begin{split}
\delta : (S_3\times S_2) \wr S_n & \xrightarrow{\raisebox{-0.7ex}[0ex][0ex]{$\;\sim\;$}} \textup{Gal}(\mathcal{K}/K(b,\mbf{m}))\\
(\overline{\sigma,e})\cdot \tau & \longmapsto \big( T_{ij} \mapsto e_{\tau(i)}T_{\tau(i),\sigma_{\tau(i)}(j)} \big),
\end{split}
\end{align}
where $(\overline{\sigma,e})$ denotes a tuple $((\sigma_1,e_1),\dotsb,(\sigma_n,e_n)) \in (S_3\times S_2)^n$.
\end{enumerate}
\end{prop}
\begin{proof}
Recall the presentation of $\mathcal{B}_n$ given in \ref{variety:cals}. Since $\mathcal{K}/K(b,\mbf{m})$ is the splitting field of the polynomial $m(g(x)^2) \in K(b,\mbf{m})[x]$ (cf. fourth line of \eqref{eq:comp factorization}), any $\sigma \in \textup{Gal}(\mathcal{K}/K(b,\mbf{m}))$ is determined by a permutation of $\{\pm T_{ij}\}$; any such permutation that preserves the identities in \eqref{eq:calsn equations} induces an automorphism of $\mathcal{B}_n$, and hence, of $\mathcal{K}$. The reader can use this to easily verify that the formulas above define honest automorphisms of $\mathcal{K}$. Using the various identities in \ref{emp:notation cals}, the reader can also verify that the maps above, regarded a priori as maps to $\aut(\mathcal{K})$, have image contained in the asserted subgroups.
First, we prove \ref{part:cals galois i}. Since $\mathcal{K}/K(b,\mbf{t})$ is the splitting field of the polynomials $g(x) - t_i = x^3 - bx - t_i$ (cf. \eqref{eq:g - ti}), any element of $\textup{Gal}(\mathcal{K}/K(b,\mbf{t}))$ is given by a simultaneous permutation of the triplets $T_{i1},T_{i2},T_{i3}$, for $i=1,\dotsc,n$. So, we have $\alpha_1((S_3)^n) = \textup{Gal}(\mathcal{K}/K(b,\mbf{t}))$. To see that $\alpha_1$ is injective, suppose for a contradiction that there exists an element $(\sigma_1,\dotsc,\sigma_n)\neq 1 \in \ker \phi_1$. Then for some $k$, we have $\sigma_k \neq 1 \in S_3$, so there exist $1\leqslant p\neq q \leqslant 3$ such that $T_{kp} = \sigma_k(T_{kp}) = T_{kq}$; but this is absurd, since $\textup{disc} (x^3 + bx - t_k) = -4b^3 - 27t_k^2 \neq 0 \in K(b,\mbf{t})$. We conclude that $\phi_1$ is an isomorphism.
Composing $\alpha_1$ with the natural restriction $\aut(\mathcal{K})\longrightarrow \aut(K(\mathcal{H}_n))$, we get $\alpha_2$. Note that $K(\mathcal{H}_n)/K(b,\mbf{u})$ is the splitting field of the polynomials $\widehat{g}(x) - u_i = x^3 + 2bx^2 + b^2x - u_i \in K(b,\mbf{u})[x]$, and $\textup{disc} (\widehat{g}(x) - u_i) = -4 b^3 u_i - 27 u_i^2 \neq 0 \in K(b,\mbf{u})$. Thus, we can repeat the above argument by replacing $T_{ij}$ with $U_{ij}$, $g(x) - t_i$ with $\widehat{g}(x) - u_i$, and $\mathcal{K}/K(b,\mbf{t})$ with $K(\mathcal{H}_n)/K(b,\mbf{u})$, to find that $\alpha_2$ is an isomorphism.
Next, we note that $\beta_2$ is obtained by composing $\beta_1$ with the restriction $\aut(\mathcal{K}) \longrightarrow \aut(K(b,\mbf{t}))$, and $\beta_2$ is clearly an isomorphism (cf. \eqref{eq:functions}). This implies that $\beta_1$ is injective, and since \ref{part:cals galois i} implies that $[\mathcal{K}/K(\mathcal{H}_n)] = [\mathcal{K}:K(b,\mbf{t})][K(b,\mbf{t}):K(b,\mbf{u})]/[K(\mathcal{H}_n)/K(b,\mbf{u})] = 2^n$, we find that $\beta_2$ is also an isomorphism. This concludes the proof of \ref{part:cals galois i}.
Part \ref{part:cals galois i} implies that $K(b,\mbf{u})$ and $K(\mathcal{H}_n)$ are linearly disjoint extensions of $K(b,\mbf{u})$, with compositum $\mathcal{K}$. So, Galois theory gives $\textup{Gal}(\mathcal{K}/K(b,\mbf{u})) \cong (S_3)^n \times (S_2)^n$, and the latter is naturally isomorphic to $(S_3\times S_2)^n$, which establishes part \ref{part:cals galois ii}.
For part \ref{part:cals galois iii}, note that if we compose $\gamma$ with the restriction $\aut(\mathcal{K}) \longrightarrow \aut(K(b,\mbf{t})$, then we get $\gamma_1$, and by composing further with the restriction $\aut(K(b,\mbf{t})) \longrightarrow \aut(K(b,\mbf{u}))$, we get $\gamma_2$. Thus, the injectivity of $\gamma$ and $\gamma_1$ would follow if $\gamma_2$ is an isomorphism; the latter fact is nothing but the Fundamental theorem of symmetric functions (cf. \cite{hungerford}*{Page~254, Theorem~2.18}).
For part \ref{part:cals galois iv}, we first note that $\delta$ is induced by $(\alpha_1,\beta_1)$ and $\gamma$ in the sense that $\delta(\overline{\sigma,e}\cdot 1) = (\alpha_1,\beta_1)(\overline{\sigma,e}) = (\alpha_1(\overline{\sigma}),\beta_1(\overline{e}))$ and $\delta(1\cdot \tau) = \gamma(\tau)$. To verify that $\delta$ is a homomorphism, we check that the map respects the semi-direct product structure; for any $\overline{\sigma,e} \in (S_3\times S_2)^n$, any $\tau \in S_n$, and any $i=1,\dotsc,n , j=1,2,3$, we compute:
\begin{align*}
\big(\delta(\tau^{-1}) \delta(\overline{\sigma,e}) \delta(\tau)\big) (T_{ij}) & = \big(\delta(\tau^{-1}) \delta(\overline{\sigma,e})\big) (T_{\tau(i),j})\\
& = \big(\delta(\tau^{-1})\big) (e_{\tau(i)}T_{\tau(i),\sigma_{\tau(i)}(j)}\\
& = e_{\tau(i)}T_{i,\sigma_{\tau(i)}(j)}\\
& = \big(q(\tau)(\overline{\sigma,e})\big)(T_{ij}),
\end{align*}
so $\delta$ is a homomorphism. That it is an isomorphism follows from parts \ref{part:cals galois ii} and \ref{part:cals galois iii}, and the following commutative diagram of exact sequences:
\begin{equation*}
\begin{tikzcd}
1 \arrow[r] & (S_3 \times S_2)^n \arrow[r] \arrow[d, "{(\alpha_1, \beta_1)}"] & (S_3\times S_2) \wr S_n \arrow[r] \arrow[d, "\delta"] & S_n \arrow[r] \arrow[d, "\gamma"] & 1\\
1 \arrow[r] & \textup{Gal}(\mathcal{K}/K(b,\mbf{u})) \arrow[r] & \textup{Gal}(\mathcal{K}/K(b,\mbf{m})) \arrow[r] & \textup{Gal}(K(b,\mbf{u})/K(b,\mbf{m})) \arrow[r] & 1.
\end{tikzcd}
\end{equation*}
\end{proof}
\begin{emp}\label{emp:sec 8 setup}
For the rest of this section, we set $n=2d$ for an integer $d\geqslant 2$, so that $\mathcal{B}_{2d}$ is a $K$-rational variety of dimension $2d+1$. The coordinate ring $S$ contains a $2d+1$-dimensional polynomial algebra $K[b,\mbf{m}]$ (cf. \ref{emp:function fields}); using the isomorphism $K[b,\Bell,\mbf{h}] \cong K[b,\mbf{m}]$ from \Cref{lem:kthroot}, where $\Bell$ and $\mbf{h}$ denote the tuples of indeterminates $l_0,\dotsc,l_{d-1}$ and $h_0,\dotsc,h_{d-1}$ respectively, we get inclusions $K[b,\Bell] \subset K[b,\Bell,\mbf{h}] \cong K[b,\mbf{m}] \subset S.$
Recall the finite, surjective $K$-morphism $f:\mathcal{B}_{2d} \longrightarrow \mathbb{A}^{2d+1}(b,\mbf{m})$ (cf. \Cref{lem:dim Bn = n+1}), and let $\Delta \subset \mathbb{A}^{2d+1}(b,\mbf{m})$ denote the \defi{branch locus} (i.e. set of branch points) of $f$. Note that $\Delta$ is non-empty; for instance, $(0,\dotsc,0) \in \mathbb{A}^{2d+1}(b,\mbf{m})$ is a branch point of $f$, since the fiber consists of the single point of $\mathcal{B}_{2d}$ with all coordinates $0$. Since $\mathcal{B}_{2d}$ is normal (cf. \Cref{thm:calg03n rational}), the \emph{Theorem of purity of branch locus} (cf. \cite{zariskipurity}*{Proposition~2} or \cite{stacks-project}*{Lemma \href{https://stacks.math.columbia.edu/tag/0BMB}{0BMB}}) then implies that $\Delta$ is a hypersurface in $\mathbb{A}^{2d+1}(b,\mbf{m})$, so it is defined by an ideal $(F) \subset K[b,\mbf{m}]$, with $F$ square-free and well-defined up to multiplication by a unit in $K[b,\mbf{m}]$.
\end{emp}
\begin{lem}\label{lem:divisbility}
There are no irreducible factors of $F \in K[b,\mbf{m}]$ contained in the subring $K[b,\Bell] \subsetneq K[b,\mbf{m}]$.
\end{lem}
\begin{proof}
We first note from the proof of \Cref{prop: cals galois group} that the Galois group $G = \textup{Gal}(\mathcal{K}/K(b,\mbf{m}))$ naturally acts on $\mathcal{B}_{2d}$ via permutations of the set $\{\pm T_{ij} \mid i=1,\dotsc,2d,j=1,2,3\}$, and moreover, $G = \aut(\mathcal{B}_{2d}/\mathbb{A}^{1+2d}(b,\mbf{m}))$ (we are abusing notation here). Let $P \coloneqq (a_{ij}) \in \mathcal{B}_{2d}(\overline{k}) $ be the point corresponding to the ring homomorphism $S \longrightarrow \overline{k}$ defined by $T_{ij} \longmapsto a_{ij}$. Then, $P$ is \emph{not} a ramification point for $f$ if and only if its orbit under $G$ is of size $\#G$, if and only if the elements of the multiset $\{\pm a_{ij}\}$ are distinct, if and only if the specialization $m_P(g_P(x)^2)$ is separable. Thus, the irreducible factors of $F$ are precisely the irreducible factors of $\textup{disc} \, m(g(x)^2)$, which we now compute.
\begin{enumerate}[(i)]
\item Suppose $a_{ij} = -a_{ij}$ for some $1\leqslant i\leqslant 2d$. Then $P$ lies in $\{T_{ij} = 0\}$, so $T_{ij}$ divides $F$ in $S$; taking the product over all $i$ and $j$, we get that $m_0$ (which is irreducible in $K[b,\mbf{m}]$) divides $F$ in $K[b,\mbf{m}]$.
\item Suppose that $a_{ij} = \pm a_{ik}$, or equivalently, that $a_{ij}^2 = a_{ik}^2$, for some $1\leqslant i\leqslant 2d$ and some $1\leqslant j\neq k \leqslant 3$. Then $P$ lies in the closed set $\{T_{ij}^2 = T_{ik}^2\} = \{U_{ij} = U_{ik}\}$, so $U_{ij} - U_{ik}$ divides $F$ in $S$. This implies that $\textup{disc} (\widehat{g}(x) - u_i)$ divides $F$ in $R$ (the coordinate ring of $\mathcal{H}_{2d}$), and so $\omega \coloneqq \prod_i \textup{disc} (\widehat{g}(x) - u_i)$ divides $F$ in $K[b,\mbf{m}]$. The group $\textup{Gal}(K(b,\mbf{u})/K(b,\mbf{m})) \cong S_{2d}$ acts transitively on $\{\textup{disc} (\widehat{g}(x)-u_1) \dotsc \textup{disc} (\widehat{g}(x) - u_{2d})\}$, which implies that $\omega$ is irreducible in $K[b,\mbf{m}]$.
\item Suppose that $a_{ij} = \pm a_{pj}$, or equivalently, that $a_{ij}^2 = a_{pj}^2$, for some $1\leqslant i\neq p\leqslant 2d$. Then, $$u_i - u_p = (x - U_{i1})(x-U_{i2})(x-U_{i3}) - (x-U_{p1})(x-U_{p2})(x-U_{p3})$$ vanishes at $P$, which implies that $\prod_{i\neq j} (u_i - u_j) = \textup{disc} \,m(x)$ divides $F$ in $K[b,\mbf{m}]$. Note that $\textup{disc} \, m(x)$ is irreducible in $K[b,\mbf{m}]$ (cf \cite{odoni}*{Lemma~8.1}).
\end{enumerate}
Thus, we have showed that $F$ factors (upto units) into irreducibles as $F = \omega \cdot m_0\cdot (\textup{disc} \,m) \in K[b,\mbf{m}]$; that these irreducible factors are co-prime follows by noting how they split in $K[b,\mbf{u}]$.
Letting $h(x),\ell(x) \in K[b,\mbf{h},\Bell] \cong K[b,\mbf{m}][x]$ denote the square-root approximation and remainder of the degree $2d$ polynomial $m(x)$ (as in \eqref{def:polys}), we re-write this factorization as:
\begin{equation*}
F = \omega \cdot (h_0^2 - \ell_0) \cdot \textup{disc} (h(x)^2 - \ell(x))\, \in K[b,\mbf{h},\Bell].
\end{equation*}
Clearly, $h_0^2 - \ell_0 \notin K[b,\Bell]$; we now show the same for the remaining two irreducible factors.
To show that $\omega \notin K[b,\Bell]$, it suffices to show that upon setting $b=0$, $\omega$ does not specialize to an element of $K[\Bell]$. Indeed, if we set $b=0$, then each $\widehat{g}(x) - u_i$ specializes to $x^3 - u_i \in K[\mbf{u}][x]$, whose discriminant has the single irreducible factor $u_i \in K[\mbf{u}]$. The product $\omega$ therefore specializes to a power of $u_1 \dotsb u_{2d} = m_0 = h_0^2 - l_0 \in K[\Bell,\mbf{h}]$, which is clearly not contained in $K[\Bell]$.
Similarly, to show $\textup{disc} (h(x)^2 - \ell(x)) \notin K[b,\Bell]$, we specialize the tuple $(b,\ell_{d-1},\dotsc,\ell_1,\ell_{0})$ to the tuple $(0,0,\dotsc,0,-1) \in K^{d+1}$, and show that $\textup{disc} \,m$ specializes to a non-constant element of $K[\mbf{h}]$. Indeed,
$m(x)$ specializes to $h(x)^2 - 1 \in K[\mbf{h}][x]$, which admits the specialization $x^{2d} - 1 \in K[x]$, which is separable, and also the specialization $(x^{d} - 1)^2 - 1 \in K[x]$, which is not separable. This establishes that $\textup{disc} (h(x)^2 - 1) \in K[\mbf{h}]$ is non-constant, and concludes the proof of the lemma.
\end{proof}
\begin{prop}\label{prop:alg closed}
The field $K(b,\Bell) = K(b,l_0,\dotsc,l_{d-1})$ is algebraically closed in $\mathcal{K}$.
\end{prop}
\begin{proof}
Assume for a contradiction that there is some finite extension $L \supsetneq K(b,\Bell)$ contained in $\mathcal{K}$, and let $Y$ be the normalization of $\mathbb{A}^{1+d}(b,\Bell) $ in $L$ (i.e. $Y = \spec A$, where $A$ is the integral closure of $K[b,\Bell]$ in $L$). By standard facts, $Y \longrightarrow \mathbb{A}^{1+d}(b,\Bell)$ is finite and surjective. We also have a morphism $\mathcal{B}_{2d} \longrightarrow \mathbb{A}^{2d+1}(b,\mbf{m}) \longrightarrow \mathbb{A}^{d+1}(b,\Bell)$.
Since $\mathcal{B}_{2d}$ is normal (cf. \Cref{thm:calg03n rational}), we have a containment $A \subset S$, which corresponds to a morphism $\mathcal{B}_{2d} \longrightarrow Y$, so we get a morphism ${\mathcal{B}_{2d} \longrightarrow Y' \coloneqq Y \times_{\mathbb{A}^{1+d}(b,\Bell)} \mathbb{A}^{1+2d}(b,\mbf{m})}$, giving a commutative diagram:
\[
\begin{tikzcd}
\mathcal{B}_{2d} \arrow[r] & Y' \arrow[r] \arrow[d] \drar[phantom, "\boxtimes"] & \mathbb{A}^{1+2d}(b,\mbf{m}) \arrow[d]\\
& Y \arrow[r] & \mathbb{A}^{1+d}(b,\Bell).
\end{tikzcd}
\]
Since the composite morphism $\mathcal{B}_{2d} \longrightarrow Y' \longrightarrow \mathbb{A}^{1+2d}(b,\mbf{m})$ has points that ramify completely in $\mathcal{B}_{2d}$ (for example, the point $P = (0,\dotsc,0) \in \mathbb{A}^{1+2d}(b,\mbf{m})(K)$), we get that $Y' \longrightarrow \mathbb{A}^{1+2d}(b,\mbf{m})$ is ramified. So, $Y \longrightarrow \mathbb{A}^{1+d}(b,\Bell)$ is also ramified, and its branch locus $\delta$ is non-empty and defined by a proper ideal $(f) \subset K[b,\Bell]$. The pre-image (call it $\delta'$) of $\delta$ under $\mathbb{A}^{1+2d}(b,\mbf{m}) \longrightarrow \mathbb{A}^{1+d}(b,\Bell)$ is the branch locus of $Y' \longrightarrow \mathbb{A}^{1+2d}(b,\mbf{m})$, and $\delta'$ is defined by $(f) \subset K[b,\mbf{m}]$. If $\Delta$ (cf. \ref{emp:sec 8 setup}) is defined by an ideal $(F)$, then the containment $\delta' \subset \Delta$ implies that $f$ divides $F$ in $K[b,\mbf{m}]$. This contradicts \Cref{lem:divisbility} above, so we are done.
\end{proof}
\section{Tools for bounding ranks of curves}\label{sec: a lemma}
In this section, we develop some tools that will be used to bound the ranks of the curves appearing in Section \ref{sec:ranks of curves}, \ref{sec:odd positive g}, and \ref{sec:kummer}.
\Cref{cor:contra rel} below is used in the proof of \Cref{lem:relations between points} by taking $(K_0,K_1,K_2) = (K(b,\Bell),K(b,\mbf{m}),\mathcal{K}).$ \Cref{lem:algebraic point} below is used in the proofs of Theorems \ref{thm:rank of lambda 2} and \ref{thm:kum curves} (we have included it in this section because it fits into the theme of this section).
We recall first that if $L/K$ is an extension of fields, a point $P:\spec L \longrightarrow X$ of a $K$-scheme $X$ is \defi{algebraic over $K$} if the image of $P$ is a \emph{closed point} of $X$.
\begin{lem}\label{lem:torsion fixed}
Let $K_0 \subset K_1 \subset K_2$ be a chain of field extensions such that $K_0$ is algebraically closed in $K_2$ and $K_2/K_1$ is finite Galois. Let $A/K_0$ be an abelian variety and $P \in A(K_2)$ a torsion point. Then, $P$ is fixed by the action of $\textup{Gal}(K_2/K_1)$ on $A(K_2)$.
\end{lem}
\begin{proof}
The key point is that \emph{torsion points of $A$ are algebraic over $K_0$}; indeed, for any $n\geqslant 2$, the multiplication by $n$ morphism $[n]:A \longrightarrow A$ is a $K_0$-isogeny, so the kernel is finite over $K_0$. Thus, the image of $P: \spec K_2 \longrightarrow A$ is a closed point $x$ with residue field $\kappa(x)$ a finite extension of $K_0$. We have an injection $\kappa(x) \lhook\joinrel\longrightarrow K_2$ given by $P$, and the hypotheses imply that $\kappa(x) = K_0$, so $P \in A(K_0) \subsetneq A(K_2)$, or more precisely, $P$ is the base change to $K_2$ of a point in $A(K_0)$. The desired conclusion then follows because $\textup{Gal}(K_2/K_1) \subset \aut_{K_0} K_2$.
\end{proof}
\begin{cor}\label{cor:contra rel}
With $K_0 \subset K_1 \subset K_2$ as above, let $X/K_0$ be a nice curve of positive genus, which has distinct $K_2$-points $P_1,\dotsc,P_n,Q_1,\dotsc,Q_m$, with $n \geqslant 3$. Suppose there exists a subgroup $H \leqslant \textup{Gal}(K_2/K_1)$ isomorphic to $S_n$, which acts by permutation on $\{P_1,\dotsc,P_n\}$ and fixes each $Q_i$. Let $D \in \Div(X)$ be a divisor of positive degree $k$, so that $\epsilon(P_i) \coloneqq [kP_i - D]$ and $\epsilon(Q_j) \coloneqq [kQ_j - D]$ are points in $J_X(K_2)$ for each $i=1,\dotsc,n$ and $j=1,\dotsc,m$ (upon regarding $D$ as a divisor on $X_{K_2}$). Then, any linear dependence relation
\begin{equation*}
\sum_{i=1}^n a_i\epsilon(P_i) + \sum_{j=1}^m b_i\epsilon(Q_i) = 0 \in J_X(K_2)
\end{equation*}
satisfies $a_1 = \dotsb = a_n$.
\end{cor}
\begin{proof}
Assume for a contradiction that we have a relation as above with $a_r \neq a_s$ for some $1 \leqslant r \neq s \leqslant n$. Let $\sigma \in H$ be a transposition that exchanges $P_r$ and $P_s$ and fixes all the other $P_i$'s and the $Q_i$'s. Applying $\sigma$ to the relation above and subtracting the result from the original, we get:
\begin{equation*}
(a_r - a_s) \cdot (\epsilon(P_r) - \epsilon(P_s)) =k(a_r - a_s)\cdot [P_r - P_s] = 0 \in J_X(K_2),
\end{equation*}
so $[P_r - P_s] \in J_X(K_2)$ is a \emph{torsion point}. Let $p \in \{1,\dotsc,n\} \setminus \{r,s\}$ (this is where we use that $n \geqslant 3$). By assumption, we can find an element $\sigma \in H\leqslant G$ that interchanges $P_r$ and $P_p$ and fixes $P_s$. Applying \Cref{lem:torsion fixed}, we have $\sigma([P_r - P_s]) = [P_r - P_s]$, and therefore
\begin{equation*}
[P_r - P_s] = \sigma ([P_r - P_s]) = [\sigma(P_r) - \sigma(P_s)] = [P_p - P_s] \in J_X(K_2).
\end{equation*}
But this implies that $[P_r - P_p] = 0 \in \pic^0(X_{K_2})$, which is absurd because $X$ is of positive genus.
\end{proof}
\begin{lem}\label{lem:algebraic point}
Let $M/L$ be an extension of fields, and let $X/L$ be a nice curve of positive genus having a point $D \in X(L)$. If $P \in X(M)$ is a point such that $[P - D] \in J_X(M)$ is torsion, then $P$ is algebraic over $L$.
\end{lem}
\begin{proof}
We can use $D$ as a base-point to embed $X$ in its Jacobian, i.e. we have a closed immersion $\mathfrak{r}:X\lhook\joinrel\longrightarrow J_X$, defined over $L$, such that for any field $F\supset L$ and point $Q \in X(F)$, we have $\mathfrak{r}(Q) = [Q-D] \in J_X(F)$. In particular, we have $\mathfrak{r}(P) = [P-D] \in J_X(M)$, and since this is a torsion point, the image of the composition
\begin{equation*}
\spec M \xlongrightarrow{P} X \xhookrightarrow{\;\, \mathfrak{r} \,\;} J_X
\end{equation*}
is a closed point $x \in J_X$. Since $\mathfrak{r}$ is a closed immersion, the pre-image $\mathfrak{r}^{-1}(x)$ (which is the image of $P$) is a closed point of $X$, so we are done.
\end{proof}
\section{Bounding the ranks of curves parametrized by $\mathcal{B}_{2d}$}\label{sec:ranks of curves}
Fix an integer $d \geqslant 2$. Recall the notation in \ref{emp:notation cals}, and in particular the various functions in $S$ (the coordinate ring of $\mathcal{B}_{2d}$) defined in \eqref{eq:functions}.
Recall from the setup in \ref{emp:sec 8 setup} that we have an equality of polynomials $m(x) = h(x)^2 - \ell(x) \in S[x]$, where $\deg \ell(x) = d-1$. Combining this with the identities in \eqref{eq:comp factorization}, we get the following identities in $S[x]$:
\begin{align}\label{eq:points identities}
\begin{split}
h(x)^2 - \ell(x) & = \prod_{i=1}^{2d} (x - u_i) ,\\
h(x^2)^2 - \ell(x^2) & = \prod_{i=1}^{2d} (x + t_i)(x- t_i),\\
h \big(\widehat{g}(x) \big)^2 - \ell \big(\widehat{g}(x) \big) & = \prod_{i=1}^{2d} (x-U_{i1})(x-U_{i2})(x-U_{i3}),\\
h \big(g(x)^2 \big)^2 - \ell \big(g(x)^2 \big) & = h \big(\widehat{g}(x^2) \big)^2 - \ell \big(\widehat{g}(x^2) \big)\\
& = \prod_{i=1}^{2d} (x + T_{i1})(x - T_{i1})(x + T_{i2})(x - T_{i2})(x + T_{i3})(x - T_{i3}).
\end{split}
\end{align}
\begin{emp}\textbf{The curves.} The table below contains a list of curves with explicit points; the first three curves (resp. last three curves) are defined over $K(\Bell)$ (resp. $K(b,\Bell)$), and the points are $K(\mbf{t})$-rational (resp. $\mathcal{K}$-rational). In the first column we denote the smooth, proper model of the corresponding affine curve $y^2 = f(x)$ in the second column (as in \ref{emp:hyperelliptic}). Note that each polynomial $f(x)$ that appears is separable; indeed, upon setting $(b,l_0,l_1,\dotsc,l_{d-2},l_{d-1}) = (0,1,0,\dotsc,0,1)$, each $f(x)$ specializes to a separable polynomial in $K[x]$. The reader can verify using \eqref{eq:functions} and \eqref{eq:points identities} that these curves come with the explicit $\mathcal{K}$-rational points listed in the table, where we recall from \ref{emp:function fields} that $\mathcal{K} = K(\mathcal{B}_{2d})$.\label{emp:curves in fact}
\end{emp}
\begin{table}[H]
\centering
\adjustbox{max width=\textwidth}{
\caption{Curves with $\mathcal{K}$-points}\label{tab:curves}
\begin{tabular}{|c|c|cc|}
\hline
Curve & Equation & $\mathcal{K}$-rational points & \\
\hline
$\Gamma_1/K(\Bell)$ & $y^2 = \ell(x)$ & $P_i \coloneqq (u_i,h(u_i))$ & $i=1,\dotsc,2d$ \\
\hline
$\Gamma_2/ K(\Bell)$ & $y^2 = x\ell(x)$ & $Q_i \coloneqq (u_i, t_i h(u_i))$ & $i=1,\dotsc,2d$ \\
\hline
$\wt{\Gamma}/K(\Bell)$ & $y^2 = \ell(x^2)$ & $\wt{P}_i^{\pm} \coloneqq (\pm t_i, h(u_i))$ & $i=1,\dotsc,2d$ \\
\hline
$\Theta_1 / K(b,\Bell)$ & $y^2 = \ell(\widehat{g}(x))$ & $P_{ij} \coloneqq (U_{ij},h(\widehat{g}(U_{ij})))$ & $i=1,\dotsc,2d, \, j=1,2,3$ \\
\hline
$\Theta_2 / K(b,\Bell)$ & $y^2 = x \ell(\widehat{g}(x))$ & $Q_{ij} \coloneqq (U_{ij},T_{ij} h(\widehat{g}(U_{ij}))$ & $i=1,\dotsc,2d, \, j=1,2,3$ \\
\hline
$\wt{\Theta}/K(b,\Bell)$ & $y^2 = \ell(\widehat{g}(x^2))$ & $\wt{P}_{ij}^{\pm} \coloneqq (\pm T_{ij}, h(\widehat{g}(U_{ij})))$ & $i=1,\dotsc,2d, \, j=1,2,3$ \\
\hline
\end{tabular}}
\end{table}
\begin{emp}\label{emp:notation for curves 6n}
By an abuse of notation, for each curve $X$ appearing in \Cref{tab:curves}, we let $D_{\infty}$ denote the degree $2$ divisor $\pi_X^*\infty$ (cf. \ref{emp:hyperelliptic}). Regarding $D_{\infty}$ as a divisor on $X_{\mathcal{K}}$, for a divisor $E \in \Div(X_{\mathcal{K}})$ of degree $k$, we let $$\epsilon(E)\coloneqq [2E - kD_{\infty}] \in J_X(\mathcal{K}).$$
If $X = \Gamma_2$ or $\Theta_2$, we let $D$ denote the closed point $(0,0)$, and for $P \in X(\mathcal{K})$, we let $$\mathfrak{r}(P) \coloneqq [P - D] \in J_X(\mathcal{K}).$$
Since $[2D - D_{\infty}] = 0 \in J_X(\mathcal{K})$, for any $P \in X(\mathcal{K})$, we have $\epsilon(P) = 2\mathfrak{r}(P)$.
\end{emp}
\begin{lem}\label{lem:relations between points}
Consider the curves $\Gamma_1,\Gamma_2,\Theta_1,$ and $\Theta_2$ from \Cref{tab:curves} above.
\begin{enumerate}[(i)]
\item If \label{part:relations i}
\begin{equation}
\sum_{i=1}^{2d}a_i\epsilon(P_i) = 0 \in J_{\Gamma_1}(K(\mbf{t}))
\end{equation}
is a linear dependence relation, then $a_1 = \dotsb = a_{2d}$. Moreover, the same is true if we replace each $\epsilon(P_i) \in J_{\Gamma_1}(K(\mbf{t}))$ with $\mathfrak{r}(Q_i) \in J_{\Gamma_2}(K(\mbf{t}))$.
\item If \label{part:relations ii}
\begin{equation}
\sum_{i=1}^{2d} \sum_{j=1}^3 a_{ij}\epsilon(P_{ij}) = 0 \in J_{\Theta_1}(\mathcal{K})
\end{equation}
is a linear dependence relation, then for each $i=1,\dotsc,2d$, we have $a_{i1} = a_{i2} = a_{i3}$. Moreover, the same is true if we replace each $\epsilon(P_{ij}) \in J_{\Theta_1}(\mathcal{K})$ with $\mathfrak{r}(Q_{ij}) \in J_{\Theta_2}(\mathcal{K})$.
\end{enumerate}
\end{lem}
\begin{proof}
The curves in question are defined over $K(b,\Bell)$, which is algebraically closed in $\mathcal{K}$ (cf. \Cref{prop:alg closed}); \Cref{prop: cals galois group} tells us that $\textup{Gal}(\mathcal{K}/K(b,\mbf{m}))$ has plenty of subgroups isomorphic to symmetric groups. So, we are well-poised to apply \Cref{cor:contra rel}.
Recall from \Cref{prop: cals galois group} \ref{part:cals galois iii} that there is a subgroup $S_n \leqslant \textup{Gal}(\mathcal{K}/K(b,\mbf{m}))$ that acts by permutation on the elements $u_1,\dotsc,u_{2d}$ (resp. $t_1,\dotsc,t_{2d}$) of $\mathcal{K}$. Visibly, this induces a permutation action of $\gamma_1(S_n)$ on $P_1,\dotsc,P_{2d} \in \Gamma_1(K(b,\mbf{t}))$ (resp. $Q_1,\dotsc,Q_{2d} \in \Gamma_2(K(b,\mbf{t}))$). Part \ref{part:relations i} now follows from \Cref{cor:contra rel}.
For each $i=1,\dotsc,2d$, let $G_i \leqslant (S_3)^{2d}$ denote the subgroup consisting of tuples with every coordinate being $1$ except for the $i$-th coordinate; note that $G_i \cong S_3$. Then, $S_3 \cong \alpha_1(G_k) \leqslant \textup{Gal}(\mathcal{K}/K(b,\mbf{m}))$ is a subgroup that acts by permutation on the elements $T_{k1},T_{k2},T_{k3}$ (resp. $U_{k1},U_{k2},U_{k3}$) and fixes all the other $T_{ij}$'s (resp. $U_{ij}$'s) (cf. \Cref{prop: cals galois group} \ref{part:cals galois i}); visibly, this subgroup acts by permutation on $P_{k1},P_{k2},P_{k3} \in \Theta_1(\mathcal{K})$ (resp. $Q_{k1},Q_{k2},Q_{k3} \in \Theta_2(\mathcal{K})$) and fixes all the other $P_{ij}$'s (resp. $Q_{ij}$'s). Part \ref{part:relations ii} now follows from \Cref{cor:contra rel}.
\end{proof}
Thus far, for $X = \Gamma_1,\Gamma_2,\Theta_1,$ or $\Theta_2$, we have produced many points in $J_X(\mathcal{K})$ and obtained restrictions on the possible dependence relations between them. In order to determine the ranks of the subgroups they generate, we need the fact that these curves fit together into a diagram:
\begin{equation}\label{diag:curves and maps}
\begin{tikzcd}
& \widetilde{\Theta} \arrow[dl, "\psi_1", swap] \arrow[dr, "\psi_2"] \arrow[d, "\widetilde{\psi}"] & \\
\Theta_1 \arrow[d, "\vartheta", swap] & \widetilde{\Gamma} \arrow[dl, "\varphi_1"] \arrow[dr, "\varphi_2"] & \Theta_2 \\
\Gamma_1 & & \Gamma_2
\end{tikzcd}
\end{equation}
in which the $K(b,\Bell)$-morphisms are defined by
\begin{equation}\label{eq: maps of curves}
\begin{alignedat}{2}
\varphi_1 : (x,y) & \longmapsto (x^2,y) \quad \quad \varphi_2: (x,y) && \longmapsto (x^2,xy)\\
\psi_1 :(x,y) & \longmapsto (x^2,y) \quad \quad \psi_2 : (x,y) && \longmapsto (x^2,xy)\\
\vartheta : (x,y) & \longmapsto (\widehat{g}(x),y) \quad \;\; \wt{\psi} : (x,y) && \longmapsto (g(x),y).
\end{alignedat}
\end{equation}
Above, we have abused notation and written $\Gamma_1$ to denote the base-change $(\Gamma_1)_{K(b,\Bell)}$, and similarly for $\Gamma_2$ and $\wt{\Gamma}$. Below, we show the effect of these maps on the explicit $\mathcal{K}$-points:
\begin{equation*}
\begin{tikzcd}
& (T_{ij},h(\widehat{g}(U_{ij}))) \arrow[dl, mapsto, "\psi_1", swap] \arrow[dr, mapsto, "\psi_2"] \arrow[d, mapsto, "\wt{\psi}"] & \\
(U_{ij},h(\widehat{g}(U_{ij}))) \arrow[d, mapsto, "\vartheta", swap] & (t_i,h(u_i)) \arrow[dl, mapsto, "\varphi_1"] \arrow[dr, mapsto, "\varphi_2"] & (U_{ij}, T_{ij}h(\widehat{g}(U_{ij}))) \\
(u_i,h(u_i)) & & (u_i, t_ih(u_i)).
\end{tikzcd}
\end{equation*}
To bound the ranks of $\wt{\Gamma}$ and $\wt{\Theta}$, we use the following (echoing Shioda's strategy in~\cite{shiodasymmetry}).
\begin{prop}[{see~\cite{shiodasymmetry}*{Proposition~3}}]\label{prop:isogenous}
Let $K$ be a field of characteristic $0$, and let $f(x) \in K[x]$ be a polynomial of degree $\geqslant 2$ such that $f(x), xf(x)$ and $f(x^2)$ are all separable. Let $X_1/K, X_2/K$ and $\wt{X}/K$ be smooth, proper models of the affine curves $y^2 = f(x), y^2 = xf(x)$, and $y^2 = f(x^2)$ respectively. Define
\begin{equation*}
\begin{alignedat}{3}
\phi_1 : \wt{X} & \longrightarrow X_1 \quad && \quad \phi_2 : \wt{X} && \longrightarrow X_2\\
(x,y) & \longmapsto (x^2,y) \quad && \quad (x,y) && \longmapsto (x^2,xy).
\end{alignedat}
\end{equation*}
Then, the induced morphism $((\phi_1)_*,(\phi_2)_*) : J_{\wt{X}} \longrightarrow J_{X_1} \times J_{X_2}$ is a $K$-isogeny. In particular, for any field $L\supset K$ for which $J_{\wt{X}}(L)$ is finitely generated, we have $\rk {\wt{X}}/L = \rk {X_1}/L + \rk X_2/L$.
\end{prop}
With this, we are finally ready to bound the ranks of the curves in \Cref{tab:curves}, beginning with the trio of curves $\Gamma_1,\Gamma_2,$ and $\wt{\Gamma}$. The reader can check that each morphism appearing in \eqref{diag:curves and maps} satisfies the condition of \Cref{lem:D infty}; below, we will use this lemma without mention.
\begin{thm}\label{thm:rank of gamma 1}
Recall from \Cref{tab:curves} the nice curve $\Gamma_1/K(\Bell)$.
\begin{enumerate}[(i)]
\item \label{part: gamma1 i} The genus of $\Gamma_1$ is $0$ if and only if $d=2$ or $3$.
\item If $d \geqslant 4$ and $d$ is odd (resp. even), then $\#\Gamma_1(\mbf{t}) \geqslant 4d$ (resp. $4d+1$). \label{part: gamma1 ii}
\item \label{part: gamma1 iii} If $d \geqslant 4$, then the points $\epsilon(P_1),\dotsc,\epsilon(P_{2d})$ satisfy the single relation $\epsilon(P_1) + \dotsb + \epsilon(P_{2d}) = 0$ and generate a subgroup of $J_{\Gamma_1}(K(\mbf{t}))$ of rank $2d - 1$.
\end{enumerate}
\end{thm}
\begin{proof}
For part \ref{part: gamma1 i}, note that if $d=2$ or $3$, then $\deg \ell(x) =d-1 = 1$ or $2$, so these are exactly the cases for which $\Gamma_1$ is of genus $0$.
Assume now that $d\geqslant 4$. If $\deg \ell(x) = d-1$ is even, then $\Gamma_1$ has the $4d$ $K(\mbf{t})$-points $P_i,\iota(P_i)$, and if $d-1$ is odd, then $\Gamma_1$ has in addition to these a $K(\mbf{t})$-point at infinity, giving a total of $4d+1$ $K(\mbf{t})$-points. This establishes \ref{part: gamma1 ii}. Continuing with the assumption $d \geqslant 4$, for \ref{part: gamma1 iii}, we note that the identity $\prod_i (x-u_i) = h(x)^2 - \ell(x)$ (cf. \eqref{eq:points identities}) implies that $\textup{div} (y-h(x)) = \sum_i P_i - dD_{\infty}$, so $$\sum_{i=1}^{2d} \epsilon(P_i) = \sum_{i=1}^{2d} [2P_i - D_\infty] = 2[ \textup{div}(y-h(x))] = 0 \in J_{\Gamma_1}(K(\mbf{t})).$$
The desired conclusion then follows from \Cref{lem:relations between points} \ref{part:relations i}.
\end{proof}
\begin{thm}\label{thm: rank of gamma 2}
Recall from \Cref{tab:curves} the nice curve $\Gamma_2/K(\Bell)$.
\begin{enumerate}[(i)]
\item \label{part: gamma2 i} The genus of $\Gamma_2$ is $0$ if and only if $d=2$.
\item If $d\geqslant 3$ and $d$ is even (resp. odd) then $\#\Gamma_2(K(\mbf{t})) \geqslant 4d+1$ (resp. $4d+2$). \label{part: gamma2 ii}
\item \label{part: gamma2 iii} If $d \geqslant 3$, then the points $\mathfrak{r}(Q_1),\dotsc, \mathfrak{r}(Q_{2d})$ are linearly independent in $J_{\Gamma_2}(K(\mbf{t}))$, so $\rk J_{\Gamma_2}(K(\mbf{t}) \geqslant 2d$.
\end{enumerate}
\end{thm}
\begin{proof}
Since $\deg x\ell(x) = d$, the case $d=2$ is the only case in which $\Gamma_2$ is of genus $0$. Assume $d\geqslant 3$. For part \ref{part: gamma2 ii}, note that if $\deg \ell(x) = d$ is even, then $\Gamma_2$ has the $4d$ $K(\mbf{t})$-points $Q_i, \iota(Q_i)$, and also the point $D = (0,0)$, for a total of $4d+1$ $K(\mbf{t})$-points. If $d$ is odd, then it has in addition to these a $K(\mbf{t})$-rational point at infinity, giving a total of $4d+2$ $K(\mbf{t})$-points.
For \ref{part: gamma2 iii}, assume $d\geqslant 3$. By \Cref{lem:relations between points}, any relation among the $\mathfrak{r}(Q_i)$'s is of the form
\begin{equation}\label{rel:gamma 2 rel}
a \cdot \sum_{i=1}^{2d} \mathfrak{r}(Q_i) = 0 \in J_{\Gamma_2}(K(\mbf{t})).
\end{equation}
Let $\overline{e} \in (S_2)^{2d} \cong \textup{Gal} (K(b,\mbf{t}) / K(b,\mbf{u}))$ (cf. \Cref{prop: cals galois group} \ref{part:cals galois i}) be an involution that sends $t_1$ to $-t_1$, and fixes all the other $t_i$'s. Then, $\overline{e}$ sends $\mathfrak{r}(Q_1)$ to $\mathfrak{r}(\iota(Q_1)) = -\mathfrak{r}(Q_1)$ and fixes all the other $\mathfrak{r}(Q_i)$'s. Applying $\overline{e}$ to the relation \eqref{rel:gamma 2 rel}, we get
\begin{equation*}
-a\mathfrak{r}(Q_1) + a\cdot \sum_{j=2}^{2d} \mathfrak{r}(Q_i) = 0 \in J_{\Gamma_2}(K(\mbf{t})).
\end{equation*}
\Cref{lem:relations between points} then forces $a=-a$, so we conclude that the points $\mathfrak{r}(Q_1),\dotsc,\mathfrak{r}(Q_{2d})$ are linearly independent in $J_{\Gamma_2}(K(\mbf{t}))$.
\end{proof}
\begin{emp} \label{emp:relations}\textbf{Relations.}
Suppose $H$ is an abelian group generated by elements $R_1,\dotsc,R_n$, and consider the homomorphism $f: \mathbb{Z}^n \longrightarrow H$ which sends each ``coordinate vector'' $e_i$ to the element $R_i$, so that $\rk_\mathbb{Z} H = n - \rk_\mathbb{Z} \ker f.$ An element $\sum_i a_i e_i$ of the kernel corresponds to an identity $\sum_i a_i R_i = 0$ in $H$; such an identity will be called a \defi{relation}. Two relations $\sum_i a_iR_i=0$ and $\sum_i b_iR_i = 0$ will be called \defi{linearly independent} if the associated elements $\sum_i a_ie_i, \sum_i b_ie_i \in \ker f$ are $\mathbb{Z}$-linearly independent. We will use without mention the following simple observation: for any disjoint subsets $S_1$ and $S_2$ of $\{1,\dotsc,n\}$, the relations $\sum_{i \in S_1} R_i = 0$ and $\sum_{j \in S_2} R_j = 0$ are linearly independent.
\end{emp}
\begin{thm}\label{cor: rank of GGamma}
Recall from \Cref{tab:curves} the nice curve $\wt{\Gamma}/K(\Bell)$. For $i=1,\dotsc,2d$, let $\mathfrak{p}_i \coloneqq \epsilon (\wt{P}_i^+) + \epsilon (\wt{P}_i^-)\in J_{\wt{\Gamma}}(K(\mbf{t}))$.
\begin{enumerate}[(i)]
\item The genus of $\wt{\Gamma}$ is $0$ if and only if $d=2$.\label{part:wtgamma i}
\item If $d\geqslant 3$, then $\#\wt{\Gamma}(K(\mbf{t})) \geqslant 8d$.\label{part:wtgamma ii}
\item If $d = 3$, then $\wt{\Gamma}$ is of genus $1$, and the points $\epsilon (\wt{P}_1^{\pm}), \dotsc, \epsilon (\wt{P}_6^{\pm})$ satisfy the linearly independent relations $\mathfrak{p}_1 = \dotsb = \mathfrak{p}_6 =0 $ and generate a subgroup of $J_{\wt{\Gamma}}(K(\mbf{t}))$ of rank $6$. \label{part:wtgamma iii}
\item If $d \geqslant 4$, then the points $\epsilon (\wt{P}_1^{\pm}), \dotsc, \epsilon (\wt{P}_{2d}^{\pm})$ satisfy the single relation $\mathfrak{p}_1 + \dotsb+ \mathfrak{p}_{2d} = 0$
and generate a subgroup of $J_{\wt{\Gamma}}(K(\mbf{t}))$ of rank $4d - 1$.\label{part:wtgamma iv}
\end{enumerate}
\end{thm}
\begin{proof}
For \ref{part:wtgamma i}, we have $\deg \ell(x^2) = 2(d-1) = 2$ if $d=2$, and clearly this is the only case when the genus of $\wt{\Gamma}$ is $0$. For \ref{part:wtgamma ii}, we note that $\wt{\Gamma}$ has the $8d$ $K(\mbf{t})$-points $\wt{P}_{i}^+,\wt{P}_i^-,\iota(\wt{P}_i^+),\iota(\wt{P}_i^-)$, for $i=1,\dotsc,2d$.
For parts \ref{part:wtgamma iii} and \ref{part:wtgamma iv}, we first let $\mathfrak{q}_i \coloneqq \epsilon (\wt{P}_i^+) - \epsilon (\wt{P}_i^-) \in J_{\wt{\Gamma}}(K(\mbf{t}))$, we let $H'$ (resp. $H$) denote the subgroup of $J_{\wt{\Gamma}}(K(\mbf{t}))$ generated by $\mathfrak{p}_i,\mathfrak{q}_i$ (resp. $\wt{P}^+_i,\wt{P}^-_i$) for $i=1,\dotsc,2d$. Clearly, $H'\leqslant H$; on the other hand, we have $2\epsilon(\wt{P}_i^+) = \mathfrak{p}_i + \mathfrak{q}_i$, and $2\epsilon(\wt{P}_i^-) = \mathfrak{q}_i - \mathfrak{p}_i$, so $H \leqslant 2\cdot H'$, which implies that $\rk_\mathbb{Z} H = \rk_\mathbb{Z} H'$.
Note that $\mathfrak{p}_i = \varphi_1^*(\epsilon(P_i))$ and $\mathfrak{q}_i = \varphi_2^*(\epsilon(Q_i))$, so under the isogeny $(\varphi_1^* + \varphi_2^*) : J_{\Gamma_1}\times J_{\Gamma_2} \longrightarrow J_{\wt{\Gamma}}$ (that is the dual isogeny of the one provided by \Cref{prop:isogenous}), the subgroup $H''$ of $J_{\Gamma_1}(K(\mbf{t}))\times J_{\Gamma_2}(K(\mbf{t}))$ generated by the $\epsilon(P_i)$'s and $\epsilon(Q_i)$'s is mapped surjectively onto $H'$, giving $\rk_{\mathbb{Z}}H = \rk_{\mathbb{Z}} H' = \rk_{\mathbb{Z}}{H''}$.
Now, if $d\geqslant 3$, then the points $\epsilon(Q_i) = 2\mathfrak{r}(Q_i)$ (cf. \ref{emp:notation for curves 6n}) are linearly independent in $J_{\Gamma_2}(K(\mbf{t}))$ (cf. \Cref{thm: rank of gamma 2} \ref{part: gamma2 iii}), so the points $\mathfrak{q}_1, \dotsc, \mathfrak{q}_{2d}$ are linearly independent in $J_{\wt{\Gamma}}(K(\mbf{t}))$.
If $d=3$, then $\Gamma_1$ is of genus $0$ (cf. \Cref{thm:rank of gamma 1} \ref{part: gamma1 i}), so we have the six linearly independent relations $\mathfrak{p}_1 = \dotsb = \mathfrak{p}_6 = 0 \in J_{\wt{\Gamma}}(K(\mbf{t}))$ (cf. \ref{emp:relations}), so we get $\rk_\mathbb{Z} H = \rk_\mathbb{Z} H'' = 6$, which establishes \ref{part:wtgamma iii}. If $d\geqslant 4$, then the $\epsilon(P_i)$'s satisfy only the single relation $\epsilon(P_i) + \dotsb + \epsilon(P_{2d})=0 \in J_{\Gamma_1}(K(\mbf{t}))$ (cf. \Cref{thm:rank of gamma 1} \ref{part: gamma1 iii}), so the $\mathfrak{p}_i$'s satisfy only the single relation $\mathfrak{p}_1 = \dotsb = \mathfrak{p}_{2d} = 0$, and we have
$\rk_\mathbb{Z} H = \rk_\mathbb{Z} H'' = (2d - 1) + 2d = 4d-1$; this concludes the proof of \ref{part:wtgamma iv}.
\end{proof}
\begin{rem}\label{rem:work of shioda}
The curves $\Gamma_1,\Gamma_2,$ and $\wt{\Gamma}$ were considered (for $d\geqslant 4$) by Shioda in \cite{shiodasymmetry} and he obtained the same bounds as we do. Our proofs of the rank bounds in Theorems \ref{thm:rank of gamma 1}, \ref{thm: rank of gamma 2}, and \ref{cor: rank of GGamma} are more elementary compared to Shioda's proofs in that we do not use the theory of the canonical height. Also, we have included some extra details in \Cref{cor: rank of GGamma} (not found in \cite{shiodasymmetry}) that will be used in the proof of \Cref{thm:rank of lambda 1}.
We also note that the present record $R(\mathbb{Q},g;\mathbb{Z}/2\mathbb{Z}) \geqslant 4g+2$ (cf. \eqref{eq:rkgt}) is obtained by taking specializations of the curve $\Gamma_2$ in the cases corresponding to the last row of \Cref{tab: genus and ranks gammas}.
\end{rem}
\begin{emp}\textbf{Summary.} \label{emp: gamma summary}
In the table below, we summarize the results in Theorems \ref{thm:rank of gamma 1}, \ref{thm: rank of gamma 2}, and \ref{cor: rank of GGamma}. For $X = \Gamma_1,\Gamma_2$, or $\wt{\Gamma}$, we denote the genus, number of explicit $K(\mbf{t})$-points, and lower bound on $\rk X / K(\mbf{t})$ by $g(X), N(X)$, and $R(X)$ respectively. We express the genera $g(\Gamma_2)$ and $g(\wt{\Gamma})$ in terms of the genus $g_0$ of $\Gamma_1$.
\end{emp}
\begin{table}[H]
\centering
\setlength{\tabcolsep}{5pt}
\adjustbox{max width=\textwidth}{
\caption{Lower bound on rank and rational points for $\Gamma_1,\Gamma_2,\wt{\Gamma}$}\label{tab: genus and ranks gammas}
\begin{tabular}{|c||c|c|c||c|c|c||c|c|c|}
\hline
& \multicolumn{3}{c||}{$\Gamma_1/K(\Bell): y^2 = \ell(x)$} & \multicolumn{3}{c||}{$\Gamma_2/K(\Bell): y^2 = x\ell(x)$} & \multicolumn{3}{c|}{$\wt{\Gamma}/K(\Bell): y^2 = \ell(x^2)$}\\
\hline
& \multicolumn{3}{c||}{$\deg \ell(x) = d-1$} & \multicolumn{3}{c||}{$\deg x\ell(x) = d$} & \multicolumn{3}{c|}{$\deg \ell(x^2) = 2(d-1)$}\\
\hline
$d$ & $g(\Gamma_1)$ & $N(\Gamma_1)$ & $R(\Gamma_1)$ & $g(\Gamma_2)$ & $N(\Gamma_2)$ & $R(\Gamma_2)$ & $g(\wt{\Gamma})$ & $N(\wt{\Gamma})$ & $R(\wt{\Gamma})$ \\
\hline
$2$ & $0$ & & & $0$ & & & $0$ & &\\
\hline
$3$ & $0$ & & & $1$ & $14$ & $6$ & $1$ & $24$ & $6$\\
\hline
$2g_0+2 \;\;(g_0\geqslant 1)$ & $g_0$ & $8g_0+9$ & $4g_0+3$ & $g_0$ & $8g(\Gamma_2)+9$ & $4g(\Gamma_2) + 4$ & $2g_0$ & $8g(\wt{\Gamma})+16$ & $4g(\wt{\Gamma}) + 7$\\
\hline
$2g_0+3 \;\;(g_0\geqslant 1)$ & $g_0$ & $8g_0 + 12$ & $4g_0 + 5$ & $g_0+1$ & $8g(\Gamma_2)+6$ & $4g(\Gamma_2) + 2$ & $2g_0+1$ & $8g(\wt{\Gamma})+16$ & $4g(\wt{\Gamma}) + 7$\\
\hline
\end{tabular}}
\end{table}
Next, we establish lower bounds on the ranks of $\Theta_1,\Theta_2,$ and $\wt{\Theta}$.
\begin{thm}\label{thm: rank of Theta 1}
Recall from \Cref{tab:curves} the nice curve $\Theta_1/K(b,\Bell)$.
\begin{enumerate}[(i)]
\item \label{part: theta1 i} If $d$ is odd (resp. even), then $\#\Theta_1(\mathcal{K}) \geqslant 12d$ (resp. $12d+1$).
\item \label{part: theta1 ii} If $d = 2,3$, then the points $\epsilon (P_{ij}), \, i=1,\dotsc,2d,\, j=1,2,3,$ satisfy the linearly independent relations $\epsilon(P_{i1}) + \epsilon(P_{i2}) + \epsilon(P_{i3}) = 0$, for $i=1,\dotsc,2d$, and generate a subgroup of $J_{\Theta_1}(\mathcal{K})$ of rank $4d$.
\item \label{part: theta1 iii} If $d \geqslant 4$, then the points $\epsilon (P_{ij}), \, i=1,\dotsc,2d,\, j=1,2,3,$ satisfy the single relation $\sum_{i,j} \epsilon (P_{ij}) = 0$ and generate a subgroup of $J_{\Theta_1}(\mathcal{K})$ of rank $6d-1$.
\end{enumerate}
\end{thm}
\begin{proof}
For \ref{part: theta1 i}, note that $\Theta_1$ has the $12d$ $\mathcal{K}$-points $P_{ij},\iota(P_{ij})$. If $d$ is odd then $\deg \ell(\widehat{g}(x)) = 3(d-1)$ is even. If $d$ is even, then $\Theta_1$ has an $\mathcal{K}$-rational point at infinity, giving a total of $12d+1$ $\mathcal{K}$-points.
For parts \ref{part: theta1 ii} and \ref{part: theta1 iii}, we first note that $\vartheta : \Theta_1 \longrightarrow \Gamma_1$ satisfies the conditions of \Cref{lem:D infty}; here, we abused notation to write $\Gamma_1$ instead of $(\Gamma_1)_{K(b,\Bell)}$. Since $\vartheta^*P_i = P_{i1} + P_{i2} + P_{i3} \in \Div(\Theta_1)$, we have $\vartheta^*(\epsilon(P_i)) = \epsilon(P_{i1}) + \epsilon(P_{i2}) + \epsilon(P_{i3})$, so we can rephrase \Cref{lem:relations between points} \ref{part:relations ii} as saying that \emph{any relation among the $\epsilon(P_{ij})$'s is of the form $\sum_i a_i \vartheta^*(\epsilon(P_i)) = 0 \in J_{\Theta_1}(\mathcal{K})$.}
If $d=2$ or $3$, then $\Gamma_1$ is of genus $0$, which implies that $\vartheta^*(\epsilon(P_i)) = 0 \in J_{\Theta_1}(\mathcal{K})$ for each $i=1,\dotsc,2d$; these relations are visibly linearly independent (cf. \ref{emp:relations}), so \ref{part: theta1 ii} follows.
If $d \geqslant 4$, then the relation $\sum_i \epsilon(P_i) = 0 \in J_{\Gamma_1}(\mathcal{K})$ pulls back to the relation $\sum_i \vartheta^*(\epsilon(P_i)) = \sum_{i,j} \epsilon(P_{ij}) = 0 \in J_{\Theta_1}(\mathcal{K})$. On the other hand, any relation $\sum_i a_i \vartheta^*(\epsilon(P_i))$ can be pushed forward to give a relation
\begin{equation*}
\sum_{i=1}^{2d} a_i\cdot (\vartheta_{*}\vartheta^*(\epsilon(P_i))) = \sum_{i=1}^{2d} 3a_i\cdot \epsilon(P_i) = 0 \in J_{\Gamma_1}(\mathcal{K}).
\end{equation*}
\Cref{thm:rank of gamma 1} \ref{part: gamma1 iii} then implies $a_1 = \dotsb = a_{2d}$, so \ref{part: theta1 iii} follows.
\end{proof}
\begin{thm}\label{thm:rank of Theta 2}
Recall from \Cref{tab:curves} the nice curve $\Theta_2/K(b,\Bell)$.
\begin{enumerate}[(i)]
\item \label{part: theta2 i} If $d$ is even (resp. odd), then $\# \Theta_2(\mathcal{K}) \geqslant 12d+1$ (resp. $12d+2$).
\item \label{part: theta2 ii} If $d = 2$, then the points $\mathfrak{r}(Q_{ij}), \, i=1,\dotsc,4,\, j=1,2,3,$ satisfy the linearly independent relations $2\big(\mathfrak{r} (Q_{i1}) + \mathfrak{r} (Q_{i2}) + \mathfrak{r} (Q_{i3})\big)= 0$, for $i=1,\dotsc,4$, and generate a subgroup of $J_{\Theta_2}(\mathcal{K})$ of rank $4d = 8$.
\item \label{part: theta2 iii} If $d \geqslant 3$, then the points $\mathfrak{r} (Q_{ij}), \, i=1,\dotsc,2d,\, j=1,2,3,$ generate a subgroup of $J_{\Theta_2}(\mathcal{K})$ of rank $6d$.
\end{enumerate}
\end{thm}
\begin{proof}
For part \ref{part: theta2 i}, we first note that $\Theta_2$ has the $12d$ $\mathcal{K}$-points $Q_{ij},\iota (Q_{ij})$, and also the point $R = (0,0)$. If $d$ is odd, then $\Theta_2$ has an additional $\mathcal{K}$-point at infinity, so we get $\# \Theta_2(\mathcal{K}) \geqslant 12d+2$.
Now, \Cref{lem:relations between points} says that any relation among the $\mathfrak{r}(Q_{ij})$'s is of the form $\sum_i a_i(\mathfrak{r}(Q_{i1}) + \mathfrak{r}(Q_{i2}) + \mathfrak{r}(Q_{i3})) = 0 \in J_{\Theta_2}(\mathcal{K})$. Parts \ref{part: theta1 ii} and \ref{part: theta2 iii} rely on the observation that for each $i=1,\dotsc,2d$, the point $\mathfrak{r}(Q_{i1}) + \mathfrak{r}(Q_{i2}) + \mathfrak{r}(Q_{i3}) \in J_{\Theta_2}(\mathcal{K})$ is ``linked'' to the point $\mathfrak{r}(Q_i) \in J_{\Gamma_2}(\mathcal{K})$ in the following way. From the diagram \eqref{diag:curves and maps}, we get group homomorphisms
\begin{align*}
(\varphi_2)_{*} \circ \wt{\psi}_{*} \circ \psi_2^{*} : \pic^0 (\Theta_2)_{\mathcal{K}} & \longrightarrow \pic^0 (\Gamma_2)_{\mathcal{K}}, \\
(\psi_2)_{*} \circ \wt{\psi}^{*} \circ (\varphi_2)^{*} : \pic^0 (\Gamma_2)_{\mathcal{K}} & \longrightarrow \pic^0 (\Theta_2)_{\mathcal{K}}.
\end{align*}
We compute
\begin{align}\label{eq:theta to gamma}
\begin{split}
\big(\mathfrak{r}(Q_{i1}) + \mathfrak{r}(Q_{i2}) + \mathfrak{r}(Q_{i3})\big)
\xmapsto{\psi_2^{*}} &\;\Big[ \sum_{j=1}^3 \big(\wt{P}_{ij}^+ + \iota(\wt{P}_{ij}^-) - 3D_\infty \big) \Big]\\
= &\;\Big[ \sum_{j=1}^3 \wt{P}_{ij}^+ + \sum_{j=1}^3 \iota\big(\wt{P}_{ij}^-\big) - 3D_\infty \Big]\\
\xmapsto{\wt{\psi}_{*}} &\;\big[ 3\wt{P}_i^+ + 3\cdot \iota \big(\wt{P}_i^-\big) - 3D_\infty \big] \\
\xmapsto{(\varphi_2)_{*}} &\;[6Q_i - 3D_\infty]\\
= &\;6\cdot \mathfrak{r}(Q_i),
\end{split}
\end{align}
and similarly,
\begin{align}\label{eq:gamma to theta}
\begin{split}
\mathfrak{r}(Q_i) \xmapsto{(\varphi_2)^*} &\; \big( \,\big[ \wt{P}_i^+ + \iota\big(\wt{P}_i^-\big) - D_\infty \big]\, \big)\\
\xmapsto{\wt{\varphi}^{*}} &\; \Big[ \sum_{j=1}^3 \wt{P}_{ij}^+ + \sum_{j=1}^3 \iota\big(\wt{P}_{ij}^-\big) - 3D_\infty \Big]\\
= &\; \Big[ \sum_{j=1}^3 \big(\wt{P}_{ij}^+ + \iota(\wt{P}_{ij}^-) - D_\infty \big) \Big]\\
\xmapsto{(\psi_2)_{*}} &\; \big[ 2Q_{i1} + 2Q_{i2} + 2Q_{i3} - 3D_\infty \big]\\
= &\; 2\big(\mathfrak{r}(Q_{i1}) + \mathfrak{r} (Q_{i2}) + \mathfrak{r} (Q_{i3})\big)
\end{split}
\end{align}
If $d=2$, then $\Gamma_2$ is of genus $0$ (cf. \Cref{thm: rank of gamma 2} \ref{part: gamma2 i}), so each $\mathfrak{r}(Q_i) = 0 \in J_{\Gamma_1}(\mathcal{K})$, and \eqref{eq:gamma to theta} implies $2(\mathfrak{r}(Q_{i1}) + \mathfrak{r} (Q_{i2}) + \mathfrak{r} (Q_{i3}))=0$. This establishes that the relations in the statement of part \ref{part: theta2 ii} are satisfied; moreover, these relations are visibly linearly independent (cf. \ref{emp:relations}), so we conclude that the $\mathfrak{r}(Q_{ij})$'s generate a subgroup of $J_{\Theta_2}(\mathcal{K})$ of rank $6d - 2d = 4d$.
For part \ref{part: theta2 iii}, if we have a relation $\sum_i a_i( \mathfrak{r}(Q_{i1}) + \mathfrak{r}(Q_{i2}) + \mathfrak{r}(Q_{i3}))$, then \eqref{eq:theta to gamma} implies that $\sum_i 6a_i \cdot \mathfrak{r}(Q_i) = 0$. Part \ref{part: gamma2 ii} of \Cref{thm: rank of gamma 2} then forces $a_i = 0$ for all $i$. We conclude that if $d \geqslant 3$, then the points $\mathfrak{r}(Q_{ij})$ are linearly independent in $J_{\Theta_2}(\mathcal{K})$, so they generate a subgroup of rank $6d$.
\end{proof}
\begin{cor}\label{cor:rank of TTheta}
Recall from \Cref{tab:curves} the nice curve $\wt{\Theta}/K(b,\Bell)$.
\begin{enumerate}[(i)]
\item We have $\# \wt{\Theta}(\mathcal{K}) \geqslant 24d$.\label{part:wttheta i}
\item If $d=2$, then $\rk \wt{\Theta}/\mathcal{K} \geqslant 8d = 16$.
\item If $d=3$, then $\rk \wt{\Theta}/\mathcal{K} \geqslant 10d = 30$.
\item If $d \geqslant 4$, then $\rk \wt{\Theta}/\mathcal{K} \geqslant 12d -1$.
\end{enumerate}
\end{cor}
\begin{proof}
Part \ref{part:wttheta i} follows from the observation that $\wt{\Theta}$ has the $\mathcal{K}$-points $\wt{P}_{ij}^+ , \wt{P}_{ij}^-, \iota(\wt{P}_{ij}^+), \iota(\wt{P}_{ij}^-)$ for $i=1,\dotsc,2d$ and $j=1,2,3$. The remaining parts follow immediately by combining \Cref{thm: rank of Theta 1}, \Cref{thm:rank of Theta 2}, and \Cref{prop:isogenous}.
\end{proof}
\begin{emp}\textbf{Summary.} \label{emp: theta summary}
In the table below, we summarize the results in Theorems \ref{thm: rank of Theta 1}, \ref{thm:rank of Theta 2}, and \Cref{cor:rank of TTheta}. For $X = \Theta_1,\Theta_2$, and $\wt{\Theta}$, we denote the genus by $g_1,g_2,$ and $\wt{g}$ respectively, the number of explicit $\mathcal{K}$-points by $N_1,N_2$, and $\wt{N}$ respectively, and the lower bound on $\rk X / \mathcal{K}$ by $R_1,R_2$, and $\wt{R}$ respectively. We express $g_1,g_2,$ and $\wt{g}$ in terms of the genus $g_0$ of $\Gamma_1$ (cf. \Cref{tab: genus and ranks gammas}).
\end{emp}
\begin{table}[H]
\centering
\setlength{\tabcolsep}{5pt}
\adjustbox{max width=\textwidth}{
\caption{Lower bound on rank and rational points for $\Theta_1,\Theta_2,\wt{\Theta}$}\label{tab: genus and ranks thetas}
\begin{tabular}{|c||c|c|c||c|c|c||c|c|c|}
\hline
& \multicolumn{3}{c||}{$\Theta_1/K(b,\Bell): y^2 = \ell(\widehat{g}(x))$} & \multicolumn{3}{c||}{$\Theta_2/K(b,\Bell): y^2 = x\ell(\widehat{g}(x))$} & \multicolumn{3}{c|}{$\wt{\Theta}/K(b,\Bell): y^2 = \ell(\widehat{g}(x^2))$}\\
\hline
& \multicolumn{3}{c||}{$\deg \ell(\widehat{g}(x)) = 3(d-1)$} & \multicolumn{3}{c||}{$\deg x\ell(\widehat{g}(x)) = 3d-2$} & \multicolumn{3}{c|}{$\deg \ell(\widehat{g}(x^2)) = 6(d-1)$}\\
\hline
$d$ & $g_1$ & $N_1$ & $R_1$ & $g_2$ & $N_2$ & $R_2$ & $\wt{g}$ & $\wt{N}$ & $\wt{R}$ \\
\hline
$2$ & $1$ & $25$ & $8$ & $1$ & $25$ & $8$ & $2$ & $48$ & $16$\\
\hline
$3$ & $2$ & $36$ & $12$ & $3$ & $38$ & $18$ & $5$ & $72$ & $30$\\
\hline
$2g_0+2 \;\;(g\geqslant 1)$ & $3g_0+1$ & $8g_1+17$ & $4g_1+7$ & $3g_0+1$ & $8g_2+17$ & $4g_2 + 8$ & $6g_0+2$ & $8\wt{g}+32$ & $4\wt{g} + 15$\\
\hline
$2g_0+3 \;\;(g\geqslant 1)$ & $3g_0+2$ & $8g_1 + 20$ & $4g_1 + 9$ & $3g_0+3$ & $8g_2+14$ & $4g_2 + 6$ & $6g_0+5$ & $8\wt{g}+32$ & $4\wt{g} + 15$\\
\hline
\end{tabular}}
\end{table}
\section{Lower bounds on $N(K,g)$ and $R(K,g)$ for $g \equiv 1,2 \pmod 3$}\label{sec:finale}
The third column of \Cref{tab: genus and ranks gammas} gives rise to the lower bounds $R(K,g) \geqslant 4g+7$ and $N(K,g) \geqslant 8g+16$ for all $g\geqslant 2$. In this section, we use the data in \Cref{tab: genus and ranks thetas} to get improved bounds for $g \equiv 1,2 \pmod{3}$. To apply \Cref{cor:rank cor}, the only ingredient remaining is the following fact.
\begin{prop}\label{prop:nonconstant}
Let $X/K(b,\Bell)$ be one of the curves $\Theta_1,\Theta_2$, or $\wt{\Theta}$. Then $X/K(b,\Bell)$ is non-constant.
\end{prop}
We recall before the proof that to any hyperelliptic curve of genus $g \geqslant 2$, we can associate a binary form in $K[x,z]$ of degree $2g+2$, which is the branch divisor of a degree $2$ morphism to $\mathbb{P}^1$ (cf. \ref{emp:hyperelliptic}). If $f_1(x,z)$ and $f_2(x,z)$ are binary forms obtained in this way from hyperelliptic curves $X_1$ and $X_2$, then $X_1$ and $X_2$ are $\overline{k}$-isomorphic if and only if $f_1$ and $f_2$ are in the same orbit of $\mbf{GL}_{2,\overline{k}}$ (see~\cite{binaryforms}*{Section~2}), which acts on the space of degree $2g+2$ binary forms by
\begin{equation*}
\mat \cdot g(x,z) = g(ax + bz, cx + dz).
\end{equation*}
We identify the space of degree $n$ binary forms over $K$ with the $K$-points of an affine space $BF_n \coloneqq \spec K[b_0,\dotsc,b_n]$. A point $P \in (a_0,\dotsc,a_n) \in BF_n(K)$ is identified with the form $\sum_i a_ix^iz^{n-i} \in K[x,z]$.
\begin{proof}
Assume for a contradiction that $X/K(b,\Bell)$ is constant. To prove non-constancy, we may assume $K=\overline{\Q}$. Note that if the explicit affine equation for $X/K(b,\Bell)$ from \Cref{tab: genus and ranks thetas} is $y^2 = f(x)$, then $f(x)$ is a polynomial in $K[b,\Bell][x]$, with $\deg f = 2g+1$ or $2g+2$, where $g$ denotes the genus of $X$. We let $h(x,z) \coloneqq z^{2g+2}f(x/z)$ be the binary form associated to this equation (cf. \ref{emp:hyperelliptic}). We now break the proof down into cases depending on $d$.
First, assume $d\geqslant 4$, and let $v$ be an indeterminate. The ring homomorphism
\begin{align*}
K[b_0,\dotsc,b_{2g+2}] & \longrightarrow K[v,b,\Bell] = K[v]\otimes_K K[b,\Bell]\\
b_i & \longmapsto \textup{ coefficient of $x^iz^{2g+2-i}$ in } h(x + vz,z)
\end{align*}
defines a $K$-morphism $\phi: \mathbb{A}^1_K \times \spec K[b,\Bell] \longrightarrow BF_{2g+2}$, which sends a $K$-point $(a,P)$ to the form $h_P(x+ az,z) \in K[x,z]$, where $h_P(x,z)$ denotes the specialization of $h(x,z)$ at $P$. It is easy to see that the morphism $\phi$ restricts to a closed immersion on the closed subset $\{v = b = 0 \}$; the image $Z$ is defined by the vanishing of some of the coefficients $b_i$, and we have $\dim Z = d \geqslant 4$. Note that $Z$ is contained in the closed subset $\{ b_{2g+1} = 0 \} \subset BF_{2g+2}$. For any $K$-point $(a,P)$ of the locally closed set $\{v \neq 0\} \cap \{b = 0\}$, the coefficient of $x^{2g+1}z$ in the form $h_P(x+az,z)$ is non-zero, i.e. $\phi(a,P) \in (BF_{2g+2} \setminus Z)(K)$ so we see that
\begin{equation*}
\dim \phi (\{ b = 0 \}) = 1 + \dim Z \geqslant 5.
\end{equation*}
The assumption that $X/K(b,\Bell)$ is non-constant implies that $\phi (\{b = 0 \})$ is contained in an orbit of $\mbf{GL}_{2,K}$, but this is a contradiction, since $\dim \mbf{GL}_{2,K} = 4 < 5$.
Next, assume $d=3$. If we set $(b,l_0,l_1,l_2) = (0,1,a,1)$, for some $a \in K$, then $\Theta_1$ (resp. $\wt{\Theta}$) specializes to (a smooth, proper model of) the curve $y^2 = x^6 + ax^3 + 1$ (resp. the curve $y^2 = x^{12} + ax^6 + 1$). We can then apply~\cite{newpts}*{Lemma~6.3.1} to conclude that as we vary $a \in K$, we get infinitely many specializations of $\Theta_1$ (resp. $\wt{\Theta}$) that are pair-wise non-isomorphic.
Next, with $d=3$, consider $\Theta_2/K(b,\Bell)$. The binary form obtained from the explicit model for $\Theta_2$ (as in \ref{emp:hyperelliptic}) is:
\begin{align*}
h(x,z) & = zx(l_2(x^3 + 2bx^2z + b^2xz^2)^2 + l_1(x^3 + 2bx^2z + b^2xz^2)z^3 + l_0z^6)\\
& = l_2x^7z + 4 bl_2 x^6z^2 + 6 b^2l_2 x^5z^3 + (l_1 + 4b^3l_2)x^4z^4 + (2l_1 + l_2b^4) x^3z^5 + l_1b^2x^2z^6 + l_0xz^7
\end{align*}
Let $\phi : \spec K[v,b,l_0,l_1,l_2] \longrightarrow BF_{8}$ be as before; we will show that the image of $\phi$ is of dimension $5$, and obtain a contradiction as before. For this, we note that the image of $\{v = b = 0\}$ is of dimension $3$ (exactly as before), and this image lands in the closed subset $\{b_{7} = 0\}$. If $(a,P)$ is a $K$-point of $\{ v = 0\} \cap \{b \neq 0 \}$, then the coefficient of $x^7z$ in $h_P(x,z)$ is non-zero, i.e. $\phi (a,P)$ is a $K$-point of $\{b \neq 0 \})$, which implies that $\dim \phi (\{ v = 0 \}) = 4$. Note moreover that $\phi (\{ v = 0 \})$ is contained in the closed subset $\{ 3b_6^2 = 8b_5b_7 \} \subset BF_8$, so to conclude, it suffices to produce a point $(a,P) \in (\spec K[v,b,\Bell])(K)$ such that the coefficients of $h_P(x + az,z)$ satisfy the (non-)identity $3b_6^2 \neq 8b_5b_7$. For this, note that if we set $(v,b,l_0,l_1,l_2) = (a,1,1,1,1) \in K^5$ for some $a \in K$, then $h(x,z)$ specializes to the binary form $x^7z + (7a + 4)x^6z^2 + (21a^2 + 28a + 6)x^5z^3 + (\textup{l.o.t})$, and the identity $3b_6^2 = 8b_5b_7$ is equivalent to the identity $21 a^2 + 56 a = 0$. The desired result follows by choosing an $a \in K$ that does not satisfy the latter equation.
Finally, assume $d=2$. In this case, $\Theta_1$ and $\Theta_2$ are elliptic curves over $K(b,\Bell)$, and the reader can check that their $j$-invariants are non-constant polynomials in $K[b,\Bell]$, so we are left with the final case of the genus $2$ and degree $6$ curve $\wt{\Theta}/K(b,l_0,l_1): y^2 = l_1 (x^6 + 2bx^4 + b^2x^2) + l_0$. To show that it is non-constant, it suffices to show that the curve $X/K(l_0) = y^2 = x^6 + 2x^4 + x^2 + l_0$ is non-constant, for which we make use of the \emph{Igusa invariants} of the polynomial $p(x) \coloneqq x^6 + 2x^4 + x^2 + l_0$ (cf. \cite{binaryforms}). We use the function \texttt{igusa\_clebsch\_invariants()} in Sage~\cite{sage} to compute the \emph{Igusa invariants} (\cite{binaryforms}) for the specializations at $l_0 = 1$ and $l_0 = 2$ , which we denote by $p_1(x)$ and $p_2(x)$ respectively. We have
\begin{align*}
(I_2(p_1),I_4(p_1),I_6(p_1),I_{10}(p_1)) & = (-272, 1060, -80792, -33856)\\
(I_2(p_2),I_4(p_2),I_6(p_2),I_{10}(p_2)) & = (-512, 5296, -799232, -1280000),
\end{align*}
and a quick computation shows that there is no $r\neq 0 \in K$ for which $I_{2i}(p_1) = r^{2i} I_{2i}(p_2)$ for $i=1,2,3,5$. Thus, the binary forms associated to the two specializations are not $\mbf{GL}_{2,K}$-conjugates, and the corresponding specializations of $X$ are also not isomorphic, which concludes the proof.
\end{proof}
\begin{thm}\label{thm:2mod3}
Let $\wt{g}$ be a positive integer congruent to $2$ modulo $3$. Then
\begin{enumerate}[(i)]
\item $N(K,\wt{g}) \geqslant 8\wt{g} + 32$. \label{part:2 mod 3 i}
\item If $\wt{g}=2$, then $R(K,\wt{g}) \geqslant 16$. \label{part:2 mod 3 ii}
\item If $\wt{g}=5$, then $R(K,\wt{g}) \geqslant 30$. \label{part:2 mod 3 iii}
\item If $\wt{g} \geqslant 8$, then $R(K,\wt{g}) \geqslant 4\wt{g} + 15$.\label{part:2 mod 3 iv}
\end{enumerate}
\end{thm}
\begin{proof}
Let $d \geqslant2$ be an integer, and consider the curve $\wt{\Theta}/\mathcal{K}$ from \Cref{tab: genus and ranks thetas}. This is a non-constant curve (cf. \Cref{prop:nonconstant}) over the function field of the $K$-rational variety $\mathcal{B}_{2d}$ (cf. \Cref{thm:calg03n rational}), which satisfies $\# \wt{\Theta}(\mathcal{K})) \geqslant \wt{N}$ and $\rk \wt{\Theta}/\mathcal{K} \geqslant \wt{R}$, so we can apply \Cref{cor:rank cor} to conclude that $N(K,\wt{g}) \geqslant \wt{N}$ and $R(K,\wt{g}) \geqslant \wt{R}$, where $\wt{g}$ is the genus of $\wt{\Theta}$.
We wrap up the proof by observing that as $d$ ranges over the integers $\{2,3,\dotsc \}$, the genus $\wt{g}$ ranges over the positive integers congruent to $2$ modulo $3$, so we immediately get \ref{part:2 mod 3 i}. Moreover, if $d=2$ (resp. $3$), then $\wt{g} = 2$ (resp. $5$), so we get parts \ref{part:2 mod 3 ii} and \ref{part:2 mod 3 iii}, and if $d\geqslant 4$, then $\wt{g} \geqslant 8$, so we get \ref{part:2 mod 3 iv}.
\end{proof}
\begin{thm}\label{thm:1mod3}
Let $g_2\geqslant 4$ be an integer congruent to $1$ modulo $3$. Then
\begin{enumerate}[(i)]
\item $N(K,g_2) \geqslant 8g_2 + 17$ \label{part:1 mod 3 i}
\item $R(K,g_2) \geqslant 4g_2 + 8$. \label{part:1 mod 3 ii}
\end{enumerate}
\end{thm}
\begin{proof}
Consider the curve $\Theta_2$ from \Cref{tab: genus and ranks thetas}. Exactly as in the proof above, we have at hand all the ingredients necessary to apply \Cref{cor:rank cor} to $\Theta_2 /\mathcal{K}$.
Note from the third line of \Cref{tab: genus and ranks thetas} that as $d$ ranges over the even integers $\{4,6,8,\dotsc \}$, the genus $g_2$ ranges over $\{4,7,10,\dotsc,\}$, that is, over the set of integers greater than $3$ that are congruent to $1$ modulo $3$, which immediately establishes both parts of the theorem.
\end{proof}
\begin{emp}\label{proof:2torsion} \emph{Proof of \Cref{thm:2torsion}.} Observe that in \Cref{tab: genus and ranks thetas}, when $d=2g_0+3$, we have $g_2 = 3(g_0+1)$, so as $g_0$ ranges over the positive integers, $g_2$ ranges over the positive integers divisible by $3$. In each of these cases, the genus $g_2$ curve $\Theta_2$ has two $\mathcal{K}$-rational Weierstrass points, namely, the point $D$ (cf. \ref{emp:notation for curves 6n}) and a point at infinity. The difference of these points gives a two torsion point in $J(\mathcal{K})$, so $J(\mathcal{K}) \supset \mathbb{Z}^{4g+6}\oplus (\mathbb{Z}/2\mathbb{Z})$. \Cref{thm:specialization} then yields the bound $R(K,g;\mathbb{Z}/2\mathbb{Z}) \geqslant 4g+6$, as desired.
\end{emp}
\begin{rem}
We conclude this section with a few words about the structure of the Jacobians of the curves of \Cref{tab: genus and ranks thetas}. None of them is simple, as evidenced by the system of morphisms \eqref{eq: maps of curves}, but $J(\wt{\Theta})$ decomposes in an interesting way. Indeed, there exist abelian varieties $A_1/K(\mathcal{B}_{2d})$ and $A_2/K(\mathcal{B}_{2d})$ such that $J(\Theta_1) \sim A_1 \times J(\Gamma_1)$, and $J(\Theta_2) \sim A_2 \times J(\Gamma_2)$, and $J(\wt{\Theta}) \sim J(\Theta_1) \times J(\Theta_2) \sim A_1 \times J(\wt{\Gamma}) \times A_2$.
The current record for $R(\mathbb{Q},g)$ with \emph{simple Jacobians} is due to Shioda and Terasoma \cite{terasoma}, who showed that the curve $\Gamma_1$ admits infinitely many specializations to curves over $\mathbb{Q}$ with the same lower bound $4g+5$ on rank, and having (absolutely) simple Jacobians.
\end{rem}
\section{The rational variety $\mathcal{Z}_n/K$}\label{sec:calz}
This section and the next comprise one block. In this section, we define and establish the $K$-rationality of a variety $\mathcal{Z}_n/K$ (see \ref{emp:way forward} \ref{part:emp cases 3}), which parametrizes $(4,n)$-composite tuples (as in \ref{emp:poa}).
\begin{emp}\label{emp:calz n}\textbf{The rational variety $\mathcal{Z}_{n}$.} Fix an integer $n\geqslant 1$. Let $G_i$ denote the polynomial $z_{i1}^2 + z_{i2}^2$, and define:
\begin{equation*}
S \coloneqq \dfrac{K[z_{11},z_{12},\dotsc,z_{n1},z_{n2},G_1^{-1}]}{(G_1 - G_2,\dotsc, G_{n-1} - G_n)}, \qquad \mathcal{Z}_n \coloneqq \spec S.
\end{equation*}
Let $L$ be the \'etale $K$-algebra $K[x] /(x^2 + 1)$, and $\zeta \in L$ the class of $x$. Following the method described in \ref{emp:cals second pres}, for each $i=1,\dotsc,n$, we identify $\spec K[z_{i1},z_{i2}]$ with the Weil restriction $\mf{R}_{L/K} \mathbb{A}^1_{L}$ by identifying, for any $K$-algebra $A$, the point $(a_{i1},a_{i2}) \in \mathbb{A}^2(A)$ with the element $a_{i1} + a_{i1}\zeta \in A\otimes_K L$. The open subscheme $\spec K[z_{i1},z_{i2},(z_{i1}^2 + z_{i2}^2)^{-1}]$ is then naturally identified with $\mf{R}_{L/K} \gm{L}$, and we have a group scheme homomorphism $N : \mf{R}_{L/K} \gm{L} \longrightarrow \gm{K}$. Observe that $\mathcal{Z}_n$ fits into a cartesian diagram:
\begin{equation}\label{diag:calz fib}
\begin{tikzcd}
\mathcal{Z}_n \arrow[d] \arrow[r, hook] \drar[phantom, "\boxtimes"] & (\mf{R}_{L/K} \gm{L})^n \arrow[d, "(N)^n"]\\
\gm{K} \arrow[r, hook, "\Delta"] & (\gm{K})^n.
\end{tikzcd}
\end{equation}
\end{emp}
\begin{prop}\label{prop:calz rational}
The scheme $\mathcal{Z}_n$ is an $n+1$-dimensional $K$-rational variety.
\end{prop}
\begin{proof}
Let $\mf{R}_{L/K}^1 \gm{L}$ denote the norm-one torus, i.e. the kernel of the homomorphism $N : \mf{R}_{L/K} \gm{L} \longrightarrow \gm{K}$. We claim that there exists a $K$-isomorphism
\begin{equation*}
\phi : \mathcal{Z}_n \longrightarrow \mf{R}_{L/K} \gm{L} \times (\mf{R}_{L/K}^1 \gm{L})^{n-1}.
\end{equation*}
We can produce such an isomorphism exactly as we did in \Cref{lem:calv rational}; we omit the details, since the proof is identical to the proof of the lemma. This isomorphism implies that $\mathcal{Z}_n$ is $K$-rational, since $\mf{R}_{L/K} \gm{L}$ is $K$-rational and one-dimensional $K$-tori are $K$-rational.
\end{proof}
\begin{emp}\label{emp:notation for calz} \textbf{Notation.} We now setup some notation for this section and the next.
\begin{enumerate}[(i)]
\item Let $\mathcal{K}$ denote the function field $K(\mathcal{Z}_n)$.
\item We define the following elements in $S$:
\begin{equation}\label{eq:functions for calz}
\begin{alignedat}{2}
t_{ij} & \coloneqq z_{ij}^2, && \qquad i=1,\dotsc,n, \; j=1,2,\\
u_i & \coloneqq -t_{i1}t_{i2} = -(z_{i1}z_{i2})^2 && \qquad i=1,\dotsc,n,\\
m_j & \coloneqq (-1)^{n-j}s_{n-j}(u_1,\dotsc,u_n), && \qquad j=0,\dotsc,n-1,\\
b & \coloneqq G_1 = t_{11} + t_{12} = z_{11}^2 + z_{12}^2. &&
\end{alignedat}
\end{equation}
\item Let $R$ denote the $K$-subalgebra of $S$ generated by the elements $t_{ij}$, and let $F$ denote the fraction field of $R$. Let $\mbf{u}$ and $\mbf{m}$ denote the tuples of elements $u_1,\dotsc,u_n$ and $m_0,\dotsc,m_{n-1}$ respectively, and abuse notation to let $K(b,\mbf{u})$ and $K(b,\mbf{m})$ denote the subfield of $\mathcal{K}$ generated by these tuples respectively, so that $K(b,\mbf{m}) \subset K(b,\mbf{u}) \subset F \subset \mathcal{K}$.
\item We define the following polynomials in $S[x]$:
\begin{align}
\begin{split}
g(x) & \coloneqq x^2 - bx\\
m(x) & \coloneqq x^n + m_{n-1}x^{n-1} + \dotsb + m_1x + m_0.
\end{split}
\end{align}
\end{enumerate}
\end{emp}
The last line of \eqref{eq:calz identities} below implies that $\mathcal{Z}_n$ parametrizes $(4,n)$-composite tuples (as in \ref{emp:poa}).
\begin{prop}
Continuing with the notation of \ref{emp:notation for calz}, we have the following identities in $S[x]$:
\begin{equation}\label{eq:calz identities}
\begin{alignedat}{2}
g(x^2) - u_i & = (x - z_{i1})(x + z_{i1})(x - z_{i2})(x + z_{i2}), && \qquad i=1,\dotsc, n,\\
g(x) - u_i & = (x - t_{i1})(x-t_{i2}), && \qquad i=1,\dotsc,n,\\
m(x) & = (x - u_1) \dotsb (x-u_n), &&\\
m(g(x)) & = \prod_{i=1}^n (x - t_{i1})(x - t_{i2}), &&\\
m(g(x^2)) & = \prod_{i=1}^n (x - z_{i1})(x + z_{i1})(x - z_{i2})(x + z_{i2}).
\end{alignedat}
\end{equation}
\end{prop}
\begin{proof}
The reader can check using the definitions in \ref{emp:notation for calz} that these identities hold.
\end{proof}
\Cref{lem:aut of Zn} below is used (in the next section) in bounding the rank of the curve $\Lambda_2/K(\mathcal{Z}_{2d})$ (defined in \Cref{tab:curves calz}).
\begin{prop}\label{lem:aut of Zn}
\begin{enumerate}[(i)]
\item The fields $F,K(b,\mbf{u})$ and $K(b,\mbf{m})$ are $n+1$-dimensional rational function fields.\label{part:aut zn i}
\item We have an isomorphism of groups:
\begin{align*}
\alpha : (V_4)^n & \xrightarrow{\raisebox{-0.7ex}[0ex][0ex]{$\;\sim\;$}} \textup{Gal}(\mathcal{K}/F)\\
((e_{11},e_{12}),\dotsc, (e_{n1},e_{n2})) & \longmapsto (z_{ij} \mapsto e_{ij}z_{ij}),
\end{align*}
where $V_4 = (S_2)^2$ denotes the Klein-four group (so each $e_{ij} \in S_2 = \{\pm 1\}$). \label{part:aut zn ii}
\end{enumerate}
\end{prop}
\begin{proof}
For \ref{part:aut zn i}, we first note that the identities in \eqref{eq:calz identities} imply that $\mathcal{K}$ is the splitting field of the polynomial $m(g(x^2)) \in K(b,\mbf{m})[x]$, so $\mathcal{K}/K(b,\mbf{m})$ is a finite extension, and all the fields in \ref{part:aut zn i} are of transcendence degree $n+1$ over $K$. Since $t_{i2} = b - t_{i1}$ for each $i=1,\dotsc,n$, the field $F$ is generated over $K$ by $b,t_{11},\dotsc,t_{n1}$. Similarly, the elements $b,\mbf{u}$ (resp. $b,\mbf{m}$) generate $K(b,\mbf{u})$ (resp. $K(b,\mbf{m})$), so we conclude all three of these are rational function fields.
For \ref{part:aut zn ii}, first note that if we regard $\alpha$ a priori as a homomorphism from $(V_4)^n$ to $\aut(K[z_{i1},z_{i2}\mid i=1,\dotsc,n])$, then the image consists of automorphisms that fix the elements $G_i = z_{i1}^2 + z_{i2}^2$, and hence, also the ideal defining $S$. Thus, $\alpha$ is indeed a homomorphism from $(V_4)^n$ to $\aut(\mathcal{K})$. It is clearly injective, and we leave it to the reader to verify that the image is $\textup{Gal}(\mathcal{K}/F)$.
\end{proof}
\section{Lower bounds on $N(K,g)$ and $R(K,g)$ for odd $g$}\label{sec:odd positive g}
We continue with the notation of the previous section.
In this section, we construct curves with many rational points over the function field $K(\mathcal{Z}_n)$ and establish lower bounds on their ranks; this results in lower bounds on $R(K,g)$ and $N(K,g)$ for odd, positive integers $g$ (see \Cref{thm:odd g result}).
For the rest of this section, we set $n=2d$, where $d \geqslant 2$ is an integer. \Cref{lem:aut of Zn} \ref{part:aut zn i} implies that the $2d+1$ elements $b,\mbf{m}$ of $S$ generate a polynomial algebra $K[b,\mbf{m}] \subset S$, and since $b$ is invertible in $S$, we have an inclusion $K[b,b^{-1},\mbf{m}] \subset S$. Let $h(x) = x^d + h_{d-1}x^{d-1} + \dotsb + h_0$ and $\ell(x) = l_{d-1}x^{d-1} + \dotsb + l_0$ denote the square-root approximation and remainder of $m(x)$ respectively (cf. \Cref{def: sqroot rem}), so that we have inclusions of $K$-algebras:
\begin{equation*}
K[b,b^{-1},\Bell] \subset K[b,b^{-1},\Bell,\mbf{h}] \cong K[b,b^{-1},\mbf{m}] \lhook\joinrel\longrightarrow R \lhook\joinrel\longrightarrow S,
\end{equation*}
where, as usual, $\mbf{h}$ and $\Bell$ denote the tuples of indeterminates $(h_0,\dotsc,h_{d-1})$ and $(l_0,\dotsc,l_{d-1})$ respectively, and the isomorphism $K[b,b^{-1},\Bell,\mbf{h}] \cong K[b,b^{-1},\mbf{m}]$ is given by \Cref{lem:kthroot}. Combining the identities in \eqref{eq:calz identities} with the identity $m(x) = h(x)^2 - \ell(x) \in S[x]$, we get the following identities in $S[x]$:
\begin{align}\label{eq:calz point identities}
\begin{split}
h(x)^2 - \ell(x) & = (x-u_1)\dotsb (x-u_{2d}),\\
h(g(x))^2 - \ell(g(x)) & = \prod_{i=1}^{2d} (x - t_{i1})(x - t_{i2}),\\
h(g(x^2))^2 - \ell(g(x^2)) & = \prod_{i=1}^{2d} (x - z_{i1})(x + z_{i1})(x - z_{i2})(x + z_{i2}).
\end{split}
\end{align}
\begin{emp}\label{emp: curves calz} \textbf{The curves.} The second column of the table below contains three affine curves over the function field $K(b,\Bell)$, given by equations of the form $y^2 = f(x)$, with $f(x) \in K(b,\Bell)$ a separable polynomial; to check separability, it suffices to note that upon setting $(b,l_0,l_1,\dotsc,l_{d-2},l_{d-1}) = (0,1,0,\dotsc,0,1)$, $f(x)$ specializes to a separable polynomial in $K[x]$. In the first column, we denote the smooth, proper model of the corresponding affine curve (see \ref{emp:hyperelliptic}). The reader can verify using \eqref{eq:calz point identities} that the curves come with the explicit $\mathcal{K}$-rational points listed in the table, where we recall that $\mathcal{K} = K(\mathcal{Z}_{2d})$.
\end{emp}
\begin{table}[H]
\centering
\adjustbox{max width=\textwidth}{
\caption{Curves with $\mathcal{K}$-points}\label{tab:curves calz}
\begin{tabular}{|c|c|cc|}
\hline
Curve & Equation & $\mathcal{K}$-rational points & \\
\hline
$\Lambda_1 / K(b,\Bell)$ & $y^2 = \ell(g(x))$ & $P_{ij} \coloneqq (t_{ij},h(g(t_{ij})))$ & $i=1,\dotsc,2d, \, j=1,2$ \\
\hline
$\Lambda_2 / K(b,\Bell)$ & $y^2 = x \ell(g(x))$ & $Q_{ij} \coloneqq (t_{ij},z_{ij} h(g(t_{ij})))$ & $i=1,\dotsc,2d, \, j=1,2$ \\
\hline
$\wt{\Lambda}/K(b,\Bell)$ & $y^2 = \ell(g(x^2))$ & $\wt{P}_{ij}^{\pm} \coloneqq (\pm z_{ij}, h(g(z_{ij}^2)))$ & $i=1,\dotsc,2d, \, j=1,2$ \\
\hline
\end{tabular}}
\end{table}
\noindent As in \eqref{diag:curves and maps}, these curves fit together into a diagram of $K(b,\Bell)$-schemes:
\begin{equation}\label{diag:curves and maps calz}
\begin{tikzcd}
& \widetilde{\Lambda} \arrow[dl, "\varphi_1", swap] \arrow[dr, "\varphi_2"] & & & (x,y) \arrow[dl, mapsto] \arrow[dr, mapsto] & \\
\Lambda_1 & & \Lambda_2 & (x^2,y) & & (x^2,xy),
\end{tikzcd}
\end{equation}
so by \Cref{prop:isogenous}, we have a $K(b,\Bell)$-isogeny
\begin{equation*}
J_{\wt{\Lambda}} \longrightarrow J_{\Lambda_1} \times J_{\Lambda_2}.
\end{equation*}
Now, we determine lower bounds for $\rk J_{\Lambda_1} / \mathcal{K}$ and $\rk J_{\Lambda_2}/\mathcal{K}$; then, we will apply \Cref{prop:isogenous} to get lower bounds for $\rk J_{\wt{\Lambda}}/\mathcal{K}$.
\begin{thm}\label{thm:rank of lambda 1}
Recall from \Cref{tab:curves calz} the nice curve $\Lambda_1/ K(b,\Bell)$.
\begin{enumerate}[(i)]
\item We have $\# \Lambda_1(\mathcal{K}) \geqslant 8d = 8g_1 + 16$, where $g_1 = d-2$ is the genus of $\Lambda_1$.\label{part:lambda1 i}
\item If $d = 3$, then $\Lambda_1$ is of genus $1$, and $\rk {\Lambda_1}/\mathcal{K} \geqslant 6$. \label{part:lambda1 iii}
\item If $d \geqslant 4$, then $\rk {\Lambda_1}/\mathcal{K} \geqslant 4d - 1 = 4g_1 + 7$.\label{part:lambda1 iv}
\end{enumerate}
\end{thm}
\begin{proof}
Part \ref{part:lambda1 i} follows from the observation that $\Lambda_1$ has the distinct $\mathcal{K}$-rational points $P_{ij}, \iota(P_{ij})$, for $i=1,\dotsc,2d$ and $j=1,2$. For parts \ref{part:lambda1 iii} and \ref{part:lambda1 iv}, we first note that the points $P_{ij}$ are actually $F$-rational points of $\Lambda_1$, where $F = K(b,t_{11},\dotsc,t_{2d,1})$ (cf. proof of \Cref{lem:aut of Zn} \ref{part:aut zn i}), so it suffices to replace $\mathcal{K}$ with $F$ in the statements of \ref{part:lambda1 iii} and \ref{part:lambda1 iv}. Let $\mathscr{O}$ denote the local ring at the generic point of the hyperplane $\{b = 0\}$ of $\spec K[b,t_{11},\dotsc,t_{2d,1}]$, and let $\mathfrak{n}$ denote its maximal ideal. Each $t_{i2} = b - t_{i1}$ reduces modulo $\mathfrak{n}$ to $-t_{i1}$, so each $u_i = -t_{i1}t_{i2} = -t_{i1}(b-t_{i1})$ reduces modulo $\mathfrak{n}$ to $t_{i1}^2$ (cf. \eqref{eq:functions for calz}); thus, we have an inclusion of fields $K(\mbf{m}) = K(m_0,\dotsc,m_{2d-1}) \lhook\joinrel\longrightarrow \mathscr{O}/\mathfrak{n} = K(t_{11},\dotsc,t_{2d,1})$, defined by
\begin{equation*}
m_j \longmapsto (-1)^{2d-j}s_{2d-j}(u_1,\dotsc,u_{2d}) = (-1)^{2d-j}s_{2d-j}(t_{i1}^2,\dotsc,t_{2d,1}^2), \qquad j=0,\dotsc,2d-1.
\end{equation*}
On the other hand, recall the field $K(\mbf{t}) = K(t_1,\dotsc,t_{2d})$ appearing in \Cref{cor: rank of GGamma}; again, we have an inclusion $K(\mbf{m}) \lhook\joinrel\longrightarrow K(\mbf{t})$ defined by $m_j \longrightarrow (-1)^{2d-j}s_{2d-j}(t_1^2,\dotsc,t_{2d}^2),\, j=0,\dotsc,2d-1$ (cf. \ref{eq:functions}). Thus, the map $t_{i1} \longrightarrow t_{i},\, i=1,\dotsc,2d,$ defines a $K(\mbf{m})$-isomorphism of fields $\mathscr{O}/\mathfrak{n} \xrightarrow{\raisebox{-0.7ex}[0ex][0ex]{$\;\sim\;$}} K(\mbf{t})$. Now, let $F(x,z) \in \mathscr{O}[x,z]$ denote the homogenization of $\ell(g(x)) = \ell(x^2-bx)$, let $\mathscr{O}[x,y,z]$ denote the graded $\mathscr{O}$-algebra in which $\deg x = \deg z = 1$ and $\deg y = g_1 +1 = d-1$, and consider the $\mathscr{O}$-scheme $\mathscr{X} \coloneqq \textup{Proj} \,\mathscr{O}[x,y,z]/(y^2 - F(x,z))$. The generic fiber of $\mathscr{X} /\spec \mathscr{O}$ is isomorphic to $\Lambda_1/F$, and the special fiber is isomorphic to $(\wt{\Gamma})_{K(\mbf{t})}/K(\mbf{t})$, where we recall from \Cref{tab:curves} that $\wt{\Gamma}/K(\Bell)$ is a smooth proper model of the affine curve $y^2 = \ell(x^2)$. Note that $\mathscr{X}$ is covered by two open affine subsets isomorphic to $\spec R_1$ and $\spec R_2$, where $R_1 = \mathscr{O}[x,y]/(y^2 - F(x,1))$ and $R_2 = \mathscr{O}[x,y]/(y^2 - F(1,z))$ are both \emph{relative global complete intersections over $\mathscr{O}$} (cf. \cite{stacks-project}*{Definition \href{https://stacks.math.columbia.edu/tag/00SP}{00SP}}). \cite{stacks-project}*{Lemma \href{https://stacks.math.columbia.edu/tag/00SW}{00SW}} then implies that $\mathscr{X} \longrightarrow \spec \mathscr{O}$ is flat, so $\mathscr{X} \longrightarrow \spec \mathscr{O}$ is smooth and projective.
Under the \defi{reduction map} $\rho :\Lambda_1(F) \longrightarrow \wt{\Gamma}(K(\mbf{t}))$, we have
\begin{equation*}
\rho (P_{i1}) = \rho(t_{i1},h(t_{i1}^2-bt_{i1})) = (t_i,h(t_i^2)) = \wt{P}_i^+ \in \wt{\Gamma}(K(\mbf{t})), \qquad i=1,\dotsc,2d,
\end{equation*}
and similarly $\rho(P_{i2}) = \wt{P}_i^-$ for $i=1,\dotsc,2d$ (cf. \Cref{tab: genus and ranks gammas}). Abusing notation to let $\rho : J_{\Lambda_1}(F) \longrightarrow J_{\wt{\Gamma}}(K(\mbf{t}))$ denote the reduction map on Jacobians, we have
\begin{equation*}
\rho (\epsilon (P_{i1})) = \epsilon (\wt{P}_i^+) \quad \textup{ and } \quad \rho (\epsilon (P_{i2})) = \epsilon (\wt{P}_i^-), \qquad i=1,\dotsc,2d,
\end{equation*}
so under the homomorphism $\rho$, the subgroup of $J_{\Lambda_1}(F)$ generated by the $\epsilon (P_{ij})$'s is mapped surjectively onto the subgroup of $J_{\wt{\Gamma}}(K(\mbf{t}))$ generated by the $\epsilon (\wt{P}_i^{\pm})$'s. Now, part \ref{part:lambda1 iii} (resp. part \ref{part:lambda1 iv}) follows from part \ref{part:wtgamma iii} (resp. \ref{part:wtgamma iv}) of \Cref{cor: rank of GGamma}, so we are done.
\end{proof}
\begin{thm}\label{thm:rank of lambda 2}
Recall from \Cref{tab:curves calz} the nice curve $\Lambda_2/K(b,\Bell)$. Let $D$ denote the closed point $(0,0)$ of $\Lambda_2$.
\begin{enumerate}[(i)]
\item The curve $\Lambda_2$ is of genus $g_2 = d-1$.\label{part:lambda2 i}
\item We have $\# \Lambda_2(\mathcal{K}) \geqslant 8d+2 = 8g_2 + 10$.\label{part:lambda2 ii}
\item The points $\mathfrak{r}(Q_{ij})\coloneqq [Q_{ij} - D],\,i=1,\dotsc,2d,\, j=1,2$, are linearly independent in $J_{\Lambda_2}(\mathcal{K})$, so they generate a subgroup of rank $4d = 4g_2 + 4$.\label{part:lambda2 iii}
\end{enumerate}
\end{thm}
\begin{proof}
See \ref{emp:hyperelliptic} for the first part. The second part follows by noting that $\Lambda_2$ has the distinct $\mathcal{K}$-points $Q_{ij}, \iota(Q_{ij})$ for $i=1,\dotsc,2d,\; j=1,2$, and two Weierstrass points ($D$ and a point at infinity). For the third part, assume for a contradiction that we have a linear dependence relation
\begin{equation}\label{eq:lambda 2 rel}
\sum_{i=1}^{2d} a_{i1} \mathfrak{r}(Q_{i1}) + a_{i2} \mathfrak{r}(Q_{i2}) = 0 \in J_{\Lambda_2}(\mathcal{K}),
\end{equation}
with $a_{pk} \neq 0$ for some $1\leqslant p \leqslant 2d$ and $1\leqslant k \leqslant 2$. Let $\sigma \in \textup{Gal}(\mathcal{K}/F)$ be the transposition that sends $z_{pk}$ to $-z_{pk}$ and fixes all the other $z_{ij}$'s (cf. the proof of \Cref{lem:aut of Zn} \ref{part:aut zn ii}). Then, $\sigma$ fixes $Q_{ij} \in \Lambda_2(\mathcal{K})$ if $(i,j)\neq (p,k)$, and $$\sigma(Q_{pk}) = (\sigma(t_{pk}),\sigma(z_{pk}h(g(t_{pk})))) = (t_{pk},-z_{pk}h(g(t_{pk}))) = \iota(Q_{pk}) \in \Lambda_2(\mathcal{K}).$$
Thus, $\sigma$ fixes $\mathfrak{r}(Q_{ij}) \in J_{\Lambda_2}(\mathcal{K})$ if $(i,j) \neq (p,k)$, and we have
$$\sigma (\mathfrak{r}(Q_{pk})) = \sigma \big([Q_{pk} - D]\big) = [\iota(Q_{pk}) - D] = -\mathfrak{r}(Q_{pk}) \in J_{\Lambda_2}(\mathcal{K}).$$ Applying $\sigma$ to the identity \eqref{eq:lambda 2 rel} and subtracting the result from the original, we find that $2a_{pk} \mathfrak{r}(Q_{pk}) = 0$, i.e. $\mathfrak{r}(Q_{pk})$ is a \emph{torsion point} of $J_{\Lambda_2}(\mathcal{K})$. \Cref{lem:algebraic point} then implies that $Q_{pk} = (t_{pk},z_{pk}h(g(t_{pk})))$ is algebraic over $K(b,\Bell)$, which implies that $t_{pk}$, and hence also $u_p = t_{pk}(b-t_{pk})$ is algebraic over $K(b,\Bell)$. By the Fundamental Theorem of symmetric functions (cf. \cite{hungerford}*{Page~254, Theorem~2.18}), the extension $K(b,\mbf{u})/K(b,\mbf{m})$ is Galois, with $\textup{Gal}(K(b,\mbf{u})/K(b,\mbf{m})) \cong S_{2d}$ acting by permutation on $u_1,\dotsc,u_{2d}$, so \emph{every} $u_i$ is algebraic over $K(b,\Bell)$, which implies that each $m_j = (-1)^{2d-j}s_{2d-j}(u_1,\dotsc,u_{2d})$ is algebraic over $K(b,\Bell)$. But this implies that $K(b,\mbf{m})/K(b,\Bell)$ is an algebraic extension, which is a contradiction because $\trdeg_{K(b,\Bell)} K(b,\mbf{m})= d>0$.
\end{proof}
\begin{cor}\label{cor:rank of wtlambda}
Recall from \Cref{tab:curves calz} the nice curve $\wt{\Lambda} / K(b,\Bell)$.
\begin{enumerate}[(i)]
\item We have $\# \wt{\Lambda}(\mathcal{K}) \geqslant 16d = 8\wt{g} + 24$, where $\wt{g} = 2d-3$ denotes the genus of $\wt{\Lambda}$.\label{part:wt lambda i}
\item If $d=2$, then $\wt{g} = 1$ and $\rk \wt{\Lambda}/\mathcal{K} \geqslant 8$.\label{part:wt lambda ii}
\item If $d=3$, then $\wt{g} =3$, and $\rk \wt{\Lambda}/\mathcal{K} \geqslant 18$.\label{part:wt lambda iii}
\item If $d\geqslant 4$, then $\rk \wt{\Lambda}/\mathcal{K} \geqslant 8d-1 = 4\wt{g} + 11$.\label{part:wt lambda iv}
\end{enumerate}
\end{cor}
\begin{proof}
For \ref{part:wt lambda i}, we note that $\wt{\Lambda}$ has the distinct $\mathcal{K}$-points $\wt{P}_{ij}^+,\wt{P}_{ij}^-, \iota (\wt{P}_{ij}^+),\iota (\wt{P}_{ij}^-)$ for $i=1,\dotsc,2d$ and $j=1,2$. The identity $\wt{g} = 2d - 3$ follows by setting $2\wt{g} +2 = \deg \ell(g(x^2)) = 4(d-1)$. When $d=2$, $\Lambda_1$ is of genus $0$ (cf. \Cref{thm:rank of lambda 1} \ref{part:lambda1 i}) and $\rk \Lambda_2 \geqslant 8$ (cf. \Cref{thm:rank of lambda 2} \ref{part:lambda2 ii}). Part \ref{part:wt lambda ii} then follows from \Cref{prop:isogenous}. When $d=3$, then $\rk \Lambda_1 /\mathcal{K} \geqslant 6$ (cf. \Cref{thm:rank of lambda 1} \ref{part:lambda1 iii}) and $\rk \Lambda_2 / \mathcal{K} \geqslant 12$, so part \ref{part:wt lambda iii} follows from \Cref{prop:isogenous}. When $d \geqslant 4$, then $\rk \Lambda_1 / \mathcal{K} \geqslant 4d -1$ (cf. \Cref{thm:rank of lambda 1} \ref{part:lambda1 iv}) and $\rk \Lambda_2 / \mathcal{K} \geqslant 4d$ (cf. \Cref{thm:rank of lambda 2} \ref{part:lambda2 iii}), so by \Cref{prop:isogenous}, we have $\rk \wt{\Lambda} / \mathcal{K} \geqslant 8d -1 = 4\wt{g} + 11$, which establishes part \ref{part:wt lambda iv}.
\end{proof}
\begin{lem}\label{lem:wtlambda nonconstant}
The curve $\wt{\Lambda}/K(b,\Bell)$ is non-constant.
\end{lem}
\begin{proof}
If $d=2$, then $\wt{\Lambda}$ is a smooth, proper model of the affine curve $y^2 = l_1(x^4 + bx^2) + l_0$, and the reader can easily verify that this is a non-constant genus $1$ curve.
Assume now that $d \geqslant 3$. If we set $b=0$, then the equation $y^2 = \ell(g(x^2))$ specializes to the equation $y^2 = \ell(x^4)$, where $\ell(x) = l_{d-1}x^{d-1} + \dotsb + l_1x + l_0 \in K(\Bell)[x]$. If we further specialize by setting $l_i = 1$ for $i=0,d-1$, and $l_i = 0$ for $i=2,\dotsc,d-3,d-2$, then the equation above becomes the equation $y^2 = (x^4)^{d-1} + l_1x^4 + 1$. \cite{newpts}*{Lemma~6.3.1} guarantees that this equation defines a non-constant curve over $K(l_1)$, so we conclude that $\widetilde{\Lambda}/K(b,\Bell)$ is non-constant.
\end{proof}
\begin{thm}\label{thm:odd g result}
Let $\wt{g}$ be an odd positive integer. Then
\begin{enumerate}[(i)]
\item $N(K,\wt{g}) \geqslant 8\wt{g} + 24$.\label{part:odd g i}
\item If $\wt{g} \geqslant 5$, then $R(K,\wt{g}) \geqslant 4\wt{g} + 11$.\label{part:odd g ii}
\end{enumerate}
\end{thm}
\begin{proof}
Let $d \geqslant 2$ be an integer, and consider the curve $\wt{\Lambda}/\mathcal{K}$ from \Cref{cor:rank of wtlambda}, where $\mathcal{K} = K(\mathcal{Z}_{2d})$. Since the genus $\wt{g}$ of $\wt{\Lambda}$ is given by $\wt{g} = 2d - 3$, we see that as $d$ ranges over the positive integers, $\wt{g}$ ranges over the positive \emph{odd} integers.
Now, \Cref{prop:calz rational} tells us that $\mathcal{Z}_{2d}$ is a $K$-rational variety, and \Cref{lem:wtlambda nonconstant} above implies that $\wt{\Lambda}/\mathcal{K}$ is non-constant (since it is non-constant over $K(b,\Bell)$). Thus, we can apply \Cref{cor:rank cor} to \Cref{cor:rank of wtlambda} \ref{part:wt lambda i} to conclude that $N(K,\wt{g}) \geqslant 8\wt{g} + 24$ if $\wt{g}$ is odd. When $d$ ranges over the integers $\{4,5,\dotsc,\}$ in the setting of \Cref{cor:rank of wtlambda}, then $\wt{g}$ ranges over the \emph{odd} integers $\{5,7,9,\dotsc \}$; applying \Cref{cor:rank cor} to \Cref{cor:rank of wtlambda} \ref{part:wt lambda iv}, we conclude that if $\wt{g} \geqslant 5$ is an odd integer then $R(K,\wt{g}) \geqslant 4\wt{g} + 11$.
\end{proof}
We conclude this section with the proof of \Cref{thm:1.2}.
\begin{emp}\label{emp:proof of 1.2}
\emph{Proof of \Cref{thm:1.2}.} Set $K= \mathbb{Q}$, and let $g$ be a positive integer not divisible by $6$.
\begin{enumerate}
\item If $g \equiv 2,5 \pmod{6}$, then $g \equiv 2 \pmod{3}$, so we get $N(K,g) \geqslant 8g + 32$ from \Cref{thm:2mod3} \ref{part:2 mod 3 i}. If moreover $g \geqslant 8$, then we get $R(K,g) \geqslant 4g + 15$ from \Cref{thm:2mod3} \ref{part:2 mod 3 iv}.
\item If $g \equiv 1,3 \pmod{6}$, then $g$ is odd, so we get $N(K,g) \geqslant 8g + 24$ from \Cref{thm:odd g result} \ref{part:odd g i}. If moreover $g \geqslant 7$, then we get $R(K,g) \geqslant
4g + 11$ from \Cref{thm:odd g result} \ref{part:odd g ii}.
\item If $g \equiv 4 \pmod{6}$, then $g \equiv 1 \pmod{3}$, so we get $N(K,g) \geqslant 8g + 17$ from \Cref{thm:1mod3} \ref{part:1 mod 3 i}. If moreover $g \geqslant 4$, then we get $R(K,g) \geqslant 4g+8$ from \Cref{thm:1mod3} \ref{part:1 mod 3 ii}. \qed
\end{enumerate}
\end{emp}
\section{Curves of high rank over cyclotomic fields}\label{sec:kummer}
In this section, we consider $(g+1,6)$-composite tuples whose coordinates satisfy a polynomial $m(x^{g+1}) \in K[x]$ with $\deg m = 6$; under the assumption that $g+1$ is an odd prime, we obtain genus $g$ curves of the form $y^2 = \ell(x^{g+1})$, with $\deg \ell(x) = 2$, having high rank over number fields that contain the $g+1$-th roots of unity.
\begin{emp}\label{emp:setup} \textbf{Setup.}
Now, and for the rest of this section, we fix an integer $g \geqslant 2$ such that \emph{$g+1$ is a prime number}. We set $e=g+1$ and $n=6$, and we let $K$ be a number field that contains a primitive $g+1$-th root of unity $\zeta$.
Let $\Bell,\mbf{h},\mbf{m},\mbf{u},\mbf{t}$ denote the tuples of indeterminates $(l_0,l_1,l_2), \,$ $(h_0,h_1,h_2), \,$ $(m_0,\dotsc,m_5),\,$ $(u_1,\dotsc,u_6),$ and $(t_1,\dotsc,t_6)$ respectively. Consider the inclusion of function fields
\begin{equation*}
K(\Bell) \subset K(\Bell,\mbf{h}) \cong K(\mbf{m}) \subset K(\mbf{u}) \subset K(\mbf{t}),
\end{equation*}
where, letting $m(x) = x^6 + m_5x^5 + \dotsb + m_0$, $h(x) = x^3 + h_2x^2 + h_1x + h_0$, and $\ell(x) = l_2x^2 + l_1x + l_0$, we define:
\begin{align*}
u_i & = t_i^{g+1}, \quad \textup{ for } i=1,\dotsc,6,\\
m(x) & = (x-u_1)\dotsb(x-u_6),\\
m(x) & = h(x)^2 - \ell(x),
\end{align*}
and the last identity gives the isomorphism $K(\Bell,\mbf{h}) \cong K(\mbf{m})$ (cf. \Cref{lem:kthroot}). Note that
\begin{equation*}\label{eq:roots of mx6}
m(x^{g+1}) = (x^{g+1} - t_1^{g+1})\dotsb (x^{g+1} - t_6^{g+1}) = \prod_{1\leqslant i \leqslant 6} \prod_{0 \leqslant j \leqslant g} (x - \zeta^jt_i) \in K[\mbf{t}][x],
\end{equation*}
so $\mathbb{A}^6(\mbf{t})$ parametrizes $(g+1,6)$-composite tuples (as in \ref{emp:poa}).
Let $X_0/K(\Bell)$ be the affine curve defined by the equation $y^2 = \ell(x^{g+1})$, and let $X/K(\Bell)$ denote its smooth, proper model (as in \ref{emp:hyperelliptic}). Note that $\ell(x^{g+1})$ is square-free, so the genus of $X$ is $g$. The curve $X_0/K$ intersects the curve $y = h(x^{g+1})$ in the $6(g+1)$ $K(\mbf{t})$-points
\begin{equation*}
P_{ij} \coloneqq (\zeta^jt_i, h(u_i)), \quad \; i=1,\dotsc,6, \;\, j=0,\dotsc,g.
\end{equation*}
Counting also the hyperelliptic conjugates, we have $\#X(K(\mbf{t})) \geqslant 12(g+1)$. As in \ref{emp:hyperelliptic}, for $P \in X(K(\mbf{t}))$, we let $\epsilon(P) \coloneqq [2P - D_\infty] \in J(K(\mbf{t}))$, where $J/K(\Bell)$ denotes the Jacobian of $X/K(\Bell)$. The curve $X/K(\Bell)$ admits an automorphism $\theta$ defined by $(x,y) \longmapsto (\zeta x,y)$, and clearly $\langle \theta \rangle \cong \mathbb{Z}/(g+1)\mathbb{Z}$. We write $\overline{\zeta} \in \aut(J)$ to denote the automorphism induced by $\theta$.
\end{emp}
\begin{lem}
The homomorphism $f: \mathbb{Z}[z] \longrightarrow \End(J)$ defined by $z \longmapsto \overline{\zeta}$ induces an injection of rings $\mathbb{Z}[\zeta] \lhook\joinrel\longrightarrow \End(J)$.
\end{lem}
\begin{proof}
Since $\theta$ is of order $g+1$, so is $\overline{\zeta}$. Thus, the kernel of $f$ contains $z^{g+1} - 1$. Since $g+1$ is prime, this polynomial factors into irreducibles as $z^{g+1} - 1 = (z-1)\Phi(z)$, where $\Phi(z) = 1 + z + \dotsb + z^g \in \mathbb{Z}[z]$ is the $(g+1)$-th cyclotomic polynomial. We want to show that $\ker f$ is generated by $\Phi(z)$, for which it suffices to show that $\Phi(z) \in \ker f$ (since $\Phi(z)$ irreducible).
Consider the nice curve $C/K(\Bell)$ defined by the equation $y^2 = \ell(x)$, which is of genus $0$ since $\ell(x)$ is quadratic. We have a natural map $\alpha: X \longrightarrow C$ defined by $(x,y) \longmapsto (x^{g+1},y)$. By abuse of notation, let $\overline{\zeta} \in \aut (\Div(X))$ denote the automorphism induced by $\theta \in \aut(X)$. Then, for a divisor $D \in \Div(X)$, we have
\begin{equation*}
\alpha^*\alpha_*D = D + \overline{\zeta}(D) + \dotsb + \overline{\zeta}^gD = \Phi(\overline{\zeta})(D) \in \Div(X).
\end{equation*}
Passing to Pic's, this implies that the endomorphism $f(\Phi(z)) = \Phi(\overline{\zeta}) \in \End(J)$ coincides with the composition $\pic^0(X) \xlongrightarrow{\alpha_*} \pic^0(C) \xlongrightarrow{\alpha^*} \pic^0(X)$; the desired result now follows since $\pic^0(C) = 0$.
\end{proof}
The above lemma allows us to abuse notation and denote by $\zeta$ (instead of $\overline{\zeta}$) the automorphism of $J$ induced by $\theta \in \aut(X)$.
\begin{thm}\label{thm:kum curves}
The points $\mathfrak{P}_i \coloneqq \epsilon(P_{i0}),\, i=1,\dotsc,6$ generate a free $\mathbb{Z}[\zeta]$-submodule $H\leqslant J(K(\mbf{t}))$ of rank $6$. Thus, we have $\rk X/K(\mbf{t}) \geqslant 6g$.
\end{thm}
\begin{proof}
The claim about $H$ immediately implies the claim about $\rk X(K(\mbf{t}))$, since
\begin{equation*}
[H:\mathbb{Z}] = [H:\mathbb{Z}[\zeta]]\cdot [\mathbb{Z}[\zeta]:\mathbb{Z}] = 6g.
\end{equation*}
Assume for a contradiction, then, that we have a linear dependence relation
\begin{equation}
a_1\mathfrak{P}_1 + \dotsb + a_6\mathfrak{P}_6 = 0 \in J(K(\mbf{t})),
\end{equation}
with each $a_i \in \mathbb{Z}[\zeta]$, and with some $a_p \neq 0$. For each $i =1,\dotsc,6$ and $j=0,\dotsc,g$, let $\sigma_{ij} \in \aut(K(\mbf{t}))$ denote the automorphism defined by
\begin{equation*}
\sigma_{ij}(t_k) =
\begin{cases}
\zeta^j t_k, & \textup{ if } k=i\\
t_k & \textup{ otherwise. }
\end{cases}
\end{equation*}
The group $G \coloneqq \langle \sigma_{ij} \rangle \cong (\mathbb{Z}/(g+1)\mathbb{Z})^6$ is nothing but the Galois group $\textup{Gal}(K(\mbf{t})/K(t_{1}^{g+1},\dotsc,t_6^{g+1}))$. In particular, $G \leqslant \aut_{K(\Bell)}K(\mbf{t})$, so $G$ acts on $X(K(\mbf{t}))$. We have
\begin{equation*}
\sigma_{ij}(P_{k0}) =
\begin{cases}
\theta^j (P_{k0}), & \textup{ if } k=i\\
P_{k0} & \textup{ otherwise. }
\end{cases}
\end{equation*}
The action of $G$ on $X(K(\mbf{t}))$ induces an action of $G$ on $J(K(\mbf{t}))$; if $\overline{\sigma}_{ij} \in \aut(J(K(\mbf{t})))$ denotes the automorphism induced by $\sigma_{ij} \in G$, we have
\begin{equation*}
\overline{\sigma}_{ij}(\mathfrak{P}_k) =
\begin{cases}
\zeta^j \mathfrak{P}_k, & \textup{ if } k=i\\
\mathfrak{P}_k & \textup{ otherwise. }
\end{cases}
\end{equation*}
Now, for each $i =1,\dotsc,6$, consider the endomorphism $\tau_i \coloneqq \overline{\sigma}_{i0} + \dotsb + \overline{\sigma}_{ig}$ of $J(K(\mbf{t}))$, which is clearly also an endomorphism of $H$.
We have
\begin{equation*}
\tau_i(\mathfrak{P}_k) =
\begin{cases}
0, & \textup{ if } k=i\\
(g+1)\mathfrak{P}_k, & \textup{ otherwise. }
\end{cases}
\end{equation*}
Thus, applying $\tau_k$ to $a_1\mathfrak{P}_1 + \dotsb + a_6\mathfrak{P}_6$ kills off the term $a_k\mathfrak{P}_k$, and multiplies all the other terms by $g+1$. By successively applying $\tau_i$ for each $i \in \{1,\dotsc,6\} \setminus \{p \}$, we end up with a relation $a\cdot \mathfrak{P}_p = 0$, where $a = a_p(g+1)^5 \neq 0 \in \mathbb{Z}[\zeta]$. This implies that \emph{$\mathfrak{P}_p$ must be torsion}; indeed, letting $N$ denote the norm $N_{\mathbb{Z}[\zeta]/\mathbb{Z}}(a) \in \mathbb{Z}$, we have $N\cdot \mathfrak{P}_p = 0$. \Cref{lem:algebraic point} then implies that the point $P_{p0} = (t_p,h(t_p)) \in X(K(\mbf{t}))$ is algebraic over $K(\Bell)$, which implies that $u_p = t_p^{g+1}$ is algebraic over $K(\Bell)$. Since $\textup{Gal}(K(\mbf{u})/K(\mbf{m})) \cong S_6$ acts by permutation on $u_1,\dotsc,u_6$, \emph{every} $u_i$ is algebraic over $K(\Bell)$, which implies that each $m_j \coloneqq (-1)^{6-j}s_{6-j}(u_1,\dotsc,u_6)$ is algebraic over $K(\Bell)$. But this is absurd, since $\trdeg_{K(\Bell)} K(\mbf{m}) = 3>0$.
\end{proof}
\begin{emp}\label{emp:proofcyc} \emph{Proof of \Cref{thm:cyc}.} Note that $X_{K(\mbf{t})}/K(\mbf{t})$ is a curve of positive genus $g$, which satisfies $\rk X_{K(\mbf{t})}/K(\mbf{t}) \geqslant 6g$. In order to apply \Cref{cor:rank cor} and conclude $R(K,g) \geqslant 6g$, it remains only to show $X_{K(\mbf{t})}/K(\mbf{t})$ is non-constant, for which it suffices to show that $X/K(\Bell)$ is non-constant. This follows, for instance, from~\cite{newpts}*{Lemma~6.3.1}, which says that the curve $X'/K(l_1)$ defined by $y^2 = x^{2g+2} + l_1x^{g+1} + 1$ is non-constant. Consequently, we have infinitely many specializations (that are nice genus $g$ curves over $K$) of the form $(l_2,l_1,l_0) = (1,a,1)$, with $a \in K$, so we are done.
\end{emp}
\begin{rem}\label{rem:kummer locus}
Recall from \ref{emp:calgen} the $\mathbb{Q}$-variety of $(e,n)$-composite tuples $\mathcal{G}(e,n)\subset \mathbb{A}^{en}_{\mathbb{Q}} = \spec \mathbb{Q}[\mbf{t}_1,\dotsc,\mbf{t}_n]$. The \defi{Kummer locus} is the closed subscheme $\mathcal{K}(e,n) \coloneqq \{s_1(\mbf{t}_1) = \dotsb = s_{e-1}(\mbf{t}_1) = 0 \} \subset \mathcal{G}(e,n).$ The reader can easily verify that for any field $K\supset \mathbb{Q}$, we have
\begin{equation*}
\mathcal{K}(e,n)(K) = \{ (a_{ij}) \in \mathcal{G}(e,n)(K) \mid \prod_{i,j} (x - a_{ij}) = M(x^e) \textup{ for some } M(x) \in K[x] \}.
\end{equation*}
If $K$ contains the $e$-th roots of unity, then $\mathcal{K}(e,n)_K$ is the union of $(e-1)!^n$ copies of $\mathbb{A}^n_K$.
\end{rem}
\begin{rem}\label{rem:brumer}
Consider the curve $X_0/K(\Bell)$ above, given by the equation $y^2 = l_2x^{2g+2} + l_1x^{g+1} + l_0$, which has a subgroup $H \leqslant \aut(X_0)$ of order $2(g+1)$. Above, we constructed \emph{six} ``full'' orbits (under $H$) of $K(\mbf{t})$-points, for a total of $6\cdot 2(g+1) = 12(g+1)$ $K(\mbf{t})$-points. We remark here that the assumption that $g+1$ be prime is used only in bounding the rank of the Jacobian.
Brumer~\cite{capratpts}*{Section~5} considered curves $C_{a,b}/K$ of the form $y^2 = ax^{2g+2} + bx^{g+1} + a$, parametrized by $(a,b) \in K^2$; here, $g\geqslant 2$ is an integer ($g+1$ need not be prime), and $K$ contains the primitive $g+1$-th roots of unity. These curves admit the extra involution $(x,y) \longmapsto (1/x , y/x^{g+1})$, so there is a subgroup $H'\leqslant \aut(C_{a,b})$ with $\#H' = 4(g+1)$. Brumer showed that there are infinitely many choices of $(a,b) \in K^2$ that give rise to an infinite set of curves $C_{a,b}/K$, each having at least \emph{four} full orbits of $K$-points under $H'$ (so $\#C_{a,b}(K)\geqslant 4\cdot 4(g+1) = 16(g+1)$). This yields the current record $N(K,g) \geqslant 16(g+1)$ for number fields $K$ containing the $g+1$-th roots of unity.
It is natural to wonder, then, if our curve $X_0/K(\mbf{t})$ admits a specialization of the form $C_{a,b}/K$. More precisely, if $P$ is a $K$-point of the locally closed subset
\begin{equation*}
W \coloneqq \{ l_0 = l_2 \} \bigcap_{1 \leqslant i,j \leqslant 6} \{((t_it_j)^{g+1} - 1)(t_i^{g+1} - t_j^{g+1}) \neq 0\} \bigcap \{\textup{disc} \;\ell(x^{g+1}) \neq 0\} \subset \mathbb{A}^6(\mbf{t}),
\end{equation*}
then the specialization of $X_0$ at $P$ would be a curve of the form $C_{a,b}/K$ having \emph{six} full orbits (under $H'$) of $K$-points, giving $\#C_{a,b}(K) \geqslant 6\cdot 4(g+1) = 24(g+1)$. So, we ask: \emph{Does $W$ have a $K$-point, or even better, does $W$ contain a $K$-rational variety}? At present, we do not know if $W(K)$ is non-empty.
\end{rem}
|
1,116,691,499,262 | arxiv | \section{Introduction \label{INTRO}}
Long-ranged dispersive interactions, such as the Casimir\cite{Casimir:1948} force or the van der Waals\cite{London:1930} force, originate from electromagnetic field fluctuations. They are present between all types of objects at any distance, regardless of whether there are permanent electric and/or magnetic moments. Despite their universal nature, the dimensionality, dielectric and magnetic response properties, and geometry of the interacting systems can influence the dispersive force in profound ways, thus offering possibilities to tailor the sign, magnitude, and distance dependences of the interaction\cite{Abrikosov:1975,Rodriguez:2011}.
Single layers of graphite have been recently isolated \cite{Novoselov:2004,Novoselov:2005}, and their dispersive forces present not only challenging theoretical problems, but they are also important for graphene based devices\cite{Lin:2010}, \cite{Stoller:2008}. The electromagnetic fluctuation forces in graphene have been considered both at low and high temperature regimes as well as with the inclusion of a chemical potential or a band gap \cite{Dobson:2011,Drosdoff:2011,Sarabadani:2011, Svetovoy:2011, Fialkovsky:2011, Sernelius:2011, Drosdoff:2010, GomezSantos:2009,Dobson:2006}. A distance power law has been obtained for the graphene/graphene Casimir force at the quantum limit, where the force per unit area($F$) goes as the inverse distance($d$) to the fourth power, $F\sim d^{-4}$. At the same time, due to the graphene gapless electronic structure, the fluctuation forces are thermal in nature at room temperature. This is in stark contrast to the Casimir forces in most matter whereupon quantum fluctuation effects are dominant even at relatively high temperatures.
Concerning Casimir interactions in general, the inclusion of spatial dispersion in the response properties of the materials has received significant attention recently, since it was shown that the dispersion may play an important role \cite{Sernelius:2006,Esquivel:2006}. In this work we investigate to what extent spatial dispersion, in combination with modifications in the graphene electronic structure through a nonzero chemical potential or an induced energy gap, affect the graphene-graphene Casimir interaction.
In practice, a finite chemical potential is induced by doping\cite{Brenner:2012,Lin:2012} or by the application of external fields\cite{Novoselov:2004}. Energy band gaps on the other hand can be brought about by the growth of graphene on certain substrates\cite{Zhou:2007}, via adsorption\cite{Yavari:2010} or hydrogenation\cite{Haberer:2010}. The response properties of graphene for the calculations here are taken into account via the conductivity obtained from the linear response to the electromagnetic field\cite{Falkovsky:2007}, from which both the small and large spatial dispersion forces are investigated.
\section{Graphene Conductivity\label{GC}}
An important factor in the Casimir force is the dielectric response properties of the materials involved. For the graphene/graphene system, these properties are described via the optical conductivity calculated within the Kubo\cite{Kubo:1957} formalism. Consider the case when there is a band gap $\Delta$ induced at the graphene Fermi level. Near the K-point of the Brillouin zone, assuming small wave vector excitations, the Hamiltonian can be written as
\begin{equation}
H=
\left(
\begin{array}{cc}
\Delta & v_0(p_x-ip_y) \\
v_0(p_x+ip_y) & -\Delta \\
\end{array}
\right),
\label{GC2}
\end{equation}
where ${\bf p}$ is the two-dimensional momentum operator and $v_0=3\gamma_0 a/2\approx 10^6 m/s$ ($a=0.142\ nm$ is the CC distance and $\gamma_0=2.4\ eV$ is the nearest neighbor tight binding overlap integral\cite{Reich:2002,Blinowski:1980}). The Hamiltonian, Eq.(\ref{GC2}), corresponds to a two-band model with eigenvalues and eigenvectors given respectively as:
\begin{equation}
E_s=s\sqrt{\Delta^2+(v_0\hbar k)^2}=s E,
\label{GC3}
\end{equation}
\begin{equation}
|{\bf k},s\rangle=\frac{E-s\Delta}{\sqrt{E_0^2+(E-s\Delta)^2}}\left(
\begin{array}{c}
\frac{E_0e^{-i\theta}}{E-s\Delta}\\
s\\
\end{array}
\right),
\label{GC4}
\end{equation}
where the wave function is
\begin{math}
\psi({\bf r},s)=|{\bf k},s\rangle e^{i{\bf k}\cdot {\bf r}},
\end{math} the two-dimenional wave vector is
${\bf k}=k_x{\bf \hat{x}}+k_y{\bf \hat{y}}$ ($\hat{x},\ \hat{y}$ are unit vectors), $e^{i\theta}= (k_x+ik_y)/k$, $E_0=v_0\hbar k$, and $s=\pm$. If $\Delta=0$, one recovers the gapless Dirac-like model suitable for perfect graphene. Using this two-band model and applying linear response theory, the dyadic two dimensional conductivity is found\cite{Falkovsky:2007} to be
\begin{widetext}
\begin{eqnarray}
\label{GC1}
&&\overleftrightarrow{\sigma}(\omega,{\bf q})=\frac{i\hbar e^2}{\pi^2}\times
\sum_{s=\pm}\int d^2k\frac{{\bf v}_{ss}{\bf v}_{ss}\left(f_0[E_s({\bf k}-{\bf q}/2)]-f_0[E_s({\bf k}+{\bf q}/2)]\right)}{[E_s({\bf k}+{\bf q}/2)-E_s({\bf k}-{\bf q}/2)][\hbar\omega-E_s({\bf k}+{\bf q}/2)+E_s({\bf k}-{\bf q}/2)]} +\nonumber \\
&&\frac{2i\hbar^2 e^2\omega}{\pi^2}\int d^2k\frac{{\bf v}_{+,-}{\bf v}_{-,+}\left(f_0[E_+({\bf k}-{\bf q}/2)]-f_0[E_-({\bf k}+{\bf q}/2)]\right)}{[E_-({\bf k}+{\bf q}/2)-E_+({\bf k}-{\bf q}/2)][\hbar^2\omega^2-[E_-({\bf k}+{\bf q}/2)-E_+({\bf k}-{\bf q}/2)]^2]},
\end{eqnarray}
\end{widetext}
\begin{floatequation}
\mbox{\textit{see eq.~\eqref{GC1}}}
\end{floatequation}
where $f_0(E_s)=1/(e^{(E_s-\mu)/k_BT}-1)$ is the equilibrium Fermi distribution function, $k_B$ is the Boltzmann constant and $T$ is the temperature. The velocity matrix elements are ${\bf v}_{ss'}=\langle s,{\bf k}|{\bf\hat{v}}|{\bf k},s'\rangle$, where ${\bf\hat{v}}=\frac{\partial H}{\partial {\bf p}}$. The first term in Eq.(\ref{GC1}) accounts for intraband transitions, while the second one accounts for interband transitions. From hence forth, imaginary frequency will be used, which is the relevant quantity for calculating Casimir forces.
In the limit when $\hbar v_0 q\rightarrow 0$, the optical conductivity of graphene, $\sigma_0(i\omega)$, is isotropic and it is found as:
\begin{eqnarray}
&&\sigma_{0,intra}(i\omega)=\nonumber \\
&&\frac{e^2\ln(2)}{\hbar^2\pi\omega\beta}+\frac{e^2}{\hbar^2\pi\omega\beta}\ln(\cosh(\Delta\beta)+\cosh(\mu\beta))-\nonumber\\
&&\frac{e^2\Delta^2}{\hbar^2\pi\omega}\int_{\Delta}^\infty\frac{dE}{E^2}\frac{\sinh(E\beta)}{\cosh(\mu\beta)+\cosh(E\beta)},
\label{GC6}
\end{eqnarray}
\begin{eqnarray}
&&\sigma_{0,inter}(i\omega)=\nonumber \\
&&\frac{e^2\omega}{\pi}\int_\Delta^\infty dE\frac{\sinh(E\beta)}{\cosh(\mu\beta)+\cosh(E\beta)}\frac{1}{(\hbar\omega)^2+4E^2}+\nonumber \\
&&\frac{e^2\omega\Delta^2}{\pi}\int_\Delta^\infty\frac{dE}{E^2}\frac{\sinh(E\beta)}{\cosh(\mu\beta)+\cosh(E\beta)}\frac{1}{(\hbar\omega)^2+4E^2},
\label{GC7}
\end{eqnarray}
where $\beta=1/k_BT$. One notes that when $\mu=0$ and $\Delta=0$, for $k_BT\ll\hbar\omega$, $\sigma_0(i\omega)$ acquires a universal value $\sigma_0(i\omega)\approx \sigma_0\equiv e^2/(4\hbar)$. In Fig(\ref{fig1a}), results are shown for $\sigma_0(i\omega)$ as a function of frequency. The largest differences appear at small $\omega$, where the conductivity is most significantly reduced when $\Delta\neq 0$ ($\mu=0$) and augmented when $\mu\neq 0$ ($\Delta=0$). For larger $\omega$, the conductivity approaches $\sigma_0$.
\begin{figure}[ht!]
\begin{center}
\subfigure{
\label{fig1a}
\includegraphics[width=0.3\textwidth]{fig1a.JPG}
}
\subfigure{
\label{fig1c}
\includegraphics[width=0.3\textwidth]{fig1c.JPG}
}
\end{center}
\caption{(a) The isotropic conductivity $\sigma_{0}(i\omega)$ as a function of frequency (in eV) for different values of $\Delta$ and $\mu$ at $T=300$ $K$. (b) $\sigma_{xx}(i\omega)$, as a function of $q$ for finite $\Delta\ \rm{and}\ \mu$ with $\hbar\omega= 0.162\ eV$ corresponding to $T=300\ K$.}
\label{fig1}
\end{figure}
If spatial dispersion is taken into account, $\overleftrightarrow{\sigma}$ is not isotropic. For graphene, however, we find that $\sigma_{xx}$ and $\sigma_{yy}$ have very similar magnitude and behavior over large wave vector and frequency regions, therefore only $\sigma_{xx}$ is shown in Fig(\ref{fig1c}). The conductivity, $\sigma_{xx}$, starts changing in a more pronounced way for $q > 10^6\ cm^{-1}$ .
It is noted that $\sigma_{xx}$ is the conductivity in the direction of the electric field parallel to the plane composing graphene sheet while $\sigma_{yy}$ is the direction perpendicular to the electric field.
\section{Graphene Casimir Force and Dispersion\label{DCF}}
The Casimir force per unit area\cite{Abrikosov:1975} between two graphene sheets separated by a distance $d$ is
\begin{widetext}
\begin{equation}
\label{DCF0}
F=-\frac{ik_BT}{2\pi}\sum_{n=-\infty}^{\infty}
\int_0^\infty h(i|\omega_n|) q dq
\left\{\left[\frac{e^{-2ih(i|\omega_n|)d}}{\rho_{E}(i|\omega_n|)^2}-1\right]^{-1}+
\left[\frac{e^{-2ih(i|\omega_n|)d}}{\rho_{B}(i|\omega_n|)^2}-1\right]^{-1}\right\},
\end{equation}
\end{widetext}
\begin{floatequation}
\mbox{\textit{see eq.~\eqref{DCF0}}}
\end{floatequation}
where $h(i|\omega_n|)=i\sqrt{\epsilon(i|\omega_n|)\mu(i|\omega_n|)(\omega_n/c)^2+q^2}$ with $\omega_n=2\pi nk_BT/\hbar$ and ${\bf q}$ is the two-dimensional wave vector in the $xy$-plane. $\rho_{E,B}$ are the generalized reflection coefficients due to the transverse electric ({\bf E}) and magnetic ({\bf B}) field modes, which are given in terms of the two dimensional conductivity by
\begin{eqnarray}
\rho_E(i\omega)&=&-\frac{2\pi\sigma_{yy}(i\omega,{\bf q})\omega/[h(i\omega)c^2]}{1+2\pi\sigma_{yy}(i\omega,{\bf q})\omega/[h(i\omega)c^2]},\nonumber \\
\rho_B(i\omega)&=&\ \ \ \frac{2\pi\sigma_{xx}(i\omega,{\bf q})h(i\omega)/\omega}{1+2\pi\sigma_{xx}(i\omega,{\bf q})h(i\omega)/\omega}
\label{DCF2}.
\end{eqnarray}
\subsection{{\bf Thermal Limit\label{FSP}}}
The thermal fluctuation forces are first considered, which correspond to the zero Matsubara term ($n=0$) of Eq.(\ref{DCF0}) in the strong dispersion limit, where $\hbar\omega\ll k_BT,\ \hbar v_0 q$. In most materials purely thermal fluctuation effects are important at distances larger than their characteristic thermal wavelength, $\lambda_T=\hbar c/(k_BT)\sim 7\ \mu m$. Instead, due to the Dirac-like Hamiltonian of graphene and its two dimensional nature, the characteristic thermal wavelength is significantly reduced - $\sim\lambda_T/200$. As a result, thermal fluctuation forces become important at relatively small scales\cite{GomezSantos:2009}, $d\approx 30\ nm$. Therefore the quasi-static response is a reasonable approximation for a large distance range.
In the quasi-static response limit of $\hbar \omega\ll\hbar v_0 q\ll k_BT$, the $\sigma_{xx}$ is
\begin{equation}
\sigma_{xx}(i\omega,{\bf q})=\frac{\omega q_s}{2\pi q^2}.
\label{FSP1}
\end{equation}
This corresponds to the longitudinal plasmon excitations in the graphene static dielectric function $\epsilon(q)=1+q_s/q$ obtained via the Random Phase Approximation(RPA) method \cite{Ando:2006}. For gapped graphene\cite{Svetovoy:2011},
\begin{eqnarray}
q_s(\Delta)&=&\frac{4\alpha c}{\hbar v_0^2}\left\{2k_BT\ln\left(2\cosh\left(\frac{\Delta}{2k_BT}\right)\right)\right\}
-\nonumber \\
&&\frac{4\alpha c}{\hbar v_0^2}\left\{\Delta\tanh\left(\frac{\Delta}{2k_BT}\right)\right\},
\label{FSP11}
\end{eqnarray}
where $\alpha\approx 1/137$ is the fine structure constant.
Using Eq.(\ref{DCF0}) and Eq.(\ref{FSP1}) the thermal fluctuation stress between two graphene sheets is obtained to be
\begin{equation}
F(d)=\frac{k_BT}{16\pi d^3}\int_0^\infty\frac{x^2dx}{e^x(x+2dq_s)^2/(2dq_s)^2-1}.
\label{FSP5}
\end{equation}
One finds that as $q_sd\gg1$, the stress approaches the thermal stress between two perfect metals, i.e., $F_T=k_BT\zeta(3)/(8\pi d^3)$.
A similar calculation can be executed for the problem of finding the thermal fluctuation force $f(d)$ between a graphene sheet and a polarizable atom\cite{Caride:2005} with polarizability $\chi(\omega)$.
\begin{widetext}
\begin{equation}
\label{FSP6}
f(d)=-k_BT\sum_{n=-\infty}^{\infty}\chi(i|\omega_n|)\int_0^\infty kdkh^2(i|\omega_n|)\left[2\rho_B(i|\omega_n|)-\left(\frac{\omega_n}{h(i|\omega_n|)c}\right)^2\left[\rho_E(i|\omega_n|)-\rho_B(i|\omega_n|)\right]\right]e^{2idh(i|\omega_n|)}.
\end{equation}
\end{widetext}
\begin{floatequation}
\mbox{\textit{see eq.~\eqref{FSP6}}}
\end{floatequation}
Using Eq.(\ref{FSP1}), the thermal fluctuation force, $n=0$ term, in this case is
\begin{equation}
f(d)=\frac{k_BT\chi(0)}{8d^4}\int_0^\infty x^3dxe^{-x}\left[\frac{2dq_s}{x+2dq_s}\right],\nonumber
\end{equation}
\begin{eqnarray}
f(d)&=&\frac{k_BT\chi(0)}{8d^4}\left[(2dq_s)^3-(2dq_s)^2+2(2dq_s)\right]-\nonumber \\
&&\frac{k_BT\chi(0)}{8d^4}\left[(2dq_s)^4e^{2dq_s}E_1(2dq_s)\right].
\label{FSP7}
\end{eqnarray}
where $E_{1}(x)$ is the exponential integral. Again as $q_sd\gg 1$, the force between atom and graphene becomes $f_T=3k_BT\chi(0)/(4d^4)$, which is the thermal interaction between an atom and a perfectly conducting plate.
The normalized forces,$F/F_T$ and $f/f_T$, as a function of $2dq_s$ are shown in Fig.(\ref{fig4}). One finds that for large distances spatial dispersion becomes less important, and the force goes to the one expected for ideal metallic surfaces. For short distances the electron screening shields the force such that it goes to zero at very small distances. From Fig.(\ref{fig5a}) it is seen that dispersion is significant when $q_sd_\approx 10$ or less. For the case of gapless graphene, this corresponds to a distance of $d\approx 20\ nm$ at room temperature, in which case higher Matsubara frequencies need to be considered. Hence, for the case of graphene, dispersion does not play a significant role in the region where the fluctuation force is thermal.
Spatial dispersion becomes important for graphene at larger distances, when $q_s$ decreases.
It has been noted that while for good conductors and dielectrics thermal Van-der-Waal/Casimir forces are not heavily influenced by dispersion effects, the latter may have a strong influence on the force for poor conductors \cite{Pitaevskii:2008,Svetovoy:2008}. A similar result occurs for graphene, where upon the conduction properties of the material are decreased by introducing a band gap.
\begin{figure}[ht]
\centering
\subfigure{
\label{fig5a}
\includegraphics[width=0.3\textwidth]{fig4.JPG}
}
\subfigure{
\label{fig5b}
\includegraphics[width=0.3\textwidth]{fig5.JPG}
}
\caption{(a) Thermal stress between two graphenes normalized to the thermal stress between two metals and thermal force between graphene and atom normalized to the thermal force between metal and an atom. (b) $q_s$ normalized to the value at zero band gap $q_{s0}$ as a function of the energy band gap $\Delta$ for different temperatures, $T=200\ K$, $T=300\ K$, $T=400\ K$.}
\label{fig4}
\end{figure}
By inducing an energy band gap, the effect of spatial dispersion in graphene is increased by effectively decreasing $q_s$, as shown in Fig.(\ref{fig5b}).
It is clear that the influence of dispersion increases as $q_s$ gets small with the effect that the thermal fluctuation forces between graphene sheets are substancially decreased. For example for $\Delta=0.05\ eV$, a typical value for the gap, one obtains a distance for dispersion to have strong effects of $d\approx 40\ nm$, within the range where thermal fluctuation forces are dominant. Of course, by inducing a sufficiently large band gap, thermal effects cease from becoming dominant. This effect is analogous to 3-D materials, whereupon the influence of dispersion in the fluctuation forces of a conductor become strong were it to become a poorer conductor.
\subsection{${\bf T=0}$ ${\bf K}$}
When the temperature is zero, the summation in $F$ (Eq.(\ref{DCF0})) is substituted by an integral, $\sum_n=[\hbar/(2\pi k_BT)]\int_0^\infty d\omega$. The $T=0$ $K$ case corresponds to purely quantum mechanical contributions to the Casimir stress. If the graphene conductivity is given by the universal value $\sigma_0=e^2/(4\hbar)$, the graphene/graphene Casimir force per unit area is found as \cite{Dobson:2006,GomezSantos:2009,Drosdoff:2010}
\begin{equation}
F_g=\frac{3e^2}{32\pi d^4}=F_0\kappa(\alpha)
\label{}
\end{equation}
where $F_0=\pi^2\hbar c/(240 d^{4})$ is the force per unit area between two perfectly conducting metallic sheets and $\kappa(\alpha)=720\alpha/(32\pi^3)$ with $\alpha=1/137$ being the fine structure constant. It is interesting that the force in this limit does not depend on any quantum mechanical characteristics nor the speed of light. Comparing $F_g$ to $F_0$ shows that the distance dependence of the graphene/graphene force is the same as for metallic plates in the retarded regime, however, the magnitude is reduced by an order of the fine structure constant. Incidentally, this differs from the non-retarded ``2D'' metal and doped graphene result--$F\sim d^{-7/2}$\cite{Sernelius:2011,Dobson:2006}
The presence of $\mu$ or $\Delta$ alters the graphene conductivity, which can change the Casimir interaction significantly. Using the isotropic $\sigma_0(i\omega)$ from Eqs.(\ref{GC6}, \ref{GC7}), it is found that $F/F_0$ increases almost linearly as a function of $d$ when $\mu\neq 0$, while for $\Delta\neq 0$ and $\mu\neq 0$ the normalized Casimir stress has nonlinear dependences - Fig.(\ref{fig4a}). This happens because at shorter distances the effect of the band gap is more pronounced, while at larger distances the finite chemical potential has a stronger effect. It is found that $F/F_0$ can be diminished by increasing the band gap of the graphene sheets ( Fig.(\ref{fig4b})) or it can be enhanced by increasing the chemical potential ( Fig.(\ref{fig4c})). These changes can be quite substantial, providing that the system supports large $\mu$ or $\Delta$.
\begin{figure}[ht!]
\begin{center}
\subfigure{
\label{fig4a}
\includegraphics[width=0.3\textwidth]{fig4a.JPG}
}
\subfigure{
\label{fig4b}
\includegraphics[width=0.3\textwidth]{Graph4b.JPG}
}
\subfigure{
\label{fig4c}
\includegraphics[width=0.3\textwidth]{fig4c.JPG}
}
\end{center}
\caption{(a) Casimir stress normalized to $F_0$ at T=0 and calculated using the isotropic conductivity $\sigma_0(i\omega)$ for different $\Delta$ and $\mu$. The flat line indicates the normalized stress obtained via the universal conductivity $\sigma_0$; (b) Casimir stress normalized to $F_0$ at T=0 as a function of $\Delta$ calculated using $\sigma_0(i\omega)$ and $\sigma(i\omega,{\bf q})$ ($\mu=0$); (c) Casimir stress normalized to $F_0$ as a function of $\mu$ calculated using $\sigma_0(i\omega)$ and ${\bf \sigma}(i\omega,{\bf q})$ ($\Delta=0$) }
\label{fig4d}
\end{figure}
It is interesting to consider to what extent spatial dispersion affects the graphene/graphene Casimir interaction. In the absence of a band gap or chemical potential, dimensional analysis dictates that the force may be written as
\begin{equation}
F=F_0g(\alpha,v_0/c),
\label{DCF2A}
\end{equation}
where the scaling factor $g(\alpha,v_0/c)$ now depends on the fine structure constant, and the ratio of $v_0/c$. If one uses Eq.(\ref{GC1}), the scaling factor is evaluated as $f(\alpha,v_0/c)\approx 0.0043$. Note that the distance dependence is still $F\sim d^{-4}$. Comparing the scaling factors $\kappa$ and $g$ shows that the inclusion of spatial dispersion reduces the graphene/graphene Casimir attraction by $\approx 20\ \%$. If one were to approximate the conductivity of graphene via RPA, then inclusion of dispersion would reduce the graphene/graphene interaction by $\approx 9\ \%$. Figs.(\ref{fig4b},\ref{fig4c}) further show that dispersion effects are rather insignificant as $\mu$ and $\Delta$ increase.
\subsection{${\bf T\neq 0 K}$}
Finally, the full finite temperature Casimir interaction is considered. Eq.(\ref{DCF0}) indicates that the largest contribution to the force occurs when $qd\sim 1$. On the other hand it was shown that for finite frequencies, the conductivity varies sharply as a function of wavenumber when $q_s>10^6\ cm^{-1}$ - Fig(\ref{fig1c}). Therefore, dispersion effects start becoming important for Matsubara terms in Eq.(\ref{DCF0}) greater than zero at a distance of $d_s\sim 1/q_s=0.01\ \mu m$. Fig.(\ref{fig3b1}) shows the Casimir interaction force per unit area between two graphene sheets where spatial dispersion is only included in the force for the $n=0$ thermal term and also where spatial dispersion is included for all Matsubara frequency terms. If the distance is reduced much further the force reduces back to the zero temperature result.
\begin{figure}[ht!]
\begin{center}
\subfigure{
\label{fig3b1}
\includegraphics[width=0.3\textwidth]{fig3a1.JPG}
}
\subfigure{
\label{fig3b}
\includegraphics[width=0.3\textwidth]{fig3a.JPG}
}
\end{center}
\caption{(a) Casimir stress normalized to the idealized stress $F_0$ at $T=300\ K$ with and without the inclusion of dispersion for Matsubara terms greater than zero. (b) Casimir stress normalized to the idealized stress $F_0$ at $T=300\ K$.}
\label{fig3}
\end{figure}
Much more influential parameters to the Casimir interaction are the induced chemical potential or energy band gap. The effects are displayed in Fig.(\ref{fig3b}), which shows the stress between graphenes for different band gaps, and chemical potentials. In addition, the role of the chemical potential is to increase the magnitude of the force while the energy band gap decreases it (not shown graphically). The characteristic behavior is similar as for the quantum $T=0$ $K$ limit.
\section{Conclusion\label{CONC}}
The Casimir interaction between two graphene sheets has been investigated at room and zero temperatures by using linear response theory with a conductivity response function in order to determine how spatial dispersion affects the interaction. It was found that these effects are not of major importance, either at large or low temperatures. The Casimir interaction may be well approximated by using a frequency only dependent conductivity. Spatial dispersion corrections start appearing at distances of $d\sim 0.01 \mu m$ at room temperature. A finite chemical potential can enhance the Casimir interaction substantially and further inhibit spatial dispersion. If a band gap is induced at finite temperature, however, graphene exhibits similar characetristics to a poor metal and dispersion effects start becoming more important, that is the thermal fluctuation forces are reduced by the presence of electronic charge screening.
In the present work, dispersion has been investigated using the 2-band model. Yet
at shorter distances, in the order of $10\ {\rm nm}$ or less, higher band transitions become important for the Van-der Waals/Casimir interaction. The inclusion of such effects without spatial dispersion have already been investigated in a previous work\cite{Drosdoff:2011}.
\section{Acknowledgements}
We acknowledge financial support from the Department of Energy under contract DE-FG02-06ER46297. I. V. B. acknowledges support from NSF-HRD-0833184 and ARO-W911NF-11-1-0189.
|
1,116,691,499,263 | arxiv | \section{Introduction}\label{RSMP3DDF_s_I}
The accurate prediction of 3-D turbulent flow in geometrically complex ducts is important in many practical applications,
including aerospace \cite{Gerolymos_Joly_Mallet_Vallet_2010a}, process \cite{AnxionnazMinvielle_Cabassud_Gourdon_Tochon_2013a}
and nuclear \cite{Chang_Tavoularis_2007a} engineering,
and agrofood industry \cite{Aloui_Berrich_Pierrat_2011a}. These flows can be particularly complex, and turbulence structure may be influenced by
various mechanisms, including 3-D boundary-layer entrainment \cite{Kovasznay_Kibens_Blackwelder_1970a}, secondary flows \cite{Bradshaw_1987a},
flow separation \cite{Simpson_1989a}, especially 3-D \cite{Delery_2001a}, and important streamline curvature \cite{So_Mellor_1978a},
associated with the presence of convex and concave bends \cite{Smits_Young_Bradshaw_1979a}. Therefore, in a \tsn{RANS} (Reynolds-averaged Navier-Stokes) framework \cite{Rumsey_2010a},
differential full Reynolds-stress models (\tsn{RSM}s) are an appropriate choice \cite{Hanjalic_1994a}, in an effort to include terms in the model that
account for all these mechanisms, especially if one considers not only the prediction of the mean flow, but also of the detailed Reynolds-stress field \cite{Yakinthos_Vlahostergios_Goulas_2008a}.
In a recent study \cite{Gerolymos_Joly_Mallet_Vallet_2010a} of a double-\tsn{S}-shaped duct intake, typical of unmanned combat air vehicles (\tsn{UCAV}s), comparison of
\tsn{RSM} predictions with available measurements highlighted the importance of the closure for the rapid part of the velocity/pressure-gradient tensor \smash{$\Pi_{ij}:=-\overline{u_i'\partial_{x_j}p'}-\overline{u_j'\partial_{x_i}p'}$}
(where $u_i\in\{u,v,w\}$ are the velocity-components in the Cartesian frame $x_i\in\{x,y,z\}$, $p$ is the pressure, \smash{$\overline{(\cdot)}$} denotes Reynolds (ensemble) averaging,
and $(\cdot)'$ denotes Reynolds-fluctuations) in successfully predicting the complex 3-D flow structure dominated by 2 pairs of contrarotating streamwise vortices.
To improve our understanding of the predictive capability, but also of limitations, of \tsn{RSM}s applied to the computation of streamwise-developing 3-D duct flows,
it seemed worthwhile to study 3 configurations, where the effects of different
mechanisms could be assessed separately, or at least sequentially: (a) developing flow in a square duct \cite{Gessner_Emery_1981a}, (b) flow in a circular-to-rectangular (\tsn{C}-to-\tsn{R}) transition duct \cite{Davis_Gessner_1992a},
and (c) separated flow in a circular diffusing \tsn{S}-duct \cite{Wellborn_Reichert_Okiishi_1994a}.
These are highly anisotropic and inhomogeneous 3-D flows, driven by mechanisms that are not modelled in linear eddy-viscosity closures, and are therefore
well suited for the assessment of anisotropy-resolving closures \cite{Leschziner_2000a}.
In turbulent fully-developed (streamwise-invariant in the mean) flow in a straight square duct \cite{Gessner_Jones_1965a} the anisotropy of the diagonal stresses,
$\overline{v'^2}$ and $\overline{w'^2}$, in the crossflow plane $yz$ \cite{Bradshaw_1987a}, but also the inhomogeneity of the gradients of the secondary shear-stress $\overline{v'w'}$ \cite[(3), p. 378]{Brundrett_Baines_1964a},
trigger secondary ($\perp x$) flow, associated with streamwise vorticity \cite{Brundrett_Baines_1964a}. The Gessner and Emery \cite{Gessner_Emery_1981a} test-case is further complicated by the streamwise evolution of the very thin inflow boundary-layers,
on the duct walls, which grow streamwise, until they interact and fill the entire duct, resulting in fully-developed (streamwise-invariant in the mean) flow.
Previous \tsn{RSM} computations of this flow \cite{So_Yuan_1999a,Gerolymos_Sauret_Vallet_2004a,Vallet_2007a}
illustrated the difficulty to correctly predict the streamwise development of the centerline velocity $\bar u_\tsn{CL}$,
but also, near the duct's exit where the flow reaches a fully-developed state,
the underestimation of the secondary velocity along the corner bisector (diagonal); this underestimation of the secondary-flow velocities is also observed in fully-developed flow predictions \cite{PetterssonReif_Andersson_2002a}.
Notice that, in fully-developed turbulent square-duct flow, secondary "{\em velocities $\cdots$ are found to be smaller than the root-mean-square turbulent velocity}" \cite[p. 376]{Brundrett_Baines_1964a},
and, furthermore, "{\em secondary-flow velocities, when nondimensionalized with either the bulk velocity {\rm ($\bar u_\tsn{B}$)} or the axial mean-flow velocity at the channel centerline {\rm ($\bar u_\tsn{CL}$)} decrease for an increase
in Reynolds number}" \cite[p. 689]{Gessner_Jones_1965a}.
The So-Yuan \cite{So_Yuan_1999a} wall-normal-free (\tsn{WNF}) \tsn{RSM} slightly underestimates the centerline velocity peak \cite[Fig. 14, p. 51]{So_Yuan_1999a},
while results with different \tsn{WNF-RSM} variants \cite{Gerolymos_Sauret_Vallet_2004a,Vallet_2007a}
demonstrated the sensitivity of the prediction of the $x$-wise development of $\bar u_\tsn{CL}$ to the closures for both $\Pi_{ij}$ \cite{Gerolymos_Sauret_Vallet_2004a} and turbulent diffusion \cite{Vallet_2007a}.
Finally, the wall-geometry-dependent Launder-Shima \cite{Launder_Shima_1989a} \tsn{RSM} was found to perform poorly for this type of flows \cite{Gerolymos_Sauret_Vallet_2004a},
despite a slight improvement when using its \tsn{WNF} version.
Contrary to turbulence-driven secondary flows \cite[Prandtl's second kind]{Bradshaw_1987a}, pressure-driven secondary flows \cite[Prandtl's first kind]{Bradshaw_1987a}
are generally much stronger \cite{Demuren_1991a}. In the \tsn{C}-to-\tsn{R} transition duct studied by Davis and Gessner \cite{Davis_Gessner_1992a}, the curvature of the walls in the transition part of the duct induces pressure-gradients
in the crossflow plane $yz$ \cite[Fig. 14, p. 373]{Davis_Gessner_1992a}, driving relatively strong secondary flows that develop into 2 contrarotating vortex pairs.
The cross-sectional area of the duct varies in the divergent/convergent transition part of the duct \cite[Fig. 4, p. 242]{Lien_Leschziner_1996a},
and this further complicates the flow, although the diverging part of the duct was sufficiently long to exclude separation \cite{Davis_Gessner_1992a}. Previous \tsn{RSM} computations for this configuration
were reported by Sotiropoulos and Patel \cite{Sotiropoulos_Patel_1994a} with a variant of the Launder-Shima \cite{Launder_Shima_1989a} \tsn{RSM},
by Lien and Leschziner \cite{Lien_Leschziner_1996a} with a zonal Gibson-Launder \cite{Gibson_Launder_1978a} \tsn{RSM} coupled with a nonlinear $\mathrm{k}$-$\varepsilon$ model near the wall, and by
Craft and Launder \cite{Craft_Launder_2001a} with their two-component limit (\tsn{TCL}) \tsn{RSM}.
The detailed comparisons of model predictions with available experimental measurements presented in \cite{Sotiropoulos_Patel_1994a} showed quite satisfactory agreement, both for the mean-flow and for the Reynolds-stresses,
with the single exception of the Reynolds-stresses at the last measurement station, located 2 inlet diameters downstream of the end of the \tsn{C}-to-\tsn{R} transition,
where computations do not predict the measured increase of turbulence levels, compared to the previous measurement station located exactly at the end of the \tsn{C}-to-\tsn{R} transition.
The diffusing \tsn{S}-duct, that was experimentally investigated by Wellborn \etal\ \cite{Wellborn_Reichert_Okiishi_1994a}, combines centerline curvature and cross-sectional area increase,
both of which induce streamline curvature, with associated crossflow pressure-gradients which generate significant secondary flows. This configuration is further complicated by the strong adverse streamwise
pressure-gradient, related to the streamwise-diverging cross-sectional area of the duct, which induces a large separated-flow zone. The presence of several interacting mechanisms renders this test-case a difficult challenge,
even for the prediction of the mean-flow velocity and total-pressure fields \cite{Harloff_Smith_Bruns_DeBonis_1993a}. Previous \tsn{RSM} computations were reported by Vallet \cite{Vallet_2007a},
who found that the predictive quality of the models depended mainly on the ability of the redistribution closure to correctly predict separation.
The second-moment closure (\tsn{SMC}) that was assessed in the present work is the \tsn{GLVY} \tsn{RSM} \cite{Gerolymos_Lo_Vallet_Younis_2012a},
which is the final result of previous research \cite{Gerolymos_Vallet_2001a,Gerolymos_Sauret_Vallet_2004a,Sauret_Vallet_2007a,Vallet_2007a}
on the development of wall-normal-free (\tsn{WNF}) \tsn{RSM}s with quasi-linear closure for the rapid part of $\Pi_{ij}$.
To put the comparisons with measurements into perspective, results were also presented for the \tsn{GV} \tsn{RSM} \cite{Gerolymos_Vallet_2001a},
the \tsn{WNF--LSS} \tsn{RSM} \cite{Gerolymos_Sauret_Vallet_2004a}, and with the baseline Launder-Sharma $\mathrm{k}$-$\varepsilon$ model \cite{Launder_Sharma_1974a}.
All of the computations were run specifically for the present assessment, carefully adjusting the boundary-conditions separately for each model,
to obtain the best possible match with the experimental data at the first available measurement plane.
The \tsn{RSM}s used in the present work are briefly reviewed in \parrefnp{RSMP3DDF_s_TCsFS}, with particular emphasis on differences between modelling choices,
and their implications. In \parrefnp{RSMP3DDF_s_A} computational results using the various models are compared with available experimental measurements.
Conclusions from the present results, and recommendations for future research, are summarized in \parrefnp{RSMP3DDF_s_Cs}.
\section{Turbulence closures and flow solver}\label{RSMP3DDF_s_TCsFS}
All measurements were performed in airflow, and a compressible aerodynamic solver was used in the computations.
The square \cite[$\bar M_\tsn{CL}\sim0.05$]{Gessner_Emery_1981a} and \tsn{C}-to-\tsn{R} \cite[$\bar M_\tsn{CL}\sim0.1$]{Davis_Gessner_1992a} ducts test-cases were at sufficiently low Mach-number for the flow
to be essentially incompressible ($\bar M_\tsn{CL}$ is a typical centerline Mach number), whereas in the \tsn{S}-duct high-subsonic flow conditions prevail \cite[$\bar M_\tsn{CL}\sim0.6$]{Wellborn_Reichert_Okiishi_1994a}.
Obviously in all of the previous cases, density fluctuations have negligible influence \cite{Bradshaw_1977a}, so that Favre (used in \parrefnp{RSMP3DDF_s_TCsFS_ss_TCs}) or Reynolds averages are, for practical purposes, equivalent.
The flow is modelled by the Reynolds-averaged Navier-Stokes (\tsn{RANS}) equations \cite{Gerolymos_Vallet_2001a,Vallet_2008a},
coupled with the appropriate modelled turbulence-transport equations \parrefsab{RSMP3DDF_s_TCsFS_ss_TCs}
{RSMP3DDF_s_TCsFS_ss_FS}.
All computations were performed for air thermodynamics \cite{Vallet_2008a}.
\subsection{Turbulence closures}\label{RSMP3DDF_s_TCsFS_ss_TCs}
Details on the development of the \tsn{RSM}s used in the present work, can be found in the original papers \cite{Gerolymos_Vallet_2001a,Gerolymos_Sauret_Vallet_2004a,Gerolymos_Lo_Vallet_Younis_2012a}.
They are summarized below for completeness, in a common representation which highlights differences in the closure choices between different models.
Define
\begin{subequations}
\label{Eq_RSMP3DDF_s_TCsFS_ss_TCs_001}
\begin{align}
r_{ij}:=\dfrac{1}{\bar\rho}\overline{\rho u_i''u_j''} \;\; ; \;\;
\mathrm{k}:=\tfrac{1}{2}r_{\ell\ell} \;\; ; \;\;
a_{ij}:=\dfrac{r_{ij}}{\mathrm{k}}-\tfrac{2}{3}\delta_{ij}
\label{Eq_RSMP3DDF_s_TCsFS_ss_TCs_001a}\\
A_2:=a_{ik}a_{ki} \;\; ; \;\;
A_3:=a_{ik}a_{kj}a_{ji} \;\; ; \;\;
A:=1-\tfrac{9}{8}(A_2-A_3)
\label{Eq_RSMP3DDF_s_TCsFS_ss_TCs_001b}\\
\varepsilon=:\varepsilon^*+2\breve\nu\dfrac{\sqrt{\mathrm{k}}}{\partial x_\ell}
\dfrac{\sqrt{\mathrm{k}}}{\partial x_\ell}\;\; ; \;\;
\ell_\tsn{T} :=\dfrac{\mathrm{k}^\frac{3}{2}}{\varepsilon} \;\; ; \;\;
\ell_\tsn{T}^*:=\dfrac{\mathrm{k}^\frac{3}{2}}{\varepsilon^*} \;\; ; \;\;
Re_\tsn{T} :=\dfrac{\mathrm{k}^2 }
{\breve\nu\varepsilon} \;\; ; \;\;
Re_\tsn{T}^* :=\dfrac{\mathrm{k}^2 }
{\breve\nu\varepsilon^*}
\label{Eq_RSMP3DDF_s_TCsFS_ss_TCs_001c}\\
\breve\mu:=\mu_{\rm Sutherland}(\tilde T) \;\; ; \;\;
\breve\nu:=\frac{\breve\mu}{\bar\rho} \;\; ; \;\;
\breve S_{ij}:=\tfrac{1}{2}\left(\dfrac{\partial\tilde u_i}{\partial x_j}
+\dfrac{\partial\tilde u_j}{\partial x_i}\right)
\label{Eq_RSMP3DDF_s_TCsFS_ss_TCs_001d}
\end{align}
\end{subequations}
where $\rho$ is the density, $r_{ij}$ are the 2-moments of velocity-fluctuations, $\mathrm{k}$ is the turbulent kinetic energy, $\delta_{ij}$ is the identity tensor, $a_{ij}$ is the deviatoric Reynolds-stress anisotropy-tensor,
with invariants $A_2$ and $A_3$, $A$ is Lumley's \cite{Lumley_1978a} flatness parameter,
$\varepsilon$ is the dissipation-rate of $\mathrm{k}$, $\varepsilon^*$ is the modified dissipation-rate \cite{Launder_Sharma_1974a},
$\ell_\tsn{T}$ ($\ell_\tsn{T}^*$) is the turbulent lengthscale and $Re_\tsn{T}$ ($Re_\tsn{T}^*$) is the turbulent Reynolds-number,
(defined using either $\varepsilon$ or $\varepsilon^*$), $\breve\mu$ is the dynamic viscosity evaluated from
Sutherland's law \cite[(6), p. 528]{Vallet_2008a} at mean temperature $\tilde T$, $\breve\nu$ is the kinematic viscosity at mean-flow conditions, $\breve S_{ij}$ is the deformation-rate tensor of the mean-velocity field,
\smash{$\widetilde{(\cdot)}$} denotes Favre averaging, \smash{$(\cdot)''$} are Favre fluctuations, and \smash{$\breve{(\cdot)}$} denotes a function of averaged quantities that
cannot be identified with a Reynolds or a Favre average \cite{Gerolymos_Vallet_1996a,Gerolymos_Vallet_2014a}.
Recall that $\varepsilon$ and $\varepsilon^*$ are significantly different only very near the wall \cite{Launder_Sharma_1974a,Gerolymos_Vallet_1996a,Gerolymos_Vallet_1997a}.
All of the 3 \tsn{RSM}s \cite{Gerolymos_Vallet_2001a,Gerolymos_Sauret_Vallet_2004a,Gerolymos_Lo_Vallet_Younis_2012a}
use the same scale-determining equation, solving for the modified dissipation-rate $\varepsilon^*$ \cite{Launder_Sharma_1974a,Gerolymos_Vallet_1997a}
\begin{subequations}
\label{Eq_RSMP3DDF_s_TCsFS_ss_TCs_002}
\begin{alignat}{6}
\dfrac{\partial\bar\rho\varepsilon^*}
{\partial t }+\dfrac{\partial\left(\bar\rho\varepsilon^*\tilde u_\ell\right)}
{\partial x_\ell }=&
\dfrac{\partial }
{\partial x_\ell}\left[C_\varepsilon\dfrac{\mathrm{k} }
{\varepsilon^*}\bar\rho r_{m\ell}\dfrac{\partial\varepsilon^*}
{\partial x_m }
+ \breve\mu\dfrac{\partial\varepsilon^*}
{\partial x_\ell }
\right]
\notag\\
+&C_{\varepsilon_1} P_\mathrm{k}\dfrac{\varepsilon^*}
{\mathrm{k} }
-C_{\varepsilon_2}\bar\rho\dfrac{{\varepsilon^*}^2}{\mathrm{k}}
+2 \breve\mu C_{\mu} \dfrac{{\rm k}^2}{\varepsilon^*} \dfrac{\partial^2 \tilde{u}_i}{\partial x_\ell \partial x_\ell} \dfrac{\partial^2 \tilde{u}_i}{\partial x_m \partial x_m}
\label{Eq_RSMP3DDF_s_TCsFS_ss_TCs_002a}
\end{alignat}
\begin{align}
P_\mathrm{k}:=\tfrac{1}{2} P_{\ell\ell} \;\; ;\;\;
C_\varepsilon=0.18 \;\; ;\;\;
C_{\varepsilon1}=1.44
\label{Eq_RSMP3DDF_s_TCsFS_ss_TCs_002b}\\
C_{\varepsilon2}=1.92(1-0.3\mathrm{e}^{-{Re^*_\tsn{T}}^2})\;\; ;\;\;
C_\mu=0.09\mathrm{e}^{-\frac{3.4 }
{(1+0.02 Re^*_\tsn{T})^2}}
\label{Eq_RSMP3DDF_s_TCsFS_ss_TCs_002c}
\end{align}
\end{subequations}
where $t$ is the time, $P_{ij}$ is the Reynolds-stress production-tensor \eqref{Eq_RSMP3DDF_s_TCsFS_ss_TCs_003} and $P_\mathrm{k}$ is the production-rate of turbulent kinetic energy $\mathrm{k}$.
The scale-determining equation \eqref{Eq_RSMP3DDF_s_TCsFS_ss_TCs_002} is solved along with the 6 transport equations for the components of the symmetric tensor $r_{ij}$ \cite[(1), p. 2849]{Gerolymos_Lo_Vallet_Younis_2012a}
\begin{align}
\underbrace{\dfrac{\partial}{\partial t}\left(\overline{\rho}r_{ij}\right)
+\dfrac{\partial}{\partial x_\ell}\left(\overline{\rho}r_{ij}\tilde{u}_\ell\right)}_{\displaystyle C_{ij}}=&
\underbrace{-\overline{\rho}r_{i\ell}\dfrac{\partial \tilde{u}_j}{\partial x_\ell}
-\overline{\rho}r_{j\ell}\dfrac{\partial \tilde{u}_i}{\partial x_\ell}}_{\displaystyle P_{ij}}+
\underbrace{\dfrac{\partial}{\partial x_\ell}\left(\breve{\mu}\dfrac{\partial r_{ij}}{\partial x_\ell}\right)}_{\displaystyle d_{ij}^{(\mu)}}
\notag\\
+&d_{ij}^{(u)}+\Pi_{ij}-\overline{\rho}\varepsilon_{ij}+K_{ij}
\label{Eq_RSMP3DDF_s_TCsFS_ss_TCs_003}
\end{align}
where convection \smash{$C_{ij}$}, production \smash{$P_{ij}$} and viscous diffusion \smash{$d_{ij}^{(\mu)}$} are computable terms, and diffusion by the fluctuating velocity field \smash{$d_{ij}^{(u)}$},
the velocity/pressure-gradient correlation \smash{$\Pi_{ij}$}, the dissipation-tensor \smash{$\varepsilon_{ij}$} and the fluctuating-density terms \smash{$K_{ij}$} require closure.
For all of the 3 \tsn{RSM}s \cite{Gerolymos_Vallet_2001a,Gerolymos_Sauret_Vallet_2004a,Gerolymos_Lo_Vallet_Younis_2012a}
the fluctuating-density terms \smash{$K_{ij}$} and the pressure-dilatation correlation \smash{$\phi_p$} \cite[(1), p. 2849]{Gerolymos_Lo_Vallet_Younis_2012a} were neglected
\begin{align}
K_{ij}=0\qquad;\qquad
\phi_p=0
\label{Eq_RSMP3DDF_s_TCsFS_ss_TCs_004}
\end{align}
this being a safe assumption for the subsonic flows that were investigated \cite{Gessner_Emery_1981a,Davis_Gessner_1992a,Wellborn_Reichert_Okiishi_1994a}.
The closure for the remaining terms ($d_{ij}^{(u)}$, $\Pi_{ij}$, $\varepsilon_{ij}$) differs between the 3 \tsn{RSM}s \cite{Gerolymos_Vallet_2001a,Gerolymos_Sauret_Vallet_2004a,Gerolymos_Lo_Vallet_Younis_2012a},
either in the functional dependence of the model coefficients on the local turbulent scales, or in the tensorial representation that was used \tabref{Tab_RSMP3DDF_s_TCsFS_ss_TCs_001}.
Diffusion by the triple velocity correlation
\begin{align}
d_{ij}^{(u)}:=\dfrac{\partial}{\partial x_\ell}\left(-\overline{\rho u_i'' u_j'' u_\ell''}\right)
\label{Eq_RSMP3DDF_s_TCsFS_ss_TCs_005}
\end{align}
is modelled \tabref{Tab_RSMP3DDF_s_TCsFS_ss_TCs_001} using either the Daly-Harlow \cite{Daly_Harlow_1970a} closure in the \tsn{WNF--LSS} \tsn{RSM} \cite{Gerolymos_Sauret_Vallet_2004a},
or the Hanjali\'c-Launder \cite{Hanjalic_Launder_1976a} closure in the \tsn{GV} \cite{Gerolymos_Vallet_2001a} and \tsn{GLVY} \cite{Gerolymos_Lo_Vallet_Younis_2012a} \tsn{RSM}s.
The dissipation-rate tensor is modelled as
\begin{align}
\bar\rho\varepsilon_{ij}=\tfrac{2}{3}\bar\rho\varepsilon\left(1-f_\varepsilon\right)\delta_{ij}+f_\varepsilon\dfrac{\varepsilon}{\mathrm{k}}\bar\rho r_{ij}
\label{Eq_RSMP3DDF_s_TCsFS_ss_TCs_006}
\end{align}
The anisotropic part modelled via $f_\varepsilon$ \tabref{Tab_RSMP3DDF_s_TCsFS_ss_TCs_001} is only present in the \tsn{GLVY} \tsn{RSM},
the \tsn{GV} and \tsn{WNF--LSS} \tsn{RSM}s following Lumley's \cite{Lumley_1978a} suggestion to include the anisotropy of $\varepsilon_{ij}$ in the closure for the slow-redistribution terms \cite{Gerolymos_Lo_Vallet_2012a}.
\begin{sidewaystable}[h!]
\vspace{4.7in}
\begin{center}
\caption{Coefficients in the closure relations \eqrefsatob{Eq_RSMP3DDF_s_TCsFS_ss_TCs_005}
{Eq_RSMP3DDF_s_TCsFS_ss_TCs_007},
for the \tsn{GLVY} \cite{Gerolymos_Lo_Vallet_Younis_2012a},
the \tsn{GV} \cite{Gerolymos_Vallet_2001a},
and the \tsn{WNF--LSS} \cite{Gerolymos_Sauret_Vallet_2004a} \tsn{RSM}s.}
\label{Tab_RSMP3DDF_s_TCsFS_ss_TCs_001}
\scalebox{.8}{
\begin{tabular}{c|l|c|c|c}\hline\hline
Eq. & term & \tsn{GLVY} & \tsn{GV} & \tsn{WNF--LSS} \\\hline
& & & & \\
\eqref{Eq_RSMP3DDF_s_TCsFS_ss_TCs_005} &$-\overline{\rho u_i'' u_j'' u_k''}$&\multicolumn{2}{|c|}{$\displaystyle C^{(\mathrm{\tiny S}u)}\dfrac{\mathrm{k} }
{\varepsilon}\left(\bar\rho r_{im}\dfrac{\partial r_{jk}}{\partial x_m}
+\bar\rho r_{jm}\dfrac{\partial r_{ki}}{\partial x_m}
+\bar\rho r_{km}\dfrac{\partial r_{ij}}{\partial x_m}\right)$}
&$\displaystyle C^{(\tsn{S}u)}\dfrac{\mathrm{k} }
{\varepsilon}\bar\rho r_{km}\dfrac{\partial r_{ij}}{\partial x_m}$\\
&$C^{(\tsn{\tiny S}u)}$ & 0.11 & 0.11 & 0.22 \\
\eqref{Eq_RSMP3DDF_s_TCsFS_ss_TCs_006} &$f_\varepsilon$ &$1-A^{[1+A^2]}\left[1-\mathrm{e}^{-\frac{Re_{\tsn{T}}}{10}}\right]$& 0 & 0 \\
\eqref{Eq_RSMP3DDF_s_TCsFS_ss_TCs_007b}&$C^{(\tsn{Sp1})}$ &-0.005 & 0 & 0 \\
\eqref{Eq_RSMP3DDF_s_TCsFS_ss_TCs_007b}&$C^{(\tsn{Sp2})}$ &+0.022 & 0 & 0 \\
\eqref{Eq_RSMP3DDF_s_TCsFS_ss_TCs_007b}&$C^{(\tsn{Rp })}$ &-0.005 & 0 & 0 \\
\eqref{Eq_RSMP3DDF_s_TCsFS_ss_TCs_007c}&$C_\phi^{(\tsn{RH})}$ &\multicolumn{2}{|c|}{$\displaystyle\min{\left[1,0.75+1.3\max{[0,A-0.55]}\right]}\;A^{[\max(0.25,0.5-1.3\max{[0,A-0.55]})]}[1-\max(0,1-{\frac{Re_\tsn{T}}
{50 }})]$}
&$\displaystyle0.75\sqrt{A}$\\
\eqref{Eq_RSMP3DDF_s_TCsFS_ss_TCs_007c}&$C_\phi^{(\tsn{RI})}$ &$\displaystyle\max{\left[{\tfrac{2}{3}-\frac{1}{6C_\phi^{(\tsn{RH})}},0}\right]}
\abs{\mathrm{grad}\Biggl(\dfrac{\ell_\tsn{T}[1-\mathrm{e}^{-\frac{Re^*_\tsn{T}}
{30 }}]}
{1+1.6A_2^{\max(0.6,A)} }\Biggr)}$
&\multicolumn{2}{c}{$\displaystyle\max{\left[{\tfrac{2}{3}-\frac{1}{6C_\phi^{(\tsn{RH})}},0}\right]}
\abs{\mathrm{grad}\Biggl(\dfrac{\ell_\tsn{T}[1-\mathrm{e}^{-\frac{Re^*_\tsn{T}}
{30 }}]}
{1+1.8A_2^{\max(0.6,A)} }\Biggr)}$}\\
\eqref{Eq_RSMP3DDF_s_TCsFS_ss_TCs_007d}&$\varepsilon_\mathrm{v}$ &$\varepsilon^*$ & $\varepsilon$ & $\varepsilon$ \\
\eqref{Eq_RSMP3DDF_s_TCsFS_ss_TCs_007d}&$C_\phi^{(\tsn{SH1})}$ &$\displaystyle3.7 A A_2^{\tfrac{1}{4}}\left[1- \mathrm{e}^{-\left(\frac{Re_{\tsn{T}}}
{130 }\right)^2}\right]$
&\multicolumn{2}{c}{$\displaystyle1+2.58A A_2^{\tfrac{1}{4}}\left[1-\mathrm{e}^{-({\frac{Re_\tsn{T}}{150}})^2}\right]$}\\
\eqref{Eq_RSMP3DDF_s_TCsFS_ss_TCs_007d}&$C_\phi^{(\tsn{SI1})}$ &$\displaystyle\left[-\tfrac{4}{9}\left(C_\phi^{(\tsn{SH1})}-\tfrac{9}{4}\right)\right]\;
\abs{\mathrm{grad}\Biggl(\dfrac{\ell_\tsn{T}[1-\mathrm{e}^{-\frac{Re^*_\tsn{T}}
{30 }}]}
{1+2.9\sqrt{A_2} }\Biggr)}$
&$\displaystyle0.83\left[1-{\tfrac{2}{3}}(C^\tsn{SH1}_\phi-1)\right]
\abs{\mathrm{grad}\left(\dfrac{\ell_\tsn{T}[1-\mathrm{e}^{-{\tfrac{Re^*_\tsn{T}}{30}}}]}{1+2 A_2^{0.8}}\right)}$
&$\displaystyle0.90\left[1-{\tfrac{2}{3}}(C^\tsn{SH1}_\phi-1)\right]
\abs{\mathrm{grad}\left(\dfrac{\ell_\tsn{T}[1-\mathrm{e}^{-{\frac{Re^*_\tsn{T}}{30}}}]}{1+1.8 A_2^{0.8}}\right)}$\\
\eqref{Eq_RSMP3DDF_s_TCsFS_ss_TCs_007d}&$C_\phi^{(\tsn{SI2})}$ &0.002 & 0 & 0 \\
\eqref{Eq_RSMP3DDF_s_TCsFS_ss_TCs_007d}&$C_\phi^{(\tsn{SI3})}$ &$\displaystyle 0.14\;
\sqrt{\dfrac{\partial\ell^*_\tsn{T}}{\partial x_\ell}
\dfrac{\partial\ell^*_\tsn{T}}{\partial x_\ell}}$ & 0 & 0 \\
\end{tabular}
}
\end{center}
{\footnotesize Notice that there is a typographic error in \cite[(22), p. 1836]{Gerolymos_Vallet_2001a},
the turbulent Reynolds-number in the definition of $C_\phi^{(\tsn{RI})}$ and $C_\phi^{(\tsn{SI1})}$ for the \tsn{GV} \tsn{RSM} being $Re_\tsn{T}^*$ as above.
The expression of $C_\phi^{(\tsn{RH})}$ for the \tsn{GLVY} and \tsn{GV} \tsn{RSM}s above is the if-less equivalent of
\begin{alignat}{6}\left[C_\phi^{(\tsn{RH})}\right]_\tsn{GLVY}=
\left[C_\phi^{(\tsn{RH})}\right]_\tsn{GV} =\Bigg(1-\max\left(0,1-\tfrac{1}{50}Re_\tsn{T}\right)\Bigg)\times
\left\{\begin{array}{lrcl} 0.75\sqrt{A} &0 &\leq A&<0.55 \\
\Big(0.75\!\!+\!\!1.3(A-0.55)\Big)A^{0.5-1.3(A-0.55)}&0.55 &\leq A&<0.55+\frac{0.25}{1.3}\\
A^{\frac{1}{4}} &0.55+\frac{0.25}{1.3}&\leq A&\leq1 \\\end{array}\right.\notag\end{alignat}
Notice that there is a typographic error of the above expression of $\left[C_\phi^{(\tsn{RH})}\right]_\tsn{GLVY}=\left[C_\phi^{(\tsn{RH})}\right]_\tsn{GV}$ for $0.55\leq A<0.55+\frac{0.25}{1.3}$
in \cite[(22), p. 419, missing parentheses]{Gerolymos_Sauret_Vallet_2004a} and
in \cite[Tab. 1, p. 1370, misplaced parenthesis]{Gerolymos_Joly_Mallet_Vallet_2010a}.}
\end{sidewaystable}
\clearpage
A general tensorial representation of the pressure terms $\Pi_{ij}$, which describes, by appropriate choice of the coefficients, all 3 models \tabref{Tab_RSMP3DDF_s_TCsFS_ss_TCs_001},
reads \cite[(4--6), pp. 2851--2854]{Gerolymos_Lo_Vallet_Younis_2012a}
\begin{subequations}
\label{Eq_RSMP3DDF_s_TCsFS_ss_TCs_007}
\begin{align}
\Pi_{ij}=&\underbrace{\overbrace{\phi_{ij}^{(\tsn{RH})}+\phi_{ij}^{(\tsn{RI})}}^{\displaystyle\phi_{ij}^{(\tsn{R})}}+
\overbrace{\phi_{ij}^{(\tsn{SH})}+\phi_{ij}^{(\tsn{SI})}}^{\displaystyle\phi_{ij}^{(\tsn{S})}}}_{\displaystyle\phi_{ij}}+\tfrac{2}{3}\phi_p\delta_{ij}+d_{ij}^{(p)}
\label{Eq_RSMP3DDF_s_TCsFS_ss_TCs_007a}\\
d_{ij}^{(p)}=&C^{(\tsn{Sp1})}\bar\rho\dfrac{\mathrm{k}^3 }
{\varepsilon^3}\dfrac{\partial\varepsilon^*}
{\partial x_i }\dfrac{\partial\varepsilon^*}
{\partial x_j }
+\dfrac{\partial }
{\partial x_\ell}\left[C^{(\tsn{Sp2})}(\overline{\rho u_m''u_m''u_j''}\delta_{i\ell}
+\overline{\rho u_m''u_m''u_i''}\delta_{j\ell})\right]
\notag\\
+&C^{(\tsn{Rp})}\bar\rho\dfrac{\mathrm{k}^2 }
{\varepsilon^2}\breve{S}_{k\ell}a_{\ell k}\dfrac{\partial\mathrm{k}}
{\partial x_i }\dfrac{\partial\mathrm{k}}
{\partial x_j }
\label{Eq_RSMP3DDF_s_TCsFS_ss_TCs_007b}\\
\phi_{ij}^{(\tsn{R})}=&\underbrace{-C_\phi^{(\tsn{RH})}\left(P_{ij}-\tfrac{1}{3}\delta_{ij} P_{mm}\right)}_{\displaystyle\phi^{(\tsn{RH})}_{ij}}
\notag\\
&\underbrace{+C_\phi^{(\tsn{RI})}\left[ \phi^{(\tsn{RH})}_{nm}e_{\tsn{I}_n}e_{\tsn{I}_m}\delta_{ij}
-\tfrac{3}{2}\phi^{(\tsn{RH})}_{in}e_{\tsn{I}_n}e_{\tsn{I}_j}
-\tfrac{3}{2}\phi^{(\tsn{RH})}_{jn}e_{\tsn{I}_n}e_{\tsn{I}_i}\right]}_{\displaystyle\phi^{(\tsn{RI})}_{ij}}
\label{Eq_RSMP3DDF_s_TCsFS_ss_TCs_007c}\\
\phi^{(\tsn{S})}_{ij}=&\underbrace{-C_\phi^{(\tsn{SH1})}\bar\rho\varepsilon_\mathrm{v} a_{ij}}_{\displaystyle\phi^{(\tsn{SH1})}_{ij}}
\underbrace{+C_\phi^{(\tsn{SI1})}\dfrac{\varepsilon_\mathrm{v}}{\mathrm{k}}\left[ \bar\rho r_{nm}e_{\tsn{I}_n}e_{\tsn{I}_m}\delta_{ij}
-\tfrac{3}{2}\bar\rho r_{ni}e_{\tsn{I}_n}e_{\tsn{I}_j}
-\tfrac{3}{2}\bar\rho r_{nj}e_{\tsn{I}_n}e_{\tsn{I}_i}\right]}_{\displaystyle\phi^{(\mathrm{\tiny SI1})}_{ij}}
\notag\\
&\underbrace{-C_\phi^{(\tsn{SI2})}\bar\rho\dfrac{\mathrm{k}}{\varepsilon}\dfrac{\partial\mathrm{k}}{\partial x_\ell}
\left[ a_{ik}\frac{\partial r_{kj}}{\partial x_\ell}
+a_{jk}\frac{\partial r_{ki}}{\partial x_\ell}
-\tfrac{2}{3}\delta_{ij}a_{mk}\frac{\partial r_{km}}{\partial x_\ell}\right]}_{\displaystyle\phi^{(\tsn{SI2})}_{ij}}
\notag\\
&\underbrace{+C_\phi^{(\tsn{SI3})}\left[ \phi^{(\tsn{SI2})}_{nm}e_{\tsn{I}_n}e_{\tsn{I}_m}\delta_{ij}
-\tfrac{3}{2}\phi^{(\tsn{SI2})}_{in}e_{\tsn{I}_n}e_{\tsn{I}_j}
-\tfrac{3}{2}\phi^{(\tsn{SI2})}_{jn}e_{\tsn{I}_n}e_{\tsn{I}_i}\right]}_{\displaystyle\phi^{(\tsn{SI3})}_{ij}}
\label{Eq_RSMP3DDF_s_TCsFS_ss_TCs_007c}\\
e_{\tsn{I}_i}:=&\dfrac{\dfrac{\partial}{\partial x_i}\Biggl(\dfrac{\ell_\tsn{T}[1-\mathrm{e}^{-{\frac{Re^*_\tsn{T}}
{ 30}}}]}
{1+2\sqrt{A_2}+2A^{16} }
\Biggr)
}
{\sqrt{\dfrac{\partial}{\partial x_\ell}\Biggl(\dfrac{\ell_\tsn{T}[1-\mathrm{e}^{-{\frac{Re^*_\tsn{T}}
{ 30}}}]}
{1+2\sqrt{A_2}+2A^{16} }
\Biggr)
\dfrac{\partial}{\partial x_\ell}\Biggl(\dfrac{\ell_\tsn{T}[1-\mathrm{e}^{-{\frac{Re^*_\tsn{T}}
{ 30}}}]}
{1+2\sqrt{A_2}+2A^{16} }
\Biggr)
}
}
\label{Eq_RSMP3DDF_s_TCsFS_ss_TCs_007d}
\end{align}
\end{subequations}
where $\phi_{ij}$ denotes the redistribution tensor, $d_{ij}^{(p)}$ denotes pressure diffusion, the superscripts \tsn{S} and \tsn{R} denote slow and rapid terms \cite{Hanjalic_1994a},
the superscripts \tsn{H} and \tsn{I} denote homogeneous and inhomogeneous terms \cite{Gerolymos_Senechal_Vallet_2013a}, and the unit-vector $\vec{e}_\tsn{I}$ was modelled \cite{Gerolymos_Vallet_2001a}
to point in the main direction of turbulence-inhomogeneity \cite{Gerolymos_Sauret_Vallet_2004a}. Notice that, although initially $\vec{e}_\tsn{I}$ was designed to mimic the wall-normal direction in wall-echo terms \cite{Gerolymos_Vallet_2001a},
it turns out that inhomogeneous terms are also active at the shear-layer edge and in regions of recirculating flow, away from or even in absence of solid walls. As a consequence, the closure \eqref{Eq_RSMP3DDF_s_TCsFS_ss_TCs_007} must be considered as a whole,
and inhomogeneous terms should be kept when computing free shear flows. Very near the walls, \smash{$\Pi_{ij}\to0\stackrel{\eqref{Eq_RSMP3DDF_s_TCsFS_ss_TCs_007a}}{\Longrightarrow}\phi_{ij}+\frac{2}{3}\phi_p\delta_{ij}\to-d_{ij}^{(p)}$},
so that all authors \cite{Gerolymos_Lo_Vallet_Younis_2012a} avoid the complexity of including terms in the model that would correctly mimic the individual behaviour of $\phi_{ij}$ and \smash{$d_{ij}^{(p)}$} as wall-distance $n\to0$,
which would cancel one another in \eqref{Eq_RSMP3DDF_s_TCsFS_ss_TCs_007a}, but rather model $\Pi_{ij}$ as a whole in that region \cite[Fig. 6, pp. 2855--2856]{Gerolymos_Lo_Vallet_Younis_2012a},
in line with the suggestion of Mansour \etal\ \cite{Mansour_Kim_Moin_1988a}. For this reason, the wall-echo-like \cite{Gibson_Launder_1978a} tensorial form of the terms containing $\vec{e}_\tsn{I}$ in \eqref{Eq_RSMP3DDF_s_TCsFS_ss_TCs_007}
is justified, because it was recently shown \cite{Gerolymos_Senechal_Vallet_2013a}, from the analysis of \tsn{DNS} data,
that it is in agreement with the near-wall behaviour of $\Pi_{ij}$ \cite[Fig. 13 p. 41]{Gerolymos_Senechal_Vallet_2013a}, unlike that of $\phi_{ij}$ \cite[Fig. 12 p. 39]{Gerolymos_Senechal_Vallet_2013a}.
The coefficients in \eqrefsatob{Eq_RSMP3DDF_s_TCsFS_ss_TCs_005}
{Eq_RSMP3DDF_s_TCsFS_ss_TCs_007}
are generally functions of the local turbulence state ($A$, $A_2$, $Re_\tsn{T}$, $\cdots$) and of its gradients, and depend on the particular \tsn {RSM} \tabref{Tab_RSMP3DDF_s_TCsFS_ss_TCs_001}.
The \tsn{WNF--LSS} \cite{Gerolymos_Sauret_Vallet_2004a} is a wall-normal-free extension of the Launder-Shima \cite{Launder_Shima_1989a} \tsn{RSM}, which, in complex flows,
performs better than the original wall-topology-dependent model, mainly because of the action of the inhomogeneous terms away from solid walls.
The main drawback of this model is that, although it quite naturally improves upon 2-equation closures, it still underestimates separation \cite{Gerolymos_Joly_Mallet_Vallet_2010a}.
The \tsn{GV} \cite{Gerolymos_Vallet_2001a} \tsn{RSM} was developed to overcome this limitation, mainly by an optimized \smash{$C_\phi^{(\tsn{RH})}$} coefficient \tabref{Tab_RSMP3DDF_s_TCsFS_ss_TCs_001}
of the isotropisation-of-production \cite{Gibson_Launder_1978a,Launder_Shima_1989a,Hanjalic_1994a}
closure of the rapid homogeneous part of redistribution \eqref{Eq_RSMP3DDF_s_TCsFS_ss_TCs_007c}.
The resulting model successfully predicted flows with large separation, but reattachment and especially relaxation were slightly slower than experimental data \cite{Sauret_Vallet_2007a,Vallet_2007a}.
The \tsn{GLVY} \cite{Gerolymos_Lo_Vallet_Younis_2012a} \tsn{RSM} improves this behaviour \cite[Fig. 9, p. 2858]{Gerolymos_Lo_Vallet_Younis_2012a}
through extended modelling of the inhomogeneous part of the slow redistribution terms \smash{$\phi_{ij}^{(\tsn{SI})}$} \eqrefsab{Eq_RSMP3DDF_s_TCsFS_ss_TCs_007a}
{Eq_RSMP3DDF_s_TCsFS_ss_TCs_007d}
and of pressure diffusion \smash{$d_{ij}^{(p)}$} \eqref{Eq_RSMP3DDF_s_TCsFS_ss_TCs_007b},
while using the same optimized closure for \smash{$\phi_{ij}^{(\tsn{RH})}$} as the \tsn{GV} \cite{Gerolymos_Vallet_2001a} \tsn{RSM}.
It was also observed that the inclusion of these modifications influences the apparent transition behaviour \cite{Rumsey_2007a} of the models,
at low external turbulence conditions \cite{icp_Gerolymos_Vallet_2013a}.
Comparing the 3 \tsn{RSM}s \tabref{Tab_RSMP3DDF_s_TCsFS_ss_TCs_001}, the \tsn{WNF--LSS} \cite{Gerolymos_Sauret_Vallet_2004a} \tsn{RSM} conceptually \cite{Hanjalic_1994a} includes pressure-diffusion in
the Daly-Harlow \cite{Daly_Harlow_1970a} closure for \smash{$d_{ij}^{(u)}$}, while the \tsn{GV} \cite{Gerolymos_Vallet_2001a} \tsn{RSM} neglects \smash{$d_{ij}^{(p)}$};
they both (\tsn{WNF--LSS} and \tsn{GV}) include the dissipation-rate anisotropy \smash{$\varepsilon_{ij}-\tfrac{2}{3}\varepsilon\delta_{ij}$} in the closure for \smash{$\phi_{ij}^{(\tsn{S})}$} \cite{Lumley_1978a,Launder_Shima_1989a}.
On the contrary, the \tsn{GLVY} \cite{Gerolymos_Lo_Vallet_Younis_2012a} \tsn{RSM} explicitly models both \smash{$d_{ij}^{(p)}$} and \smash{$\varepsilon_{ij}-\tfrac{2}{3}\varepsilon\delta_{ij}$}.
Computations were also compared with the baseline linear Launder-Sharma \cite{Launder_Sharma_1974a} $\mathrm{k}$-$\varepsilon$ closure, as implemented in \cite{Gerolymos_Vallet_1996a}.
\subsection{Flow solver} \label{RSMP3DDF_s_TCsFS_ss_FS}
Computations were performed using a structured multiblock solver \cite{Gerolymos_Vallet_2005a}, with \tsn{WENO3} \cite{Gerolymos_Senechal_Vallet_2009a}
reconstruction of the primitive variables, both mean-flow and turbulent, an \tsn{HLLC}$_\mathrm{h}$ approximate Riemann solver \cite{BenNasr_Gerolymos_Vallet_2014a},
and implicit multigrid dual-time-stepping pseudo-time-marching integration \cite{Gerolymos_Vallet_2009a}.
All of the computations presented in the paper were run using $L_\tsn{GRD}=3$ levels of multigrid with a $\tsn{V(2,0)}$ sawtooth cycle \cite{Gerolymos_Vallet_2009a}
and dual-time-stepping parameters \cite{Gerolymos_Vallet_2005a} $[\tsn{CFL},\tsn{CFL}^*;M_\mathrm{it},r_\tsn{TRG}]=[100,10;\text{---},-1]$
(where $\tsn{CFL}$ is the \tsn{CFL}-number for the pseudo-time-step, $\tsn{CFL}^*$ is the \tsn{CFL}-number for the dual pseudo-time-step, $M_\mathrm{it}$ is the number of dual subiterations,
and $r_\tsn{TRG}<0$ is the target-reduction in orders-of-magnitude of the nonlinear pseudo-time-evolution system solution).
This methodology is implemented in the open source software \texttt{aerodynamics} \cite{sft_Gerolymos_Vallet_aerodynamics_2009} with which the present results were obtained.
In all instances, a subsonic reservoir condition was applied at inflow \cite[(24), p. 1324]{Gerolymos_Vallet_1996a},
a subsonic pressure condition \cite[(26), p. 1324]{Gerolymos_Vallet_1996a} was applied at outflow (uniform static pressure at outflow),
and the no-slip walls were considered adiabatic \cite[(25), p. 1324]{Gerolymos_Vallet_1996a}.
The inflow boundary condition was implemented using the method of finite waves \cite{Atkins_Casper_1994a}.
Note that in this approach, the inflow boundary-layers are prescribed through the initial total pressure and total temperature profiles \cite{Gerolymos_Sauret_Vallet_2004c},
but the streamwise mean-flow velocity $\tilde u$ at inflow is also influenced by the outgoing pressure-wave \cite{Chakravarthy_1984a},
and may therefore evolve differently for different turbulence closures \cite[(Fig. 6), p. 209]{BenNasr_Gerolymos_Vallet_2014a}.
\begin{figure}[h!]
\begin{center}
\begin{picture}(340,420)
\put(0,-5){\includegraphics[angle=0,width=340pt,bb=62 154 561 781]{Fig_FlowTurbComb_01_grids.pdf}}
\end{picture}
\end{center}
\caption{Computational grid topology \tabref{Tab_RSMP3DDF_s_A_001} for the square duct \cite{Gessner_Emery_1981a}, the \tsn{C}-to-\tsn{R} transition duct \cite{Davis_Gessner_1992a} and the \tsn{S}-duct \cite{Wellborn_Reichert_Okiishi_1994a}
test-cases (in all cases the $i=\const$ grid-surfaces are $\perp x$ planes.}
\label{Fig_RSMP3DDF_s_A_001}
\end{figure}
\section{Assessment}\label{RSMP3DDF_s_A}
The predictive capability of the 4 turbulence models \parref{RSMP3DDF_s_TCsFS_ss_TCs} was assessed by systematic comparison with experimental data for 3 duct-flow configurations \cite{Gessner_Emery_1981a,Davis_Gessner_1992a,Wellborn_Reichert_Okiishi_1994a}.
Hereafter, the abbreviations \tsn{GLVY RSM} \cite{Gerolymos_Lo_Vallet_Younis_2012a}, \tsn{GV RSM} \cite{Gerolymos_Vallet_2001a},
\tsn{WNF--LSS RSM} \cite{Gerolymos_Sauret_Vallet_2004a} and \tsn{LS} $\mathrm{k}$--$\varepsilon$ \cite{Launder_Sharma_1974a},
are used consistently to denote each model.
\begin{table}[ht!]
\begin{center}
\caption{Computational grids and mesh-generation parameters \cite{Gerolymos_Sauret_Vallet_2004b,Gerolymos_Tsanga_1999a}
for the square duct \cite{Gessner_Emery_1981a}, the \tsn{C}-to-\tsn{R} transition duct \cite{Davis_Gessner_1992a} and the \tsn{S}-duct \cite{Wellborn_Reichert_Okiishi_1994a}
test-cases.}
\label{Tab_RSMP3DDF_s_A_001}
\input{Tab_FlowTurbComb_01_grids}
\end{center}
{\footnotesize $N_\mathrm{p}$: number of grid-points;
$n_\tsn{D}$: domain index;
type: domain grid-type;
\tsn{H}: \tsn{H}-type grid ($x\times y\times z$);
\tsn{O}$_\tsn{$\square$}$: circumferentially \tsn{O}-type grid ($x\times\;-\theta\times R$) between the duct-casing and the inner square domain ($n_\tsn{D}=2$) around the centerline;
\tsn{H}$_\tsn{$\square$}$: \tsn{H}-type grid for the inner square domain ($n_\tsn{D}=2$) around the centerline;
$N_i\times N_j\times N_k$: grid-points;
$N_{j_\mathrm{s}}$, $N_{k_\mathrm{s}}$: number of points geometrically stretched near the solid walls;
$r_j$, $r_k$: geometric progression ratio;
$r_\tsn{$\square$}$: ratio of the side of the square domain around the centerline to the size of the cross-section (defined as the average of its projections on the $y$ and $z$ axes);
$\Delta n_w^+$ nondimensional wall-normal size of the first grid-cell in wall-units \cite{Gerolymos_Sauret_Vallet_2004b}.
}
\begin{center}
\caption{Initial (\tsn{IC}s) and boundary-conditions (\tsn{BC}s) for the square duct \cite{Gessner_Emery_1981a}, the \tsn{C}-to-\tsn{R} transition duct \cite{Davis_Gessner_1992a} and the \tsn{S}-duct \cite{Wellborn_Reichert_Okiishi_1994a}
test-cases, using the \tsn{GLVY RSM} \cite{Gerolymos_Lo_Vallet_Younis_2012a}, the \tsn{GV RSM} \cite{Gerolymos_Vallet_2001a}, the \tsn{WNF--LSS RSM} \cite{Gerolymos_Sauret_Vallet_2004a}
and the \tsn{LS} $\mathrm{k}$-$\varepsilon$ \cite{Launder_Sharma_1974a}.}
\label{Tab_RSMP3DDF_s_A_002}
\input{Tab_FlowTurbComb_01_BCs_ICs}
\end{center}
{\footnotesize $\delta_\mathrm{i}$: boundary-layer thickness at inflow (\tsn{IC});
$\Pi_{\tsn{C}_\mathrm{i}}$: inflow boundary-layer Coles-parameter \cite{Gerolymos_Sauret_Vallet_2004c} (\tsn{IC});
$M_{\tsn{CL}_\mathrm{i}}$: inflow centerline Mach-number (\tsn{IC});
$T_{u_{\tsn{CL}_\mathrm{i}}}$: turbulent intensity outside of the boundary-layers at inflow \cite{Gerolymos_Sauret_Vallet_2004c};
$\ell_{\tsn{T}_{\tsn{CL}_\mathrm{i}}}$: turbulent lengthscale outside of the boundary-layers at inflow \cite{Gerolymos_Sauret_Vallet_2004c};
$p_{t_{\tsn{CL}_\mathrm{i}}}$: inflow centerline total pressure (\tsn{BC});
$T_{t_{\tsn{CL}_\mathrm{i}}}$: inflow centerline total temperature (\tsn{BC});
$q_w$: wall heat-flux (\tsn{BC});
$p_\mathrm{o}$: outflow static pressure (\tsn{BC}).
}
\end{table}
\subsection{Developing turbulent flow in a square duct \cite{Gessner_Emery_1981a}}\label{RSMP3DDF_s_A_ss_DTFSDGE1981}
The experimental data described by Gessner and Emery \cite{Gessner_Emery_1981a} were obtained \cite{Po_1975a,Gessner_Po_Emery_1979a,Gessner_Emery_1981a}
in a duct of square cross-section \figrefsab{Fig_RSMP3DDF_s_A_001}
{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_001}.
The duct's height, which at incompressible flow conditions is also the duct's hydraulic diameter \cite[(3.55), p. 123]{White_1991a},
is $D_\mathrm{h}=2\mathrm{a}=0.254\;\mathrm{m}$, and the length of the straight working section is $87D_\mathrm{h}$ \cite[Fig. 2, p. 121]{Gessner_Po_Emery_1979a}.
The flow \cite{Gessner_Emery_1981a} is essentially incompressible (centerline Mach number $\bar M_\tsn{CL}\approxeq0.05$) at bulk Reynolds number $Re_\tsn{B}\approxeq250000$
($Re_\tsn{B}=\bar u_\tsn{B} D_\mathrm{h} \nu^{-1}$, where $\bar u_\tsn{B}$ is the bulk velocity and $\nu$ is the practically constant kinematic viscosity).
The flow at the duct's inlet is nearly uniform, with very thin boundary-layers, whose virtual origin was estimated experimentally at $x\approxeq-0.65D_\mathrm{h}$ \cite[p. 122]{Gessner_Po_Emery_1979a}.
These very thin boundary-layers grow until they fill the entire duct at $x\approxeq32D_\mathrm{h}$ \cite[p. 123]{Gessner_Po_Emery_1979a} and interact
to reach practically fully developed flow conditions at the last measurement station located at $x=84D_\mathrm{h}$, near the exit of the duct's working section \cite[Fig. 2, p. 121]{Gessner_Po_Emery_1979a}.
Measurements \cite{Po_1975a,Gessner_Po_Emery_1979a,Gessner_Emery_1981a},
taken at 5 axial planes \figref{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_002}, include mean-flow $x$-wise velocities (Kiel probes in conjunction with a wall static pressure tap),
and secondary mean-flow velocities and Reynolds-stresses (hot-wire). They also include the detailed $x$-wise evolution of the centerline velocity (Kiel probe) and limited
skin-friction data (Preston tubes) only at the last measurement station ($x=84D_\mathrm{h}$) where the flow is considered fully developed \cite[Fig. 2, p. 448]{Gessner_Emery_1981a}.
\begin{figure}[h!]
\begin{center}
\begin{picture}(340,170)
\put(0,-5){\includegraphics[angle=0,width=340pt,bb=95 481 483 684]{Fig_FlowTurbComb_01_GE_1981_sqduct_uCL.pdf}}
\end{picture}
\end{center}
\caption{Comparison of measured \cite{Gessner_Emery_1981a} streamwise evolution of $x$-wise centerline ($y=z=\mathrm{a}$) velocity $\bar u_\tsn{CL}$
with
computations ($18\times10^6$ points grid discretizing $\tfrac{1}{4}$ of the square duct; \tabrefnp{Tab_RSMP3DDF_s_A_001})
using \parref{RSMP3DDF_s_TCsFS_ss_TCs} the \tsn{GV} \cite{Gerolymos_Vallet_2001a}, the \tsn{WNF--LSS} \cite{Gerolymos_Sauret_Vallet_2004a}
and the \tsn{GLVY} \cite{Gerolymos_Lo_Vallet_Younis_2012a} \tsn{RSM}s, and the \tsn{LS} \cite{Launder_Sharma_1974a} linear $\mathrm{k}$--$\varepsilon$ model, for developing turbulent flow in a square duct
($Re_\tsn{B}=250000$, $\bar M_\tsn{CL}\approxeq0.05$; \tabrefnp{Tab_RSMP3DDF_s_A_002}).}
\label{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_001}
\end{figure}
\begin{figure}[h!]
\begin{center}
\begin{picture}(340,360)
\put(0,-5){\includegraphics[angle=0,width=340pt,bb=86 260 524 724]{Fig_FlowTurbComb_01_GE_1981_sqduct_u.pdf}}
\end{picture}
\end{center}
\caption{Comparison of measured \cite{Gessner_Emery_1981a} streamwise ($x$-wise) velocity $\bar u$, along the wall-bisector ($z=\mathrm{a}$) and along the corner-bisector ($z=y$),
at the 5 experimental measurement stations, with
computations ($18\times10^6$ points grid discretizing $\tfrac{1}{4}$ of the square duct; \tabrefnp{Tab_RSMP3DDF_s_A_001})
using \parref{RSMP3DDF_s_TCsFS_ss_TCs} the \tsn{GV} \cite{Gerolymos_Vallet_2001a}, the \tsn{WNF--LSS} \cite{Gerolymos_Sauret_Vallet_2004a}
and the \tsn{GLVY} \cite{Gerolymos_Lo_Vallet_Younis_2012a} \tsn{RSM}s, and the \tsn{LS} \cite{Launder_Sharma_1974a} linear $\mathrm{k}$--$\varepsilon$ model, for developing turbulent flow in a square duct
($Re_\tsn{B}=250000$, $\bar M_\tsn{CL}\approxeq0.05$; \tabrefnp{Tab_RSMP3DDF_s_A_002}).}
\label{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_002}
\end{figure}
\begin{figure}[h!]
\begin{center}
\begin{picture}(340,360)
\put(0,-5){\includegraphics[angle=0,width=340pt,bb=86 260 527 724]{Fig_FlowTurbComb_01_GE_1981_sqduct_v.pdf}}
\end{picture}
\end{center}
\caption{Comparison of measured \cite{Gessner_Emery_1981a} wall-normal ($y$-wise) velocity $\bar v$, along the wall-bisector ($z=\mathrm{a}$) and along the corner-bisector ($z=y$, where by symmetry $\bar w=\bar v$),
at the 5 experimental measurement stations, with
computations ($18\times10^6$ points grid discretizing $\tfrac{1}{4}$ of the square duct; \tabrefnp{Tab_RSMP3DDF_s_A_001})
using \parref{RSMP3DDF_s_TCsFS_ss_TCs} the \tsn{GV} \cite{Gerolymos_Vallet_2001a}, the \tsn{WNF--LSS} \cite{Gerolymos_Sauret_Vallet_2004a}
and the \tsn{GLVY} \cite{Gerolymos_Lo_Vallet_Younis_2012a} \tsn{RSM}s, and the \tsn{LS} \cite{Launder_Sharma_1974a} linear $\mathrm{k}$--$\varepsilon$ model, for developing turbulent flow in a square duct
($Re_\tsn{B}=250000$, $\bar M_\tsn{CL}\approxeq0.05$; \tabrefnp{Tab_RSMP3DDF_s_A_002}).}
\label{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_003}
\end{figure}
\begin{figure}[h!]
\begin{center}
\begin{picture}(340,360)
\put(0,-5){\includegraphics[angle=0,width=340pt,bb= 85 258 525 724]{Fig_FlowTurbComb_01_GE_1981_sqduct_uv_wu.pdf}}
\end{picture}
\end{center}
\caption{Comparison of measured \cite{Gessner_Emery_1981a} Reynolds shear-stresses, $\overline{u'v'}$ along the wall-bisector ($z=\mathrm{a}$) and along the corner-bisector ($z=y$),
and $\overline{u'w'}$ along the corner-bisector ($z=y$, where by symmetry $\overline{u'w'}=\overline{u'v'}$),
at the 5 experimental measurement stations, with
computations ($18\times10^6$ points grid discretizing $\tfrac{1}{4}$ of the square duct; \tabrefnp{Tab_RSMP3DDF_s_A_001})
using \parref{RSMP3DDF_s_TCsFS_ss_TCs} the \tsn{GV} \cite{Gerolymos_Vallet_2001a}, the \tsn{WNF--LSS} \cite{Gerolymos_Sauret_Vallet_2004a}
and the \tsn{GLVY} \cite{Gerolymos_Lo_Vallet_Younis_2012a} \tsn{RSM}s, and the \tsn{LS} \cite{Launder_Sharma_1974a} linear $\mathrm{k}$--$\varepsilon$ model, for developing turbulent flow in a square duct
($Re_\tsn{B}=250000$, $\bar M_\tsn{CL}\approxeq0.05$; \tabrefnp{Tab_RSMP3DDF_s_A_002}).}
\label{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_004}
\end{figure}
\begin{figure}[h!]
\begin{center}
\begin{picture}(340,360)
\put(0,-5){\includegraphics[angle=0,width=340pt,bb= 86 258 524 724]{Fig_FlowTurbComb_01_GE_1981_sqduct_uu.pdf}}
\end{picture}
\end{center}
\caption{Comparison of measured \cite{Gessner_Emery_1981a} streamwise diagonal Reynolds-stress $\overline{u'u'}$, along the wall-bisector ($z=\mathrm{a}$) and along the corner-bisector ($z=y$),
at the 5 experimental measurement stations, with
computations ($18\times10^6$ points grid discretizing $\tfrac{1}{4}$ of the square duct; \tabrefnp{Tab_RSMP3DDF_s_A_001})
using \parref{RSMP3DDF_s_TCsFS_ss_TCs} the \tsn{GV} \cite{Gerolymos_Vallet_2001a}, the \tsn{WNF--LSS} \cite{Gerolymos_Sauret_Vallet_2004a}
and the \tsn{GLVY} \cite{Gerolymos_Lo_Vallet_Younis_2012a} \tsn{RSM}s, and the \tsn{LS} \cite{Launder_Sharma_1974a} linear $\mathrm{k}$--$\varepsilon$ model, for developing turbulent flow in a square duct
($Re_\tsn{B}=250000$, $\bar M_\tsn{CL}\approxeq0.05$; \tabrefnp{Tab_RSMP3DDF_s_A_002}).}
\label{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_005}
\end{figure}
In the Gessner and Emery \cite{Gessner_Emery_1981a} square duct, the main mechanisms are the interaction of stress-induced secondary flows, typical of the square cross-section \cite{Gessner_Jones_1965a},
with boundary-layer entrainment \cite{Kovasznay_Kibens_Blackwelder_1970a}. The streamwise thickening of the wall-layers induces blockage, resulting in flow acceleration,
which overshoots before stabilizing at the streamwise-inva\-riant fully developed level \figref{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_001}.
Sufficient grid resolution is therefore required, both near the walls and at the centerline, to correctly reproduce the development and interaction of the boundary-layers,
and as a consequence to obtain grid-convergence of the streamwise evolution of centerline velocity \figref{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_001}.
Results are presented for an $18\times10^6$ points grid \tabref{Tab_RSMP3DDF_s_A_001} discretizing one quadrant of the duct, with symmetry-conditions at the $y$- and $z$-wise symmetry-planes.
The computational domain $L_x\times L_y\times L_z=98.43D_\mathrm{h}\times\tfrac{1}{2}D_\mathrm{h}\times\tfrac{1}{2}D_\mathrm{h}$ was slightly longer ($0\leq x\leq25\;\mathrm{m}\approxeq98D_\mathrm{h}>87D_\mathrm{h}$)
than the actual duct \cite{Gessner_Po_Emery_1979a} to avoid interaction between the uniform outflow pressure boundary-condition and computed results at the last
measurement station ($x=84D_\mathrm{h}$). The grid \figref{Fig_RSMP3DDF_s_A_001} is uniform in the streamwise ($x$) direction,
while in the $y$ and $z$ directions, $65\%$ of the $N_j=N_k=149$ points are stretched geometrically near the walls \cite{Gerolymos_Sauret_Vallet_2004b}
with ratio $r_j=r_k=1.067$ \tabref{Tab_RSMP3DDF_s_A_001}, the remaining $35\%$ being uniformly distributed in the centerline region. For the investigated flow conditions, the
first node at the walls is located at $\Delta y_w^+=\Delta z_w^+\lessapprox\tfrac{1}{2}$ \tabref{Tab_RSMP3DDF_s_A_001}.
At inflow \tabref{Tab_RSMP3DDF_s_A_002}, standard atmospheric total conditions ($p_{t_{\tsn{CL}_\mathrm{i}}}=101325\;\mathrm{Pa}$, $T_{t_{\tsn{CL}_\mathrm{i}}}=288\;\mathrm{K}$),
with a turbulent intensity $T_{u_{\tsn{CL}_\mathrm{i}}}=1\%$ and turbulent lengthscale $\ell_{\tsn{T}_{\tsn{CL}_\mathrm{i}}}=50\;\mathrm{mm}$, were assumed at the centerline.
The outflow pressure was adjusted to obtain the correct $Re_\tsn{B}=250000$ ($p_\mathrm{o}=0.995p_{t_{\tsn{CL}_\mathrm{i}}}$) corresponding to an inlet Mach number
at centerline $M_{\tsn{CL}_\mathrm{i}}\approxeq0.0516$ \tabref{Tab_RSMP3DDF_s_A_002}.
The initial inflow boundary-layer was adjusted to a different value for each turbulence model ($0.1\;\mathrm{mm}\leq\delta_\mathrm{i}\leq0.875\;\mathrm{mm}$)
to obtain a close fit to the experimental centerline velocity $\bar u_\tsn{CL}$ in the entry region of the duct ($x\in [0,10 D_\mathrm{h}]$; \figrefnp{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_001}).
Another approach would have been to start the computations with $0$ initial inflow boundary-layer thickness and to adjust in the post-processing phase the virtual origin of the developing boundary-layers ($x$-shift the results)
to best fit the experimental centerline velocity $\bar u_\tsn{CL}$ in the entry region.
The computations using the 3 \tsn{RSM}s and the $\mathrm{k}$--$\varepsilon$ model \parref{RSMP3DDF_s_TCsFS_ss_TCs} highlight \figrefsatob{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_001}
{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_006}
the great sensitivity of the predictions to the turbulence model.
It should be stated from the outset that the underlying Boussinesq hypothesis \cite[pp. 273--278]{Wilcox_1998a} renders the linear \tsn{LS} $\mathrm{k}$--$\varepsilon$ model
ill-adapted for the present Reynolds-stress-anisotropy-driven flow \cite{Gessner_Emery_1981a}; results with the baseline \tsn{LS} $\mathrm{k}$--$\varepsilon$ model are only included as a reference to the limitations of standard Boussinesq models.
In the initial part of the duct ($0\lessapprox x\lessapprox30D_\mathrm{h}$; \figrefnp{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_001}), all of the 4 models \parref{RSMP3DDF_s_TCsFS_ss_TCs} correctly predict the
thickening of and associated blockage by the developing wall-layers, which determine, because of massflow conservation, the centerline velocity $\bar u_\tsn{CL}$.
Recall that the initial conditions for the boundary-layers at inflow ($x=0$) were independently adjusted for each turbulence model \tabref{Tab_RSMP3DDF_s_A_002},
precisely to obtain the best fit of $\bar u_\tsn{CL}$ in this region ($0\lessapprox x\lessapprox30D_\mathrm{h}$; \figrefnp{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_001}).
The best prediction is obtained by the \tsn{GLVY} and \tsn{GV} \tsn{RSM}s (whose results are quite similar; \figrefnp{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_001}),
both of which correctly simulate the $\bar u_\tsn{CL}$-peak ($30D_\mathrm{h}\lessapprox x\lessapprox50D_\mathrm{h}$; \figrefnp{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_001})
and the final fully developed level at $x=84D_\mathrm{h}$ \figref{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_001}. However, the results of the \tsn{GLVY} and \tsn{GV} \tsn{RSM}s do not
tend to this final level monotonically, as the experimental data in the region $50D_\mathrm{h}\lessapprox x\lessapprox80D_\mathrm{h}$ seem to indicate, but exhibit a $\sim\!\!2.5\%$ undershoot before reaching the correct
fully developed level at $x=84D_\mathrm{h}$ \figref{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_001}.
In contrast with the \tsn{GLVY} and \tsn{GV} \tsn{RSM}s, the \tsn{WNF--LSS RSM} severely underpredicts the experimentally observed $\bar u_\tsn{CL}$-peak ($30D_\mathrm{h}\lessapprox x\lessapprox60D_\mathrm{h}$; \figrefnp{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_001})
and also underpredicts by $\sim\!\!2.5\%$ the final fully developed value ($x=84D_\mathrm{h}$; \figrefnp{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_001}).
On the other hand, the \tsn{WNF--LSS RSM} tends to this final value in a less oscillatory fashion ($40D_\mathrm{h}\lessapprox x\lessapprox80D_\mathrm{h}$; \figrefnp{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_001}).
Finally, the \tsn{LS} $\mathrm{k}$--$\varepsilon$ model also underestimates the $\bar u_\tsn{CL}$-peak ($30D_\mathrm{h}\lessapprox x\lessapprox60D_\mathrm{h}$; \figrefnp{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_001})
and tends monotonically to an $\sim\!\!1.5\%$ overestimated value of the final fully developed level ($x=84D_\mathrm{h}$; \figrefnp{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_001}).
The detailed evolution of the streamwise mean-flow velocity $\bar u$ profiles \figref{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_002} provides insight into the predictions of centerline velocity $\bar u_\tsn{CL}$ \figref{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_001}
by the different models. The term wall-bisector was used by Gessner and Emery \cite{Gessner_Emery_1981a} to denote the symmetry-plane at $z=\mathrm{a}=\tfrac{1}{2}D_\mathrm{h}$
and the term corner-bisector to denote the diagonal with distance \smash{$y_c:=\tfrac{\sqrt{2}}{2}(y-y_w)+\tfrac{\sqrt{2}}{2}(z-z_w)$} from the corner (notice that $\mathrm{a}_c^{-1}y_c=\mathrm{a}^{-1}y=\mathrm{a}^{-1}z$
along the diagonal whose length between the corner and the centerline is $\mathrm{a}_c:=\mathrm{a}\sqrt{2}$).
The flowfield along the corner-bisector $y_c$ is strongly influenced by the secondary flows. The \tsn{GLVY} and \tsn{GV} \tsn{RSM}s yield quite accurate results for the $\bar u$ profiles \figref{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_002},
both along the wall-bisector $y$ ($z=\mathrm{a}$; \figrefnp{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_002}) and along the corner-bisector $y_c$ ($z=y$; \figrefnp{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_002}).
Notice, nonetheless, that no experimental data are available in the region $50D_\mathrm{h}\lessapprox x\lessapprox80D_\mathrm{h}$ where the slight undershoot in centerline velocity $\bar u_\tsn{CL}$
was observed \figref{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_001}. The predictions of the \tsn{WNF--LSS RSM} for the streamwise velocity $\bar u$ \figref{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_001}
are similar to those of the \tsn{GLVY} and \tsn{GV} \tsn{RSM}s, except for the outer part (wake region) of the boundary-layer, especially at $x=40D_\mathrm{h}$ along the wall-bisector $y$ ($z=\mathrm{a}$; \figrefnp{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_002}) and at
$x\in\{8D_\mathrm{h},16D_\mathrm{h},24D_\mathrm{h},40D_\mathrm{h}\}$ along the corner-bisector $y_c$ ($z=y$; \figrefnp{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_002}).
Finally, expectedly, the linear \tsn{LS} $\mathrm{k}$--$\varepsilon$ model makes the worst prediction, especially at the last 2 stations
$x\in\{40D_\mathrm{h},84D_\mathrm{h}\}$, where it overpredicts $\bar u$ in the lower part of the boundary-layer along the wall-bisector $y$ ($z=\mathrm{a}$; \figrefnp{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_002})
and rather severely underpredicts it in the lower part of the boundary-layer along the corner-bisector $y_c$ ($z=y$; \figrefnp{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_002}).
By the continuity equation the $x$-wise development of the streamwise velocity $\bar u$ is related to the profiles of the in-plane velocity components, $\bar v$ and $\bar w$. Measurements of the $y$-wise component $\bar v$
are only available at the last 2 measurement planes ($x\in\{40D_\mathrm{h},84D_\mathrm{h}\}$; \figrefnp{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_003}). Notice first that, along the wall-bisector $y$ ($z=\mathrm{a}$) we have $\bar w=0$ by symmetry,
while along the corner-bisector $y_c$ ($z=y$) we have $\bar w=\bar v$ again by symmetry. Contrary to the results for the profiles of streamwise velocity $\bar u$ \figref{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_002},
the predictions of the $y$-wise component $\bar v$ have noticeable differences between the various models \figref{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_003}. The \tsn{GLVY RSM} gives the best prediction of secondary velocities,
both along the wall-bisector $y$ ($z=\mathrm{a}$; \figrefnp{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_003}) and along the corner-bisector $y_c$ ($z=y$; \figrefnp{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_003}).
Although the agreement of the \tsn{GLVY RSM} results with measurements is quite satisfactory at $x=40D_\mathrm{h}$, the secondary velocities are underestimated at the last measurement station $x=84D_\mathrm{h}$ \figref{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_003}.
The \tsn{GV RSM} gives results very close to those of the \tsn{GLVY RSM} along the corner-bisector $y_c$ ($z=y$; \figrefnp{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_003}),
some discrepancies very near the corner ($y_c\lessapprox0.1\mathrm{a}_c$; \figrefnp{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_003}) notwithstanding, but underestimates $\bar v$ along the wall-bisector $y$ ($z=\mathrm{a}$; \figrefnp{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_003})
at the outer part of the boundary-layer. The \tsn{WNF--LSS RSM}, does predict secondary flows, less intense than the \tsn{GLVY} and \tsn{GV} \tsn{RSM}s \figref{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_003},
while the linear \tsn{LS} $\mathrm{k}$--$\varepsilon$ model completely fails \figref{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_003},
implying that the strong values of $\bar v$ at the last measurement stations ($x\in\{40D_\mathrm{h},84D_\mathrm{h}\}$; \figrefnp{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_003}) are the consequence of secondary turbulence-driven flows,
in a region where the flow approaches the fully developed state \cite{Gessner_Jones_1965a,Gessner_Emery_1981a,Bradshaw_1987a}.
\vspace{-.05in}
The comparison of computational results with measured Reynolds-stresses \figrefsatob{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_004}
{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_006}
is consistent with the comparisons of the mean-flow velocity field \figrefsab{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_002}
{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_003}.
The \tsn{GLVY} and \tsn{GV} \tsn{RSM}s give the best overall prediction of the shear Reynolds-stresses, $\overline{u'v'}$ along the wall-bisector $y$ ($z=\mathrm{a}$; \figrefnp{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_004})
and $\overline{u'v'}=\overline{u'w'}$ (by symmetry) along the corner-bisector $y_c$ ($z=y$; \figrefnp{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_004}),
but overestimate their magnitude, especially along the corner-bisector $y_c$ ($z=y$; \figrefnp{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_004}).
The experimental data are generally consistent with the symmetry condition $\overline{u'v'}=\overline{u'w'}$ along the corner-bisector $y_c$ ($z=y$; \figrefnp{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_004}),
except at $x=40D_\mathrm{h}$ in the outer part of the boundary-layer ($y_c\gtrapprox0.4\mathrm{a}_c$; $z=y$; \figrefnp{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_004}).
The overprediction of the shear Reynolds-stress $\overline{u'v'}=\overline{u'w'}$ along the corner-bisector $y_c$ at $x=40D_\mathrm{h}$ ($z=y$; \figrefnp{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_004}) is not consistent
with the satisfactory prediction of the mean-flow velocity field at this station ($x=40D_\mathrm{h}$; $z=y$; \figrefsnpab{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_002}
{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_003}),
especially as the $x$-wise gradients predicted by the \tsn{GLVY} and \tsn{GV} \tsn{RSM}s are in good agreement with experimental data at $x=40D_\mathrm{h}$ \figref{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_001}.
Regarding the last measurement station at $x=84D_\mathrm{h}$, the shear Reynolds-stress $\overline{u'v'}=\overline{u'w'}$ predicted by the \tsn{GLVY} and \tsn{GV} \tsn{RSM}s
along the corner-bisector $y_c$ ($x=84D_\mathrm{h}$; $z=y$; \figrefnp{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_004}) is closer to the experimental data than at $x=40D_\mathrm{h}$,
but computed values are still larger in magnitude by $\sim\!\!30\%$. The \tsn{WNF--LSS RSM} and \tsn{LS} $\mathrm{k}$--$\varepsilon$ models predictions of the shear Reynolds-stress $\overline{u'v'}$
along the wall-bisector $y$ ($z=\mathrm{a}$; \figrefnp{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_004}) are generally similar with those of the \tsn{GLVY} and \tsn{GV} \tsn{RSM}s, in satisfactory agreement with measurements.
On the other hand, the \tsn{WNF--LSS RSM} and the \tsn{LS} $\mathrm{k}$--$\varepsilon$ model perform less satisfactorily than the \tsn{GLVY} and \tsn{GV} \tsn{RSM}s regarding the prediction
of the shear Reynolds-stresses $\overline{u'v'}=\overline{u'w'}$ (by symmetry) along the corner-bisector $y_c$ ($z=y$; \figrefnp{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_004}),
the \tsn{WNF--LSS RSM}, expectedly, performing better than the linear \tsn{LS} $\mathrm{k}$--$\varepsilon$ model.
The \tsn{GLVY} and \tsn{GV} \tsn{RSM}s predict quite accurately the streamwise normal Reynolds-stress $\overline{u'^2}$ \figref{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_005}
both along the wall-bisector $y$ ($z=\mathrm{a}$) and the corner-bisector $y_c$ ($z=y$), some slight discrepancies
along the corner-bisector $y_c$ ($x\in\{16D_\mathrm{h},24D_\mathrm{h},40D_\mathrm{h}\}$; $z=y$; \figrefnp{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_005}) notwithstanding.
The predictions of the streamwise normal Reynolds-stress $\overline{u'^2}$ by the \tsn{WNF--LSS RSM} and the \tsn{LS} $\mathrm{k}$--$\varepsilon$ model are, again,
less satisfactory \figref{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_005}, especially along the wall-bisector $y$ ($z=\mathrm{a}$; \figrefnp{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_005}).
Regarding the prediction of the other normal Reynolds-stresses, wall-normal $\overline{v'^2}$ along the wall-bisector $y$ ($z=\mathrm{a}$), transverse $\overline{w'^2}$ along the wall-bisector $y$ ($z=\mathrm{a}$),
and secondary $\overline{v'^2}=\overline{w'^2}$ along the corner-bisector $y_c$ ($z=y$), all 3 \tsn{RSM}s (\tsn{GLVY}, \tsn{GV} and \tsn{WNF--LSS}) are in good agreement with experimental data \figref{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_006},
in contrast with the linear \tsn{LS} $\mathrm{k}$--$\varepsilon$ model, which completely fails in predicting the Reynolds-stress tensor anisotropy \figref{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_006},
because of the pathological shortcomings of the Boussinesq hypothesis \cite[pp. 273--278]{Wilcox_1998a}.
\begin{figure}[h!]
\begin{center}
\begin{picture}(340,480)
\put(0,-5){\includegraphics[angle=0,width=340pt,bb=85 99 524 724]{Fig_FlowTurbComb_01_GE_1981_sqduct_vv_ww.pdf}}
\end{picture}
\end{center}
\caption{Comparison of measured \cite{Gessner_Emery_1981a} normal Reynolds-stresses $\overline{v'v'}$ and $\overline{w'w'}$, along the wall-bisector ($z=\mathrm{a}$) and along the corner-bisector ($z=y$, where by symmetry $\overline{w'w'}=\overline{v'v'}$),
at the 5 experimental measurement stations, with
computations ($18\times10^6$ points grid discretizing $\tfrac{1}{4}$ of the square duct; \tabrefnp{Tab_RSMP3DDF_s_A_001})
using \parref{RSMP3DDF_s_TCsFS_ss_TCs} the \tsn{GV} \cite{Gerolymos_Vallet_2001a}, the \tsn{WNF--LSS} \cite{Gerolymos_Sauret_Vallet_2004a}
and the \tsn{GLVY} \cite{Gerolymos_Lo_Vallet_Younis_2012a} \tsn{RSM}s, and the \tsn{LS} \cite{Launder_Sharma_1974a} linear $\mathrm{k}$--$\varepsilon$ model, for developing turbulent flow in a square duct
($Re_\tsn{B}=250000$, $\bar M_\tsn{CL}\approxeq0.05$; \tabrefnp{Tab_RSMP3DDF_s_A_002}).}
\label{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_006}
\end{figure}
\clearpage
To explain the better agreement with experimental data of the \tsn{GLVY} and \tsn{GV} \tsn{RSM}s, compared to the \tsn{WNF--LSS RSM} \figrefsatob{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_001}
{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_006},
it is interesting to examine the differences between the closures \tabref{Tab_RSMP3DDF_s_TCsFS_ss_TCs_001}.
The differences \tabref{Tab_RSMP3DDF_s_TCsFS_ss_TCs_001} between the \tsn{GLVY} and \tsn{GV} \tsn{RSM}s (pressure diffusion $d_{ij}^{(p)}$, explicit algebraic modelling for $\varepsilon_{ij}$,
extra inhomogeneous terms in $\Pi_{ij}$) do not have any substantial influence on the prediction of the \cite{Gessner_Emery_1981a} square duct flow, the only noticeable difference
being the better prediction by the \tsn{GLVY RSM} of the wall-normal velocity $\bar v$ along the wall-bisector $y$ ($x\in\{40D_\mathrm{h},84D_\mathrm{h}\}$; $z=\mathrm{a}$; \figrefnp{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_003}),
especially in the outer part of the boundary-layer ($y\gtrapprox0.6\mathrm{a}$). There are 2 main differences between the \tsn{WNF--LSS RSM} and the \tsn{GV RSM} \tabref{Tab_RSMP3DDF_s_TCsFS_ss_TCs_001},
the coefficient-function $C_\phi^{(\tsn{RH})}$ of the isotropisation-of-production model for the homogeneous rapid part of redistribution \eqref{Eq_RSMP3DDF_s_TCsFS_ss_TCs_007c},
and the closure for the triple velocity correlations \eqref{Eq_RSMP3DDF_s_TCsFS_ss_TCs_005}. The coefficient $C_\phi^{(\tsn{RH})}$ \tabref{Tab_RSMP3DDF_s_TCsFS_ss_TCs_001} in the \tsn{GLVY} and \tsn{GV} \tsn{RSM}s was designed
\cite[Fig. 4, p. 1838]{Gerolymos_Vallet_2001a} to increase rapidly towards a value of $1$ when Lumley's \cite{Lumley_1978a} flatness parameter $A$ \eqref{Eq_RSMP3DDF_s_TCsFS_ss_TCs_001b} increases
beyond $0.55$ (which is approximately the value of $A$ just before the beginning of the logarithmic zone of the flat-plate boundary-layer velocity profile). Regarding turbulent diffusion \tabref{Tab_RSMP3DDF_s_TCsFS_ss_TCs_001},
the \tsn{GLVY} and \tsn{GV} \tsn{RSM}s use the Hanjali\'c-Launder \cite{Hanjalic_Launder_1976a}
closure, whereas the \tsn{WNF--LSS RSM} uses the Daly-Harlow \cite{Daly_Harlow_1970a} closure \parref{RSMP3DDF_s_TCsFS_ss_TCs}.
Obviously \figrefsatob{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_001}
{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_006},
the combined use of these 2 modelling choices in the \tsn{GLVY} and \tsn{GV} \tsn{RSM}s substantially improves the prediction of the \cite{Gessner_Emery_1981a} square duct flow
compared to the \tsn{WNF--LSS RSM}. To put into perspective the specific influence of each of the 2 mechanisms, a test-model (not recommended for practical use), hereafter \tsn{GV--DH RSM},
which combines the $C_\phi^{(\tsn{RH})}$ coefficient-function of the \tsn{GLVY} and \tsn{GV} \tsn{RSM}s \tabref{Tab_RSMP3DDF_s_TCsFS_ss_TCs_001} with the Daly-Harlow \cite{Daly_Harlow_1970a} closure
for turbulent diffusion (with appropriate recalibration of various coefficients to get the correct log-law in flat-plate boundary-layer flow; \cite[Tab. 2, p. 418]{Gerolymos_Sauret_Vallet_2004a}),
has been developed \cite{Gerolymos_Sauret_Vallet_2004a,Vallet_2007a}.
Calculations of the \cite{Gessner_Emery_1981a} square duct flow with the \tsn{GV}, \tsn{GV--DH} and \tsn{WNF--LSS} \tsn{RSM}s, using the same inflow boundary-layer-thickness
\cite[$\delta_{y_i}=\delta_{z_i}=0.5\;\mathrm{mm}$, Fig. 7, p. 422]{Gerolymos_Sauret_Vallet_2004a} for all of the models, indicate that $C_\phi^{(\tsn{RH})}$ influences the initial part of the
region where the boundary-layers on the 4 walls first merge ($30D_\mathrm{h}\lessapprox x\lessapprox40D_\mathrm{h}$; \figrefnp{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_001}) whereas turbulent diffusion is active especially in the region
after the centerline velocity peak ($40D_\mathrm{h}\lessapprox x\lessapprox60D_\mathrm{h}$; \figrefnp{Fig_RSMP3DDF_s_A_ss_DTFSDGE1981_001}).
\subsection{Circular-to-rectangular transition duct \cite{Davis_Gessner_1992a}}\label{RSMP3DDF_s_A_ss_CtoRTDDG1992}
This configuration \cite{Davis_1991a,Davis_Gessner_1992a}
\begin{figure}[h!]
\begin{center}
\begin{picture}(340,300)
\put(0,-5){\includegraphics[angle=0,width=340pt,bb= 99 204 522 586]{Fig_FlowTurbComb_01_DG_1992_CtoR_duct_Cp_cf.pdf}}
\end{picture}
\end{center}
\caption{Comparison of measured \cite{Davis_Gessner_1992a} wall-pressure coefficient $C_{p_\tsn{B}}$ and skin-friction coefficient $c_{f_\tsn{B}}$,
at 4 experimental measurement stations, with
computations ($10\times10^6$ points grid discretizing the entire duct; \tabrefnp{Tab_RSMP3DDF_s_A_001})
using \parref{RSMP3DDF_s_TCsFS_ss_TCs} the \tsn{GV} \cite{Gerolymos_Vallet_2001a}, the \tsn{WNF--LSS} \cite{Gerolymos_Sauret_Vallet_2004a}
and the \tsn{GLVY} \cite{Gerolymos_Lo_Vallet_Younis_2012a} \tsn{RSM}s, and the \tsn{LS} \cite{Launder_Sharma_1974a} linear $\mathrm{k}$--$\varepsilon$ model, for turbulent flow in a \tsn{C}-to-\tsn{R} transition duct
($Re_\tsn{B}=390000$, $\bar M_\tsn{CL}\approxeq0.09$; \tabrefnp{Tab_RSMP3DDF_s_A_002}; $\mathrm{s}$ is the curvilinear coordinate of the duct contour in the $yz$-plane normalized by $\tfrac{1}{4}$ of the circumference $\mathrm{s}_{\frac{1}{4}}$).}
\label{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_001}
\end{figure}
\begin{table}[h!]
\begin{center}
\caption{Comparison of experimental inflow boundary-layer parameters, for the \tsn{C}-to-\tsn{R} transition duct \cite[Tab. 1, p. 370]{Davis_Gessner_1992a} at station 1 located at $x=-R_{\tsn{CSG}_1}$
(cross-sectional transition starts at $x=+R_{\tsn{CSG}_1}$, one diameter downstream; \figrefnp{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_001}) and for
the \tsn{S}-duct \cite[Tab. 2, p. 671]{Wellborn_Reichert_Okiishi_1994a} at station \tsn{A} located at $x=-R_{\tsn{CSG}_\tsn{A}}$
(centerline and cross-section evolution starts at $x=0$, half a diameter downstream; \figrefsnpab{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_001}{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_002}),
with computational results (grids \tabrefnp{Tab_RSMP3DDF_s_A_001}) using the \tsn{GLVY RSM} \cite{Gerolymos_Lo_Vallet_Younis_2012a}, the \tsn{GV RSM} \cite{Gerolymos_Vallet_2001a}, the \tsn{WNF--LSS RSM} \cite{Gerolymos_Sauret_Vallet_2004a}
and the \tsn{LS} $\mathrm{k}$-$\varepsilon$ \cite{Launder_Sharma_1974a}.}
\label{Tab_RSMP3DDF_s_A_003}
\input{Tab_FlowTurbComb_01_inflow_BL_data}
\end{center}
{\footnotesize $\bar u_\tsn{B}$: bulk velocity;
$R_\tsn{CSG}$ ($D_\tsn{CSG}$): casing radius (diameter);
$Re_\tsn{B}=\bar u_\tsn{B} D_\tsn{CSG} \nu^{-1}$: bulk Reynolds number ($\nu$ is the practically constant kinematic viscosity);
$\delta_{995}$ ($\delta_{95}$): boundary-layer thickness measured from the wall to the location where the velocity is $99.5\%$ ($95\%$) of $\bar u_\tsn{CL}$;
$\delta_{{1_k}_\mathrm{axi}}=\int^{\delta}_0(1-\bar{u}/\bar{u}_\tsn{CL})(R/R_\tsn{CSG})d(R_\tsn{CSG}-R)$: axisymmetric kinematic boundary-layer displacement-thickness;
$\delta_{{2_k}_\mathrm{axi}}=\int^{\delta}_0(1-\bar{u}/\bar{u}_\tsn{CL})(\bar{u}/\bar{u}_\tsn{CL})(R/R_\tsn{CSG})d(R_\tsn{CSG}-R)$: axisymmetric kinematic boundary-layer momentum-thickness;
$\delta_{{3_k}_\mathrm{axi}}=\int^{\delta}_0(1-\bar{u}^2/\bar{u}_\tsn{CL}^2)(\bar{u}/\bar{u}_\tsn{CL})(R/R_\tsn{CSG})d(R_\tsn{CSG}-R)$: axisymmetric kinematic boundary-layer energy-thickness;
$H_{{12_k}_\mathrm{axi}}=\delta_{{1_k}_\mathrm{axi}}/\delta_{{2_k}_\mathrm{axi}}$ ($H_{{32_k}_\mathrm{axi}}=\delta_{{3_k}_\mathrm{axi}}/\delta_{{2_k}_\mathrm{axi}}$): axisymmetric kinematic boundary-layer shape-factors;
$\breve M_\tsn{CL}=\tilde u_\tsn{CL} \breve a_\tsn{CL}$: centerline Mach number ($\breve a_\tsn{CL}$ is the centerline sound-speed);
$Re_\tsn{CL}=\tilde u_\tsn{CL} D_\tsn{CSG} \nu_\tsn{CL}^{-1}$: Reynolds number based on centerline flow quantities;
axisymmetric integral boundary-layer thicknesses defined following Reichert \cite[p. 67]{Reichert_1991a}
}
\end{table}
is a transition duct where the cross-section changes \figref{Fig_RSMP3DDF_s_A_001} from circular at the inlet to quasi-rectangular at the exit (rectangle aspect-ratio of $3$ at the exit section).
Such geometries are typical of the transition section necessary to connect an aircraft engine exit to a rectangular nozzle \cite{Reichert_1991a}.
The precise geometrical specification of the duct's cross-section is superelliptical \cite[(A.1), p. 136]{Davis_1991a} so that
the exit section has slightly rounded corners with a "{\em variable radius fillet}" \cite[p. 2]{Davis_1991a}. The diameter of the circular inlet section is $D_{\tsn{CSG}_1}=2R_{\tsn{CSG}_1}=0.2043\;\mathrm{m}$ \cite[p. 137]{Davis_1991a},
and the length of the transition section (from inlet station 2 to exit station 5; \figrefnp{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_001}) is $\tfrac{3}{2}D_{\tsn{CSG}_1}$. Although there is no net cross-sectional area change,
between inflow and outflow, locally \cite[Fig. 4, p. 242]{Lien_Leschziner_1996a}, the transition section is divergent \cite[p. 2]{Davis_1991a} from inlet to midpoint (cross-sectional area increase of $15\%$)
and then convergent from midpoint to exit (cross-sectional area decreases back to the inlet area).
The duct is cylindrical upstream (circular cross-section of diameter $D_{\tsn{CSG}_1}$ for several diameters upstream of station 2) and
downstream (quasi-rectangular superelliptical constant cross-section for several inlet-diameters $D_{\tsn{CSG}_1}$ downstream of station 5) of the transition section \cite[Fig. 3.1, p. 22]{Davis_1991a}.
The flow \cite{Davis_Gessner_1992a} is essentially incompressible (centerline Mach number $\bar M_\tsn{CL}\approxeq0.10$) at bulk Reynolds number $Re_\tsn{B}\approxeq390000$
($Re_\tsn{B}=\bar u_\tsn{B} D_{\tsn{CSG}_1} \nu^{-1}$, where $\bar u_\tsn{B}$ is the bulk velocity and $\nu$ is the practically constant kinematic viscosity).
Measurements, taken at 6 axial stations \figref{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_001}, include total pressure (circular and flattened Pitot tubes and Kiel probes),
static pressures (static pressure probes and wall pressure taps), mean-velocities and Reynolds-stresses (hot wires) and skin-friction (Preston tubes resting on the duct walls).
These data are available \cite{Davis_Gessner_1992a} in digital form \cite{ERCOFTAC_1999a}.
Because of the combined streamwise evolution of both cross-sectional form and area \figrefsab{Fig_RSMP3DDF_s_A_001}
{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_001},
the curvature of the duct's walls changes sign $x$-wise \cite[Fig. 3, p. 241]{Lien_Leschziner_1996a}.
The upper and lower ($z$-wise) walls are concave from the inlet (station 2) to approximately midpoint (located between stations 3 and 4; \figrefnp{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_001})
and then convex from approximately midpoint to exit (station 5; \figrefnp{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_001}). The opposite applies to the sidewalls ($y$-wise) which are convex in the first part (from station 2 to approximately midpoint)
then switching \cite[Fig. 3, p. 241]{Lien_Leschziner_1996a} to concave (from approximately midpoint to exit station 5). This streamwise evolution of the duct's geometry directly affects the mean pressure field,
inducing strong pressure gradients, both streamwise (area change) and crossflow (wall curvature), generating intense ($\gtrapprox10\%\;\bar u_\tsn{B}$; \cite[Fig. 7, p. 371]{Davis_Gessner_1992a})
pressure-driven secondary flows \cite[Prandtl's first kind]{Bradshaw_1987a}
which rapidly form 2 contrarotating ($y$-wise symmetry) pairs of contrarotating ($z$-wise symmetry) vortices \cite[Fig. 7, p. 371]{Davis_Gessner_1992a}, one pair near the $z=0$ midplane of each sidewall.
Downstream of station 5 (exit of the transition section; \figrefnp{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_001}) the vortex system persists, evolving streamwise, and is clearly visible at the last measurement station 6, 2 inlet diameters ($2D_{\tsn{CSG}_1}$)
downstream of station 5 \figref{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_001}.
The in-depth analysis of the experimental data by Davis \cite{Davis_1991a} has largely contributed to our understanding of the dynamics of the mean-flow and associated Reynolds-stresses.
Careful computations of the Davis and Gessner \cite{Davis_Gessner_1992a} \tsn{C}-to-\tsn{R} transition duct, in quite satisfactory agreement with experimental measurements, have been reported
by Sotiropoulos and Patel \cite{Sotiropoulos_Patel_1994a}, using a 7-equation \tsn{RSM}, which is a variant of the Launder-Shima \cite{Launder_Shima_1989a} \tsn{RSM} in 2 respects, (a) the use of the
Hanjali\'c-Launder \cite{Hanjalic_Launder_1976a} closure for the triple velocity correlations in lieu of the Daly-Harlow \cite{Daly_Harlow_1970a} closure adopted for turbulent diffusion in the original model \cite{Launder_Shima_1989a},
and (b) the use of the modified coefficient-functions in the $\varepsilon$-equation introduced by Shima \cite{Shima_1993a} to improve the prediction of skin-friction.
Notice the the closure used for turbulent diffusion has a strong influence on the predicted secondary flows \cite{Vallet_2007a}.
Sotiropoulos and Patel \cite{Sotiropoulos_Patel_1995b} have further exploited their computational results to analyse the streamwise ($x$-wise component of) mean-flow vorticity equation \cite[(1), p. 504]{Sotiropoulos_Patel_1995b},
and have shown that all of the vorticity-production mechanisms (vortex stretching and skewing, turbulence-induced production) are important in different regions of the flow.
\begin{figure}[h!]
\begin{center}
\begin{picture}(340,330)
\put(0,-5){\includegraphics[angle=0,width=340pt,bb=59 140 513 579]{Fig_FlowTurbComb_01_DG_1992_CtoR_duct_u.pdf}}
\end{picture}
\end{center}
\caption{Comparison of measured \cite{Davis_Gessner_1992a} streamwise ($x$-wise) velocity $\bar u$, along the $y$-wise ($z=0$ symmetry plane) and the $z$-wise ($y=0$ symmetry plane) directions,
at 5 experimental measurement stations, with
computations ($10\times10^6$ points grid discretizing the entire duct; \tabrefnp{Tab_RSMP3DDF_s_A_001})
using \parref{RSMP3DDF_s_TCsFS_ss_TCs} the \tsn{GV} \cite{Gerolymos_Vallet_2001a}, the \tsn{WNF--LSS} \cite{Gerolymos_Sauret_Vallet_2004a}
and the \tsn{GLVY} \cite{Gerolymos_Lo_Vallet_Younis_2012a} \tsn{RSM}s, and the \tsn{LS} \cite{Launder_Sharma_1974a} linear $\mathrm{k}$--$\varepsilon$ model, for turbulent flow in a \tsn{C}-to-\tsn{R} transition duct
($Re_\tsn{B}=390000$, $\bar M_\tsn{CL}\approxeq0.09$; \tabrefnp{Tab_RSMP3DDF_s_A_002}; contour plots \tsn{GLVY RSM}).}
\label{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_002}
\end{figure}
The computations were run on a $10\times10^6$ grid \tabref{Tab_RSMP3DDF_s_A_001} discretizing the entire duct without symmetry conditions \figref{Fig_RSMP3DDF_s_A_001}. Based on the grid-convergence studies of Sotiropoulos and Patel \cite{Sotiropoulos_Patel_1994a},
who used an $O(\Delta\ell^2)$ upwind numerical scheme for the incompressible \tsn{RSM--RANS} equations, this grid \tabref{Tab_RSMP3DDF_s_A_001} is sufficiently fine. As defined in the experimental study \cite{Davis_1991a,Davis_Gessner_1992a},
the origin of the coordinates system, $x=0$, is located at mid-distance between station 1 and station 2, located at the beginning of the transition section ($x_1=-R_{\tsn{CSG}_1}$, $x_2=+R_{\tsn{CSG}_1}$; \figrefnp{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_001}).
The computational domain ($-D_{\tsn{CSG}_1}\leq x\leq5D_{\tsn{CSG}_1}$) starts $\tfrac{3}{2}$ inlet-diameters ($\tfrac{3}{2}D_{\tsn{CSG}_1}$) upstream of the transition section inlet (station 2; \figrefnp{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_001})
and extends 3 inlet-diameters ($3D_{\tsn{CSG}_1}$) downstream of the transition section exit (station 5; \figrefnp{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_001}), to avoid interaction between the uniform outflow pressure boundary-condition and computed results at the last
measurement station 6 located 2 inlet-diameters ($2D_{\tsn{CSG}_1}$) downstream of the transition section exit \figref{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_001}.
The grid is uniform in the streamwise ($x$) direction and consists of 2 blocks (\figrefnp{Fig_RSMP3DDF_s_A_001}; \tabrefnp{Tab_RSMP3DDF_s_A_001}).
The inner block (\tsn{H}$_\tsn{$\square$}$; \tabrefnp{Tab_RSMP3DDF_s_A_001}) is an \tsn{H}-grid of $x$-wise varying square cross-section with uniform $yz$-spacing, introduced to avoid the axis-singularity of an axisymmetric-type grid.
The outer block (\tsn{O}$_\tsn{$\square$}$; \tabrefnp{Tab_RSMP3DDF_s_A_001}) is stretched geometrically near the wall with ratio $r_k$ \tabref{Tab_RSMP3DDF_s_A_001}.
For the investigated flow conditions, the first node at the walls is located at $\Delta n_w^+\lessapprox\tfrac{2}{10}$ \tabref{Tab_RSMP3DDF_s_A_001}, $n$ being the wall-normal direction.
\begin{figure}[h!]
\begin{center}
\begin{picture}(340,210)
\put(0,-5){\includegraphics[angle=0,width=340pt,bb= 56 130 515 410]{Fig_FlowTurbComb_01_DG_1992_CtoR_duct_v.pdf}}
\end{picture}
\end{center}
\caption{Comparison of measured \cite{Davis_Gessner_1992a} wall-normal velocity $\bar v$, along the $y$-wise ($z=0$ symmetry plane) direction,
at 4 experimental measurement stations, with
computations ($10\times10^6$ points grid discretizing the entire duct; \tabrefnp{Tab_RSMP3DDF_s_A_001})
using \parref{RSMP3DDF_s_TCsFS_ss_TCs} the \tsn{GV} \cite{Gerolymos_Vallet_2001a}, the \tsn{WNF--LSS} \cite{Gerolymos_Sauret_Vallet_2004a}
and the \tsn{GLVY} \cite{Gerolymos_Lo_Vallet_Younis_2012a} \tsn{RSM}s, and the \tsn{LS} \cite{Launder_Sharma_1974a} linear $\mathrm{k}$--$\varepsilon$ model, for turbulent flow in a \tsn{C}-to-\tsn{R} transition duct
($Re_\tsn{B}=390000$, $\bar M_\tsn{CL}\approxeq0.09$; \tabrefnp{Tab_RSMP3DDF_s_A_002}; contour plots \tsn{GLVY RSM}).}
\label{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_003}
\end{figure}
\begin{figure}[h!]
\begin{center}
\begin{picture}(340,210)
\put(0,-5){\includegraphics[angle=0,width=340pt,bb= 60 293 520 564]{Fig_FlowTurbComb_01_DG_1992_CtoR_duct_w.pdf}}
\end{picture}
\end{center}
\caption{Comparison of measured \cite{Davis_Gessner_1992a} wall-normal velocity $\bar w$, along the $z$-wise ($y=0$ symmetry plane) direction,
at 4 experimental measurement stations, with
computations ($10\times10^6$ points grid discretizing the entire duct; \tabrefnp{Tab_RSMP3DDF_s_A_001})
using \parref{RSMP3DDF_s_TCsFS_ss_TCs} the \tsn{GV} \cite{Gerolymos_Vallet_2001a}, the \tsn{WNF--LSS} \cite{Gerolymos_Sauret_Vallet_2004a}
and the \tsn{GLVY} \cite{Gerolymos_Lo_Vallet_Younis_2012a} \tsn{RSM}s, and the \tsn{LS} \cite{Launder_Sharma_1974a} linear $\mathrm{k}$--$\varepsilon$ model, for turbulent flow in a \tsn{C}-to-\tsn{R} transition duct
($Re_\tsn{B}=390000$, $\bar M_\tsn{CL}\approxeq0.09$; \tabrefnp{Tab_RSMP3DDF_s_A_002}; contour plots \tsn{GLVY RSM}).}
\label{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_004}
\end{figure}
\begin{figure}[h!]
\begin{center}
\begin{picture}(340,360)
\put(0,-5){\includegraphics[angle=0,width=340pt,bb=106 150 511 557]{Fig_FlowTurbComb_01_DG_1992_CtoR_duct_uu.pdf}}
\end{picture}
\end{center}
\caption{Comparison of measured \cite{Davis_Gessner_1992a} streamwise ($x$-wise) velocity-variance $\overline{u'u'}$, along the $y$-wise ($z=0$ symmetry plane) and the $z$-wise ($y=0$ symmetry plane) directions,
at 3 experimental measurement stations, with
computations ($10\times10^6$ points grid discretizing the entire duct; \tabrefnp{Tab_RSMP3DDF_s_A_001})
using \parref{RSMP3DDF_s_TCsFS_ss_TCs} the \tsn{GV} \cite{Gerolymos_Vallet_2001a}, the \tsn{WNF--LSS} \cite{Gerolymos_Sauret_Vallet_2004a}
and the \tsn{GLVY} \cite{Gerolymos_Lo_Vallet_Younis_2012a} \tsn{RSM}s, and the \tsn{LS} \cite{Launder_Sharma_1974a} linear $\mathrm{k}$--$\varepsilon$ model, for turbulent flow in a \tsn{C}-to-\tsn{R} transition duct
($Re_\tsn{B}=390000$, $\bar M_\tsn{CL}\approxeq0.09$; \tabrefnp{Tab_RSMP3DDF_s_A_002}; contour plots \tsn{GLVY RSM}).}
\label{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_005}
\end{figure}
\begin{figure}[h!]
\begin{center}
\begin{picture}(340,360)
\put(0,-5){\includegraphics[angle=0,width=340pt,bb=106 150 511 557]{Fig_FlowTurbComb_01_DG_1992_CtoR_duct_vv.pdf}}
\end{picture}
\end{center}
\caption{Comparison of measured \cite{Davis_Gessner_1992a} $y$-wise velocity-variance $\overline{v'v'}$, wall-normal along the $y$-wise ($z=0$ symmetry plane) direction and transverse along the $z$-wise ($y=0$ symmetry plane) direction,
at 3 experimental measurement stations, with
computations ($10\times10^6$ points grid discretizing the entire duct; \tabrefnp{Tab_RSMP3DDF_s_A_001})
using \parref{RSMP3DDF_s_TCsFS_ss_TCs} the \tsn{GV} \cite{Gerolymos_Vallet_2001a}, the \tsn{WNF--LSS} \cite{Gerolymos_Sauret_Vallet_2004a}
and the \tsn{GLVY} \cite{Gerolymos_Lo_Vallet_Younis_2012a} \tsn{RSM}s, and the \tsn{LS} \cite{Launder_Sharma_1974a} linear $\mathrm{k}$--$\varepsilon$ model, for turbulent flow in a \tsn{C}-to-\tsn{R} transition duct
($Re_\tsn{B}=390000$, $\bar M_\tsn{CL}\approxeq0.09$; \tabrefnp{Tab_RSMP3DDF_s_A_002}; contour plots \tsn{GLVY RSM}).}
\label{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_006}
\end{figure}
\begin{figure}[h!]
\begin{center}
\begin{picture}(340,360)
\put(0,-5){\includegraphics[angle=0,width=340pt,bb=106 150 511 557]{Fig_FlowTurbComb_01_DG_1992_CtoR_duct_ww.pdf}}
\end{picture}
\end{center}
\caption{Comparison of measured \cite{Davis_Gessner_1992a} $z$-wise velocity-variance $\overline{w'w'}$, transverse along the $y$-wise ($z=0$ symmetry plane) direction and wall-normal along the $z$-wise ($y=0$ symmetry plane) direction,
at 3 experimental measurement stations, with
computations ($10\times10^6$ points grid discretizing the entire duct; \tabrefnp{Tab_RSMP3DDF_s_A_001})
using \parref{RSMP3DDF_s_TCsFS_ss_TCs} the \tsn{GV} \cite{Gerolymos_Vallet_2001a}, the \tsn{WNF--LSS} \cite{Gerolymos_Sauret_Vallet_2004a}
and the \tsn{GLVY} \cite{Gerolymos_Lo_Vallet_Younis_2012a} \tsn{RSM}s, and the \tsn{LS} \cite{Launder_Sharma_1974a} linear $\mathrm{k}$--$\varepsilon$ model, for turbulent flow in a \tsn{C}-to-\tsn{R} transition duct
($Re_\tsn{B}=390000$, $\bar M_\tsn{CL}\approxeq0.09$; \tabrefnp{Tab_RSMP3DDF_s_A_002}; contour plots \tsn{GLVY RSM}).}
\label{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_007}
\end{figure}
\begin{figure}[h!]
\begin{center}
\begin{picture}(340,440)
\put(0,-5){\includegraphics[angle=0,width=340pt,bb=102 127 513 669]{Fig_FlowTurbComb_01_DG_1992_CtoR_duct_uv_wu.pdf}}
\end{picture}
\end{center}
\caption{Comparison of measured \cite{Davis_Gessner_1992a} shear Reynolds-stress, $\overline{u'v'}$ along the $y$-wise ($z=0$ symmetry plane) and $\overline{u'w'}$ the $z$-wise ($y=0$ symmetry plane) directions,
at 3 experimental measurement stations, with
computations ($10\times10^6$ points grid discretizing the entire duct; \tabrefnp{Tab_RSMP3DDF_s_A_001})
using \parref{RSMP3DDF_s_TCsFS_ss_TCs} the \tsn{GV} \cite{Gerolymos_Vallet_2001a}, the \tsn{WNF--LSS} \cite{Gerolymos_Sauret_Vallet_2004a}
and the \tsn{GLVY} \cite{Gerolymos_Lo_Vallet_Younis_2012a} \tsn{RSM}s, and the \tsn{LS} \cite{Launder_Sharma_1974a} linear $\mathrm{k}$--$\varepsilon$ model, for turbulent flow in a \tsn{C}-to-\tsn{R} transition duct
($Re_\tsn{B}=390000$, $\bar M_\tsn{CL}\approxeq0.09$; \tabrefnp{Tab_RSMP3DDF_s_A_002}; contour plots \tsn{GLVY RSM}).}
\label{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_008}
\end{figure}
At inflow \tabref{Tab_RSMP3DDF_s_A_002}, measured \cite{ERCOFTAC_1999a,Davis_1991a,Davis_Gessner_1992a}
total conditions ($p_{t_{\tsn{CL}_\mathrm{i}}}=101325\;\mathrm{Pa}$, $T_{t_{\tsn{CL}_\mathrm{i}}}=298.3\;\mathrm{K}$), with a turbulent intensity $T_{u_{\tsn{CL}_\mathrm{i}}}=0.3\%$, were applied at the centerline.
In the absence of experimental data, a turbulent lengthscale $\ell_{\tsn{T}_{\tsn{CL}_\mathrm{i}}}=50\;\mathrm{mm}$ was assumed at the centerline,
with reference to the duct radius ($R_{\tsn{CSG}_1}=0.10215\;\mathrm{m}$) and the measured boundary-layer thickness $\delta_{995}=30.85\;\mathrm{mm}$ at station 1 (\figrefnp{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_001}; \tabrefnp{Tab_RSMP3DDF_s_A_003}).
Detailed measurements of the boundary-layer profiles of mean-flow and Reynolds-stresses are available \cite{ERCOFTAC_1999a,Davis_1991a,Davis_Gessner_1992a},
and were interpolated onto the computational grid to define the inflow conditions. These data were extended to the wall, in the region where experimental data were not available, using semi-analytical profiles \cite{Gerolymos_Sauret_Vallet_2004c},
and used to define, by assuming local equilibrium in the boundary-layer and matching to the prescribed centerline $\ell_{\tsn{T}_{\tsn{CL}_\mathrm{i}}}$ \cite{Gerolymos_Sauret_Vallet_2004c}, the $\varepsilon$ profiles.
The outflow pressure was adjusted to obtain the correct $Re_\tsn{B}=390000$ ($p_\mathrm{o}=100627\;\mathrm{Pa}$) corresponding to an inlet Mach number at centerline $M_{\tsn{CL}_\mathrm{i}}\approxeq0.0940$ \tabref{Tab_RSMP3DDF_s_A_002}.
Computational results for the integral axisymmetric \cite{Fujii_Okiishi_1972a} boundary-layer thicknesses and associated shape-factors at the first measurement station 1
\figref{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_001}, where the flow is still practically axisymmetric, are in good agreement \tabref{Tab_RSMP3DDF_s_A_003} with those determined from the experimental data
\cite[Tab. 1, p. 370]{Davis_Gessner_1992a}. Following Davis and Gessner \cite{Davis_Gessner_1992a} the approximate (linearized; $\delta\ll R_\tsn{CSG}$) definitions of the axisymmetric integral boundary-layer thicknesses \cite[(3.5--3.7), p. 20]{Davis_1991a},
as defined by Fujii and Okiishi \cite{Fujii_Okiishi_1972a}, were applied.
Predicted wall-pressures are quite similar for all 4 turbulence models and are in quite satisfactory agreement with available measurements \figref{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_001}. Skin-friction was measured by Preston tubes aligned with the $x$-wise
direction and "{\em presumes that the 2-D form of the law-of-the-wall is valid and that streamwise pressure-gradients are small}" \cite[p. 19]{Davis_1991a}.
Computed skin-friction was determined by the wall-normal gradient of streamwise velocity $\bar{u}$, at each measurement plane. At stations 5 (exit of the transition section) and 6 (2 inlet diameters further downstream), the \tsn{GLVY} and
\tsn{GV} \tsn{RSM}s predict quite well the evolution of skin-friction along the peripheral wall-coordinate $\mathrm{s}$ \figref{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_001}, yielding the correct $\mathrm{s}$-gradient of $c_{f_\tsn{B}}$ everywhere.
The small differences in absolute level at station 5 \figref{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_001}, where the streamwise pressure-gradient is not negligible, are of the same order as the differences between measurements with Preston tubes of various diameters
\cite[Fig. 15, p. 374]{Davis_Gessner_1992a}, and can also be attributed to the error introduced by the log-law assumption in the measurements \cite{Sotiropoulos_Patel_1995a}.
On the other hand, the linear \tsn{LS} $\mathrm{k}$--$\varepsilon$ model is unsatisfactory predicting a peculiar inverted $\mathrm{s}$-curvature of $c_{f_\tsn{B}}$
around $\mathrm{s}\approxeq 0.6\mathrm{s}_{\frac{1}{4}}$ at both stations \figref{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_001}
and a nearly constant level of $c_{f_\tsn{B}}$ on the sidewall ($0.9\mathrm{s}_{\frac{1}{4}}\lessapprox s \lessapprox\mathrm{s}_{\frac{1}{4}}$; \figrefnp{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_001}).
The negative $\mathrm{s}$-gradient of $c_{f_\tsn{B}}$ on the sidewall ($\mathrm{s}=\mathrm{s}_{\frac{1}{4}}$ correspond to the middle of the sidewall at $z=0$; \figrefnp{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_001})
is an important feature of the flow, as it is directly related
\cite[pp. 50--51]{Davis_1991a} to the presence of the secondary flow streamwise vortices \cite[Fig. 7, p. 371]{Davis_Gessner_1992a}. The overall prediction of $c_{f_\tsn{B}}$ by the \tsn{WNF--LSS RSM} is satisfactory, except for the too weak negative
$\mathrm{s}$-gradient of $c_{f_\tsn{B}}$ for $\mathrm{s}\gtrapprox 0.9 \mathrm{s}_{\frac{1}{4}}$ \figref{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_001} which is indicative of an underestimation of the strength of the streamwise vortices.
All 4 turbulence models predict quite accurately the streamwise mean-flow velocity $\bar {u}$ along the $z$-traverse on the $y$-symmetry plane at all measurement stations \figref{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_002}. Along the $y$-traverse on the
$z$-symmetry plane \figref{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_002}, differences between turbulence models start appearing at station 4, where the linear \tsn{LS} $\mathrm{k}$--$\varepsilon$ model does not reproduce the experimentally observed inflection
of the velocity profile at $y_w-y\approxeq 0.1 R_{\tsn{CSG}_1}$ \figref{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_002}.
Further downstream, at stations 5 and 6, the linear \tsn{LS} $\mathrm{k}$--$\varepsilon$ model fails to predict the experimentally observed double inflection of the velocity profile along the $y$-traverse \figref{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_002},
returning instead a more filled 2-D-boundary-layer-like profile. Davis \cite[pp. 50--51]{Davis_1991a} has identified this feature of the $\bar {u}$ velocity profile as the result of a "{\em transfer of low-momentum fluid from the boundary-layer toward the
centerline creating a flat spot in the velocity field}", which "{\em is seen to be much larger at station 6 than at station 5}" (contour plots of $\bar {u}$; \figrefnp{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_002}).
This transfer, along the sidewall, is directly related to the presence of the secondary flow vortex-pair near the $z=0$ symmetry plane \cite[Fig. 7, p. 371]{Davis_Gessner_1992a}. The 3 \tsn{RSM}s successfully predict the double inflection
of the $\bar {u}$ profile along the $y$-traverses at planes 5 and 6 \figref{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_002}. The \tsn{GLVY} and \tsn{GV} \tsn{RSM}s agree quite well with measurements along the $y$-traverses at planes 5 and 6,
indeed everywhere \figref{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_002}. Although the \tsn{WNF--LSS RSM} predicts the double inflection shape of the $\bar {u}$ profile along the $y$-traverses at stations 5 and 6, it overpredicts $\bar {u}$, implying
a slight underprediction of secondary flows.
Differences between turbulence closures in predicting the wall-normal velocity $\bar{v}$ along the $y$-traverses at the $z=0$ symmetry plane (where $\bar{w}=0$ by symmetry) appear already at station 3 \figref{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_003}.
The \tsn{GLVY} and \tsn{GV} \tsn{RSM}s predict $\bar{v}$ quite accurately at stations 3 and 4, where the linear \tsn{LS} $\mathrm{k}$--$\varepsilon$ model and to a lesser extent the \tsn{WNF--LSS RSM}, slightly overestimate it near the sidewall
($y_w-y\lessapprox 0.4 R_{\tsn{CSG}_1}$; \figrefnp{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_003}).
At station 5, the 3 \tsn{RSM}s perform quite well in the outer part of the boundary-layer ($y_w-y\gtrapprox 0.2 R_{\tsn{CSG}_1}$; \figrefnp{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_003}) but overestimate $\bar{v}$ near the sidewall
($y_w-y\lessapprox 0.2 R_{\tsn{CSG}_1}$; \figrefnp{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_003}) by $\sim\!\!50$\% at the peak. They are nonetheless in much better agreement with experimental data than the linear \tsn{LS} $\mathrm{k}$--$\varepsilon$ model which predicts
levels that are 5-fold lower ($y\lessapprox 0.4 R_{\tsn{CSG}_1}$; station 5; \figrefnp{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_003}). At station 6, 2 inlet diameters further downstream, the $\bar{v}$ velocity along the $y$-traverse is severely underestimated by
the 3 \tsn{RSM}s \figref{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_003} which underpredict the strength of the secondary flows at this station. Nonetheless, the 3 \tsn{RSM}s largely outperform the linear \tsn{LS} $\mathrm{k}$--$\varepsilon$,
which completely fails, returning negligible small levels of $\bar v$ at station 6 \figref{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_003}.
The pair of contrarotating vortices observed at stations 5 and 6 near the intersection between the $z$-symmetry plane and the sidewall \cite[Fig. 7, p. 371]{Davis_Gessner_1992a} induces velocities away from the sidewall ($\bar{v}<0$ on the near-sidewall
along the $y$-traverse; \figrefnp{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_003}), whose measured peak value remains approximately constant ($\sim\!\!-0.1$) between stations 5 and 6 \figref{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_003}.
The failure of the \tsn{RSM} computations to correctly predict the relaxation of the flow in the straight constant cross-section duct between stations 5 and 6, possibly reveals an inadequacy of the models. Nonetheless, grid-resolution on the cross-section
at these stations is rather poor \figref{Fig_RSMP3DDF_s_A_001}, containing only a few cells across the vortices \cite[Fig. 7, p. 371]{Davis_Gessner_1992a}.
For this reason, computations using finer ($j$-wise and $k$-wise; \tabrefnp{Tab_RSMP3DDF_s_A_001}) grids are required to determine computational grid-convergence of the flow in the contrarotating vortex pair region, and this will be the subject of future work.
The wall-normal velocity $\bar{w}$ along the $z$-traverses at the $y=0$ symmetry plane (where $\bar{v}=0$ by symmetry) is very well predicted at all stations by all 4 turbulence closures \figref{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_004}.
All 3 \tsn{RSM}s predict quite accurately the streamwise Reynolds-stress $\overline{u'u'}$ along the $z$-traverse on the $y=0$ symmetry plane at station 5 \figref{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_005}, and also, despite a slight underestimation, at station 6
\figref{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_005} further downstream.
Along the $y$-traverse on the $z=0$ symmetry plane, except for station 1 \figref{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_005} near the computational inflow where the measured Reynolds-stresses were interpolated onto the grid and applied as boundary conditions,
the 3 \tsn{RSM}s predict correctly the profile shape but underestimate by $\sim\!\!50$\% the peak value at stations 5 and 6 \figref{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_005}. All 3 \tsn{RSM}s predict quite accurately the in-plane diagonal Reynolds-stresses
$\overline{v'v'}$ \figref{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_006} and $\overline{w'w'}$ \figref{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_007}, with the exception of $\overline{v'v'}$ at station 6 along the $y$-traverse on the $z$-symmetry
plane \figref{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_007} where the peak value is underestimated by $\sim\!\!50$\%.
The predictions of the diagonal Reynolds-stresses ($\overline{u'u'}$, $\overline{v'v'}$, $\overline{w'w'}$) by the \tsn{GLVY} and \tsn{GV} \tsn{RSM}s are in very close agreement
\figrefsatob{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_005}
{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_007},
and also with those predicted by \tsn{WNF--LSS RSM} \figrefsatob{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_005}
{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_007}
except at station 6 along the $y$-traverse
on the $z$-symmetry plane where the \tsn{GLVY} and \tsn{GV} \tsn{RSM}s are in closer agreement with measurements.
Expectedly, the linear $\mathrm{k}$--$\varepsilon$ model completely fails in predicting the Reynolds-stress tensor anisotropy, yielding unsatisfactory results for the diagonal Reynolds-stresses
\figrefsatob{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_005}
{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_007},
because of the pathological shortcomings of the Boussinesq hypothesis \cite[pp. 273--278]{Wilcox_1998a}.
The prediction of the shear Reynolds-stress $\overline{u'v'}$ along the $z$-traverse on the $y$-symmetry plane (where $\overline{u'v'}=0$ by symmetry) at stations 5 and 6 by the 3 \tsn{RSM}s is quite satisfactory
\figref{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_008}. On the contrary, the \tsn{LS} \cite{Launder_Sharma_1974a} linear $\mathrm{k}$--$\varepsilon$ model does not reproduce as well the shape of the $\overline{u'v'}$ profile at station 5
\figref{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_008}, a deficiency which does not appear to have a substantial influence on the prediction of the streamwise mean-velocity profile $\bar{u}$ ($z$-traverse, station 5, \figrefnp{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_002}).
The prediction of the shear Reynolds-stress $\overline{u'v'}$ \figref{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_008} along the $y$-traverse on the $z$-symmetry plane (where $\overline{u'w'}=0$ by symmetry) should be analyzed in relation to the prediction of the streamwise
mean-velocity $\bar{u}$ \figref{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_002}. At station 6, along the $y$-traverse on the $z$-symmetry plane, all turbulence models underestimate by $\sim\!\!50$\% the outer peak of $\overline{u'v'}$
at $y_w-y\approxeq 0.45 R_{\tsn{CSG}_1}$ \figref{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_008}. The grid-resolution issues mentioned above not withstanding, notice that the \tsn{GLVY} and \tsn{GV} \tsn{RSM}s predict quite well $\overline{u'v'}$
at station 6 for $0\lessapprox y_w-y\lessapprox 0.3 R_{\tsn{CSG}_1}$ \figref{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_008}, and this is obviously related to the satisfactory prediction of $\bar{u}$ by these models
($y$-traverse, station 6; \figrefnp{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_002}). On the contrary, the linear \tsn{LS} $\mathrm{k}$--$\varepsilon$ model which strongly overpredicts $\overline{u'v'}$ in this range
($y$-traverse, $0\lessapprox y_w-y\lessapprox 0.3 R_{\tsn{CSG}_1}$, station 5; \figrefnp{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_008}) fails to correctly predict the streamwise mean-velocity $\bar{u}$ at this location \figref{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_002}.
Notice that the \tsn{WNF--LSS RSM} which performs much better than the \tsn{LS} $\mathrm{k}$--$\varepsilon$ model in predicting the shear Reynolds-stress $\overline{u'v'}$
($y$-traverse, $0\lessapprox y_w-y\lessapprox 0.3 R_{\tsn{CSG}_1}$, station 6; \figrefnp{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_008}) also predicts the correct double inflection shape of the $\bar{u}$-profile
($y$-traverse, station 6; \figrefnp{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_002}), albeit less accurately than the \tsn{GLVY} and \tsn{GV} \tsn{RSM}s.
Despite the grid-convergence issues raised above (which can only be resolved by additional calculations on much finer grids), the systematic comparison of the computations of the Davis and Gessner \cite{Davis_Gessner_1992a} \tsn{C}-to-\tsn{R} duct configuration
with the experimental data \figrefsatob{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_001}
{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_008}
yields useful conclusions. The linear \tsn{LS} $\mathrm{k}$--$\varepsilon$ model, handicapped by Boussinesq's hypothesis
\cite[pp. 273--279]{Wilcox_1998a} fails to predict with sufficient accuracy the regions of the flowfield that are dominated by secondary flows \figrefsatob{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_001}
{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_008}.
The 3 \tsn{RSM}s perform much better, capturing several complex features of the flow \figrefsatob{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_001}{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_008}, although they are not sufficiently accurate on the
$10\times10^6$ points grid used \tabref{Tab_RSMP3DDF_s_A_001} in predicting all the details of the flow near the intersection of the sidewall with the $z=0$ symmetry plane \figrefsatob{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_001}
{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_008}.
As for the Gessner and Emery \cite{Gessner_Emery_1981a} square duct case \parref{RSMP3DDF_s_A_ss_DTFSDGE1981}, the \tsn{GLVY} and \tsn{GV} \tsn{RSM}s (which yield very similar results) perform sometimes better than the \tsn{WNF--LSS RSM},
especially near the sidewall in the region of strong secondary flows.
\subsection{Diffusing 3-D S-Duct \cite{Wellborn_Reichert_Okiishi_1994a}}\label{RSMP3DDF_s_A_ss_D3DSDWRO1994}
The previously studied square duct \parref{RSMP3DDF_s_A_ss_DTFSDGE1981} and \tsn{C}-to-\tsn{R} transition duct \parref{RSMP3DDF_s_A_ss_CtoRTDDG1992} test-cases have a straight centerline ($x$-axis of the coordinates system).
Furthermore, the diverging part of the \tsn{C}-to-\tsn{R} transition duct (from station 2 to midpoint of the transition section; \figrefnp{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_001}) was sufficiently long to avoid separation.
The \tsn{S}-duct test-case \cite{Wellborn_Okiishi_Reichert_1993a,Wellborn_Reichert_Okiishi_1994a}
includes these 2 features, \viz it has a serpentine centerline (\tsn{S}-duct) combined with substantial ($52\%$) area increase, from inflow to outflow \cite{Wellborn_Reichert_Okiishi_1994a},
inducing a large region of separated flow near the duct floor, immediately after the beginning of the \tsn{S}-bend \figref{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_001}.
The serpentine centerline of the \tsn{S}-duct lies on the $xz$-plane (no off-plane skewing; \figrefnp{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_002}) and consists of 2 circular arcs of opposite curvature smoothly joined at a common tangency point
\cite[Fig. 2, p. 670]{Wellborn_Reichert_Okiishi_1994a}. Planes $\perp$ to the centerline define stations of circular cross-section, with varying radius, whose dependence on the angular coordinate (equivalently the curvilinear length) along the centerline
\cite[(2), p. 670]{Wellborn_Reichert_Okiishi_1994a} defines the geometry of the duct. The origin of the curvilinear coordinates along the centerline $\mathrm{s}_\tsn{CL}$ is at the beginning of the \tsn{S}-bend, which also corresponds to $x=0$.
The inlet-diameter is $D_{\tsn{CSG}_\tsn{A}}=2R_{\tsn{CSG}_\tsn{A}}=0.2042\;\mathrm{m}$ (this is also the diameter at the first measurement plane \tsn{A},
located at $x=-\tfrac{1}{2}D_{\tsn{CSG}_\tsn{A}}$, 1 inlet-radius upstream of the beginning of the \tsn{S}-bend; \figrefnp{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_002}) while the exit diameter is $D_{\tsn{CSG}_\tsn{E}}=0.2514\;\mathrm{m}$
(this is also the diameter at the last measurement plane \tsn{E},
located at $x\approxeq5.61D_{\tsn{CSG}_\tsn{A}}$, $0.61$ inlet-diameters downstream of the exit of the \tsn{S}-bend located at $x=5D_{\tsn{CSG}_\tsn{A}}$; \figrefnp{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_002}).
The flow \cite{Wellborn_Reichert_Okiishi_1994a} is subsonic (centerline Mach number at measurement plane \tsn{A} $\bar M_{\tsn{CL}_\tsn{A}}\approxeq0.60$) and the centerline Reynolds number is $Re_{\tsn{CL}_\tsn{A}}\approxeq2.6\times10^6$
($Re_{\tsn{CL}_\tsn{A}}=\bar u_{\tsn{CL}_\tsn{A}} D_{\tsn{CSG}_\tsn{A}} \bar\nu^{-1}_{\tsn{CL}_\tsn{A}}$, where $\bar u_{\tsn{CL}_\tsn{A}}$ is the centerline velocity and $\bar\nu_{\tsn{CL}_\tsn{A}}$ is the kinematic viscosity at centerline).
Available field measurements, taken at 5 axial planes $\perp$ to the centerline (circular cross-section; \figrefnp{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_002}), using calibrated 3-hole
and 5-hole pneumatic probes \cite{Wellborn_Okiishi_Reichert_1993a,Wellborn_Reichert_Okiishi_1994a},
provide pressures (total and static) and the mean-flow velocity vectors. Wall-pressure measurements are also available \cite{Wellborn_Okiishi_Reichert_1993a,Wellborn_Reichert_Okiishi_1994a},
both around the circumference of 4 of the measurement planes \figref{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_003},
and streamwise, at 3 angular locations \figref{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_002}.
\begin{figure}[h!]
\begin{center}
\begin{picture}(340,220)
\put(0,-5){\includegraphics[angle=0,width=340pt,bb= 12 323 590 705]{Fig_FlowTurbComb_01_WRO_1994_S_duct_Mkpt_RSMs.pdf}}
\end{picture}
\end{center}
\caption{Level plots of Mach number $\breve M$ and of turbulent kinetic energy $\mathrm{k}$ on the $y=0$ symmetry plane of the Wellborn \etal\ \cite{Wellborn_Reichert_Okiishi_1994a} diffusing \tsn{S}-duct
($Re_{\tsn{CL}_\tsn{A}}=2.6\times10^6$, $\bar M_{\tsn{CL}_\tsn{A}}\approxeq0.6$; \tabrefnp{Tab_RSMP3DDF_s_A_002}) obtained from computations
($2\times10^6$ points grid discretizing the entire duct; \tabrefnp{Tab_RSMP3DDF_s_A_001})
using \parref{RSMP3DDF_s_TCsFS_ss_TCs} the \tsn{GV} \cite{Gerolymos_Vallet_2001a}, the \tsn{WNF--LSS} \cite{Gerolymos_Sauret_Vallet_2004a}
and the \tsn{GLVY} \cite{Gerolymos_Lo_Vallet_Younis_2012a} \tsn{RSM}s, and the \tsn{LS} \cite{Launder_Sharma_1974a} linear $\mathrm{k}$--$\varepsilon$ model.}
\label{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_001}
\end{figure}
\begin{figure}[h!]
\begin{center}
\begin{picture}(340,350)
\put(0,-5){\includegraphics[angle=0,width=340pt,bb= 86 105 571 598]{Fig_FlowTurbComb_01_WRO_1994_S_duct_Cp_vs_x.pdf}}
\end{picture}
\end{center}
\caption{Comparison of measured \cite{Wellborn_Reichert_Okiishi_1994a} wall-pressure coefficient $C_p$ (based on centerline quantities at plane \tsn{A}),
plotted against the curvilinear coordinate $\mathrm{s}_\tsn{CL}$ along the duct centerline (planes $\perp$ to the centerline define stations of circular cross-section),
at 3 azimuthal locations, with computations ($2\times10^6$ points grid discretizing the entire duct; \tabrefnp{Tab_RSMP3DDF_s_A_001})
using \parref{RSMP3DDF_s_TCsFS_ss_TCs} the \tsn{GV} \cite{Gerolymos_Vallet_2001a}, the \tsn{WNF--LSS} \cite{Gerolymos_Sauret_Vallet_2004a}
and the \tsn{GLVY} \cite{Gerolymos_Lo_Vallet_Younis_2012a} \tsn{RSM}s, and the \tsn{LS} \cite{Launder_Sharma_1974a} linear $\mathrm{k}$--$\varepsilon$ model, for turbulent flow in a diffusing \tsn{S}-duct
($Re_{\tsn{CL}_\tsn{A}}=2.6\times10^6$, $\bar M_{\tsn{CL}_\tsn{A}}\approxeq0.6$; \tabrefnp{Tab_RSMP3DDF_s_A_002}; contour plots \tsn{GLVY RSM}).}
\label{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_002}
\end{figure}
\begin{figure}[h!]
\begin{center}
\begin{picture}(340,280)
\put(0,-5){\includegraphics[angle=0,width=340pt,bb= 52 270 574 699]{Fig_FlowTurbComb_01_WRO_1994_S_duct_Cp_vs_phi_planes.pdf}}
\end{picture}
\end{center}
\caption{Comparison of measured \cite{Wellborn_Reichert_Okiishi_1994a} wall-pressure coefficient $C_p$ (based on centerline quantities at plane \tsn{A}),
at 4 experimental measurement stations (planes $\perp$ to the centerline whose intersection with the duct defines circular cross-sections),
plotted against the azimuthal location along the circumference, with computations ($2\times10^6$ points grid discretizing the entire duct; \tabrefnp{Tab_RSMP3DDF_s_A_001})
using \parref{RSMP3DDF_s_TCsFS_ss_TCs} the \tsn{GV} \cite{Gerolymos_Vallet_2001a}, the \tsn{WNF--LSS} \cite{Gerolymos_Sauret_Vallet_2004a}
and the \tsn{GLVY} \cite{Gerolymos_Lo_Vallet_Younis_2012a} \tsn{RSM}s, and the \tsn{LS} \cite{Launder_Sharma_1974a} linear $\mathrm{k}$--$\varepsilon$ model, for turbulent flow in a diffusing \tsn{S}-duct
($Re_{\tsn{CL}_\tsn{A}}=2.6\times10^6$, $\bar M_{\tsn{CL}_\tsn{A}}\approxeq0.6$; \tabrefnp{Tab_RSMP3DDF_s_A_002}; $\mathrm{s}_\tsn{CL}$ is the curvilinear coordinate along the duct centerline; contour plots \tsn{GLVY RSM}).}
\label{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_003}
\end{figure}
The computations were run on a $2\times10^6$ grid \tabref{Tab_RSMP3DDF_s_A_001} discretizing the entire duct without symmetry conditions \figref{Fig_RSMP3DDF_s_A_001}. Based on previous grid-convergence studies \cite{Gerolymos_Joly_Mallet_Vallet_2010a},
on a similar \tsn{2S}-duct configuration, this grid \tabref{Tab_RSMP3DDF_s_A_001} is sufficient to obtain accurate results for comparison between the different models \parref{RSMP3DDF_s_TCsFS_ss_TCs}.
The computational domain ($-0.98D_{\tsn{CSG}_\tsn{A}}\lessapprox x\lessapprox9.8D_{\tsn{CSG}_\tsn{A}}$) starts approximately 1 inlet-diameter ($D_{\tsn{CSG}_\tsn{A}}$) upstream of the start of the \tsn{S}-bend and
extends approximately 5 inlet-diameters ($5D_{\tsn{CSG}_\tsn{A}}$) downstream of the \tsn{S}-bend exit, thus avoiding any
interaction between the uniform outflow pressure boundary-condition and computed results at the last
measurement station \tsn{E} \figref{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_001}.
The grid is uniform in the streamwise ($x$) direction (the $i=\const$ grid-surfaces are $\perp x$ planes) and consists of 2 blocks (\figrefnp{Fig_RSMP3DDF_s_A_001}; \tabrefnp{Tab_RSMP3DDF_s_A_001}).
The inner block (\tsn{H}$_\tsn{$\square$}$; \tabrefnp{Tab_RSMP3DDF_s_A_001}) is an \tsn{H}-grid of $x$-wise varying square cross-section with uniform $yz$-spacing, introduced to avoid the axis-singularity of an axisymmetric-type grid.
The outer block (\tsn{O}$_\tsn{$\square$}$; \tabrefnp{Tab_RSMP3DDF_s_A_001}) is stretched geometrically near the wall with ratio $r_k$ \tabref{Tab_RSMP3DDF_s_A_001}.
For the investigated flow conditions, the first node at the walls is located at $\Delta n_w^+\lessapprox\tfrac{4}{10}$ \tabref{Tab_RSMP3DDF_s_A_001}, $n$ being the wall-normal direction.
At inflow \tabref{Tab_RSMP3DDF_s_A_002} total conditions ($p_{t_{\tsn{CL}_\mathrm{i}}}=111330\;\mathrm{Pa}$, $T_{t_{\tsn{CL}_\mathrm{i}}}=296.4\;\mathrm{K}$) were assumed at the centerline,
corresponding to the Mach ($M_{\tsn{CL}_\mathrm{i}}=0.60$) and Reynolds ($Re_{\tsn{CL}_\mathrm{i}}=2.6\times10^6$) number values reported in the measurements \cite{Wellborn_Okiishi_Reichert_1993a,Wellborn_Reichert_Okiishi_1994a}.
A turbulent intensity $T_{u_{\tsn{CL}_\mathrm{i}}}=0.63\%$ was applied at the centerline; Wellborn \etal\ \cite[p. 29]{Wellborn_Okiishi_Reichert_1993a} report this value from measurements of Reichert \cite{Reichert_1991a} on the same facility.
In the absence of experimental data, a turbulent lengthscale $\ell_{\tsn{T}_{\tsn{CL}_\mathrm{i}}}=50\;\mathrm{mm}$ was assumed at the centerline,
with reference to the duct radius ($R_{\tsn{CSG}_\tsn{A}}=0.1021\;\mathrm{m}$). The initial inflow boundary-layer thickness and Coles-parameter \cite{Gerolymos_Sauret_Vallet_2004c} were adjusted, independently for each model \tabref{Tab_RSMP3DDF_s_A_002},
to match the experimental boundary-layer data at the first measurement plane \tsn{A}. Finally the outflow pressure was also adjusted, independently for each model \tabref{Tab_RSMP3DDF_s_A_002},
to obtain the correct $\breve M_{\tsn{CL}_\tsn{A}}\approxeq0.60$ \tabref{Tab_RSMP3DDF_s_A_003}.
Computational results for the integral axisymmetric \cite{Fujii_Okiishi_1972a} boundary-layer thicknesses and associated shape-factors at the first measurement plane \tsn{A}
\figref{Fig_RSMP3DDF_s_A_ss_CtoRTDDG1992_001}, where the flow is still practically axisymmetric, are in good agreement \tabref{Tab_RSMP3DDF_s_A_003} with those determined from the experimental data
\cite[Tab. 2, p. 671]{Wellborn_Reichert_Okiishi_1994a}. Following Wellborn \etal\ \cite{Wellborn_Okiishi_Reichert_1993a} the approximate (linearized; $\delta\ll R_\tsn{CSG}$) definitions of the axisymmetric integral boundary-layer thicknesses
\cite[(V.1--V.2), p.29]{Wellborn_Okiishi_Reichert_1993a}, as defined by Fujii and Okiishi \cite{Fujii_Okiishi_1972a}, were applied.
The definitions given by Wellborn \etal\ \cite[(V.1--V.2), p.29]{Wellborn_Okiishi_Reichert_1993a} concern compressible integral thicknesses, but the actual shape-factor values ($\sim\!\!1.38$) imply that
the thicknesses provided in the experimental database \cite{Wellborn_Okiishi_Reichert_1993a,Wellborn_Reichert_Okiishi_1994a}
are kinematic (the corresponding compressible value would be $\sim\!\!1.65$), as defined in the associated study (from which the inlet turbulent intensity was determined, on the same experimental facility, by Reichert \cite[(V.7--V.8), p. 67]{Reichert_1991a}.
This is implied by the statement that "{\em comparisons indicate little deviation from a conventional turbulent boundary-layer}" \cite[p. 29]{Wellborn_Okiishi_Reichert_1993a}.
All 4 turbulence closures predict separation near the duct floor \figref{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_001} in agreement with experiment \cite{Wellborn_Reichert_Okiishi_1994a}, but differ in the location of separation and reattachment, in the extent
($x$-wise) and thickness ($z$-wise) of the separated flow region, and in the predicted structure of the recirculating flow \figref{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_001}. The \tsn{GLVY} and \tsn{GV} \tsn{RSM}s yield very similar results
\figref{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_001}, and are in quite satisfactory agreement with available measurements \figrefsatob{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_002}
{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_008}.
The \tsn{WNF--LSS RSM} predicts separation further downstream (with respect to the \tsn{GLVY} and \tsn{GV} \tsn{RSM}s; \figrefnp{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_001}) and the linear \tsn{LS} $\mathrm{k}$--$\varepsilon$ model, which is known to
underestimate flow detachment \cite{Gerolymos_1990c}, separates a little further downstream still. Even more important, there are noticeable differences in the separated flow structure \figref{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_001}
between the \tsn{GLVY} and \tsn{GV} \tsn{RSM}s on the one hand, and the \tsn{WNF--LSS RSM} and the linear \tsn{LS} $\mathrm{k}$--$\varepsilon$ model on the other.
The \tsn{GLVY} and \tsn{GV} \tsn{RSM}s predict a much thicker ($z$-wise) low-speed region with a stronger recirculation zone near the wall just downstream of separation \figref{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_001}.
This flow structure contains strong mean-velocity gradients producing high levels of turbulent kinetic energy $\mathrm{k}$, which presents 2 local maxima, one in the post-separation wake-region (dark blue levels, \tsn{GLVY} and \tsn{GV RSM}s;
\figrefnp{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_001}) and another near the wall in the pre-reattachment region (light green levels of $\mathrm{k}$, \tsn{GLVY} and \tsn{GV RSM}s; \figrefnp{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_001}).
On the other hand, the \tsn{WNF--LSS RSM} and the linear \tsn{LS} $\mathrm{k}$--$\varepsilon$ model predict a thinner ($z$-wise) low-speed region, with weak recirculation near the wall, and lower levels of $\mathrm{k}$ \figref{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_001}.
The \tsn{GLVY} and \tsn{GV} \tsn{RSM}s' predictions compare quite well with experimental wall-pressure data \figrefsab{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_002}
{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_003},
correctly predicting the pressure-plateau on the duct floor ($\phi_\tsn{EXP}=170\;\mathrm{deg}$; \figrefnp{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_002})
and the significant $z$-wise extent of the low-speed region indicated by the presence of the pressure-plateau at duct midplane ($\phi_\tsn{EXP}=90\;\mathrm{deg}$; \figrefnp{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_002}).
This large separated flow region induces substantial flow blockage \cite[pp. 310--311]{Cumpsty_1989a}, accelerating the flow in the duct's ceiling area ($\phi_\tsn{EXP}=10\;\mathrm{deg}$; \figrefnp{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_002}).
The satisfactory agreement of the \tsn{GLVY} and \tsn{GV RSM}s' predictions with measurements near the duct ceiling ($\phi_\tsn{EXP}=10\;\mathrm{deg}$; \figrefnp{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_002}) indicates that
the \tsn{GLVY} and \tsn{GV} \tsn{RSM}s yield a satisfactory prediction of the blockage induced by the large separation on the duct floor \figref{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_001}.
Near the beginning of the \tsn{S}-bend, at planes \tsn{A} (one inlet radius $R_{\tsn{CSG}_1}$ upstream) and \tsn{B} (approximately one inlet diameter $D_{\tsn{CSG}_1}$ downstream), all 4 turbulence models are in excellent agreement with measurements
\figref{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_003}, correctly predicting in plane \tsn{B} the circumferential pressure-gradient that drives the boundary-layer fluid along the duct's circumference \figref{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_007}
from ceiling (higher pressure due to the streamwise-concave wall; \figrefnp{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_003}) to floor (lower pressure due to the streamwise-convex wall; \figrefnp{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_003}).
At plane \tsn{C}, in the separated flow region \figrefsab{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_001}{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_002}, the \tsn{GLVY} and \tsn{GV} \tsn{RSM}s are again in excellent agreement with measurements, correctly predicting the
circumferential evolution of $C_p$ \figref{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_003} both in level and shape. The \tsn{WNF--LSS RSM} predicts the correct shape of the circumferential evolution of $C_p$ at plane \tsn{C}, but largely overestimates its value
by $\sim\!\!50$\%, whereas the linear \tsn{LS} $\mathrm{k}$--$\varepsilon$ model which overestimates $C_p$ even more fails to predict the inversion the circumferential pressure-gradient \figref{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_003} from channel
mid-height ($\phi_{\tsn{EXP}}\approxeq 110\;\mathrm{deg}$) to floor ($\phi_{\tsn{EXP}}\approxeq 180\;\mathrm{deg}$).
At plane \tsn{D}, where the flow reattaches in the experiment \figref{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_002}, the \tsn{GLVY} and \tsn{GV} \tsn{RSM}s again provide the best prediction, compared to the \tsn{WNF--LSS RSM}
and the linear \tsn{LS} $\mathrm{k}$--$\varepsilon$ model, but they slightly overestimate $C_p$, especially near the floor ($130\;\mathrm{deg}\lessapprox \phi_{\tsn{EXP}}\lessapprox 180\;\mathrm{deg}$; \figrefnp{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_003}).
Field pneumatic-probe measurements of $C_p$ \figref{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_004} at plane \tsn{B} indicate a slight static-pressure distortion which is not predicted by the computations \figref{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_004}
and is not observed in the wall-pressure measurements \figref{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_003}. At plane \tsn{C}, the \tsn{GLVY} and \tsn{GV} \tsn{RSM}s are in reasonable agreement with measurements, correctly predicting the flow acceleration near the
ceiling \figref{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_004} induced by the floor-separation blockage \figrefsab{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_001}
{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_002}.
The \tsn{WNF--LSS RSM} which predicts separation downstream of experiment \figrefsab{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_001}
{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_002}
severely overestimates $C_p$ at plane \tsn{C} \figref{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_004}, the linear \tsn{LS} $\mathrm{k}$--$\varepsilon$ model performing worse.
At the near-reattachment plane \tsn{D}, the \tsn{GLVY} and \tsn{GV} \tsn{RSM}s are in good agreement with measurements, substantially outperforming the \tsn{WNF--LSS RSM}
and the \tsn{LS} $\mathrm{k}$--$\varepsilon$ closure \figref{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_004}.
\begin{figure}[h!]
\begin{center}
\begin{picture}(340,490)
\put(50,-15){\includegraphics[angle=0,height=510pt,bb=131 51 468 752]{Fig_FlowTurbComb_01_WRO_1994_S_duct_Cp_RSMs.pdf}}
\end{picture}
\end{center}
\caption{Comparison, at 3 measurement planes (planes $\perp$ to the centerline define stations of circular cross-section),
of experimental \cite{Wellborn_Reichert_Okiishi_1994a} contours of pressure coefficient $C_p$ (based on centerline quantities at plane \tsn{A}; contour step 0.025),
with computations ($2\times10^6$ points grid discretizing the entire duct; \tabrefnp{Tab_RSMP3DDF_s_A_001})
using \parref{RSMP3DDF_s_TCsFS_ss_TCs} the \tsn{GV} \cite{Gerolymos_Vallet_2001a}, the \tsn{WNF--LSS} \cite{Gerolymos_Sauret_Vallet_2004a}
and the \tsn{GLVY} \cite{Gerolymos_Lo_Vallet_Younis_2012a} \tsn{RSM}s, and the \tsn{LS} \cite{Launder_Sharma_1974a} linear $\mathrm{k}$--$\varepsilon$ model
($Re_{\tsn{CL}_\tsn{A}}=2.6\times10^6$, $\bar M_{\tsn{CL}_\tsn{A}}\approxeq0.6$; \tabrefnp{Tab_RSMP3DDF_s_A_002}).}
\label{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_004}
\end{figure}
\begin{figure}[h!]
\begin{center}
\begin{picture}(340,490)
\put(15,-15){\includegraphics[angle=0,height=510pt,bb= 72 33 512 752]{Fig_FlowTurbComb_01_WRO_1994_S_duct_VnoVref_RSMs.pdf}}
\end{picture}
\end{center}
\caption{Comparison, at 4 measurement planes (planes $\perp$ to the centerline define stations of circular cross-section),
of experimental \cite{Wellborn_Reichert_Okiishi_1994a} contours of normal-to-the-plane (streamwise) velocity $\breve V_\perp$ (made nondimensional
by the centerline velocity at plane \tsn{A}, $\breve V_{\tsn{CL}_\tsn{A}}$; contour step 0.05),
with computations ($2\times10^6$ points grid discretizing the entire duct; \tabrefnp{Tab_RSMP3DDF_s_A_001})
using \parref{RSMP3DDF_s_TCsFS_ss_TCs} the \tsn{GV} \cite{Gerolymos_Vallet_2001a}, the \tsn{WNF--LSS} \cite{Gerolymos_Sauret_Vallet_2004a}
and the \tsn{GLVY} \cite{Gerolymos_Lo_Vallet_Younis_2012a} \tsn{RSM}s, and the \tsn{LS} \cite{Launder_Sharma_1974a} linear $\mathrm{k}$--$\varepsilon$ model
($Re_{\tsn{CL}_\tsn{A}}=2.6\times10^6$, $\bar M_{\tsn{CL}_\tsn{A}}\approxeq0.6$; \tabrefnp{Tab_RSMP3DDF_s_A_002}).}
\label{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_005}
\end{figure}
\begin{figure}[h!]
\begin{center}
\begin{picture}(340,490)
\put(15,-15){\includegraphics[angle=0,height=510pt,bb= 72 51 512 752]{Fig_FlowTurbComb_01_WRO_1994_S_duct_Cpt_RSMs.pdf}}
\end{picture}
\end{center}
\caption{Comparison, at 4 measurement planes (planes $\perp$ to the centerline define stations of circular cross-section),
of experimental \cite{Wellborn_Reichert_Okiishi_1994a} contours of total pressure coefficient $C_{p_t}$ (based on centerline quantities at plane \tsn{A}; contour step 0.05),
with computations ($2\times10^6$ points grid discretizing the entire duct; \tabrefnp{Tab_RSMP3DDF_s_A_001})
using \parref{RSMP3DDF_s_TCsFS_ss_TCs} the \tsn{GV} \cite{Gerolymos_Vallet_2001a}, the \tsn{WNF--LSS} \cite{Gerolymos_Sauret_Vallet_2004a}
and the \tsn{GLVY} \cite{Gerolymos_Lo_Vallet_Younis_2012a} \tsn{RSM}s, and the \tsn{LS} \cite{Launder_Sharma_1974a} linear $\mathrm{k}$--$\varepsilon$ model
($Re_{\tsn{CL}_\tsn{A}}=2.6\times10^6$, $\bar M_{\tsn{CL}_\tsn{A}}\approxeq0.6$; \tabrefnp{Tab_RSMP3DDF_s_A_002}).}
\label{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_006}
\end{figure}
\clearpage
The velocity field, at each measurement plane, can be decomposed into a plane-normal component $\vec{V}_{\perp\tsn{PLN}}$ and an in-plane (parallel) component $\vec{V}_{\parallel\tsn{PLN}}$, $\vec{V}=\vec{V}_{\perp\tsn{PLN}}+\vec{V}_{\parallel\tsn{PLN}}$,
where \tsn{PLN} $\in \{\tsn{A},\tsn{B},\tsn{C},\tsn{D},\tsn{E}\}$. The plane-normal mean-velocities $\breve{V}_{\perp}$ \figref{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_005} indicate the regions of separated and low-speed flow, which also
correspond to the high-loss regions (low $C_{p_t}$; \figrefnp{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_006}).
\begin{figure}[h!]
\begin{center}
\begin{picture}(340,380)
\put(0,-5){\includegraphics[angle=0,width=340pt,bb= 71 74 558 623]{Fig_FlowTurbComb_01_WRO_1994_S_duct_Vpvectors_GLVY_BC.pdf}}
\end{picture}
\end{center}
\caption{Comparison, at 2 measurement planes (planes $\perp$ to the centerline define stations of circular cross-section),
of experimental \cite{Wellborn_Reichert_Okiishi_1994a} vectors (unknown scale) of in-plane (secondary) velocity $\bar{\vec{V}}_\parallel$ (made nondimensional
by the centerline velocity at plane \tsn{A}, $\breve V_{\tsn{CL}_\tsn{A}}$),
with computations ($2\times10^6$ points grid discretizing the entire duct; \tabrefnp{Tab_RSMP3DDF_s_A_001})
using \parref{RSMP3DDF_s_TCsFS_ss_TCs} the \tsn{GLVY} \cite{Gerolymos_Lo_Vallet_Younis_2012a} \tsn{RSM}
($Re_{\tsn{CL}_\tsn{A}}=2.6\times10^6$, $\bar M_{\tsn{CL}_\tsn{A}}\approxeq0.6$; \tabrefnp{Tab_RSMP3DDF_s_A_002}).}
\label{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_007}
\end{figure}
\begin{figure}[h!]
\begin{center}
\begin{picture}(340,380)
\put(0,-5){\includegraphics[angle=0,width=340pt,bb= 65 74 564 623]{Fig_FlowTurbComb_01_WRO_1994_S_duct_Vpvectors_GLVY_DE.pdf}}
\end{picture}
\end{center}
\caption{Comparison, at 2 measurement planes (planes $\perp$ to the centerline define stations of circular cross-section),
of experimental \cite{Wellborn_Reichert_Okiishi_1994a} vectors (unknown scale) of in-plane (secondary) velocity $\bar{\vec{V}}_\parallel$ (made nondimensional
by the centerline velocity at plane \tsn{A}, $\breve V_{\tsn{CL}_\tsn{A}}$),
with computations ($2\times10^6$ points grid discretizing the entire duct; \tabrefnp{Tab_RSMP3DDF_s_A_001})
using \parref{RSMP3DDF_s_TCsFS_ss_TCs} the \tsn{GLVY} \cite{Gerolymos_Lo_Vallet_Younis_2012a} \tsn{RSM}
($Re_{\tsn{CL}_\tsn{A}}=2.6\times10^6$, $\bar M_{\tsn{CL}_\tsn{A}}\approxeq0.6$; \tabrefnp{Tab_RSMP3DDF_s_A_002}).}
\label{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_008}
\end{figure}
The \tsn{GLVY} and \tsn{GV} \tsn{RSM}s are in overall satisfactory agreement with measurements \figrefsab{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_005}
{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_006}
correctly predicting the backflow region at plane \tsn{C} and the flow blockage at the reattachment plane \tsn{D} and at the exit plane \tsn{E} \figref{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_005}.
As a consequence, the \tsn{GLVY} and \tsn{GV} \tsn{RSM}s also predict correctly the high level of loss in the backflow region (low $C_{p_t}$; plane \tsn{C}; \figrefnp{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_006})
and the subsequent streamwise evolution of the high-loss region (planes \tsn{D} and \tsn{E}; \figrefnp{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_006}).
On the contrary, the linear \tsn{LS} $\mathrm{k}$--$\varepsilon$ model, and to a lesser extent the \tsn{WNF--LSS RSM}, underpredict both backflow \figref{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_005}
and losses \figref{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_006}, predicting a less thick low-speed high-loss region everywhere \figrefsab{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_005}
{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_006}.
\begin{figure}[h!]
\begin{center}
\begin{picture}(340,460)
\put(0,-5){\includegraphics[angle=0,width=340pt,bb= 71 33 509 642]{Fig_FlowTurbComb_01_WRO_1994_S_duct_VpoVref_RSMs.pdf}}
\end{picture}
\end{center}
\caption{Level plots of the module of in-plane (secondary) velocity $\bar{\vec{V}}_\parallel$ (made nondimensional by the centerline velocity at plane \tsn{A}, $\breve V_{\tsn{CL}_\tsn{A}}$),
at 4 measurement planes (planes $\perp$ to the centerline define stations of circular cross-section),
computed ($2\times10^6$ points grid discretizing the entire duct; \tabrefnp{Tab_RSMP3DDF_s_A_001})
using \parref{RSMP3DDF_s_TCsFS_ss_TCs} the \tsn{GV} \cite{Gerolymos_Vallet_2001a}, the \tsn{WNF--LSS} \cite{Gerolymos_Sauret_Vallet_2004a}
and the \tsn{GLVY} \cite{Gerolymos_Lo_Vallet_Younis_2012a} \tsn{RSM}s, and the \tsn{LS} \cite{Launder_Sharma_1974a} linear $\mathrm{k}$--$\varepsilon$ model
($Re_{\tsn{CL}_\tsn{A}}=2.6\times10^6$, $\bar M_{\tsn{CL}_\tsn{A}}\approxeq0.6$; \tabrefnp{Tab_RSMP3DDF_s_A_002}).}
\label{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_009}
\end{figure}
\clearpage
The \tsn{GLVY RSM} (whose results are very close to those obtained with the \tsn{GV RSM}; \figrefsnpatob{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_001}
{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_006}),
predicts quite satisfactorily the structure of secondary (in-plane $\vec{V}_{\parallel}$) flows \figrefsab{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_007}
{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_008}.
At plane \tsn{B} \figref{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_007}, the circumferential pressure-gradient \figref{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_003} drives the boundary-layer flow
from ceiling to floor along the duct walls \figref{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_007}.
At the separated-flow plane \tsn{C} \figref{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_007} this downward flow interacts with the large separation at the duct's floor \figref{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_001} forming 2 contrarotating vortices
\figref{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_007}, which lift off the floor as they are convected downstream (planes \tsn{D} and \tsn{E}; \figrefnp{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_008}).
The differences in predictive accuracy between the 4 turbulence models \figrefsatob{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_001}
{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_006}
is directly related to differences in the secondary-flow structure \figref{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_009}.
At plane \tsn{B}, where the flow is still attached \figref{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_005}, all 4 turbulence closures yield quite similar results \figref{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_009}.
At the separated-flow plane \tsn{C}, \tsn{GLVY} and \tsn{GV RSM}s predict a thick low-$\vec{V}_{\parallel}$ region \figref{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_009}, with distinct tails roughly marking the centers of 2 contrarotating vortices
\figref{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_007}, in good agreement with measurements.
The \tsn{WNF--LSS RSM} predicts too thin a low-speed region \figref{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_009} and the 2 tails are less sharp, these 2 defaults being even more pronounced for the linear \tsn{LS} $\mathrm{k}$--$\varepsilon$ model. The differences
between the \tsn{GLVY} and \tsn{GV} \tsn{RSM}s on the one hand and the \tsn{WNF--LSS RSM} and the linear \tsn{LS} $\mathrm{k}$--$\varepsilon$ closure on the other,
are much more pronounced at the reattachment plane \tsn{D} \figref{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_009},
where the 2 vortices have lifted off the floor in the \tsn{GLVY} and \tsn{GV} \tsn{RSM}s predictions \figref{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_009}, in quite satisfactory agreement with measurements \figref{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_008},
whereas they are more diffuse and closer to the wall in the \tsn{WNF--LSS RSM} predictions, which also underestimate the 2 symmetric high-$\breve{V}_\parallel$ regions near the duct floor
(plane \tsn{D}; \figrefsnpab{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_008}
{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_009}).
These high-$V_{\parallel}$ regions are simply absent in the linear \tsn{LS} $\mathrm{k}$--$\varepsilon$ model predictions \figref{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_009}.
At the exit plane \tsn{E}, the \tsn{GLVY} and \tsn{GV} \tsn{RSM}s predict sharp regions of low speed \figref{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_009} which correspond to the centers of the vortices \figref{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_008}, with
regions of high-$V_\parallel$ near the ducts floor \figrefsab{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_008}
{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_009}
and in the region between the 2 contrarotating vortices \figref{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_009},
in good agreement with measurements. The vortices predicted by the \tsn{WNF--LSS RSM} and the linear $\mathrm{k}$--$\varepsilon$ model are closer to the duct floor and their centers are less sharp \figref{Fig_RSMP3DDF_s_A_ss_D3DSDWRO1994_009}.
For the Wellborn \etal\ \cite{Wellborn_Reichert_Okiishi_1994a} test-case as for the previous ones \parrefsab{RSMP3DDF_s_A_ss_DTFSDGE1981}
{RSMP3DDF_s_A_ss_CtoRTDDG1992},
the \tsn{GLVY} and \tsn{GV} \tsn{RSM}s yield very similar results, and are in quite satisfactory agreement with
measurements, showing that properly calibrated \tsn{RSM--RANS} closures can predict flows with large separation and wall-curvature effects. The \tsn{GLVY} and \tsn{GV} \tsn{RSM}s considerably outperform the
\tsn{WNF--LSS RSM}, and this is again attributed to the $C_\phi^{(\tsn{RH})}$ coefficient-function used \tabref{Tab_RSMP3DDF_s_TCsFS_ss_TCs_001},
because predictions of the Wellborn \etal\ \cite{Wellborn_Reichert_Okiishi_1994a} test-case using the \tsn{GV} and \tsn{GV--DH} (\cf\ \parrefnp{RSMP3DDF_s_A_ss_DTFSDGE1981}) \tsn{RSM}s are very similar one with another \cite[Figs. 11--12, pp. 1153--1154]{Vallet_2007a},
implying that the turbulent diffusion closure is less influential than pressure-strain redistribution in this flow.
On the other hand, the improvement of the \tsn{WNF--LSS RSM} over the linear \tsn{LS} $\mathrm{k}$--$\varepsilon$ model for this separation-dominated flow is weak.
\section{Conclusions}\label{RSMP3DDF_s_Cs}
In the present work, 3 wall-normal-free \tsn{RSM}s were assessed through comparison with experimental data for complex 3-D duct flows,
highlighting the impact of the closure used for the velocity/pressure-gradient tensor $\Pi_{ij}$ \eqref{Eq_RSMP3DDF_s_TCsFS_ss_TCs_007}
and for turbulent diffusion by the fluctuating velocities $d_{ij}^{(u)}$ \eqref{Eq_RSMP3DDF_s_TCsFS_ss_TCs_005} on the predictive accuracy of the models.
The Gessner and Emery \cite{Gessner_Emery_1981a} square duct flow is dominated by turbulence-anisotropy-driven secondary flows
whereas the Davis and Gessner \cite{Davis_Gessner_1992a} \tsn{C}-to-\tsn{R} transition duct flow combines pressure-driven secondary flows in the transition section
with turbulence-anisotropy-driven secondary flows in the straight constant cross-section exit part. Therefore, these test-cases are particularly useful in evaluating the predictive accuracy of turbulence closures for secondary flows where
streamwise vorticity is important.
Finally, the Wellborn \etal\ \cite{Wellborn_Reichert_Okiishi_1994a} diffusing \tsn{S}-duct contains a large region of separated flow and tests the ability of the turbulence models to accurately predict 3-D separation and reattachment in presence of blockage due to
confinement and of secondary flows.
Results with the baseline \tsn{LS} \cite{Launder_Sharma_1974a} linear $\mathrm{k}$--$\varepsilon$ closure were included as a reference for comparison with the more advanced differential \tsn{RSM}s. The underlying Boussinesq's hypothesis
pathologically returns negligible levels of normal-stress anisotropy \cite[pp. 273--279]{Wilcox_1998a}
and for this reason the \tsn{LS} $\mathrm{k}$--$\varepsilon$ predicts negligibly weak ($\sim\!\!0$) secondary velocities both in the square-duct \cite{Gessner_Emery_1981a}
and in the straight exit part of the \tsn{C}-to-\tsn{R} duct \cite{Davis_Gessner_1992a}. Furthermore, in the \tsn{S}-duct \cite{Wellborn_Reichert_Okiishi_1994a} test-case,
the \tsn{LS} $\mathrm{k}$--$\varepsilon$ model, which has been calibrated for equilibrium shear flows, severely underestimates separation. For all of the 3 test-cases the
\tsn{LS} $\mathrm{k}$--$\varepsilon$ closure compares very poorly with experimental data.
The \tsn{WNF--LSS RSM} \cite{Gerolymos_Sauret_Vallet_2004a} adopts the Launder-Shima \cite{Launder_Shima_1989a} closure for the homogeneous part of $\Pi_{ij}$ and is therefore calibrated in zero-pressure-gradient flat-plate boundary-layer flow. As a consequence,
it underestimates separation in the \tsn{S}-duct \cite{Wellborn_Reichert_Okiishi_1994a} test-case. On the other hand it has the differential \tsn{RSM}s' inherent ability to predict normal-stress anisotropy and performs quite well for the
\tsn{C}-to-\tsn{R} duct \cite{Davis_Gessner_1992a} but underestimates the centerline velocity peak in the developing square-duct flow \cite{Gessner_Emery_1981a}; this inadequacy was traced to the cumulative influence of the homogeneous
rapid redistribution isotropisation-of-production closure \eqref{Eq_RSMP3DDF_s_TCsFS_ss_TCs_007} $C_\phi^{(\tsn{RH})}$ \tabref{Tab_RSMP3DDF_s_TCsFS_ss_TCs_001} and the Daly-Harlow turbulent-diffusion model \tabref{Tab_RSMP3DDF_s_TCsFS_ss_TCs_001}.
For all of the 3 test-cases that were examined \cite{Gessner_Emery_1981a,Davis_Gessner_1992a,Wellborn_Reichert_Okiishi_1994a},
the \tsn{GLVY} \cite{Gerolymos_Lo_Vallet_Younis_2012a} and \tsn{GV} \cite{Gerolymos_Vallet_2001a} \tsn{RSM}s yield very similar results in quite satisfactory agreement with measurements,
implying that the extra terms in the $\Pi_{ij}$ closure \eqref{Eq_RSMP3DDF_s_TCsFS_ss_TCs_007} used in the \tsn{GLVY RSM} \tabref{Tab_RSMP3DDF_s_TCsFS_ss_TCs_001}
have little influence for the secondary and/or separated 3-D flows studied in this paper; however, these extra terms were found to substantially improve the apparent transition behaviour of the model \cite{icp_Gerolymos_Vallet_2013a}.
The coefficient-function $C_\phi^{(\tsn{RH})}$ used in the \tsn{GLVY} and \tsn{GV} \tsn{RSM}s \tabref{Tab_RSMP3DDF_s_TCsFS_ss_TCs_001} was calibrated
with reference to flows with large separation \cite{Gerolymos_Vallet_2001a,Gerolymos_Vallet_2002a,Gerolymos_Sauret_Vallet_2004b}.
As a result, the \tsn{GLVY} and \tsn{GV} \tsn{RSM}s perform quite well in the \tsn{S}-duct \cite{Wellborn_Reichert_Okiishi_1994a} flow. They predict quite satisfactorily
the other 2 test-cases \cite{Gessner_Emery_1981a,Davis_Gessner_1992a} as well, although they underpredict the strength of the secondary flow velocities and the level of the Reynolds-stress tensor anisotropy.
The results presented in the paper suggest that \tsn{RSM} \tsn{RANS} has the potential to predict complex 3-D flows with streamwise vorticity and separation. Further improvements in the prediction of the Reynolds-stress tensor anisotropy can be
achieved by the use of a differential model for the full Reynolds-stress-dissipation tensor $\varepsilon_{ij}$ \cite{Lumley_Yang_Shih_1999a,Gerolymos_Lo_Vallet_Younis_2012a}. Furthermore, the turbulence structure in separated
and reattaching/relaxing flows exhibits strong hysteresis \cite{Gerolymos_Kallas_Papailiou_1989a} whose inclusion in the model should be investigated \cite{Olsen_Coakley_2001a}.
\section*{Acknowledgments}
The present work was initiated within the \tsn{DGA}-funded research project \tsn{CACV} with Dassault-Aviation.
The computations were performed using \tsn{HPC} ressources allocated at \tsn{GENCI--IDRIS} (Grants 2013-- and 2014--020218) from \tsn{ICS--UPMC} (\tsn{ANR--10--EQPX--29--01}).
The authors are listed alphabetically.
\bibliographystyle{spmpsci.bst}\footnotesize
|
1,116,691,499,264 | arxiv | \section{Introduction}
\noindent
Understanding algebraic relations between solutions of a differential equation is, perhaps along with understanding solutions of a particular form, the central problem in algebraic differential equations.
Characterizing algebraic relations between solutions is an important output of various approaches to differential Galois theory~\cite{kolchin1968algebraic, singer1988algebraic, dreyfus2018hypertranscendence}, while work using the model theory of differential fields frequently applies techniques from stability theory to obtain conclusions in this vein~\cite{nagloo2014algebraic, nagloo2017algebraic, jaoui2019generic}.
A number of recent transcendence results for analytic functions (equivalently, problems of \emph{bi-algebraic geometry}~\cite{klingler2018bi}) have played an important role in diophantine geometry and have natural interpretations in terms of characterizing algebraic relations between solutions of differential equations and their derivatives~\cite{pila2016ax, freitag2017strong, casale2020ax}.
To state our main theorem let us fix a differential field $(k,\delta)$ of characteristic zero, as well as an order $n$ algebraic differential equation
\begin{equation}
\label{ode}
P(y,\delta y,\dots,\delta^ny)=0,
\end{equation}
where $P\in k[X_0,\dots,X_n]$ is irreducible.
For each $m\geq 1$, consider the following condition:
\begin{itemize}
\item[$(C_m)$]
For any $m$ distinct solutions $a_1,\dots,a_m\notin k^{\operatorname{alg}}$ of~(\ref{ode}), the sequence
$$(\delta^{(i)}a_j:i=0,\dots,n-1,j=1,\dots,m)$$
is algebraically independent over $k$.
\end{itemize}
In~\cite{nmdeg}, as a byproduct of working on a related problem, we showed that as long as $n>1$, one always has $(C_{n+2})\implies(C_m)$ for all $m$.
We can now do much better:
\begin{customthm}{A}
\label{A}
If $n>1$ then $(C_3)\implies(C_m)$ for all $m$.
\end{customthm}
The theorem is sharp in both of the natural possible ways: $(C_2)$ is in general not enough and when $n=1$ then $(C_3)$ is not enough.
See Examples~\ref{exa} and~\ref{exb} below.
However, if $n=1$ then we do have $(C_4)\implies(C_m)$ for all $m$, as we will explain.
Let us consider some specific examples that have been studied recently.
\begin{exam}
Consider a Painlev\'e equation from the families $P_{II}$ through $P_{V}$ with generic coefficients.
For instance, one might take $y''=2y^3+ty+\pi$ over the differential field $(\mathbb C(t),\frac{d}{dt})$.
Nagloo and Pillay \cite{nagloo2017algebraic} show that $(C_m)$ holds of all $m$.
\end{exam}
\begin{exam}
In~\cite{jaoui2019generic}, Jaoui studied the case when $P(y,\delta y,\delta^2y)=0$ is a general order two and degree at least 3 algebraic differential equation over a constant field of parameters.
He showed that in that case $(C_2)\implies(C_m)$ for all $m$.
Indeed, it is his phrasing of the problem in the Introduction to~\cite{jaoui2019generic} that lead us to consider the question in these terms.
\end{exam}
\begin{exam}
Consider the following equation satisfied by the $j$-function
$$S(y) + \frac{y^2 - 1968 y + 2654208}{2 y^2 (y - 1728)^2} (y')^2=0,$$ where $S$ is the Schwarzian derivative:
$S(y)=\left( \frac{y''}{y'}\right) '-\frac{1}{2} \left(\frac{y''}{y'}\right)^2$.
It follows from results of \cite{freitag2017strong} that $(C_2)$ fails of this equation.
In fact, if $y_1,y_2$ are nonalgebraic solutions then $(y_1,y_1', y_1'',y_2,y_2',y_2'')$ are algebraically dependent if and only if a modular polynomial vanishes at $(y_1,y_2 )$.
A generalisation of this equation and setting is considered in \cite{casale2020ax}, where the $j$-function is replaced by the uniformisation function of the quotient of the upper half plane by an arbitrary genus zero Fuchsian group $\Gamma$ of the first kind.
For such equations, it is shown that $(C_2)$ fails if and only if $(C_m)$ fails for some $m$, if and only if $\Gamma$ is a proper subgroup of its commensurator.
This analysis of the algebraic relations amongst the solutions and their derivatives, in~\cite{casale2020ax}, is key to proving the Ax-Lindemann-Weierstrass theorem for uniformising functions of Fuchsian groups and answering an old open question of Painlev\'e (1895).
Ax-Lindemann-Weierstrass results for the uniformizing functions of Shimura varieties are an essential component of the resolution of a number of recent diophantine conjectures and can naturally be phrased in terms of algebraic relations between solutions of differential equations (see \cite{klingler2018bi} and the references therein).
\end{exam}
Another way of talking about algebraic differential equations is in terms of algebraic vector fields; namely algebraic varieties $V$ equipped with a polynomial section~$s$ to the tangent bundle.
To study the algebraic relations between the solutions of an equation amounts to looking at subvarieties of cartesian powers of $V$ that are invariant under the vector field.
(Here we put on $V^m$ the canonical vector field induced by $s$, obtained by identifying $T(V^m)$ with the $m$-fold fibred product of $TV$ with itself.)
We can formulate our main theorem as follows:
\begin{customthm}{B}
\label{B}
Suppose $(V,s)$ is an algebraic vector field over a field $k$ of characteristic zero, and $\dim V>1$.
If the third cartesian power $V^3$ admits no invariant subvarieties over $k$ projecting dominantly onto each co-ordinate, other than the diagonals, then the same holds of $V^m$ for all $m\geq 0$.
\end{customthm}
\begin{remark*}
We have restricted our attention here to algebraic vector fields only to keep the exposition concise.
In fact, the theorem holds more generally of ``$D$-varieties" in the sense of Buium~\cite{Buium1}.
That is, our results work identically in the non-autonomous situation where the derivation is not assumed to be trivial on the base field $k$.
The tangent bundlle must then be replaced with the {\em prolongation}, a natural torsor of the tangent bundle that takes into account the twisting induced by~$\delta$.
See~$\S$\ref{sectdvar} below for a discussion, and Corollary~\ref{dvarcor} for the general statement of Theorem~\ref{B} in the non-autonomous setting.
\end{remark*}
There is an apparent discrepancy between Theorems~\ref{A} and~\ref{B}.
The latter is really about relations among {\em generic} solutions to the differential equation -- this is the meaning of only considering invariant subvarieties that project {\em dominantly} on each co-ordinate -- while Theorem~\ref{A} seems to be stronger, making conclusions about all nonalgebraic solutions.
In fact, however, $(C_1)$ already rules out the possibility of lower order equations, so that all nonalgebraic solutions are generic.
These theorems are applications of a more fundamental theorem about the model theory of differentially closed fields ($\operatorname{DCF}_0$), which we now state.
We work model-theoretically, assuming familiarity with geometric stability theory as well as the particular case of $\operatorname{DCF}_0$.
We suggest~\cite{gst} as a general reference for the former, and~\cite{Marker96modeltheory} as an introduction to the latter.
The following is the main clause of Theorem~\ref{c3min} below:
\begin{customthm}{C}
\label{C}
Suppose $p$ is a stationary nonalgebraic type in $\operatorname{DCF}_0$ such that any three distinct realisations of $p$ are independent.
Then $p$ is minimal.
\end{customthm}
Once we have a minimal generic type we can apply the Zilber trichotomy in differentially closed fields and deduce Theorems~\ref{A} and~\ref{B} quite readily.
The proof of minimality itself goes via reducing to the case when $p$ is internal to the constants and with a {\em binding group} that acts $3$-transitively.
As the binding group is an algebraic group action, a theorem of Knop~\cite{knop1983mehrfach} implying that the only $3$-transitive algebraic group action is that of $\operatorname{PGL}_2$ on $\mathbb P$ now applies, and settles the matter.
Other than standard facts from geometric stability theory and the model theory of differentially closed fields, as well as the theorem of Knop just mentioned, the paper is self-contained.
In particular, the results of~\cite{nmdeg} are not used here, though the approach taken there does inform what we do.
The proof of Theorem~\ref{C} works in any totally transcendental theory where one has the following strong form of the Zilber dichotomy: every non locally modular minimal type over $A$ is nonorthogonal to an $A$-definable pure algebraically closed field.
So, for example, we could just as easily have worked with several commuting derivations in partial differentially closed fields ($\operatorname{DCF}_{0,m}$).
Or in the theory $\operatorname{CCM}$ of compact complex manifolds.
Indeed, concerning the latter, a short final section is dedicated to articulating the result as a theorem in bimeromorphic geometry that may be of independent interest, see Theorem~\ref{ccm3min} below.
\medskip
Throughout, we work in a sufficiently saturated model $\mathcal U\models\operatorname{DCF}_0$ with field of constants $\mathcal C$.
\bigskip
\section{Generic transitivity and degree of nonminimality}
\noindent
Recall from~\cite{BC2008} that a definable group action $(G,S)$ of finite Morley rank is \emph{generically transitive} if the action of $G$ on $S$ admits an orbit $\mathcal O$ such that
$$\operatorname{RM}(S\setminus \mathcal O)<\operatorname{RM}(S).$$
Assuming that $\operatorname{dM}(S)=1$, this is equivalent to asking that any two generic elements of $S$ over $A$ are conjugate.
Actually, we are interested in the multiple-transitivity version:
\begin{definition}
A finite rank definable group action $(G,S)$ is {\em generically $k$-transitive} if the diagonal action $(G,S^k)$ is generically transitive.
\end{definition}
Generic $k$-transitivity, and the Borovik-Cherlin conjecture about such group actions (articulated in~\cite{BC2008}), played an important role in our earlier work in~\cite{nmdeg} on bounding the size of a witness to the nonminimality of a type in $\operatorname{DCF}_0$.
Recall:
\begin{definition}
The {\em degree of nonminimality} of $p\in S(A)$ is the least $d$ such that there are distinct realisations $a_1,\dots,a_d$ of $p$ and a nonalgebraic forking extension of $p$ to $Aa_1\cdots a_d$.
\end{definition}
The main result of~\cite{nmdeg} was that $\operatorname{nmdeg}(p)\leq U(p)+1$.
One step in that proof was the following observation, which we repeat here for the sake of completeness.
\begin{proposition}
\label{gendtran}
Suppose $p$ is a stationary type with $U(p)>1$ that is $\mathcal C$-internal and weakly $\mathcal C$-orthogonal.
Let $d:=\operatorname{nmdeg}(p)$.
Then the binding group of $p$ acts transitively and generically $d$-transitively on $p(\mathcal U)$.
\end{proposition}
\begin{proof}
Denote the binding group by $G:=\operatorname{Aut}_A(p/\mathcal C)$.
It acts definably and faithfully on $S:=p(\mathcal U)$.
Weak orthogonality implies that the action is transitive and $p$ is isolated.
So $(G,S)$ is a definable homogeneous space.
To show that the action is generically $d$-transitive is precisely to show that $G$ acts transitively on $p^{(d)}$.
As $G$ is also the binding group of the $\mathcal C$-internal type $p^{(d)}$, it suffices to show that $p^{(d)}$ is weakly orthogonal to $\mathcal C$.
Let $r>1$ be least such that $p^{(r)}$ is not weakly orthogonal to $\mathcal C$.
(This exists by $\mathcal C$-internality.)
Fix $\overline a=(a_1,\dots,a_{r-1})\models p^{(r-1)}$ and let $q$ be the nonforking extension of $p$ to $A\overline a$.
Then $q$ is not weakly orthogonal to $\mathcal C$.
It follows (using elimimination of imaginaries for the structure induced on $\mathcal C$; namely that of a pure algebraically closed field) that there is $a\models q$ and a constant $c\in\mathcal C$ such that $c\in\operatorname{dcl}(A\overline a a)\setminus\operatorname{acl}(A\overline a)$.
Write $c=f(a)$ where $f$ is a partial $A\overline a$-definable function from $S$ to $\mathcal C$.
The fibre $f^{-1}(c)$ is infinite as $U(q)>1$, and so we may assume that all the fibres are infinite.
Let $c_0\in\operatorname{Im}(f)\cap\mathbb Q$, which exists as $\operatorname{Im}(f)$ is an infinite and hence cofinite subset of $\mathcal C$.
Let $a_0$ be generic in $f^{-1}(c_0)$ over $A\overline a$.
Then $U(a_0/A\overline a)>0$ as $f^{-1}(c_0)$ is infinite.
On the other hand, $U(a_0/A\overline a)<U(p)$, as otherwise $a_0\models q$, which is impossible as $f(a)\notin\operatorname{acl}(A\overline a)$ while $f(a_0)\in\mathbb Q=\operatorname{dcl}(\emptyset)$.
Hence $\operatorname{tp}(a_0/A\overline a)$ is a nonalgebraic forking extension of $p$ witnessing that $d\leq r-1$.
By minimal choice of $r$, it follows that $p^{(d)}$ is weakly $\mathcal C$-orthogonal, as desired.
\end{proof}
\bigskip
\section{Distinct solutions being independent}
\noindent
We are interested in the following condition on a stationary type $p\in S(A)$.
\begin{definition}[The condition $D_m$]
Given $m>1$, let us say that {\em $p\in S(A)$ satisfies~$D_m$} if every $m$-tuple of distinct realisations of $p$ is independent over $A$.
\end{definition}
We begin with a few straightforward consequences of $D_2$.
\begin{lemma}
Every type satisfying $D_2$ is of finite rank.
\end{lemma}
\begin{proof}
As some co-ordinate of any realisation of an infinite rank type must be $\delta$-transcendental, it suffices to observe that the $1$-type of a $\delta$-transcendental element does not satisfy $D_2$.
Indeed, if $a$ is $\delta$-transcendental over a differential field $k$ then so is $\delta(a)$, which is distinct for $a$, and $a\mathop{\mathpalette\Notind{}}_k\delta a$.
\end{proof}
\begin{lemma}
\label{notfinitecover}
No type satisfying $D_2$ can be a nontrivial finite cover.
That is, if $p=\operatorname{tp}(a/A)$ is a nonalgebraic stationary type satisfying $D_2$, and $a$ is interalgebraic with $b$ over $A$, then $a\in\operatorname{dcl}(Ab)$.
\end{lemma}
\begin{proof}
If $a\notin\operatorname{dcl}(Ab)$ then we can find $a'\neq a$ such that $\operatorname{tp}(a'/Ab)=\operatorname{tp}(a/Ab)$.
By $D_2$ we must have $a'\mathop{\mathpalette\Ind{}}_Aa$.
But $a\in\operatorname{acl}(Ab)=\operatorname{acl}(Aa')$, which forces $a\in\operatorname{acl}(A)$ and contradicts nonalgebraicity of $p$.
\end{proof}
\begin{lemma}
\label{noweakfibrations}
If $p=\operatorname{tp}(a/A)$ is a nonalgebraic stationary type satisfying $D_2$ then it satisfies the following form of exchange: if $b\in\operatorname{acl}(Aa)\setminus\operatorname{acl}(A)$ then $a\in\operatorname{dcl}(Ab)$.
\end{lemma}
\begin{proof}
Note that the conclusion is a strengthening of the property called {\em admitting no proper fibrations} in~\cite{moosa2014some}.
In any case, by Lemma~\ref{notfinitecover}, it suffices to show that $a\in\operatorname{acl}(Ab)$.
Suppose $a\notin\operatorname{acl}(Ab)$, and seek a contradiction.
It follows that if $a'$ realises the nonforking extension of $\operatorname{tp}(a/Ab)$ to $Aa$ then $a\neq a'$.
On the other hand, using the Lascar inequality, $U(a/Aa')=U(a/Ab)<U(a/A)$ as $b\in\operatorname{acl}(Aa)\setminus\operatorname{acl}(A)$.
So $a$ and $a'$ are dependent over $A$, contradicting $D_2$.
\end{proof}
\begin{proposition}
\label{c2internal}
Every stationary type of $U$-rank $>1$ satisfying $D_2$ is internal and weakly orthogonal to the constants.
\end{proposition}
\begin{proof}
Suppose $p=\operatorname{tp}(a/A)$ is stationary, nonminimal, and satisfies $D_2$.
As $p$ is of finite rank it is nonorthogonal to some minimal type $r \in S(B)$ for some $B\supseteq A$.
We first show that $r$ is not locally modular.
Indeed, in~\cite[Proposition~2.3]{moosa2014some}, it is shown that, because $p$ has no proper fibrations (Lemma~\ref{noweakfibrations}), if $r$ is locally modular than in fact $p$ is interalgebraic with some $q^{(k)}$ where $q\in S(A)$ is a locally modular minimal type and $k\geq 1$.
Now, $k=1$ is impossible by the assumption that $U(p)>1$.
But $k>1$ is also impossible: taking $(b_1,\dots,b_k)\models q^{(k)}$, we obtain
$$b_1\in\operatorname{acl}(Ab_1\cdots b_k)\setminus\operatorname{acl}(A)=\operatorname{acl}(Aa)\setminus\operatorname{acl}(A)$$
while
$$\operatorname{acl}(Aa)=\operatorname{acl}(Ab_1\cdots b_k)\not\subseteq\operatorname{acl}(Ab_1)$$
contradicting Lemma~\ref{noweakfibrations}.
We have shown that $p$ is nonorthogonal to a non locally modular minimal type.
In $\operatorname{DCF}_0$ this means that $p$ is nonorthogonal to the constants $\mathcal C$.
It follows by~\cite[Corollary~7.4.6]{gst} that there is $c\in\operatorname{dcl}(Aa)\setminus\operatorname{acl}(A)$ such that $q:=\operatorname{tp}(c/A)$ is $\mathcal C$-internal.
By Lemma~\ref{noweakfibrations}, $a\in\operatorname{dcl}(Ac)$.
That is, $p$ and $q$ are interdefinable, and hence $p$ is also $\mathcal C$-internal.
If $p$ were not weakly orthogonal to $\mathcal C$ then there would be $c\in\mathcal C$ such that $c\in\operatorname{dcl}(Aa)\setminus\operatorname{acl}(A)$.
Lemma~\ref{noweakfibrations} would then imply that $a\in\operatorname{dcl}(Ac)$, contradicting nonminimality of $p$ (and also $D_2$).
\end{proof}
Here is our main theorem.
\begin{theorem}
\label{c3min}
Suppose $p$ is a stationary nonalgebraic type satisfying~$D_3$.
Then $p$ is minimal.
Moreover, there are in this case exactly two possibilities:
\begin{itemize}
\item[(a)]
Either $p$ is trivial and satisfies $D_m$ for all $m>1$, or,
\item[(b)]
$p$ is nonorthogonal to the constants and $D_4$ fails.
\end{itemize}
\end{theorem}
\begin{proof}
We assume $p$ is not minimal and seek a contradiction.
By Proposition~\ref{c2internal} we have that $p$ is $\mathcal C$-internal and weakly $\mathcal C$-orthogonal.
Denote the binding group by $G:=\operatorname{Aut}_A(p/\mathcal C)$ and set $S:=p(\mathcal U)$.
Proposition~\ref{gendtran} tells us that $(G,S)$ is a generically $d$-transitive definable homogeneous space, where~$d$ is the degree of nonminimality of $p$.
Let $a_1,\dots,a_d$ be distinct realisations of $p$ and $q$ a nonalgebraic forking extension of $p$ to $Aa_1\cdots a_d$.
Let $a_{d+1}\models q$.
Then $a_1,\dots,a_{d+1}$ witnesses the failure of $D_{d+1}$.
Since $D_3$ holds, we must have $d\geq 3$.
In particular, $G$ acts generically $3$-transitively on $S$.
But $D_3$ tells us that every tuple of three distinct elements is generic.
Hence $(G,S)$ is outright $3$-transitive.
As $(G,S)$ is the binding group action of a $\mathcal C$-internal type, it is definably isomorphic (over possibly additional parameters) with a definable homogeneous space in the induced structure on $\mathcal C$.
That induced structure being a pure algebraically closed field, and using the Weil group-chunk theorem, we have that $(G,S)$ is definably isomorphic to an algebraic group action $(\overline G,\overline S)$ in the constants.
So $(\overline G,\overline S)$ is $3$-transitive.
But the only $3$-transitive algebraic group action is $\operatorname{PGL}_2$ acting naturally on $\mathbb P$.
Indeed, Knop~\cite[Satz~2]{knop1983mehrfach} classifies the (faithful) $2$-transitive actions of algebraic groups as being of only two kinds:
\begin{itemize}
\item The standard action of $\operatorname{PGL}_{n+1}$ on $\mathbb P^n.$
\item Certain subgroups of the affine group of transformations on a vector space.
\end{itemize}
In both cases, the action preserves colinearity of three points, and hence the only possibility for $3$-transitivity is if we are in the first case and $n=1$.
So $(\overline G,\overline S)$ is isomorphic to $(\operatorname{PGL}_2(\mathcal C),\mathbb P(\mathcal C))$.
This contradicts nonminimality of $p$.
We have proven that $p$ is minimal.
The ``moreover" clause now follows by applying the Zilber trichotomy as it is manifested in $\operatorname{DCF}_0$.
That trichotomy says that exactly one of the following must hold:
\begin{itemize}
\item[(i)]
$p$ is trivial;
\item[(ii)]
$p$ is nonorthogonal to the generic type of the Manin kernel $\mathcal G$ of a simple abelian variety over $\operatorname{acl}(A)$ that does not descend to the constants; or,
\item[(iii)]
$p$ is nonorthogonal to the constants.
\end{itemize}
This is originally an unpublished theorem of Hrushovski and Sokolovic~\cite{HrSo}.
Published references and explanations can be found in various places, see for example the discussion around Fact~4.1 of~\cite{isolated}.
We first point out that $D_3$ rules out case~(ii).
Indeed, assuming we are in case~(ii), and
setting $q$ to be the nonforking extension of $p$ to $Aa$, it would follow that $q$ is interalgebraic with the generic type $r$ of $\mathcal G$ over $Aa$.
This is because both~$q$ and~$r$ would be modular minimal types, and for such types weak orthogonality implies orthogonality (see~\cite[Corollary~2.5.5]{gst}).
As $p$ satisfies $D_3$, $q$ satisfies $D_2$, and hence Lemma~\ref{noweakfibrations} implies that $r$ is a finite cover of~$q$, say $\pi:r\to q$.
Given $g\models r$, since $\mathcal G$ has finite $n$-torsion, $ng$ also realises $r$ for all $n$, and hence we can find $n>1$ such that $\pi(g)\neq\pi(ng)$.
But these two elements are distinct realisations of $q$ that are dependent over $Aa$.
It follows that $\{a,\pi(g),\pi(ng)\}$ witnesses the failure of $p$ to satisfy~$D_3$.
We must, therefore, be in case~(i) or~(iii); either $p$ is trivial or it is nonorthogonal to the constants.
In the trivial case, $D_2$ already implies $D_m$ for all $m$, as desired.
So it remains to consider the case of a minimal type nonorthogonal to the constants.
We will show, and this will suffice, that in this case $D_4$ fails.
Indeed, from nonorthogonality to the constants, and using $D_2$, exactly as in the proof of Proposition~\ref{c2internal}, we conclude that $p$ is actually $\mathcal C$-internal and either interdefinable with the generic type of the constants or weakly $\mathcal C$-orthogonal.
In the former case $D_2$ would clearly fail, so $p$ is weakly $\mathcal C$-orthogonal.
It follows that $S:=p(\mathcal U)$ is a strongly minimal homogeneous space for the binding group action.
These are completely classified (see, for example,~\cite[Fact~1.6.25]{gst}), and it follows from that classification that if $\overline a=(a_1,a_2,a_3)\models p^{(3)}$ then the nonforking extension of $p$ to $A\overline a$ is not weakly orthogonal to $\mathcal C$.
This gives rise to an $A\overline a$-definable function $f:X\to\mathcal C$ where $X\subseteq S\setminus\{a_1,a_2,a_3\}$ is cofinite,
$f(X)$ is cofinite, and the fibres of~$f$ are finite.
Let $c\in f(X)\cap\mathbb Q$ and choose any $a_4\in f^{-1}(c)$.
Then $a_4\in\operatorname{acl}(A\overline a)$, and hence $(a_1,a_2,a_3,a_4)$ witnesses the failure of $D_4$.
\end{proof}
The following example shows that Theorem~\ref{c3min} is sharp in the sense that $D_2$ would not suffice for minimality.
\begin{example}
\label{exa}
In~\cite[$\S4.2$]{nmdeg} we gave an example, for each $n\geq 3$, of a $\mathcal C$-internal type $p$ of $U$-rank $n-1$ whose binding group action is that of $\operatorname{PGL}_n$ on $\mathbb P^{n-1}$. In particular, the binding group action is $2$-transitive, and hence $D_2$ holds for these nonminimal types.
The construction is based on one from~\cite{jin-moosa} when $n=2$.
Essentially, $p$ is the generic type of the projectivisation of the space of solutions to $\delta X=BX$ where $B$ is an $n\times n$ matrix whose entries form a differentially transcendental set, and $X$ is an $n$-column of variables.
\end{example}
Let us point out that both cases of the ``moreover" clause of Theorem~\ref{c3min} actually occur.
That there are minimal trivial types satisfying $D_m$ for all $m>1$ is well known, with constructions appearing in~\cite{HrIt, mcgrail2000search, notmin}.
The following shows that case~(b) also occurs: there are minimal types nonorthogonal to the constants satisfying $D_3$.
\begin{example}
\label{exb}
Consider the order~$1$ Riccati equation $\delta y=y^2+c$ with $c\in\mathbb C(t)$ a transcendental rational function.
The generic type $p$ of this equation is minimal and $\mathcal C$-internal.
Nagloo~\cite{nagloo2020algebraic} has shown that it does satisfy~$D_3$.
(Another argument is by~\cite[$\S4.3$]{nmdeg} where we show that the binding group acts $3$-transitively on $p$.)
\end{example}
\medskip
\subsection{Deducing Theorem~\ref{A}}
Let us spell out how Theorem~\ref{c3min} implies Theorem~\ref{A} of the Introduction.
This is a standard translation from model-theoretic to differential-algebraic language.
We are working over a differential field $(k,\delta)$ of characteristic zero, with an order $n>1$ algebraic differential equation
$$P(y,\delta y,\dots,\delta^ny)=0$$
where $P\in k[X_0,\dots,X_n]$ is irreducible.
Denote by $Y\subseteq\mathcal U$ the Kolchin closed set defined by this equation.
For $m>1$, the condition $(C_m)$ on the equation translates precisely into saying that any distinct $m$ elements of $Y\setminus k^{\operatorname{alg}}$ are independent (in the stability-theoretic sense).
Suppose that $(C_3)$ holds.
In particular, $(C_1)$ holds, and this already implies that $Y$ has no infinite Kolchin closed subsets over $k$ of order less than $n$.
Together with irreducibility of $P$, this implies that $Y$ is Kolchin irreducible and that all points of $Y\setminus k^{\operatorname{alg}}$ realise the Kolchin generic type $p\in S(k)$.
So condition $(C_3)$ implies that $p$ satisfies $D_3$, and hence is minimal by Theorem~\ref{c3min}.
That $n>1$ now implies that $p$ cannot be nonorthogonal to the constants.
We thus conclude that we are in case~(a) of Theorem~\ref{c3min}, and so $D_m$ holds of $p$ for all $m$.
This in turn implies that $(C_m)$ holds of the equation, for all~$m$.\qed
\bigskip
\section{A formulation for $D$-varieties}
\label{sectdvar}
\noindent
There is a well studied correspondence between types in $\operatorname{DCF}_0$ and so called ``$D$-varieties", and it is worth reformulating Theorem~\ref{c3min} in these terms.
The idea goes back to Buium~\cite{Buium1}, who considered algebraic varieties over a differential field $(k,\delta)$ equipped with an extension of the derivation $\delta$ to the structure sheaf of the variety.
Later, an equivalent formulation, paralleling more closely the language of algebraic vector fields, came into use.
We point the reader to~\cite[$\S2$]{kowalski2006quantifier} for a more detailed survey with references.
We restrict ourselves here to a very concise explanation.
Fix a differential field $(k,\delta)$ of characteristic zero.
If $V\subseteq \mathbb A^n$ is an irreducible affine algebraic variety over $k$, then the {\em prolongation} of $V$ is the algebraic variety $\tau V\subseteq\mathbb A^{2n}$ over~$k$ whose defining equations are
\begin{eqnarray*}
P(x_1,\dots,x_n)&=&0\\
P^\delta(x_1,\dots,x_n)+\sum_{i=1}^n\frac{\partial P}{\partial x_i}(x_1,\dots,x_n)\cdot y_i&=&0
\end{eqnarray*}
for each $P$ vanishing on $V$.
Here $P^\delta$ denotes the polynomial obtained by applying $\delta$ to the coefficients of $P$.
The projection onto the first $n$ coordinates gives us a surjective morphism $\pi:\tau V\to V$.
\begin{definition}
\label{defn-dvar}
A {\em $D$-variety over $k$} is a pair $(V,s)$ where $V$ is an irreducible algebraic variety over $k$ and $s:V\to\tau V$ is a regular section to the prolongation defined over $k$.
A {\em $D$-subvariety} of $(V,s)$ is then a $D$-variety $(W,t)$ where $W$ is a closed subvariety of $V$ over $k$ and $t=s|_W$.
\end{definition}
Note that if $\delta=0$ on $k$, then the above equations for $\tau V$ reduce to the familiar equations for the tangent bundle $TV$.
A $D$-variety in that case is nothing other than an algebraic vector field, and a $D$-subvariety is an invariant subvariety of the vector field.
In general, $\tau V$ will be a torsor for $TV$; for each $a\in V$ the fibre $\tau_a V$ is an affine translate of the tangent space $T_aV$.
Associated to a $D$-variety $(V,s)$ over $k$ is a certain complete type over $k$ in $\operatorname{DCF}_0$, namely the {\em generic type of $(V,s)$ over $k$}.
This type, $p(x)$, is axiomatised by asserting that $x$ is Zariski-generic in $V$ over $k$, and that the following system of order $1$ algebraic differential equations holds: $s(x)=(x,\delta(x))$.
In fact, up to interdefinability, all finite rank types in $\operatorname{DCF}_0$ arise in this way; as the generic types of $D$-varieties.
Results about finite rank types in $\operatorname{DCF}_0$, such as Theorem~\ref{c3min} above, can therefore be translated into algebro-geometric statements about $D$-varieties.
For example, the condition we called $D_m$ in the previous section, namely that of having every distinct $m$-tuple of realisations being independent, when applied to the generic type of $(V,s)$ translates to the absence of any proper $D$-subvarieties of $(V^m,s^m)$ projecting dominantly onto each co-ordinate, other than the diagonals.
In particular, we obtain
\begin{corollary}
\label{dvarcor}
Suppose $(V,s)$ is a $D$-variety over $(k,\delta)$ with $\dim V>1$.
If the third cartesian power $(V^3,s^3)$ admits no proper $D$-subvarieties over $k$ projecting dominantly onto each co-ordinate, other than the diagonals, then the same holds of $(V^m,s^m)$ for all $m$.
\end{corollary}
\begin{proof}
Let $p\in S(k)$ be the generic type of $(V,s)$ over $k$.
The assumption on $(V^3,s^3)$ tells us that $p$ satisfies $D_3$.
Hence, by Theorem~\ref{c3min}, $p$ is minimal.
Now, the fact that $\dim V>1$ rules out the possibility of $p$ being nonorthogonal to the constants.
So we are in case~(a) of the Theorem, and $p$ satisfies $D_m$ for all $m$.
This implies the desired conclusion about $D$-subvarieties of $(V^m,s^m)$ over $k$.
\end{proof}
Specialising to the case when $\delta=0$ on $k$ yeilds Theorem~\ref{B} of the Introduction.
When $\dim V=1$, then one has to make the assumption on $(V^4,s^4)$ in order to rule out case~(b) of Theorem~\ref{c3min} and thus conclude it for all $m$.
\bigskip
\section{The case of compact complex manifolds}
\noindent
As mentioned in the introduction, the main part of Theorem~\ref{c3min} holds also in $\operatorname{CCM}$, the theory of compact complex manifolds.
This translates into the following statement in bimeromorphic geometry that may be of independent interest:
\begin{theorem}
\label{ccm3min}
Suppose $f:X\to Y$ is a fibre space of compact K\"ahler manifolds\footnote{More generally our results apply to compact complex anlaytic varieties in Fujiki's class $\mathscr C$. In fact, all that is needed is that they be {\em essentially saturated} in the sense of~\cite{saturated}.}.
Suppose that the three-fold fibred product $X\times_YX\times_YX$ contains no proper complex analytic subvarieties that projects onto each co-ordinate, other than the diagonals.
Then the `general' fibre of $f$ is simple.
\end{theorem}
Let us define the terminology used here.
That $f$ is a {\em fibre space} means that off a proper complex analytic subset of~$Y$ the fibres are irreducible.
A property holds of the {\em `general'} fibre of $f$ if it holds of all fibres off a countable union of proper complex analytic subsets of $Y$.
Finally, we mean {\em simple} in the sense of~\cite{fujiki83}; namely, not covered by an analytic family of proper positive-dimensional analytic subsets.
We will not give a proper proof of this theorem, as it is really the same as (part of) Theorem~\ref{c3min}.
But let us give some explanations.
First of all, we work in $\operatorname{CCM}$ which is the theory of the multisorted structure $\mathcal A$ where there is a sort for each compact complex analytic variety and a predicate for every complex analytic subset of a cartesian product of sorts.
The role of the constants is played by the projective line which is a sort on which the induced structure is bi-interpretable with that of a pure algebraically closed field.
The truth of the Zilber dichotomy in $\operatorname{CCM}$ tells us that every non locally modular minimal type is nonorthogonal to the projective line.
We suggest~\cite{moosapillay-survey} for an introduction to, and survey of, the subject.
Theorem~\ref{ccm3min} is proved by considering the generic type~$p$ of the generic fibre of~$f$, observing that the assumption on the three-fold fibred product expresses precisely that $p$ satisfies $D_3$, and then deducing, exactly as in the proof of Theorem~\ref{c3min}, that~$p$ is minimal.
But for a compact complex analytic variety to be simple is equivalent to its generic type in $\operatorname{CCM}$ being minimal.
But there is a subtlety: the structure $\mathcal A$ is not saturated and so ``generic" in the above paragraph has to be understood, {\em a priori}, in the sense of a nonstandard elementary extension of $\mathcal A$.
See~\cite[$\S2$]{ret} for details about the Zariski geometry in nonstandard models.
The reason for restricting to the K\"ahler case is then precisely so that we can, following~\cite{saturated}, find a suitable countable language in which the standard model is saturated.
We can thus conclude simplicity of the `general' fibre in $\mathcal A$ from simplicity of the generic fibre in an elementary extension.
Note that we only concluded minimality, and did not state an analogue of the ``moreover" clause in Theorem~\ref{c3min}.
The reason for this is that there remains some ambiguity about the characterisation of nontrivial locally modular types in $\operatorname{CCM}$.
It is known that they are nonorthogonal to simple complex tori, but it seems as yet unclear whether one has the same control over parameters that one does in $\operatorname{DCF}_0$.
This is related to Question~5.1 of~\cite{isolated}.
\bigskip
|
1,116,691,499,265 | arxiv | \section{Introduction}
In 1982, R. S. Hamilton introduced the intrinsic geometric flows, \textit
Ricci flow} \cite{ham-82}
\begin{equation}
\frac{\partial }{\partial t}g(t)=-2\func{Ric}(g(t))
\end{equation}
and \textit{Yamabe flow} \cite{ham}
\begin{equation}
\frac{\partial }{\partial t}g(t)=-\func{scal}(t)\cdot g(t)
\end{equation}
which are evolution equations for Riemannian metrics. In a $2$-dimensional
manifold, Ricci flow and Yamabe flow are equivalent, but for higher
dimensions, we do not have such a relation.
Ricci solitons and Yamabe solitons correspond to self-similar solutions of
Ricci flow and Yamabe flow, respectively. Therefore, on an $n$-dimensional
smooth manifold $M$, a Riemannian metric $g$ and a non-vanishing vector
field $V$ is said to define \textit{a Ricci soliton} \cite{ham} if there
exists a real constant $\lambda$ such that
\begin{equation} \label{1}
\frac{1}{2}\pounds _{V}g+\func{Ric}=\lambda g,
\end{equation}
respectively, \textit{a Yamabe soliton} \cite{ham} if there exists a real
constant $\lambda$ such that
\begin{equation} \label{2}
\frac{1}{2}\pounds _{V}g=(\func{scal}-\lambda) g,
\end{equation}
where $\pounds _{V}$ denotes the Lie derivative operator in the direction of
the vector field $V$, $\func{Ric}$ and $\func{scal}$ denote the Ricci
curvature tensor field and respectively the scalar curvature of $g$. A Ricci
soliton (or a Yamabe soliton) $(V,\lambda)$ on a Riemannian manifold $(M,g)$
is said to be \textit{shrinking}, \textit{steady} or \textit{expanding}
according as $\lambda$ is positive, zero or negative, respectively.
Remark that Ricci solitons are natural generalizations of Einstein metrics,
any Einstein metric giving a trivial Ricci soliton.
\bigskip
Different generalizations of Ricci and Yamabe solitons have been lately
considered. If $\lambda$ is a smooth function on $M$, then (\ref{1}) defines
\textit{an almost Ricci soliton} \cite{pi} and (\ref{2}) defines \textit{an
almost Yamabe soliton} \cite{BB13}. Moreover, for a given $1$-form $\eta$ on
$M$, if there exist two real constants $\lambda$ and $\mu$ such that
(i)
\begin{equation} \label{11}
\frac{1}{2}\pounds _{V}g+\func{Ric}=\lambda g+\mu \eta\otimes \eta
\end{equation}
we call $(V,\lambda,\mu)$ \textit{an $\eta$-Ricci soliton} \cite{ch}, and if
(ii)
\begin{equation} \label{111}
\frac{1}{2}\pounds _{V}g=(\func{scal}-\lambda) g+\mu \eta\otimes \eta
\end{equation}
we call $(V,\lambda,\mu)$ \textit{an $\eta$-Yamabe soliton} \cite{cd}. If
\lambda$ and $\mu$ are smooth functions on $M$, then (\ref{11}) defines
\textit{an almost $\eta$-Ricci soliton} \cite{b} and (\ref{111}) defines
\textit{an almost $\eta$-Yamabe soliton} \cite{neto}.
\bigskip
If the potential vector field $V$ is of gradient type, $V=\func{grad}(\sigma
)$, for $\sigma $ a smooth function on $M$, then $(V,\lambda ,\mu )$ is
called a \textit{gradient soliton} (see \cite{ham}, \cite{pi}, \cite{ch} and
\cite{b}). If $\sigma $ is a constant, then the gradient soliton $(V,\lambda
,\mu )$ is trivial.
In \cite{ch}, J. T. Cho and M. Kimura classified a real hypersurface
admitting $\eta $-Ricci soliton in a non-flat complex space form. In \cit
{Ch-10}, the same authors studied compact Lagrangian submanifolds in a K\"{a
hler manifold such that the induced metric on the Lagrangian submanifold is
a Ricci soliton with potential field $JH$, where $J$ is the complex
structure of the manifold and $H$ is the mean curvature vector field of the
submanifold. In \cite{Ch-12}, they considered complete Ricci solitons on
conformally flat hypersurfaces in Euclidean spaces and spheres. The
classification of all Ricci solitons on Euclidean hypersurfaces arisen from
the position vector field was given in \cite{Cd-14}, by B.-Y. Chen and S.
Deshmukh. A necessary and sufficient condition for an $n$-dimensional
submanifold $M$ to be a Ricci soliton with potential field $V^{T}$, where
V^{T}$ is the tangential component of $V$, was given in \cite{Chen-15}, by
B.-Y. Chen. Some necessary and sufficient conditions for a hypersurface of a
Euclidean space to be a gradient Ricci soliton were obtained in \cite{h}, by
{H. Al-Sodais, H. Alodan and S. Deshmukh}. In \cite{cd}, B.-Y. Chen and S.
Deshmukh considered Yamabe and quasi-Yamabe solitons on Euclidean
submanifolds whose soliton fields are the tangential components of their
position vector fields. In \cite{Eken-19}, \c{S}. E. Meri\c{c} and E. Kil
\c{c} studied the conditions under which a submanifold of a Ricci soliton is
also a Ricci soliton or an almost Ricci soliton. The classification of
almost Yamabe solitons on hypersurfaces in Euclidean spaces arisen from the
position vector field was given in \cite{SM19}, by T. Seko and S. Maeta.
\bigskip
Motivated by the above studies, in the present paper, we establish some
properties of almost $\eta $-Ricci and almost $\eta $-Yamabe solitons on
Riemannian manifolds and on submanifolds isometrically immersed into a
Riemannian manifold $\left( \widetilde{M},\widetilde{g}\right) $ whose
potential vector field is the tangential component of a torse-forming (in
particular, concircular or recurrent) vector field on $\widetilde{M}$,
treating also the case of a minimal or pseudo quasi-umbilical hypersurface.
Moreover, we provide necessary and sufficient conditions for an orientable
hypersurface of the unit sphere to be an almost $\eta $-Ricci or an almost
\eta $-Yamabe soliton in terms of the second fundamental tensor field. A
partial study on this topic has been begun in \cite{bl}. Remark also that
almost Yamabe solitons on submanifolds were studied in \cite{SM19}.
\pagebreak
\section{Preliminaries}
\medskip A non-flat Riemannian manifold $(M,g)$ $(n\geq 3)$ is called
a) \textit{mixed generalized quasi-Einstein manifold} \cite{bhat} if its
Ricci tensor field is not identically zero and verifie
\begin{equation}
\func{Ric}=\alpha g+\beta A\otimes A+\gamma B\otimes B+\delta (A\otimes
B+B\otimes A), \label{GQE}
\end{equation
where $\alpha ,\beta ,\gamma $ and $\delta $ are smooth functions and $A,$
B $ are $1$-forms on $M$ such that the corresponding vector fields to the $1
-forms $A$ and $B$ are $g$-orthogonal. In particular, the manifold $M$ is
called:
\ \ \ i) \textit{generalized quasi-Einstein} \cite{Chaki-2001} if $\delta=0$;
\ \ \ ii) \textit{almost quasi-Einstein} \cite{chen-17} if $\beta=\gamma=0$;
\ \ \ iii) \textit{quasi-Einstein} \cite{ChakiMaity-2000} if $\beta=\delta=0$
or $\gamma=\delta=0$;
\ \ \ iv) \textit{Einstein} \cite{besse} if $\beta=\gamma=\delta=0$;
\medskip
b) \textit{pseudo quasi-Einstein} \cite{DeMa-2018} if its Ricci tensor field
is not identically zero and verifies
\begin{equation*}
\func{Ric}=\alpha g+\beta A\otimes A+\gamma E,
\end{equation*
where $\alpha ,\beta$ and $\gamma $ are smooth functions, $A$ is a $1$-form
and $E$ is a symmetric $(0,2)$-tensor field with vanishing trace on $M$. In
particular, if $\gamma=0$, then the manifold $M$ is \textit{quasi-Einstein}.
\medskip A vector field $V$ on a (pseudo)-Riemannian manifold $\left(
M,g\right) $ is called \textit{torse-forming }\cite{Yano-44} if
\begin{equation*}
\nabla _{X}V=aX+\psi (X)V,
\end{equation*
where $a$ is a smooth function, $\psi $ is a $1$-form and $\nabla $ is the
Levi-Civita connection of $g.$ Moreover, if $\psi (V)=0$, then $V$ is called
a \textit{torqued} vector field.
In particular, if $\psi =0$, $V$ is called a \textit{concircular} \textit
vector field }\cite{Fialkow-39} and if $a=0$, $V $ is called a \textit
recurrent} \textit{vector field }\cite{Shou}.
\section{Solitons with torse-forming potential vector field}
\subsection{Almost $\protect\eta$-Ricci solitons}
\bigskip Remark that any concircular vector field with $a(x)\neq 0$, for any
$x\in M$, is of gradient type, namely
\begin{equation*}
V=\frac{1}{2a}\func{grad}(|V|^{2})
\end{equation*
whose divergence is $\func{div}(V)=an$, with $n=\dim (M)$. Moreover,
\begin{equation*}
R(X,V)V=X(a)V-V(a)X
\end{equation*
for any $X\in \chi (M)$ and
\begin{equation*}
\func{Ric}(V,V)=(1-n)V(a).
\end{equation*}
If $(M,g)$ is an almost $\eta $-Ricci soliton with the potential vector
field $V$ and $\eta $ is the $g$-dual of $V$, then
\begin{equation*}
\func{Ric}=(\lambda -a)g+\mu \eta \otimes \eta ,
\end{equation*
\begin{equation*}
\func{scal}=(n-1)\left[ (\lambda -a)-\frac{V(a)}{|V|^{2}}\right]
\end{equation*
and we can state:
\begin{proposition}
\label{p} If a Riemannian manifold $(M,g)$ is an almost $\eta $-Ricci
soliton $(V,\lambda ,\mu )$ with concircular potential vector field $V$ and
a$ is a non zero constant, $\eta$ is the $g$-dual of $V$, then
i) $M$ is a quasi-Einstein manifold with associated functions $(\lambda-a)$
and $\mu$;
ii) $\func{grad}(\lambda)$, $\func{grad}(\mu)$ and $\func{grad}(\func{scal})$
are collinear with $V $.
\end{proposition}
\begin{proof}
Since $R(X,V)V=0$, for any $X\in \chi(M)$, we get $\Ric(V,V)=0$ and
$$(\lambda-a)+|V|^2\mu=0.$$
Differentiating the previous relation, using $d\scal =2\div(\Ric)$ and taking into account that
$$(\div(\mu(\eta\otimes \eta)))(X)=\mu g(\nabla_VV,X)+\mu\eta(X)\div(V)+\eta(X)d\mu(V),$$
we get:
$$(n-3)\grad (\lambda)=2[(n+1)a\mu+g(\grad(\mu),V)]V$$
and applying the gradient to the same relation, we have:
$$\grad(\lambda)=-2a\mu V-\grad(\mu)|V|^2,$$
which combined give:
$$\grad(\mu)=-\frac{2}{(n-3)|V|^2}[2(n-2)a\mu+g(\grad(\mu),V)]V.$$
Taking now the inner product with $V$, we get:
$$g(\grad(\mu),V)=-\frac{4(n-2)}{n-1}a\mu,$$
therefore
$$\grad(\mu)=-\frac{4(n-2)}{n-1}a\mu\frac{V}{|V|^2}.$$
Also, we get
$$\grad(\lambda)=\frac{2(n-3)}{n-1}a\mu V$$
and
$$\grad(\scal)=2(n-3)a\mu V.$$
\end{proof}
If $M$ has constant scalar curvature and $n>3$, then $\mu=0$, $\lambda=a$,
\func{Ric}=0$ and $\func{scal}=0$, therefore:
\begin{corollary}
Under the hypotheses of Proposition \ref{p}, if $M$ is of constant scalar
curvature and $n>3$, then $M$ is a Ricci-flat manifold.
\end{corollary}
For the almost Ricci solitons, we can state:
\begin{proposition}
\label{pp} If an $n$-dimensional Riemannian manifold $(M,g)$, with $n>3$, is
an almost Ricci soliton $(V,\lambda )$ with concircular potential vector
field $V$ and $a$ is a non zero constant, then $M$ is a Ricci-flat manifold.
\end{proposition}
\begin{proposition}
\label{p1} If a Riemannian manifold $(M,g)$ is an almost $\eta$-Ricci
soliton $(V,\lambda,\mu)$ with torqued potential vector field $V$ and $\eta$
is the $g$-dual of $V$, then $M$ is a mixed generalized quasi-Einstein
manifold with associated functions $(\lambda-a)$, $\mu$, $0$ and $-\frac{1}{
}$.
\end{proposition}
\begin{proof}
From the condition for $V$ to be torqued, we get
$$\frac{1}{2}(\pounds _{V}g)(X,Y)=\frac{1}{2}[g(\nabla_XV,Y)+g(\nabla_YV,X)]=ag(X,Y)+\frac{1}{2}[\psi(X)\eta(Y)+\eta(X)\psi(Y)]$$
and from the soliton equation, we obtain
$$\Ric(X,Y)=(\lambda-a)g(X,Y)+\mu \eta(X)\eta(Y)-\frac{1}{2}[\psi(X)\eta(Y)+\eta(X)\psi(Y)].$$
Let $U$ be the $g$-dual vector field of $\psi$. Then
$$\eta(U)=\psi(V)=0,$$
hence the conclusion.
\end{proof}
\begin{corollary}
Under the hypotheses of Proposition \ref{p1}, if $\psi=\mu \eta$, then $M$
is an Einstein manifold. In this case:
i) $V$ is a geodesic vector field if and only if $\mu=-\frac{a}{|V|^2}$;
ii) $V$ is concircular and the soliton is given by $(V,a,0)$.
\end{corollary}
\begin{proof}
From $\psi(V)=\mu \eta(V)$, we have
$$\nabla_VV=(a+\mu|V|^2)V,$$
hence the two statements.
\end{proof}
\medskip
If the potential vector field is torse-forming, we get the following two
results for almost $\eta $-Ricci solitons similar to those proved in \cit
{chende} for Ricci solitons with concurrent potential vector field.
\begin{theorem}
Let $(M,g)$ be an $n$-dimensional Riemannian manifold and let $V$ be a
torse-forming vector field satisfying $\func{Ric}_{M}(V,V)=0$. Then
(V,\lambda ,\mu )$ is an almost $\eta $-Ricci soliton with $\eta$ the $g
-dual of $V$ if and only if the following conditions hold:
i) the soliton is given by $\left(\lambda, \psi(V)-\frac{\lambda-a}{|V|^2
\right)$;
ii) $M$ is an open part of a warped product manifold $(I\times
_{s}F,ds^{2}+s^{2}g_{F})$, where $I$ is an open real interval with arclength
$s$ and $F$ is an $(n-1)$-dimensional Einstein manifold with $\func{Ric
_{F}=(n-2)g_{F}$.
\end{theorem}
\begin{proof}
Following the same steps like in \cite{chende}, we get ii). In our case
$$\frac{1}{2}\pounds _{V}g=ag+\frac{1}{2}(\psi\otimes \eta+\eta\otimes \psi)$$
and from (\ref{11}), we have:
$$\Ric_M=(\lambda-a) g+\mu \eta \otimes \eta-\frac{1}{2}(\psi\otimes \eta+\eta\otimes \psi).$$
By using $\Ric_M(V,V)=0$, we obtain i).
\end{proof}
For almost Ricci solitons with concurrent potential vector field, we can
state:
\begin{proposition}
An $n$-dimensional Riemannian manifold $(M,g)$ is an almost Ricci soliton
(V,\lambda)$ with concurrent potential vector field $V$ if and only if the
following conditions hold:
i) the soliton is a shrinking Ricci soliton with $\lambda=1$;
ii) $M$ is an open part of a warped product manifold $(I\times
_{s}F,ds^{2}+s^{2}g_{F})$, where $I$ is an open real interval with arclength
$s$ and $F$ is an $(n-1)$-dimensional Einstein manifold with $\func{Ric
_{F}=(n-2)g_{F}$.
\end{proposition}
Therefore, there do not exist proper almost Ricci solitons (i.e. with
non-constant $\lambda $) with concurrent potential vector field.
\bigskip
It was proved in \cite{c} that the gradient of a non-constant smooth
function $\sigma $ on a Riemannian manifold $(M,g)$ is a torse-forming
vector field with
\begin{equation*}
\nabla _{X}(\func{grad}(\sigma ))=aX+\psi (X)\func{grad}(\sigma )
\end{equation*
if and only if
\begin{equation*}
\func{Hess}(\sigma )=ag+\delta d\sigma \otimes d\sigma ,
\end{equation*
where $\psi =\delta d\sigma $ and we can state:
\begin{proposition}
If a Riemannian manifold $(M,g)$ is an almost $\eta$-Ricci soliton
(V,\lambda,\mu)$ with torse-forming potential vector field $V=\func{grad
(\sigma)$ and $\eta$ is the $g$-dual of $V$, then $M$ is a quasi-Einstein
manifold with associated functions $(\lambda-a)$ and $(\mu-\delta)$.
\end{proposition}
\begin{proof}
From (\ref{11}), we get
$$\func{Ric}=(\lambda-a) g+(\mu-\delta) d\sigma \otimes d\sigma.$$ So $M$ is a quasi-Einstein manifold with
associated functions $(\lambda-a)$ and $(\mu-\delta)$.
\end{proof}
As a consequence, we obtain:
\begin{corollary}
With the above notations, if a Riemannian manifold $(M,g)$ is an almost
\eta $-Ricci soliton $(V,a,\delta)$ with torse-forming potential vector
field $V$ of gradient type and $\eta$ is the $g$-dual of $V$, then $M$ is a
Ricci-flat manifold.
\end{corollary}
\begin{example}
\label{ex1} \label{wexa} Let $M=\{(x,y,z)\in\mathbb{R}^3, z> 0\}$, where
(x,y,z)$ are the standard coordinates in $\mathbb{R}^3$. Set
\begin{equation*}
V:=-z\frac{\partial}{\partial z}, \ \ \eta:=-\frac{1}{z}dz,
\end{equation*}
\begin{equation*}
g:=\frac{1}{z^2}(dx\otimes dx+dy\otimes dy+dz\otimes dz).
\end{equation*}
Consider the linearly independent system of vector fields:
\begin{equation*}
E_1:=z\frac{\partial}{\partial x}, \ \ E_2:=z\frac{\partial}{\partial y}, \
\ E_3:=-z\frac{\partial}{\partial z}.
\end{equation*}
Then:
\begin{equation*}
\eta(E_1)=0, \ \ \eta(E_2)=0, \ \ \eta(E_3)=1,
\end{equation*}
\begin{equation*}
[E_1,E_2]=0, \ \ [E_2,E_3]=E_2, \ \ [E_3,E_1]=-E_1
\end{equation*}
and the Levi-Civita connection $\nabla$ is deduced from Koszul's formula
\begin{equation*}
2g(\nabla_XY,Z)=X(g(Y,Z))+Y(g(Z,X))-Z(g(X,Y))-
\end{equation*
\begin{equation*}
-g(X,[Y,Z])+g(Y,[Z,X])+g(Z,[X,Y]),
\end{equation*}
precisely
\begin{equation*}
\nabla_{E_1}E_1=-E_3, \ \ \nabla_{E_1}E_2=0, \ \ \nabla_{E_1}E_3=E_1, \ \
\nabla_{E_2}E_1=0,
\end{equation*}
\begin{equation*}
\nabla_{E_2}E_2=-E_3, \ \ \nabla_{E_2}E_3=E_2, \ \ \nabla_{E_3}E_1=0, \ \
\nabla_{E_3}E_2=0, \ \ \nabla_{E_3}E_3=0.
\end{equation*}
Then the Riemann and the Ricci curvature tensor fields are given by:
\begin{equation*}
R(E_1,E_2)E_2=-E_1, \ \ R(E_1,E_3)E_3=-E_1, \ \ R(E_2,E_1)E_1=-E_2,
\end{equation*}
\begin{equation*}
R(E_2,E_3)E_3=-E_2, \ \ R(E_3,E_1)E_1=-E_3, \ \ R(E_3,E_2)E_2=-E_3,
\end{equation*}
\begin{equation*}
\func{Ric}(E_1,E_1)=\func{Ric}(E_2,E_2)=\func{Ric}(E_3,E_3)=-2.
\end{equation*}
Writing the $\eta$-Ricci soliton equation in $(E_i,E_i)$ we obtain:
\begin{equation*}
g(\nabla_{E_i}E_3,E_i)+\func{Ric}(E_i,E_i)=\lambda
g(E_i,E_i)+\mu\eta(E_i)\eta(E_i),
\end{equation*}
for all $i\in\{1,2,3\}$ and we get that for $\lambda=\mu=-1$, the data
(V,\lambda,\mu)$ define an $\eta$-Ricci soliton on $(M, g)$. Moreover, it is
a gradient $\eta$-Ricci soliton, since the potential vector field $V$ is a
torse-forming vector field of gradient type, $V=\func{grad}(f)$, where
f(x,y,z):=-\ln z$.
\end{example}
\begin{example}
\label{ex2} Let $M=\mathbb{R}^3$, $(x,y,z)$ be the standard coordinates in
\mathbb{R}^3$ and $g$ be the Lorentzian metric:
\begin{equation*}
g:=e^{-2z}dx\otimes dx+e^{2x-2z}dy\otimes dy-dz\otimes dz.
\end{equation*}
Consider the vector field $V$ and the $1$-form $\eta$:
\begin{equation*}
V:=\frac{\partial}{\partial z}, \ \ \eta:=dz.
\end{equation*}
For the orthonormal vector fields:
\begin{equation*}
E_1=e^z\frac{\partial}{\partial x}, \ \ E_2=e^{z-x}\frac{\partial}{\partial
}, \ \ E_3=\frac{\partial}{\partial z},
\end{equation*}
we get:
\begin{equation*}
\nabla_{E_1}E_1=-E_3, \ \ \nabla_{E_1}E_2=0, \ \ \nabla_{E_1}E_3=-E_1, \ \
\nabla_{E_2}E_1=e^zE_2,
\end{equation*
\begin{equation*}
\nabla_{E_2}E_2=-e^zE_1-E_3, \ \ \nabla_{E_2}E_3=-E_2, \ \
\nabla_{E_3}E_1=0, \ \ \nabla_{E_3}E_2=0, \ \ \nabla_{E_3}E_3=0
\end{equation*}
and the Riemann tensor field and the Ricci tensor field are given by:
\begin{equation*}
R(E_1,E_2)E_2=(1-e^{2z})E_1, \ \ R(E_1,E_3)E_3=-E_1, \ \
R(E_2,E_1)E_1=(1-e^{2z})E_2,
\end{equation*}
\begin{equation*}
R(E_2,E_3)E_3=-E_2, \ \ R(E_3,E_1)E_1=E_3, \ \ R(E_3,E_2)E_2=E_3,
\end{equation*}
\begin{equation*}
\func{Ric}(E_1,E_1)=\func{Ric}(E_2,E_2)=2-e^{2z}, \ \ \func{Ric}(E_3,E_3)=-2.
\end{equation*}
Then the data $(V,\lambda,\mu)$, for $\lambda=1-e^{2z}$ and $\mu=-1-e^{2z}$,
define an almost $\eta$-Ricci soliton on $(M,g)$. Moreover, it is a gradient
almost $\eta$-Ricci soliton, since the potential vector field $V$ is of
gradient type, $V=\func{grad}(f)$, where $f(x,y,z):=-z$.
\end{example}
\subsection{Almost $\protect\eta$-Yamabe solitons}
Assume that $V$ is a torse-forming vector field on an $n$-dimensional
Riemannian manifold $(M,g)$, $\nabla_XV=aX+\psi(X)V$, for any $X\in \chi(M)
, where $\nabla$ is the Levi-Civita connection of $g$. If $\eta$ is the $g
-dual $1$-form of $V$ and $U$ is the $g$-dual vector field of $\psi$, then
\begin{equation*}
\eta(U)=\psi(V)
\end{equation*}
and
\begin{equation*}
\div(V)=an+\eta(U)=an+\psi(V).
\end{equation*}
Also
\begin{equation*}
\div(\psi\otimes \eta+\eta\otimes
\psi)=\div(V)\psi+\div(U)\eta+i_{\nabla_UV}g+i_{\nabla_VU}g.
\end{equation*}
Since
\begin{equation*}
\nabla_UV=aU+|U|^2V,
\end{equation*}
we obtain:
\begin{equation*}
\div(\psi\otimes \eta+\eta\otimes
\psi)=[(n+1)a+\psi(V)]\psi+[|U|^2+\div(U)]\eta+i_{\nabla_VU}g.
\end{equation*}
\bigskip
Let $(V,\lambda ,\mu)$ be an almost $\eta$-Yamabe soliton with $\eta$ the $g
-dual $1$-form of the torse-forming vector field $V$. Then
\begin{equation*}
\frac{1}{2}(\psi\otimes \eta+\eta\otimes \psi)=(\func{scal}-\lambda-a) g+\mu
\eta \otimes \eta
\end{equation*}
and taking the divergence, we get
\begin{equation} \label{e3}
d(\func{scal}-\lambda-a)=\frac{1}{2}i_{\nabla_VU}g
\end{equation}
\begin{equation*}
+\frac{1}{2}[(n+1)a+\psi(V)]\psi+\left[\frac{|U|^2}{2}+\frac{\div(U)}{2
-(\mu+n)a-(\mu+1)\psi(V)-V(\mu)\right]\eta.
\end{equation*}
\begin{proposition}
Let $(M,g)$ be an $n$-dimensional Riemannian manifold and let $(V,\lambda
,\mu)$ be an almost $\eta$-Yamabe soliton with $\eta$ the $g$-dual $1$-form
of the torse-forming vector field $V$. Then $(V,\lambda ,\mu)$ is an almost
Yamabe soliton with $\lambda=\func{scal}-\frac{1}{2|V|^2}V(|V|^2)$ or
[\psi(V)]^2=|V|^2\cdot|U|^2$.
\end{proposition}
\begin{proof}
From the soliton equation (\ref{111}), taking $X=Y=V$, we get
\begin{equation}\label{e1}
\scal-\lambda-a-\psi(V)+\mu|V|^2=0
\end{equation}
and taking $X=Y=U$, we obtain
\begin{equation}\label{e2}
[\scal-\lambda-a-\psi(V)]|U|^2+\mu [\psi(V)]^2=0.
\end{equation}
Replacing (\ref{e1}) in (\ref{e2}), we get:
$$-\mu|V|^2|U|^2+\mu [\psi(V)]^2=0,$$
which implies $\mu=0$ (which yields an almost Yamabe soliton) or $[\psi(V)]^2=|V|^2\cdot|U|^2$.
If $\mu=0$, from (\ref{e1}), we get
$$\scal-\lambda-a-\psi(V)=0$$ and since
$$\frac{1}{2}V(|V|^2)=g(\nabla_VV,V)=[a+\psi(V)]|V|^2,$$
we obtain $\lambda=\scal-\frac{1}{2|V|^2}V(|V|^2)$.
\end{proof}
\begin{proposition}
Let $(M,g)$ be an $n$-dimensional Riemannian manifold and let $(V,\lambda
,\mu)$ be an almost $\eta$-Yamabe soliton with $\eta$ the $g$-dual $1$-form
of the concircular vector field $V$. Then $V$ is $\nabla$-parallel or the
soliton is given by
\begin{equation*}
(\lambda,\mu)=(\func{scal}-a+n|V|^2, n).
\end{equation*}
\end{proposition}
\begin{proof}
Taking $\psi=0$ and $U=0$ in (\ref{e3}), we get
$$d(\scal-\lambda-a)=[-(\mu+n)a-V(\mu)]\eta$$
and by differentiating (\ref{e1}), we obtain:
$$d(\scal-\lambda-a)=-\mu d(|V|^2)-|V|^2d\mu.$$
Replacing the second relation in the previous one and computing it in $V,$ we get
$$\mu V(|V|^2)=(\mu+n)a|V|^2.$$
Also
$$V(|V|^2)=2g(\nabla_VV,V)=2a|V|^2,$$
which combined with the previous relation implies either $a=0$ (i.e. $V$ is $\nabla$-parallel) or $\mu=n$, which from (\ref{e1}) gives $\lambda=\scal-a+n|V|^2$.
\end{proof}
\begin{proposition}
Let $(M,g)$ be an $n$-dimensional mixed generalized quasi-Einstein manifold
and let $(V,\lambda ,\mu )$ be an almost $\eta $-Yamabe soliton with $\eta $
the $g$-dual $1$-form of the torqued vector field $V$. If the Ricci tensor
field of $M$ is of the form $\func{Ric}=\alpha g+\beta
\eta\otimes\eta+\gamma ( \eta\otimes \psi+\psi\otimes\eta)$, then
\begin{equation}
\lambda =\left(\beta+\frac{\mu}{n}\right)|V|^2+\alpha n-a. \label{e4}
\end{equation}
\end{proposition}
\begin{proof}
Since $V$ is a torqued vector field and $\eta $ the $g$-dual $1
-form of $V,$ we have
$$\left( \pounds
_{V}g\right) (X,Y)=2ag(X,Y)+\eta (X)\psi(Y)+\psi(X)\eta (Y).$$ On the other
hand
\begin{equation*}
\func{Ric}(X,Y)=\alpha g(X,Y)+\beta \eta(X)\eta(Y)+\gamma [ \eta(X)\psi(Y)+\psi(X)\eta(Y)],
\end{equation*
gives us
\begin{equation*}
\scal=\alpha n+\beta |V|^2+2\gamma \psi(V)=\alpha n+\beta |V|^2.
\end{equation*
Then (\ref{111}) turns into
\begin{equation*}
ag(X,Y)+\frac{1}{2}\left[ \eta (X)\psi(Y)+\psi(X)\eta (Y)\right]
=[ \alpha n+\beta |V|^2-\lambda ] g(X,Y)+\mu \eta (X)\eta (Y).
\end{equation*
So by a contraction in the last equation, we obtain (\ref{e4}).
\end{proof}
\begin{example}
\label{exh} Let $M=\{(x,y,z)\in\mathbb{R}^3, z>0\}$, where $(x,y,z)$ are the
standard coordinates in $\mathbb{R}^3$. Set
\begin{equation*}
V:=-z\frac{\partial}{\partial z}, \ \ \eta:=-\frac{1}{z}dz,
\end{equation*}
\begin{equation*}
g:=\frac{1}{z^2}(dx\otimes dx+dy\otimes dy+dz\otimes dz)
\end{equation*}
and consider the linearly independent system of vector fields:
\begin{equation*}
E_1:=z\frac{\partial}{\partial x}, \ \ E_2:=z\frac{\partial}{\partial y}, \
\ E_3:=-z\frac{\partial}{\partial z}.
\end{equation*}
According to Example \ref{ex1}, the Riemann and the Ricci curvature tensor
fields are given by:
\begin{equation*}
R(E_1,E_2)E_2=-E_1, \ \ R(E_1,E_3)E_3=-E_1, \ \ R(E_2,E_1)E_1=-E_2,
\end{equation*}
\begin{equation*}
R(E_2,E_3)E_3=-E_2, \ \ R(E_3,E_1)E_1=-E_3, \ \ R(E_3,E_2)E_2=-E_3,
\end{equation*}
\begin{equation*}
\func{Ric}(E_1,E_1)=\func{Ric}(E_2,E_2)=\func{Ric}(E_3,E_3)=-2
\end{equation*}
and the scalar curvature is $\func{scal}=-6$. \newline
Writing the $\eta$-Yamabe soliton equation in $(E_i,E_i)$ we obtain:
\begin{equation*}
g(\nabla_{E_i}E_3,E_i)=(-6-\lambda)g(E_i,E_i)+\mu\eta(E_i)\eta(E_i),
\end{equation*}
for all $i\in\{1,2,3\}$ and we get that for $\lambda=-7$ and $\mu=-1$, the
data $(V,\lambda,\mu)$ define an $\eta$-Yamabe soliton on $(M, g)$.
Moreover, it is a gradient $\eta$-Yamabe soliton, since the potential vector
field $V$ is a torse-forming vector field of gradient type, $V=\func{grad
(f) $, where $f(x,y,z):=-\ln z$.
\end{example}
\begin{example}
Let $M=\mathbb{R}^3$, $(x,y,z)$ be the standard coordinates in $\mathbb{R}^3
, let $g$ be the Lorentzian metric:
\begin{equation*}
g:=e^{-2z}dx\otimes dx+e^{2x-2z}dy\otimes dy-dz\otimes dz.
\end{equation*}
Consider the vector field $V$ and the $1$-form $\eta$:
\begin{equation*}
V:=\frac{\partial}{\partial z}, \ \ \eta:=dz.
\end{equation*}
For the orthonormal vector fields:
\begin{equation*}
E_1=e^z\frac{\partial}{\partial x}, \ \ E_2=e^{z-x}\frac{\partial}{\partial
}, \ \ E_3=\frac{\partial}{\partial z},
\end{equation*}
according to Example \ref{ex2}, the Riemann tensor field, the Ricci tensor
field and the scalar curvature are given by:
\begin{equation*}
R(E_1,E_2)E_2=(1-e^{2z})E_1, \ \ R(E_1,E_3)E_3=-E_1, \ \
R(E_2,E_1)E_1=(1-e^{2z})E_2,
\end{equation*}
\begin{equation*}
R(E_2,E_3)E_3=-E_2, \ \ R(E_3,E_1)E_1=E_3, \ \ R(E_3,E_2)E_2=E_3,
\end{equation*}
\begin{equation*}
\func{Ric}(E_1,E_1)=\func{Ric}(E_2,E_2)=2-e^{2z}, \ \ \func{Ric}(E_3,E_3)=-2,
\end{equation*}
\begin{equation*}
\func{scal}=2(3-e^{2z}).
\end{equation*}
Then the data $(V,\lambda,\mu)$, for $\lambda=7-2e^{2z}$ and $\mu=-1$,
define an almost $\eta$-Yamabe soliton on $(M,g)$. Moreover, it is a
gradient almost $\eta$-Yamabe soliton, since the potential vector field $V$
is of gradient type, $V=\func{grad}(f)$, where $f(x,y,z):=-z$.
\end{example}
\section{Almost $\protect\eta$-Ricci and almost $\protect\eta$-Yamabe
solitons on submanifolds}
Let $M$ be a submanifold isometrically immersed into a Riemannian manifold
\left( \widetilde{M},\widetilde{g}\right) $. Denote by $g$ the Riemannian
metric induced on $M$, by $\nabla $ and $\widetilde{\nabla }$ the
Levi-Civita connections on $(M,g)$ and $\left( \widetilde{M},\widetilde{g
\right) $. The Gauss and Weingarten formulas corresponding to $M$ are given
by:
\begin{equation}
\widetilde{\nabla }_{X}Y=\nabla _{X}Y+h(X,Y), \label{16}
\end{equation
\begin{equation}
\widetilde{\nabla }_{X}N=-B_{N}(X)+\nabla _{X}^{\bot }N, \label{e5}
\end{equation
where $h$ is the second fundamental form and $B_{N}$ is the shape operator
in the direction of the normal vector field $N$ defined by $\widetilde{g
(B_{N}(X),Y)=\widetilde{g}(h(X,Y),N)$ for $X$, $Y\in \chi (M)$ \cite{Chen-73
.
\medskip
A Riemannian submanifold $M$ is called \textit{minimal} if its mean
curvature vanishes.
\medskip
A submanifold $M$ in a Riemannian manifold $\left( \widetilde{M},\widetilde{
}\right) $ is called
i) \textit{$\xi $-umbilical }\cite{Chen-16} (with respect to a normal vector
field $\xi $) if its shape operator satisfies $B_{\xi }=\varphi I,$ where
\varphi $ is a function on $M $ and $I$ is the identity map;
ii) \textit{totally umbilical }\cite{Chen-73} if it is umbilical with
respect to every unit normal vector field.
\medskip
An $n$-dimensional hypersurface $M$, $n\geq 4,$ in a Riemannian manifold
\left( \widetilde{M},\widetilde{g}\right) $ is called
i) $2$\textit{-quasi-umbilical }\cite{DeVers-79} if its second fundamental
tensor field $H$ satisfies
\begin{equation*}
H=\alpha g+\beta \varpi \otimes \varpi +\gamma \eta \otimes \eta ,
\end{equation*
where $\varpi $ and $\eta $ are $1$-forms and $\alpha $, $\beta $ and
\gamma $ are smooth functions on $M$ such that the corresponding vector
fields to the $1$-forms $\varpi $ and $\eta $ are $g$-orthogonal. In
particular, if $\gamma =0$, then $M$ is called \textit{quasi-umbilical} \cit
{Chen-73};
ii) \textit{pseudo quasi-umbilical} \cite{DeMa-2018}, if its second
fundamental tensor field $H$ satisfie
\begin{equation*}
H=\alpha g+\beta \varpi \otimes \varpi +E,
\end{equation*
where $\varpi $ is a $1$-form, $\alpha $ and $\beta $ are smooth functions
on $M$ and $E$ is a symmetric $(0,2)$-tensor field with vanishing trace. If
E$ vanishes, then $M$ is quasi-umbilical.
\medskip If $V$ is a concurrent vector field on $\widetilde{M}$, then for
any $X\in \chi (M)$, we have
\begin{equation*}
\nabla _{X}V^{T}=X+B_{V^{\bot }}(X),\ \ \nabla _{X}^{\bot }V^{\bot
}=-h(X,V^{T}),
\end{equation*
therefore
\begin{equation*}
\frac{1}{2}\func{grad}(|V^{\bot }|^{2})=-B_{V^{\bot }}(V^{T}),\ \ \frac{1}{2
\func{grad}(|V|^{2})=V^{T}
\end{equation*
and we can state:
\begin{proposition}
Every almost $\eta$-Ricci and every almost $\eta$-Yamabe soliton
(V^T,\lambda,\mu)$ on a submanifold $M$, which is isometrically immersed
into a Riemannian manifold $\left( \widetilde{M},\widetilde{g}\right) $, is
of gradient type with the potential function $\frac{1}{2}\func{grad}(|V|^2)
, where $V^T\in \chi(M)$ is the tangential component of the concurrent
vector field $V\in \chi(\widetilde{M})$ and $\eta$ is the $g$-dual of $V^T$.
\end{proposition}
In particular, if $V$ is of constant length, then the almost $\eta$-Ricci
soliton $(M,g)$ is a quasi-Einstein manifold with associated functions
\lambda$ and $\mu$.
\bigskip
Next, we shall characterize almost $\eta$-Ricci and almost $\eta$-Yamabe
solitons on $M$ whose potential vector field is the tangential component of
a torse-forming vector field on $\widetilde{M}$.
\begin{theorem}
\label{thm-sub}Let $M$ be a submanifold isometrically immersed into a
Riemannian manifold $\left( \widetilde{M},\widetilde{g}\right) $, let $V$ be
a torse-forming vector field on $\widetilde{M}$ and let $\eta $ be the $g
-dual of $V^T$. Then
i) $(M,g)$ is an almost $\eta $-Ricci soliton $(V^{T},\lambda ,\mu )$ if and
only if the Ricci tensor field of $M$ satisfies:
\begin{equation*}
\func{Ric}_{M}(X,Y)=(\lambda -a)g(X,Y)-\widetilde{g}(h(X,Y),V^{\perp })+\mu
\eta (X)\eta (Y)-\frac{1}{2}[\psi (X)\eta (Y)+\eta (X)\psi (Y)],
\end{equation*
for any $X,Y\in \chi (M)$;
ii) $(M,g)$ is an almost $\eta $-Yamabe soliton $(V^{T},\lambda ,\mu )$ if
and only if:
\begin{equation*}
(\func{scal}-\lambda-a)g(X,Y)-\widetilde{g}(h(X,Y),V^{\perp })+\mu \eta
(X)\eta (Y)-\frac{1}{2}[\psi (X)\eta (Y)+\eta (X)\psi (Y)]=0,
\end{equation*
for any $X,Y\in \chi (M)$.
\end{theorem}
\begin{proof}
For any $X\in \chi(M)$, we have:
$$aX+\psi(X)V^T+\psi(X)V^\bot=\widetilde{\nabla}_XV=\nabla_XV^T+h(X,V^T)-B_{V^{\bot}}(X)+\nabla^{\perp}_XV^{\bot}$$
and by the equality of the tangent components, we get:
$$\nabla_XV^T=aX+\psi(X)V^T+B_{V^{\bot}}(X).$$
Then
$$(\pounds _{V^T}g)(X,Y)=g(\nabla_XV^T,Y)+g(\nabla_YV^T,X)=2[ag(X,Y)+\widetilde{g}(h(X,Y),V^\bot)]$$$$+\psi(X)\eta(Y)+\eta(X)\psi(Y).$$
i) Suppose that there exist smooth functions $\lambda$ and $\mu$ on $M$ such that the condition in the hypotheses holds. Then we obtain
$$\frac{1}{2}(\pounds _{V^T}g)(X,Y)+\Ric_M(X,Y)=\lambda g(X,Y)+\mu \eta(X)\eta(Y).$$
Hence the submanifold $(M,g)$ is an almost $\eta$-Ricci soliton. The converse is trivial.
\pagebreak
ii) Suppose that there exist smooth functions $\lambda$ and $\mu$ on $M$ such that the condition in the hypotheses holds. Then we obtain
$$\frac{1}{2}(\pounds _{V^T}g)(X,Y)=(\scal-\lambda) g(X,Y)+\mu \eta(X)\eta(Y).$$
Hence the submanifold $(M,g)$ is an almost $\eta$-Yamabe soliton. The converse is trivial.
\end{proof}
If $M$ is a minimal submanifold, then we can state the following corollary:
\begin{corollary}
Let $M$ be an $n$-dimensional isometrically immersed minimal submanifold of
a Riemannian manifold $\left( \widetilde{M},\widetilde{g}\right) $, let $V$
be a concircular vector field on $\widetilde{M}$ and let $\eta $ be the $g
-dual of $V^{T}$.
i) If $(V^T,\lambda,\mu)$ is an almost $\eta$-Ricci soliton on $M$, then
\func{scal}_M=n(\lambda-a)+\mu |V^T|^2$.
ii) If $(V^T,\lambda,\mu)$ is an almost $\eta$-Yamabe soliton on $M$, then
\func{scal}_M=\lambda+a-\frac{\mu}{n}|V^T|^2$.
\end{corollary}
When $M$ is a $V^{\bot }$-umbilical submanifold, we have:
\begin{corollary}
Let $M$ be an $n$-dimensional $V^{\bot }$-umbilical submanifold
isometrically immersed into an $(n+d)$-dimensional Riemannian manifold
\left( \widetilde{M},\widetilde{g}\right) .$ If $V$ is a concircular vector
field on $\widetilde{M},$ then $M$ is an almost $\eta $-Ricci soliton with
potential vector field $V^{T}$, for $\eta $ the $g$-dual of $V^{T}$ if and
only if $M$ is a quasi-Einstein submanifold with associated functions
\left( \lambda -a-\varphi \right) $ and $\mu .$
\end{corollary}
For a hypersurface, since
\begin{equation*}
\widetilde{g}\left( h(X,Y),V^{\perp }\right) =g(B(X),Y)g(N,V^{\perp
})=H(X,Y)g(N,V^{\perp }),
\end{equation*}
where $N$ is the unit normal vector field of $M$ and $H$ is the second
fundamental tensor field, if we denote by $\rho =g(N,V^{\perp }),$ then we
can state:
\begin{corollary}
\label{corA}Let $M$ be an $n$-dimensional hypersurface isometrically
immersed into an $(n+1)$-dimensional Riemannian manifold $\left( \widetilde{
},\widetilde{g}\right).$ If $V$ is a torse-forming vector field on
\widetilde{M}$ and $\eta$ is the $g$-dual of $V^{T},$ then
i) $(M,g)$ is an almost $\eta $-Ricci soliton with potential vector field
V^{T}$ if and only if there exist two smooth functions $\lambda $ and $\mu $
on $M$ such that the Ricci tensor field of $M$ satisfies
\begin{equation}
\func{Ric}_M=\left( \lambda -a\right) g-\rho H+\mu \eta \otimes \eta-\frac{
}{2}(\psi\otimes \eta+\eta\otimes \psi); \label{B2}
\end{equation}
ii) $(M,g)$ is an almost $\eta $-Yamabe soliton with potential vector field
V^{T}$ if and only if there exist two smooth functions $\lambda $ and $\mu $
on $M$ such that
\begin{equation}
(\func{scal}-\lambda -a)g-\rho H+\mu \eta \otimes \eta-\frac{1}{2
(\psi\otimes \eta+\eta\otimes \psi)=0. \label{B23}
\end{equation}
\end{corollary}
\begin{theorem}
Let $M$ be an $n$-dimensional hypersurface isometrically immersed into an
(n+1)$-dimensional Riemannian manifold $\left( \widetilde{M}(c),\widetilde{g
\right) $ of constant curvature $c.$ If $V$ is a torse-forming vector field
on $\widetilde{M}$ and $\eta$ is the $g$-dual of $V^{T},$ then
i) $(M,g)$ is an almost $\eta $-Ricci soliton with potential vector field
V^{T}$ if and only if there exist two smooth functions $\lambda $ and $\mu $
on $M$ such that the second fundamental tensor field $H$ of $M$ satisfies
\begin{equation}
H^{2}=\left[ \rho +tr(H)\right] H+\left[ (n-1)c-\lambda +a) \right] g-\mu
\eta \otimes \eta +\frac{1}{2}(\psi \otimes \eta +\eta \otimes \psi);
\label{B3}
\end{equation}
ii) $(M,g)$ is an almost $\eta $-Yamabe soliton with potential vector field
V^{T}$ if and only if there exist two smooth functions $\lambda $ and $\mu $
on $M$ such that the second fundamental tensor field $H$ of $M$ satisfies
\begin{equation}
\rho H=[n(n-1)c+(tr(H))^2-tr(H^2)-\lambda-a]g+\mu\eta\otimes \eta-\frac{1}{2
(\psi \otimes\eta+\eta \otimes\psi). \label{B55}
\end{equation}
\end{theorem}
\begin{proof}
From the Gauss equation, we have
\begin{equation}
\Ric_M(X,Y)=tr(H)H(X,Y)-H^{2}(X,Y)+(n-1)cg(X,Y). \label{B4}
\end{equation
i) Then comparing (\ref{B4}) and (\ref{B2}), we get
\begin{equation*}
\left( \lambda -a\right) g(X,Y)-\rho H(X,Y)+\mu \eta (X)\eta
(Y)-\frac{1}{2}[\psi(X) \eta(Y)+\eta(X)\psi(Y)]$$$$=tr(H)H(X,Y)-H^{2}(X,Y)+(n-1)cg(X,Y),
\end{equation*
which gives us
\begin{equation*}
H^{2}(X,Y)=\left[ \rho +tr(H)\right] H(X,Y)+\left[ (n-1)c-\lambda
+a\right] g(X,Y)-\mu \eta (X)\eta (Y)$$$$+\frac{1}{2}[\psi(X) \eta(Y)+\eta(X)\psi(Y)].
\end{equation*
Conversely, assume that (\ref{B3}) is satisfied. Then by the Gauss equation, we
have
\begin{equation}
\Ric_M(X,Y)=\left( \lambda -a\right) g(X,Y)-\rho H(X,Y)+\mu \eta (X)\eta (Y)-\frac{1}{2}[\psi(X) \eta(Y)+\eta(X)\psi(Y)],
\label{B5}
\end{equation
so by Corollary \ref{corA}, $(M,g)$ is an almost $\eta $-Ricci soliton with
potential vector field $V^{T}.$
ii) By a contraction in (\ref{B4}), we find
$$\scal=(tr(H))^2-tr(H^2)+n(n-1)c$$
and replacing $\scal$ in (\ref{B23}), we get
$$[n(n-1)c+(tr(H))^2-tr(H^2)-\lambda-a]g-\rho H+\mu\eta\otimes \eta-\frac{1}{2
(\psi \otimes\eta+\eta \otimes\psi)=0.$$
Conversely, assume that (\ref{B55}) is satisfied. Then by a contraction, we
find
\begin{equation}
\scal=(tr(H))^2-tr(H^2)+n(n-1)c,
\end{equation
so by Corollary \ref{corA}, $(M,g)$ is an almost $\eta $-Yamabe soliton with
potential vector field $V^{T}.$
\end{proof}
\begin{proposition}
Let $M$ be an $n$-dimensional quasi-Einstein hypersurface isometrically
immersed into an $(n+1)$-dimensional Riemannian manifold $\left( \widetilde{
},\widetilde{g}\right) .$ Assume that the Ricci tensor field of $M$ is of
the form $\func{Ric}=\alpha g+\beta \eta \otimes \eta .$ If $V$ is a
concircular vector field on $\widetilde{M},$ then $(M,g)$ is an almost $\eta
$-Ricci soliton with potential vector field $V^{T}$, for $\eta $ the $g
-dual of $V^{T},$ if and only if it is a quasi-umbilical hypersurface with
associated functions $\frac{\lambda -a-\alpha }{\rho }$ and $\frac{\mu
-\beta }{\rho }.$
\end{proposition}
\begin{proof}
Assume that $M$ is a quasi-Einstein hypersurface whose Ricci tensor field $\Ric$ is
of the form $\Ric=\alpha g+\beta \eta \otimes \eta .$ If $V$ is a concircular
vector field on $\widetilde{M},$ then from Theorem \ref{thm-sub}, we can
write
\begin{equation*}
\alpha g(X,Y)+\beta \eta (X)\eta (Y)=\left( \lambda -a\right) g(X,Y)-\rho
H(X,Y)+\mu \eta (X)\eta (Y),
\end{equation*
which gives us
\begin{equation*}
H(X,Y)=\frac{\lambda -a-\alpha }{\rho }g(X,Y)+\frac{\mu -\beta }{\rho }\eta
(X)\eta (Y).
\end{equation*
Hence $M$ is a quasi-umbilical hypersurface with associated functions $\frac
\lambda -a-\alpha }{\rho }$ and $\frac{\mu -\beta }{\rho }.$
The converse is trivial.
\end{proof}
It is known that an $n$-dimensional hypersurface $M$, $n\geq 4,$ in a
Riemannian manifold $\left( \widetilde{M}(c),\widetilde{g}\right) $ of
constant curvature $c$ is conformally flat if and only if it is
quasi-umbilical \cite{Shou-21}. So we have:
\begin{corollary}
Let $M$ be an $n$-dimensional quasi-Einstein hypersurface isometrically
immersed into an $(n+1)$-dimensional Riemannian manifold $\left( \widetilde{
},\widetilde{g}\right) .$ Assume that the Ricci tensor field of $M$ is of
the form $\func{Ric}=\alpha g+\beta \eta \otimes \eta .$ If $V$ is a
concircular vector field on $\widetilde{M},$ then $(M,g)$ is an almost $\eta
$-Ricci soliton with potential vector field $V^{T}$, for $\eta $ the $g
-dual of $V^{T}$ and $M$ is a conformally flat hypersurface.
\end{corollary}
\bigskip
Let $\varphi :M\rightarrow \mathbb{S}^{n+1}(1)$ be an immersion. We denote
by $g$ the induced metric on the hypersurface $M$ as well as that on the
unit sphere $\mathbb{S}^{n+1}(1).$ Let $N$ and $B$ be the unit normal vector
field and the shape operator of the hypersurface $M$ in the unit sphere
\mathbb{S}^{n+1}(1)$ and we denote by $\left\langle ,\right\rangle $ the
Euclidean metric on the Euclidean space $\mathbb{E}^{n+2}.$ Assume that $V$
is a torse-forming vector field on $\mathbb{E}^{n+2}$. If we denote by $N_
\mathbb{S}}$ the unit normal vector field of the unit sphere $\mathbb{S
^{n+1}(1)$ in the Euclidean space $\mathbb{E}^{n+2}$, we can define the
smooth functions $\delta ,\varrho $ on the hypersurface $M$ by $\delta =$
\left\langle V,N\right\rangle \mid _{M}$ and $\varrho $ $=\left\langle V,N_
\mathbb{S}}\right\rangle \mid _{M}.$ Hence the restriction of the
torse-forming vector field $V$ to the hypersurface $M$ can be written as
V\mid _{M}=U+\delta N+$ $\varrho N_{\mathbb{S}}$, where $U\in \chi (M).$
Then as an extension of Theorem 3.3 given in \cite{h}, we can state:
\begin{theorem}
\label{ThmB}Let $M$ be an orientable hypersurface of the unit sphere
\mathbb{S}^{n+1}(1),$ with immersion $\varphi :M\rightarrow \mathbb{S
^{n+1}(1)$ and let $V$ be a torse-forming vector field on the Euclidean
space $\mathbb{E}^{n+2}.$ Denote by $\xi $ the tangential component of $V$
to the unit sphere $\mathbb{S}^{n+1}(1)$ and by $U$ the tangential component
of $\xi $ on $M.$ Then
i) the hypersurface $(M,g)$ is an almost $\eta $-Ricci soliton $(U,\lambda
,\mu )$ if and only if
\begin{equation*}
H^{2}(X,Y)=\left[ tr(H)+\delta \right] H(X,Y)+(n-1-\varrho -\lambda +a)g(X,Y)
\end{equation*
\begin{equation*}
-\mu \eta (X)\eta (Y)+\frac{1}{2}\left[ \psi (X)g(U,Y)+\psi (Y)g(U,X)\right]
,
\end{equation*
for any $X,Y\in \chi (M)$;
ii) the hypersurface $(M,g)$ is an almost $\eta $-Yamabe soliton $(U,\lambda
,\mu )$ if and only if \
\begin{equation*}
\delta H(X,Y)=[n(n-1)+(tr(H))^{2}-tr(H^{2})+\varrho -\lambda -a]g(X,Y)
\end{equation*
\begin{equation*}
+\mu \eta (X)\eta (Y)-\frac{1}{2}\left[ \psi (X)g(U,Y)+\psi (Y)g(U,X)\right]
,
\end{equation*
for any $X,Y\in \chi (M)$. In this case, $M$ is a pseudo quasi-umbilical
hypersurface.
\end{theorem}
\begin{proof}
Let $\nabla ,\overline{\nabla }$ and $D$ denote the Levi-Civita connections
on $M$, $\mathbb{S}^{n+1}(1)$ and $\mathbb{E}^{n+2},$ respectively. Then we
can write
\begin{equation*}
V\mid _{\mathbb{S}^{n+1}(1)}=\xi +\varrho N_{\mathbb{S}},
\end{equation*
and for any $X\in \chi (M)$, by taking the covariant differential w.r.t. $X
, we have
\begin{equation*}
aX+\psi (X)V=aX+\psi (X)\xi +\psi (X)\varrho N_{\mathbb{S}}=D_{X}V
=D_{X}\xi +X(\varrho )N_{\mathbb{S}}+\varrho D_{X}N_{\mathbb{S}}.
\end{equation*
By using the Gauss and Weingarten formulas, we find
\begin{equation*}
aX+\psi (X)\xi +\psi (X)\varrho N_{\mathbb{S}}=\overline{\nabla }_{X}\xi
-g(X,\xi )N_{\mathbb{S}}+X(\varrho )N_{\mathbb{S}}+\varrho X.
\end{equation*
By the equality of the tangent and the normal components, we ge
\begin{equation}
\overline{\nabla }_{X}\xi +(\varrho -a)X=\psi (X)\xi \label{recur1}
\end{equation
and
\begin{equation*}
X(\varrho )-g(X,\xi )=\psi (X)\varrho .
\end{equation*
The vector field $\xi $ on $\mathbb{S}^{n+1}(1)$ can be written as
\begin{equation*}
\xi =U+\delta N.
\end{equation*
So from (\ref{recur1}), we have
\begin{equation*}
\overline{\nabla }_{X}\left( U+\delta N\right) +(\varrho -a)X=\psi (X)\left(
U+\delta N\right) .
\end{equation*
By using Gauss and Weingarten formulas again, we fin
\begin{equation*}
\psi (X)U+\psi (X)\delta N-(\varrho -a)X=\nabla _{X}U+g(B(X),U)N+X(\delta
)N-\delta B(X).
\end{equation*
Then by the equality of the tangent and the normal components, we hav
\begin{equation}
\nabla _{X}U=\psi (X)U-(\varrho -a)X+\delta B(X) \label{nablaU}
\end{equation
and
\begin{equation*}
\psi (X)\delta =g(B(X),U)+X(\delta ).
\end{equation*
So
\begin{equation*}
\left( \pounds _{U}g\right) (X,Y)=g(\nabla _{X}U,Y)+g(\nabla _{Y}U,X)
\end{equation*
\begin{equation}
=\psi (X)g(U,Y)+\psi (Y)g(U,X)-2(\varrho -a)g(X,Y)+2\delta H(X,Y).
\label{LieU}
\end{equation
On the other hand, the Gauss equation for a hypersurface $M$ in $\mathbb{S
^{n+1}(1)$ gives us
\begin{equation}
\func{Ric}(X,Y)=(n-1)g(X,Y)+tr(H)H(X,Y)-H^{2}(X,Y). \label{Ric}
\end{equation
i) Then combining (\ref{Ric}) and (\ref{LieU}), we find
\begin{equation*}
\frac{1}{2}\left( \pounds _{U}g\right) \left( X,Y\right) +\func{Ric
(X,Y)=\left( n-1-\varrho +a\right) g(X,Y)+\left[ tr(H)+\delta \right] H(X,Y)
\end{equation*
\begin{equation*}
-H^{2}(X,Y)+\frac{1}{2}\left[ \psi (X)g(U,Y)+\psi (Y)g(U,X)\right] .
\end{equation*
\pagebreak
Suppose that there exist smooth functions $\lambda $ and $\mu $ on $M$ such
that the condition in the hypothesis holds. Then we obtain $\frac{1}{2
\left( \pounds _{U}g\right) \left( X,Y\right) +\func{Ric}(X,Y)=\lambda
g\left( X,Y\right) +\mu \eta (X)\eta (Y).$ Hence the hypersurface $M$ is an
almost $\eta $-Ricci soliton. The converse is trivial.
ii) By a contraction in (\ref{Ric}), we find
\begin{equation}
\scal=n(n-1)+\left( tr(H\right) )^{2}-tr( H^{2}) . \label{scal}
\end{equation
Then combining (\ref{scal}) and (\ref{LieU}), we get
\begin{equation*}
\frac{1}{2}\left( \pounds _{U}g\right) \left( X,Y\right) -(\func{scal
-\lambda )g\left( X,Y\right) =\frac{1}{2}\left[ \psi (X)g(U,Y)+\psi (Y)g(U,X
\right]
\end{equation*
\begin{equation*}
-\left[ n(n-1)+\left( tr(H\right) )^{2}-tr( H^{2}) +\varrho -\lambda
-a\right] g(X,Y)+\delta H(X,Y).
\end{equation*
Suppose that there exist smooth functions $\lambda $ and $\mu $ on $M$ such
that the condition in the hypothesis holds. Then we obtain $\frac{1}{2
\left( \pounds _{U}g\right) (X,Y)=(\func{scal}-\lambda )g(X,Y)+\mu \eta
(X)\eta (Y)$. Hence the hypersurface $M$ is an almost $\eta $-Yamabe soliton
and moreover, $M$ is a pseudo quasi-umbilical hypersurface. The converse is
trivial.
\end{proof}
\begin{corollary}
Under the conditions of Theorem \ref{ThmB}, if the $1$-form $\psi $ is the
g $-dual of $U$, then
i) the hypersurface $(M,g)$ is an almost $\eta $-Ricci soliton $(U,\lambda
,\mu )$ if and only if
\begin{equation*}
H^{2}=\left[ tr(H)+\delta \right] H+(n-1-\varrho -\lambda +a)g-\mu \eta
\otimes \eta +\psi \otimes \psi ;
\end{equation*}
ii) the hypersurface $(M,g)$ is an almost $\eta $-Yamabe soliton $(U,\lambda
,\mu )$ if and only if
\begin{equation*}
\delta H=[n(n-1)+\left( tr(H\right) )^{2}-tr(H^{2})+\varrho -\lambda
-a]g+\mu \eta \otimes \eta -\psi \otimes \psi .
\end{equation*}
\end{corollary}
\begin{corollary}
Let $M$ be an orientable hypersurface of the unit sphere $\mathbb{S
^{n+1}(1) $ and let $V$ be a $\nabla $-parallel or constant vector field on
the Euclidean space $\mathbb{E}^{n+2}.$ Then
i) the hypersurface $(M,g)$ is an almost $\eta $-Ricci soliton $(U,\lambda
,\mu )$ if and only if
\begin{equation*}
H^{2}=\left[ tr(H)+\delta \right] H+(n-1-\varrho -\lambda )g-\mu \eta
\otimes \eta ;
\end{equation*}
ii) the hypersurface $(M,g)$ is an almost $\eta $-Yamabe soliton $(U,\lambda
,\mu )$ if and only if
\begin{equation*}
\delta H=[n(n-1)+\left( tr(H\right) )^{2}-tr(H^{2})+\varrho -\lambda ]g+\mu
\eta \otimes \eta .
\end{equation*}
\end{corollary}
It is known that a pseudo quasi-umbilical hypersurface of a Riemannian
manifold of constant curvature $\left( \widetilde{M}(c),\widetilde{g}\right)
$ is a pseudo quasi-Einstein hypersurface \cite{DeMa-2018}. So we have:
\begin{corollary}
\label{ThmB copy(3)}Let $M$ be an orientable hypersurface of the unit sphere
$\mathbb{S}^{n+1}(1)$ and let $V$ be a torse-forming vector field on the
Euclidean space $\mathbb{E}^{n+2}.$ If the hypersurface $(M,g)$ is an almost
$\eta $-Yamabe soliton $(U,\lambda ,\mu )$, then it is a pseudo
quasi-Einstein hypersurface.
\end{corollary}
|
1,116,691,499,266 | arxiv | \section{Introduction}\label{section:Intro}
Quantifier prefix classes have for long been a dominating paradigm for the classification of first-order sentences into decidable and undecidable fragments.
The \emph{Bernays--Sch\"onfinkel--Ramsey fragment (BSR)}---$\exists^*\forall^*$ sentences---and the \emph{Ackermann fragment}---$\exists^*\forall\exists^*$ sentences---are two prefix classes that are well-known for their decidable satisfiability problem.
We show in this paper how both of them can be generalized to substantially larger, decidable fragments in which quantifier prefixes are not restricted anymore.
Instead, we formulate certain conditions on how existentially and universally quantified variables may be interlinked by joint occurrences in atoms.
This means that the classes of relational first-order sentences characterized by the quantifier prefixes $\exists^*\forall^*$ and $\exists^*\forall\exists^*$ are merely the tips of two icebergs that we shall call the \emph{generalized BSR fragment (GBSR)} and the \emph{generalized Ackermann fragment (GAF)}, respectively.
To the best knowledge of the author, both fragments are novel.
Interestingly, the \emph{relational monadic fragment}---relational first-order sentences in which all predicate symbols have arity one---is contained in both GBSR and GAF.
Recently, the \emph{separated fragment (SF)} has been introduced \cite{Voigt2016}.
It can be considered as an intermediate step between BSR and GBSR.
Its defining principle is that universally and existentially quantified variables may not occur together in atoms.
In~\cite{Voigt2017} it is shown that the satisfiability problem for SF is non-elementary.
Yet, SF as well as GBSR enjoy the finite model property.
However, the size of smallest models of a satisfiable GBSR sentences grows at least $k$-fold exponentially with the length of the sentences for arbitrarily large $k \geq 0$.
This is in contrast with BSR, where the size of smallest models grows at most linearly with the length of satisfiable sentences.
Since every GBSR sentence is equivalent to some BSR sentence,
one could say that GBSR is as expressive as BSR but admits a non-elementarily more compact representation of models.
It remains an open question whether a similar result holds for GAF relative to the Ackermann fragment.
The following example gives a taste of the kind of sentences treated in this paper and outlines a key method that we shall employ later.
The sentence $\varphi_1$ is a sample taken from GAF and $\varphi_2$ belongs to GBSR and to GAF.
\begin{example}\label{example:GAFandGBSR}
Consider the first-order sentence $\varphi_1 := \exists u \forall x \exists v \forall z \exists y_1 y_2 . \bigl(\neg P(u, x) \vee \bigl(Q(x,v) \wedge R(u,z,$ $y_1)\bigr)\bigr) \;\wedge\; \bigl(P(u, x) \vee \bigl(\neg Q(x,v) \wedge \neg R(u,z,y_2)\bigr)\bigr)$.
Due to the Boolean structure of $\varphi_1$, the quantifiers $\exists y_2$, $\exists y_1$, and $\forall z$ can be moved inwards immediately but $\exists v$ cannot.
Because of the two universal quantifiers $\forall x$ and $\forall z$, which are even interspersed with an existential one, $\varphi_1$ does not lie in the Ackermann fragment or in BSR.
%
Skolemization of $\varphi_1$ leads to
$\forall x z.
\bigl(\neg P(c, x) \vee \bigl(Q\bigl(x,f(x)\bigr) \wedge R\bigl(c,z,g(x,z)\bigr)\bigr)\bigr)
\wedge \bigl(P(c, x) \vee \bigl(\neg Q\bigl(x,f(x)\bigr) \wedge \neg R\bigl(c,z,h(x,z)\bigr)\bigr)\bigr)$
and thus explicitly fixes the dependency of $y_1$ on the universally quantified variables $x$ and $z$, as $y_1$ is replaced with the term $g(x,z)$.
However, the shape of the original $\varphi_1$ did not immediately indicate such a dependency of $y_1$ on $x$, since $x$ and $y_1$ do not occur together in any atom.
Moreover, there are no other variables that depend on $x$ and establish a connection between $x$ and $y_1$ by means of joint occurrences in atoms.
One may say that it is the Boolean structure of $\varphi_1$ alone which causes a dependency of $y_1$ on $x$, and that such a form of dependency has only a \emph{finite character}.
(This finite character will be made explicit by the notions of \emph{fingerprints} and \emph{uniform} strategies that we introduce in Section~\ref{section:ModeltheoreticApproachToGBSRsat}.)
The described point of view is supported by the existence of an equivalent sentence $\varphi_1'$, in which the dependency of $y_1$ on $x$ has vanished. The price we have to pay, however, is an increase in the size of the formula.\\
\strut\hspace{2ex}
$\varphi'_1 :=
\exists u.
\bigl(\forall x.\bigl(\neg P(u, x) \vee \exists v. Q(x,v) \bigr)\bigr)
\wedge \bigl(\bigl(\forall x.\neg P(u, x)\bigr) \vee \forall z \exists y_1. R(u,z,y_1) \bigr)$ \\
\strut\hspace{12ex}
$\wedge \bigl(\forall x.\bigl(\exists v. \neg Q(x,v)\bigr) \vee P(u,x) \bigr)
\wedge \bigl(\bigl(\forall x.\exists v. \neg Q(x,v)\bigr) \vee \forall z \exists y_1. R(u,z,y_1) \bigr)$ \\
\strut\hspace{12ex}
$\wedge \bigl(\bigl(\forall z \exists y_2. \neg R(u,z,y_2)\bigr) \vee \forall x.P(u,x) \bigr)
\wedge \bigl(\bigl(\forall z \exists y_2. \neg R(u,z,y_2)\bigr) \vee \forall x \exists v. Q(x,v) \bigr)$ \\
\strut\hspace{12ex}
$\wedge \bigl(\bigl(\forall z \exists y_2. \neg R(u,z,y_2)\bigr) \vee \forall z \exists y_1. R(u,z,y_1) \bigr)$\\
Transforming $\varphi_1$ into $\varphi'_1$ requires only basic logical laws (details can be found in the appendix):
first, we push the quantifiers $\exists y_2, \exists y_1, \forall z$ inwards as far as possible.
Then, we construct a disjunction of conjunctions of certain subformulas using distributivity.
This allows us to move the quantifier $\exists v$ inwards.
Afterwards, we apply the laws of distributivity again to obtain a conjunction of disjunctions of certain subformulas.
This step enables us to push the universal quantifier $\forall x$ inwards.
In the resulting sentence every occurrence of an existential quantifier lies in the scope of at most one universal quantifier.
Moreover, every atom in the original formula $\varphi_1$ contains at most one universally quantified variable.
Skolemization of $\varphi_1'$ leads to a sentence whose shape is quite close to the shape of a Skolemized sentence from the Ackermann fragment.
More precisely, every atom contains at most one variable, possibly with multiple occurrences.
The only difference is that we allow for more than only one universally quantified variable in the sentence as a whole, but at most one in every atom.
It is this particular form that one can exploit to construct an equisatisfiable monadic sentence.
As another example, consider the sentence $\varphi_2 := \exists u \forall x \exists y \forall z. \bigl(P(u,z) \wedge Q(u,x)\bigr) \vee \bigl(P(y,z) \wedge Q(u,y) \bigr)$. This sentence can be transformed in the same spirit, leading to the equivalent \\
\noindent
\strut\hspace{2ex}
$\varphi'_2 :=
\exists u \exists y \forall x z v.\,
\bigl(\bigl(P(u,x) \vee P(y,x)\bigr) \wedge P(u,x) \wedge Q(u,x) \bigr)$ \\
\strut\hspace{20ex}
$\vee \bigl(\bigl(P(u,z) \vee P(y,z)\bigr) \wedge Q(u,y) \wedge Q(u,z) \bigr)$ \\
\strut\hspace{20ex}
$\vee \bigl(\bigl( P(u,v) \vee P(y,v)\bigr) \wedge Q(u,y) \wedge P(y,v) \bigr)$.
While Skolemization of $\varphi_2$ introduces terms $f(x)$ to replace $y$, $\varphi'_2$ is a much nicer target for Skolemization, since all introduced symbols are constants. This second approach is so attractive, because it leads to a BSR sentence.
\end{example}
As a matter of fact, the sentence $\varphi_2$ belongs to GBSR and GAF at the same time, while it does not belong to the Ackermann fragment, SF, BSR, or the monadic fragment. Hence, even the intersection of GBSR and GAF contains sentences which do not fall into the categories offered by standard fragments.
The transformation technique outlined in Example~\ref{example:GAFandGBSR} is one tool with which we establish the decidability of GBSR and GAF.
An interesting model-theoretic approach to establishing a small model property for GBSR sentences is presented as well.
Moreover, we employ a proof-theoretic result to argue that satisfiability of GAF sentences with equality is decidable.
In short, the main contributions of the present paper are the following.
In Section~\ref{section:GeneralizedBSR} we define GBSR and outline an effective equivalence-preserving transformation from GBSR into BSR (Lemma~\ref{lemma:TransformationGBSR}), which entails decidability of GBSR-satisfiability (Theorem~\ref{theorem:DecidabilityGBSR}).
Using this translation, we moreover derive a Craig--Lyndon interpolation theorem for GBSR (Theorem~\ref{theorem:InterpolationGBSR}).
In Section~\ref{section:ModeltheoreticApproachToGBSRsat} we develop a model-theoretic point of view, which eventually leads to a small model property for GBSR (Theorem~\ref{theorem:GBSRsmallModelProperty}).
The computational hardness of GBSR-satisfiability is derived from the hardness of SF-satisfiability (cf.\ Theorems~\ref{theorem:ComputationalHardnessGBSR} and~\ref{theorem:ComputationalComplexityGBSR}).
In Section~\ref{section:GeneralizedAckermann} we introduce GAF.
Decidability of GAF-satisfiability is shown
(a) for GAF sentences with equality but without non-constant function symbols by employing a proof-theoretic result (Theorem~\ref{theorem:DecidabilityGAFwithEquality}) and
(b) for GAF sentences without equality but with arbitrarily nested unary function symbols via an effective, (un)satisfiability-preserving transformation from GAF into the monadic fragment with unary function symbols (Theorem~\ref{theorem:DecidabilityGAFwithUnaryFunctions}).
In order to facilitate smooth reading, long proofs are only sketched in the main text and presented in full in the appendix.
\section{Notation and preliminaries}\label{section:Preliminaries}
We consider first-order logic formulas with equality.
We call a first-order formula \emph{relational} if it contains neither function nor constant symbols.
We use $\varphi(x_1, \ldots, x_m)$ to denote a formula $\varphi$ whose free variables form a subset of $\{x_1, \ldots, x_m\}$.
In all formulas, if not explicitly stated otherwise, we tacitly assume that no variable occurs freely and bound at the same time and that no variable is bound by two different occurrences of quantifiers.
For convenience, we sometimes identify tuples $\vec{\mathbf{x}}$ of variables with the set containing all the variables that occur in $\vec{\mathbf{x}}$.
By $\text{\upshape{vars}}(\varphi)$ we denote the set of all variables occurring in $\varphi$.
Similar notation is used for other syntactic objects.
A sentence $\varphi := \forall \vec{\mathbf{x}}_1 \exists \vec{\mathbf{y}}_1 \ldots \forall \vec{\mathbf{x}}_n \exists \vec{\mathbf{y}}_n. \psi$ is said to be in \emph{standard form} if and only if $\psi$ is quantifier free, in negation normal form, and exclusively contains the connectives $\wedge, \vee, \neg$.
In addition, we assume that all variables bound in the quantifier prefix actually occur in $\psi$.
The tuples $\vec{\mathbf{x}}_1$ and $\vec{\mathbf{y}}_n$ may be empty, i.e.\ the quantifier prefix does not have to start with a universal quantifier, and it does not have to end with an existential quantifier.
As usual, we interpret a formula $\varphi$ with respect to given structures. A \emph{structure} $\mathcal{A}$ consists of a nonempty universe $\mathfrak{U}_\mathcal{A}$ and interpretations $f^\mathcal{A}$ and $P^\mathcal{A}$ of all considered function and predicate symbols, in the usual way.
Given a formula $\varphi$, a structure $\mathcal{A}$, and a variable assignment $\beta$, we write $\mathcal{A}, \beta \models \varphi$ if $\varphi$ evaluates to \emph{true} under $\mathcal{A}$ and $\beta$. We write $\mathcal{A} \models \varphi$ if $\mathcal{A}, \beta \models \varphi$ holds for every $\beta$.
The symbol $\models$ also denotes semantic entailment of formulas, i.e.\ $\varphi \models \psi$ holds whenever for every structure $\mathcal{A}$ and every variable assignment $\beta$, $\mathcal{A},\beta \models \varphi$ entails $\mathcal{A},\beta \models \psi$.
The symbol $\mathrel{|}\joinrel\Relbar\joinrel\mathrel{|}$ denotes \emph{semantic equivalence} of formulas, i.e.\ $\varphi \mathrel{|}\joinrel\Relbar\joinrel\mathrel{|} \psi$ holds whenever $\varphi \models \psi$ and $\psi \models \varphi$.
We call two sentences $\varphi$ and $\psi$ \emph{equisatisfiable} if $\varphi$ has a model if and only if $\psi$ has one.
A structure $\mathcal{A}$ is a \emph{substructure} of a structure $\mathcal{B}$ (over the same signature) if (1) $\mathfrak{U}_\mathcal{A} \subseteq \mathfrak{U}_\mathcal{B}$, (2) $c^\mathcal{A} = c^\mathcal{B}$ for every constant symbol $c$, (3) $P^\mathcal{A} = P^\mathcal{B} \cap \mathfrak{U}_\mathcal{A}^m$ for every $m$-ary predicate symbol $P$, and (4) $f^\mathcal{A}(\mathfrak{a}_1, \ldots, \mathfrak{a}_m) = f^\mathcal{B}(\mathfrak{a}_1, \ldots, \mathfrak{a}_m)$ for every $m$-ary function symbol $f$ and every $m$-tuple $\<\mathfrak{a}_1, \ldots, \mathfrak{a}_m\> \in \mathfrak{U}_\mathcal{A}^m$.
The following are standard lemmas, see, e.g., \cite{Ebbinghaus1994} for a proof.
\begin{lemma}[Substructure Lemma]
Let $\varphi$ be a first-order sentence in prenex normal form without existential quantifiers and let $\mathcal{A}$ be a substructure of $\mathcal{B}$.
%
$\mathcal{B} \models \varphi$ entails $\mathcal{A} \models \varphi$.
\end{lemma}
\begin{lemma}[Miniscoping]\label{lemma:Miniscoping}
Let $\varphi, \psi, \chi$ be arbitrary first-order formulas, and assume that $x$ and $y$ do not occur freely in $\chi$.
We have the following equivalences, where $\circ \in \{\wedge, \vee\}$:\\
\begin{tabular}{cl@{\hspace{2ex}}c@{\hspace{2ex}}l@{\hspace{8ex}}cl@{\hspace{2ex}}c@{\hspace{2ex}}l}
(i) & $\exists y. (\varphi \vee \psi)$ &$\mathrel{|}\joinrel\Relbar\joinrel\mathrel{|}$& $(\exists y. \varphi) \vee (\exists y. \psi)$
& (ii) & $\forall x. (\varphi \wedge \psi)$ &$\mathrel{|}\joinrel\Relbar\joinrel\mathrel{|}$& $(\forall x. \varphi) \wedge (\forall x. \psi)$ \\
(iii) & $\exists y. (\varphi \circ \chi)$ &$\mathrel{|}\joinrel\Relbar\joinrel\mathrel{|}$& $(\exists y. \varphi) \circ \chi$
& (iv) & $\forall x. (\varphi \circ \chi)$ &$\mathrel{|}\joinrel\Relbar\joinrel\mathrel{|}$& $(\forall x. \varphi) \circ \chi$
\end{tabular}
\end{lemma}
\section{The generalized Bernays--Sch\"onfinkel--Ramsey fragment}\label{section:GeneralizedBSR}
In this section we generalize the Bernays--Sch\"onfinkel--Ramsey fragment---$\exists^* \forall^*$ sentences with equality but without non-constant function symbols.
We show in two ways that the satisfiability problem for the new fragment is decidable.
The first approach (Section~\ref{section:SyntacticApproachToGBSRsat}) is of a syntactic nature, as it is based on an effective translation into BSR.
The second approach (Section~\ref{section:ModeltheoreticApproachToGBSRsat}) uses model-theoretic techniques to establish a small model property.
For the considerations in this section we fix a first-order sentence $\varphi := \forall \vec{\mathbf{x}}_1 \exists \vec{\mathbf{y}}_1 \ldots \forall \vec{\mathbf{x}}_n \exists \vec{\mathbf{y}}_n. \psi$ in standard form that may contain the distinguished equality predicate and constant symbols but no non-constant function symbols.
Let $\text{At}$ be the set of all atoms occurring in $\varphi$ and let $\vec{\mathbf{x}} := \vec{\mathbf{x}}_1 \cup \ldots \cup \vec{\mathbf{x}}_n$ and $\vec{\mathbf{y}} := \vec{\mathbf{y}}_1 \cup \ldots \cup \vec{\mathbf{y}}_n$.
\begin{definition}[GBSR fragment, axiomatically]\label{definition:GBSRaxiomatic}
The sentence $\varphi$ belongs to the \emph{generalized Bernays--Sch\"onfinkel--Ramsey fragment (GBSR)} if and only if
we can partition $\text{At}$ into sets $\text{At}_0, \text{At}_1, \ldots, \text{At}_n$ such that
\begin{enumerate}[label=(\roman{*}), ref=(\roman{*})]
\item\label{enum:GBSRaxiomatic:I} for every $i$, $0 \leq i \leq n$, it holds $\text{\upshape{vars}}(\text{At}_i) \subseteq \vec{\mathbf{y}}_1 \cup \ldots \cup \vec{\mathbf{y}}_i \cup \vec{\mathbf{x}}_{i+1} \cup \ldots \cup \vec{\mathbf{x}}_n$, and
\item\label{enum:GBSRaxiomatic:II} for all distinct $i,j$, $0 \leq i < j \leq n$, it holds $\text{\upshape{vars}}(\text{At}_i) \cap \text{\upshape{vars}}(\text{At}_j) \cap \vec{\mathbf{x}} = \emptyset$.
\end{enumerate}
\end{definition}
We shall see in Section~\ref{section:ModeltheoreticApproachToGBSRsat} how the described way of partitioning the atoms in GBSR sentences facilitates a model-theoretic approach to proving decidability of the satisfiability problem for GBSR sentences \emph{(GBSR-satisfiability)}.
We complement the axiomatic definition of GBSR with an alternative definition of an algorithmic flavor.
For one thing, the algorithmic definition shows that membership in GBSR is easily decidable.
Moreover, the notions used in the algorithmic definition will be of use in the syntactic approach to decidability of GBSR-satisfiability outlined in Section~\ref{section:SyntacticApproachToGBSRsat}.
We shall see in Lemma~\ref{lemma:AlternativeDefinitionOfGBSR} that both definitions are equivalent.
But first, we need additional notation.
Given $\varphi$, we define the undirected graph $\mathcal{G}_{\varphi} := \<V, E\>$ by setting $V := \vec{\mathbf{x}}$ and
$E := \{ \<x, x'\> \mid \text{there is an atom in $\varphi$ containing both $x$ and $x'$} \}$.
A \emph{connected component in $\mathcal{G}_{\varphi}$} is a maximal subset $C \subseteq V$ such that for all distinct variables $x, x' \in C$ the transitive closure of $E$ contains the pair $\<x,x'\>$. The set of all connected components in $\mathcal{G}_{\varphi}$ forms a partition of $V$.
For every connected component $C$ in $\mathcal{G}_{\varphi}$ we denote by $\mathcal{L}(C)$ the \emph{set of all literals in $\varphi$ which contain at least one variable from $C$}.
For every index $k$, $1\leq k\leq n$, we denote by $\mathcal{L}_k$ the smallest set of literals such that
$\mathcal{L}_k$ contains all literals taken from $\varphi$ in which variables from $\vec{\mathbf{y}}_k$ occur, and
for every connected component $C$ in $\mathcal{G}_{\varphi}$ containing a variable $x \in \text{\upshape{vars}}(\mathcal{L}_k)$ it holds $\mathcal{L}(C) \subseteq \mathcal{L}_k$.
Intuitively, every $\mathcal{L}_k$ constitutes a reasonably small superset of the literals in $\varphi$ which remain in the scope of the quantifier block $\exists \vec{\mathbf{y}}_k$ when the rules of miniscoping (cf.\ Lemma~\ref{lemma:Miniscoping}) are applied from left to right.
\begin{definition}[GBSR fragment, algorithmically]\label{definition:GBSRalgorithmic}
The sentence $\varphi$ is in \emph{GBSR} if and only if for all $k,\ell$ with $1\leq \ell \leq k \leq n$ it holds $\text{\upshape{vars}}(\mathcal{L}_k) \cap \vec{\mathbf{x}}_{\ell} = \emptyset$.
\end{definition}
By $\widetilde{\cL}_k$ we denote the set $\mathcal{L}_k \setminus \bigcup_{\ell > k} \mathcal{L}_{\ell}$.
Moreover, $\widetilde{\cL}_0$ stands for the set of all literals in $\varphi$ which do not belong to any of the $\widetilde{\cL}_k$.
Note that the sets $\widetilde{\cL}_0, \ldots, \widetilde{\cL}_n$ form a partition of the set of all literals in $\varphi$.
For every $k$, $0 \leq k \leq n$, we write $\widetilde{X}_{k}$ to address the set $\text{\upshape{vars}}(\widetilde{\cL}_k) \cap \vec{\mathbf{x}}$.
\begin{lemma}\label{lemma:LiteralsDecomposition}~
\begin{enumerate}[label=(\roman{*}), ref=(\roman{*})]
\item\label{enum:LiteralsDecomposition:I}
For all distinct indices $k, \ell$ we have $\widetilde{X}_{k} \cap \widetilde{X}_{\ell} = \emptyset$.
\item\label{enum:LiteralsDecomposition:II}
For every $k$ it holds $\text{\upshape{vars}}(\widetilde{\cL}_{k}) \cap \vec{\mathbf{y}} \subseteq \bigcup_{1 \leq \ell \leq k} \vec{\mathbf{y}}_{\ell}$.
\item\label{enum:LiteralsDecomposition:III}
If $\varphi$ satisfies Definition~\ref{definition:GBSRalgorithmic}, then
for every $k$ it holds $\widetilde{X}_{k} \subseteq \bigcup_{k < \ell \leq n} \vec{\mathbf{x}}_{\ell}$.
\end{enumerate}
\end{lemma}
\begin{proof}~
\begin{description}
\item{Ad \ref{enum:LiteralsDecomposition:I}:}
Suppose there are distinct indices $k, \ell$, $k < \ell$, and a variable $x \in \vec{\mathbf{x}}$ such that $x \in \widetilde{X}_k \cap \widetilde{X}_{\ell}$.
But then, there must be literals $L_k \in \widetilde{\cL}_k \subseteq \mathcal{L}_k$ and $L_{\ell} \in \widetilde{\cL}_{\ell} \subseteq \mathcal{L}_{\ell}$, both containing $x$.
Let $C$ denote the (unique) connected component in $\mathcal{G}_\psi$ to which $x$ belongs.
By definition of $\mathcal{L}(C)$, both $L_k$ and $L_{\ell}$ belong to $\mathcal{L}(C)$.
Therefore, it holds $\{L_k, L_{\ell}\} \subseteq \mathcal{L}(C) \subseteq \mathcal{L}_{\ell}$.
But since $\widetilde{\cL}_k \subseteq \mathcal{L}_k \setminus \mathcal{L}_{\ell}$, $L_k$ cannot belong to $\mathcal{L}_k$. A contradiction.
\item{Ad \ref{enum:LiteralsDecomposition:II}:}
Let $k \leq n$ be some non-negative integer. Since for any $\ell > k$ $\mathcal{L}_\ell$ contains all literals in which a variable $y \in \vec{\mathbf{y}}_\ell$ occurs, $\widetilde{\cL}_k \subseteq \mathcal{L}_k \setminus \mathcal{L}_\ell$ cannot contain any occurrence of such a variable $y$.
\item{Ad \ref{enum:LiteralsDecomposition:III}:}
Let $k \leq n$ be some non-negative integer. $\varphi$'s belonging to GBSR means $\text{\upshape{vars}}(\mathcal{L}_k) \cap \bigcup_{\ell' \leq k} \vec{\mathbf{x}}_{\ell'} = \emptyset$. Because of $\widetilde{X}_k = \text{\upshape{vars}}(\widetilde{\cL}_k) \cap \vec{\mathbf{x}} \subseteq \text{\upshape{vars}}(\mathcal{L}_k) \cap \vec{\mathbf{x}}$, we conclude $\widetilde{X}_k \cap \bigcup_{\ell' \leq k} \vec{\mathbf{x}}_{\ell'} = \emptyset$. Hence, we have $\widetilde{X}_k \subseteq \bigcup_{\ell > k} \vec{\mathbf{x}}_{\ell}$.
\qedhere
\end{description}
\end{proof}
We now have the right notions at hand to show that the axiomatic and the algorithmic definitions of GBSR sentences yield the same fragment of first-order logic.
\begin{lemma}\label{lemma:AlternativeDefinitionOfGBSR}
The sentence $\varphi$ satisfies Definition~\ref{definition:GBSRaxiomatic} if and only if it satisfies Definition~\ref{definition:GBSRalgorithmic}.
\end{lemma}
\begin{proof}
The \emph{if}-direction follows immediately from Lemma~\ref{lemma:LiteralsDecomposition}.
We just define the $\text{At}_k \subseteq \text{At}$ such that $A \in \text{At}_k$ if and only if either $A$ or $\neg A$ or both belong to $\widetilde{\cL}_k$.
The \emph{only if}-direction can be argued as follows.
For every $i$, $0 \leq i \leq n$, let $X_i := \text{\upshape{vars}}(\text{At}_i) \cap \vec{\mathbf{x}}$.
Consider the graph $\mathcal{G}_\varphi$.
Since the $X_1, \ldots, X_n$ are pairwise disjoint, they induce subgraphs of $\mathcal{G}_\varphi$ that are not connected to one another.
Moreover, for every connected component $C$ in $\mathcal{G}_\varphi$, there is one $X_i$ such that $C \subseteq X_i$.
When we write $\text{At}(C)$ to denote all atoms in $\psi$ that contain variables from $C$, then the previous observation entails that for every connected component $C$ in $\mathcal{G}_\varphi$ there is some $\text{At}_i$ such that $\text{At}(C) \subseteq \text{At}_i$.
Let $\text{At}'_k$ be the set of atoms occurring in $\mathcal{L}_k$.
In other words, let $\text{At}'_k$ be the smallest set of all atoms in $\psi$ that contain variables from $\vec{\mathbf{y}}_k$ and for every connected component $C$ in $\mathcal{G}_\varphi$ containing a variable $x \in \text{\upshape{vars}}(\text{At}_k)$, $\text{At}'_k$ contains $\text{At}(C)$.
Hence, $\text{At}'_k \subseteq \text{At}_k \cup \ldots \cup \text{At}_n$ and, moreover, $\text{\upshape{vars}}(\text{At}'_k) \cap \vec{\mathbf{x}} \subseteq \vec{\mathbf{x}}_{k+1} \cup \ldots \cup \vec{\mathbf{x}}_n$.
This means, $\text{\upshape{vars}}(\text{At}'_k) \cap \vec{\mathbf{x}}_\ell = \text{\upshape{vars}}(\mathcal{L}_k) \cap \vec{\mathbf{x}}_\ell = \emptyset$ for every $\ell$, $1 \leq \ell \leq k$.
It follows that Definition~\ref{definition:GBSRalgorithmic} is satisfied.
\end{proof}
In \cite{Voigt2016} the \emph{separated fragment (SF)} is defined to be the set of first-order sentences $\chi := \exists \vec{\mathbf{z}}\, \forall\vec{\mathbf{u}}_1 \exists\vec{\mathbf{v}}_1 \ldots \forall\vec{\mathbf{u}}_n \exists\vec{\mathbf{v}}_n. \chi'$ with quantifier-free $\chi'$ that may contain equality but no non-constant function symbols. Moreover, for every atom $A$ in $\chi$ it is required that either $\text{\upshape{vars}}(A) \cap \bigcup_i \vec{\mathbf{u}}_i = \emptyset$ or $\text{\upshape{vars}}(A) \cap \bigcup_i \vec{\mathbf{v}}_i = \emptyset$ or both hold, i.e.\ variables $u \in \vec{\mathbf{u}}_k$ and $v \in \vec{\mathbf{v}}_\ell$ may not occur together in any atom in $\chi$ for arbitrary $k, \ell$.
We have advertised GBSR as an extension of SF.
Indeed, we can partition the set of $\chi$'s atoms into two sets $\text{At}_0, \text{At}_n$ such that $\text{\upshape{vars}}(\text{At}_0) \subseteq \vec{\mathbf{z}} \cup \vec{\mathbf{u}}_1 \cup \ldots \cup \vec{\mathbf{u}}_n$ and $\text{\upshape{vars}}(\text{At}_n) \subseteq \vec{\mathbf{z}} \cup \vec{\mathbf{v}}_1 \cup \ldots \cup \vec{\mathbf{v}}_n$.
This partition obviously satisfies the requirements of Definition~\ref{definition:GBSRaxiomatic}.
Moreover, it is known that SF contains the BSR fragment as well as the relational monadic fragment without equality \cite{Voigt2016}.
On the other hand, Example~\ref{example:GAFandGBSR} contains the sentence $\varphi_2$ which belongs to GBSR but not to SF, thus showing that GBSR is a proper extension of SF.
\begin{proposition}\label{theorem:InclusionGBSR}
GBSR properly contains the separated fragment, the BSR fragment, and the relational monadic fragment without equality.
\end{proposition}
One of the main results of the present paper is that GBSR-satisfiability is decidable.
\begin{theorem}\label{theorem:DecidabilityGBSR}
Satisfiability of GBSR sentences is decidable.
\end{theorem}
We prove this theorem in two ways.
The first route is a syntactic one (Section~\ref{section:SyntacticApproachToGBSRsat}): we show that every GBSR sentence can be effectively transformed into an equivalent BSR sentence (see the proof Lemma~\ref{lemma:TransformationGBSR}).
Our second approach is a model-theoretic one: we devise a method that, given any model $\mathcal{A}$ of a GBSR sentence $\varphi$, constructs a model $\mathcal{B}$ of $\varphi$ with a domain of a bounded size.
In other words, we show that GBSR enjoys a \emph{small model property} (Theorem~\ref{theorem:GBSRsmallModelProperty}).
Although already the existence of an effective translation of GBSR sentences into equivalent BSR sentences entails that GBSR inherits the small model property from BSR, our approach does not exploit this fact.
It rather relies on a technique that emphasizes the finite character of the dependencies between universal variables and the existential variables that lie within their scope in a given GBSR sentence.
Concerning computational complexity, the hierarchy of $k$-\textsc{NExpTime}-complete subproblems of \emph{SF-satisfiability} presented in \cite{Voigt2017} together with the containment of SF in GBSR leads to the observation that GBSR-satisfiability is non-elementary.
\begin{theorem}\label{theorem:ComputationalHardnessGBSR}
GBSR-satisfiability is $k$-\textsc{NExpTime}-hard for every positive integer $k$.
\end{theorem}
The small model property that we derive for GBSR sentences in Section~\ref{section:ModeltheoreticApproachToGBSRsat} entails the existence of a similar hierarchy of complete problems for GBSR as there exists for SF.
\begin{theorem}\label{theorem:ComputationalComplexityGBSR}
We can divide the GBSR fragment into an increasing sequence of subclasses $\text{GBSR}_k$ with $k = 1, 2, 3, \ldots$ such that for every $k$ the set of all satisfiable sentences from $\text{GBSR}_k$ forms a $k$-\textsc{NExpTime}-complete set.
\end{theorem}
\begin{proof}[Proof sketch]
The division of GBSR into subfragments $\text{GBSR}_1 \subseteq \text{GBSR}_2 \subseteq \ldots$ is based on the degree $\partial_\varphi$ of GBSR sentences $\varphi$, which we shall define right after Corollary~\ref{corollary:UniformStrategies}.
More precisely, we define $\text{GBSR}_k$ to be the subfragment of GBSR that contains all GBSR sentences $\varphi$ with $\partial_\varphi = k-1$.
%
By Theorem~\ref{theorem:GBSRsmallModelProperty}, every satisfiable GBSR sentence $\varphi$ has a model with at most $\text{len}(\varphi)^2 \cdot \bigl( \twoup{\partial_\varphi+1}{\text{len}(\varphi)} \bigr)^{\text{len}(\varphi)^2}$ domain elements.
Hence, for any $k \geq 1$ the satisfiability problem for $\text{GBSR}_k$ is decidable in nondeterministic $k$-fold-exponential time.
On the other hand, the lower bound proof for SF-satisfiability in \cite{Voigt2017} is based on SF-formulas that encode computationally hard \emph{domino problems}.
One can use the same formulas to show that every set $\text{GBSR}_k$ contains an infinite subset $\text{SF}_k$ of SF formulas of degree\footnote{Notice that the notion of \emph{degree} used in the present paper differs from the notion of \emph{degree} used in \cite{Voigt2017}. The proof of Theorem~\ref{theorem:ComputationalComplexityGBSR} solely refers to the notions from the present paper.} $k$ such that there is a polynomial reduction from some $k$-\textsc{NExpTime}-hard domino problem to the satisfiability problem for $\text{SF}_k$ (see Lemma~20 in \cite{Voigt2017}).
Hence, the satisfiability problem for $\text{GBSR}_k$ is $k$-\textsc{NExpTime}-hard as well.
\end{proof}
Every GBSR sentence with $n$ $\forall \exists$ quantifier alternations occurs in $\text{GBSR}_n$ at the latest.
It might occur earlier in the sequence $\text{GBSR}_1 \subseteq \text{GBSR}_2 \subseteq \ldots$, though.
For instance, every relational monadic sentence $\varphi$ (in prenex form) without equality belongs to $\text{GBSR}_1$ and thus also to every later set in the sequence, no matter what form $\varphi$'s quantifier prefix has.
\subsection{A syntactic approach to decidability of GBSR-satisfiability}\label{section:SyntacticApproachToGBSRsat}
The next lemma provides the key ingredient to show decidability of GBSR-satisfiability by a reduction to the satisfiability problem for the BSR fragment.
Moreover, it will help proving that relational GBSR without equality is closed under Craig--Lyndon interpolation.
\begin{lemma}\label{lemma:TransformationGBSR}
Let $\varphi := \forall \vec{\mathbf{x}}_1 \exists \vec{\mathbf{y}}_1 \ldots \forall \vec{\mathbf{x}}_n \exists \vec{\mathbf{y}}_n.\,\psi$ be a GBSR sentence in standard form.
%
There exists a quantifier-free first-order formula $\psi'(\vec{\mathbf{u}}, \vec{\mathbf{v}})$ such that $\varphi' := \exists \vec{\mathbf{u}}\, \forall \vec{\mathbf{v}}.\, \psi'(\vec{\mathbf{u}}, \vec{\mathbf{v}})$ is in standard form and semantically equivalent to $\varphi$ and all literals in $\varphi'$ also occur in $\varphi$ (modulo variable renaming).
\end{lemma}
\begin{proof}[Proof sketch]
The following transformations mainly use the standard laws of Boolean algebra and the miniscoping rules (Lemma~\ref{lemma:Miniscoping}) to (re-)transform $\psi$ into particular syntactic shapes.
Note that this does not change the set of literals occurring in the intermediate steps (modulo variable renaming), since we start from a formula in negation normal form restricted to the connectives $\wedge, \vee, \neg$.
We make use of the partition of the literals in $\psi$ into the sets $\widetilde{\cL}_0, \widetilde{\cL}_1, \ldots, \widetilde{\cL}_n$ (defined right before Lemma~\ref{lemma:LiteralsDecomposition}) to group exactly those literals in each step of our transformation.
To begin with, we transform the matrix $\psi$ into a disjunction of conjunctions of literals $\bigvee_i \psi_i$.
In addition, we rewrite every $\psi_i$ into $\psi_i = \widetilde{\chi}_{i,0}^{(1)} \wedge \ldots \wedge \widetilde{\chi}_{i,n}^{(1)}$.
Every $\widetilde{\chi}_{i,\ell}^{(1)}$ is a conjunction of literals and comprises exactly the literals from the $\psi_i$ which belong to $\widetilde{\cL}_\ell$.
Moreover, by Lemma~\ref{lemma:LiteralsDecomposition}\ref{enum:LiteralsDecomposition:II} and \ref{enum:LiteralsDecomposition:III}, we know that $\text{\upshape{vars}}(\widetilde{\chi}_{i,\ell}^{(1)}) \subseteq \vec{\mathbf{y}}_1 \cup \ldots \cup \vec{\mathbf{y}}_\ell \cup \vec{\mathbf{x}}_{\ell+1} \cup \ldots \cup \vec{\mathbf{x}}_n$.
%
Using the rules of miniscoping, we move the existential quantifier block $\exists \vec{\mathbf{y}}_n$ inwards such that it binds the $\widetilde{\chi}_{i,n}^{(1)}$ alone.
The thus obtained sentence $\varphi''$ has the form $\forall\vec{\mathbf{x}}_1 \exists\vec{\mathbf{y}}_1 \ldots \forall\vec{\mathbf{x}}_n . \bigvee_i \widetilde{\chi}_{i,0}^{(1)} \wedge \ldots \wedge \widetilde{\chi}_{i,n-1}^{(1)} \wedge \exists\vec{\mathbf{y}}_n. \widetilde{\chi}_{i,n}^{(1)}$.
In further transformations, we treat the subformulas $\bigl(\exists\vec{\mathbf{y}}_n. \widetilde{\chi}_{i,n}^{(1)}\bigr)$ as indivisible units.
Next, we transform the big disjunction in $\varphi''$ into a conjunction of disjunctions $\bigwedge_k \psi'_k$, group the disjunctions $\psi'_k$ into subformulas $\widetilde{\eta}_{k,\ell}^{(1)}$ as before, and move the universal quantifier block $\forall \vec{\mathbf{x}}_n$ inwards.
Due to Lemma~\ref{lemma:LiteralsDecomposition}\ref{enum:LiteralsDecomposition:I}, we can split the quantifier block $\forall \vec{\mathbf{x}}_n$ so that universal quantifiers can be moved directly before the $\widetilde{\eta}_{k,\ell}^{(1)}$.
The result is of the form $\forall\vec{\mathbf{x}}_1 \exists\vec{\mathbf{y}}_1 \ldots \forall\vec{\mathbf{x}}_{n-1} \exists\vec{\mathbf{y}}_{n-1} . \bigwedge_k \bigl(\forall(\vec{\mathbf{x}}_n \cap \widetilde{X}_0).\, \widetilde{\eta}_{k,0}^{(1)}\bigr) \vee \ldots \vee \bigl(\forall(\vec{\mathbf{x}}_n \cap \widetilde{X}_{n-1}).\, \widetilde{\eta}_{k,n-1}^{(1)}\bigr) \vee \widetilde{\eta}_{k,n}^{(1)}$.
We reiterate this process until all quantifier blocks have been moved inwards as described.
In the resulting formula we observe that no existential quantifier occurs within the scope of any universal quantifier.
%
Using the miniscoping rules, we can move all quantifiers outwards again---existential quantifiers first---, renaming variables as necessary. In the end, we obtain a prenex formula of the form $\varphi' :=\;\exists \vec{\mathbf{u}}\, \forall \vec{\mathbf{v}}. \psi'$, where $\psi'$ is quantifier free and contains exclusively literals that are renamed variants of literals occurring in the original $\psi$.
\end{proof}
The just proven lemma shows that every GBSR sentence $\varphi$ is equivalent to some BSR sentence $\varphi' := \exists \vec{\mathbf{u}}\, \forall \vec{\mathbf{v}}.\, \psi'$.
This immediately entails decidability of GBSR-satisfiability.
The number of leading existential quantifiers in BSR sentences induces an upper bound on the size of \emph{small models}---every satisfiable BSR sentences has such a small model.
One can adapt the methods applied in~\cite{Voigt2017} to facilitate the derivation of a tight upper on the number of leading existential quantifiers.
To this end, the notion of \emph{degree of interaction of existential variables} used in that paper needs to be extended so that it also covers the interaction of universally and existentially quantified variables caused by joint occurrences in atoms.
In Section~\ref{section:ModeltheoreticApproachToGBSRsat}, we present a different, a model-theoretic approach to deriving an upper bound on the size of small models.
In order to formulate this bound accurately, we introduce a related, yet complementary notion of \emph{degree} based on the interaction of universally quantified variables in atoms.
We conclude the present section by applying the above lemma to show that relational GBSR without equality is closed under Craig--Lyndon interpolation~\cite{Craig1957a, Lyndon1959}.
Hence, relational GBSR without equality additionally enjoys Beth's definability property, which is well-known to be a consequence of the Craig--Lyndon interpolation property (see, e.g., Chapter 20 in \cite{Boolos2002}).
Given a formula $\varphi$ that exclusively contains the connectives $\wedge, \vee, \neg$, we say that a predicate symbol $P$ \emph{occurs positively in $\varphi$} if there is an occurrence of some atom $P(\ldots)$ in $\varphi$ such that
the number of subformulas of $\varphi$ that contain this occurrence and have a negation sign as topmost connective is even.
Analogously, we say that a predicate symbol $P$ \emph{occurs negatively in $\varphi$} if there is an occurrence of some atom $P(\ldots)$ in $\varphi$ such that
the number of subformulas of $\varphi$ that contain this occurrence and have a negation sign as topmost connective is odd.
\begin{theorem}[Interpolation Theorem for GBSR]\label{theorem:InterpolationGBSR}
Let $\varphi$ and $\psi$ be relational GBSR sentences in standard form without equality.
If $\varphi \models \psi$, then there exists a relational BSR sentence $\chi$ without equality such that
\begin{enumerate}[label=(\roman{*}), ref=(\roman{*})]
\item $\varphi \models \chi$ and $\chi \models \psi$, and
\item any predicate symbol $P$ occurs positively (negatively) in $\chi$ only if it occurs positively (negatively) in $\varphi$ and in $\psi$.
\end{enumerate}
\end{theorem}
This is a consequence of Lemma~\ref{lemma:TransformationGBSR} combined with the following lemma.
\begin{lemma}\label{lemma:LyndonInterpolationBSR}
Let $\varphi$ and $\psi$ be relational BSR sentences without equality in which the only Boolean connectives are $\wedge, \vee, \neg$.
If $\varphi \models \psi$, then there exists a relational BSR sentence $\chi$ without equality such that
\begin{enumerate}[label=(\roman{*}), ref=(\roman{*})]
\item\label{enum:LyndonInterpolationBSR:I} $\varphi \models \chi$ and $\chi \models \psi$, and
\item\label{enum:LyndonInterpolationBSR:II} any predicate symbol $P$ occurs positively (negatively) in $\chi$ only if it occurs positively (negatively) in $\varphi$ and in $\psi$.
\end{enumerate}
\end{lemma}
Since the proof of Lemma~\ref{lemma:LyndonInterpolationBSR} requires techniques that are largely unrelated to the rest of the present paper, the proof sketch has been moved to the appendix.
\subsection{A model-theoretic approach to decidability of GBSR-satisfiability}\label{section:ModeltheoreticApproachToGBSRsat}
In this section we investigate the \emph{finite character} of the dependency of existential variables $y$ on universal variables $x$ that appear earlier in the quantifier prefix $\ldots \forall x \ldots \exists y \ldots$ of a GBSR sentence.
This finite character will be made explicit with the following concepts:
\emph{fingerprints} are sets of sets of \ldots sets of atoms that characterize certain tuples of domain elements by finite means;
\emph{fingerprint functions} $\mu$ assign fingerprints to tuples of domain elements;
\emph{$\mu$-uniform strategies} select domain elements for existentially quantified variables exclusively depending on the fingerprints of the domain elements that have been assigned to preceding universally quantified variables.
Again, for the considerations in this section we fix a GBSR sentence $\varphi :=\, \forall \vec{\mathbf{x}}_1 \exists \vec{\mathbf{y}}_1 \ldots \forall \vec{\mathbf{x}}_n$ $\exists \vec{\mathbf{y}}_n.\psi$ in standard form in which $\psi$ is quantifier free.
Without loss of generality, we assume that $\varphi$ is relational and, hence, does not contain any function or constant symbols.
Let $\text{At}$ denote the set of all atoms occurring in $\varphi$ and let $\vec{\mathbf{x}} := \vec{\mathbf{x}}_1 \cup \ldots \cup \vec{\mathbf{x}}_n$ and $\vec{\mathbf{y}} := \vec{\mathbf{y}}_1 \cup \ldots \cup \vec{\mathbf{y}}_n$.
By Definition~\ref{definition:GBSRaxiomatic}, we may assume that $\text{At}$ can be partitioned into (possibly empty) sets $\text{At}_0, \ldots, \text{At}_n$ such
that Conditions~\ref{enum:GBSRaxiomatic:I} and~\ref{enum:GBSRaxiomatic:II} of Definition~\ref{definition:GBSRaxiomatic} are met.
Let $\mathcal{A}$ be any structure over the vocabulary of $\psi$.
We next define the semantic equivalent of Skolem functions.
\begin{definition}
A \emph{strategy $\sigma$} comprises a tuple of $n$ mappings $\<\sigma_1, \ldots, \sigma_n\>$ with signatures $\sigma_i : \mathfrak{U}_\mathcal{A}^{|\vec{\mathbf{x}}_{1}|} \times \ldots \times \mathfrak{U}_\mathcal{A}^{|\vec{\mathbf{x}}_i|} \to \mathfrak{U}_\mathcal{A}^{|\vec{\mathbf{y}}_i|}$.
%
A strategy $\sigma$ is \emph{satisfying for $\varphi$} if and only if \\
\centerline{$\mathcal{A}, [\vec{\mathbf{x}}_1 {\mathop{\mapsto}} \vec{\fb}_1, \ldots, \vec{\mathbf{x}}_n {\mathop{\mapsto}} \vec{\fb}_n, \vec{\mathbf{y}}_1 {\mathop{\mapsto}} \sigma_1(\vec{\fb}_1), \ldots, \vec{\mathbf{y}}_n {\mathop{\mapsto}} \sigma_n(\vec{\fb}_1, \ldots, \vec{\fb}_n)] \models \psi$}
holds for every choice of tuples $\vec{\fb}_1 \in \mathfrak{U}_\mathcal{A}^{|\vec{\mathbf{x}}_1|}, \ldots, \vec{\fb}_n \in \mathfrak{U}_\mathcal{A}^{|\vec{\mathbf{x}}_n|}$.
\end{definition}
\begin{definition}
Let $\vec{\fb}_1 \in \mathfrak{U}_\mathcal{A}^{|\vec{\mathbf{x}}_1|}, \ldots, \vec{\fb}_n \in \mathfrak{U}_\mathcal{A}^{|\vec{\mathbf{x}}_n|}$ be tuples of domain elements. By $\text{out}_{\text{At},\sigma}(\vec{\fb}_1, \ldots, \vec{\fb}_n)$ we denote the set $\bigl\{ A \in \text{At} \bigm| \mathcal{A},[\vec{\mathbf{x}}_1 {\mathop{\mapsto}} \vec{\fb}_1, \ldots, \vec{\mathbf{x}}_n {\mathop{\mapsto}} \vec{\fb}_n, \vec{\mathbf{y}}_1 {\mathop{\mapsto}} \sigma_1(\vec{\fb}_1), \ldots,$ $\vec{\mathbf{y}}_n {\mathop{\mapsto}} \sigma_n(\vec{\fb}_1, \ldots, \vec{\fb}_n)] \models A \bigr\}$, dubbed the \emph{outcome of $\vec{\fb}_1, \ldots, \vec{\fb}_n$ under $\sigma$ with respect to the atoms in $\text{At}$}.
%
By $\text{Out}_{\text{At},\sigma}$ we denote the \emph{set of all possible outcomes of $\sigma$ with respect to the atoms in $\text{At}$}, i.e.\
$\text{Out}_{\text{At},\sigma} := \bigl\{ \text{out}_{\text{At},\sigma}(\vec{\fb}_1, \ldots, \vec{\fb}_n) \bigm| \vec{\fb}_1 \in \mathfrak{U}_\mathcal{A}^{|\vec{\mathbf{x}}_1|}, \ldots, \vec{\fb}_n \in \mathfrak{U}_\mathcal{A}^{|\vec{\mathbf{x}}_n|} \bigr\}$.
\end{definition}
The \emph{Boolean abstraction $\psi_\text{bool}$ of $\psi$} is the propositional formula that results from $\psi$ if we conceive every atom $A$ in $\psi$ as propositional variable $p_A$.
A subset $S \subseteq \text{At}$ can be conceived as a valuation of $\psi_\text{bool}$ by setting $S \models p_A$ if and only if $A \in S$.
Clearly, a strategy $\sigma$ is satisfying for $\varphi$ if and only if for every outcome $S \in \text{Out}_{\text{At},\sigma}$ it holds $S \models \psi_\text{bool}$.
\begin{proposition}
The structure $\mathcal{A}$ is a model of $\varphi$ if and only if there exists a strategy $\sigma$ which is satisfying for $\varphi$.
\end{proposition}
If $\varphi$ is satisfied by $\mathcal{A}$, we are interested in special satisfying strategies whose image only covers a finite portion of $\mathcal{A}$'s domain.
Such a strategy induces a finite substructure of $\mathcal{A}$ that also satisfies $\varphi$.
In order to find such strategies, we need to identify the key features of domain elements that make them distinguishable by the formula $\varphi$.
We express these features by the already mentioned \emph{fingerprints}.
We use $\mathfrak{P}$ to denote the power set operator, i.e.\ $\mathfrak{P} S$ stands for the set of all subsets of a given set $S$.
The iterated application of $\mathfrak{P}$ is denoted by $\mathfrak{P}^k$, meaning $\mathfrak{P}^0 S := S$ and $\mathfrak{P}^{k+1} S := \mathfrak{P}^k (\mathfrak{P} S)$ for every $k \geq 0$.
\begin{definition}
We define the family of \emph{fingerprint functions $\mu_{\ell, k}$} with $0 \leq \ell < k \leq n$ as follows
\begin{description}
\item $\mu_{\ell, n} : \mathfrak{U}_\mathcal{A}^{|\vec{\mathbf{y}}_1|} \times \ldots \times \mathfrak{U}_\mathcal{A}^{|\vec{\mathbf{y}}_\ell|} \times \mathfrak{U}_\mathcal{A}^{|\vec{\mathbf{x}}_{\ell+1}|} \times \ldots \times \mathfrak{U}_\mathcal{A}^{|\vec{\mathbf{x}}_n|} \to \mathfrak{P} \text{At}_{\ell}$
such that for all tuples $\vec{\fa}_1 \in \mathfrak{U}_\mathcal{A}^{|\vec{\mathbf{y}}_1|}, \ldots, \vec{\fa}_{\ell} \in \mathfrak{U}_\mathcal{A}^{|\vec{\mathbf{y}}_{\ell}|}, \vec{\fb}_{\ell+1} \in \mathfrak{U}_\mathcal{A}^{|\vec{\mathbf{x}}_{\ell+1}|}, \ldots, \vec{\fb}_{n} \in \mathfrak{U}_\mathcal{A}^{|\vec{\mathbf{x}}_{n}|}$ and every $A \in \text{At}_{\ell}$ we have
$A \in \mu_{\ell,n}(\vec{\fa}_1, \ldots, \vec{\fa}_{\ell}, \vec{\fb}_{\ell+1}, \ldots,$ $\vec{\fb}_{n})$ if and only if
$\mathcal{A}, [\vec{\mathbf{y}}_1 {\mathop{\mapsto}} \vec{\fa}_1, \ldots, \vec{\mathbf{y}}_{\ell} {\mathop{\mapsto}} \vec{\fa}_{\ell}, \vec{\mathbf{x}}_{\ell+1} {\mathop{\mapsto}} \vec{\fb}_{\ell+1}, \ldots,$ $\vec{\mathbf{x}}_{n} {\mathop{\mapsto}} \vec{\fb}_{n}] \models A$;
\item $\mu_{\ell, n-1} :\mathfrak{U}_\mathcal{A}^{|\vec{\mathbf{y}}_1|} \times \ldots \times \mathfrak{U}_\mathcal{A}^{|\vec{\mathbf{y}}_\ell|} \times \mathfrak{U}_\mathcal{A}^{|\vec{\mathbf{x}}_{\ell+1}|} \times \ldots \times \mathfrak{U}_\mathcal{A}^{|\vec{\mathbf{x}}_{n-1}|} \to \mathfrak{P}^2 \text{At}_{\ell}$
such that for all tuples $\vec{\fa}_1 \in \mathfrak{U}_\mathcal{A}^{|\vec{\mathbf{y}}_1|}, \ldots, \vec{\fa}_{\ell} \in \mathfrak{U}_\mathcal{A}^{|\vec{\mathbf{y}}_{\ell}|}, \vec{\fb}_{\ell+1} \in \mathfrak{U}_\mathcal{A}^{|\vec{\mathbf{x}}_{\ell+1}|}, \ldots, \vec{\fb}_{n-1} \in \mathfrak{U}_\mathcal{A}^{|\vec{\mathbf{x}}_{n-1}|}$ and every $S \in \mathfrak{P} \text{At}_{\ell}$ we have
$S \in \mu_{\ell,n-1}(\vec{\fa}_1, \ldots, \vec{\fa}_{\ell}, \vec{\fb}_{\ell+1}, \ldots, \vec{\fb}_{n-1})$ if and only if there exists some $\vec{\fb}_{n}$ such that $\mu_{\ell,n}(\vec{\fa}_1, \ldots, \vec{\fa}_{\ell},$ $\vec{\fb}_{\ell+1}, \ldots, \vec{\fb}_{n-1}, \vec{\fb}_{n}) = S$;
\item[] \qquad$\vdots$
\item $\mu_{\ell, \ell+1} : \mathfrak{U}_\mathcal{A}^{|\vec{\mathbf{y}}_1|} \times \ldots \times \mathfrak{U}_\mathcal{A}^{|\vec{\mathbf{y}}_\ell|} \times \mathfrak{U}_\mathcal{A}^{|\vec{\mathbf{x}}_{\ell+1}|} \to \mathfrak{P}^{n-\ell} \text{At}_{\ell}$
such that for all tuples $\vec{\fa}_1 \in \mathfrak{U}_\mathcal{A}^{|\vec{\mathbf{y}}_1|}, \ldots, \vec{\fa}_{\ell} \in \mathfrak{U}_\mathcal{A}^{|\vec{\mathbf{y}}_{\ell}|}, \vec{\fb}_{\ell+1} \in \mathfrak{U}_\mathcal{A}^{|\vec{\mathbf{x}}_{\ell+1}|}$ and every $S \in \mathfrak{P}^{n-\ell-1} \text{At}_{\ell}$ we have
$S \in \mu_{\ell,\ell+1}(\vec{\fa}_1, \ldots, \vec{\fa}_{\ell}, \vec{\fb}_{\ell+1})$ if and only if there exists $\vec{\fb}_{\ell+2}$ such that $\mu_{\ell,\ell+2}(\vec{\fa}_1, \ldots, \vec{\fa}_{\ell}, \vec{\fb}_{\ell+1}, \vec{\fb}_{\ell+2}) = S$.
\end{description}
%
We denote the \emph{image} of a fingerprint function $\mu_{\ell,k}$ under a strategy $\sigma = \<\sigma_1, \ldots, \sigma_n\>$ by \\
\centerline{$\text{im}_\sigma(\mu_{\ell, k}) := \bigl\{ \mu_{\ell,k}\bigl(\sigma_1(\vec{\fb}_1), \ldots, \sigma_\ell(\vec{\fb}_1, \ldots, \vec{\fb}_\ell), \vec{\fb}_{\ell+1}, \ldots, \vec{\fb}_k\bigr) \bigm| \vec{\fb}_1 \in \mathfrak{U}_\mathcal{A}^{|\vec{\mathbf{x}}_1|}, \ldots, \vec{\fb}_k \in \mathfrak{U}_\mathcal{A}^{|\vec{\mathbf{x}}_k|} \bigr\}$.}
\end{definition}
Having a suitable notion of fingerprints at hand, we next define strategies that induce finite substructures of $\mathcal{A}$.
\begin{definition}
A strategy $\sigma = \<\sigma_1, \ldots, \sigma_n\>$ is \emph{$\mu$-uniform} if for every $k$, $1 \leq k\leq n$, the following holds.
For all tuples $\vec{\fb}_{1}, \vec{\fb}'_{1} \in \mathfrak{U}_\mathcal{A}^{|\vec{\mathbf{x}}_{1}|}, \ldots, \vec{\fb}_k, \vec{\fb}'_k \in \mathfrak{U}_\mathcal{A}^{|\vec{\mathbf{x}}_k|}$ we have $\sigma_k(\vec{\fb}_1, \ldots, \vec{\fb}_k) = \sigma_k(\vec{\fb}'_1, \ldots, \vec{\fb}'_k)$ whenever for every $k'$, $1 \leq k' \leq k$, all of the following conditions are met:\\
\centerline{$\mu_{0,k'}\bigl( \vec{\fb}_1, \ldots, \vec{\fb}_{k'}) = \mu_{0,k'}(\vec{\fb}'_1, \ldots, \vec{\fb}'_{k'} \bigr)$,}
\centerline{$\mu_{1,k'}\bigl( \sigma_1(\vec{\fb}_1), \vec{\fb}_2, \ldots, \vec{\fb}_{k'} \bigr) = \mu_{1,k'}\bigl( \sigma_1(\vec{\fb}'_1), \vec{\fb}'_2, \ldots, \vec{\fb}'_{k'} \bigr)$,}
\centerline{$\vdots$}
\centerline{$\mu_{k'-1,k'}\bigl( \sigma_1(\vec{\fb}_1), \ldots, \sigma_{k'-1}(\vec{\fb}_1, \ldots, \vec{\fb}_{k'-1}), \vec{\fb}_{k'} \bigr) = \mu_{k'-1,k'}\bigl( \sigma_1(\vec{\fb}'_1), \ldots, \sigma_{k'-1}(\vec{\fb}'_1, \ldots, \vec{\fb}'_{k'-1}), \vec{\fb}'_{k'} \bigr)$.}
\end{definition}
Intuitively, $\mu$-uniformity of a strategy $\sigma$ means that $\sigma$ reacts in the same way on inputs that have identical fingerprints.
The next lemma provides the key argument to infer the existence of some satisfying $\mu$-uniform strategy from the existence of any satisfying strategy.
\begin{lemma}\label{lemma:UniformStrategies}
For every strategy $\sigma = \<\sigma_1, \ldots, \sigma_n\>$ there is a $\mu$-uniform strategy $\widehat{\sigma} = \< \widehat{\sigma}_1, \ldots, \widehat{\sigma}_n \>$ such that $\text{Out}_{\text{At},\widehat{\sigma}} \subseteq \text{Out}_{\text{At},\sigma}$.
\end{lemma}
\begin{proof}[Proof sketch]
For $i = 1, \ldots, n$ we define $\mathfrak{U}_i$ as abbreviation of $\mathfrak{U}_\mathcal{A}^{|\vec{\mathbf{x}}_1|} \times \ldots \times \mathfrak{U}_\mathcal{A}^{|\vec{\mathbf{x}}_i|}$.
We construct certain representatives $\alpha_{k, \<\overline{S}_0^{(k)}, \ldots, \overline{S}_{k-1}^{(k)}\>} \in \mathfrak{U}_k$ inductively as follows.
The $\overline{S}_i^{(k)}$ stand for sequences $S_{i,i+1}^{(k)} \ldots S_{i,k}^{(k)}$ of fingerprints satisfying $S_{i,k}^{(k)} \in S_{i,k-1}^{(k)} \in \ldots \in S_{i,i+1}^{(k)}$. \\
%
Let $k=1$.
We partition
$\mathfrak{U}_1$ into sets $\mathfrak{U}_{1,\<S_0^{(1)}\>}$ with $S_0^{(1)} \in \text{im}_\sigma(\mu_{0,1})$ by setting
$\mathfrak{U}_{1,\<S_0^{(1)}\>} := \bigl\{ \vec{\fb}_1 \in \mathfrak{U}_\mathcal{A}^{|\vec{\mathbf{x}}_1|} \mid \mu_{0,1}(\vec{\fb}_1) = S_0^{(1)} \bigr\}$.
We pick one representative $\alpha_{1,\<S_0^{(1)}\>}$ from each $\mathfrak{U}_{1,\<S_0^{(1)}\>}$. \\
Let $k > 1$.
We construct subsets $\mathfrak{U}_{k,\<\overline{S}_0^{(k)}, \ldots, \overline{S}_{k-1}^{(k)}\>} \subseteq \mathfrak{U}_k$ with
$S_{0,j}^{(k)} \in \text{im}_\sigma(\mu_{0,j}),
\ldots,
S_{k-1,j}^{(k)} \in \text{im}_\sigma(\mu_{k-1,j})$,
for every $j \leq k$,
by setting $\mathfrak{U}_{k,\<\overline{S}_0^{(k)}, \ldots, \overline{S}_{k-1}^{(k)}\>} :=$\\
$\bigl\{ \bigl\<\vec{\fc}_1, \ldots, \vec{\fc}_{k-1}, \vec{\fb}_k\bigr\> \bigm|\; \text{$\vec{\fb}_k \in \mathfrak{U}_\mathcal{A}^{|\vec{\mathbf{x}}_k|}$ and there is some $\alpha_{k-1,\<\overline{S}_0^{(k-1)}, \ldots, \overline{S}_{k-2}^{(k-1)}\>} = \bigl\< \vec{\fc}_1, \ldots, \vec{\fc}_{k-1} \bigr\>$}$ \\
\strut\hspace{2ex}
$\text{with $\vec{\fc}_i \in \mathfrak{U}_\mathcal{A}^{|\vec{\mathbf{x}}_i|}$, for every $i$, such that $\mu_{0,k}\bigl(\vec{\fc}_1, \ldots, \vec{\fc}_{k-1}, \vec{\fb}_k\bigr) = S_0,\; \mu_{1,k}\bigl(\sigma_1(\vec{\fc}_1), \vec{\fc}_2, \ldots,$}$ \\
\strut\hspace{2ex}
$\vec{\fc}_{k-1}, \vec{\fb}_k\bigr) = S_1, \ldots, \mu_{k-1,k}\bigl(\sigma_1(\vec{\fc}_1), \ldots, \sigma_{k-1}(\vec{\fc}_1, \ldots, \vec{\fc}_{k-1}), \vec{\fb}_k\bigr) = S_{k-1}$, and \\
\strut\hspace{2ex}
$\overline{S}_0^{(k)} = \overline{S}_0^{(k-1)} S_0;\; \ldots ;\; \overline{S}_{k-2}^{(k)} = \overline{S}_{k-2}^{(k-1)} S_{k-2};\;\; \overline{S}_{k-1}^{(k)} = S_{k-1} \bigr\}$.
We pick one representative $\alpha_{k,\<\overline{S}_0^{(k)}, \ldots, \overline{S}_{k-1}^{(k)}\>}$ from each nonempty $\mathfrak{U}_{k,\<\overline{S}_0^{(k)}, \ldots, \overline{S}_{k-1}^{(k)}\>}$.
Having all the representatives $\alpha_{k,\<\overline{S}_0^{(k)}, \ldots, \overline{S}_{k-1}^{(k)}\>}$ at hand, we inductively construct $\widehat{\sigma}$, starting from $\widehat{\sigma}_1$ and going to $\widehat{\sigma}_n$. \\
%
Let $k=1$.
For every $\vec{\fb}_1 \in \mathfrak{U}_\mathcal{A}^{|\vec{\mathbf{x}}_1|}$ we set $\widehat{\sigma}_1(\vec{\fb}_1) := \sigma_1(\alpha_{1,\<S_0\>})$, where $S_0 := \mu_{0,1}\bigl( \vec{\fb}_1 \bigr)$.\\
%
Let $k>1$.
For all tuples $\vec{\fb}_1 \in \mathfrak{U}_\mathcal{A}^{|\vec{\mathbf{x}}_1|}, \ldots, \vec{\fb}_k \in \mathfrak{U}_\mathcal{A}^{|\vec{\mathbf{x}}_k|}$ we set $\widehat{\sigma}_k(\vec{\fb}_1, \ldots, \vec{\fb}_k) := \sigma_k(\vec{\fc}_1, \ldots, \vec{\fc}_k)$, where
$\<\vec{\fc}_1, \ldots, \vec{\fc}_k\> : = \alpha_{k, \<\overline{S}_0^{(k)}, \ldots, \overline{S}_{k-1}^{(k)}\>}$ with $\vec{\fc}_i \in \mathfrak{U}_\mathcal{A}^{|\vec{\mathbf{x}}_i|}$, for every $i$,
and we have \\
$S_{0,j}^{(k)} = \mu_{0,j}\bigl(\vec{\fb}_1, \ldots, \vec{\fb}_j\bigr)$ for every $j$, $0 < j \leq k$, \\
$S_{1,j}^{(k)} = \mu_{1,j}\bigl(\widehat{\sigma}_1(\vec{\fb}_1), \vec{\fb}_2, \ldots, \vec{\fb}_j\bigr)$ for every $j$, $1 < j \leq k$, \\[-1ex]
\strut\hspace{4ex} $\vdots$ \\[-1ex]
$S_{k-2,j}^{(k)} = \mu_{k-2,j}\bigl(\widehat{\sigma}_1(\vec{\fb}_1), \ldots, \widehat{\sigma}_{k-2}(\vec{\fb}_1, \ldots, \vec{\fb}_{k-2}), \vec{\fb}_{k-1}, \ldots, \vec{\fb}_j \bigr)$ for every $j$, $k-2 < j \leq k$, \\
$S_{k-1,k}^{(k)} = \mu_{k-1,k}\bigl(\widehat{\sigma}_1(\vec{\fb}_1), \ldots, \widehat{\sigma}_{k-1}(\vec{\fb}_1, \ldots, \vec{\fb}_{k-1}), \vec{\fb}_k \bigr)$,\\
if such an $\alpha_{k, \<\overline{S}_0^{(k)}, \ldots, \overline{S}_{k-1}^{(k)}\>}$ exists.
\begin{description}
\item\underline{Claim I:} For every $k$, $1 \leq k \leq n$, and all tuples $\vec{\fb}_1 \in \mathfrak{U}_\mathcal{A}^{|\vec{\mathbf{x}}_1|}, \ldots, \vec{\fb}_k \in \mathfrak{U}_\mathcal{A}^{|\vec{\mathbf{x}}_k|}$ there is a representative $\alpha_{k, \<\overline{S}_0, \ldots, \overline{S}_{k-1}\>}$ such that \\
$S_{0,j} = \mu_{0,j}\bigl( \vec{\fb}_1, \ldots, \vec{\fb}_{j} \bigr)$ for every $j$, $0 < j \leq k$, \\
$S_{1,j} = \mu_{1,j}\bigl(\widehat{\sigma}_1(\vec{\fb}_1), \vec{\fb}_2, \ldots, \vec{\fb}_{j}\bigr)$ for every $j$, $1 < j \leq k$, \\[-1ex]
\strut\hspace{4ex} $\vdots$ \\[-1ex]
$S_{k-2,j} = \mu_{k-2,j}\bigl(\widehat{\sigma}_1(\vec{\fb}_1), \ldots, \widehat{\sigma}_{k-2}(\vec{\fb}_1, \ldots, \vec{\fb}_{k-2}), \vec{\fb}_{k-1}, \ldots, \vec{\fb}_j \bigr)$ for every $j$, $k-2 < j \leq k$, \\
$S_{k-1,k} = \mu_{k-1,k}\bigl( \widehat{\sigma}_1(\vec{\fb}_1), \ldots, \widehat{\sigma}_{k-1}(\vec{\fb}_1, \ldots, \vec{\fb}_{k-1}), \vec{\fb}_k \bigr)$.
\item\underline{Proof:} We proceed by induction on $k$. Details can be found in the appendix. \hfill$\Diamond$
\end{description}
By construction, $\widehat{\sigma}$ is $\mu$-uniform.
Now let $S \in \text{Out}_{\text{At},\widehat{\sigma}}$, i.e.\ there exist tuples $\vec{\fb}_1 \in \mathfrak{U}_\mathcal{A}^{|\vec{\mathbf{x}}_1|}, \ldots, \vec{\fb}_n \in \mathfrak{U}_\mathcal{A}^{|\vec{\mathbf{x}}_n|}$ such that
$S = \text{out}_{\text{At},\widehat{\sigma}}(\vec{\fb}_1, \ldots, \vec{\fb}_n)$.
%
We partition $S$ into sets $S_0 := S \cap \text{At}_0, \ldots, S_n := S \cap \text{At}_n$ and thus obtain the fingerprints
$S_\ell = \mu_{\ell,n}\bigl(\widehat{\sigma}_{1}(\vec{\fb}_1), \ldots, \widehat{\sigma}_\ell(\vec{\fb}_1, \ldots, \vec{\fb}_\ell), \vec{\fb}_{\ell+1}, \ldots, \vec{\fb}_n \bigr) \subseteq \text{At}_\ell$ for every $\ell$, $0 \leq \ell < n$.
Claim I guarantees the existence of some representative $\alpha_{n,\<\overline{S}'_0, \ldots, \overline{S}'_{n-1}\>} = \<\vec{\fc}_1, \ldots, \vec{\fc}_n\>$, with $\vec{\fc}_i \in \mathfrak{U}_\mathcal{A}^{|\vec{\mathbf{x}}_i|}$, for every $i$, such that
$S_\ell = \mu_{\ell,n}\bigl(\sigma_{1}(\vec{\fc}_1), \ldots, \sigma_\ell(\vec{\fc}_1, \ldots, \vec{\fc}_\ell), \vec{\fc}_{\ell+1}, \ldots, \vec{\fc}_n \bigr)$ for every $\ell$, $0 \leq \ell < n$.
Consider any $A \in \text{At}$, and fix the $\ell$ for which $A \in \text{At}_\ell$.
We distinguish two cases.\\
%
Suppose that $\ell < n$.
The definition of $\alpha_{n, \<\overline{S}'_0, \ldots, \overline{S}'_{n-1}\>}$ and the fingerprint functions $\mu_{\ell, n}$ entail that $A \in S_\ell$ if and only if
$\mathcal{A},[\vec{\mathbf{y}}_1 {\mathop{\mapsto}} \widehat{\sigma}_1(\vec{\fb}_1), \ldots, \vec{\mathbf{y}}_\ell {\mathop{\mapsto}} \widehat{\sigma}_\ell(\vec{\fb}_1, \ldots, \vec{\fb}_{\ell}), \vec{\mathbf{x}}_{\ell+1} {\mathop{\mapsto}} \vec{\fb}_{\ell+1}, \ldots, \vec{\mathbf{x}}_n {\mathop{\mapsto}} \vec{\fb}_n] \models A$
if and only if
$\mathcal{A},[\vec{\mathbf{y}}_1 {\mathop{\mapsto}} \sigma_1(\vec{\fc}_1), \ldots, \vec{\mathbf{y}}_\ell {\mathop{\mapsto}} \sigma_\ell(\vec{\fc}_1, \ldots, \vec{\fc}_{\ell}), \vec{\mathbf{x}}_{\ell+1} {\mathop{\mapsto}} \vec{\fc}_{\ell+1}, \ldots, \vec{\mathbf{x}}_n {\mathop{\mapsto}} \vec{\fc}_n] \models A$.\\
In case of $\ell = n$, we have $A \in S_n$ if and only if
$\mathcal{A},[\vec{\mathbf{y}}_1 {\mathop{\mapsto}} \widehat{\sigma}_1(\vec{\fb}_1), \ldots, \vec{\mathbf{y}}_n {\mathop{\mapsto}} \widehat{\sigma}_n(\vec{\fb}_1, \ldots, \vec{\fb}_n)] \models A$
if and only if
$\mathcal{A},[\vec{\mathbf{y}}_1 {\mathop{\mapsto}} \sigma_1(\vec{\fc}_1), \ldots, \vec{\mathbf{y}}_n {\mathop{\mapsto}} \sigma_n(\vec{\fc}_1, \ldots, \vec{\fc}_n)] \models A$.
In both cases, we get $A \in \text{out}_{\text{At},\widehat{\sigma}}(\vec{\fb}_1, \ldots, \vec{\fb}_n)$ if and only if $A \in \text{out}_{\text{At},\sigma}(\vec{\fc}_1, \ldots, \vec{\fc}_n)$.
%
Consequently, we have $S = \text{out}_{\text{At},\widehat{\sigma}}(\vec{\fb}_1, \ldots, \vec{\fb}_n) = \text{out}_{\text{At},\sigma}(\vec{\fc}_1, \ldots,$ $\vec{\fc}_n) \in \text{Out}_{\text{At}, \sigma}$.
Altogether, it follows that $\text{Out}_{\text{At}, \widehat{\sigma}} \subseteq \text{Out}_{\text{At}, \sigma}$.
\end{proof}
\begin{corollary}\label{corollary:UniformStrategies}
If there is a satisfying strategy $\sigma$ for $\varphi$, then there is also a $\mu$-uniform strategy $\widehat{\sigma}$ that is satisfying for $\varphi$.
\end{corollary}
\begin{proof}
Let $\sigma$ be a satisfying strategy for $\varphi$.
By Lemma~\ref{lemma:UniformStrategies}, there is a $\mu$-uniform strategy $\widehat{\sigma}$ such that for every $S \in \text{Out}_{\text{At},\widehat{\sigma}}$ we have $S \in \text{Out}_{\text{At},\sigma}$.
Since $\sigma$ is satisfying for $\varphi$, we get $S \models \psi_{\text{bool}}$ for every $S \in \text{Out}_{\text{At},\sigma}$.
Hence, we observe $S \models \psi_{\text{bool}}$ for every $S \in \text{Out}_{\text{At},\widehat{\sigma}} \subseteq \text{Out}_{\text{At},\sigma}$.
This means that $\widehat{\sigma}$ is also satisfying for $\varphi$.
\end{proof}
In order to derive a small model property for GBSR sentences, it remains to show that any satisfying $\mu$-uniform strategy induces a finite substructure of $\mathcal{A}$ that satisfies $\varphi$.
In what follows, we denote by $\kappa$ the smallest positive integer meeting the following condition.
For every $\text{At}_i$, $0 \leq i < n$, there are at most $\kappa$ distinct indices $i+1 < j_1 < \ldots < j_\kappa \leq n$ such that $\vec{\mathbf{x}}_{j_\ell} \cap \text{\upshape{vars}}(\text{At}_i) \neq \emptyset$.
We call $\kappa$ the \emph{degree} of $\varphi$ and write $\partial_\varphi = \kappa$.
Notice that we have $0 \leq \kappa < n$.
To formulate the upper bound on the size of the domain, we need some notation for the \emph{tetration operation}.
We define $\twoup{k}{m}$ inductively: $\twoup{0}{m} := m$ and $\twoup{k+1}{m} := 2^{\left(\twoup{k}{m}\right)}$.
\begin{lemma}\label{lemma:FiniteModelsSemanticArgumentsGBSR}
If there is a satisfying $\mu$-uniform strategy $\sigma$ for $\varphi$, then there is a model $\mathcal{B} \models \varphi$ such that $\mathfrak{U}_\mathcal{B}$ contains at most $n \cdot |\vec{\mathbf{y}}| \cdot \bigl( \twoup{\kappa+1}{|\text{\upshape{\text{At}}}|} \bigr)^{n^2}$ domain elements.
\end{lemma}
\begin{proof}[Proof sketch]~
One can derive the following observations for all integers $\ell, k$, $0 \leq \ell < k < n$:
(a) we have $|\text{im}_\sigma(\mu_{\ell,k})| \leq 2^{|\text{im}_\sigma(\mu_{\ell,k+1})|}$, and
(b) for all tuples $\vec{\fa}_1, \ldots, \vec{\fa}_\ell, \vec{\fb}_{\ell+1}, \ldots, \vec{\fb}_k$ with $\vec{\fa}_i \in \mathfrak{U}_\mathcal{A}^{|\vec{\mathbf{y}}_i|}$ and $\vec{\fb}_i \in \mathfrak{U}_\mathcal{A}^{|\vec{\mathbf{x}}_i|}$, for every $i$, we observe that, if $\text{\upshape{vars}}(\text{At}_\ell) \cap \vec{\mathbf{x}}_{k+1} = \emptyset$, then
$\bigl| \mu_{\ell,k}(\vec{\fa}_1, \ldots, \vec{\fa}_\ell, \vec{\fb}_{\ell+1}, \ldots, \vec{\fb}_k) \bigr|$ $= 1$
and, consequently,
$|\text{im}_\sigma(\mu_{\ell,k})| \leq |\text{im}_\sigma(\mu_{\ell,k+1})|$.
Due to (a) and (b), our assumptions about $\kappa$ entail that \\
(c) for all integers $\ell, k$ with $0 \leq \ell < k \leq n$ we have $|\text{im}_\sigma(\mu_{\ell,k})| \leq \twoup{\kappa+1}{\text{At}_\ell}$.
Let $\mathcal{T}_\sigma$ be the \emph{target set} of $\sigma$, defined by $\mathcal{T}_\sigma := \bigcup_{k = 1}^n \mathcal{T}_k$, where \\
$\mathcal{T}_k := \bigl\{ \mathfrak{a} \in \mathfrak{U}_\mathcal{A} \bigm|\; \text{there are tuples $\vec{\fb}_1, \ldots, \vec{\fb}_k$ with $\vec{\fb}_i \in \mathfrak{U}_\mathcal{A}^{|\vec{\mathbf{x}}_i|}$, for every $i$,}$ \\
\strut\hspace{17ex}
$\text{such that $\sigma_k(\vec{\fb}_1, \ldots, \vec{\fb}_k) = \<\ldots, \mathfrak{a}, \ldots\>$} \bigr\}$. \\
Since $\sigma$ is $\mu$-uniform, $\mathcal{T}_\sigma$ is a finite set.
By definition of the fingerprint functions $\mu_{\ell, k}$, and by virtue of (c), we derive the following upper bounds, where we write $\widehat{\im}_\sigma(\mu_{i,j})$ to abbreviate $\text{im}_\sigma(\mu_{i,i+1}) \times \text{im}_\sigma(\mu_{i,i+2}) \times \ldots \times \text{im}_\sigma(\mu_{i,j})$ for all $i,j$, $0\leq i < j \leq n$.\\
\strut\hspace{1ex}\!
$\bigl| \mathcal{T}_1 \bigr| \;\;\leq\;\; |\vec{\mathbf{y}}_1| \cdot \bigl|\widehat{\im}_\sigma(\mu_{0,1})\bigr| \;\;\leq\;\; |\vec{\mathbf{y}}_1| \cdot \twoup{\kappa+1}{|\text{At}|}$, \\
\strut\hspace{1ex}
$\bigl| \mathcal{T}_2 \bigr| \;\;\leq\;\; |\vec{\mathbf{y}}_2| \cdot \bigl|\widehat{\im}_\sigma(\mu_{0,2}) \times \widehat{\im}_\sigma(\mu_{1,2})\bigr| \;\;\leq\;\; |\vec{\mathbf{y}}_2| \cdot \bigl( \twoup{\kappa+1}{|\text{At}|} \bigr)^3$, \\[-1ex]
\strut\hspace{8ex}
$\vdots$ \\[-1ex]
\strut\hspace{1ex}
$\bigl| \mathcal{T}_n \bigr| \;\;\leq\;\; |\vec{\mathbf{y}}_n| \cdot \bigl|\widehat{\im}_\sigma(\mu_{0,n}) \times \ldots \times \widehat{\im}_\sigma(\mu_{n-1,n})\bigr| \;\;\leq\;\; |\vec{\mathbf{y}}_n| \cdot \bigl( \twoup{\kappa+1}{|\text{At}|} \bigr)^{n^2}$.\\
Consequently, $\mathcal{T}_\sigma$ contains at most $n \cdot |\vec{\mathbf{y}}| \cdot \bigl( \twoup{\kappa+1}{|\text{At}|} \bigr)^{n^2}$ domain elements.
Let $\varphi_\text{Sk}$ be the result of exhaustive Skolemization of $\varphi$, i.e.\ every existentially quantified variable $y \in \vec{\mathbf{y}}_k$ in $\varphi$ is replaced by the Skolem function $f_y(\vec{\mathbf{x}}_1, \ldots, \vec{\mathbf{x}}_k)$.
Clearly, $\sigma$ induces interpretations for all the Skolem functions $f_y$ such that $\mathcal{A}$ can be extended to a model $\mathcal{A}'$ of $\varphi_\text{Sk}$ for which we have $f_y^{\mathcal{A}'}(\vec{\fb}_1, \ldots, \vec{\fb}_k) \in \mathcal{T}_k$ for every $f_y$ with $y \in \vec{\mathbf{y}}_k$ and all tuples $\vec{\fb}_1, \ldots, \vec{\fb}_k$.
We define $\mathcal{B}$ to be the substructure of $\mathcal{A}'$ (with respect to $\varphi_\text{Sk}$'s vocabulary) with universe $\mathfrak{U}_\mathcal{B} := \mathcal{T}_\sigma$.
By the Substructure Lemma, $\mathcal{B}$ satisfies $\varphi_\text{Sk}$ and thus also the original $\varphi$.
Moreover, we can bound the number of elements in $\mathcal{B}$'s domain from above by $n \cdot |\vec{\mathbf{y}}| \cdot \bigl( \twoup{\kappa+1}{|\text{At}|} \bigr)^{n^2}$.
\end{proof}
\begin{theorem}\label{theorem:GBSRsmallModelProperty}
Every satisfiable GBSR sentence $\varphi := \forall \vec{\mathbf{x}}_1 \exists \vec{\mathbf{y}}_1 \ldots \forall \vec{\mathbf{x}}_n \exists \vec{\mathbf{y}}_n.\psi$ in standard form with quantifier-free $\psi$ and degree $\partial_\varphi \geq 1$ has a model with at most $\text{len}(\varphi)^2 \cdot \bigl( \twoup{\partial_\varphi+1}{\text{len}(\varphi)} \bigr)^{n^2}$ domain elements.
\end{theorem}
Note that the notion of degree used in this paper depends on a fixed partition of $\text{At}$ into the sets $\text{At}_0, \ldots, \text{At}_n$.
Consider, for instance, a relational monadic formula $\varphi_\text{Mon}$.
We could partition its atoms into two parts $\text{At}_0, \text{At}_n$, where $\text{At}_0$ contains all atoms that contain a universally quantified variable and $\text{At}_n$ comprises all atoms with existentially quantified variables.
Clearly, $\text{At}_0$ will cause the highest possible degree for $\varphi_\text{Mon}$, since all universally quantified variables occur in atoms in $\text{At}_0$.
To obtain a lower degree, we could partition $\text{At}$ as follows.
For every $i$, $0 \leq i < n$, we set $\text{At}_i := \bigl\{ P(x) \in \text{At} \bigm| x \in \vec{\mathbf{x}}_{i+1} \bigr\}$, and the set $\text{At}_n$ again contains all atoms with existentially quantified variables.
Clearly, this partition induces a degree $\partial_{\varphi_\text{Mon}} = 0$ and thus a potentially much lower degree.
\section{The generalized Ackermann fragment}\label{section:GeneralizedAckermann}
In this section we generalize the Ackermann fragment (relational $\exists^* \forall \exists^*$ sentences without equality) in the same spirit as we have generalized the BSR fragment in Section~\ref{section:GeneralizedBSR}. We even go farther than Ackermann's original result, and, doing so, diverge in two directions:
In one direction we allow for unary function symbols to appear in formulas, even in a nested fashion, but do not allow equality.
In the other direction we allow equality but no non-constant function symbols.
In the former case, we devise an effective (un)satisfiability-preserving translation from the new fragment into the monadic fragment with unary function symbols.
In the latter case, we employ a result due to Ferm\"uller and Salzer~\cite{Fermuller1993b} to argue decidability of the satisfiability problem.
For the remainder of this section we fix a sentence $\varphi := \forall \vec{\mathbf{x}}_1 \exists \vec{\mathbf{y}}_1 \ldots \forall \vec{\mathbf{x}}_n \exists \vec{\mathbf{y}}_n. \psi$ in standard form without equality and without non-constant function symbols.
Before Lemma~\ref{lemma:GAFSkolemizedProperties} we do not have to pose any restrictions on the function symbols occurring in $\varphi$, and even the equality predicate would not do any harm.
Still, for the sake of clear definitions, we add the restrictions here and soften them where appropriate.
Let $\text{At}$ be the set of all atoms occurring in $\varphi$ and let $\vec{\mathbf{x}} := \vec{\mathbf{x}}_1 \cup \ldots \cup \vec{\mathbf{x}}_n$ and $\vec{\mathbf{y}} := \vec{\mathbf{y}}_1 \cup \ldots \cup \vec{\mathbf{y}}_n$.
We define the \emph{index of a variable $v \in \vec{\mathbf{x}} \cup \vec{\mathbf{y}}$} by $\text{idx}(v) := k$ if and only if $v \in \vec{\mathbf{x}}_k$ or $v \in \vec{\mathbf{y}}_k$.
\begin{definition}[GAF, axiomatically]\label{definition:GAFaxiomatic}
The sentence $\varphi$ belongs to the \emph{generalized Ackermann fragment (GAF)} if and only if
we can partition $\text{At}$ into sets $\text{At}_0$ and $\text{At}_x$, $x \in \vec{\mathbf{x}}$, such that the following conditions are met.
\begin{enumerate}[label=(\alph{*}), ref=\alph{*}]
\item\label{enum:definitionGAFaxiomatic:I} $\text{\upshape{vars}}(\text{At}_0) \cap \vec{\mathbf{x}} = \emptyset$.
\item\label{enum:definitionGAFaxiomatic:II} For every $x \in \vec{\mathbf{x}}$ we have $\text{\upshape{vars}}(\text{At}_x) \cap \vec{\mathbf{x}} = \{x\}$.
\item\label{enum:definitionGAFaxiomatic:III} For every $y \in \vec{\mathbf{y}}$ occurring in some $\text{At}_x$ there are two (mutually exclusive) options
\begin{enumerate}[label=(\ref{enum:definitionGAFaxiomatic:III}.\arabic{*}), ref=(\ref{enum:definitionGAFaxiomatic:III}.\arabic{*})]
\item\label{enum:definitionGAFaxiomatic:III:I} either for every $\text{At}_x$ in which $y$ occurs we have $\text{idx}(y) < \text{idx}(x)$,
\item\label{enum:definitionGAFaxiomatic:III:II} or there is exactly one $\text{At}_x$ in which $y$ occurs and $y$ does not occur in $\text{At}_0$ and we have $\text{idx}(y) \geq \text{idx}(x)$.
\end{enumerate}
\end{enumerate}
\end{definition}
Intuitively speaking, Conditions~(\ref{enum:definitionGAFaxiomatic:II}) and~(\ref{enum:definitionGAFaxiomatic:III}) entail that, although a quantifier $\exists y$ may lie within the scope of two different quantifiers $\forall x$ and $\forall x'$ in $\varphi$, we can move $\exists y$ out of the scope of at least one of $\forall x$ and $\forall x'$ by suitable equivalence-preserving transformations.
This will be the essence of the proof of Lemma~\ref{lemma:GAFTransformation}.
In Example~\ref{example:GAFandGBSR} we have sketched the transformation procedure for the GAF sentence $\varphi_1$.
As in the case of GBSR, we complement the axiomatic definition of GAF with an alternative definition of an algorithmic flavor.
We shall see in Lemma~\ref{lemma:AlternativeDefinitionOfGAF} that both definitions are equivalent.
For the algorithmic point of view we need additional notation.
Let $\mathcal{G}_\varphi := \<V,E\>$ be a directed graph such that $V := \vec{\mathbf{y}}$ and
$E := \bigl\{ \<y,y'\> \bigm| \text{idx}(y) \leq \text{idx}(y') \text{ and there is an atom $A$ in $\psi$ in which both $y$ and $y'$ occur}\bigr\}$.
For any variable $y \in \vec{\mathbf{y}}$ the \emph{upward closure} $\Cup_y$ is the smallest subset of $\vec{\mathbf{y}}$ such that $y \in \Cup_y$ and for every $y' \in \Cup_y$ the existence of an edge $\<y', y''\>$ in $\mathcal{G}_\varphi$ entails $y'' \in \Cup_y$.
$\mathcal{L}(\Cup_y)$ denotes the set of all literals in $\psi$, in which a variable from $\Cup_y$ occurs.
For any $x \in \vec{\mathbf{x}}$ let $\mathcal{L}_x$ be the smallest set of literals such that
(a) every literal in $\psi$ in which $x$ occurs belongs to $\mathcal{L}_x$, and
(b) for every $y \in \text{\upshape{vars}}(\mathcal{L}_x) \cap \vec{\mathbf{y}}$ with $\text{idx}(y) \geq \text{idx}(x)$ we have $\mathcal{L}(\Cup_y) \subseteq \mathcal{L}_x$.
Intuitively, $\mathcal{L}_x$ collects all literals from $\psi$ which will remain in the scope of the quantifier $\forall x$ when we apply the laws of Boolean algebra and the rules of miniscoping to $\varphi$.
By $\mathcal{L}_0$ we denote the set of all literals that occur in $\psi$ but in none of the $\mathcal{L}_x$ with $x \in \vec{\mathbf{x}}$.
Moreover, we use the notation $Y_x := \text{\upshape{vars}}(\mathcal{L}_x) \cap \bigcup_{i\geq \text{idx}(x)} \vec{\mathbf{y}}_i$.
\begin{definition}[GAF, algorithmically]\label{definition:GAFalgorithmic}
The sentence $\varphi$ belongs to GAF if and only if
\begin{enumerate}[label=(\roman{*}), ref=(\roman{*})]
\item\label{enum:definitionGAFalgorithmic:II} every atom in $\psi$ contains at most one variable from $\vec{\mathbf{x}}$, and
\item\label{enum:definitionGAFalgorithmic:III} for all distinct variables $x, x' \in \vec{\mathbf{x}}$ with $\text{idx}(x) \leq \text{idx}(x')$ and any variable $y \in \bigcup_{i\geq \text{idx}(x)} \vec{\mathbf{y}}_i$ it holds $y \not\in \text{\upshape{vars}}(\mathcal{L}_x) \cap \text{\upshape{vars}}(\mathcal{L}_{x'})$.
\end{enumerate}
\end{definition}
\begin{lemma}\label{lemma:GAFPropertiesOne}
If $\varphi$ satisfies Definition~\ref{definition:GAFalgorithmic}, then the following properties hold.
\begin{enumerate}[label=(\roman{*}), ref=(\roman{*})]
\item\label{enum:GAFPropertiesOne:I}
For all distinct $x,x' \in \vec{\mathbf{x}}$ it holds $\mathcal{L}_x \cap \mathcal{L}_{x'} = \emptyset$.
\item\label{enum:GAFPropertiesOne:II}
For every $x \in \vec{\mathbf{x}}$ it holds $\text{\upshape{vars}}(\mathcal{L}_x) \cap \vec{\mathbf{x}} = \{x\}$.
\item\label{enum:GAFPropertiesOne:III}
For every $x \in \vec{\mathbf{x}}$ we have $Y_x \cap \text{\upshape{vars}}(\mathcal{L}_0) = \emptyset$.
\item\label{enum:GAFPropertiesOne:IV}
For all distinct $x, x' \in \vec{\mathbf{x}}$ with $\text{idx}(x) \leq \text{idx}(x')$ we have $Y_x \cap (\text{\upshape{vars}}(\mathcal{L}_{x'})) = \emptyset$.
\end{enumerate}
\end{lemma}
\begin{proof}~
\begin{description}
\item{Ad \ref{enum:GAFPropertiesOne:I}:}
Suppose there are variables $x, x' \in \vec{\mathbf{x}}$ and there is a literal $L \in \mathcal{L}_x \cap \mathcal{L}_{x'}$.
$L$ must belong to $\mathcal{L}(\Cup_y)$ for some variable $y \in \vec{\mathbf{y}}$ with $\text{idx}(y) \geq \text{idx}(x)$ or $\text{idx}(y) \geq \text{idx}(x')$, since otherwise it would hold $\{x,x'\} \subseteq \text{\upshape{vars}}(L)$ which contradicts part \ref{enum:definitionGAFalgorithmic:II} of Definition~\ref{definition:GAFalgorithmic}.
This in turn means that some variable $y' \in \vec{\mathbf{y}}$ occurs in $L$ for which $\text{idx}(y') \geq \text{idx}(y)$.
Hence, $y' \in \text{\upshape{vars}}(L) \subseteq \bigl(\text{\upshape{vars}}(\mathcal{L}_x) \cap \text{\upshape{vars}}(\mathcal{L}_{x'})\bigr)$ with $\text{idx}(y') \geq \text{idx}(x)$ or $\text{idx}(y') \geq \text{idx}(x')$.
This constitutes a contradiction to the second part of Definition~\ref{definition:GAFalgorithmic}.
\item{Ad \ref{enum:GAFPropertiesOne:II}:}
This is a direct consequence of \ref{enum:GAFPropertiesOne:I} and the definition of $\mathcal{L}_x$.
\item{Ad \ref{enum:GAFPropertiesOne:III}:}
From $y \in Y_x \subseteq \bigcup_{i\geq \text{idx}(x)} \vec{\mathbf{y}}_i$ it follows that $\mathcal{L}(\Cup_y) \subseteq \mathcal{L}_x$, by definition of $\mathcal{L}_x$.
Suppose there is a $y \in Y_x \cap \text{\upshape{vars}}(\mathcal{L}_0)$, i.e.\ there is some $L \in \mathcal{L}_0$ with $y \in \text{\upshape{vars}}(L)$.
Since $L \in \mathcal{L}(\Cup_y) \subseteq \mathcal{L}_x$, $L$ cannot occur in $\mathcal{L}_0$, by definition of $\mathcal{L}_0$.
\item{Ad \ref{enum:GAFPropertiesOne:IV}:}
From $y \in Y_x \subseteq \bigcup_{i\geq \text{idx}(x)} \vec{\mathbf{y}}_i$ it follows that $\mathcal{L}(\Cup_y) \subseteq \mathcal{L}_x$, by definition of $\mathcal{L}_x$.
Suppose there is some $y \in Y_x \cap \text{\upshape{vars}}(\mathcal{L}_{x'})$. Hence, there must be some literal $L \in \mathcal{L}_{x'}$ in which $y$ occurs. But since $L$ belongs to $\mathcal{L}(\Cup_y)$, we know that $L \in \mathcal{L}_x$. This contradicts~\ref{enum:GAFPropertiesOne:I}.
\qedhere
\end{description}
\end{proof}
We now have the right notions at hand to show equivalence of Definitions~\ref{definition:GAFaxiomatic} and~\ref{definition:GAFalgorithmic}.
\begin{lemma}\label{lemma:AlternativeDefinitionOfGAF}
The sentence $\varphi$ satisfies Definition~\ref{definition:GAFaxiomatic} if and only if it satisfies Definition~\ref{definition:GAFalgorithmic}.
\end{lemma}
\begin{proof}
Regarding the \emph{if}-direction, we define $\text{At}_0$ to be the set of all atoms in $\mathcal{L}_0$ and, analogously, for every $x \in \vec{\mathbf{x}}$ we define $\text{At}_x$ to be the set of atoms in $\mathcal{L}_x$.
Condition~(\ref{enum:definitionGAFaxiomatic:I}) of Definition~\ref{definition:GAFaxiomatic} holds due to the definition of $\mathcal{L}_0$ and the $\mathcal{L}_x$.
%
(\ref{enum:definitionGAFaxiomatic:II}) is an immediate consequence of Lemma~\ref{lemma:GAFPropertiesOne}\ref{enum:GAFPropertiesOne:II}.
%
(\ref{enum:definitionGAFaxiomatic:III}) is entailed by Lemma~\ref{lemma:GAFPropertiesOne} \ref{enum:GAFPropertiesOne:III} together with \ref{enum:GAFPropertiesOne:IV}.
More precisely, we observe that, if a variable $y$ occurs in a set $Y_x$, then condition \ref{enum:definitionGAFaxiomatic:III:II} applies.
If, on the other hand, $y$ does not belong to any $Y_x$, then it satisfies condition \ref{enum:definitionGAFaxiomatic:III:I}.
For the \emph{only if}-direction we argue as follows.
Condition \ref{enum:definitionGAFalgorithmic:II} of Definition~\ref{definition:GAFalgorithmic} is certainly satisfied, if Condition~(\ref{enum:definitionGAFaxiomatic:I}) of Definition~\ref{definition:GAFaxiomatic} is met by $\varphi$.
Consider any variable $y \in \vec{\mathbf{y}}$ to which Option \ref{enum:definitionGAFaxiomatic:III:II} applies and
let $\text{At}_{x_y}$ be the set in which $y$ occurs.
By definition of upward closure, every variable $y'$ in $\Cup$ must also be an Option-\ref{enum:definitionGAFaxiomatic:III:II} variable that occurs exclusively in $\text{At}_{x_y}$.
Hence, we conclude $\Cup_y \subseteq \text{\upshape{vars}}(\text{At}_{x_y})$.
This in turn entails that every atom occurring in $\mathcal{L}(\Cup_y)$ belongs to $\text{At}_{x_y}$.
Consequently, for every $x \in \vec{\mathbf{x}}$ and for every literal $[\neg]A$ in $\mathcal{L}_x$ we have $A \in \text{At}_x$.
Now consider two distinct variables $x, x' \in \vec{\mathbf{x}}$ with $\text{idx}(x) \leq \text{idx}(x')$ and let $y$ be some variable in $Y_x$.
Since $y$ occurs in $\mathcal{L}_x$ and thus also in $\text{At}_x$, and since $\text{idx}(y) \geq \text{idx}(x)$, $y$ must be an Option-\ref{enum:definitionGAFaxiomatic:III:II} variable.
Hence, it does not occur in any atom in $\text{At}_{x'}$.
And, due to the observation that for every literal $[\neg]A \in \mathcal{L}_{x'}$ it also holds $A \in \text{At}_{x'}$, $y$ cannot occur in any literal in $\mathcal{L}_{x'}$.
This shows that $\varphi$ satisfies Condition~\ref{enum:definitionGAFalgorithmic:III} of Definition~\ref{definition:GAFalgorithmic}.
\end{proof}
The name \emph{generalized} Ackermann fragment suggests that it properly contains the Ackermann fragment.
This is confirmed by the next proposition.
But we also observe that the relational monadic fragment without equality is a proper subfragment of GAF.
Since neither the Ackermann fragment contains the monadic fragment nor vice versa, it is immediately clear that GAF constitutes a proper extension of both. Moreover, the sentence $\varphi_1$ treated in Example~\ref{example:GAFandGBSR} belongs to GAF but lies in neither of the two other fragments.
\begin{proposition}\label{proposition:InclusionGAF}
GAF properly contains the Ackermann fragment and the monadic first-order fragment, both without equality and non-constant function symbols.
\end{proposition}
\begin{proof}
Let $\varphi' := \exists \vec{\mathbf{y}}\, \forall x \exists \vec{\mathbf{z}}. \psi'$ be an Ackermann sentence in standard form, which does neither contain equality nor any non-constant function symbols.
Any atom in $\varphi'$ contains at most one universally quantified variable, namely $x$.
Let $\text{At}_x$ be the set of all atoms occurring in $\varphi'$.
Condition~(\ref{enum:definitionGAFaxiomatic:II}) of Definition~\ref{definition:GAFaxiomatic} is satisfied by $\text{At}_x$.
Moreover, every $y \in \vec{\mathbf{y}}$ is a $\ref{enum:definitionGAFaxiomatic:III:I}$ variable and every $z \in \vec{\mathbf{z}}$ is a $\ref{enum:definitionGAFaxiomatic:III:II}$ variable.
%
Consequently, $\varphi'$ belongs to GAF.
Let $\varphi'' := \forall \vec{\mathbf{x}}_1 \exists \vec{\mathbf{y}}_1 \ldots \forall \vec{\mathbf{x}}_n \exists \vec{\mathbf{y}}_n.\psi''$ be a monadic first-order sentence without equality that is in standard form.
%
For every $x \in \vec{\mathbf{x}}$ define $\text{At}_x$ to be the set containing exactly the atoms in $\varphi''$ that contain $x$.
Let $\text{At}_0$ be the set of all atoms in $\varphi''$ that do not belong to any $\text{At}_x$.
Clearly, this partition of $\varphi''$'s atoms meets all the conditions posed in Definition~\ref{definition:GAFaxiomatic}.
Hence, $\varphi''$ belongs to GAF.
\end{proof}
We shall now work towards showing that the satisfiability problem for GAF sentences (\emph{GAF-satisfiability}) is decidable, even if we extend it with equality or unary function symbols.
But first, we need additional notation.
For every $x \in \vec{\mathbf{x}}$ we refine the set $\mathcal{L}_x$ into subsets $\mathcal{L}_{x,0}, \mathcal{L}_{x,\text{idx}(x)}, \mathcal{L}_{x,\text{idx}(x)+1},\ldots, \mathcal{L}_{x,n}$: \\
\begin{tabular}{r@{\hspace{1ex}}c@{\hspace{1ex}}l}
$\mathcal{L}_{x,n}$ &$:=$ &$\bigcup_{y \in \text{\upshape{vars}}(\mathcal{L}_x) \cap \vec{\mathbf{y}}_n} \mathcal{L}(\Cup_y)$,\\
$\mathcal{L}_{x,k}$ &$:=$ &$\bigcup_{y \in \text{\upshape{vars}}(\mathcal{L}_x) \cap \vec{\mathbf{y}}_{k}} \mathcal{L}(\Cup_y) \setminus \bigcup_{\ell > k} \mathcal{L}_{x,\ell}$ for every $k$ satisfying $\text{idx}(x) \leq k < n$, and\\
$\mathcal{L}_{x,0}$ &$:=$ &$\mathcal{L}_x \setminus \bigcup_{\ell \geq \text{idx}(x)} \mathcal{L}_{x,\ell}$.
\end{tabular}\\
Similarly, we define $Y_{x,k} := \text{\upshape{vars}}(\mathcal{L}_{x,k}) \cap Y_x$ for every $k$, $0\leq k\leq n$.
\begin{lemma}\label{lemma:GAFPropertiesTwo}
If $\varphi$ belongs to GAF, then the following properties hold for every $x \in \vec{\mathbf{x}}$.
\begin{enumerate}[label=(\roman{*}), ref=(\roman{*})]
\item\label{enum:GAFPropertiesTwo:I}
For every $k$ we have $\mathcal{L}_{x,k} \subseteq \mathcal{L}_x$.
\item\label{enum:GAFPropertiesTwo:II}
For all distinct $k, \ell$ we have $\mathcal{L}_{x,k} \cap \mathcal{L}_{x,\ell} = \emptyset$.
\item\label{enum:GAFPropertiesTwo:IV}
For every $k > 0$ it holds $\text{\upshape{vars}}(\mathcal{L}_{x,k}) \cap \vec{\mathbf{y}} \subseteq \bigcup_{i \leq k} \vec{\mathbf{y}}_i$.
\item\label{enum:GAFPropertiesTwo:V}
We have $\text{\upshape{vars}}(\mathcal{L}_{x,0}) \cap \vec{\mathbf{y}} \subseteq \bigcup_{i < \text{idx}(x)} \vec{\mathbf{y}}_i$.
\end{enumerate}
\end{lemma}
\begin{proof}~
\begin{description}
\item{Ad \ref{enum:GAFPropertiesTwo:I}:}
The claim holds by definition of $\mathcal{L}_x$ and $\mathcal{L}_{x,k}$.
\item{Ad \ref{enum:GAFPropertiesTwo:II}:}
Assume, without loss of generality, that $k < \ell$.
By definition of $\mathcal{L}_{x,k}$ and by \ref{enum:GAFPropertiesTwo:I}, we have $\mathcal{L}_{x,k} \subseteq \mathcal{L}_x \setminus \mathcal{L}_{x,\ell}$.
\item{Ad \ref{enum:GAFPropertiesTwo:IV}:}
Suppose there is some index $\ell > k$ and a variable $y \in \vec{\mathbf{y}}_\ell$ which occurs in some literal $L \in \mathcal{L}_{x,k}$.
By definition of $\mathcal{L}(\Cup_y)$, we observe $L \in \mathcal{L}(\Cup_y)$.
But, by definition of $\mathcal{L}_{x,\ell}$, we conclude $\mathcal{L}(\Cup_y) \setminus \bigcup_{j > \ell} \mathcal{L}_{x,j} \subseteq \mathcal{L}_{x,\ell}$. Hence, it either holds $L \in \mathcal{L}_{x,\ell}$ or $L \in \bigcup_{j > \ell} \mathcal{L}_{x,j}$.
This yields a contradiction, because $\mathcal{L}_{x,k} \subseteq \mathcal{L}_x \setminus \bigcup_{j \geq \ell} \mathcal{L}_{x,j}$ and thus $\mathcal{L}_{x,k}$ cannot contain $L$.
\item{Ad \ref{enum:GAFPropertiesTwo:V}:}
Suppose there is some index $\ell \geq k$ and a variable $y \in \vec{\mathbf{y}}_\ell$ which occurs in some literal $L \in \mathcal{L}_{x,0}$.
By definition of $\mathcal{L}(\Cup_y)$, we observe $L \in \mathcal{L}(\Cup_y)$.
But, by definition of $\mathcal{L}_{x,\ell}$, we conclude $\mathcal{L}(\Cup_y) \setminus \bigcup_{j > \ell} \mathcal{L}_{x,j} \subseteq \mathcal{L}_{x,\ell}$. Hence, it either holds $L \in \mathcal{L}_{x,\ell}$ or $L \in \bigcup_{j > \ell} \mathcal{L}_{x,j}$.
This yields a contradiction, because $\mathcal{L}_{x,0} \subseteq \mathcal{L}_x \setminus \bigcup_{j \geq \ell} \mathcal{L}_{x,j}$ and thus $\mathcal{L}_{x,0}$ cannot contain $L$.
\qedhere
\end{description}
\end{proof}
The next lemma provides the key ingredient to show decidability of GAF-satisfiability.
Similar to the transformation described in Lemma~\ref{lemma:TransformationGBSR}, the following lemma describes a transformation of GAF sentences into a nicer syntactic form.
However, this transformation constitutes only the first stage of the decidability proof.
\begin{lemma}\label{lemma:GAFTransformation}
If $\varphi$ belongs to GAF, we can effectively construct an equivalent sentence $\varphi'$ in standard form, in which every subformula lies within the scope of at most one universal quantifier.
Moreover, all literals in $\varphi'$ already occur in $\varphi$ (modulo variable renaming).
\end{lemma}
\begin{proof}[Proof sketch]
Similarly to the proof of Lemma~\ref{lemma:TransformationGBSR}, we (re-)transform parts of $\varphi$ repeatedly into a disjunction of conjunctions (or a conjunction of disjunctions) of subformulas which we treat as indivisible units.
The literals and indivisible units in the respective conjunctions (disjunctions) will be grouped in accordance with the sets $\mathcal{L}_0, \mathcal{L}_x$, and $\mathcal{L}_{x,\text{idx}(x)}, \ldots, \mathcal{L}_{x,n}$, where needed.
For this purpose, it is important to note that Lemma~\ref{lemma:GAFPropertiesOne}\ref{enum:GAFPropertiesOne:I} and the definition of $\mathcal{L}_0$ entail that $\mathcal{L}_0$ together with the sets $\mathcal{L}_x$ partition the set of all literals occurring in $\varphi$.
Moreover, every $\mathcal{L}_x$ is partitioned by the sets $\mathcal{L}_{x,0}, \mathcal{L}_{x, \text{idx}(x)}, \ldots, \mathcal{L}_{x,n}$, by virtue of Lemma~\ref{lemma:GAFPropertiesTwo}\ref{enum:GAFPropertiesTwo:I}, \ref{enum:GAFPropertiesTwo:II} and the definition of $\mathcal{L}_{x,0}$.
At the beginning, we transform $\psi$ into a disjunction of conjunctions of literals $\bigvee_i \psi_i$.
At this point, we move the existential quantifier block $\exists \vec{\mathbf{y}}_n$ inwards.
Lemmas~\ref{lemma:GAFPropertiesOne} and~\ref{lemma:GAFPropertiesTwo} guarantee that the quantifiers from this quantifier block can be distributed over the constituents of the $\psi_i$ in a beneficial way.
%
The thus obtained sentence $\varphi''$ has the form
$\forall \vec{\mathbf{x}}_1 \exists \vec{\mathbf{y}}_1 \ldots \forall \vec{\mathbf{x}}_n. \bigvee_{i} \bigl( \exists \vec{\mathbf{y}}_n. \chi_{i,0}^{(1)} \bigr)
\wedge \bigwedge_{k=1}^{n} \bigwedge _{x \in \vec{\mathbf{x}}_k}$ $\bigl( \chi_{i,x,0}^{(1)}
\wedge \bigl( \bigwedge_{j=\text{idx}(x)}^{n-1} \chi_{i,x,j}^{(1)} \bigr)
\wedge \exists(\vec{\mathbf{y}}_n \cap Y_{x,n}). \chi_{i,x,n}^{(1)} \bigr)$,
where $\chi_{i,0}^{(1)}$ comprises all literals in $\psi_i$ which belong to $\mathcal{L}_0$, and for every $k$ the $\chi_{i,x,k}^{(1)}$ group the literals which occur in $\psi_i$ and belong to $\mathcal{L}_{x,k}$, respectively.
Next, we transform the big disjunction in $\varphi''$ into a conjunction of disjunctions $\bigwedge_i \psi'_i$,
and move the universal quantifier block $\forall \vec{\mathbf{x}}_n$ inwards.
The resulting formula has the shape $\forall \vec{\mathbf{x}}_1 \exists \vec{\mathbf{y}}_1 \ldots \forall \vec{\mathbf{x}}_{n-1}$ $\exists \vec{\mathbf{y}}_{n-1}. \bigwedge_{i} \eta_{i,0}^{(1)} \vee \bigl( \bigvee_{k=1}^{n-1} \bigvee _{x \in \vec{\mathbf{x}}_k} \eta_{i,x}^{(1)} \bigr) \vee \bigvee _{x \in \vec{\mathbf{x}}_n} \forall x.\, \eta_{i,x}^{(1)}$,
where grouping of the constituents of each $\psi'_i$ is similar to what we have done above, but this time in accordance with the more coarse-grained sets $\mathcal{L}_0$ and $\mathcal{L}_x$.
We reiterate the described process until all the quantifiers have been moved inwards in the outlined way.
%
The final result of this transformation is the sought $\varphi'$ and it does not contain any nested occurrences of universal quantifiers.
\end{proof}
The above proof still works if $\varphi$ contains the equality predicate or non-constant function symbols.
Moreover, the sentence $\varphi'$ in Lemma~\ref{lemma:GAFTransformation} has a very particular shape.
For one part, it does not contain nested universal quantifiers.
In addition, the vocabulary in $\varphi'$ is identical to the vocabulary of the original $\varphi$.
If, for instance, $\varphi$ does not contain function symbols of arity larger then one, then the same holds true for $\varphi'$.
These two properties have implications for the outcome $\varphi_\text{Sk}$ of Skolemizing $\varphi'$.
\begin{lemma}\label{lemma:GAFSkolemizedProperties}
The Skolemized variant $\varphi_{\text{Sk}}$ of $\varphi'$ satisfies the following properties:
\begin{enumerate}[label=(\roman{*}), ref=(\roman{*})]
\item\label{enum:propositionGAFSkolemizedProperties:I} $\varphi_{\text{Sk}}$ does not contain any function symbol of arity larger than one.
\item\label{enum:propositionGAFSkolemizedProperties:II} Every atom $A$ in $\varphi_{\text{Sk}}$ is either ground or contains exactly one variable.
\end{enumerate}
\end{lemma}
\begin{proof}
Property~\ref{enum:propositionGAFSkolemizedProperties:I} is a direct consequence of the fact that $\varphi'$ does neither contain free variables nor nested occurrences of universal quantifiers.
%
Concerning \ref{enum:propositionGAFSkolemizedProperties:II}, consider an atom $A$ in $\varphi_\text{Sk}$ and suppose that $A$ is not ground.
Since $\varphi_\text{Sk}$ resulted from Skolemization, it can only contain universally quantified variables.
Suppose $A$ contains two distinct variables $x, x'$.
Because of $\varphi_\text{Sk}$ being closed, $A$ must lie within the scope of two distinct universal quantifiers $\forall x$ and $\forall x'$. But this contradicts Lemma~\ref{lemma:GAFTransformation}.
\end{proof}
These observations lead to the first decidability result with respect to GAF-satisfiability.
\begin{theorem}\label{theorem:DecidabilityGAFwithEquality}
Satisfiability of a given GAF sentence $\varphi$ is decidable, even if $\varphi$ contains equality (but no non-constant function symbols).
\end{theorem}
This result follows from Theorem~2 in~\cite{Fermuller1993b}, where Ferm\"uller and Salzer show that the satisfiability for a clausal fragment called $\mathcal{A}^=$ is decidable.
Roughly speaking, in any clause set in $\mathcal{A}^=$ every literal of the form $P(t_1, \ldots, t_m)$ or $t_1 \approx t_2$ contains at most one variable (possibly with multiple occurrences) and each $t_i$ is either a constant symbol, a variable, or a term of the form $f(v)$ for some variable $v$.
Clause sets corresponding to GAF sentences with equality but without non-constant function symbols, fall exactly into the syntactic category of $\mathcal{A}^=$.
In order to see this more clearly, we can strengthen \ref{enum:propositionGAFSkolemizedProperties:I} in Lemma~\ref{lemma:GAFSkolemizedProperties} to the following property:
\ref{enum:propositionGAFSkolemizedProperties:I}$'$ Every term $t$ in $\varphi_\text{Sk}$ is either a constant symbol, a variable, or of the form $t = f(v)$ for some variable $v$.
On the other hand, we can show decidability of the satisfiability problem for GAF sentences $\varphi$ without equality in which we allow unary function symbols to occur in an arbitrarily nested fashion.
Given the Skolemized variant $\varphi_\text{Sk}$ of the result $\varphi'$ of Lemma~\ref{lemma:GAFTransformation}, the following lemma entails that we can effectively construct a sentence $\varphi'_\text{Sk}$ which is equisatisfiable to $\varphi_\text{Sk}$ and belongs to the \emph{full monadic fragment} (monadic first-order sentences with unary function symbols but without equality).
Decidability of the satisfiability problem for the full monadic fragment has first been shown by L\"ob and Gurevich~\cite{Lob1967, Gurevich1969}.
\begin{lemma}\label{lemma:TransformToFullMonadic:UnaryFunctions}
Let $\varphi_\text{Sk}$ be a first-order sentence without equality, in which
\begin{enumerate}[label=(\alph{*}), ref=(\alph{*})]
\item function symbol of arity larger than one do not occur,
\item existential quantifiers do not occur, and
\item every atom contains at most one variable (possibly with multiple occurrences).
\end{enumerate}
Then we can effectively construct an equisatisfiable sentence $\varphi'_\text{Sk}$ in which all occurring predicate symbols are unary, and no other constant symbols and function symbols appear than the ones that occur in $\varphi_\text{Sk}$. Moreover, the length of $\varphi'_\text{Sk}$ lies in $\mathcal{O}\bigl(\text{len}(\varphi_\text{Sk})^4\bigr)$.
\end{lemma}
Section~\ref{section:proofGAFtoMonadic} is devoted to the proof of this lemma.
Putting Lemmas~\ref{lemma:GAFTransformation}, \ref{lemma:GAFSkolemizedProperties} and \ref{lemma:TransformToFullMonadic:UnaryFunctions} together, we can prove decidability of GAF-satisfiability.
\begin{theorem}\label{theorem:DecidabilityGAFwithUnaryFunctions}
Satisfiability of a given GAF sentence $\varphi$ without equality is decidable, even if $\varphi$ contains arbitrarily nested unary function symbols.
\end{theorem}
\subsection{Translating GAF sentences into monadic first-order sentences}\label{section:proofGAFtoMonadic}
We start with an auxiliary lemma.
Two atoms $A$ and $B$ are considered to be \emph{variable disjoint} if and only if $\text{\upshape{vars}}(A) \cap \text{\upshape{vars}}(B) = \emptyset$.
We say $A$ is \emph{more general} than $B$, denoted $A \lesssim B$, if and only if there is a substitution $\theta$ such that $A\theta = B$.
Moreover, we write $A\simeq B$ if and only if $A\lesssim B$ and $A \gtrsim B$.
A substitution $\theta$ for which $A\theta$ equals $B\theta$ is called a \emph{unifier of $A$ and $B$}.
If such a unifier exists, then we say \emph{$A$ and $B$ are unifiable}.
A unifier $\theta$ of $A$ and $B$ is considered to be \emph{most general} if and only if for every unifier $\theta'$ of $A$ and $B$ it holds $A\theta \lesssim A\theta'$.
As usual, we shall abbreviate the term \emph{most general unifier} with the acronym \emph{mgu}.
\begin{lemma}\label{lemma:UnifiabilityForSingleVariableAtoms}
Let $A$ and $B$ be two variable-disjoint atoms, and assume each of them contains at most one variable (possibly with multiple occurrences).
%
If $A$ and $B$ are unifiable, and $\theta$ is an mgu of the two, then either $A\theta \simeq A$ or $B\theta \simeq B$ or $A\theta = B\theta$ is ground.
\end{lemma}
\begin{proof}
Suppose $A\theta \not\simeq A$ and $B\theta \not\simeq B$.
Hence, there are distinct variables $x_1, x_2$ such that $\text{\upshape{vars}}(A) = \{x_1\}$ and $\text{\upshape{vars}}(B) = \{x_2\}$.
%
Because of $A\theta = B\theta$, $A\theta \not\simeq A$, and $B\theta \not\simeq B$, and since $\theta$ is most general, there must be two term positions $\pi_1, \pi_2$ such that
\begin{itemize}
\item $A|_{\pi_1} = x_1$ and $B|_{\pi_1} = t_2 \neq x_2$, and
\item $A|_{\pi_2} = t_1 \neq x_1$ and $B|_{\pi_2} = x_2$.
\end{itemize}
Consequently, we know $x_1\theta = t_2\theta$ and $x_2\theta = t_1\theta$.
If $A\theta = B\theta$ were not ground, then $\text{\upshape{vars}}(t_1) = \{x_1\}$ and $\text{\upshape{vars}}(t_2) = \{x_2\}$ would hold true.
%
While $x_1\theta = t_2\theta$ thus entails that the term depth of $x_1\theta$ is strictly larger than that of $x_2\theta$, the observation of $x_2\theta = t_1\theta$ implies that the term depth of $x_1\theta$ is strictly smaller then that of $x_2\theta$. This contradiction means that $A\theta = B\theta$ must be ground.
\end{proof}
In order to show decidability of $\exists^* \forall \exists^*$ sentences, Ackermann translated $\exists^*\forall\exists^*$ sentences into equisatisfiable monadic ones \cite{Ackermann1954}. F\"urer adopted Ackermann's method to give an upper bound on the complexity of the decision problem for the Ackermann Fragment \cite{Furer1981}. We shall employ a generalization of F\"urer's reduction approach to prove Lemma~\ref{lemma:TransformToFullMonadic:UnaryFunctions}.
Without loss of generality, we assume that $\varphi_\text{Sk}$ contains at least the constant symbol $d$.
%
Let $\text{At}$ be the set of all atoms occurring in $\varphi_\text{Sk}$.
%
Consider the set $\text{At}'$ which we define to be the smallest set of variable-disjoint atoms such that
(a) for every $A\in \text{At}$ there is some $B \in \text{At}'$ such that $B \simeq A$,
(b) for all $A, B \in \text{At}'$, for which there is an mgu $\theta$, we find some atom $C \in \text{At}'$ such that $C \simeq A\theta = B\theta$, and
(c) for all $A, B \in \text{At}'$ we have $A \not\simeq B$.
By Lemma~\ref{lemma:UnifiabilityForSingleVariableAtoms}, the set $\text{At}'$ is finite. More precisely: $\text{At}'$ contains at most $\tfrac{|\text{At}| \cdot (|\text{At}|-1)}{2} + |\text{At}| \leq |\text{At}|^2$ elements.
%
Let $A_1, \ldots, A_q$ be an enumeration of all the atoms in $\text{At}'$, and let $P_1, \ldots, P_q$ be distinct unary predicate symbols which do not occur in $\varphi_\text{Sk}$.
We construct the sentence $\varphi_\text{Mon}$ from $\varphi_\text{Sk}$ as follows:
(a) replace every occurrence of a non-ground atom $B(x)$ in $\varphi_\text{Sk}$ with the atom $P_i(x)$ which corresponds to the (unique) atom $A_i \in \text{At}'$ with $A_i \simeq B$, and
(b) replace every occurrence of a ground atom $B'$ in $\varphi_\text{Sk}$ with $P_j(d)$ corresponding to the unique $A_j \in \text{At}'$ with $A_j = B'$.
Consider two distinct atoms $A_i, A_j \in \text{At}'$. If there is a unifier $\theta$ of $A_i$ and $A_j$, we must make sure that the instances of $P_i(x)$ and $P_j(x')$ corresponding to $A_i\theta$ and $A_j\theta$ are interpreted in the same way by any model of $\varphi_\text{Mon}$.
In order to do so, we define the sets $\Psi, \Psi'$ of formulas as follows.
Let $x_*$ be a fresh variable which does not occur in $\varphi_\text{Mon}$, and let $\tau_*$ be the substitution mapping all variables to $x_*$.
We set \\
\centerline{$\Psi := \bigl\{ P_i\bigl((x\theta)\tau_*\bigr) \leftrightarrow P_j\bigl(x_*\bigr) \!\bigm|\,
\text{$A_i, A_j\in \text{At}'$ are distinct, $A_i \lesssim A_j$, $\text{\upshape{vars}}(A_i) = \{x\}$, and $A_i\theta = A_j$} \bigr\}$.}
%
In addition, we define $\Psi' := \bigl\{ P_j(x_*) \leftrightarrow P_j(d) \bigm| \text{$A_j \in \text{At}'$ is ground} \bigr\}$.
We now set $\varphi'_\text{Sk} := \varphi_\text{Mon} \wedge \forall x_*. \bigwedge_{\psi \in \Psi \cup \Psi'} \psi$.
By construction, $\varphi'_\text{Sk}$ exclusively contains predicate symbols which are unary.
Its constituent $\varphi_\text{Mon}$ may contain the constant symbol $d$ but no non-constant function symbols.
The formulas in $\Psi \cup \Psi'$, on the other hand, may contain non-constant function symbols.
However, all of these symbols have already occurred in $\varphi_\text{Sk}$.
By virtue of Lemma~\ref{lemma:UnifiabilityForSingleVariableAtoms}, the number of formulas in $\Psi$ is at most $|\text{At}| \cdot |\text{At}'| \leq |\text{At}|^3$.
The length of the formulas in $\Psi$ is upper bounded by the length of $x\theta$ in $\Psi$'s definition plus some constant value, and thus can exceed the length of the longest atom in $\varphi_\text{Sk}$ only by this constant value.
The cardinality of $\Psi'$ is upper bounded by $|\text{At}'| \leq |\text{At}|^2$ and the length of the formulas therein is constant.
All in all, the length of $\varphi'_\text{Sk}$ lies in $\mathcal{O}\bigl(\text{len}(\varphi_\text{Sk})^4\bigr)$.
In what follows, we tacitly assume that the signatures underlying $\varphi_\text{Sk}$ and $\varphi'_\text{Sk}$ share the same constant symbols and function symbols---namely, the ones occurring in $\varphi_\text{Sk}$. Consequently, when we refer to Herbrand structures with respect to $\varphi_\text{Sk}$ and $\varphi'_\text{Sk}$, we base these structures on exactly the same universe of ground terms. However, the sets of occurring predicate symbols are disjoint (as stipulated above).
Since $\varphi_\text{Sk}$ and $\varphi'_\text{Sk}$ do neither contain equality nor existential quantifiers, we know that there are Herbrand models for them, if they are satisfiable at all.
\begin{lemma}\label{lemma:GAFequisat:I}
Given any Herbrand model $\mathcal{A} \models \varphi_\text{Sk}$, we can construct a model $\mathcal{B} \models \varphi'_\text{Sk}$.
\end{lemma}
\begin{proof}
Since $\mathcal{A}$ is a Herbrand model of $\varphi_\text{Sk}$, the universe $\mathfrak{U}_\mathcal{A}$ contains all ground terms constructed from the constant symbols and function symbols occurring in $\varphi_\text{Sk}$.
%
We define $\mathcal{B}$ by taking over $\mathcal{A}$'s universe and its interpretations of the constant and function symbols.
Moreover, we set $P_i^\mathcal{B} := \bigl\{ t\in \mathfrak{U}_\mathcal{B} \bigm| \mathcal{A}, [x \mapsto t] \models A_i \bigr\}$ if there is some $x \in \text{\upshape{vars}}(A_i)$.
For any $P_i$, for which $A_i$ is ground, we set $P_i^\mathcal{B} := \mathfrak{U}^\mathcal{B}$ if $\mathcal{A} \models A_i$ and $P_i^\mathcal{B} := \emptyset$ otherwise.
We first prove $\mathcal{B} \models \varphi_\text{Mon}$. $\varphi_\text{Sk}$ differs from $\varphi_\text{Mon}$ only in the occurrences of atoms. It thus suffices to show that for two corresponding atom occurrences $A$ in $\varphi_\text{Sk}$ and $B$ in $\varphi_\text{Mon}$ and for an arbitrary variable assignment $\beta$ it holds $\mathcal{A}, \beta \models A$ if and only if $\mathcal{B}, \beta \models B$. But this is guaranteed by construction of $\varphi_\text{Mon}$ and $\mathcal{B}$:
%
We need to consider two cases.
\begin{description}
\item If $A$ is ground, then there is some $j$ such that $A = A_j \in \text{At}'$ and $B = P_j(d)$. By construction of $\mathcal{B}$, we observe $\mathcal{A} \models A_j$ if and only if $\mathcal{B} \models P(d)$.
\item If $A$ contains a variable $x$, then there is some $A_j \in \text{At}'$ such that $A \simeq A_j$. Moreover, we know that $B = P_j(x)$. Because of $A \simeq A_j$, there must be some $x' \in \text{\upshape{vars}}(A_j)$ such that for every $t\in \mathfrak{U}_\mathcal{A} = \mathfrak{U}_\mathcal{B}$ it holds $\mathcal{A}, [x \mapsto t] \models A$ if and only if $\mathcal{A}, [x' \mapsto t] \models A_j$.
In addition, we have constructed $\mathcal{B}$ in a way leading to $\mathcal{A}, [x' \mapsto t] \models A_j$ if and only if $\mathcal{B}, [x \mapsto t] \models P_j(x)$.
Put togther, this yields $\mathcal{A}, [x \mapsto t] \models A$ if and only if $\mathcal{B}, [x \mapsto t] \models P_j(x)$.
\end{description}
\noindent
Next, we have to show $\mathcal{B} \models \forall x_*. \psi$ for every $\psi \in \Psi \cup \Psi'$.
\begin{description}
\item If $\psi \in \Psi$, then $\psi = P_i\bigl( (x\theta)\tau_* \bigr) \leftrightarrow P_j(x_*)$ with $\text{\upshape{vars}}(A_i) = \{x\}$ and $A_i\theta = A_j$.
Assume, without loss of generality, that $\text{\upshape{vars}}(A_j) = \text{\upshape{vars}}(A_j\theta) = \{x'\}$. (If $A_j$ is ground, the argument is still valid.)
By construction of $\mathcal{B}$, for any $t$ we have $\mathcal{B},[x_*\mapsto t] \models P_i\bigl( (x\theta)\tau_* \bigr)$ if and only if $\mathcal{A},[x' \mapsto t] \models A_i\theta$ and, moreover, $\mathcal{B},[x_*\mapsto t] \models P_j(x_*)$ if and only of $\mathcal{A},[x'\mapsto t] \models A_j$.
Consequently, $A_i\theta = A_j$ leads to $\mathcal{B},[x_*\mapsto t] \models P_i\bigl( (x\theta)\tau_* \bigr)$ if and only if $\mathcal{B},[x_*\mapsto t] \models P_j(x_*)$.
\item If $\psi \in \Psi'$, then $\psi = P_j(x_*) \leftrightarrow P_j(d)$ for some ground atom $A_j \in \text{At}'$. The structure $\mathcal{B}$ is constructed so that $P_j^\mathcal{B} = \mathfrak{U}_\mathcal{B}$ if $\mathcal{A} \models A_j$ and $P_j^\mathcal{B} = \emptyset$ otherwise. Hence, we have $\mathcal{B}, \beta \models P_j(x_*)$ if and only if $\mathcal{B}, \beta \models P_j(d)$.
\end{description}
Hence, $\mathcal{B} \models \forall x_*. \psi$ follows in both cases.
\end{proof}
\begin{lemma}\label{lemma:GAFequisat:II}
Given any Herbrand model $\mathcal{B} \models \varphi'_\text{Sk}$, we can construct a model $\mathcal{A} \models \varphi_\text{Sk}$.
\end{lemma}
\begin{proof}
Since $\mathcal{B}$ is a Herbrand model of $\varphi'_\text{Sk}$, and due to our assumption that the signature underlying $\varphi'_\text{Sk}$ contains the same constant symbols and function symbols as the signature underlying $\varphi_\text{Sk}$ does, the universe $\mathfrak{U}_\mathcal{B}$ contains all ground terms constructed from the constant symbols and function symbols occurring in $\varphi_\text{Sk}$.
We now define $\mathcal{A}$ by taking over $\mathcal{B}$'s universe and its interpretations of the constant and function symbols.
Moreover, for any predicate symbol $Q$ of arity $m$ occurring in $\varphi_\text{Sk}$ we define $Q^\mathcal{A} := S_1 \cup S_2$, where
\begin{align*}
\hspace{-3ex}
S_1 := \bigl\{ \<t_1, \ldots, t_m\> \in \mathfrak{U}_\mathcal{A}^m \bigm|\;
&\text{there is a non-ground atom $Q(s_1, \ldots, s_m) \in \text{At}$ and some}\\
&\text{$A_j \in \text{At}'$ with $A_j \simeq Q(s_1, \ldots, s_m)$ and there is a ground term $t$}\\
&\text{ such that $Q(t_1, \ldots, t_m) = (Q(s_1, \ldots, s_1)\tau_*)\subst{x_*}{t}$}\\
&\text{ and $\mathcal{B} \models P_j(t)$} \bigr\}
\end{align*}
and
\begin{align*}
S_2 :=\; \bigl\{ \<t_1, \ldots, t_m\> \in \mathfrak{U}_\mathcal{A}^m \bigm|\;
&\text{there is a ground atom $A_j \in \text{At}'$ such that}\\
&\text{$A_j = Q(t_1, \ldots, t_m)$ and $\mathcal{B} \models P_j(d)$} \bigr\} ~.
\end{align*}
Again, since $\varphi_\text{Sk}$ differs from $\varphi_\text{Mon}$ only in the occurrences of atoms, it suffices to show that for two corresponding atom occurrences $A$ in $\varphi_\text{Sk}$ and $B$ in $\varphi_\text{Mon}$ and for an arbitrary variable assignment $\beta$ it holds $\mathcal{A}, \beta \models A$ if and only if $\mathcal{B}, \beta \models B$.
Let $A = Q(s_1, \ldots, s_m)$. There must be some $A_j \in \text{At}'$ such that $A_j \simeq A$ and $B = P_j(t)$.
\begin{description}
\item If $t = d$, then $A_j$ must be ground.
Hence, we have $\<s_1, \ldots, s_m\> \in S_2$ if and only if $\mathcal{B} \models P_j(d)$.
It remains to show that $\mathcal{B} \not\models P_j(d)$ entails $\<s_1, \ldots, s_m\> \not\in S_1$.
Suppose $\<s_1, \ldots, s_m\> \in S_1$, i.e.\ there is some non-ground atom $A_i \in \text{At}'$, some $x \in \text{\upshape{vars}}(A_i)$, and some ground term $t'$ such that $A_i\subst{x}{t'} = A_j$ and $\mathcal{B} \models P_i(t)$.
Because of $A_i \subst{x}{t'} = A_j$, we find the formula $P_i(t') \leftrightarrow P_j(x_*)$ in $\Psi$. Moreover, the formula $P_j(x_*) \leftrightarrow P_j(d)$ belongs to $\Psi'$.
Hence, $\mathcal{B}$ is a model of both $\forall x_*. P_i(t') \leftrightarrow P_j(x_*)$ and $\forall x_*. P_j(x_*) \leftrightarrow P_j(d)$.
But then $\mathcal{B} \models P_i(t')$ contradicts $\mathcal{B} \not\models P_j(d)$.
Consequently, $\<s_1, \ldots, s_m\>$ cannot belong to $S_1$.
\item If $t = x$ for some variable $x$, then $\text{\upshape{vars}}(A) = \{x\}$ and there is some $x'\in \text{\upshape{vars}}(A_j)$.
Let $\theta := \subst{x}{\beta(x)}$.
If $\mathcal{B} \models P_j(x\theta)$, then we get $\<s_1, \ldots, s_m\>\theta \in S_1 \subseteq Q^\mathcal{A}$, and thus $\mathcal{A},\beta \models Q(s_1, \ldots, s_m)$.
If $\mathcal{B} \not\models P_j(x\theta)$, then we may conclude $\<s_1, \ldots, s_m\>\theta \not\in S_1 \cup S_2 = Q^\mathcal{A}$ and thus $\mathcal{A},\beta \not\models Q(s_1, \ldots, s_m)$ due to the following arguments.
Suppose $\<s_1, \ldots, s_m\>\theta \in S_1$.
By definition of $\mathcal{B}$, it holds $\mathcal{B} \not\models A_j(x\theta)$.
Hence, there must be some non-ground $A_i(x'') \in \text{At}'$ with $i \neq j$, for which we can find a substitution $\rho$ such that $A_i\rho = Q(s_1, \ldots, s_m)\theta$ and it holds $\mathcal{B} \models P_i(x''\rho)$.
But then, $A_i$ and $A_j$ are unifiable, and there must exist an mgu $\sigma_{ij}$ of $A_i$ and $A_j$.
By Lemma~\ref{lemma:UnifiabilityForSingleVariableAtoms} and due to the definition of $At'$, we have to consider the following cases:
\begin{enumerate}[label=(\arabic{*}), ref=(\arabic{*})]
\item\label{enum:proofGAFequisat:I:I} $A_i \lesssim A_j$,
\item\label{enum:proofGAFequisat:I:II} $A_j \lesssim A_i$, or
\item\label{enum:proofGAFequisat:I:III} $A_i\sigma_{ij} = A_j\sigma_{ij} = Q(s_1, \ldots, s_m)\theta$.
\end{enumerate}
In case \ref{enum:proofGAFequisat:I:I} there must be some $\tau_1$ such that $A_i\tau_1 = A_j$ and $\Psi$ contains a formula $\psi_1 := P_i\bigl((x''\tau_1)\tau_*\bigr) \leftrightarrow P_j\bigl(x_*\bigr)$.
Hence, we get $\mathcal{B} \models P_j(x\theta)$ if and only if $\mathcal{B}, [x_* \mapsto x\theta] \models P_j(x_*)$ if and only if $\mathcal{B}, [x_* \mapsto x\theta] \models P_i((x''\tau_1)\tau_*)$ if and only if $\mathcal{B} \models P_i(x''\rho)$. But this is contradictory to our assumptions.
In case \ref{enum:proofGAFequisat:I:II} there must be some $\tau_2$ such that $A_j\tau_2 = A_i$ and $\Psi$ contains a formula $\psi_2 := P_j\bigl((x'\tau_2)\tau_*\bigr) \leftrightarrow P_i\bigl(x_*\bigr)$.
Therefore, it holds $\mathcal{B} \models P_i(x''\rho)$ if and only if $\mathcal{B}, [x_* \mapsto x''\rho] \models P_i\bigl(x_*\bigr)$ if and only if $\mathcal{B}, [x_* \mapsto x''\rho] \models P_j\bigl((x'\tau_2)\tau_*\bigr)$ if and only if $\mathcal{B} \models P_j(x\theta)$. Again, this contradicts our assumptions.
In case \ref{enum:proofGAFequisat:I:III} the set $\text{At}'$ contains some atom $A_k := A\theta$ and $\Psi$ contains the formulas $\psi_3 := P_j\bigl(x\theta\bigr) \leftrightarrow P_k \bigl(x_*\bigr)$ and $\psi_4 := P_i\bigl(x''\rho\bigr) \leftrightarrow P_k \bigl(x_*\bigr)$. In addition, $\Psi'$ contains the formula $\psi_5 := P_k(x_*) \leftrightarrow P_k(d)$.
Since $\mathcal{B}$ is a model of $\forall x_*. \psi_3 \wedge \psi_4 \wedge \psi_5$, we get $\mathcal{B} \models P_j(x\theta)$ if and only if $\mathcal{B} \models P_i(x''\rho)$. This is on contradiction with our assumptions $\mathcal{B}\not\models P_j(x\theta)$ and $\mathcal{B}\models P_i(x''\rho)$ as well.
In all three cases, we conclude $\<s_1, \ldots, s_m\>\theta \not\in S_1$.
Suppose $\<s_1, \ldots, s_m\>\theta \in S_2$. Hence, there must be some ground atom $A_k \in \text{At}'$ with $k \neq i$ and $A_k = Q(s_1, \ldots, s_m)\theta$.
Moreover, $P_k(d)$ must hold true under$\mathcal{B}$.
Since we then observe $A_j \lesssim A_k$, $\Psi$ must contain the formula $\psi'_1 := P_j(x\theta) \leftrightarrow P_k(x_*)$.
Hence, it holds $\mathcal{B} \models \forall x_*. P_k(x_*)$ if and only if $\mathcal{B} \models P_j(x\theta)$, because $x\theta$ is ground. In particular, our assumption $\mathcal{B}\not\models P_j(x\theta)$ entails $\mathcal{B},[x_*\mapsto d] \not\models P_k(d)$. Thus, we have reached a contradiction.
Consequently, $\<s_1, \ldots, s_m\>\theta \not\in S_2$.
\qedhere
\end{description}
\end{proof}
This finishes the proof of Lemma~\ref{lemma:TransformToFullMonadic:UnaryFunctions}.
\section{Related and future work} \label{section:furthrelwork}
In \cite{Dreben1979} (page 65) Dreben and Goldfarb extend the relational monadic fragment to a certain extent and call the result \emph{Initially-extended Essentially Monadic Fragment}.
This class lies in the intersection of GBSR and GAF.
Hence, this result could be considered as a first step away from the standard classification based on quantifier prefixes or the arity of used predicate symbols.
In \cite{Fermuller1993a} (page 152) Ferm\"uller et al.\ combine the just described formula class with the Ackermann fragment and call the result \emph{AM}.
AM is properly contained in GAF.
Moreover, Ferm\"uller et al.\ argue that AM itself is a special case of Maslov's fragment K~\cite{Maslov1968}.
K is incomparable to GBSR and GAF.
In the last couple of years several new fragments have been discovered \cite{Kieronski2014, Barany2015, Mogavero2015}, all of which are incomparable to GBSR and GAF.
There exist a number of results showing decidability of $\exists^* \forall \exists^*$ sentences with arbitrary function symbols~\cite{Gurevich1973, Maslov1972, Gradel1990a}---see \cite{Borger1997}, chapter 6.3, for an overview.
The methods therein may help to also show decidability of GAF with arbitrary function symbols.
The methods and results described in the present paper may be extended in various directions:
generalizing other known decidable prefix classes,
more liberal conditions regarding function symbols,
scenarios in interpreted theories such as arithmetic.
Moreover, we have not yet thoroughly investigated the complexity of deciding GAF-satisfiability.
|
1,116,691,499,267 | arxiv | \section{Introduction}
Video-based face recognition has received increasing interest in recent years as a growing number of videos are continually being captured by mobile devices and CCTV systems for surveillance. Due to requirements in law enforcement and the fact that videos contain rich temporal and multi-view information, it is necessary to develop robust and accurate face recognition techniques for surveillance systems. Although the ubiquity of deep learning algorithms has advanced face recognition technology for static face images, video-based face recognition still poses a significant research challenge. Compared to static photos which are generally taken under controlled conditions (in terms of pose, illumination, expression, and occlusion) and with subject's cooperation, individual video frames have relatively low image quality because of unconstrained capture
environments \cite{Kalka2018IJBS}.
\begin{figure}
\captionsetup{font=footnotesize}
\centering
\begin{subfigure}[b]{0.33\linewidth}
\includegraphics[width=\linewidth]{figs/pooling_image.pdf}
\caption{{\footnotesize Image-level}}
\label{fig:image_fusion}
\end{subfigure}\hfill
\begin{subfigure}[b]{0.33\linewidth}
\includegraphics[width=\linewidth]{figs/pooling_instance.pdf}
\caption{{\footnotesize Representation-level}}
\label{fig:feature_vector_fusion}
\end{subfigure}\hfill
\begin{subfigure}[b]{0.33\linewidth}
\includegraphics[width=\linewidth]{figs/pooling_component.pdf}
\caption{{\footnotesize Component-wise}}
\label{fig:dimension_fusion}
\end{subfigure}\\
\caption{Methods to aggregate identity information from a set of video frames of the same person. Figure \ref{fig:image_fusion}: individual frames are merged (e.g., pixel wise pooling or generative model to obtain a fused image); Figure \ref{fig:feature_vector_fusion}: deep feature vector for each video frame is assigned a single weight, $W_i$; Figure \ref{fig:dimension_fusion}: weighted average of feature components, where weights (quality value) are learned by the same network which gives the feature vector (see Figure \ref{fig: framework})}
\label{fig:fusion}
\end{figure}
\begin{figure*}
\captionsetup{font=footnotesize}
\centering
\newcommand{\vspace{-18px}}{\vspace{-10px}}
\begin{subfigure}[b]{0.24\linewidth}
\includegraphics[width=0.25\linewidth]{figs/dataset/msceleb/m_0b1dys/15-FaceId-0.jpg}\hfill
\includegraphics[width=0.25\linewidth]{figs/dataset/msceleb/m_0b1dys/35-FaceId-1.jpg}\hfill
\includegraphics[width=0.25\linewidth]{figs/dataset/msceleb/m_0b1dys/62-FaceId-0.jpg}\hfill
\includegraphics[width=0.25\linewidth]{figs/dataset/msceleb/m_0b1dys/70-FaceId-0.jpg}\\
\includegraphics[width=0.25\linewidth]{figs/dataset/msceleb/m_0b2bxw/1-FaceId-0.jpg}\hfill
\includegraphics[width=0.25\linewidth]{figs/dataset/msceleb/m_0b2bxw/3-FaceId-0.jpg}\hfill
\includegraphics[width=0.25\linewidth]{figs/dataset/msceleb/m_0b2bxw/10-FaceId-0.jpg}\hfill
\includegraphics[width=0.25\linewidth]{figs/dataset/msceleb/m_0b2bxw/12-FaceId-0.jpg}\\
\includegraphics[width=0.25\linewidth]{figs/dataset/msceleb/m_0b2g5r/0-FaceId-0.jpg}\hfill
\includegraphics[width=0.25\linewidth]{figs/dataset/msceleb/m_0b2g5r/10-FaceId-0.jpg}\hfill
\includegraphics[width=0.25\linewidth]{figs/dataset/msceleb/m_0b2g5r/22-FaceId-0.jpg}\hfill
\includegraphics[width=0.25\linewidth]{figs/dataset/msceleb/m_0b2g5r/43-FaceId-0.jpg}\\
\vspace{-18px}\caption{MS-Celeb-1M \cite{guo2016msceleb}}
\label{fig:img_msceleb}
\end{subfigure}\hfill
\begin{subfigure}[b]{0.24\linewidth}
\includegraphics[width=0.25\linewidth]{figs/dataset/ytf/{aligned_detect_0.551}.jpg}\hfill
\includegraphics[width=0.25\linewidth]{figs/dataset/ytf/{aligned_detect_0.565}.jpg}\hfill
\includegraphics[width=0.25\linewidth]{figs/dataset/ytf/{aligned_detect_0.579}.jpg}\hfill
\includegraphics[width=0.25\linewidth]{figs/dataset/ytf/{aligned_detect_0.593}.jpg}\\
\includegraphics[width=0.25\linewidth]{figs/dataset/ytf/{aligned_detect_0.594}.jpg}\hfill
\includegraphics[width=0.25\linewidth]{figs/dataset/ytf/{aligned_detect_0.608}.jpg}\hfill
\includegraphics[width=0.25\linewidth]{figs/dataset/ytf/{aligned_detect_0.636}.jpg}\hfill
\includegraphics[width=0.25\linewidth]{figs/dataset/ytf/{aligned_detect_0.664}.jpg}\\
\includegraphics[width=0.25\linewidth]{figs/dataset/ytf/{aligned_detect_0.681}.jpg}\hfill
\includegraphics[width=0.25\linewidth]{figs/dataset/ytf/{aligned_detect_0.709}.jpg}\hfill
\includegraphics[width=0.25\linewidth]{figs/dataset/ytf/{aligned_detect_0.774}.jpg}\hfill
\includegraphics[width=0.25\linewidth]{figs/dataset/ytf/{aligned_detect_0.840}.jpg}\\
\vspace{-18px}\caption{YoutubeFaces \cite{wolf2011face}}
\end{subfigure}\hfill
\begin{subfigure}[b]{0.24\linewidth}
\includegraphics[width=0.25\linewidth]{figs/dataset/ijba/6/img_4566.jpg}\hfill
\includegraphics[width=0.25\linewidth]{figs/dataset/ijba/6/frame_32761_00495.png}\hfill
\includegraphics[width=0.25\linewidth]{figs/dataset/ijba/6/frame_32761_00555.png}\hfill
\includegraphics[width=0.25\linewidth]{figs/dataset/ijba/6/frame_32761_00615.png}\\
\includegraphics[width=0.25\linewidth]{figs/dataset/ijba/411/img_2512.jpg}\hfill
\includegraphics[width=0.25\linewidth]{figs/dataset/ijba/411/frame_28841_00000.png}\hfill
\includegraphics[width=0.25\linewidth]{figs/dataset/ijba/411/frame_28844_00120.png}\hfill
\includegraphics[width=0.25\linewidth]{figs/dataset/ijba/411/frame_28842_00175.png}\\
\includegraphics[width=0.25\linewidth]{figs/dataset/ijba/2498/img_33683.jpg}\hfill
\includegraphics[width=0.25\linewidth]{figs/dataset/ijba/2498/frame_33690_00390.png}\hfill
\includegraphics[width=0.25\linewidth]{figs/dataset/ijba/2498/frame_33690_00450.png}\hfill
\includegraphics[width=0.25\linewidth]{figs/dataset/ijba/2498/frame_33690_00660.png}\\
\vspace{-18px}\caption{IJB-A \cite{klare2015pushing}}
\end{subfigure}\hfill
\begin{subfigure}[b]{0.24\linewidth}
\includegraphics[width=0.25\linewidth]{figs/dataset/ijbs/208/img_100819.jpg}\hfill
\includegraphics[width=0.25\linewidth]{figs/dataset/ijbs/208/videos_107_1037.jpg}\hfill
\includegraphics[width=0.25\linewidth]{figs/dataset/ijbs/208/videos_107_1074.jpg}\hfill
\includegraphics[width=0.25\linewidth]{figs/dataset/ijbs/208/videos_107_1137.jpg}\\
\includegraphics[width=0.25\linewidth]{figs/dataset/ijbs/240/img_100594.jpg}\hfill
\includegraphics[width=0.25\linewidth]{figs/dataset/ijbs/240/videos_111_2851.jpg}\hfill
\includegraphics[width=0.25\linewidth]{figs/dataset/ijbs/240/videos_111_2932.jpg}\hfill
\includegraphics[width=0.25\linewidth]{figs/dataset/ijbs/240/videos_111_3497.jpg}\\
\includegraphics[width=0.25\linewidth]{figs/dataset/ijbs/241/img_100587.jpg}\hfill
\includegraphics[width=0.25\linewidth]{figs/dataset/ijbs/241/videos_5076_18327.jpg}\hfill
\includegraphics[width=0.25\linewidth]{figs/dataset/ijbs/241/videos_5078_22067.jpg}\hfill
\includegraphics[width=0.25\linewidth]{figs/dataset/ijbs/241/videos_5078_48476.jpg}\\
\vspace{-18px}\caption{IJB-S \cite{Kalka2018IJBS}}
\label{fig:img_ijbs}
\end{subfigure}\\
\caption{Example images from different datasets. MS-Celeb-1M and YoutubeFaces contain only still images or video frames, respectively. IJB-A and IJB-S include both still images and videos. The first columns of IJB-A and IJB-S show still images, followed by video frames of the respective subjects in the next three columns.}
\label{fig:dataset}
\end{figure*}
There are two main challenges in face recognition in surveillance videos. First, subjects captured by surveillance cameras are usually moving without directly looking at the cameras, leading to large pose variations and severe motion blur. Second, the video consists of multiple frames of the same subject, providing both noisy frames with poor quality and unfavorable viewing angles, but useful temporal and multi-view information (see Figure \ref{fig:img_ijbs}). Therefore, efficiently using a set of frames to generate a compact face representation is another challenge in video-based face recognition.
State-of-the-art face recognition algorithms for still images strive to generate robust face representations by deep neural networks trained on a large face dataset \cite{Schroff_2015_CVPR}, \cite{taigman2014deepface}, \cite{liu2017sphereface}, \cite{sun2015deeply}. Although the problems of image blur, pose variations, and occlusions can be partially solved by data augmentation \cite{ding2018trunk} and ensemble convolutional neural networks \cite{ding2018trunk}, \cite{sun2015deeply}, \cite{liu2015targeting}, such strategies may not be directly applicable to video frames. Furthermore, the majority of existing deep face recognition approaches cast network predictions as point estimates, which means every face image has an explicit representation regardless of quality. However, there are many noisy video frames with low facial information content. For those frames, the corresponding face representation should have lower reliability. Also, in many practical surveillance applications, video clips of a range of frames for each identity can be collected. It is crucial to effectively integrate the information across different frames together from frame-level features \cite{liu2017quality}, \cite{yang2017neural} or raw input frames \cite{hassner2016pooling}, \cite{rao2017learning} by assigning each instance\footnote{An image or a feature vector.} with an adaptive instance-level weight. However, since each component of an instance encodes different subsets of facial features, noise could also be integrated when we emphasize or suppress all components simultaneously.
In this work, we propose a new approach for video face recognition by considering component-wise feature aggregation. We cast video face recognition as a template (a set of images from the same person) matching problem \footnote{The same setting as the IARPA Janus Benchmark~\cite{klare2015pushing}}. Figure \ref{fig:fusion} shows the proposed component-wise feature aggregation, compared to the other two instance-level aggregation methods.
In component-wise aggregation, each component of the feature vectors is aggregated separately by learning the corresponding quality weight vectors as shown in Figure \ref{fig:dimension_fusion}.
Based on the hypothesis that
different covariates only lead to a variation in a subset of deep feature components, we train the network to predict an adaptive quality score for each deep feature component. The network can be trained on a still image dataset (e.g. MS-Celeb-1M \cite{guo2016msceleb}, Figure \ref{fig:img_msceleb}) with randomly generated templates. During testing, the features of each video or template are aggregated independently on each component using the respective quality scores. Experimental results on the IJB-S dataset \cite{Kalka2018IJBS} as well as other template/video matching benchmarks show that the proposed method significantly boosts the performance compared with average pooling and outperforms instance-level feature aggregation. The contributions of the paper are summarized below:
\begin{itemize}
\item A component-wise feature aggregation network (C-FAN) that aggregates each component of deep feature vectors separately, leading to better representation for video face recognition. \vspace{-5px}
\item The quality scores predicted by C-FAN correlate with the visual image quality. \vspace{-5px}
\item Evaluating the proposed C-FAN on a challenging surveillance dataset IJB-S~\cite{Kalka2018IJBS}. Consequently, we achieve state-of-the-art results on the five face identification protocols.
\item We also attain comparable performance on the other two face recognition benchmarks, YouTube Faces \cite{wolf2011face}, and IJB-A \cite{klare2015pushing}.
\end{itemize}
\section{Related Work}
\subsection{Deep Face Recognition}
Deep neural networks dominate the ongoing research in face recognition ~\cite{taigman2014deepface},~\cite{deepid2},~\cite{Schroff_2015_CVPR},~\cite{masi2016we},~\cite{liu2017sphereface},~\cite{hasnat2017deepvisage},~\cite{ranjan2017l2},~\cite{wang2018additive}, given their success in the ImageNet competition ~\cite{krizhevsky2012imagenet}. Taigman~\etal~\cite{taigman2014deepface} proposed the first face recognition application using deep neural networks. The following works have been exploring different loss functions to improve the discriminability of the embedding vector. Wen~\etal ~\cite{wen2016discriminative} proposed center loss to reduce the intra-class variation. Other work proposed new metric learning methods for face recognition ~\cite{Schroff_2015_CVPR}~\cite{sohn2016improved}. More recent work has been trying to close the gap between metric learning and identification learning by learning classification on a spherical embedding space~\cite{liu2017sphereface}~\cite{wang2018additive},~\cite{wang2018cosface},~\cite{ranjan2017l2}.
\subsection{Video Face Recognition}
Previous work on video face recognition can mainly be categorized into three categories: \emph{space}-based, \emph{classifier}-based method and \emph{aggregation}-based. \emph{Space}-based methods aim to model the feature space spanned by the instances in the video, such as probabilistic distributions~\cite{shakhnarovich2002face}, ~\cite{arandjelovic2005face}, affine hulls~\cite{shakhnarovich2002face},~\cite{yang2013face},~\cite{cevikalp2010face}, SPD matrices~\cite{huang2015log}, $n$-order statistics~\cite{lu2013image} and manifolds~\cite{lee2003video},~\cite{harandi2011graph}. \emph{Classifier}-based methods train a supervised classifier on each image set or video to obtain a representation~\cite{wolf2011face},~\cite{crosswhite2017template}.
Because deep convolution neural networks are shown to be able to lead to highly discriminative compact face representations from a collection of images, recent works on video face recognition mainly use an aggregation-based method to fuse a set of feature vectors or images into a single vector. Compared to previous methods as well as score-level fusion~\cite{best2014unconstrained}, aggregation methods are more efficient in terms of both computation and storage. In particular, image-aggregation methods fuse a set of images to a single image~\cite{hassner2016pooling},~\cite{rao2017learning} for feature extraction while feature-aggregation methods fuses the features extracted from different images into one vector. Although some feature-aggregation methods achieve good performance by simply using an average pooling~\cite{ding2018trunk},~\cite{chen2018unconstrained}, a number of recent work has focused on fusing features with visual qualities, such as detection score~\cite{ranjan2018crystal}, or predicted quality scores~\cite{liu2017quality},~\cite{yang2017neural},~\cite{sankaran2018metadata}. However, all of them only consider an instance-level aggregation while our work tries to aggregate the feature vectors in each component separately.
\section{Component-wise Feature Aggregation}
\subsection{Motivation}
Having multiple frames of a face in a video clip can be both advantageous and challenging at the same time. On the one hand, multiple frames incorporate temporal and multi-view information; on the other hand, large pose variations and motion blur of the frames will incur data noise and may impair the accuracy of video-based face recognition. Therefore, a key issue is how to aggregate the rich information in video frames to generate a face representation with stronger discriminative power than of individual frames.
Hassner~\etal~\cite{hassner2016pooling} found that simply eliminating poor quality frames will not necessarily help improve the face recognition accuracy.
One of the possible reasons is that even poor quality frames carry some valuable information. Efforts have been made to aggregate multiple frames into one single image \cite{hassner2016pooling}, \cite{rao2017learning}. A more straightforward solution might be extracting a feature vector for each frame and averaging the feature vectors to obtain a single video representation, which is known as average pooling. Let $T = \{I_1, I_2, \cdots, I_N\}$ be a template of face images. Assume that each subject has a noiseless embedding $\boldsymbol{\mu}$, and a feature vector of the $i^{th}$ image in the template $T$ generated by face representation model is
\begin{equation}
\mathbf{f}_i = \boldsymbol{\mu} + \boldsymbol{\epsilon}_i,
\end{equation}
where $\boldsymbol{\epsilon}_i$ is the noise caused by the variations in the image. The template representation of average pooling can thus be explained as
\begin{equation}
\mathbf{r}^{avg} = \frac{1}{N} \sum_{i}(\boldsymbol{\mu} + \boldsymbol{\epsilon}_i) = \boldsymbol{\mu} + \frac{1}{N} \sum_{i}{\boldsymbol{\epsilon}}_i,
\end{equation}
where it assumes the expectation of noise $E(\boldsymbol{\epsilon}) = \mathbf{0}$. If the number of images $N$ is big enough, the approximation of expectation $\hat{E}(\boldsymbol{\epsilon}) = \frac{1}{N} \sum_{i}{\boldsymbol{\epsilon}}_i$ will converge to $\mathbf{0}$. However, the assumption cannot hold when $N$ is small. Previous work \cite{liu2017quality}, \cite{yang2017neural} attempts to solve this problem by learning a weight scalar for each feature vector (referred to as instance) and their template representation can formulated as:
\begin{equation}
\mathbf{r}^{inst} = \frac{1}{N} \sum_{i} w_i (\boldsymbol{\mu} + \boldsymbol{\epsilon}_i) = \boldsymbol{\mu} + \frac{1}{N} \sum_{i} w_i \boldsymbol{\epsilon}_i,
\end{equation}
where $w_i$ is the weight for the feature vector of the $i^{th}$ image. Although it can reduce noise to some extent, it still assumes that individual components of a feature vector have correlated noise such that $\frac{1}{N} \sum_{i} w_i \epsilon_{ij} < \frac{1}{N} \sum_{i}{\epsilon_{ij}}$ for all $j$, where $j$ is the index of the $j^{th}$ feature component. However, such assumption does not hold if each component contains different amount of identity information and their noise values are weakly correlated to each other. In Figure~\ref{fig:heatmap} we can observe that the intra-class correlations between individual components (off-diagonal values) of the deep feature vectors are quite small (close to $0$). This inspires us to learn component-wise feature aggregation to further minimize noise on each component.
\begin{figure}
\captionsetup{font=footnotesize}
\centering
\includegraphics[width=\linewidth]{figs/corr_mat.pdf}
\caption{Heat map of the intra-class deep features extracted from the feature embedding module of C-FAN on IJB-S dataset~\cite{Kalka2018IJBS}. Instead of original correlation values, we show their absolute values because we are only interested in their magnitude. }
\label{fig:heatmap}
\end{figure}
The proposed fusion will generate different weights for each component of the deep feature vectors, and each component of the template representation is
\begin{equation}
r_{j}^{C-FAN} = \mu_j + \frac{1}{N} \sum_{i} w_{ij} \epsilon_{ij}.
\end{equation}
The proposed method will make better use of the components with more identity information instead of enhancing or weakening the whole feature vector if different components of a feature vector contain different identity information.
\subsection{Overall Framework}
\begin{figure}
\captionsetup{font=footnotesize}
\centering
\includegraphics[width=\linewidth]{figs/framework.pdf}
\caption{Framework of the proposed C-FAN.}
\label{fig: framework}
\end{figure}
The overall framework of the proposed C-FAN is presented in Figure \ref{fig: framework}. C-FAN incorporates a base CNN model for extracting face representation and a feature aggregation module. The base model is first trained on a large scale face dataset (MS-Celeb-1M~\cite{guo2016msceleb}). Then we fix the base model and train the aggregation module to learn the quality score for each single feature component on the same dataset using a different objective function. The aggregation module is attached to the last convolutional layer of the base model as a second branch besides the feature extraction layer. The features of a template are then pooled with the component-wise quality scores into a compact feature vector.
\subsection{Feature Aggregation Module}
Let $T = \{I_1, I_2, \cdots, I_N\}$ be a template of face images. Notice that $N$ here is not a fixed number but changes with template size. Let $\mathcal{H}(\cdot)$ denote deep convolutional layers before the feature extraction layer, and $\mathcal{F}(\cdot)$ denote the feature extraction layer.
The feature map of $i^\text{th}$ image in the template is generated by: $\mathcal{H}(I_i) = \mathbf{m}_i$, whose feature vector is then extracted by: $\mathcal{F}(\mathbf{m}_i) = \mathbf{f}_i$, where $\mathbf{f}_i$ is a $D$-dimensional feature vector. The corresponding quality vector is obtained by $\mathcal{Q}(\mathbf{m}_i) = \mathbf{q}_i$, where $\mathcal{Q}(\cdot)$ is the feature aggregation module, and $\mathbf{q}_i$ has the same dimension $D$ as the feature vector. The quality vectors are then normalized with a softmax operator.
Formally, given a set of quality vectors $\{\mathbf{q}_1, \mathbf{q}_2, \ldots, \mathbf{q}_N\}$, the $j^{\text{th}}$ component of the $i^{\text{th}}$ vector is normalized by:
\begin{equation}
w_{ij} = \frac{exp(q_{ij})}{\sum_{k=1}^{N}{exp(q_{kj})}}
\end{equation}
The final face representation of a template is obtained by pooling the feature vectors using the normalized quality vectors
\begin{equation}
\label{eq:fusion}
\mathbf{r} = \sum_{i=1}^{N}{\mathbf{f}_i \odot \mathbf{w}_i},
\end{equation}
where $\odot$ represents the operation of element-wise multiplication. The aggregated feature vector of a template $\mathbf{r}$ has the same dimensionality as a single face image frame extracted by the embedding module.
\subsection{Training C-FAN}
Given a pre-trained CNN for face feature extraction, C-FAN adds an aggregation module to the original network to predict the quality for each component. Unlike previous work~\cite{yang2017neural} which takes the feature vector as input, we use the last feature map as the input of the aggregation layer since it contains more information. The feature aggregation module includes only one batch normalization layer and one fully connected layer. The batch normalization layer is added here to alleviate the difficulty of hyper-parameter tuning by normalizing the input of the aggregation module. During training, we fix the base CNN model and only update the aggregation layer. That is, the new model only learns how to aggregate the original features without changing them. To train the aggregation layer, we use a template triplet loss, where each triplet consists of three random templates. The templates are randomly constructed online for each mini-batch. The template features are obtained by Equation~(\ref{eq:fusion}). Online hard triplet mining is adopted and the loss function is given by:
\begin{equation}
\mathcal{L}_{triplet} = \sum_{i=1}^{M}{[\alpha+d(\mathbf{r}^{a},\mathbf{r}^{+}) - d(\mathbf{r}^{a},\mathbf{r}^{-})]_+},
\end{equation}
where $M$ is the number of triplets, $\mathbf{r}^{a}$, $\mathbf{r}^{+}$ and $\mathbf{r}^{-}$ are the fused features of anchor template, positive template and negative template, respectively. $[x]_+=\max\{0,x\}$, $\alpha$ is a margin parameter and $d(\cdot)$ is the squared euclidean distance.
\section{Experiments}
\begin{figure*}
\newcolumntype{Y}{>{\centering\arraybackslash}X}
\captionsetup{font=footnotesize}
\centering
\setlength{\arrayrulewidth}{1px}
\begin{tabularx}{\linewidth}{Y|Y|Y}
\includegraphics[height=43px]{figs/dim_visualize/2999.jpg}\;\includegraphics[height=43px,clip,trim={0 5px 0 5px}]{figs/dim_visualize/2999_confidence.pdf}
& \includegraphics[height=43px]{figs/dim_visualize/4933.jpg}\;\includegraphics[height=43px,clip,trim={0 5px 0 5px}]{figs/dim_visualize/4933_confidence.pdf}
& \includegraphics[height=43px]{figs/dim_visualize/6024.jpg}\;\includegraphics[height=43px,clip,trim={0 5px 0 5px}]{figs/dim_visualize/6024_confidence.pdf} \\
\includegraphics[height=43px]{figs/dim_visualize/1800.jpg}\;\includegraphics[height=43px,clip,trim={0 5px 0 5px}]{figs/dim_visualize/1800_confidence.pdf}
& \includegraphics[height=43px]{figs/dim_visualize/4600.jpg}\;\includegraphics[height=43px,clip,trim={0 5px 0 5px}]{figs/dim_visualize/4600_confidence.pdf}
& \includegraphics[height=43px]{figs/dim_visualize/5100.jpg}\;\includegraphics[height=43px,clip,trim={0 5px 0 5px}]{figs/dim_visualize/5100_confidence.pdf} \\
\includegraphics[height=43px]{figs/dim_visualize/1600.jpg}\;\includegraphics[height=43px,clip,trim={0 5px 0 5px}]{figs/dim_visualize/1600_confidence.pdf}
& \includegraphics[height=43px]{figs/dim_visualize/3700.jpg}\;\includegraphics[height=43px,clip,trim={0 5px 0 5px}]{figs/dim_visualize/3700_confidence.pdf}
& \includegraphics[height=43px]{figs/dim_visualize/6000.jpg}\;\includegraphics[height=43px,clip,trim={0 5px 0 5px}]{figs/dim_visualize/6000_confidence.pdf} \\
\end{tabularx}
\caption{Component-wise quality scores given by C-FAN for images of three subjects in the IJB-S dataset. Unnormalized quality scores are shown here for fair comparison. In the plots, quality values are shown for each of the 512 components. Each column shows a high-quality photo, a medium-quality frame and a low-quality frame from the same person, respectively.}
\label{fig:quality-scores}
\end{figure*}
\subsection{Datasets and Protocols}
We evaluated C-FAN on three datasets including two video face datasets, the IARPA Janus Benchmark -- Surveillance (IJB-S) \cite{Kalka2018IJBS}, the YouTube Face datast (YTF) \cite{wolf2011face}; and a template-based face dataset, the IARPA Janus Benchmark A (IJB-A) \cite{klare2015pushing}.
\textbf{\textit{IJB-S}}: The IJB-S dataset \cite{Kalka2018IJBS} is a new dataset released by IARPA for evaluating face recognition systems on surveillance videos. The dataset includes $350$ surveillance videos spanning $30$ hours in total, $5,656$ enrollment images, and $202$ enrollment videos. This dataset provides a number of challenging face videos that are captured in real-world environments with a variety of subject activities and interactions. There are five 1:N face identification evaluations (supporting both open- and closed-set evaluation) involving surveillance video, including (i) surveillance-to-still \footnote{"Still" refers to single frontal still images.}, (ii) surveillance-to-booking \footnote{The "booking" reference template comprises the full set of images captured of a single subject at enrollment time.}, (iii) Multi-view surveillance-to-booking, (iv) surveillance-to-surveillance, and (v) UAV \footnote{UAV is a small fixed-wing unmanned aerial vehicle that was flown to collect images and videos.} surveillance-to-booking. In our experiments, we follow all five identification protocols to evaluate our method. Because of the poor quality of the images in IJB-S, we are only able to detect around $9$M out of $16$M faces. In the original protocol of IJB-S, the recognition performance is normalized by the detection rate~\cite{Kalka2018IJBS}. \emph{However, since our work focuses on recognition and not detection, we do not follow the original metric. Instead, we report the standard Identification Rate (IR) and TPIR@FPIR\footnote{True positive identification and false positive identification rate.}.} Failure-to-enroll images are ignored during feature aggregation. For templates that do not contain any detected faces, we set its representation as a zero-vector.
\textbf{\textit{YTF}}: The YouTube Face dataset is a video face dataset released in 2011~\cite{wolf2011face}. It contains $3,425$ videos of $1,595$ different subjects. The number of frames in the YTF face videos ranges from $48$ to $6,070$, and the average number of frames is $181$. Compared with IJB-S, the media of YTF is more photojournalistic \cite{Kalka2018IJBS}. In experiments, we follow the 1:1 face verification protocol with the given $5,00$0 video pairs.
\textbf{\textit{IJB-A}}: IJB-A\cite{klare2015pushing} is a template-based unconstrained face recognition benchmark. Although its images present similar challenges as IJB-S, the templates in the IJB-A include images from mixed media sources, with average image quality being better than IJB-S. The benchmark provides template-based 1:1 verification and 1:N identification protocols. IJB-A contains $500$ subjects with a total of $25,813$ images, and has been widely used by a number of both still image and video-based face recognition algorithms.
\begin{figure*}
\newcolumntype{Y}{>{\centering\arraybackslash}X}
\newcommand{0.2\linewidth}{0.2\linewidth}
\captionsetup{font=footnotesize}
\centering
\setlength{\arrayrulewidth}{1px}
\setlength{\tabcolsep}{0.7px}
\begin{tabularx}{\linewidth}{YYY}
Probe & Retrieved & Ground-truth \\
\includegraphics[width=0.2\linewidth]{figs/failure_surveillance/229_196/videos_109_2330.jpg}\hfill
\includegraphics[width=0.2\linewidth]{figs/failure_surveillance/229_196/videos_109_2434.jpg}\hfill
\includegraphics[width=0.2\linewidth]{figs/failure_surveillance/229_196/videos_109_2436.jpg}\hfill
\includegraphics[width=0.2\linewidth]{figs/failure_surveillance/229_196/videos_109_2437.jpg}\hfill
\includegraphics[width=0.2\linewidth]{figs/failure_surveillance/229_196/videos_109_2441.jpg}&
\includegraphics[width=0.2\linewidth]{figs/failure_surveillance/229_196/videos_1101_32704.jpg}\hfill
\includegraphics[width=0.2\linewidth]{figs/failure_surveillance/229_196/videos_1101_75249.jpg}\hfill
\includegraphics[width=0.2\linewidth]{figs/failure_surveillance/229_196/videos_1101_75276.jpg}\hfill
\includegraphics[width=0.2\linewidth]{figs/failure_surveillance/229_196/videos_1101_75288.jpg}\hfill
\includegraphics[width=0.2\linewidth]{figs/failure_surveillance/229_196/videos_1101_75638.jpg}&
\includegraphics[width=0.2\linewidth]{figs/failure_surveillance/229_196/videos_1101_129952.jpg}\hfill
\includegraphics[width=0.2\linewidth]{figs/failure_surveillance/229_196/videos_1101_129960.jpg}\hfill
\includegraphics[width=0.2\linewidth]{figs/failure_surveillance/229_196/videos_1101_130036.jpg}\hfill
\includegraphics[width=0.2\linewidth]{figs/failure_surveillance/229_196/videos_1101_130058.jpg}\hfill
\includegraphics[width=0.2\linewidth]{figs/failure_surveillance/229_196/videos_1101_129952.jpg} \\
\includegraphics[width=0.2\linewidth]{figs/failure_surveillance/112_89/videos_5047_5369.jpg}\hfill
\includegraphics[width=0.2\linewidth]{figs/failure_surveillance/112_89/videos_5047_5375.jpg}\hfill
\includegraphics[width=0.2\linewidth]{figs/failure_surveillance/112_89/videos_5047_5730.jpg}\hfill
\includegraphics[width=0.2\linewidth]{figs/failure_surveillance/112_89/videos_5047_6454.jpg}\hfill
\includegraphics[width=0.2\linewidth]{figs/failure_surveillance/112_89/videos_5047_6469.jpg} &
\includegraphics[width=0.2\linewidth]{figs/failure_surveillance/112_89/videos_1001_59466.jpg}\hfill
\includegraphics[width=0.2\linewidth]{figs/failure_surveillance/112_89/videos_1001_59497.jpg}\hfill
\includegraphics[width=0.2\linewidth]{figs/failure_surveillance/112_89/videos_1001_59589.jpg}\hfill
\includegraphics[width=0.2\linewidth]{figs/failure_surveillance/112_89/videos_1001_59630.jpg}\hfill
\includegraphics[width=0.2\linewidth]{figs/failure_surveillance/112_89/videos_1001_60261.jpg}&
\includegraphics[width=0.2\linewidth]{figs/failure_surveillance/112_89/videos_904_50501.jpg}\hfill
\includegraphics[width=0.2\linewidth]{figs/failure_surveillance/112_89/videos_904_51169.jpg}\hfill
\includegraphics[width=0.2\linewidth]{figs/failure_surveillance/112_89/videos_904_51334.jpg}\hfill
\includegraphics[width=0.2\linewidth]{figs/failure_surveillance/112_89/videos_904_61155.jpg}\hfill
\includegraphics[width=0.2\linewidth]{figs/failure_surveillance/112_89/videos_904_63212.jpg} \\
\includegraphics[width=0.2\linewidth]{figs/failure_booking/173_109/videos_28_716.jpg}\hfill
\includegraphics[width=0.2\linewidth]{figs/failure_booking/173_109/videos_28_725.jpg}\hfill
\includegraphics[width=0.2\linewidth]{figs/failure_booking/173_109/videos_28_809.jpg}\hfill
\includegraphics[width=0.2\linewidth]{figs/failure_booking/173_109/videos_28_827.jpg}\hfill
\includegraphics[width=0.2\linewidth]{figs/failure_booking/173_109/videos_28_842.jpg}&
\includegraphics[width=0.2\linewidth]{figs/failure_booking/173_109/img_100202.jpg}\hfill
\includegraphics[width=0.2\linewidth]{figs/failure_booking/173_109/img_100203.jpg}\hfill
\includegraphics[width=0.2\linewidth]{figs/failure_booking/173_109/img_100204.jpg}\hfill
\includegraphics[width=0.2\linewidth]{figs/failure_booking/173_109/img_100205.jpg}\hfill
\includegraphics[width=0.2\linewidth]{figs/failure_booking/173_109/img_100206.jpg}&
\includegraphics[width=0.2\linewidth]{figs/failure_booking/173_109/img_101062.jpg}\hfill
\includegraphics[width=0.2\linewidth]{figs/failure_booking/173_109/img_101063.jpg}\hfill
\includegraphics[width=0.2\linewidth]{figs/failure_booking/173_109/img_101064.jpg}\hfill
\includegraphics[width=0.2\linewidth]{figs/failure_booking/173_109/img_101065.jpg}\hfill
\includegraphics[width=0.2\linewidth]{figs/failure_booking/173_109/img_101066.jpg}\\
\includegraphics[width=0.2\linewidth]{figs/failure_booking/183_112/videos_5081_8336.jpg}\hfill
\includegraphics[width=0.2\linewidth]{figs/failure_booking/183_112/videos_5081_11684.jpg}\hfill
\includegraphics[width=0.2\linewidth]{figs/failure_booking/183_112/videos_5081_11747.jpg}\hfill
\includegraphics[width=0.2\linewidth]{figs/failure_booking/183_112/videos_5081_11747.jpg}\hfill
\includegraphics[width=0.2\linewidth]{figs/failure_booking/183_112/videos_5081_11806.jpg}&
\includegraphics[width=0.2\linewidth]{figs/failure_booking/183_112/img_100223.jpg}\hfill
\includegraphics[width=0.2\linewidth]{figs/failure_booking/183_112/img_100224.jpg}\hfill
\includegraphics[width=0.2\linewidth]{figs/failure_booking/183_112/img_100225.jpg}\hfill
\includegraphics[width=0.2\linewidth]{figs/failure_booking/183_112/img_100226.jpg}\hfill
\includegraphics[width=0.2\linewidth]{figs/failure_booking/183_112/img_100227.jpg}&
\includegraphics[width=0.2\linewidth]{figs/failure_booking/183_112/img_100992.jpg}\hfill
\includegraphics[width=0.2\linewidth]{figs/failure_booking/183_112/img_100993.jpg}\hfill
\includegraphics[width=0.2\linewidth]{figs/failure_booking/183_112/img_100994.jpg}\hfill
\includegraphics[width=0.2\linewidth]{figs/failure_booking/183_112/img_100995.jpg}\hfill
\includegraphics[width=0.2\linewidth]{figs/failure_booking/183_112/img_100996.jpg}\\
\end{tabularx}
\caption{Examples of failed templates on the IJB-S dataset. The first two rows show the failure cases from "surveillance-to-surveillance" protocol. The last two show the failure cases from "surveillance-to-booking" protocol. Because of space limit, we only show 5 randomly selected images for each template.}
\label{fig:failure-examples}
\end{figure*}
\subsection{Implementation Details}
\textbf{\textit{Pre-processing}}: We employ a facial landmark detection algorithm MTCNN \cite{zhang2016joint} to detect and align all faces in training and testing. Each face is cropped from the detected face region and resized to $112 \times 96$ pixels using a similarity transformation based on the detected five facial landmarks.
\textbf{\textit{Training}}: We first train a base CNN model on a cleaned version\footnote{\url{https://github.com/AlfredXiangWu/face\_verification\_experiment}.} of MS-Celeb-1M dataset~\cite{guo2016msceleb} to learn face recognition of still images, for which we adopts the Face-ResNet (DeepVisage) architecture~\cite{hasnat2017deepvisage}. The component-wise quality module is then trained on the same training dataset using the template triplet loss. The parameters of the aggregation layer are optimized using stochastic gradient descent with a momentum of $0.9$ and a weight decay of $0.01$. Gaussian blur and motion blur are randomly applied to augment the training data. We set $\alpha=1.0$. Each mini-batch samples $240$ images of $80$ templates from $20$ random subjects, $3$ images per template. The module is trained for $4,000$ steps in all. All these hyper-parameters are chosen by testing on a validation set, which is a small subset of IJB-S. All the experiments are conducted on a Nvidia Geforce GTX 1080 Ti GPU. The average feature extraction speed on this GPU is $1$ms per image. After training on MS-Celeb-1M, the feature network and aggregation module are applied to the testing datasets without fine-tuning.
\subsection{Baseline}
We compare the proposed component-wise pooling with the other two pooling types, average pooling and instance-level pooling, in particular we compare between three methods:
\begin{itemize}
\vspace{-0.6em}
\item \textbf{Average} directly applies the average pooling to the base CNN features to generate the template representation.\vspace{-0.6em}
\item \textbf{Instance} is trained similarly as C-FAN, but it only outputs one quality scalar for each image and aggregates the base CNN feature at the instance-level.\vspace{-0.6em}
\item \textbf{C-FAN} generates a quality vector for each image and does component-wise aggregation.\vspace{-0.6em}
\end{itemize}
After the aggregation, cosine similarity is used to compare the template representations.
\subsection{Qualitative Analysis on IJB-S}
To explore the relationship between visual quality and the quality scores generated by C-FAN, we visualize the component-wise quality scores of images from IJB-S dataset. Figure \ref{fig:quality-scores} shows the images of three subjects with the corresponding 512-dimensional quality vectors. We can observe that images with high visual quality have higher quality scores than those with low visual quality. The images with motion blur, occlusion, and deformity tend to be assigned with negative quality values. We also observe that each component of a deep feature vector presents different response to the noise in images.
Figure \ref{fig:failure-examples} shows some failure examples of face identification on IJB-S dataset. For the ``Surveillance-to-Surveillance'' protocol, both probe templates and gallery templates are frames from surveillance videos, making it very difficult to discriminate identities from the images with such low quality and significant variations. However, we observe that the proposed C-FAN tends to retrieve templates with good visual quality. The ground-truth images are more blurry than the retrieved ones in the top two rows of Figure \ref{fig:failure-examples}. Even though probe frames have low visual quality in the ``Surveillance-to-booking'' protocol, we can still observe some facial patterns shared by the probe and the retrieved templates, e.g., the same hair style, the blurry cap and the hair bang.
\subsection{Quantitative Analysis on IJB-S}
\begin{table*}[!h]
\centering
\footnotesize
\caption{Performance comparisons on IJB-S dataset.}
\label{table: ijbs}
\centering
\scalebox{1.0}{
\begin{tabularx}{0.9\linewidth}{X c c c c c c c c}
\toprule
\multirow{2}{*}{Test Name} & \multirow{2}{*}{Method} && \multicolumn{3}{c}{Closed-set (\%)} && \multicolumn{2}{c}{Open-set (\%)} \\
\cline{4-6} \cline{8-9}
& && Rank-1 & Rank-5 & Rank-10 && 1 \% FPIR & 10 \% FPIR \\
\midrule
\multirow{3}{*}{Surveillance-to-still} & Average && $47.93$ & $58.86$ & $63.20$ && $15.22$ & $21.77$ \\
& Instance && $49.57$ & $59.58$ & $64.07$ && $\mathbf{16.48}$ & $23.14$ \\
& C-FAN && $\mathbf{50.82}$ & $\mathbf{61.16}$ & $\mathbf{64.95}$ && $16.44$ & $\mathbf{24.19}$ \\
\midrule
\multirow{3}{*}{Surveillance-to-booking} & Average&& $49.31$ & $59.92$ & $64.44$ && $26.27$ & $27.05$ \\
& Instance && $51.14$ & $61.43$ & $65.37$ && $26.64$ & $28.16$ \\
& C-FAN && $\mathbf{53.04}$ & $\mathbf{62.67}$ & $\mathbf{66.35}$ && $\mathbf{27.40}$ & $\mathbf{29.70}$ \\
\midrule
Multi-view & Average && $92.57$ & $97.52$ & $99.01$ && $58.42$ & $77.72$ \\
\multirow{2}{*}{Surveillance-to-booking} & Instance && $94.55$ & $99.01$ & $99.50$ && $61.39$ & $83.17$ \\
& C-FAN && $\mathbf{96.04}$ & $\mathbf{99.50}$ & $\mathbf{99.50}$ && $\mathbf{70.79}$ & $\mathbf{85.15}$ \\
\midrule
\multirow{3}{*}{Surveillance-to-Surveillance} & Average && $9.38$ & $17.44$ & $\mathbf{22.14}$ && $0.03$ & $0.54$ \\
& Instance && $8.90$ & $16.61$ & $21.38$ && $0.06$ & $0.54$ \\
& C-FAN && $\mathbf{10.05}$ & $\mathbf{17.55}$ & $21.06$ && $\mathbf{0.11}$ & $\mathbf{0.68}$ \\
\midrule
UAV & Average && $1.27$ & $8.86$ & $13.92$ && $0.00$ & $0.00$ \\
\multirow{2}{*}{Surveillance-to-booking} & Instance && $5.06$ & $11.39$ & $15.19$ && $0.00$ & $0.00$ \\
& C-FAN && $\mathbf{7.59}$ & $\mathbf{12.66}$ & $\mathbf{20.25}$ && $0.00$ & $0.00$ \\
\bottomrule
\end{tabularx}}
\end{table*}
Table \ref{table: ijbs} reports identification results on IJB-S dataset. We compare C-FAN with average pooling and instance-level aggregation. Both instance-level aggregation and C-FAN outperforms average pooling on almost all protocols of IJB-S, showing that the features aggregated by quality weights are more discriminative than by simply computing the mean representations. Furthermore, compared with instance-level aggregation, C-FAN achieves higher identification rates on most of the protocols, showing the advantage of the aggregating different features components separately. With the expectation of the Multi-view protocol, the performance of our model on IJB-S dataset is significantly lower than on other datasets (See Section~\ref{sec:ytf_ijba}), showing that this dataset is indeed much more challenging. Notice that the open-set TPIR of ``UAV Surveillance-to-booking'' are all $0\%$. This is because $29$ out of $79$ probe templates are empty (no faces detected) and thus it is impossible to control the FPIR under $36.71\%$.
\begin{table}[t]
\centering
\footnotesize
\caption{Verification performance on YouTube Face dataset, compared with baseline methods and other state-of-the-art methods.}
\label{table: ytf}
\scalebox{1.0}{
\begin{tabularx}{\linewidth}{X c X c}
\toprule
Method & Accuracy (\%) & Method & Accuracy (\%)\\
\midrule
EigenPEP~\cite{li2014eigen} & $84.8 \pm 1.4$ & DeepFace \cite{taigman2014deepface} & $91.4 \pm 1.1$ \\
DeepID2+~\cite{sun2015deeply} & $93.2 \pm 0.2$ & Wen~\etal~\cite{wen2016discriminative} & $94.9$ \\
FaceNet~\cite{Schroff_2015_CVPR} & $95.52 \pm 0.06$ & DAN~\cite{rao2017learning} & $94.28 \pm 0.69$ \\
NAN~\cite{yang2017neural} & $95.72 \pm 0.64$ & QAN~\cite{liu2017quality} & $96.17 \pm 0.09$ \\
\midrule
\textit{Average} & $96.36 \pm 1.01$ & \textit{Instance} & $96.42 \pm 0.95$ \\
&& \textit{C-FAN} & $\mathbf{96.50} \pm \mathbf{0.90}$ \\
\bottomrule
\end{tabularx}}
\end{table}
\subsection{Performance Comparison on YTF and IJB-A}
\label{sec:ytf_ijba}
\begin{table*}
\centering
\caption{Comparisons of verification and identification on IJB-A dataset.}
\label{table: ijba}
\footnotesize
\begin{tabularx}{\linewidth}{X c c c c c c c c}
\toprule
\multirow{2}{*}{Method} & \multicolumn{2}{c}{Verification TAR (\%)} && \multicolumn{2}{c}{Close-set Identification (\%)} && \multicolumn{2}{c}{Open-set Identification (\%)} \\
\cline{2-3} \cline{5-6} \cline{8-9}
& 0.1\% FAR & 1\% FAR && Rank-1 & Rank-5 && 1\% FPIR & 10\% FPIR \\
\midrule
LSFS \cite{wang2017face} & $51.4 \pm 6.0$ & $73.3 \pm 3.4$ && $82.0 \pm 2.4$ & $92.9 \pm 1.3$ && $38.3 \pm 6.3$ & $61.3\pm 3.2$ \\
Pose-aware Models \cite{masi2016pose} & $65.2 \pm 3.7$ & $82.6 \pm 1.8$ && $84.0 \pm 1.2$ & $92.5 \pm 0.8$ && - & - \\
Masi~\etal~ \cite{masi2016we} & $72.5$ & $88.6$ && $90.6$ & $96.2$ && - & - \\
Triplet Embedding \cite{sankaranarayanan2016triplet} & $81.3 \pm 2$ & $90.0 \pm 1.0$ && $93.2 \pm 1.0$ & - && $75.3 \pm 3.0$ & $86.3 \pm 1.4$ \\
Shi~\etal~ \cite{shi2018improving} & $60.2 \pm 6.9$ & $82.3 \pm 2.2$ && $89.8 \pm 0.92$ & $96.0 \pm 0.6$ && $58.9 \pm 1.6 $ & $64.7 \pm 1.4$ \\
QAN \cite{liu2017quality} & $89.3 \pm 3.9$ & $94.2 \pm 1.5$ && - & - && - & - \\
NAN \cite{yang2017neural} & $88.1 \pm 1.1$ & $94.1 \pm 0.8$ && $95.8 \pm 0.5$ & $98.0 \pm 0.5$ && $81.7 \pm 4.1$ & $91.7 \pm 0.9$ \\
Pooling Faces \cite{hassner2016pooling} & $63.1$ & $81.9$ && $84.6$ & $93.3$ && - & - \\
\midrule
\textit{Average} & $91.14 \pm 1.13$ & $\mathbf{93.99} \pm \mathbf{0.79}$ && $94.44 \pm 0.88$ & $96.22 \pm 0.53$ && $86.23 \pm 4.30$ & $92.36 \pm 1.00$ \\
\textit{Instance} & $91.52 \pm 0.73$ & $93.78 \pm 0.77$ && $94.54 \pm 0.87$ & $96.24 \pm 0.57$ && $86.58 \pm 4.76$ & $92.45 \pm 0.99$ \\
\textit{C-FAN} & $\mathbf{91.59} \pm \mathbf{0.99}$ & $93.97 \pm 0.78$ && $\mathbf{94.57} \pm \mathbf{0.82}$ & $\mathbf{96.27} \pm \mathbf{0.53}$ && $\mathbf{86.88} \pm \mathbf{4.70}$ & $\mathbf{92.85} \pm \mathbf{1.02}$ \\
\bottomrule
\end{tabularx}
\end{table*}
Table \ref{table: ytf} and Table \ref{table: ijba} report the results of C-FAN, the corresponding baseline methods, and other state-of-the-art on YTF and IJB-A, respectively. C-FAN achieves the best performance on YTF dataset. In particular, C-FAN outperforms the state-of-the-art algorithm NAN \cite{yang2017neural} by $0.78\%$ and QAN by $0.33\%$. On IJB-A dataset, C-FAN achieves the best performance on 1:1 verification and open-set identification tasks, compared to previous methods. It can be seen that C-FAN outperforms NAN by $3.49\%$ at $0.1\%$ FAR, and by $5.18\%$ at $1\%$ FPIR, respectively. Notice that the gaps between C-FAN and two baseline pooling methods are relatively small. This is because the images in YTF and IJB-A datasets are not typical video frames from real-world surveillance scenarios and environments. Most images in these datasets contain rich information for face recognition. Thus, a smaller improvement is expected for quality-aware feature aggregation.
\section{Conclusion}
In this paper, we propose a component-wise feature aggregation network (C-FAN) for video face recognition. It adaptively predicts quality values for each component of a deep feature vector and aggregates the most discriminative components to generate a single feature vector for a set of face images. We empirically demonstrate that the quality scores predicted by C-FAN fit the visual quality of images and are also beneficial in template representation by retaining discriminative components of feature vectors. Our future work will explore an aggregation network that combines different levels of fusion.
{\footnotesize
\bibliographystyle{ieee}
|
1,116,691,499,268 | arxiv | \section{Introduction}
\IEEEPARstart{W}ith the development of the Internet, we have entered the era of big data. It is consequently a natural idea to leverage the large scale yet noisy data on the web for various vision tasks \cite{li2015weakly,zhang2014weakly,shen2013approximate,tang2016weakly,shen2016fast,shen2016face}. Methods of exploiting web images for automatic image dataset construction have recently become a hot topic \cite{hua2015prajna,schroff2011harvesting,icme2016yao,li2010optimol} in the field of multimedia processing. Existing methods \cite{hua2015prajna,schroff2011harvesting,li2010optimol} usually use an iterative mechanism in the process of image selection. However, due to the visual feature distribution of images selected in this way, these datasets tend to have the dataset bias problem \cite{niu2015visual,torralba2011unbiased,mm2016yao}.
Fig. \ref{fig1} shows the ``airplane" images from four different image datasets. We can observe some significant differences in these datasets: PASCAL \cite{everingham2010pascal} shows ``airplanes" from the flying viewpoint, while SUN \cite{xiao2010sun} tends to show distant views at the airport; Caltech \cite{griffin2007caltech} has a strong preference for side views and
\begin{figure}[tbp]
\centering
\includegraphics[width=0.43\textwidth]{1.pdf}
\caption{Most discriminative images for ``airplane" from four different datasets. Each dataset has their preference for image selection.}
\label{fig1}
\end{figure}
ImageNet \cite{deng2009imagenet} is rich in diversity, but mainly contains close-range views. Classifiers learned from these datasets usually perform poorly in domain adaptation tasks \cite{niu2015visual}.
To address the dataset bias problem, a large number of domain-robust approaches have been proposed for various vision tasks \cite{vijayanarasimhan2008keywords,duan2011improving,shen2015hashing}. The images in these methods are partitioned into a set of clusters; each cluster is treated as a ``bag" and the images in each bag as ``instances". As a result, these tasks can be formulated as a multi-instance learning (MIL) problem. Different MIL methods have been proposed in \cite{vijayanarasimhan2008keywords,duan2011improving}. However, the yield for all of these methods is limited by the diversity of training data which was collected with a single query.
To obtain highly accurate and diverse candidate images, as well as to overcome the download restrictions of the image search engine, Divvala et al. \cite{divvala2014learning} and Yao et al. \cite{icme2016yao,yao2016new} proposed the use of multiple query expansions instead of a single query to collect candidate images from the image search engine. The issue remains that these methods still use iterative mechanisms in the process of image selection, which leads to the dataset bias problem \cite{niu2015visual,torralba2011unbiased,mm2016yao}.
Motivated by the situation described above, we target the construction of an image dataset in a scalable way, while ensuring accuracy and robustness. The basic idea is to leverage multiple query expansions for initial candidate images collection and to use MIL methods for selecting images from different distributions. We first expand each query to a set of query expansions, from which the visually non-salient and less relevant expansions are filtered out. Then we set the rest as selected query expansions and construct the raw image dataset with these selected query expansions. By treating each selected query expansion as a ``bag" and the images therein as ``instances", we formulate image selection and noise removal as a multi-instance learning problem. In this way, images from different distributions will be kept while noise is filtered out.
To verify the effectiveness of our proposed approach, we build an image dataset with 20 categories, which we refer to as
DRID-20. We compare the image classification ability, cross-dataset generalization ability and diversity of our dataset with three manually labelled datasets and three automated datasets, to demonstrate the domain robustness of our dataset. We also report the results of object detection on PASCAL VOC 2007, and then compare the object detection ability of our method with weakly supervised and web-supervised methods.
The main contributions of this work are summarized as follows:
[1.] To the best of our knowledge, this is the first proposal for automatic domain-robust image dataset construction. Our proposed approach, based on multiple query expansions and multi-instance learning, considers the source of candidate images and retains images from different distributions. The dataset constructed by our approach thus efficiently alleviates the dataset bias problem.
[2.] To suppress the search error and noisy query expansions induced noisy images, we formulate image selection as a multi-instance learning problem and propose to solve the associated optimization problems by the cutting-plane and concave-convex procedure (CCCP) algorithm, respectively.
[3.] We have released our image dataset DRID-20 on \href{https://drive.google.com/drive/folders/0B7dS7AFpUzt1bmpwcFRKcDZwUUE?usp=sharing}{Google Drive}. We hope the diversity of DRID-20 will offer unparalleled opportunities to researchers in the multi-instance learning, transfer learning, image dataset construction and other related fields.
This paper is an extended version of \cite{mm2016yao}. The extensions include: Taking both bag level and instance level noisy images into account in the process of image selection instead of only instance level noisy images, we use a combination of bag level and instance level selection mechanisms and achieve better results; comparing the image classification ability and diversity of our dataset DRID-20 with three manually labelled datasets (STL-10, CIFAR-10 and ImageNet) and three automated datasets (Optimol, Harvesting and AutoSet); and increasing the number of categories in the dataset from 10 to 20, so that our dataset DRID-20 covers all categories in the PASCAL VOC 2007 dataset.
The rest of the paper is organized as follows: In Section \uppercase\expandafter{\romannumeral2}, a brief discussion of related works is given. The proposed algorithm including query expanding, noisy expansions filtering and noisy images filtering is described in Section \uppercase\expandafter{\romannumeral3}. We evaluate the performance of the proposed algorithm against several other methods in Section \uppercase\expandafter{\romannumeral4}. Lastly, the conclusion and future work are offered in Section \uppercase\expandafter{\romannumeral5}.
\section{Related works}
Given the importance of labelled image datasets in the area of high-level image understanding, many efforts have been directed toward image dataset construction. In general, these efforts can be divided into three principal categories: manual annotation, active learning and automatic methods.
\subsection{Manual Annotation and Active Learning methods}
In the early years, manual annotation was the most important way to construct image datasets. (e.g., STL-10 \cite{coates2011analysis}, CIFAR-10 \cite{krizhevsky2009learning}, PASCAL VOC 2007 \cite{everingham2010pascal}, ImageNet \cite{deng2009imagenet} and Caltech-101 \cite{griffin2007caltech}). The process of constructing these datasets mainly consists of submitting keywords to an image search engine to download candidate images, then cleaning these candidate images by manual annotation. This method usually has a high accuracy, but is labor intensive.
To reduce the cost of manual annotation, a large number of works have focused on active learning (a special case of semi-supervised method). Li et al. \cite{collins2008towards} randomly labelled some seed images to learn visual classifiers. The learned visual classifiers were then implemented to conduct image classification on unlabelled images, to find low confidence images for manual labelling. Here low confidence images are those whose probability is classified into positive and negative close to 0.5. The process is iterated until sufficient classification accuracy is achieved. Siddiquie et al. \cite{siddiquie2010beyond} presented an active learning framework to simultaneously learn contextual models for scene understanding tasks (multi-class classification). Grauman et al. \cite{vijayanarasimhan2014large} presented an approach for on-line learning of object detectors, in which the system automatically refines its models by actively requesting crowd-sourced annotations on images crawled from the web.
However, both manual annotation and active learning require pre-existing annotations, which often results in one of the most significant limitations to construct a large scale image dataset.
\subsection{Automatic Methods}
To further reduce the cost of manual annotation, automatic methods have attracted more and more people's attention. Schroff et al. \cite{schroff2011harvesting} adopted text information to rank images retrieved from a web search and used these top-ranked images to learn visual models to re-rank images once again. Li et al. \cite{li2010optimol} leveraged the first few images returned from an image search engine to train the image classifier, classifying images as positive or negative. When the image is classified as a positive sample, the classifier uses incremental learning to refine its model. With the increase in the number of positive images accepted by the classifier, the trained classifier will reach a robust level for this query. Hua et al. \cite{hua2015prajna} proposed to use clustering based method to filter ``group'' noisy images and propagation based method to filter individual noisy images. The advantage of these methods is that the need for manual intervention is eliminated. However, for methods \cite{schroff2011harvesting,li2010optimol,hua2015prajna}, the domain adaptation ability is limited by the initial candidate images and the iterative mechanism in the process of image selection. In order to obtain a variety of candidate images, Yao et al. \cite{icme2016yao} proposed the use of multiple query expansions instead of a single query in the process
\begin{figure*}[htb]
\centering
\includegraphics[width=0.95\textwidth]{2.pdf}
\caption{Domain robust image dataset construction framework. The input is text query that we would like to build a image dataset for. The outputs are a set of selected images corresponding to the given query.}
\label{fig2}
\end{figure*}
of initial candidate images collection, then using an iterative mechanism to filter noisy images. The automatic works discussed here mainly focus on accuracy and scale in the process of image dataset construction, which often results in a poor performance on domain adaptation.
\subsection{Other Related Works}
There are many works related to the generation of query expansions and noisy images filtering, though they are not aimed at image dataset construction. Since most image search engines restrict the number of images returned for each query, WordNet \cite{miller1995wordnet} and ConceptNet \cite{speer2013conceptnet} are often used to obtain synonyms to overcome the download restriction of these search engines. The advantage of WordNet and ConceptNet is that synonyms are usually relevant to the given query and almost do not need to be purified. The disadvantage of WordNet and ConceptNet is that both of them are usually not comprehensive enough for query expanding. Worse, the images returned from image search engine using synonyms tend to experience the homogenization problem, which results in poor performance on domain adaptation.
In order to alleviate the homogenization problem, recent works \cite{icme2016yao,divvala2014learning} proposed the use of Google Books Ngram Corpus (GBNC) \cite{lin2012syntactic} instead of WordNet and ConceptNet to obtain query expansions for candidate images collection. The Google Books Ngrams Corpus covers almost all related queries at the text level. It is much more general and richer than WordNet and ConceptNet. The disadvantage of using GBNC for query expanding is that it may also bring noisy query expansions. Recently, word embedding \cite{cilibrasi2007google} provides a learning-based method for computing the word-word similarity distance which can be used to filter noisy query expansions. In this paper, we use GBNC to expand the query to a set of query expansions, and then take both word-word and visual-visual similarity to filter noisy query expansions.
To efficiently ease the dataset bias problem, several authors have developed domain-robust approaches for vision tasks. Duan et al. \cite{duan2011improving} clustered relevant images using both textual and visual features. By treating each cluster as a ``bag" and the images in the bag as ``instances", the authors formulated this problem as a multi-instance learning problem (MIL) which learns a target decision function for image re-ranking. However, the yield is limited by the initial candidate images obtained from image search engine with a single query. In this paper, we focus on the MIL method, as it can retain images from different distributions with noisy images filtered out.
It can be anticipated that there will be more visual patterns (corresponding to different query expansions) in our work to represent the given query. In addition, MIL methods are applied to filter noisy images and keep images from different distributions. In return, the constructed dataset could achieve a better domain adaptation ability than traditional datasets constructed by a single query and an iterative mechanism.
\section{Domain robust image dataset construction}
We seek to construct a domain-robust image dataset that can generalize to unseen target domains. As shown in Fig. \ref{fig2}, we propose our web-supervised image dataset construction framework by three major steps: query expanding, noisy query expansions filtering and noisy images filtering. We expand the query to a set of semantically rich expansions by searching Google Books Ngram Corpus, from which the visually non-salient and less relevant expansions are filtered out. After obtaining the candidate images by retrieving the selected expansions with an image search engine, we treat each selected expansion as a ``bag" and the images in each bag as ``instances". We then formulate image selection and noisy images filtration as an MIL problem with constrained positive bags. In particular, the learned classifiers are used to filter individual noisy images (corresponding to the top-ranked images for selected expansions) and group noisy images (corresponding to the positive bags). Using this approach, images from different distributions will be kept while noisy images are filtered out, and a domain-robust image dataset will be constructed.
\subsection{Query Expanding}
Image datasets constructed by existing methods tend to be highly accurate but usually have weak domain adaptation ability \cite{niu2015visual,torralba2011unbiased,mm2016yao}. To construct a domain-robust image dataset, we expand given query (e.g., ``horse'') to a set of query expansions (e.g., ``jumping horse, walking horse, roaring horse'') and then use these different query expansions (corresponding images) to reflect the different ``visual distributions" of the query. We use GBNC to discover query expansions for the given query with Parts-Of-Speech (POS), specifically with NOUN, VERB, ADJECTIVE and ADVERB. GBNC is much more general and richer than WordNet \cite{miller1995wordnet} and ConceptNet \cite{speer2013conceptnet}. Using GBNC can help us to find all the expansions ever published for any possible query from the text semantics \cite{divvala2014learning}.
\subsection{Noisy Expansions Filtering}
Through query expanding, we obtain a comprehensive semantic description for the given query. However, query expanding not only provides all the useful query expansions, but also some noise. These noisy query expansions can be roughly divided into two types: (1) visually non-salient (e.g., ``betting horse'') and (2) less relevant (e.g., ``sea horse''). Using these noisy query expansions to retrieve images will have a negative effect on dataset accuracy and robustness.
\subsubsection{Visual non-salient expansions filtering}
From the visual perspective, we aim to identify visually salient and eliminate non-salient query expansions in this step. The intuition is that visually salient expansions should exhibit predictable visual patterns. Hence, we can use the image classifier-based filtering method.
For each query expansion, we directly download the top $N$ images from the Google image search engine as positive images ( based on the fact that the top few images returned from image search engine tend to be positive), then randomly split these images into a training set and validation set $I_i=\{I_i^t, I_i^v\}$. We gather a random pool of negative images and split them into a training set and validation set $\overline I=\{\overline I^t, \overline I^v\}$. We train a linear support vector machine (SVM) classifier $C_i$ with $I_i^t$ and $\overline I^t$ using dense histogram of oriented gradients (HOG) features \cite{dalal2005histograms}. We then use $\{I_i^v,\overline I^v\}$ as validation images to calculate the classification results. We declare a query expansion $i$ to be visually salient if the classification results $S_i$ give a relatively high score.
\subsubsection{Less relevant expansions filtering}
From the relevance perspective, we want to identify both semantically and visually relevant expansions for the given query. The intuition is that relevant expansions should have a relatively small semantic and visual distance; therefore, we use a combined word-word and visual-visual similarity distance-based filtering method.
Words and phrases acquire meaning from the way they are used in society. For computers, the equivalent of ``society'' is ``database'', and the equivalent of ``use'' is ``a way to search the database'' \cite{cilibrasi2007google}. Normalized Google Distance (NGD) constructs a method to extract semantic similarity distance from the World Wide Web (WWW) using Google page counts. For a search term $x$ and search term $y$, NGD is defined by:
\begin{equation}\label{eq1}
\mathrm{NGD}(x,y)=\frac{\max\{\log f(x),\log f(y)\}-\log f(x,y)}{\log N-\min\{\log f(x),\log f(y)\}}
\end{equation}
where $f(x)$ denotes the number of pages containing $x$, $f(x,y)$ denotes the number of pages containing both $x$ and $y$ and $N$ is the total number of web pages searched by Google.
We denote the semantic distance of all query expansions by a graph $G_{semantic} = \{N, D\}$ in which each node represents a query expansion and its edge represents the NGD between two nodes. We set the target query as center $ y $ and other expansions have a score $D_{xy}$ which corresponds the NGD to the target query.
Similarly, we represent the visual distance of query and expansions by a graph $G_{visual} = \{C,E\}$ in which each node represents a query expansion and each edge represents the visual distance between the query and the expansions. We denote the visual distribution of each query expansion by the compound feature $\phi_{k}=\frac{1}{k}\sum_{i=1}^{k}x_{i}$ of its first $k$ images from the image search engine. We set the target query as center $ y $ and other query expansions have a score $E_{xy}$ which corresponds to the Euclidean distance to the target query.
The semantic distance $D_{xy}$ and visual distance $E_{xy}$ will be used to construct a new two-dimensional feature $V = [D_{xy};E_{xy}]$. The problem is to calculate the importance weight $w$ and bias penalty $b$ in decision function $f(x) = w^{T}x+b$ to determine whether or not the expansion is relevant. There are many methods of obtaining these coefficients $w$ and $b$. Here we take the linear SVM to work around this problem. Although the linear SVM is not the prevailing state-of-the-art method for classification, we find our method to be effective in pruning irrelevant query expansions.
We set the remainder which is not filtered out as the selected expansions and construct raw image dataset by retrieving the top $N$ images from image search engine with these selected query expansions.
Regardless of the fact that our method is not able to remove noisy expansions thoroughly in most cases, the raw image dataset constructed by our method still achieves much higher accuracy than directly using the Google image data. Besides, the raw image dataset constructed through the selected query expansions has much richer visual distributions.
\subsection{Noisy Images Filtering}
Although the Google image search engine has ranked the returned images, some noisy images may still be included. In addition, a few noisy expansions which are not filtered out will also bring noisy images to the raw image dataset. In general, these noisy images can be divided into two types: group noisy images (caused by noisy query expansions) and individual noisy images (as a result of the error index of the image search engine). To filter these group and individual noisy images while retaining the images from different distributions, we use MIL methods instead of iterative methods in the process of image selection and noise removal.
By treating each selected expansion as a ``bag" and the images corresponding to the expansion as ``instances", we formulate a multi-instance learning problem by selecting a subset of bags and a subset of images from each bag to construct the domain-robust image dataset. Since the precision of images returned from image search engine tends to be relatively high, we define each positive bag as at least having a portion of $\delta $ positive instances which effectively filter group noisy images caused by noisy query expansions.
We denote each instance as $x_{i}$ with its label $y_{i} \in \left \{ 0,1 \right \}$, where $ i $ =1,...,$ n $. We also denote the label of each bag $B_{I}$ as $Y_{I} \in \left \{ 0,1 \right \}$. The transpose of a vector or matrix is represented by superscript $'$ and the element-wise product between two matrices is represented by $\odot$. We define the identity matrix as $\mathbf{I}$ and $\mathbf{0}$, $\mathbf{1}$ $\in \Re ^{n}$ denote the column vectors of all zeros and ones, respectively. The inequality $\mathbf{u}=\left [ u_{1},u_{2}...u_{n} \right ]{}'\geq \mathbf{0}$ means that $u_{i}\geq 0$ for $ i $ =1,...,$ n $.
\subsubsection{Filtering individual noisy images}
The decision function for filtering individual noisy images is assumed in the form of $ f(x)={w}' \varphi (x)+b$ and has to be learned from the raw image dataset. We employ the formulation of Lagrangian SVM, in which the square bias penalty $b^{2}$ and the square hinge loss for each instance are used in the objective function. The decision function can be learned by minimizing the following structural risk function:
\begin{equation}\label{eq2}
\min_{y,w,b,\rho ,\varepsilon_{i}} \:
\frac{1}{2}\left ( \left \| w \right \|^{2}+b^{2}+C\sum_{i=1}^{n}\varepsilon _{i}^{2} \right )-\rho \\
\end{equation}
\begin{equation}\label{eq3}
\text{s.t.} \:\: y_{i}({w}'\varphi \left ( x_{i} \right )+b)\geq \rho -\varepsilon_{i}, i=1,...n. \\
\end{equation}
\begin{equation}\label{eq4}
\begin{aligned}
& & \sum_{i:x_{i}\in B_{I}} \frac{y_{i}+1}{2}\geq \delta \left | B_{I} \right | \quad for \quad Y_{I}=1, \\
& & \quad y_{i}=0 \quad \quad for \quad Y_{I}=0 \\
\end{aligned}
\end{equation}
where $\varphi$ is a mapping function that maps $x$ from the original space into a high dimensional space $\varphi (x)$, $C>0$ is a regularization parameter and $\varepsilon _{i}$ values are slack variables. The margin separation is defined as $\rho /\left \| w \right \|$. $ y=\left [ y_{1}...y_{n} \right ]'$ means the vector of instance labels, $\lambda =\{ \mathbf{y} |y_{i}\in \left \{ 0,1 \right \} \}$ and $\mathbf{y}$ satisfies constraint (\ref{eq4}). By introducing a dual variable $\alpha _{i}$ for inequality constraint (\ref{eq3}) and kernel trick $k_{ij}=\varphi (x_{i}){}'\varphi (x_{j})$, we arrive at the optimization problem below:
\begin{equation}\label{eq5}
\min_{\mathbf{y}\in \lambda} \max_{\alpha} \:
-\frac{1}{2}(\sum_{i,j=1}^{n}\alpha_{i}\alpha_{j}y_{i}y_{j}k_{ij}+\sum_{i,j=1}^{n}\alpha_{i}\alpha_{j}y_{i}y_{j}+\frac{1}{C}) \\
\end{equation}
where $\alpha _{i}\geq 0$, $\sum_{i=1}^{n}\alpha _{i}=1$ and $\mathbf{\alpha }=\left [ \alpha_{1},\alpha_{2}...\alpha_{n} \right ]{}'$. By defining $\mathbf{K}=\left [ k_{ij} \right ]$ as a $n\times n$ kernel matrix, $\mathbf{\nu }=\left \{ \mathbf{\alpha }|\mathbf{\alpha \geq 0, \alpha {}'1}=1 \right \}$ and $\widetilde{\mathbf{K}}=\mathbf{K}+\mathbf{1}\mathbf{1}{}'$ as a $n\times n$ transformed kernel matrix for the augmented feature mapping $\widetilde{\varphi }(x)= [\varphi (x){}'. \: 1]{}'$
of kernel $\widetilde{k_{ij}}=\widetilde{\varphi}(x_{i}){}'\widetilde{\varphi}(x_{j})$. (\ref{eq5}) can be rewritten as follows:
\begin{equation}\label{eq6}
\min_{\mathbf{y}\in \lambda}\max_{\alpha \in \nu} \:
-\frac{1}{2}\alpha {}'(\widetilde{\mathbf{K}}\odot \mathbf{y}\mathbf{y}{}'+\frac{1}{C}\mathbf{I})\alpha. \\
\end{equation}
\begin{algorithm}[thb]
\caption{Cutting-plane algorithm for solving (\ref{eq10})}
\label{alg1}
\begin{algorithmic}[1]
\STATE Initialize $y_{i}=Y_{I}$ for $x_{i}\in B_{I}$ as $\mathbf{y}^1$, and set $ \zeta =\left \{ \mathbf{y}^1 \right \}$; \\
\STATE Use MKL to solve $\alpha $ and $\mathbf{u}$ in (\ref{eq10}) with $\zeta$; \\
\STATE Select most violated $\mathbf{y}^{t}$ with $\alpha$ and set $\zeta =\mathbf{y}^{t}\cup \zeta$;\\
\STATE Repeat step 2 and step 3 until convergence. \\
\end{algorithmic}
\end{algorithm}
(\ref{eq6}) is a mixed integer programming problem with respect to the instance labels $y_{i}\in \left \{ 0,1 \right \}$. We take the Label-Generating MMC (LG-MMC) algorithm proposed in \cite{li2009tighter} to solve this mixed integer programming problem.
We first consider interchanging the order of $\max_{\alpha \in \upsilon }$ and $\min_{\mathbf{y}\in \lambda}$ in (\ref{eq6}) and obtain:
\begin{equation}\label{eq7}
\max_{\alpha \in \nu}\min_{\mathbf{y}\in \lambda} \:
-\frac{1}{2}\alpha'(\widetilde{\mathbf{K}}\odot \mathbf{y}\mathbf{y}{}'+\frac{1}{C}\mathbf{I})\alpha. \\
\end{equation}
According to the minmax theorem \cite{kim2008minimax}, the optimal objective of (\ref{eq6}) is an upper bound of (\ref{eq7}). We rewrite (\ref{eq7}) as:
\begin{equation}\label{eq8}
\max_{\alpha \in \nu}\left \{\max_{\theta}-\theta |\theta \geq \frac{1}{2} \alpha '(\widetilde{\mathbf{K}}\odot \mathbf{y}^{t}\mathbf{y}^{t^{'}}+\frac{1}{C}\mathbf{I})\alpha , \forall \mathbf{y}^{t}\in \lambda \right \}
\end{equation}
$\mathbf{y}^{t}$ is any feasible solution in $\lambda$. For the inner optimization sub-problem, let $u_{t}\geq 0$ be the dual variable for inequality constraint. Its Lagrangian can be obtained as:
\begin{equation}\label{eq9}
-\theta +\sum_{t:\mathbf{y}_{t}\in \lambda }u_{t}(\theta - \frac{1}{2} \alpha {}'(\widetilde{\mathbf{K}}\odot \mathbf{y}^{t}\mathbf{y}^{t^{'}}+\frac{1}{C}\mathbf{I})\alpha ).
\end{equation}
Setting the derivative of (\ref{eq9}) with respect to $\theta$ to zero, we have $\sum u_{t}=1$. $\mathbf{M}=\left \{ \mathbf{u} | \sum u_{t}=1,u_{t}\geq 0 \right \}$ is denoted as the domain of $\mathbf{u}$, where $\mathbf{u}$ is the vector of $u_{t}$. The inner optimization sub-problem is replaced by its dual and (\ref{eq8}) can be rewritten as:
\begin{equation*}
\max_{\alpha \in \nu}\min_{\mathbf{u}\in \mathbf{M}} \:
-\frac{1}{2}\alpha {}'(\sum _{t:\mathbf{y}^{t}\in \lambda }u_{t} \widetilde{\mathbf{K}}\odot \mathbf{y}\mathbf{y}{}'+\frac{1}{C}\mathbf{I})\alpha \\
\end{equation*}
or
\begin{equation}\label{eq10}
\min_{\mathbf{u}\in \mathbf{M}}\max_{\alpha \in \nu} \:
-\frac{1}{2}\alpha {}'(\sum _{t:\mathbf{y}^{t}\in \lambda }u_{t} \widetilde{\mathbf{K}}\odot \mathbf{y}\mathbf{y}{}'+\frac{1}{C}\mathbf{I})\alpha. \\
\end{equation}
Here, we can interchange the order of $\max_{\alpha \in \nu}$ and $\min_{\mathbf{u}\in \mathbf{M}}$ because the objective function is concave in $\mathbf{\alpha }$ and convex in $\mathbf{u}$. Additionally, (\ref{eq10}) can be regarded as a multiple kernel learning (MKL) problem \cite{bach2004multiple}, and the target kernel matrix is a convex combination of base kernel matrices $\left \{ \widetilde{\mathbf{K}}\odot \mathbf{y_{t}}\mathbf{y_{t}}{}' \right \}$.
Although $\lambda$ is finite and (\ref{eq10}) is an MKL problem, we can not directly use existing MKL techniques like \cite{rakotomamonjy2008simplemkl} to solve this problem. The reason is that the exponential number of possible labellings $\mathbf{y}_{t}\in \lambda$ and the fact that the base kernels are exponential in size make direct MKL computations intractable.
Fortunately, not all the constraints in (\ref{eq8}) are active at optimality, thus we can employ a cutting-plane algorithm \cite{kelley1960cutting} to find a subset $\zeta \in \lambda$ of the constraints that can well approximate the original optimization problem. The detailed solutions of the cutting-plane algorithm for (\ref{eq10}) are described in Algorithm \ref{alg1}. Finding the most violated constraint $\mathbf{y}^{t}$ is the most challenging aspect of the cutting-plane algorithm.
According to (\ref{eq5}), the most violated $\mathbf{y}^{t}$ is equivalent to the following optimization problem:
\begin{equation}\label{eq11}
\max_{\mathbf{y}\in \lambda} \:
\sum_{i,j=1}^{n}\alpha _{i}\alpha _{j}y_{i}y_{j}k_{ij}. \\
\end{equation}
We solve this integer optimization problem by enumerating all possible candidates of $\mathbf{y}^{t}$. Here we only enumerate the possible labelling candidates of the instances in positive bags as all instances in the negative bags are assumed to be negative in our paper. Lastly, we can derive the decision function from the raw image dataset for the given query as:
\begin{equation}\label{eq12}
f(x)=\sum _{i:\alpha _{i}\neq 0}\alpha _{i}\widetilde{y}_{i}\widetilde{k}(x,x_{i}) \\
\end{equation}
where $\widetilde{y}_{i}= \sum _{t:\mathbf{y}^{t}\in \lambda }u_{t}y_{i}^{t}$ and $\widetilde{k}(x,x_{i})={k}(x,x_{i})+1$. The decision function will be used to filter individual noisy images in each bag which correspond to selected query expansions.
\subsubsection{Filtering group noisy images}
To filter group noisy images, we represent bag $B_{I}$ with the compound feature $\phi_{f,k}$ of its first $k$ positive instances:
\begin{equation}\label{eq13}
\phi_{f,k}(B_{I})=\frac{1}{k}\sum_{x_{i}\in \Psi_{f,k}^{*}(B_{I})}x_{i} \\
\end{equation}
with
\begin{equation}\label{eq14}
\Psi_{f,k}^{*}(B_{I})=_{\Psi\subseteq B_{I},|\Psi|=k}^{\quad \arg \max}\sum_{x_{i}\in \Psi}f(x_{i}). \\
\end{equation}
We refer to the instances in $\Psi_{f,k}^{*}(B_{I})$ as the first $k$ instances of $B_{I}$ according to classifier $f$ (see Equation \ref{eq12}). Since the closer of images in $B_{I}$ from the bag center, the higher probability of these images to be relevant to the bag. The assignment of relatively heavier weights to images which have short distance to bag center would increase the accuracy of classifying bag $B_{I}$ to be positive or negative, then increase the efficiency of filtering noisy group images. Following \cite{carneiro2007supervised}, we assume $\xi_{i}=[1+\exp(\alpha \log d(x_{i})+\beta )]^{-1}$ to be a weighting function, $d(x_{i})$ represents the Euclidean distance of images $x_{i}$ from the bag center, $\alpha \in \mathbb{R}_{++}$ and $\beta$ are scaling and offset parameters which can be determined by cross-validation. The representation of (\ref{eq13}) for bag $B_{I}$ can be generalized to a weighted compound feature:
\begin{equation}\label{eq15}
\phi_{f,k}(B_{I})=\phi(X,h^{*})=\frac{Xh^{*}}{\xi^{T}h^{*}} \\
\end{equation}
with
\begin{equation}\label{eq16}
h^{*}= _{\:\:\: h \in H}^{\arg \max}f(\frac{Xh}{\xi^{T}h}), \:\:\: \text{s.t.} \:\: \sum _{i}h_{i}= k\\
\end{equation}
where $X=\left [ x_{1},x_{2},x_{3}..,x_{i} \right ]\in \mathbb{R}^{D\times i}$ is a matrix whose columns are the instances of bag $B_{I}$, $\xi = \left [ \xi_{1},\xi_{2},\xi_{3}...\xi_{i} \right ]^{T}\in \mathbb{R}_{++}^{i}$ are the vectors of weights, and $h^{*}\in H=\left \{ 0,1 \right \}^{i}\setminus \left \{ 0 \right \}\left ( \ \sum _{i}h_{i}= k \right )$ is an indicator function for the first k positive instances of bag $B_{I}$.
Then classifying rule of bag $B_{I}$ to be selected or not is:
\begin{equation}\label{eq17}
f_{\omega}(X)=_{h\in H}^{\max}\omega^{T}\phi(X,h), \:\:\: \sum_{i}h_{i}=k\\
\end{equation}
where $\omega \in \mathbb{R}^{D}$ is the vector of classifying coefficients, $\phi(X,h)\in \mathbb{R}^{D}$ is the feature vector of (\ref{eq15}), $h$ is a vector of
\begin{algorithm}[thb]
\caption{Concave-convex procedure for solving (\ref{eq21})}
\label{alg2}
\begin{algorithmic}[1]
\STATE Initialize $\omega$ with SVM by setting $h=\mathbf{1}\in \mathbb{R}^{i}$; \\
\STATE Compute a convex upper bound using the current model for the second term of (\ref{eq21}); \\
\STATE Minimize this upper bound by solving a structural SVM problem via the proximal bundle method \cite{kiwiel1990proximity};\\
\STATE Repeat step 2 and step 3 until convergence. \\
\end{algorithmic}
\end{algorithm}
latent variables and $H$ is the hypothesis space $\left \{ 0,1 \right \}^{i}\setminus \left \{ 0 \right \}$.
The learning problem is to determine the parameter vector $\omega$.
Given a training set $\tau = \left \{ B_{I},Y_{I} \right \}_{I=1}^{n}$, this is a latent SVM learning problem:
\begin{equation}\label{eq18}
\min_{\mathbf{\omega }}\frac{1}{2}\left \| \omega \right \|^2+C\sum_{I=1}^{n}\max\left ( 0,1-Y_{I}f_\omega \left ( X_{B_I} \right ) \right ).\
\end{equation}
Before solving (\ref{eq18}), we first solve the classifying rule of (\ref{eq17}). It is necessary to solve the below following problem:
\begin{equation}\label{eq19}
_{h \in H}^{\max}\textrm{}\frac{\omega ^{T}Xh}{\xi^{T}h}, \:\:\: \text{s.t.} \:\: \sum_{i}h_{i}=k.\
\end{equation}
This is an integer linear-fractional programming problem. Since $\xi \in \mathbb{R}_{++}^{i}$, (\ref{eq19}) is identical to the relaxed problem:
\begin{equation}\label{eq20}
_{h\in \ss ^i}^{\max}\frac{\omega ^{T}Xh}{\xi^{T}h}, \:\:\: \text{s.t.} \:\: \sum_{i}h_{i}=k\
\end{equation}
where $\ss ^i=\left [ 0,1 \right ]^i$ is a unit box in $\mathbb{R}^i$. (\ref{eq20}) is a linear-fractional programming problem and can be reduced to a linear programming problem of $i+1$ variables and $i+2$ constraints \cite{boyd2004convex}.
In this work, we take the concave-convex procedure (CCCP) \cite{yuille2003concave} algorithm to solve (\ref{eq18}). We rewrite the objective of (\ref{eq18}) as two convex functions:
\begin{equation}\label{eq21}
\begin{aligned}
& & \min_{\mathbf{\omega }}
\left [ \frac{1}{2}\left \| \omega \right \|^2+C\sum_{I\in D_{n}}\max\left ( 0,1+f_\omega \left ( X_{B_I} \right ) \right ) + \right. \\
& & \left. C\sum_{I\in D_{p}}\max\left ( f_\omega \left ( X_{B_I} \right ),1 \right ) \right ]-\left [ C\sum_{I\in D_{p}} f_\omega \left ( X_{B_I} \right ) \right ] \\
\end{aligned}
\end{equation}
where $D_{p}$ and $D_{n}$ are positive and negative training sets respectively. The detailed solutions of the CCCP algorithm for (\ref{eq21}) are described in Algorithm \ref{alg2}. Lastly, we obtain the bag classifying rule as (\ref{eq17}) to filter group noisy images which correspond to noisy query expansions.
In summary, the existing automatic methods reduce the cost of manual annotation by leveraging the generalization ability of machine learning models. However, this generalization ability is affected by both the quality of the initial candidate images and the capability of models to retain images from different distributions. Previous works primarily focus on
accuracy and scale, and most use an iterative mechanism for the image selection process which often results in a dataset bias problem. To the best of our knowledge, this is the first proposal for automatic domain-robust image dataset construction. We achieve the domain adaptation ability of our dataset by maximizing both the initial candidate images and the final selected images from different distributions.
\section{Experiments}
To demonstrate the effectiveness of our approach, we have constructed an image dataset with 20 categories. We compare the image classification ability, cross-dataset generalization ability and diversity of our dataset with three manually labelled and three automated datasets. The motivation is to verify that a domain-robust image dataset has a better image classification ability on a third-party dataset; and to confirm that a domain-robust image dataset has better cross-dataset generalization ability and dataset diversity. We also report the object detection ability of our dataset and compare it with weakly supervised and web-supervised state-of-the-art methods.
\subsection{Image Dataset DRID-20 Construction}
Since most existing weakly supervised and web-supervised learning methods were evaluated on the PASCAL VOC 2007 dataset, we choose the 20 categories in PASCAL VOC 2007 as the target categories for the construction of DRID-20.
For each given query (e.g.,``horse"), we first expand the given query to a set of query expansions with POS. To filter
visual non-salient expansions, we retrieve the top $\bold{N} = 100$ images from the image search engine as positive images (in spite of the fact that noisy images might be included). Set the training set and validation set $I_i=\{I_i^t=75, I_i^v=25\}$, $\overline I=\{\overline I^t=25, \overline I^v=25\}$. By experimentation, we declare a query expansion $i$ to be visually salient if the classification result ($S_i \geq 0.7$) returns a relatively high score. We have released the query expansions for 20 categories in DRID-20 and the corresponding images (original image URL) on: \href{https://drive.google.com/drive/folders/0B7dS7AFpUzt1bmpwcFRKcDZwUUE?usp=sharing}{Google Drive}.
To filter the less relevant expansions, we select $n_{+}$ positive training samples from these expansions that have a small semantic or visual distance. We calculate the semantic distance and visual distance between the different queries (e.g., ``horse'' and ``cow'') to obtain the $n_{-}$ negative training samples. Here, we set $n=1000$ and train a classifier based on linear SVM to filter less relevant expansions.
The first $\bold{N}=100$ (for category ``plant'' expansions, $\bold{N}=350$) images are retrieved from image search engine for each selected query expansion to construct the raw image dataset. We treat the selected query expansions as positive bags and images therein as instances. Specifically, we define each positive bag as having at least a portion of $\delta =0.7 $ positive instances. Negative bags can be obtained by randomly sampling a few irrelevant images. MIL methods are applied to learn the decision function (\ref{eq12}) for individual noisy images filtering.
The decision function (\ref{eq12}) is also used to select the most $k$ positive instances in each bag, representing this bag for group noisy images filtering. The value of $k$ for different categories may be different. In general, categories with larger query expansions tend to select a smaller value. There are multiple methods for learning the weighting function (e.g., logistic regression or cross-validation), here we follow \cite{carneiro2007supervised} and use cross-validation to learn the weighting function. To this end, we label 10 datasets, each containing 100 positive bags and 100 negative bags. The positive bags and negative bags each have 50 images. Labelling only needs to be carried out once to learn the weighting function and the weighted bag classification rule (\ref{eq17}). The learned weighted bag classification rule (\ref{eq17}) will be used to filter noisy bags (corresponding to group noisy images). For better comparison with other datasets, we evenly select positive images from positive bags to construct the dataset DRID-20. Each category in DRID-20 has 1000 images and this dataset has been released publicly on \href{https://drive.google.com/drive/folders/0B7dS7AFpUzt1bmpwcFRKcDZwUUE?usp=sharing}{Google Drive}.
\subsection{Comparison of Image Classification Ability, Cross-dataset Generalization Ability and Dataset Diversity}
The goal of these experiments is to evaluate the domain robustness of our dataset.
\subsubsection{Experimental setting}
We chose PASCAL VOC 2007 as the third-party testing benchmark dataset for comparing the image classification ability of our dataset with other baseline datasets. For this experiment, the same categories between various datasets are compared. Specifically, we compare the category ``airplane", ``bird", ``cat", ``dog", ``horse" and ``car/automobile" between STL-10\cite{coates2011analysis}, CIFAR-10 \cite{krizhevsky2009learning} and DRID-20. We sequentially select [200,400,600,800,1000] training images from CIFAR-10, STL-10 and DRID-20 as the positive training images, and use 1000 fixed irrelevant images as the negative training images to learn the image classifiers.
For comparison with ImageNet \cite{deng2009imagenet}, Optimol \cite{li2010optimol}, Harvesting \cite{schroff2011harvesting} and AutoSet \cite{icme2016yao}, we use all the 20 categories among these datasets. In specific, we randomly select 500 training images for each category from these datasets as the positive training images. Similarly, we use 1000 fixed irrelevant images as the negative training images to learn the image classifiers. We then test the performance of these classifiers on the corresponding categories of the PASCAL VOC 2007 dataset. We repeat the above experiment ten times and use the average performance as the final performance for each dataset. The image classification ability of all datasets for each category is shown in Fig. \ref{fig3} and Fig. \ref{fig4}.
For the comparison of cross-dataset generalization ability, we randomly select 200 images for each category as the testing data. For the choice of training data, we sequentially select [200,300,400,500,600,700,800] images per category from various datasets as the positive training samples, and use 1000 fixed irrelevant images as the negative training samples to learn the image classifiers. The training images in each category are selected randomly. In addition, the training data and testing data have no duplicates. Like the comparison of image classification ability, we also compare the category ``airplane", ``bird", ``cat", ``dog", ``horse" and ``car/automobile" among STL-10 \cite{coates2011analysis}, CIFAR-10
\cite{krizhevsky2009learning} and DRID-20. For comparison with ImageNet \cite{deng2009imagenet}, Optimol \cite{li2010optimol}, Harvesting \cite{schroff2011harvesting} and AutoSet \cite{icme2016yao}, we also use all the 20 categories among these datasets. The average classification accuracy represents the cross-dataset generalization ability of one dataset on another dataset. The experimental results are shown in Fig. \ref{fig5} and Fig. \ref{fig6} respectively.
For the comparison of dataset diversity, we select five common categories ``airplane", ``bird",``cat",``dog" and ``horse" in STL-10, ImageNet and DRID-20 as testing examples.
\begin{figure*} [thb]
\centering
\subfloat[]{%
\includegraphics[width=1.7in, height=1.6in]{exp1-airplane.pdf}}
\subfloat[]{%
\includegraphics[width=1.7in, height=1.6in]{exp1-bird.pdf}}
\subfloat[]{%
\includegraphics[width=1.7in, height=1.6in]{exp1-cat.pdf}}
\subfloat[]{%
\includegraphics[width=1.7in, height=1.6in]{exp1-dog.pdf}}\
\subfloat[]{%
\includegraphics[width=1.7in, height=1.6in]{exp1-horse.pdf}}
\hspace{0.9cm}
\subfloat[]{%
\includegraphics[width=1.7in, height=1.6in]{exp1-car.pdf}}
\hspace{0.9cm}
\subfloat[]{%
\includegraphics[width=1.7in, height=1.6in]{exp1-average.pdf}}
\caption{Image classification ability of CIFAR-10, STL-10 and DRID-20 on PASCAL VOC 2007 dataset: (a) airplane, (b) bird, (c) cat, (d) dog, (e) horse, (f) car/automobile and (g) average.}
\label{fig3}
\end{figure*}
\begin{figure*}[thb]
\centering
\includegraphics[width=7 in, height=2.08 in]{exp1-2.pdf}
\caption{Image classification ability of Optimol, Harvesting, ImageNet, AutoSet and DRID-20 on PASCAL VOC 2007 dataset.}
\label{fig4}
\end{figure*}
Following method \cite{deng2009imagenet} and \cite{collins2008towards}, we compute the average image of each category and measure the lossless JPG file size. In particular, we resize all images in STL-10, ImageNet, DRID-20 to 32$\times$32 images, and create average images for each category from 100 randomly sampled images. Fig. \ref{fig7} (a) presents the lossless JPG file sizes of five common categories in dataset DRID-20, ImageNet and STL-10. The example and average images for five categories in three datasets are shown in Fig. \ref{fig7} (b).
For image classification ability and cross-dataset generalization ability comparison, we set the same options for all datasets. Particularly, we set the type of SVM as C-SVC, the type of kernel as a radial basis function and all other options as the default LIBSVM options. For all datasets, we extract the same dense histogram of oriented gradients (HOG) feature \cite{dalal2005histograms} and train one-versus-all classifiers.
\subsubsection{Baselines}
In order to validate the performance of our dataset, we compare the image classification ability, cross-dataset generalization ability and dataset diversity of our dataset DRID-20 with two sets of baselines:
$\bullet$ Manually labelled datasets. The manually labelled datasets include STL-10 \cite{coates2011analysis}, CIFAR-10 \cite{krizhevsky2009learning} and ImageNet \cite{deng2009imagenet}. The STL-10 dataset has ten categories, and each category of which contains 500 training images and 800 test images. All of the images are color 96 $\times$ 96 pixels. The CIFAR-10 dataset consists of 32$\times$32 images in 10 categories, with 6000 images per category. ImageNet is an image dataset organized according to the WordNet hierarchy. It provides an average of 1000 images to illustrate each category.
$\bullet$ Automated datasets. The automated datasets contain
\begin{figure*}[t]
\centering
\subfloat[]{%
\includegraphics[width=1.7in, height=1.6in]{exp21-CIFAR.pdf}}
\subfloat[]{%
\includegraphics[width=1.7in, height=1.6in]{exp21-STL.pdf}}\
\subfloat[]{%
\includegraphics[width=1.7in, height=1.6in]{exp21-DRID.pdf}}
\subfloat[]{%
\includegraphics[width=1.7in, height=1.6in]{exp21-AVER.pdf}}
\caption{Cross-dataset generalization ability of classifiers learned from CIFAR-10, STL-10, DRID-20 and then tested on: (a) CIFAR-10, (b) STL-10, (c) DRID-20, (d) Average.}
\label{fig5}
\end{figure*}
\begin{figure*}[t]
\centering
\subfloat[]{%
\includegraphics[width=1.7in, height=1.6in]{exp22-OPT.pdf}}
\hspace{0.8cm}
\subfloat[]{%
\includegraphics[width=1.7in, height=1.6in]{exp22-HAR.pdf}}
\hspace{0.8cm}
\subfloat[]{%
\includegraphics[width=1.7in, height=1.6in]{exp22-INT.pdf}}\
\subfloat[]{%
\includegraphics[width=1.7in, height=1.6in]{exp22-AutoSet.pdf}}
\hspace{0.8cm}
\subfloat[]{%
\includegraphics[width=1.7in, height=1.6in]{exp22-DRID.pdf}}
\hspace{0.8cm}
\subfloat[]{%
\includegraphics[width=1.7in, height=1.6in]{exp22-AVER.pdf}}
\caption{Cross-dataset generalization ability of classifiers learned from Optimol, Harvesting, ImageNet, AutoSet, DRID-20 and then tested on: (a) Optimol, (b) Harvesting, (c) ImageNet, (d) AutoSet, (e) DRID-20, (f) Average.}
\label{fig6}
\end{figure*}
Optimol \cite{li2010optimol}, Harvesting \cite{schroff2011harvesting} and AutoSet \cite{icme2016yao}. For \cite{li2010optimol}, 1000 images for each category are collected by using the incremental learning method. Following \cite{schroff2011harvesting}, we firstly obtain the candidate images from the web search and rank the returned images by the text information. Then we use the top-ranked images to learn visual classifiers to re-rank the images once again.
We select the categories in DRID-20 as the target queries and accordingly obtain the multiple textual metadata. Following the proposed method in \cite{icme2016yao}, we take iterative mechanisms for noisy images filtering and construct the dataset. In total, we construct 20 same categories as DRID-20 for Optimol, Harvesting and AutoSet.
\subsubsection{Experimental results for image classification}
By observing Fig. \ref{fig3} and Fig. \ref{fig4}, we make the following conclusions:
It is interesting to observe that the categories ``airplane", ``tv" and ``plant" have a relatively higher classification accuracy than other categories with a small amount of training data. A possible explanation is that the scenes and visual patterns of ``airplane", ``tv" and ``plant" are relatively simpler than other categories. Even with a small amount of training data, there is still a large number of positive patterns in both auxiliary and target domains. That is to say, the samples are densely distributed in the feature space, and the distribution of the
two domains overlaps much more easily.
\begin{figure*} [t]
\centering
\subfloat[]{\includegraphics[width=1.6in, height=1.6in]{exp3-1.pdf}}
\subfloat[]{\includegraphics[width=5.4in, height=1.6in]{exp3-2.pdf}}
\caption{(a) Comparison of the lossless JPG file sizes of average images for five different categories in DRID-20, ImageNet and STL-10. (b) Example images from DRID-20, ImageNet, STL-10 and average images for each category indicated by (a).}
\label{fig7}
\end{figure*}
CIFAR-10 exhibits a much worse performance on image classification than STL-10 and DRID-20 according to its accuracy over six common categories. This demonstrates that the classifier learned with the training data from the auxiliary domain performs poorly on the target domain. The explanation is perhaps that the data distributions of CIFAR-10 are quite
different from those of the PASCAL VOC 2007 dataset. The CIFAR-10 dataset has a more serious dataset bias problem than STL-10 and DRID-20.
STL-10 performs much better on category ``dog" than CIFAR-10 and DRID-20 when the number of training data is 400. The explanation is that STL-10 may have more effective visual patterns than CIFAR-10 and DRID-20 on category ``dog" with 400 training data. On the other hand, the positive samples from CIFAR-10 and DRID-20 are distributed sparsely in the feature space with 400 training images. It is likely that there are less overlap between the auxiliary and target domains for CIFAR-10 and DRID-20.
DRID-20 outperforms the automated datasets in terms of average accuracy in 20 categories, which demonstrates the domain robustness of DRID-20. A possible explanation is that our DRID-20 dataset, being constructed by multiple query expansions, has many more visual patterns or feature distributions than Harvesting and Optimol. At the same time, compared to AutoSet which uses iterative mechanisms in the process of image selection, MIL mechanisms can maximize the retention of useful visual patterns. Thus, our dataset has a better image classification ability.
\subsubsection{Experimental results for cross-dataset generalization}
Cross-dataset generalization measures the performance of classifiers learned from one dataset and tested on another dataset. It indicates the robustness of dataset \cite{torralba2011unbiased,mm2016yao}.
By observing Fig. \ref{fig5} and Fig. \ref{fig6}, we draw the following conclusions:
Compared to STL-10 and DRID-20, CIFAR-10 has a poor cross-dataset generalization ability except on its own dataset. The explanation is that the data distributions of its auxiliary domain and target domain are strongly related, making it difficult for other datasets to exceed its performance when tested on CIFAR-10. All images in CIFAR-10 are cut to 32$\times$32 and the objects in these images are located in the middle of the image. Besides, these images contain relatively fewer other objects and scenes. The images in STL-10 are 96$\times$96 and are full size in DRID-20. These images not only contain target objects, but also include a large number of other scenarios and objects. Based on these conditions, CIFAR-10 has a serious dataset bias problem which coincides with its average cross-dataset generalization performance.
AutoSet is better than Optimol, Harvesting and ImageNet but slightly worse than DRID-20, possibly because the distribution of samples is relatively rich. AutoSet is constructed using multiple textual meta-data and the objects of its images have variable appearances, positions, viewpoints, and poses.
DRID-20 outperforms CIFAR-10, STL-10, ImageNet, Optimol, Harvesting and AutoSet in terms of average cross-dataset performance, which demonstrates the domain robustness of DRID-20. This may be because DRID-20 constructed by multiple query expansions and MIL selection mechanisms has much more effective visual patterns than other datasets given the same number of training samples. In other words, DRID-20 has a much richer feature distribution and is more easily overlapped with unknown target domains.
\subsubsection{Experimental results for dataset diversity}
The lossless JPG file size of the average image for each category reflects the amount of information in an image. The basic idea is that a diverse image dataset will result in a blurrier average image, the extreme being a gray image. Meanwhile, an image dataset with limited diversity will result in a more structured, sharper average image. Therefore, we expect the average image of a more diverse image dataset to have a smaller JPG file size. By observing Fig. \ref{fig7}:
DRID-20 has a slightly smaller JPG file size than ImageNet and STL-10 which indicates the diversity of our dataset. This phenomenon is universal for all five categories. It can be seen that the average image of DRID-20 is blurred and it is difficult to recognize the object, while the average image of ImageNet and STL-10 is relatively more structured and sharper.
DRID-20 is constructed with the goal that images in this dataset should exhibit domain robustness and be able to effectively alleviate the dataset bias problem. To achieve domain robustness, we not only consider the source of the candidate images, but also retain the images from different distributions.
\subsection{Comparison of Object Detection Ability}
The idea of training detection models without bounding boxes has received renewed attention due to the success of the DPM \cite{felzenszwalb2010object} detector. To compare the object detection ability of our collected data with other baseline methods \cite{divvala2014learning,felzenszwalb2010object,siva2011weakly,prest2012learning,icme2016yao}, we selected PASCAL VOC 2007 as the test data. The reason is recent state-of-the-art weakly supervised and web-supervised methods have been evaluated on this dataset.
\subsubsection{Experimental setting}
For each query expansion, we train a separate DPM to constrain the visual variance. We resize images to a maximum of 500 pixels and ignore images with extreme aspect ratios (aspect ratio $>$ 2.5 or $<$ 0.4). To avoid getting stuck to the image boundary during the latent re-clustering step, we initialize our bounding box to a sub-image within the image that ignores the image boundaries. Following \cite{felzenszwalb2010object}, we also initialize components using the aspect-ratio heuristic. Some of the components across different query expansion detectors ultimately learn the same visual pattern. For example, the images corresponding to the query expansion ``walking horse" are similar to the images corresponding to ``standing horse". In order to select a representative subset of the components and merge similar components, we represent the space of all query expansions components by a graph $G=\left \{ C,E \right \}$, in which each node represents a component and each edge represents the visual similarity between them.
The score $d_{i}$ for each node corresponds to the average precision. The weight on each edge $e_{i,j}$ is obtained by running the $j\mathrm{th}$ component detector on the $i\mathrm{th}$ component set. We solve the same objective function proposed in \cite{divvala2014learning} to select the representative components $ S $ ($S\subseteq V$) :
\begin{equation}\label{eq22}
\max_{S}\sum_{i\in V}d_{i}\cdot \vartheta (i,S) \\
\end{equation}
where $\vartheta$ is a soft coverage function that implicitly pushes for diversity:
\begin{equation}\label{eq23}
\vartheta(i,S)=
\begin{cases}
1 &\mbox{$i\in S$}\\
1-\prod _{j\in S}(1-e_{i,j}) &\mbox{$ i \notin S $}.
\end{cases}
\end{equation}
After the representative subset of components has been obtained, we augment them with the method described in \cite{felzenszwalb2010object} and subsequently merge all the components to produce the final detector.
\subsubsection{Baselines}
In order to validate the object detection ability of our collected data, we compare our approach with three sets of baselines:
$\bullet$ Weakly supervised methods. The weakly supervised learning methods include WSL \cite{siva2011weakly} and SPM-VID \cite{prest2012learning}. WSL uses weak human supervision (VOC data with image-level labels for training) and initialization from objectness. SPM-VID is trained on manually selected videos without bounding boxes and shows results in 10 out of 20 categories.
$\bullet$ Web-supervised methods. Such methods include WSVCL \cite{divvala2014learning} and IDC-MTM \cite{icme2016yao}. WSVCL takes web supervision and then trains a mixture DPM detector for the object. IDC-MTM collects candidate images with multiple textual metadata and filters these images using an iterative method. Images which are not filtered out are then selected as positive training images for mixture DPM detector learning.
$\bullet$ Fully supervised method. The fully supervised method includes OD-DPM \cite{felzenszwalb2010object}. OD-DPM is a fully supervised object detection method and it is a possible upper bound for weakly supervised and web-supervised approaches.
\subsubsection{Experimental results for object detection}
We report the performance of object detection on PASCAL VOC 2007 test set. Table \ref{tab2} shows the results of our proposed method and
\begin{table}[htb]
\centering
\caption{OBJECT DETECTION RESULTS (A.P.) (\%) ON PASCAL VOC 2007 (TEST).}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
Method & \cite{siva2011weakly} & \cite{prest2012learning} & \cite{divvala2014learning} & \cite{icme2016yao} & $\mathbf{Our}$ & \cite{felzenszwalb2010object} \\
\hline
Supervision & weak & weak & web & web & web & full\\
\hline
\hline
airplane & 13.4 & $\mathbf{17.4}$ & 14.0 & 14.8 & 15.5 & 33.2\\
\hline
bike & $\mathbf{44.0}$ & - & 36.2 & 38.4 & 40.6 & 59.0\\
\hline
bird & 3.1 & 9.3 & 12.5 & \textbf{16.5} & 16.1 & 10.3\\
\hline
boat & 3.1 & 9.2 & $\mathbf{10.3}$ & 7.4 & 9.69 & 15.7 \\
\hline
bottle & 0.0 & - & 9.2 & 12.6 & $\mathbf{13.7}$ & 26.6\\
\hline
bus & 31.2 & - & 35.0 & 39.5 & $\mathbf{42.0}$ & 52.0 \\
\hline
car & $\mathbf{43.9}$ & 35.7 & 35.9 & 38.1 & 37.9 & 53.7\\
\hline
cat & 7.1 & 9.4 & 8.4 & 8.9 & $\mathbf{9.8}$ & 22.5 \\
\hline
chair & 0.1 & - & $\mathbf{10.0}$ & 9.3 & 9.6 & 20.2\\
\hline
cow & 9.3 & 9.7 & 17.5 & 17.9 & $\mathbf{18.4}$ & 24.3 \\
\hline
table & 9.9 & - & 6.5 & 10.2 & $\mathbf{10.6}$ & 26.9\\
\hline
dog & 1.5 & 3.3 & $\mathbf{12.9}$ & 11.5 & 11.6& 12.6\\
\hline
horse & 29.4 & 16.2 & 30.6 & 31.8 & $\mathbf{36.1}$& 56.5 \\
\hline
motorcycle& $\mathbf{38.3}$ & 27.3 & 27.5 & 29.7 & 36.9 & 48.5 \\
\hline
person & 4.6 & - & 6.0 & 7.2 & $\mathbf{7.9}$ & 43.3\\
\hline
plant & 0.1 & - & $\mathbf{1.5}$ & 1.1 & 1.3 & 13.4\\
\hline
sheep & 0.4 & - & 18.8 & 19.5 & $\mathbf{20.4}$ & 20.9 \\
\hline
sofa & 3.8 & - & 10.3 & 10.3 & $\mathbf{10.8}$ & 35.9 \\
\hline
train & $\mathbf{34.2}$ & 15.0 & 23.5 & 24.2 & 27.6 & 45.2 \\
\hline
tv/monitor & 0.0 & - & 16.4 & 15.6 & $\mathbf{18.4}$ & 42.1\\
\hline
average & 13.87 & 15.25 & 17.15 & 18.22 & $\mathbf{19.74}$ & 33.14 \\
\hline
\end{tabular}
\label{tab2}
\end{table}
compares it to the state-of-the-art weakly supervised and web-supervised methods \cite{siva2011weakly,prest2012learning,divvala2014learning,icme2016yao}. By observing the Table \ref{tab2}, we draw the following conclusions:
Compared to WSL and SPM-VID (which use weak supervision) and OD-DPM (which uses full supervision), the training sets of our proposed approach and WSVCL, IDC-MTM do not need to be labelled manually. Nonetheless, the results of our proposed approach and WSVCL, IDC-MTM surpass the previous best results of weakly supervised object detection methods WSL, SPM-VID. A possible explanation is perhaps that both our approach and that of WSVCL, IDC-MTM use multiple query expansions for candidate image collection, and the training data collected by our approach and WSVCL, IDC-MTM are richer and contain more effective visual patterns.
In most cases, our method surpasses the results obtained from WSVCL, IDC-MTM, which also uses web supervision and multiple query expansions for candidate images collection. The explanation for this is that we use different mechanisms for the removal of noisy images. Compared to WSVCL, IDC-MTM which uses iterative mechanisms in the process of noisy images filtering, our approach applies an MIL method for removing noisy images. This maximizes the ability to retain images from different data distributions while filtering out the noisy images.
Our approach outperforms the weakly supervised and web-supervised methods \cite{siva2011weakly,prest2012learning,divvala2014learning,icme2016yao}. The main reason being that our training data is generated using multiple expansions and MIL filtering mechanisms. Thus, our data contains much richer and more accurate visual descriptions for these categories. In other words, our approach discovers much more useful linkages to visual patterns for the given category.
\begin{figure} [t]
\centering
\subfloat[]{%
\includegraphics[width=1.7in, height=1.6in]{exp4-1.pdf}}
\subfloat[]{%
\includegraphics[width=1.7in, height=1.6in]{exp4-2.pdf}}
\caption{The image classification ability of different steps on category (a) dog, (b) horse.}
\label{fig8}
\end{figure}
\subsection{Different Steps Analysis}
Our proposed framework involves three major steps: query expanding, noisy expansions filtering and noisy images filtering. In order to quantify the role of different steps contributing to the final results, we build two new datasets. One is based on noisy expansions filtering and another one is based on noisy images filtering. In particular, we construct the noisy expansions filtering based image dataset by query expanding and noisy expansions filtering. After the noisy expansions are filtered out, we retrieve the top images from image search engine for selected expansions to construct the dataset. We build the noisy images filtering based image dataset by query expanding and noisy images filtering. After we obtain the query expansions through query expanding, we take MIL based methods for noisy images filtering and construct the dataset.
We compare the image classification ability of these two new datasets with our DRID-20. By selecting ``dog" and ``horse" as two target categories to construct the dataset, we sequentially collect [200,400,600,800,1000] images for each category and test the image classification ability on PASCAL VOC 2007 dataset. The results are shown in Fig. \ref{fig8}. It can be seen that:
When the number of images in each category is below 600, the noisy expansions filtering based method tends to have a better image classification ability. This is possibly because the top images returned from the image search engine have a relatively high accuracy. The noisy images induced by noisy expansions are more serious than those caused by the image search engine. With the increased number of images in each category, the images returned from the image search engine have more and more noisy images. The noisy images caused by image search engine have a worse effect than those induced by noisy expansions.
Our method outperforms both noisy expansions filtering based and noisy images filtering based method. This is because our method, which takes a combination of noisy expansions filtering and noisy images filtering, can effectively remove the noisy images induced by both noisy expansions and the image search engine.
\begin{figure} [t]
\centering
\subfloat[]{%
\includegraphics[width=1.7in, height=1.6in]{exp5-1.pdf}}
\subfloat[]
\includegraphics[width=1.7in, height=1.6in]{exp5-2.pdf}}
\caption{The training time and image classification accuracies with respect to various numbers of training images.}
\label{fig9}
\end{figure}
\subsection{Time Complexity And Space Complexity Analysis}
For the time complexity analysis, we mainly focus on multi-instance learning time. During the process of MIL learning, we solve the convex problem in \eqref{eq10} by using the cutting-plane algorithm. By identifying the most violating candidate and solving the MKL sub-problem at each iteration, the time complexity of \eqref{eq10} can be approximately computed as $ T\cdot O $(MKL), where \textit{T} is the number of iterations and $ O $(MKL) is the time complexity of the MKL sub-problem. According to [29], the time complexity of MKL is between $ t \cdot O(LCM) $ and $ t \cdot O((LCM)^{2.3}) $, where $ M, L, C $ are the numbers of latent domains, bags and categories respectively. $ t $ is the number of iterations in MKL.
We have chosen PASCAL VOC 2007 as the testing set for the evaluation of our method. In particular, we use various numbers of training images for each category to learn the classifier. PASCAL VOC 2007 has 20 categories and we use $ n $ training images for each category, so we have a total of 20$ n $ training images. Fig. \ref{fig9} shows the training time and image classification accuracies with respect to the various numbers of training images. We can observe that both training time and image classification accuracies increase as the number of training images grows.
\section{Conclusion and future work}
In this paper, we presented a new framework for domain-robust image dataset construction with web images. Three successive modules were employed in the framework, namely query expanding, noisy expansion filtering and noisy image filtering. To verify the effectiveness of our proposed method, we constructed an image dataset DRID-20. Extensive experiments have shown that our dataset not only has better domain adaptation ability than the traditional manually labelled datasets STL-10, CIFAR-10 and ImageNet, but also has better domain adaptation ability than the automated datasets Optimol, Harvesting and AutoSet. In addition, our data was successfully applied to help improve object detection on PASCAL VOC 2007, and the results demonstrated the superiority of our method to several weakly supervised and web-supervised state-of-the-art methods. We have publicly released the DRID-20 dataset to facilitate the research in this field.
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
|
1,116,691,499,269 | arxiv | \section{I. Introduction}
In the field of three-dimensionally frustrated magnets, the predominant research focus has centered on the magnetic diamond and pyrochlore lattices \cite{bergman-balents, lee-balents, buessen-trebst, gchen, bernier2008quantum, savary2011impurity, bramwell1998frustration, harris1996frustration, harris1998liquid, moessner1998properties, canals1998pyrochlore, ramirez1999zero, bramwellspin, bramwell2001spinice, gardner2010magnetic, ross2011quantum}. Both of these frameworks appear within the family of transition-metal spinels of the form $AB_2X_4$ ($A,B$ = transition metal or metalloid, $X$ = chalcogenide), where the diamond and pyrochlore lattices appear on the $A$- and $B$-site sublattices, respectively. Strong magnetic frustration within each of these sublattice types is known to suppress typical Ne\'el order and instead favor the manifestation of unconventional ground states, including classical spin liquids \cite{moessner1998properties, canals1998pyrochlore}, (quantum) spin ices \cite{ramirez1999zero, bramwellspin, bramwell2001spinice, ross2011quantum}, and (quantum) spiral spin liquids \cite{bergman-balents, lee-balents, buessen-trebst}.
Quantum fluctuations that manifest in the small spin limit on these lattices further suppress magnetic order and can formulate the basis for highly entangled ground states \cite{Lee, balents4, balents5, witczak6, zhou7, cava-broholm}. At this limit, the magnetic diamond lattice has been less thoroughly studied in comparison to the magnetic pyrochlore lattice, as the magnetic pyrochlore lattice also manifests in a large, well-studied family of rare-earth $Ln_2M_2O_7$ ($Ln$ = lanthanide, $M$ = metal or metalloid) compounds \cite{bernier2008quantum, savary2011impurity, bramwell1998frustration, harris1996frustration, harris1998liquid, moessner1998properties, canals1998pyrochlore, ramirez1999zero, bramwellspin, bramwell2001spinice, gardner2010magnetic, ross2011quantum}. Furthermore, while introducing model $J_{eff}=1/2$ lanthanide moments within frustrated magnetic motifs has shown promise in realizing intrinsically quantum disordered states (e.g. Yb$_2$Ti$_2$O$_7$ pyrochlore \cite{ross2011quantum, gaudet-gaulin} and triangular lattice NaYbO$_2$ \cite{bordelon2019field, bordelon2020, ding-tsirlin, ranjith-baenitz}), isolating materials that comparably incorporate model $f$-electron moments within a diamond lattice framework is a challenge.
\begin{figure*}[t]
\includegraphics[width=\textwidth*10/10]{Figure1.png}
\caption{a) Crystal structure of LiYbO$_2$ with YbO$_6$ octahedra shaded in green and black spheres noting the positions of Li ions. b) The frustrated $J_1 – J_2$ model on the diamond lattice consists of two interpenetrating face centered cubic (FCC) sublattices, A and B, with a $J_1$ (black) magnetic interaction connecting the two sublattices and a $J_2$ (orange) spanning interactions within an FCC sublattice. When this structure is stretched along one of the cubic axes, the $I4_1/amd$ lattice of LiYbO$_2$ is reproduced where the dashed green line represents the unit cell origin of LiYbO$_2$ shown in panel c). In LiYbO$_2$, the stretched bond (5.909 \AA, dashed orange) is 1.527 \AA \ longer than the in-plane $J_2$ (4.382 \AA, solid orange). In the present model for LiYbO$_2$, the stretched bond is assumed negligible in strength relative to the shorter $J_2$. c) NN ($J_1$) and NNN ($J_2$) exchange pathways between Yb-ions in LiYbO$_2$ with Yb ions in the $A$ and $B$ sublattices shaded differently for clarity.}
\label{fig:fig1}
\end{figure*}
Frustration within the diamond lattice is best envisioned by dividing the lattice into two interpenetrating face centered cubic (FCC) lattices with two exchange interactions, $J_1$ and $J_2$, where in the Heisenberg limit (Figure \ref{fig:fig1}) \cite{bergman-balents, lee-balents, buessen-trebst}.
\begin{equation}
\label{eq:J1J2}
H = J_1\sum_{<i,j>} {\bf{S_i}}\cdot{\bf{S_j}} + J_2\sum_{<<i,j>>} {\bf{S_i}}\cdot{\bf{S_j}}
\end{equation}
In the two limits where either $J_1$ or $J_2$ is zero, this bipartite system is unfrustrated with a conventional Ne\'el ordered ground state. However, when $J_2 > 0$ and $|J_1| > 0$, ordering becomes frustrated. When $J_2 /|J_1| \ge 1/8$, the classical interpretation of this model develops a degenerate ground state manifold of coplanar spin spirals \cite{bergman-balents, lee-balents, buessen-trebst}. Each of these spirals can be described by a unique momentum vector, and together the degenerate momentum vectors formulate a spin spiral surface in reciprocal space \cite{bergman-balents, lee-balents, buessen-trebst}. The degeneracy of these spin spirals can be lifted entropically via an order-by-disorder mechanism that selects a unique spin spiral state \cite{bergman-balents, lee-balents, buessen-trebst}, but in the presence of strong quantum fluctuations ($S \le 1$), long-range order is quenched and a spiral spin liquid ground state manifests that fluctuates about the spiral surface \cite{buessen-trebst}.
Identifying materials exhibiting (quantum) spiral spin liquid states derived from this $J_1$\---$J_2$ model remains an outstanding goal. Transition-metal-based $AB_2X_4$ spinels have been primarily investigated as potential hosts; however two vexing problems typically occur: (1) non-negligible further neighbor interactions beyond the $J_1$\---$J_2$ limit arise and lift the degeneracy and (2) weak tetragonal distortions from the ideal $Fm\bar{3}m$ spinel structure appear. For example, detailed investigations of the spinels MgCr$_2$O$_4$ \cite{tomiyasu-yamada, bai-mourigal}, MnSc$_2$S$_4$ \cite{gao-ruegg, iqbal-reuther, krimmel-loidl}, NiRh$_2$O$_4$ \cite{buessen-trebst, chamorro-mcqueen}, and CoRh$_2$O$_4$ \cite{flynn-mourigal} have all required expanding the model Hamiltonian to include up to third-neighbor interactions, originating from the large spatial extent of $d$-orbitals, to describe the generation of their helical magnetic ground states. Within some materials like NiRh$_2$O$_4$ \cite{buessen-trebst, chamorro-mcqueen}, single ion anisotropies must also be incorporated to digest the experimental results. Complexities with extended interactions beyond the $J_1$\---$J_2$ limit may also compound with inequivalent exchange pathways that form as the cubic $Fm\bar{3}m$ spinel structure undergoes a distortion to a tetragonal $I4_1/amd$ or $I\bar{4}2d$ space group prior to magnetic ordering (e.g. NiRh$_2$O$_4$ \cite{buessen-trebst, chamorro-mcqueen} and CoRh$_2$O$_4$ \cite{flynn-mourigal}).
The tetragonal distortion in spinels can be viewed as a compression of the diamond lattice along one of its cubic axes (opposite to that illustrated in Figure \ref{fig:fig1}), and it splits the nominal $J_2$ of the ideal diamond lattice structure into two different pathways. This disrupts the reciprocal space spiral surface generated in the $J_1$\---$J_2$ model's cubic limit. Despite these complications common to $A$-site transition metal spinels, the predictions born from the model Hamiltonian show substantial promise as materials such as MnSc$_2$S$_4$ \cite{gao-ruegg, iqbal-reuther, krimmel-loidl}, CoAl$_2$O$_4$ \cite{zaharko-loidl, roy-furukawa, macdougall-nagler}, and NiRh$_2$O$_4$ \cite{buessen-trebst, chamorro-mcqueen} are nevertheless either close to or partially manifest degenerate spiral spin states. Identifying other crystal structures that realize comparable physics but with more localized $f$-electron moments is an appealing path forward.
Here we present an investigation of an alternative, frustrated diamond lattice framework in the material LiYbO$_2$. This material can be viewed as containing a stretched diamond lattice of Yb$^{3+}$ moments (Figure \ref{fig:fig1}), and it falls within a broader family of $ALnX_2$ ($A$ = alkali, $Ln$ = lanthanide, $X$ = chalcogenide) materials where the lattice structure is dictated by the ratio of lanthanide ion radius to alkali plus chalcogenide radii (Figure \ref{fig:fig2}). Our results show that LiYbO$_2$ realizes the expected ground state derived from a $J_1$\---$J_2$ Heisenberg model on a tetragonally-elongated diamond lattice and that $J_{eff} = 1/2$ Yb$^{3+}$ ions in related materials may act as the basis for applying the Heisenberg $J_1$\---$J_2$ model to $Ln$-ion diamond-like materials. Notably, however, variance between the observed and predicted phasing of Yb moments on the bipartite lattice as well as the emergence of an intermediate, partially disordered state suggests the presence of interactions/fluctuation effects not captured in the classical $J_1$\---$J_2$ Heisenberg framework.
\section{II. Methods}
\subsection{Sample preparation}
Polycrystalline LiYbO$_2$ was prepared from Yb$_2$O$_3$ (99.99\%, Alfa Aesar) and Li$_2$CO$_3$ (99.997\%, Alfa Aesar) via a solid-state reaction in a 1:1.10 molar ratio. This off-stoichiometric ratio was used to compensate for the partial loss of Li$_2$CO$_3$ during the open crucible reaction. The constituent precursors were ground together, heated to 1000 $^{\circ}$C for three days in air, reground, and then reheated to 1000 $^{\circ}$C for 24 hrs. Samples were kept in a dry, inert environment to prevent moisture absorption. Measurements were conducted with minimal atmospheric exposure to maintain sample integrity. Sample composition was verified via x-ray diffraction measurements on a Panalytical Empyrean powder diffractometer with Cu-K$\alpha$ radiation, and data were analyzed using the Rietveld method in the Fullprof software suite \cite{rodriguezfullprof}.
\subsection{Magnetic susceptibility}
The bulk magnetization and static spin susceptibility of LiYbO$_2$ were measured using three different instruments. Low-field d.c. magnetization data from 2 to 300 K were collected on a Quantum Design Magnetic Properties Measurement System (MPMS3) with a 7 T magnet, and isothermal d.c. magnetization data between 2 to 300 K were collected on a Quantum Design Physical Properties Measurement System (PPMS) equipped with a vibrating sample magnetometer insert and a 14 T magnet. Low-temperature a.c. susceptibility data between 2 K and 330 mK were collected on an a.c. susceptometer at 711.4 Hz with a 0.1 Oe (7.96 A m$^{-1}$) drive field) in a $^{3}$He insert. The background generated by the sample holder in this low temperature a.c. measurement is subtracted from the data presented.
\subsection{Heat capacity}
Specific heat measurements were collected between 100 mK and 300 K on sintered samples of LiYbO$_2$ in external magnetic fields of 0, 3, 5, and 9 T. Specific heat data between 2 to 300 K were collected on a Quantum Design PPMS with the standard heat capacity module, while specific heat data below 2 K was obtained with a dilution refrigerator insert. The lattice contribution to the specific heat of LiYbO$_2$ was modeled with a Debye function using two Debye temperatures of $\Theta_{D1} = 230.5$ K and $\Theta_{D2} = 615.3$ K. The magnetic specific heat was then obtained by subtracting out the modeled lattice contribution from the data, and $C_{mag}/T$ was integrated from 100 mK to 40 K to determine magnetic entropy of LiYbO$_2$ at 0, 3, 5, and 9 T.
\subsection{Neutron diffraction}
Neutron powder diffraction data were collected on the HB-2A diffractometer at the High Flux Isotope Reactor (HFIR) in Oak Ridge National Laboratory. The sample was placed inside a cryostat with a $^{3}$He insert and a 5 T vertical field magnet, and data were collected between 270 mK and 1.5 K. Sintered pellets of LiYbO$_2$ were loaded into Cu canisters, and incident neutrons of wavelength $\lambda=2.41$ \AA ~were selected using a Ge(113) monochromator. Rietveld refinement of diffraction patterns was conducted using the FullProf software suite \cite{rodriguezfullprof}, and magnetic symmetry analysis was performed with the program $SARAh$ \cite{wills2000new}. The structural parameters were determined using data collected at 1.5 K and then fixed for the analysis of the temperature-subtracted data used for magnetic refinements.
Inelastic neutron scattering (INS) data were collected on two instruments. High-energy inelastic data were obtained on the wide Angular-Range Chopper Spectrometer (ARCS) at the Spallation Neutron Source in Oak Ridge National Laboratory. Two incident neutron energies of $E_i$ = 150 meV (Fermi 2, Fermi frequency 600 Hz) and 300 meV (Fermi 1, Fermi frequency 600 Hz) were used, and data were collected at 5 K and 300 K \cite{lin2019energy}. Background contributions from the aluminum sample can were subtracted out by measuring an empty cannister under the same conditions. Crystalline electric field (CEF) analysis was conducted by integrating energy cuts ($E$-cuts) of the 300 meV data between $|Q| = [4, 6]$ \AA$^{-1}$. Integrated $E$-cuts of the 150 meV data between $|Q| = [2, 3]$ \AA$^{-1}$ are shown in the Supplementary Materials \cite{supplementalmaterial}. Peaks were fit with a Gaussian function that approximates the beam shape of the instrument. Low-energy inelastic scattering data were collected on the Disc Chopper Spectrometer (DCS) instrument at the NIST Center for Neutron Research (NCNR), National Institute of Standards and Technology (NIST). Neutrons of incident energy $E_i = 3.32$ meV in the medium-resolution setting were used, and the sample was loaded into a cryostat with a 10 T vertical field magnet and a dilution insert.
\subsection{Crystalline electric field analysis}
The crystalline electric field (CEF) of LiYbO$_2$ was fit following a procedure outlined in Bordelon et al. \cite{bordelon2020}, and a rough overview is reviewed here.
In LiYbO$_2$, magnetic Yb$^{3+}$ with total angular momentum $J = 7/2$ ($L = 3$, $S = 1/2$) is split into a series of four Kramers doublets in the local $D_{2d}$ CEF point group symmetry. Estimations of the splitting can be modeled with a point charge (PC) model of varying coordination shells in the crystal field interface of Mantid Plot \cite{Mantid}. Three coordination-shell variants with increasing distance from a central Yb ion are displayed as PC 1, PC 2, and PC 3 in Table \ref{tab:tab2}. The minimal Hamiltonian with CEF parameters $B_n^m$ and Steven's operators $\hat{O}_m^n$ \cite{StevensOperators} in $D_{2d}$ symmetry is written as follows:
\begin{multline}
\label{eq:CEF}
H_{CEF} = B_2^0 \hat{O}_2^0 + B_4^0 \hat{O}_4^0 + B_4^4 \hat{O}_4^4 + B_6^0 \hat{O}_6^0 + B_6^4 \hat{O}_6^4
\end{multline}
The diagonalized CEF Hamiltonian was used to calculate energy eigenvalues, relative transition intensities, a powder-averaged $g_{avg}$ factor, and corresponding wave functions. These values were compared with data obtained from integrated $E$-cuts of ARCS 300 meV data and bulk magnetic property measurements. The deviation was minimized with a combination of Mantid Plot \cite{Mantid}, SPECTRE \cite{spectre}, and numerical error minimization according to the procedure in Bordelon et al. \cite{bordelon2020, gaudet-gaulin} to approach a global minimum that represents the Yb CEF environment in LiYbO$_2$.
\section{III. Experimental Results}
\begin{figure}[t]
\includegraphics[scale=.55]{Figure2.png}
\caption{The series of $ALnX_2$ ($A$ = alkali, $Ln$ = lanthanide, $X$ = chalcogenide) compounds crystallize in several structures governed by the ratio of lanthanide radius divided by the sum of the alkali and chalcogenide radii derived from tabulated ionic radii and reported crystal structures \cite{shannon-prewitt, shannon-prewitt-revised, hashimoto2002structures, hashimoto2003magnetic, bronger1993synthesis, dong2008structure, cantwell2011crystal, liu-zhangAMX2, xing-sefat2}. Crossover between differing phases occurs at the dashed lines, and materials on these lines can crystallize in both neighboring space groups (e.g. NaErO$_2$ $R\bar{3}m$ and $C2/c$ \cite{hashimoto2003magnetic}).}
\label{fig:fig2}
\end{figure}
\subsection{Radius ratio rule in $\bf{ALnX_2}$ materials}
The $I4_1/amd$ space group is one of the seven major space groups ($R\bar{3}m$ and $P6_3/mmc$, $C2/c$, $\alpha$-$I4_1/amd$, $\beta$-$P2_1/c$ $\gamma$-$Pbmn$, $\delta$-$P2_1/c$) that represent the $ALnX_2$ compounds as shown in Figure \ref{fig:fig2}. The structure types adopted by this family of compounds switch depending on the relative sizes between alkali and lanthanide radii. An empirical relationship between the radii of all three chemical constituents of the $ALnX_2$ family and the major space groups reported in this series is shown in Figure \ref{fig:fig2} by comparing reported structures in the literature \cite{hashimoto2002structures, hashimoto2003magnetic, bronger1993synthesis, dong2008structure, cantwell2011crystal, liu-zhangAMX2, xing-sefat2} to tabulated ionic radii \cite{shannon-prewitt, shannon-prewitt-revised}. The $\alpha, \beta, \gamma, \delta$ follow the nomenclature of Hashimoto et al. \cite{hashimoto2002structures}, and the $R\bar{3}m$ space group is the $\alpha$-NaFeO$_2$ structure type. Plots of the radius ratio relationships for varying chalcogenides are also displayed in the Supplemental Material section \cite{supplementalmaterial}.
Compounds residing close to or on the dashed lines separating two space groups can crystallize in either space group depending on synthesis conditions or temperature. For example, NaErO$_2$ crystallizes in $R\bar{3}m$ and $C2/c$ \cite{hashimoto2003magnetic} structures at room temperature, and LiErO$_2$ goes through a structural phase transition from $\alpha$-$I4_1/amd$ at 300 K to $\beta$-$P2_1/c$ at 15 K \cite{hashimoto2002structures}. Two related crystal structures are possible in the $R\bar{3}m$ and $P6_3/mmc$ area of Figure \ref{fig:fig2}, and both of these space groups contain sheets of equilateral triangles comprised of lanthanide ions and vary only in the stacking sequence of the triangular sheets ($ABC$ for $R\bar{3}m$ and $AAA$ for $P6_3/mmc$). Previous reports also indicate that the $P6_3/mmc$ phase is favored with large Cs$^+$ ions \cite{xing-sefat2, bronger1993synthesis}. We note here that this empirical radius-ratio rule excludes one of the known $ALnX_2$ phases: the chemically-disordered $Fm\bar{3}m$ NaCl phase that is primarily present at high temperatures when the alkali radius is close to that of the lanthanide radius \cite{liu-zhangAMX2, bronger1973ueber, verheijen1975flux, tromme, ohtani1987synthesis, fabry2014structure}. This chemically-disordered phase goes through a first order phase transition to the $R\bar{3}m$ phase in materials such as NaNdS$_2$ \cite{ohtani1987synthesis, fabry2014structure}.
\begin{table}[t]
\caption{Rietveld refinement structural parameters at 1.5\,K from elastic neutron scattering measurements on LiYbO$_2$ on HB2A in the $I4_1/amd$ space group with origin setting two. Within error, all ions refined to full occupation and no quantifiable site mixing is present.}
\begin{tabular}{cc|ccccc}
\hline
\multicolumn{2}{c|}{$T$} & \multicolumn{5}{c}{1.5 K} \\ \hline
\multicolumn{2}{c|}{\ $\chi^{2}$} & \multicolumn{5}{c}{3.421} \\
\multicolumn{2}{c|}{$\lambda$} & \multicolumn{5}{c}{2.41 \AA} \\
\multicolumn{2}{c|}{$a=b$} & \multicolumn{5}{c}{4.3824(2) \AA} \\
\multicolumn{2}{c|}{$c$} & \multicolumn{5}{c}{10.0625(2) \AA} \\ \hline
Atom & Wyckoff & x & y & z & $B_{iso}$ (\AA$^{2}$) & Occupancy \\ \hline
Yb & 4a & 0 & 0 & 0 & 0.28(9) & 1.000(6) \\
Li & 4b & 0 & 0 & 0.5 & 2.02(30) & 1.00(5) \\
O & 8e & 0 & 0 & 0.22546(7) & 0.74(9) & 1.00(3) \\ \hline
\end{tabular}
\label{tab:tab2}
\end{table}
\begin{figure*}[t]
\includegraphics[width=\textwidth*10/10]{Figure3.png}
\caption{a) Inelastic neutron scattering (INS) spectrum $S(\bf{Q},\hbar \omega)$ collected at 5 K and $E_i = 300$ meV on the ARCS spectrometer with full width half max resolution in the elastic line of 12.8 meV. Three CEF levels indicated by dashed black lines were observed. b) $\bf{Q}$-integrated cut from panel a) overplotted with model lineshapes derived from the CEF fits in Table 1 and convolved with the instrumental resolution in MantidPlot \cite{Mantid}. c) The $D_{2d}$ $J = 7/2$ Yb$^{3+}$ ion generates four Kramers doublets centered at 45, 63, and 128 meV determined via CEF fits to the INS data. Errors shown correspond to instrumental resolution, and the well-separated ground state doublet has an average g-factor $g_{avg}=3.0$. The intensity ratios $I_2/I_1$ and $I_3/I_1$ were determined relative to the 45 meV CEF excitation. d) INS spectrum $S(\bf{Q},\hbar \omega)$ collected at 5 K and $E_i = 150$ meV from the ARCS spectrometer. The first two CEF excitations observed with $E_i = 300$ meV show splitting of $\approx7$ meV, highlighted by the dashed black lines. At $E_i = 300$ meV, this splitting is not resolvable due to the poorer energy resolution. e-f) Integrating the split CEF excitations shows that the ratio of integrated intensities of CEF mode 2 (I$_{2a}+I_{2b}$) to CEF mode 1 (I$_{1a}+I_{1b}$) correlates to the ratios of the integrated intensities in b). The splitting of CEF excitations may represent two distinct chemical environments within LiYbO$_2$ that are not resolvable within structural neutron powder diffraction experiments. }
\label{fig:fig5}
\end{figure*}
\begin{table*}[]
\caption{Point charge (PC) models and CEF fit for LiYbO$_2$ obtained by minimizing observed parameters from $E_i$ = 300 meV INS data and powder averaged $g_{avg}$ factor from isothermal magnetization. The three PC models of increasing size incorporate one (O$^{2-}$ ions), two (O$^{2-}$ and Li$^{+}$ ions) and three (O$^{2-}$, Li$^{+}$, and Yb$^{3+}$ ions) coordination shells surrounding a central Yb$^{3+}$ ion, respectively.}
\begin{tabular*}{14.5cm}{l|llllll|l|llllll}
\hline
& $E_1$ & $E_2$ & $E_3$ & $\frac{I_2}{I_1}$ & $\frac{I_3}{I_1}$ & $g_{avg}$ & $\chi^2$ & $B_2^0$ & $B_4^0$ & $B_6^0$ & $B_4^4$ & $B_6^4$ \\ \hline
PC (2.5 \AA) & 33.3 & 33.8 & 69.0 & 0.98 & 0.08 & 3.6 & 51.0 & -0.67210 & -0.031153 & 0.000064591 & -0.17420 & -0.0012000 \\
PC (3.1 \AA) & 30.9 & 86.2 & 87.5 & 0.10 & 0.06 & 3.7 & 27.8 & 2.1336 & -0.029755 & 0.000069724 & -0.19050 & -0.0012660
\\
PC (3.5 \AA) & 108.6 & 149.9 & 156.6 & 0.07 & 0.08 & 4.5 & 232.0 & -4.2146 & -0.033288 & 0.000081398 & -0.18211 & -0.0014202 \\
Fit & 45.0 & 62.8 & 127.9 & 1.74 & 0.10 & 3.0 & 0.002 & 0.31777 & -0.072378 & 0.0010483 & -0.27051 & 0.0015364 \\ \hline
Observed & 45.0 & 63.0 & 128.0 & 1.76 & 0.10 & 3.0 & & & & & & &
\end{tabular*}
\hskip+0.1cm
\begin{tabular*}{14.525cm}{ll}
\hline
Fit wave functions: & \\
$| \omega_{0, \pm}\rangle =$ $0.901|\mp1/2\rangle +0.434|\pm7/2\rangle$ &\\
$| \omega_{1, \pm}\rangle =$ $-0.434|\mp1/2\rangle +0.901|\pm7/2\rangle$ &\\
$| \omega_{2, \pm}\rangle =$ $0.849|\pm3/2\rangle +0.529|\mp5/2\rangle$ &\\
$| \omega_{3, \pm}\rangle =$ $-0.529|\mp3/2\rangle +0.849|\pm5/2\rangle$ &\\ \hline
\end{tabular*}
\label{tab:tab1}
\end{table*}
\subsection{Chemical structure}
Elastic neutron powder diffraction data collected from LiYbO$_2$ are shown in Figure \ref{fig:fig6}. The crystal structure was fit at 1.5 K to the $I4_1/amd$ structure previously reported \cite{hashimoto2002structures}, and this structural fit was used as the basis for analyzing the magnetic peaks observed below 1.5 K as a function of magnetic field. Details of the structural fit are presented in Table \ref{tab:tab2}. Within resolution of this experiment, all chemical sites are fully occupied without site-mixing, and no impurity phases are present.
LiYbO$_2$ consists of $D_{2d}$ edge-sharing YbO$_6$ octahedra that are connected three-dimensionally within a bipartite magnetic lattice (Figure \ref{fig:fig1}). Each sublattice of trivalent Yb ions ($A$ or $B$ sublattice in Figure \ref{fig:fig1}) connects to the neighboring sublattice's layers with two bonds above and two bonds below with a nearest-neighbor Yb$_{A/B}$-Yb$_{B/A}$ distance of 3.336 \AA \ ($J_1$). This forms a stretched tetrahedron with a Yb ion at its center. The next-nearest-neighbor bond is within the same Yb sublattice where four bonds within the $ab$-plane are connected at 4.4382 \AA \ ($J_2$). Despite this nearest-neighbor and next-nearest-neighbor interaction appearing significantly different in length, superexchange is likely promoted along $J_2$ due to the more favorable Yb-O-Yb bond angle, making the longer next-nearest neighbor exchange comparably relevant to the nearest-neighbor $J_1$. Exchange pathways through oxygen anions along $J_1$ and $J_2$ are nearly equivalent at 4.473 \AA \ and 4.410 \AA, respectively. Therefore, the two magnetic exchange interactions are likely similar in magnitude, and when $|J_1| > 0$ and $J_2 > 0$, this lattice is expected to be geometrically frustrated.
The Yb$^{3+}$ magnetic lattice can be visualized as an extreme limit of tetragonal elongation of the diamond lattice as shown in Figure \ref{fig:fig1}. The diamond lattice originally contains two magnetic interactions, $J_1$ and $J_2$, where $J_2$ interactions within any face of the diamond lattice are equivalent. Stretching the lattice in LiYbO$_2$ breaks the $J_2$ degeneracy, creating a $J_{2a}$ interaction along the elongated direction 5.9090 \AA \ and an in-plane $J_{2b}$ of 4.438 \AA. In the full chemical unit cell of LiYbO$_2$, the elongated $J_{2a}$ interaction necessitates two O$^{2-}$ ion superexchange links relative to the single O$^{2-}$ superexchange in the in-plane $J_{2b}$ and the $J_1$ interaction. As it is likely negligible in strength relative to the other two interactions, the elongated $J_{2a}$ interaction is therefore neglected in this paper and $J_{2b}$ is simply referred to as $J_2$.
\subsection{Crystalline electric field excitations}
Inelastic neutron scattering (INS) data were collected at $T$ = 5 K and $E_i$ = 300 meV to map the intramultiplet CEF excitations in LiYbO$_2$. Figures \ref{fig:fig5}a-c) show three CEF excitations that are centered around 45, 63, and 128 meV. A cut through $S(\bf{Q},\hbar \omega)$ shows the energy-widths of the transitions in Fig. 3b) are limited by the instrumental resolution at $E_i$ = 300 meV. As expected, the lowest-energy CEF transition is high enough to render the ground state Kramers doublet a well-separated $J_{eff} = 1/2$ state at low temperatures. An analysis of the CEF splitting of the $J = 7/2$ Yb$^{3+}$ manifold is detailed in Figure \ref{fig:fig5} and Table \ref{tab:tab1}. With the extracted parameters from the $S(\bf{Q},\hbar \omega)$ cut, the best level scheme fit to the data is shown in Table \ref{tab:tab1}. The calculated CEF $g_{avg}$ is split into two anisotropic components of $g_{//} = 0.58$ and $g_{\perp} = 3.71$, where $g_{avg} = \sqrt (1/3( g_{//}^2 + 2g_{\perp}^2))$. The fit diverges from point charge models of varying coordination size presented in Table \ref{tab:tab1} and is closest in sign to the parameters $B_m^n$ generated from a point charge model incorporating two ionic shells (3.1 \AA \ with O$^{2-}$ and Li$^{+}$ ions).
The first two CEF excitations were further analyzed with lower $E_i $ = 150 meV INS data presented in Figure \ref{fig:fig5}d-f). Within this higher resolution window, the lower two CEF excitations at 45 meV and 63 meV show new features, and the two CEF excitations are asymmetrically split into peaks centered at 39.5 meV + 47.0 meV (excitation 1) and 55.6 meV + 62.6 meV (excitation 2). At $E_i $ = 300 meV, this splitting is below the instrumental resolution and is not readily apparent. The relative integrated intensities of the split modes in excitation 1 and excitation 2 at $E_i $ = 150 meV however agree with the ratios of the single/convolved modes observed in the $E_i $ = 300 meV. The most likely explanation for the observed splitting at $E_i $ = 150 meV is the presence of two distinct chemical environments surrounding Yb ions that are outside of the resolution of the current neutron powder diffraction measurements.
LiYbO$_2$ indeed contains two sublattices of Yb ions ($A$ and $B$ in \ref{fig:fig1}c), and, in the ideal $I4_1/amd$ structure, Yb ions within each sublattice reside in chemically-equivalent environments. Since the CEF fit is closest to a point charge model including both nearest O$^{2-}$ and Li$^{+}$ ions, the observed splitting could arise from these non-magnetic ions residing slightly off of their ideal Wyckoff positions. A similar chemical feature has been observed in tetragonally-distorted spinels, such as CuRh$_2$O$_4$ \cite{flynn-mourigal} where Cu ions are displaced off of their ideal Wyckoff site. While such a feature is outside of the resolution of the average structural refinement for LiYbO$_2$, the large isotropic thermal parameter of the Li ions suggests this as a possibility. We note here that this distortion is necessarily small and should not significantly affect the $J_1$\---$J_2$ model of the LiYbO$_2$ magnetic lattice. For this reason, analysis of the CEF environment was calculated in the limit assuming only one CEF environment using the $E_i $ = 300 meV data.
\begin{figure}[t]
\includegraphics[scale=0.275]{Figure4.png}
\caption{a) Temperature dependence of the inverse magnetic susceptibility of LiYbO$_2$. Solid line shows the a Curie-Weiss fit to the data between $20<T<100$ K. b) Field dependence of the magnetization collected at a variety of temperatures. c) 2 K isothermal magnetization curve with a linear fit in the saturated state above 10 T. The 0T intercept ($g_{avg}\mu_{B}/2$) provides a powder-averaged $g_{avg,VV}$ and the slope provides $\chi_{VV}$. d) a.c. magnetic susceptibility $\chi `(T)$ data collected for $330$ mK$<T<3.5$ K at zero-field. The two dashes lines at 1.13 K and 0.45 K mark the onset of peaks observed in zero-field heat capacity data.}
\label{fig:fig3}
\end{figure}
\begin{figure}[]
\includegraphics[scale=0.45]{Figure5.png}
\caption{a-d) Specific heat $C(T)$ of LiYbO$_2$ collected as a function of temperature under $\mu_0 H=$ 0, 3, 4, and 9 T. The integrated magnetic entropy $\delta S_M$ is overplotted with the data as a black line. Results from a Debye model of lattice contributions to $C(T)$ are shown as orange lines. The horizontal dashed lines represent $Rln(2)$.}
\label{fig:fig4}
\end{figure}
\subsection{Magnetization, susceptibility, and heat capacity results}
Figure \ref{fig:fig3} shows the magnetic susceptibility, isothermal magnetization, and a.c. susceptibility measured on powders of LiYbO$_2$. In the low temperature regime where the ground state Kramers doublet is primarily occupied ($T < 100$ K), data were fit to a Curie-Weiss-type behavior with a $\Theta_{CW} = -3.4$ K and an effective moment $\mu_{eff} = 2.74 \ \mu_B$. This implies a powder-averaged $g$-factor $g_{avg,CW} = 3.13$ assuming $J_{eff} = 1/2$ Yb ions. The nonlinearity of the Curie-Weiss fit above 100 K arises due to Van Vleck contributions to the susceptibility that derive from the CEF splitting of the $J = 7/2$ Yb manifold. In order to independently determine $g_{avg}$, the $\chi_{VV}$ contribution to the total susceptibility was fit in the saturated regime ($\mu_0H > 10$ T) of the 2 K isothermal magnetization data shown in Figure \ref{fig:fig3}. In the near-saturated state, the slope of isothermal magnetization yields $\chi_{VV} = 0.0206$ cm$^{3}$ mol$_{Yb}^{-1}$ \cite{li2015rare}, and the intercept of this linear fit with $\mu_0H = 0$ T was utilized to determine the saturated magnetic moment ($g\mu_B/2$) that corresponds to a powder-averaged $g_{avg,VV} = 2.98$. As the Curie-Weiss fit is more susceptible to minor perturbations and background terms, the $g_{avg,VV} = 2.98$ derived from isothermal magnetization data was used for fitting the CEF scheme in Figure \ref{fig:fig5} and Table \ref{tab:tab1}.
Magnetic susceptibility data in Figure \ref{fig:fig3} explore the low temperature magnetic behavior of LiYbO$_2$. Two low-temperature ($T < 10$ K) features appear: The first is a broad cusp in susceptibility centered near 1.5 K and is an indication of the likely onset of magnetic correlations. The second feature is a small upturn below 0.45 K. When compared with specific heat measurements in Figure \ref{fig:fig4}, these two features in $\chi^\prime(T)$ coincide with the two sharp anomalies in $C_p(T)$ at $T_{N1}=1.13$ K and $T_{N2}=0.45$ K. An additional broad peak also appears in $C(T)$ centered near 2 K, likely indicative of the likely onset of short-range correlations. As discussed later in this manuscript, the two lower temperature peaks in $C_p(T)$ mark the staged onset of long-range magnetic order with $T_{N1}$ marking the onset of partial order with disordered relative phases between the $A$ and $B$ Yb-ion sublattices and with $T_{N2}$ marking the onset of complete order between the two sublattices.
Figure \ref{fig:fig4}a) also displays the total magnetic entropy released upon cooling down to 100 mK. Below 200 mK, a nuclear Schottky feature arises from Yb nuclei as similarly observed in NaYbO$_2$ \cite{bordelon2019field}. Integrating $C_p/T$ between 100 mK and 40 K shows that 98\% of $Rln(2)$ is reached at 0 T, showing that the ordering is complete by 100 mK. Approximately half of $Rln(2)$ is released upon cooling through the broad 2 K peak representing the onset of short range correlations. $C_p(T)$ data were also collected under a series of applied magnetic fields. The onset of $T_{N1}$ stays fixed at 1.13 K from 0 T to 5 T and shifts up to 1.40 K at 9 T. The 0 T heat capacity anomaly at $T_{N2}=0.45$ K begins to broaden at 3 T into a small shoulder of the initial 1.13 K transition and vanishes by 5 T. The broad $C_p(T)$ peak marking the onset of short-range correlations near 2 K shifts to higher temperatures with increasing magnetic field, consistent with a number of other frustrated spin systems \cite{bordelon2019field, li2015rare}. The suppression of the staged $T_{N1}$-$T_{N2}$ ordering under modest magnetic field strengths suggests that zero-field fluctuations/remnant degeneracy likely influence the ordering behavior.
\begin{figure*}[t]
\includegraphics[width=\textwidth*9/10]{Figure6.png}
\caption{Neutron powder diffraction data collected for LiYbO$_2$ at HB-2A at the High Flux Isotope Reactor. a) Fits to the elastic scattering data at 1.5 K reveal only one structural phase. b) Temperature-subtracted diffraction data ($T-1.5$ K) revealing a series of new magnetic peaks upon cooling. Additionally, at 270 mK and 3 T, another set of magnetic peaks arise. Intensity near 1.5 \AA$^{-1}$ results from slight under/over subtraction of the structural peak at that position in a) and is not a magnetic Bragg reflection. c) Helical magnetic structure fit below the ordering transition $T_{N2}$. d) 270 mK data collected under zero field with the 1.5 K structural data subtracted. Green line shows the resulting fit using the magnetic structure described in the text. e) 830 mK data collected under zero field with the 1.5 K structural data subtracted. The orange line shows the partially disordered, intermediate helical state described in the text and the green line shows a fit using the fully ordered helical structure for comparison. f) 270 mK data collected under $\mu H=3$ T with the 1.5 K structural data subtracted. The red line shows the fit to the commensurate magnetic structure describe in the text.}
\label{fig:fig6}
\end{figure*}
\begin{table}[]
\caption{Coefficients of the magnetic basis vectors creating the helical models of the base temperature magnetic structure of LiYbO$_2$ at 0 T and 3 T, where $bv_1 = (100)$, $bv_2 = (010)$, and $bv_3 = (001)$.}
\begin{tabular}{c|ccc|ccc}
\hline
& \multicolumn{3}{c|}{270 mK, 0 T} & \multicolumn{3}{c}{270 mK, 3 T} \\
& \multicolumn{3}{c|}{${\bf{k}} = (0.384, \pm 0.384, 0)$} & \multicolumn{3}{c}{${\bf{k}} = (1/3, \pm 1/3, 0)$} \\ \hline
atom ($x$, $y$, $z$) & $bv_1$ & $bv_2$ & $bv_3$ & $bv_1$ & $bv_2$ & $bv_3$ \\ \hline
Yb$_1$ (0, 0.75, 0.125) & 0 & -1.26$i$ & 1.26 & 0 & -1.26$i$ & 1.26 \\
Yb$_2$ (0, 0.25, 0.875) & 0 & -1.26$i$ & 1.26 & 0 & -1.26$i$ & 1.26
\end{tabular}
\label{tab:tab3}
\end{table}
\subsection{Neutron diffraction results}
To further investigate the low-temperature, ordered state, neutron powder diffraction measurements were performed. Figure \ref{fig:fig6} details the field- and temperature-evolution of magnetic order in LiYbO$_2$ about the $T_{N1}$ and $T_{N2}$ transitions identified in specific heat measurements (Figure \ref{fig:fig4}). Magnetic peaks appear in the powder neutron diffraction data below 1 K, and three regions of ordering were analyzed: (1) In the zero-field low-temperature, fully ordered state ($T<450$ mK); (2) in the zero-field, intermediate ordered state ($450$ mK$<T< 1$ K); and (3) in the field-modified ordered state ($T<450$ mK and $\mu_0 H=3$ T). Figure \ref{fig:fig6}a) shows the data and structural refinement collected at 1.5 K in the high temperature paramagnetic regime---this is used as nonmagnetic background that is subtracted from the low-temperature data. Figure \ref{fig:fig6}b) shows the subtracted data in each of the above regions overplotted with one another, and each magnetic profile is discussed separately in the following subsections. We note here that in each region, the large difference signal observed slightly above 1.5~\AA$^{-1}$ ~is due to the slight under/over subtraction of a nuclear reflection.
\subsubsection{Region 1: $\mu_0 H=0$ T , $T<450$ mK}
At 270 mK, well below $T_{N2}$, a series of peaks appear at incommensurate momentum transfers. These new magnetic reflections are described by a doubly-degenerate ordering wave vector of ${\bf{k}} = (0.384, \pm 0.384, 0)$. The best fit to the data in this regime corresponds to a helical magnetic structure shown in \ref{fig:fig6}c) that is produced from the $\Gamma_1$ irreducible representation (Kovalev scheme) of this space group with the three basis vectors $bv_1 = (1, 0, 0)$, $bv_2 = (0, 1, 0)$, and $bv_3 = (0, 0, 1)$. The helical state is defined by a combination of the ordering wave vector $\bf{k}$ and the helical propagation direction. The latter defines a vector that moments rotate in the plane perpendicular. Best fits for the refinement data were achieved when the helical propagation vector is restricted to the $ab$-plane. However, all helical propagation directions within the $ab$-plane produce equivalent fits to the data.
The fit presented in Figure \ref{fig:fig6}d) corresponds to the instance where helices propagate along the $b$-axis with moments rotating within the $ac$-plane depicted in in Figure \ref{fig:fig6}c). Coefficients of the basis vector representation of this fit are shown in Table \ref{tab:tab3}. Due to the bipartite nature of this lattice, two magnetic Yb$^{3+}$ atoms are defined in the system (denoted as sublattices $A$ and $B$), and in effect, this creates a relative phase difference in the moment rotation between the two sites that is experimentally fit at $0.58\pi$. Additional simulations provided in the Supplemental Material section detail how altering the phasing of the sublattices affects the refinement \cite{supplementalmaterial}. The ordered magnetic moment refined with this fit is $\mu=1.26(10)$ $\mu_B$, comprising 84\% of the expected 1.5 $\mu_B$ moment in a $J_{eff} = 1/2$ system with $g_{avg} = 3$.
\subsubsection{Region 2: $\mu_0 H=0$ T , $450$ mK $<T<1.13$ K}
As the temperature is increased above $T_{N2}$ to 830 mK into the intermediate ordered state, incommensurate magnetic reflections with the same ordering wave vector of ${\bf{k}} = (0.384, \pm 0.384, 0)$ persist (Figure \ref{fig:fig6}e)). Order in this $T_{N1}$ state is seemingly still long-range and the lowest angle reflection can be fit to a Lorentzian peak shape to extract an estimated, minimum correlation length. In both the 270 mK base temperature and 830 mK intermediate temperature regimes, the minimum correlation length corresponds to $\approx$ 364 \AA. Modeling the pattern of magnetic peaks in this intermediate temperature regime using the same $T_{N2}$ structure as described above however fails to fully capture the data. As seen in Figure \ref{fig:fig6}e), the $T_{N2}$ (green) structure overestimates reflections near 1.2 \AA$^{-1}$.
One potential model for the magnetic order in this intermediate temperature regime is to allow the relative phasing of the $A$ and $B$ magnetic sublattices to become disordered upon warming into the $T_{N1}$ state. In other words, helical magnetic order could establish with ${\bf{k}} = (0.384, \pm 0.384, 0)$; however the phasing between Yb-sites would remain disordered prior to selecting a specific phase below $T_{N2}$. This conjecture was modeled by averaging over ten fits using equally-spaced relative phases from zero to $2\pi$ between Yb-sites, and where each fit was calculated using an identical moment size (1.26 $\mu_b$). This averaged phasing model (Figure \ref{fig:fig6}d) orange) captures the relative peak intensities better than the single-phase model used below $T_{N2}$ and is supported by $C(T)$ data showing that additional entropy freezes out below $T_{N2}$.
\subsubsection{Region 3: $\mu_0 H=3$ T , $T<450$ mK}
Upon applying a magnetic field to the low-temperature ordered state below $T_{N2}$, the magnetic ordering of the system changes. Figure \ref{fig:fig6}f) shows that a $\mu_0 H=3$ T field drives commensurate peaks to appear in place of the incommensurate reflections in the zero-field ordered state. The modified propagation vector corresponds to the doubly-degenerate ${\bf{k}} = (1/3, \pm 1/3, 0)$. Although the modified ${\bf{k}}$ reflects a locking into a commensurate structure, qualitatively, the details of the ordered state remain similar to the zero-field $T_{N2}$ model. The commensurate 3 T state is still best represented by an $ab$-plane helical magnetic structure with basis vector coefficients displayed in Table \ref{tab:tab3}. The magnetic moment is refined to be $\mu=1.26(9)$ $\mu_B$ and the two Yb-sublattices differ by a relative phase of $0.42\pi$.
\subsection{Low-energy magnetic fluctuations}
The low-energy spin dynamics of Yb moments in LiYbO$_2$ were investigated in all three ordered regimes described in the previous section via inelastic neutron scattering measurements. While the powder-averaged data is difficult to interpret given the complexity of the ordered state, Figure \ref{fig:fig8} plots a series of background-subtracted inelastic spectra that qualitatively illustrate a few key points. Below $T_{N2}$ and in zero-field, the bandwidth of spin excitations extends to roughly 1 meV. Spectral weight appears to originate from the magnetic zone centers of $k = (q, \pm q, 0)$ (where $q = 0.384$ at 0 T and $q = 1/3$ at 3 T) and the $\Gamma$ point. As the ordered does not change appreciably under moderate fields, the low-energy spectra remain qualitatively similar for both 0 T and 3 T data below $T_{N2}$. Similarly, upon heating from $T_{N2}$ into the $T_{N1}$ state, minimal changes are observed in the inelastic spectra. At 10 T and 36 mK however, LiYbO$_2$ enters a field-polarized state where the low energy spin fluctuations are dramatically suppressed. The removal of low-energy fluctuations in this high-field data was used to subtract out background contributions in the data shown in Figure \ref{fig:fig8}. There are slight differences in the dynamics of the 0 T and 3 T states in Figure \ref{fig:fig8} that will require future experiments to detail their differences with higher statistics. The raw data for each field and temperature setting are plotted in the Supplemental Material section for reference \cite{supplementalmaterial}.
\begin{figure}[t]
\includegraphics[scale=0.7]{Figure7.png}
\caption{Low-energy inelastic neutron scattering (INS) spectra $S(|Q|,\hbar \omega)$ collected on the DCS spectrometer at a) $\mu_0 H= 0$ T and 36 mK, b) $\mu_0 H= 0$ T and 800 mK, and c) $\mu_0 H= 3$ T and 36 mK. All data have data collected at 36 mK and 10 T subtracted, where LiYbO$_2$ enters a field-polarized state, indicated by isothermal magnetization data from Figure 4b).}
\label{fig:fig8}
\end{figure}
\section{IV. Theoretical analysis}
In the following subsections, we construct a classical Heisenberg Hamiltonian to describe the interactions of Yb ions in LiYbO$_2$. We then use this Hamiltonian, extended out to next-nearest neighbors, to model the potential magnetic ground states in LiYbO$_2$ for comparison with experimental data. Spin excitations are then also modeled in the parameter space predicting magnetic order most closely matching that experimentally observed.
\subsection{LiYbO$_2$ symmetry analysis}
A minimal Hamiltonian describing the nearest-neighbor (NN) interactions in LiYbO$_2$ ($I4_1/amd$) following symmetry analysis \cite{supplementalmaterial} can be written as
\begin{equation}
\begin{aligned}
\label{eq:H1}
H_1=&\sum_{\langle i,j\rangle}J_z S^z_i S^z_j+ J_{xy} (S^x_i S^x_j+ S^y_i S^y_j) \\
&+ J_\delta (\bm{S}_i\cdot \bm{f}_{ij})(\bm{S}_j\cdot \bm{f}_{ij})
+J_{cz}(\bm{S}_i\cdot \bm{f}_{ij} S^z_j + S^z_i\cdot \bm{f}_{ij} \hat{\bm{z}}),
\end{aligned}
\end{equation}
where $\bm{f}_{ij}$ is the projection of {the bond vector} $\bm{e}_{ij}$ onto the basal plane. The symmetry-allowed next nearest-neighbor (NNN) interactions are written as
\begin{equation}
\begin{aligned}
\label{eq:H2}
H_2=&\sum_{\langle\langle i,j\rangle\rangle} J'_z S^z_i S^z_j + J'_{xy}(S^x_iS^x_j+S^y_iS^y_j) \\
&+ J'_\delta(\bm{S}_i\cdot \bm{e}_{ij})(\bm{S}_j\cdot \bm{e}_{ij})
+ \bm{D}_{ij} \cdot \bm{S}_i\times \bm{S}_j,
\end{aligned}
\end{equation}
where the Dzyaloshinskii-Moriya (DM) vectors for the NNN bonds $\langle ij\rangle$ along $\bm{a}$ and $\bm{b}$ are $\bm{D}_{ij} = (-1)^{\mu(i)} D \bm{a}\times \hat{\bm{z}}$ and $\bm{D}_{ij} =(-1)^{\mu(i)} D\bm{b}\times \hat{\bm{z}}$, respectively. Here $\mu(i)=0,1$ for the sublattice $i=A,B$, respectively, indicating that the sign of the DM vector alternates between layers.
We hereby restrict our study to the Hamiltonian up to NNN: $H = H_1+H_2$. For $f$-orbital ions such as Yb, the anisotropies $J_\delta$ and $J'_\delta$ are usually negligible, and as a good approximation we take the Heisenberg limit $J_z=J_{xy}=J_1$, and $J'_z=J'_{xy}=J_2$ (see \cite{supplementalmaterial} for a discussion on the effect of $J_z\neq J_{xy}$ and $J'_z\neq J'_{xy}$). This generates as a physical model the $J_1$\---$J_2$ Heisenberg Hamiltonian
\begin{equation}
\label{eq:Htot}
H = J_1\sum_{\langle ij\rangle} \bm{S}_i\cdot \bm{S}_j + J_2 \sum_{\langle \langle ij\rangle\rangle} \bm{S}_i \cdot \bm{S}_j + \bm{D}_{ij}\cdot \bm{S}_i\times \bm{S}_j.
\end{equation}
\subsection{The $\bf{J_1}$\---$\bf{J_2}$ model and spiral order}
We first look at the $J_1$\---$J_2$ Heisenberg model on the stretched diamond lattice without the DM term. The classical ground state of this model can be solved exactly. In momentum space, the $J_1$\---$J_2$ Heisenberg model is written as
\begin{equation}\label{shs}
H = \sum_{\bm{q},\mu,\nu} \bm{S}_{\bm{q},\mu} J^{\mu\nu}_{\bm{q}} \bm{S}_{-\bm{q},\nu},
\end{equation}
with
\begin{equation}
\nonumber
\begin{aligned}
J^{11}_{\bm{q}} &= J^{22}_{\bm{q}} = J_2 (\cos \bm{q}\cdot \bm{a} + \cos \bm{q}\cdot \bm{b}),\\
J^{12}_{\bm{q}} &= J^{21*}_{\bm{q}}=J_1 \left(e^{-i\frac{\bm{q}\cdot \bm{c}}{4}}\cos \frac{\bm{q}\cdot \bm{a}}{2}+e^{i\frac{\bm{q}\cdot \bm{c}}{4}}\cos \frac{\bm{q}\cdot \bm{b}}{2}\right).
\end{aligned}
\end{equation}
Therefore the lower branch of the band is
\begin{equation}
\lambda_{\bm{q}} = J^{11}_{\bm{q}} - | J^{12}_{\bm{q}}|.
\end{equation}
Solving for the minimum of $\lambda_{\bm{q}}$, the classical ground state is an incommensurate spiral, with wave vector
\begin{equation}
\bm{q} = \frac{2\pi}{a}(q,q,0)\quad \text{or}\quad \bm{q}= \frac{2\pi}{a}(q,-q,0),
\end{equation}
\begin{figure}[!t]
\centering
\includegraphics[width=0.4\textwidth]{Figure_j1j2cartoon.png}
\caption{Phase diagram of magnetic order in the $J_1$\---$J_2$ Heisenberg model, assuming $J_2>0$, where ferromagnetic (FM), incommensurate (IC) spiral, and antiferromagnetic (AFM) N\'eel order exist.}\label{j1j2cartoon}
\end{figure}
where
\begin{equation}
\nonumber
q \equiv \left\{\begin{array}{ll} \pm \frac{1}{\pi} \arccos \frac{|J_1|}{4J_2},\\
0,\end{array}\right.
\end{equation}
\begin{equation}
\nonumber
\text{respectively for} \left\{\begin{array}{ll} |J_1|\leq 4J_2,\\
|J_1|>4J_2. \end{array} \right.
\end{equation}
Note that due to the sublattice structure, both the FM and AFM N\'eel orders have $q=0$. From now on we assume $J_2>0$ since spiral order can appear only for a positive $J_2$ (Figure \ref{j1j2cartoon}).
The experimental value for the doubly-degenerate spiral wave vector is $\frac{2\pi}{a}(0.384, \pm 0.384,0)$,
which gives
\begin{equation}
J_1 = \pm 4 \cos (0.384\pi)J_2 = \pm 1.426 J_2.
\end{equation}
The eigenvector corresponding to $\lambda_{\bm{q}}$ is $u_{\bm{q}} = \frac{1}{\sqrt{2}} (e^{i \phi_{\bm{q}}},1)^T$, where the phase $\phi_{\bm{q}} = \pi + \mathrm{Arg} J^{12}_{\bm{q}}$ determines the relative angle or phase between the spins of the two sublattices. The magnetic order then is
\begin{equation}
\label{mag}
\bm{S}_{\bm{r}_i} = \left(0, \cos \bm{q}\cdot \bm{r}_i, \sin \bm{q}\cdot \bm{r}_i\right)
\end{equation}
or any coplanar configuration that is related to Eq.~\eqref{mag} by a global SO(3) rotation.
\begin{figure}[!b]
\centering
\includegraphics[width=0.19\textwidth]{Figure8.png}
\caption{The classical ground state condition $\bm{S}_{\triangle,1} + \bm{S}_{\triangle,2} + \frac{J_1}{2J_2}\bm{S}_{\triangle,3}=0$.}\label{triangle}
\end{figure}
A more intuitive, geometrical way to obtain the ground state of the Heisenberg $J_1$\---$J_2$ Hamiltonian is to rewrite it as the sum over all the ``elementary'' triangles $\triangle$ that are enclosed by two NN bonds and one NNN bond, where each NNN bond belongs to only one ``elementary'' triangle while each NN bond is shared by two ``elementary'' triangles. Concretely, for each $\triangle$, label the two spins connected with an NNN bond as $\bm{S}_{\triangle,1}$ and $\bm{S}_{\triangle,2}$, and the third spin as $\bm{S}_{\triangle,3}$, we then have:
\begin{equation}
H = \text{Constant} + \frac{J_2}{2} \sum_{\triangle} \left(\bm{S}_{\triangle,1} + \bm{S}_{\triangle,2} + \frac{J_1}{2J_2}\bm{S}_{\triangle,3}\right)^2.
\end{equation}
Written in this way, the classical ground state is the spin configuration that satisfies $\bm{S}_{\triangle,1} + \bm{S}_{\triangle,2} + \frac{J_1}{2J_2}\bm{S}_{\triangle,3}=0$ for all $\triangle$. Denote the (orientationless) angle between two vectors $\bm{S}_1$ and $\bm{S}_2$ by $\langle \bm{S}_1,\bm{S}_2\rangle$. One easily infers from Figure \ref{triangle} that
\begin{equation}
\begin{aligned}
& \langle \bm{S}_{\triangle,1},\bm{S}_{\triangle,3}\rangle = \langle \bm{S}_{\triangle,2},\bm{S}_{\triangle,3}\rangle \\
&=
\left\{\begin{array}{ll}\pi - \arccos \frac{J_1}{4 J_2}>\frac{\pi}{2},& 4J_2\geq J_1>0 \\ \arccos \frac{|J_1|}{4 J_2}<\frac{\pi}{2},& 4J_2\geq -J_1 >0\end{array}\right.,\\
&\langle \bm{S}_{\triangle,1},\bm{S}_{\triangle,2}\rangle = 2\arccos \frac{|J_1|}{4J_2}.
\end{aligned}
\end{equation}
This result agrees with the exact diagonalization result above. When $J_1 = 1.426 J_2>0$ with a sublattice phasing of $\pi$, the angle between the two spins in a primitive cell is expected to be $\pi - \arccos(1.426/4) = 1.935 \sim 111^\circ$.
\subsection{Effect of other terms; phasing and lattice distortion}
The $J_1$\---$J_2$ model reproduces the spiral phase and the incommensurate wave vector in the ground state of LiYbO$_2$. The angle difference between the nearest spins ($111^\circ$), however, does not agree with the best experimental fitting (staggered in alternating $34^\circ$ and $172^\circ$ angles). One plausible explanation is a small lattice distortion that is outside of resolution of the neutron powder diffraction data.
In this subsection, we study the effect of a lattice distortion on the magnetic order. We assume a simple scenario in which the lattice distortion results in a displacement between two sublattices: suppose the $\mu=1$ sublattice, originally $\delta = \bm{a}/2 + \bm{c}/4$ part from the $\mu=0$ sublattice, is offset by $\bm{\varepsilon}$ from the original position, where $\bm{\varepsilon} = (\epsilon,\epsilon,0)$. In this case the NN vectors from the Yb ion at the origin become $\frac{\bm{a}}{2} + \frac{\bm{c}}{4} + \bm{\varepsilon}$, $-\frac{\bm{a}}{2} + \frac{\bm{c}}{4} + \bm{\varepsilon}$, $\frac{\bm{b}}{2} - \frac{\bm{c}}{4} + \bm{\varepsilon}$, and $-\frac{\bm{b}}{2} - \frac{\bm{c}}{4} + \bm{\varepsilon}$, which correspond to $J'_1,J''_1,J'_1,J''_1$, respectively. Here we assume antiferromagnetic exchange $J'_1,J''_1>0$ in order to agree with experiment. We can again write down the Hamiltonian in momentum space in the form of Eq.~\eqref{shs}, with modified off-diagonal element
\begin{equation}
\begin{aligned}
J^{12}_{\bm{q}} = J^{21*}_{\bm{q}}&=\frac{J'_1}{2}\left( e^{i \bm{q}\cdot\left(\frac{\bm{a}}{2} + \frac{\bm{c}}{4} + \bm{\varepsilon}\right)} + e^{i \bm{q}\cdot\left(\frac{\bm{b}}{2} - \frac{\bm{c}}{4} + \bm{\varepsilon}\right)}\right)\\
&+\frac{J''_1}{2}\left( e^{i \bm{q}\cdot\left(-\frac{\bm{a}}{2} + \frac{\bm{c}}{4} + \bm{\varepsilon}\right)} + e^{i \bm{q}\cdot\left(-\frac{\bm{b}}{2} - \frac{\bm{c}}{4} + \bm{\varepsilon}\right)}\right)\\
&=\frac{1}{2} e^{i \bm{q}\cdot \bm{\varepsilon}}\left(J'_1 e^{i \bm{q}\cdot\left(\frac{\bm{a}}{2} + \frac{\bm{c}}{4}\right)} + J'_1 e^{i \bm{q}\cdot\left(\frac{\bm{b}}{2} - \frac{\bm{c}}{4}\right)}\right.\\
&\left.+ J''_1 e^{i \bm{q}\cdot\left(-\frac{\bm{a}}{2} + \frac{\bm{c}}{4}\right)} + J''_1e^{i \bm{q}\cdot\left(-\frac{\bm{b}}{2} - \frac{\bm{c}}{4}\right)}\right),
\end{aligned}
\end{equation}
where we denote $q_x = \bm{q}\cdot \bm{a}$, $q_y = \bm{q}\cdot \bm{b}$, and $q_z = \bm{q} \cdot \bm{c}$. It is easy to show that
\begin{equation}
\begin{aligned}
\lambda_{\bm{q}} \geq J_2 ( \cos q_x + \cos q_y)&-\sqrt{\frac{J'^2_1}{4} + \frac{J''^2_1}{4} + \frac{1}{2}J'_1 J''_1 \cos q_x}\\
& - \sqrt{\frac{J'^2_1}{4} + \frac{J''^2_1}{4} + \frac{1}{2}J'_1 J''_1 \cos q_y},
\end{aligned}
\end{equation}
hence the energy minimum is reached at $q_x = q_y \equiv q_0 $ and $q_z = 0$. Here $q_0=0.384\times 2\pi$ is the required experimental value to minimize $f(q) = J_2 \cos q - \sqrt{ \frac{J'^2_1}{2} + \frac{J''^2_1}{2} + J'_1 J''_1 \cos q}$, and we get
$$ \cos q_0 = \frac{J'^2_1 J''^2_1 - 4 J_2^2 (J'^2_1 + J''^2_1)}{8 J_2^2 J'_1J''_1},$$
This equation restricts the value between $J'_1/J_2$ and $J''_1/J_2$. Setting $J'_1 = J''_1=J_1$ recovers the previous undistorted result, $J_1 = 4 \cos\frac{q_0}{2} = 4 \cos \pi q$. The eigenvector corresponding to $\lambda_{\bm{q}}$ is again $u_{\bm{q}} = \frac{1}{\sqrt{2}} (e^{i \phi_{\bm{q}}},1)^T$, where we now have
\begin{equation}\label{eq:phase}
\begin{aligned}
\phi_{\bm{q}_0} &= \pi +\bm{q}_0\cdot \bm{\varepsilon} + \arctan \left( \tan \left( \frac{\pi}{4} - \beta\right) \tan \frac {q_0}{2}\right)\\
& \approx \pi + \arctan \left( \tan \left( \frac{\pi}{4} - \beta\right) \tan \frac {q_0}{2}\right),
\end{aligned}
\end{equation}
and we define $\tan \beta = J''_1 /J'_1$. The term $\bm{q}\cdot \bm{\varepsilon}$ is small and can be ignored. Eq.~\eqref{eq:phase} suggests that the angle difference between NN spins (which is $\phi_{\bm{q}_0}+ q_0/2$) depends on the spiral wave vector and the ratio of NN bond exchange energies. If we plug in $\phi_{\bm{q}_0} = 360^{\circ}-34^{\circ} = 172^{\circ}$, then we get $\tan \beta \approx 6$. This means that in our simple lattice distortion scenario, a large exchange ratio is needed in order to reproduce the experimentally observed order.
We note that the DM contribution vanishes if different layers are assumed to have the same order: assume $D\ll J_1,J_2$; suppose the coplanar order is normal to $\bm{n}$, then the DM interaction in layer $l$ is proportional to $(-1)^{\mu(l)}D (\bm{a}-\bm{b}) \cdot \bm{n} \sin qa$. The sign $(-1)^{\mu(l)}$ indicates that neighboring layers (belonging to different sublattices $A$ and $B$) have opposite contributions, leading to a vanshing DM energy.
\subsection{Linear spin wave theory}
In this subsection, we present simulations of the dynamical structure factor using linear spin wave theory. An undistorted lattice is assumed. Introducing Holstein-Primakoff (HP) bosons
\begin{equation}
\bm{S}_i \cdot \bm{a}_i = \sqrt{s}\frac{a_i+a^\dag_i}{\sqrt{2}},\,\,
\bm{S}_i \cdot \bm{b}_i = \sqrt{s} \frac{a_i - a^\dag_i}{\sqrt{2}i},\,\,
\bm{S}_i \cdot \bm{c}_i = s - n_i
\end{equation}
where $\bm{c}_i = \bm{u}\cos \widetilde{\bm{q}}\cdot \bm{r}_i + \bm{v} \sin \widetilde{\bm{q}}\cdot \bm{r}_i$ is the spin order $(\bm{u}$ and $\bm{v}$ are orthogonal unit vectors spanning the order plane), $\bm{b}_i = \bm{u}\times \bm{v}$, and $\bm{a}_i = \bm{b}_i\times \bm{c}_i$. We define $\widetilde{\bm{q}} = \frac{2\pi}{a}(1-q,1-q,0)$ to remind that the angle between NN spins is obtuse in the $J_1$\---$J_2$ model. The spin wave Hamiltonian is then
\begin{equation}
H = \sum_{\bm{k}\in \text{BZ}^+}\Phi^\dag_{\bm{k}}\mathcal{H}(\bm{k}) \Phi_{\bm{k}},
\end{equation}
where $\Phi_{\bm{k}} = \left(a_{\bm{k},0}, a_{\bm{k},1},a^\dag_{-\bm{k},0}, a^\dag_{-\bm{k},1}\right)^T$ are the HP bosons in momentum space, and
\begin{equation}\label{hs}
\mathcal{H}(\bm{k})
=
2\left(\begin{array}{cccc}
h_{11} &h_{12}&p_{11}& p_{12}\\
h_{12}^*& h_{11} &p_{12}^*&p_{11}\\
p_{11}&p_{12}& h_{11} &h_{12}\\
p_{12}^*& p_{11}& h_{12}^*& h_{11}
\end{array}\right),
\end{equation}
with
\begin{subequations}\label{entries}
\begin{eqnarray}
h_{11} &=& J_2\sum_{\bm{\delta} = \bm{a},\bm{b}} \left(2s\cos \bm{k}\cdot \bm{\delta} \left[\frac{1}{4}(c_{\bm{\delta}}+1)\right]-s c_{\bm{\delta}}\right)\notag\\
&& - J_1\sum_{\bm{\delta} = \pm\frac{\bm{a}}{2}-\frac{\bm{c}}{4},\pm\frac{\bm{b}}{2}+\frac{\bm{c}}{4}} \frac{s}{2} c_{\bm{\delta}},\\
h_{12} &=& J_1\sum_{\bm{\delta} = \pm\frac{\bm{a}}{2}-\frac{\bm{c}}{4},\pm\frac{\bm{b}}{2}+\frac{\bm{c}}{4}} s e^{i\bm{k}\cdot \bm{\delta}} \left[\frac{1}{4}(c_{\bm{\delta}}+1)\right],\\
p_{11} &=& J_2\sum_{\bm{\delta} = \bm{a},\bm{b}} 2s \cos \bm{k}\cdot \bm{\delta} \left[\frac{1}{4}(c_{\bm{\delta}}-1)\right],\\
p_{12} &=& J_1\sum_{\bm{\delta} = \pm\frac{\bm{a}}{2}-\frac{\bm{c}}{4},\pm\frac{\bm{b}}{2}+\frac{\bm{c}}{4}} s e^{i \bm{k}\cdot \bm{\delta}} \left[\frac{1}{4}(c_{\bm{\delta}}-1)\right],
\end{eqnarray}
\end{subequations}
where we defined
$$c_{\bm{\delta}} \equiv \cos \widetilde{\bm{q}}\cdot \bm{\delta} = \left\{\begin{array}{ll}- J_1/4J_2,& \bm{\delta} \in \text{NN},\\
2 \left(\frac{J_1}{4 J_2}\right)^2 - 1,& \bm{\delta} \in \text{NNN}.\end{array}\right.$$
The boson canonical commutation relation is preserved by the diagonalization $V^\dag_{\bm{k}} \mathcal{H}(\bm{k}) V_{\bm{k}} = \Lambda_{\bm{k}}$, $\Phi_{\bm{k}} = V_{\bm{k}} \Psi_{\bm{k}}$, where $V^\dag_{\bm{k}}J V_{\bm{k}} = J\equiv\mathrm{Diag}(1,1,-1,-1)$. Diagonalizing $J\mathcal{H}(\bm{k})$ then gives the spin wave spectrum $\Lambda=(\lambda_1,\lambda_2,-\lambda_1,-\lambda_2)$, with
\begin{equation}\label{sp}
\lambda_{1,2} = \sqrt{(h_{11} \pm |h_{12}|)^2 - (p_{11} \mp |q_{12}|)^2}.
\end{equation}
The spin wave spectrum \eqref{sp} along the (110) direction is shown in Figure \ref{spectrum_110}a. One observes that the spectrum is gapless at
\begin{equation}
\bm{q} = (0,0,0),\quad \pm\frac{2\pi}{a}(q,q,0),\quad \text{and}\quad \pm \frac{2\pi}{a}(1-q,1-q,0),
\end{equation}
and the momenta that are related to $\bm{q}$ by a $C_4$ rotation along $(001)$ or translation by reciprocal lattice vectors.
\begin{figure}
\centering
\includegraphics[width=0.4\textwidth]{Figure9.png}
\caption{a) Spin wave spectrum (red lines) and the structure factor simulation for $J_1=1.42565 J_2>0$. Both along the (110) direction. b) Angular averaged structure factor for $J_1=1.42565 J_2>0$.}
\label{spectrum_110}
\end{figure}
We then derive an expression for the dynamical structure factor, which is the Fourier transform of spin-spin correlation function. One obtains
\begin{equation}\label{skw}
\begin{aligned}
&\mathcal{S}(\bm{k},\omega)\\
&=\sum_{i,j=1}^3(\delta_{ij} -(\hat{\bm{k}})_i(\hat{\bm{k}})_j)
\sum_{\mu,\nu=0}^1\langle m^i_\mu(-\bm{k},-\omega)m^j_\nu(\bm{k},\omega)\rangle\\
&= 2s \mu^2_B \sum_{e=1}^4
\delta(\omega - J \lambda_{\bm{k}-\widetilde{\bm{q}},e})
\left[V_{\bm{k}-\widetilde{\bm{q}}}^\dag {K}_1^\dag g^\dag P_{\bm{k}} g {K}_1 V_{\bm{k}-\widetilde{\bm{q}}}\right]_{e,e} \\
&\qquad\qquad + \delta(\omega - J \lambda_{\bm{k}+\widetilde{\bm{q}},e})
\left[V_{\bm{k}+\widetilde{\bm{q}}}^\dag {K}^T_1 g^\dag P_{\bm{k}} g {K}^*_1 V_{\bm{k}+\widetilde{\bm{q}}}\right]_{e,e}\\
& \qquad\qquad + \delta(\omega - J \lambda_{\bm{k},e}) \left[V^\dag_{\bm{k}} {K}_2^\dag g^\dag P_{\bm{k}}g {K}_2 V_{\bm{k}}\right]_{e,e},
\end{aligned}
\end{equation}
where we defined projector $P_{\bm{k}} = 1_{3\times 3}- \hat{\bm{k}}\hat{\bm{k}}^T$. The derivation and the notation for ${K}_{1,2}$ and $g$ can be found in the Supplementary Material \cite{supplementalmaterial}. From Eq.~\eqref{skw}, it is clear that the structure factor intensity at one $\bm{k}$ receives contributions from three momenta: $\bm{k}\pm \widetilde{\bm{q}}$ and $\bm{k}$. The simulated structure factor according to Eq.~\eqref{skw} is shown in Figure \ref{spectrum_110}a) for a specific $(1,1,0)$ direction, and in Figure \ref{spectrum_110}b) for the angular averaged result. One of the main features at low-energy is the vanishing intensity at $\Gamma$ and $|\bm{q}| = \frac{2\sqrt{2}\pi}{a}0.384$, where the spin wave spectrum is gapless, and one would naively expect a strong intensity peak at zero energy due to singular BdG Hamiltonian at these momenta. Physically the ``missing'' intensity is a consequence of the destructive interference of the two sublattices at $\Gamma$ and $\bm{q}$ that leads to vanishing contribution to the structure factor. The same interference pattern is also true for the static structure factor. The perfect cancellation is really a consequence of the (undistorted) $J_1$\---$J_2$ Heisenberg model. On the other hand, the persistence of high intensities at $\Gamma$ and $\bm{q}$ from the neutron experiment suggests this cancellation is partially lifted in the real material due to other effects not captured by the $J_1$\---$J_2$ Heisenberg model.
\subsection{Free energy analysis}
The classical ground state of the $J_1$\---$J_2$ Heisenberg model has a global SO(3) symmetry due to the freedom in choosing the spiral plane. Since the lattice only has discrete symmetries, it is likely that this continuous symmetry is lifted due to other effects, such as spin-orbit coupling and fluctuations, and it is the goal of this section to address this issue energetically from a symmetry point of view. Specifically, we will examine the symmetry constraints on the free energy. We first write down the spiral order parameter. Assuming the spiral plane is spanned by two orthogonal vectors $\bm{u}$ and $\bm{v}$, the order parameter can be chosen as the Fourier transform of the magnetic order, which can be written as
\begin{equation}
\bm{d} = e^{i \theta(\bm{r})} (l\bm{u} + im \bm{v}),
\end{equation}
where $\theta(\bm{r})$ determines the direction of the spins in the spiral plane. While it is a constant in the spiral phase, spatial fluctuation of $\theta$ must be considered near the incommensurate-to commensurate (IC-C) transition. Note we have introduced $l$ and $m$ to account for either perfect circular ($l=m$, no net magnetization), elliptical ($m\neq l >0$) or linear ($m=0$) polarization, which correspond to zero, low and high magnetic fields, respectively.
We first look at the zero-field case, $l=m$. Following Lee and Balents \cite{lee-balents}, we seek to write down the free energy for the order parameter to quadratic order using symmetry considerations. Out of the symmetry generators $T_{1,2,3}$, $S_{4z}$, $C_{2y}$ and $P$, the little group of the wave vector $\widetilde{\bm{q}}$ contains $P$, $T_{1,2,3}$, $S_{4z}^2$, and $S^3_{4z}C_{2y}\colon (x,y,z)\rightarrow (y-1/2,x-1/2,3/2-z)$. Under these symmetries, the order parameter transforms as
\begin{subequations}\label{pstts}
\begin{eqnarray}
P\colon&& \bm{d}\rightarrow e^{i\pi \widetilde{q}} \bm{d}^*,\\
S_{4z}^2\colon&& \bm{d}\rightarrow \mathrm{Diag}(-1,-1,1)\bm{d}^*,\\
T_{1,2}\colon&& \bm{d}\rightarrow \bm{d},\\
T_3\colon&& \bm{d}\rightarrow e^{- 2i \pi \widetilde{q}} \bm{d},\\
S^3_{4z}C_{2y}\colon&& \bm{d}\rightarrow \left(\begin{array}{ccc} &1 &\\1 &&\\ && 1\end{array}\right)e^{i 2\pi\widetilde{q}} \bm{d},
\end{eqnarray}
\end{subequations}
where the last symmetry operation can be composed with $T_3$ to get $T_3S^3_{4z}C_{2y}\colon \bm{d}\rightarrow (d_y,d_x,d_z)$. From this, one can write down a free energy density that is quadratic in $\bm{d}$:
\begin{equation}\label{f110}
f(\bm{d}) = c_0 |\bm{d}|^2 + c_1 (d^*_1d_2+c.c.)+c_2 d^*_3 d_3.
\end{equation}
By minimizing this free energy one finds there are three choices for the spiral plane depending on the value of $c_1$ and $c_2$ \cite{lee-balents}: the normal of the order plane can be along either $(001)$, $(1\bar{1}0)$, or $(110)$.
The result above applies to a generally incommensurate wave vector $\widetilde{q}$ at zero magnetic field. As the field is switched on, the spiral order ceases to be circularly polarized, and the unequal components $l\neq m$ allow for nonzero net magnetization. As a consequence, some of the symmetry transformations in \eqref{pstts} are no longer valid and need to be modified. Nevertheless, we assume that all the symmetry transformations in \eqref{pstts} remain approximately valid at small field. Under these assumptions, we proceed to an explanation of the IC-C transition at 3\,T. The commensurate phase has a three-unit cell order with corresponding wave vector $\bm{q} = 2\pi(\frac{1}{3},\frac{1}{3},0)$. In this phase, another term can be added to the free energy density:
\begin{equation}\label{fC}
f_{\text{C}} = f(\bm{d}) - \widetilde{c}_6 \left((\bm{d}\cdot \bm{d})^3 + c.c.\right).
\end{equation}
The development of unequal $l$ and $m$ can be further modeled phenomenologically by fourth-order terms in the free energy such as $\beta_2|\bm{d}\cdot \bm{d}|^2 + \chi_1 H^2(\bm{d}^*\cdot \bm{d}) + \chi_2 | \bm{H}\cdot \bm{d}|^2$, which we do not discuss here but instead refer to Ref. \cite{zhitom}.
In the following, we show that the IC-C transition can be described phenomenologically by a sine-Gordon model. For given $J_1$ and $J_2$, assume $\bm{q}$ is the (generally incommensurate) ground state spiral wave vector, while $\bm{k}$ is a nearby commensurate wave vector. Assume $\bm{q} = \bm{k}+\bm{\delta k}+\bm{\nabla} \theta$, where $\bm{\nabla} \theta$ denotes the spatial fluctuation of the order parameter. The classical energy can be expanded around $\bm{k}$:
\begin{equation}
\lambda = \lambda_0 + 2 \bm{\delta}\cdot \bm{\nabla}\theta + \frac{\kappa_{xy}}{2}((\partial_x \theta)^2+(\partial_y \theta)^2) + \frac{\kappa_z}{2} (\partial_z \theta)^2,
\end{equation}
where $\lambda_0 = - \frac{J_1^2}{4J_2}-2J_2$, and the rigidity for $\theta$ is
\begin{equation}
\kappa_{xy} = -\frac{a^2}{16 J_2}(J_1^2-16 J_2^2),\qquad \kappa_z = \frac{c^2J_1^2}{32 J_2}.
\end{equation}
importantly, a term linear in the gradient of $\theta$ exists, with coefficient $\bm{\delta} = \kappa_{xy}\bm{\delta k}$. A full theory for $\theta$ then appears as
\begin{equation}\label{ftf}
F[\theta] = A \int d^3x \left( \frac{\kappa}{2}(\bm{\nabla}\theta)^2 + 2 \bm{\delta}\cdot \bm{\nabla} \theta
- c_6 \cos 6 \theta\right),
\end{equation}
where the last term comes from Eq.~\eqref{fC} with $c_6 \sim (l^2-m^2)^3 c_6$. This is the sine-Gordon model that has been analyzed in numerous works; see e.g. Ref. \cite{zhitom}. The basic physics is that the soliton number $N$ of the lowest energy solution to the free energy functional \eqref{ftf} distinguishes commensurate phase ($N=0$) and incommensurate phase ($N = \pm 1$); the C-IC transition then is determined by the energetics of $N=0$ and $N\neq 0$ configurations, with critical relation $\kappa^2 c_6/4\kappa \delta k = \pi^2/32$ ($\kappa^2 c_6/4\kappa \delta k < \pi^2/32$ gives the incommensurate phase). Since the elliptic polarization is induced by magnetic field, following Ref.~\cite{zhitom} we conclude that the coefficient $c_6\propto (l^2-m^2)^3 \propto H^6$, and that increasing the magnetic field will inevitably induce an IC-C transition.
\section{V. Discussion}
LiYbO$_2$ shows a rich magnetic phase diagram (see Figure \ref{fig:fig10}) with inherent similarities to the $A$-site transition metal spinels and the $J_1$\---$J_2$ diamond lattice model, indicating that the underlying physics of both systems arises from the same bipartite frustration. The $J_1$\---$J_2$ model on the ideal diamond lattice with $J_2/|J_1| > 1/8$, produces frustrated spiral order with wave vectors directed along the high-symmetry directions of the lattice (e.g. $(q, q, q)$, $(q, q, 0)$, $(0, 0, q)$) and simliar spiral order also appears in tetragonaly elongated diamond lattice of LiYbO$_2$ near $|J_1|\leq 4J_2$. Spiral wave vectors in the distorted case are however limited to $(q, \pm q, 0)$, and tetragonal distortion lifts the degeneracy of the spiral spin liquid surface predicted for the perfect diamond lattice \cite{bergman-balents, lee-balents,buessen-trebst}.
Curiously, in zero-field, the long-range helical ground state forms through two successive magnetic transitions upon cooling. An intermediate state formed upon cooling below $T_{N1}$ is best fit by modeling a spiral state on each Yb-site but with disordered relative phasing between the two spirals. This apparent frustration in the relative phase between magnetic sublattices and the formation of a partially ordered state is also likely reflected in the departure of the relative phasing between Yb-ions within the fully ordered state (below $T_{N2}$) from the predictions of the Heisenberg $J_1$\---$J_2$ model. Specifically, the model predicts that moments rotate along all $A$-to-$B$ sublattice bonds equivalently (i.e. the angle difference between every NN spin is $111^\circ$), while the experimental data suggests that moments rotate in a staggered fashion, where the first $A$-to-$B$ sublattice bond is $34^\circ$ and the second is $172^\circ$. This generates a magnetic structure in which pairs of spins between the $A$ and $B$ sublattices are nearly aligned antiparallel.
While CEF data suggest the presence of two Yb environments in the lattice, this is not readily apparent in the average structural data, suggesting that the distortion responsible for this is reasonably subtle. Given the large distortion required for the model to produce the experimentally observed phasing between Yb-moments,
the possible origin for the phase difference instead lies in the presence of anisotropic exchange interactions in LiYbO$_2$. We note however that, assuming spiral order with a single wave vector $q$, including Ising type of anisotropy at NN and NNN level does not help in explaining the disagreement between theory and experiment (further details in Supplementary Materials \cite{supplementalmaterial}). Resolving the possibility of other anisotropic terms in the Hamiltonian as well as the precise nature of the anomalous state between 0.45 K $< T <$ 1.13 K will require future single crystal studies.
The incommensurate helical structure in LiYbO$_2$ evolves into a commensurate helical structure when $\mu_0 H=3$ T is applied. A similar type of ``lock-in'' incommensurate-to-commensurate (IC-C) phase transition occurs in the $A$-site spinels, originating from magnetic anisotropy on top of the $J_1$\---$J_2$ model \cite{lee-balents}. Anisotropy accounts for the change from an incommensurate $(q, \pm q, 0)$ helical phase to a commensurate one in MnSc$_2$S$_4$ \cite{lee-balents, gao-ruegg, iqbal-reuther} and CoCr$_2$O$_4$ \cite{chang2009crossover, lawes2006dielectric, chen2013coexistence} with decreasing temperature. In LiYbO$_2$ however, the field-driven ``lock-in'' phase transition is captured within the sine-Gordon model in Eq.~\eqref{fC} without the need to perturb the Heisenberg $J_1$\---$J_2$ model.
In fact, a considerable amount of the zero-field magnetic behavior of LiYbO$_2$ is captured at the ideal Heisenberg $J_1$\---$J_2$ limit. The doubly-degenerate ordering wave vector $(q, \pm q, 0)$ predicted by the model is reproduced in the fits to elastic neutron diffraction data, and the theory predicts that the spiral structure's ordering plane should be along $(0,0,1)$, $(1,1,0)$, or $(1,\bar{1},0)$. Experimental fits in Figure \ref{fig:fig6} and Table \ref{tab:tab3} rule out the $(0,0,1)$ ordering plane and the remaining planes of $(a,b,0)$ can not be distinguished with the present powder data. Future single crystal neutron experiments could reveal if the ordering plane aligns with the energy minimization in the $(1,1,0)$ or $(1,\bar{1},0)$ planes.
Additionally, the extracted value of $|J_1|/J_2$ = 1.426 from the $J_1$\---$J_2$ model makes intuitive sense within the chemical lattice. It is unsurprising that the two magnetic interactions would be comparable in strength due to their relative superexchange pathways. In comparison, materials such as KRuO$_4$ \cite{marjerrison2016structure} and KOsO$_4$ \cite{song2014unquenched, injac2019structural} share the same $I4_1/amd$ magnetic sublattice comprised of Ru and Os ions, but break the oxygen-based superexchange connection along $J_2$. In these systems, magnetic order resides in the $J_2 = 0$ limit of the Heisenberg $J_1$\---$J_2$ model, where moments order within a Ne\'el antiferromagnetic state and an unfrustrated $J_1$ \cite{marjerrison2016structure, song2014unquenched, injac2019structural}.
Calculations of low-energy spin excitations with the parameters obtained from the $J_1$\---$J_2$ model largely reproduce the low-energy INS spectrum in Figures \ref{fig:fig8} and \ref{spectrum_110} with $J_2 \approx 1/3$ meV and $J_1 \approx 0.475$ meV. One difference appears in the spectral weight at the $\Gamma$ and $|\bm{q}| = \frac{2\sqrt{2}\pi}{a}\times0.384$ positions, where a cancellation of the simulated structure factor intensity occurs due to destructive interference of the two sublattices at these momenta.
This cancellation does not occur in the experimental data due to the difference in phasing between Yb-moments relative to the predictions of the $J_1$\---$J_2$ model .
Despite this minor deviation, rooted in the relative phasing between the Yb-sublattices, our work establishes that LiYbO$_2$ contains a tetragonally-elongated diamond lattice largely captured by the Heisenberg $J_1$\---$J_2$ model. To the best of our knowledge, reports of diamond lattices decorated with trivalent lanthanide ions are rare, and, based upon our results, we expect that an ideal diamond lattice decorated with Yb$^{3+}$ moments may reside close to the ideal Heisenberg limit. Such an ideal cubic $Ln$-ion diamond lattice would be a promising platform for manifesting (quantum) spiral spin liquid states, similar to transition metal spinels, while potentially avoiding the complications of extended exchange interactions born from $d$-electron systems.
\begin{figure}[t]
\includegraphics[width=0.45\textwidth]{Figure10.png}
\caption{Proposed powder-averaged, low-temperature ($H$, $T$) diagram of LiYbO$_2$ extracted from a combination of specific heat ($C_p$) measurements and elastic neutron powder diffraction data. At high temperature, LiYbO$_2$ is in the paramagnetic ($PM$) phase. Below approximately 10 K, specific heat shows a broad feature where roughly half of the magnetic entropy of $Rln(2)$ is released and signifies the onset of short-range magnetic correlations. A sharp anomaly at 1.13 K at 0, 3, and 5 T and 1.40 K at 9 T in specific heat measurements shows where long-range magnetic order sets in. Combining specific heat data with neutron powder diffraction data suggests that the temperature regime between 0.45 K and 1.13 K consists of a helical magnetic structure with disordered phasing between the two interpenetrating Yb sublattices. The system undergoes a lock-in phase transition from an incommensurate helical structure at zero field to a commensurate structure at 3 T.}
\label{fig:fig10}
\end{figure}
\section{VI. Conclusions}
LiYbO$_2$ provides an interesting material manifestation of localized
$f$-electron moments decorating a frustrated diamond-like lattice.
Long-range incommensurate spiral magnetic order of
${\bf{k}} = (0.384, \pm 0.384, 0)$ forms in the ground state, which
seemingly manifests through a two-step ordering process via a
partially ordered intermediate state. Upon applying an external
magnetic field, magnetic order becomes commensurate with the lattice
with ${\bf{k}} = (1/3, \pm 1/3, 0)$ through a ``lock-in'' phase
transition. Remarkably, the majority of this behavior in LiYbO$_2$ can
be captured in the Heisenberg $J_1$\---$J_2$ limit where the magnetic
Yb$^{3+}$ ions are split into two interpenetrating $A$-$B$
sublattices. This model was explicitly re-derived and tuned for
LiYbO$_2$, and it is directly related to a physical elongation of the
diamond lattice Heisenberg $J_1$\---$J_2$ model. Differences in the
relative phasing of $A$-$B$ sublattices between the Heisenberg model
and the observed magnetic structure suggest additional interactions
and quantum effects may be present in LiYbO$_2$. This is possibly related to
the observation of crystal field splittings suggesting two Yb
environments. Exploring these as well as the nature of the
intermediate ordered state are promising future steps in
single-crystal studies.
\section{Acknowledgments}
\begin{acknowledgments}
This work was supported by the US Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under award DE-SC0017752 (S.D.W. and M.B.). M.B. acknowledges partial support by the National Science Foundation Graduate Research Fellowship Program under grant no. 1650114. Work by L.B. and C.L. was supported by the DOE, Office of Science, Basic Energy Sciences under award no. DE-FG02-08ER46524. Identification of commercial equipment does not imply recommendation or endorsement by NIST. A portion of this research used resources at the High Flux Isotope Reactor and Spallation Neutron Source a DOE Office of Science User Facility operated by the Oak Ridge National Laboratory.
\end{acknowledgments}
|
1,116,691,499,270 | arxiv | \section{Motivation}
\label{Section_Introduction}
Students of gravitational wave (GW) physics are taught that the key observable -- the fractional distortion of the arms of laser interferometers employed by detectors such as LIGO, Virgo, Kagra, etc. -- induced by the passage of a GW train generated by a distant astrophysical source, is directly proportional to the `transverse-traceless' portion of the metric perturbation. Specifically, in a weakly curved spacetime\footnote{The Greek indices $\mu, \nu,\ldots,$ run from $0$ to $d-1$, while the Latin ones $i, j, \ldots,$ run over spatial coordinates from $1$ to $d-1$, and the ``mostly plus'' sign convention for the metric is used, namely $\eta_{\mu\nu} = \text{diag}[-1,+1,\ldots,+1]$. Throughout this paper, the symmetrization and anti-symmetrization of indices are denoted by the symbols $(\ldots)$ and $[\ldots]$, respectively, e.g., $T_{(\mu\nu)} \equiv \frac12 \left( T_{\mu\nu} + T_{\nu\mu}\right)$ and $T_{[\mu\nu]} \equiv \frac12 \left( T_{\mu\nu} - T_{\nu\mu} \right)$.}
\begin{align}
\label{PerturbedMinkowski}
g_{\mu\nu}[t,\vec{x}] = \eta_{\mu\nu} + h_{\mu\nu}[t,\vec{x}],
\qquad\qquad
|h_{\mu\nu}| \ll 1 ;
\end{align}
if $X^i$ denotes the Cartesian coordinate vector joining one end of an interferometer arm to another, its change $\delta X^i$ due to a GW signal impinging upon the detector is often claimed to be\footnote{See, for example, eq.~(27.26) of Thorne and Blandford \cite{ThorneBlandfordBook}.}
\begin{align}
\label{FractionalDistortion_Textbooks}
\delta X^i = \frac{1}{2} h_{ij}^{\text{TT}} X^j ,
\end{align}
where $h_{ij}^{\text{TT}}$ is the `transverse-traceless' portion of the space-space components of $h_{\mu\nu}$ in eq.~\eqref{PerturbedMinkowski}. What does `transverse-traceless' really mean in this context? R\'{a}cz \cite{Racz:2009nq} and -- more recently -- Ashtekar and Bonga \cite{Ashtekar:2017wgq,Ashtekar:2017ydh} have pointed out, the GW literature erroneously uses two distinct notions of `transverse-traceless' interchangeably.\footnote{Frenkel and R\'{a}cz \cite{Frenkel:2014cra} have also pointed out a similar error within the electromagnetic context.} (We shall adopt Ashtekar and Bonga's notation of `TT' and `tt'.) On the one hand, there is one involving the divergence-free condition,
\begin{align}
\label{TT}
\partial_i h_{ij}^{\text{TT}} = \partial_i h_{ji}^{\text{TT}} = 0 = \delta^{ij} h_{ij}^{\text{TT}} ;
\end{align}
while on the other hand there is one involving a transverse-projection in position space,
\begin{align}
\label{tt}
h_{ij}^{\text{tt}} &\equiv P_{ij ab} h_{ab} .
\end{align}
The definition of transverse-projection in eq.~\eqref{tt} is based on the unit radial vector $\widehat{r} \equiv \vec{x}/|\vec{x}|$ pointing from the isolated astrophysical source centered at $\vec{0}$ to the observer at $\vec{x}$, namely
\begin{align}
\label{tt_Projector}
P_{ij ab} &\equiv P_{a(i} P_{j)b} - \frac{1}{d-2} P_{ij} P_{ab} , \\
P_{ij} &\equiv \delta_{ij} - \widehat{r}_i \widehat{r}_j
\label{t_Projector} .
\end{align}
Because the rank-2 object is a projector, in the sense that
\begin{align}
P_{ia} P_{aj} = P_{ij} ,
\end{align}
and is also transverse to the radial direction,
\begin{align}
\widehat{r}^i P_{ij} = 0 = P_{ij} \widehat{r}^j ,
\end{align}
we see that the `tt' GW in eq.~\eqref{tt} enjoys the same traceless condition as its `TT' counterpart in eq.~\eqref{TT} (i.e., $\delta^{ij} h_{ij}^{\text{tt}} = 0$) but is transverse to the unit radial vector
\begin{align}
\widehat{r}^i h_{ij}^{\text{tt}} = 0 = h_{ij}^{\text{tt}} \widehat{r}^j
\end{align}
instead of being divergence-free.
We believe the intent of much of the contemporary gravitational literature is to claim the TT GW, obeying eq.~\eqref{TT}, to be the
observable; while the tt one in eq.~\eqref{tt} to be only an approximate expression of the same gravitational signal when the observer is very far from the source.\footnote{The exception appears to be Thorne and Blandford \cite{ThorneBlandfordBook}, where they went straight to the tt form of the GW (see Box 27.2) without any discussion of gauge invariance whatsoever.} To our knowledge, the clearest enunciation of this stance may be found in the review by Flanagan and Hughes \cite{Flanagan:2005yc}. After describing how the TT piece of the gravitational perturbation of flat spacetime is the only gauge invariant portion that obeys a wave equation in \S 2.2 -- the remaining 2 scalars and one vector obey Poisson equations -- and after attempting to justify how the TT GW is the one appearing in eq.~\eqref{FractionalDistortion_Textbooks} (cf.~eq.~(3.12) of \cite{Flanagan:2005yc}) they went on in \S 4.2 to assert, albeit without justification, that the far zone version of this TT GW in fact reduces to the tt one. In eq.~(4.23), they then expressed the final GW quadrupole formula in the latter tt form.
Other pedagogical discussions of GWs usually begin with the homogeneous TT wave solutions in perturbed Minkowski spacetime completely devoid of matter:
\begin{align}
\label{TT_Spin2Waves}
h_{ij}^{\text{TT}}[t,\vec{x}]
&= \int_{\mathbb{R}^{d-1}} \frac{\text{d}^{d-1}\vec{k}}{(2\pi)^{d-1}} \left( \widehat{P}_{ij ab}[\vec{k}] \epsilon_{ab}[\vec{k}] e^{ik_\mu x^\mu} + \text{c.c.} \right) , \\
k_\mu &\equiv \left( - |\vec{k}|, \, k_i \right), \qquad\qquad
k^2 \equiv k_\sigma k^\sigma = 0 ;
\end{align}
\footnote{See, for instance, \S 35.2-35.4 of Misner, Thorne, Wheeler \cite{Misner:1974qy}; \S 10.2 of Weinberg \cite{Weinberg:1972kfs}; or \S 9.1 of Schutz \cite{Schutz:1985jx}. Briefly, one may start with the de Donder gauge condition $\partial^\mu h_{\mu\nu} = (1/2) \partial_\nu h$, where $h \equiv \eta^{\mu\nu} h_{\mu\nu}$, and solve the linearized vacuum Einstein's equations $\partial^2 (h_{\mu\nu} - (1/2) \eta_{\mu\nu} h) = 0$. By performing a gauge transformation in Fourier spacetime to set to zero the $h_{\mu 0} = h_{0 \mu}$ components, one would arrive at eq.~\eqref{TT_Spin2Waves}. Often, eq.~\eqref{TT} is called the `TT gauge' but texts often do not caution the reader that this gauge condition can no longer be imposed once matter is introduced into the setup, i.e., once the very source of GWs is present. (Maggiore \cite{Maggiore:1900zz} does note that `TT gauge' does not exist inside the source, but goes on to impose it anyway in the far zone.) We wish to reiterate the remarks already made in \S C of \cite{Chu:2016ngc}: if the `TT gauge' were to exist, that would mean the initially non-trivial gauge invariant scalars and vector variables may be coordinate-transformed to zero. Furthermore, the misleading use of the phrase `TT gauge' suggests one may even choose a different gauge to compute GW patterns -- after all, one ought to be able to use any desired coordinate system -- but this cannot be the case, as the GW pattern is an observable and must therefore yield a unique result.}where ``c.c." denotes the complex conjugate of the preceding term; $\epsilon_{ab}$ can be viewed as the purely spatial gravitational wave amplitude tensor; and the projector is now one in momentum/Fourier space,
\begin{align}
\label{GravitonProjector}
\widehat{P}_{ij ab}[\vec{k}]
&\equiv \widehat{P}_{a(i} \widehat{P}_{j)b} - \frac{1}{d-2} \widehat{P}_{ij} \widehat{P}_{ab} , \\
\widehat{P}_{ij}
&\equiv \delta_{ij} - \widehat{k}_i \widehat{k}_j,
\qquad\qquad \widehat{k}_i \equiv \frac{k_i}{|\vec{k}|} .
\label{TransverseProjector}
\end{align}
Because in Fourier space a spatial derivative is replaced with a momentum vector, $\partial_j \to i k_j$, and because of the transverse-traceless properties
\begin{align}
k^i \widehat{P}_{ij ab} = k^i \widehat{P}_{ji ab} = 0 = \delta^{ij} \widehat{P}_{ij ab} ;
\end{align}
the perturbations in eq.~\eqref{TT_Spin2Waves} do indeed satisfy the `TT' conditions in eq.~\eqref{TT}.
These GW discussions typically go on to justify eq.~\eqref{FractionalDistortion_Textbooks} in vacuum before, as opposed to after, solving the perturbations engendered by a non-trivial source. The excuse is that, one expects these perturbations from an isolated system to approach TT plane GWs in the asymptotic far zone limit. As we shall see below, the TT and tt GWs do indeed coincide in this $r \equiv |\vec{x}| \to \infty$ limit. Hence, one might reasonably question: why bother with the distinction at all? To this end, Ashtekar and Bonga point out that the tt GWs miss the `Coulombic aspects' that are contained in the TT ones. Moreover, in Quantum Field Theory, each mode of the superposition of TT GWs in eq.~\eqref{TT_Spin2Waves} and not those of $h_{ij}^{\text{tt}}$ -- because it is the former that is fully gauge-invariant -- would be regarded as an irreducible spin-2 graviton. Therefore, one may be led to a principled stance and insist that it is $h_{ij}^{\text{TT}}$ that is physical.
But is the TT GW really a standalone observable? One of us (YZC) has been confused by this issue since several years ago, when he began developing a program to explore novel ways to understand the causal structure of gravitational signals in curved spacetimes -- i.e., how they propagate both on and within the null cone. As highlighted in \cite{Flanagan:2005yc}, the gauge invariant TT GW is a nonlocal function of the metric perturbation $h_{\mu\nu}$ in eq.~\eqref{PerturbedMinkowski}, because the TT projection process takes place in Fourier space. Since, at least about a Minkowski background, the de Donder gauge gravitational perturbation depends on its matter source in a causal manner, this suggests the TT GW may thus depend on the same matter source in an acausal manner due to this nonlocal character. This in turn would render it unphysical, as no classical physical observable should arise from outside the past light cone of the observer.
In this paper, we wish to clarify how the gauge-invariant forms of the vector potential and metric perturbations of, respectively, electromagnetism and linearized gravitation contribute to the observables of these theories. This will include understanding how all their gauge-invariant field variables, not just the dynamical massless spin-1 and spin-2 ones, play crucial roles in ensuring that their physical observables depend on their progenitors -- namely, the electric current and matter stress tensor -- in a causal manner. Through a concrete evaluation of the massless spin-1 photon and spin-2 graviton Green's functions, in Minkowski and spatially flat cosmological spacetimes, we will show that they are indeed acausally dependent on these sources and therefore cannot be standalone observables. However, by ensuring the rest of the gauge-invariant variables are included in the computation of the electromagnetic Faraday tensor $F_{\mu\nu}$ as well as the $\delta_1 R_{0i0j}$ components of the linearized Riemann tensor in Minkowski and the $\delta_1 C^i{}_{0j0}$ components of the linearized Weyl tensor in spatially flat cosmologies, the electromagnetic and gravitational tidal forces become strictly causal ones. In particular, we gain the following insight into the gauge-invariant content of electromagnetism and linearized General Relativity. While the magnetic field $F_{ij}$ does depend only on the massless spin-1 photon, the electric field $F_{0i}$ depends on the causal portion of the velocity of the photon, with its acausal portion canceled by the gauge-invariant scalar of the vector potential, in all spacetime dimensions $d \geq 3$. For the gravitational case, tidal forces in a flat spacetime background are encoded within the causal part of the acceleration of the massless spin-2 graviton, with the acausal portion eliminated by the two gauge-invariant scalars and one vector potential for all $d \geq 4$. Additionally, about a cosmological background, if the Weyl tensor describes the dominant contributions to tidal forces, then the latter appear to depend on the causal portions of both the massless spin-2 and the two gauge-invariant Bardeen scalar potentials. We view this latter analysis as a first step towards an understanding of whether the two Bardeen gauge-invariant scalar potentials ought to be considered an integral portion of gravitational waves and their associated memories in cosmological settings -- even though the dynamics of General Relativity (in $3+1$ dimensions) is usually attributed exclusively to its two spin-2 degrees of freedom.
In \S \eqref{Chapter_Observables} we will define the electromagnetic and gravitational gauge invariant variables; and proceed to clarify what the relevant (classical) electromagnetic and gravitational observables are. In \S \eqref{WhyAcausal} we will use the non-local character of the transverse projection in momentum space to argue why these gauge-invariant variables are expected to be acausal. Following which, we begin in \S \eqref{MinkowskiSpacetime} to compute the explicit forms of the transverse-photon and TT graviton Minkowski Green's functions, confirming their acausal nature. We also compute the solutions to the gauge-invariant scalars and vectors; and combine the results to study how the electromagnetic Faraday tensor and gravitational linearized Riemann are causally dependent on their respective sources. The far zone and stationary limit are examined; and micro-causality violated is pointed out. In \S \eqref{SpatiallyFlatw}, we move on to study similar issues but in a cosmology dominated by a cosmological constant or driven by a relativistic fluid with equation of state $0 < w \leq 1$. Finally, we summarize our findings and outline future directions in \S \eqref{Section_Summary}.
\section{Gauge-Invariance and Observables}
\label{Chapter_Observables}
{\bf Setup} \qquad Throughout the rest of this paper, we will be studying the $d-$dimensional perturbed Friedmann-Lema\^{i}tre-Robertson-Walker (FLRW)-like metric
\begin{align}
\label{PerturbedFLRW}
g_{\mu\nu}[x] = a[\eta]^2 \left( \eta_{\mu\nu} + \chi_{\mu\nu}[x] \right) ,
\qquad\qquad
x^\mu \equiv (\eta,\vec{x}) ;
\end{align}
where $a=1$ for a flat background or
\begin{align}
\label{ScaleFactor}
a[\eta] = \left(\frac{\eta}{\eta_0}\right)^{\frac{2}{q_w}}, \qquad\qquad
q_w \equiv (d-3) + (d-1) w .
\end{align}
In eq.~\eqref{ScaleFactor}, if the perturbations $\chi_{\mu\nu}$ were not present, setting $w=-1$ with $\eta<0$ yields de Sitter spacetime and $0 \leq w \leq 1$ with $\eta>0$ a spatially flat cosmology driven by a single perfect fluid with equation-of-state $w$. The non-trivial perturbations $\chi_{\mu\nu}$ satisfy Einstein's equations coupled to the fluid plus a compact astrophysical system, linearized about the corresponding backgrounds. The detailed analysis can be found in \S III and \S IV in \cite{Chu:2016ngc}, and we will cite the relevant results below.
Let us consider an infinitesimal coordinate transformation
\begin{align}
\label{GaugeTransformation_Gravity}
x^\mu = x'^\mu + \xi^\mu[x'] ,
\end{align}
where $x' \equiv (\eta',\vec{x}')$ and $\xi^\mu$ is small in the same sense that $h_{\mu\nu}$ is small. Then up to first order in perturbations, the metric tensor in the primed coordinate system may be written as
\begin{align}
\label{PerturbedFLRW_Prime}
g_{\mu'\nu'}[\eta',\vec{x}'] = a[\eta']^2 \left( \eta_{\mu\nu} + \chi_{\mu'\nu'}[\eta',\vec{x}'] \right) ,
\end{align}
with all the coordinate transformation induced by eq.~\eqref{GaugeTransformation_Gravity} attributed to that of the metric perturbation in the following way:
\begin{align}
\label{GaugeTransformation_Perturbation}
\chi_{\mu'\nu'}[x']
&= \chi_{\mu\nu}[x'] + 2\eta_{\sigma(\mu}\partial_{\nu')} \xi^\sigma[x'] + 2\frac{\dot{a}[\eta']}{a[\eta']} \xi^0[x'] \eta_{\mu\nu} .
\end{align}
(The $\chi_{\mu\nu}[\eta',\vec{x}']$ and $\xi^{\mu}[x']$ on the right hand side of eq.~\eqref{GaugeTransformation_Perturbation} are the perturbation and gauge-transformation vector components in the `old' $x^\mu \equiv (\eta,\vec{x})$ coordinate basis, but with the replacement $x \to x'$.) Next, we perform a scalar-vector-tensor decomposition of both the metric perturbations\footnote{See, for example, \S IV in \cite{Chu:2016ngc} for a discussion of the scalar-vector-tensor decomposition and the gauge-invariant formalism of linearized gravitation.}
\begin{align}
\label{Bardeen_SVTDecomposition}
\chi_{00} &\equiv E, \qquad \qquad \chi_{0i} \equiv \partial_i F + F_i, \nonumber\\
\chi_{ij} &\equiv D_{ij} + \partial_{( i} D_{j) } + \frac{D}{d-1} \delta_{ij} + \left( \partial_i \partial_j - \frac{\delta_{ij}}{d-1} \vec{\nabla}^2 \right) K ,
\end{align}
as well as the astrophysical stress tensor
\begin{align}
\label{Astro_SVT_IofII}
\,^{(\text{a})}T_{00} & \equiv \rho, \qquad\qquad
\,^{(\text{a})}T_{0i} \equiv \Sigma_i + \partial_i \Sigma, \\
\label{Astro_SVT_IIofII}
\,^{(\text{a})}T_{ij} & \equiv \sigma_{ij} + \partial_{ (i } \sigma_{ j ) } + \frac{\sigma}{d-1} \delta_{ij}
+ \left( \partial_i \partial_j - \frac{\delta_{ij}}{d-1} \vec{\nabla}^2 \right) \Upsilon ;
\end{align}
where these variables subject to the following constraints
\begin{align}
\label{Bardeen_SVTConstraints}
\partial_i F_i &= \partial_i D_i = 0 = \delta^{ij} D_{ij} = \partial_i D_{ij} , \\
\label{AstroTmunu_SVTConstraints}
\partial_i \Sigma_i &= \partial_i \sigma_i = 0 = \delta^{ij} \sigma_{ij} = \partial_i \sigma_{ij} .
\end{align}
We may then gather the following are gauge-invariant at first order in perturbations:
\begin{align}
\label{Bardeen_Psi}
\Phi &\equiv -\frac E2 + \frac{1}{a} \partial_0 \left\{ a \left( F - \frac{\dot{K}}{2} \right) \right\} , \\
\label{Bardeen_Phi}
\Psi &\equiv -\frac{D - \vec{\nabla}^2 K}{2(d-1)} - \frac{\dot{a}}{a} \left( F - \frac{\dot{K}}{2} \right) , \\
\label{Bardeen_VandDij}
V_i &\equiv F_i - \frac{\dot{D}_i}2
\qquad \qquad \text{ and }\qquad \qquad
D_{ij} \equiv \chi_{ij}^\text{TT} .
\end{align}
\footnote{Notice that the sign convention for the metric as well as certain gauge-invariant variables are defined differently in \cite{Chu:2016ngc}. To change the gauge-invariant notations into those employed in \cite{Chu:2016ngc}, we follow the conversions: $\Phi[\text{here}] \to \Psi[\text{\cite{Chu:2016ngc}}]/2$, $\Psi[\text{here}] \to \Phi[\text{\cite{Chu:2016ngc}}]/2$, $V_i[\text{here}] \to -V_i[\text{\cite{Chu:2016ngc}}]$, and $D_{ij}[\text{here}] \to - D_{ij}[\text{\cite{Chu:2016ngc}}]$.}Within the cosmological case, the solution of $D_{ij}$ can be found in eq.~(111), that of $V_i$ in eq.~(119) and those of $\Psi$ in equations (123), (125) and (130) of \cite{Chu:2016ngc}. $\Phi$ and $\Psi$ are related through
\begin{align}
\label{PhiPsiRelationship}
(d-3) \Psi - \Phi = 8\pi G_{\rm N} \Upsilon .
\end{align}
Within the Minkowski case, on the other hand, eq.~\eqref{PhiPsiRelationship} still holds but the solution of $\Psi$ can be found in eq.~(A27); that of $V_i$ in eq.~(A28); and that of $D_{ij}$ in eq.~(A29) of \cite{Chu:2016ngc}. As already alluded to, of all the gauge invariant variables in a flat background, only the tensor $D_{ij}$ admits wave solutions.
Because of the TT constraints in eq.~\eqref{Bardeen_SVTConstraints}, note that the tensor mode $D_{ij}$ exists only for $d \geq 4$. The apparent physical importance of these field variables in equations \eqref{Bardeen_Psi}, \eqref{Bardeen_Phi}, and \eqref{Bardeen_VandDij} lies in the fact that, if some observable can be expressed in terms of them, then the same observable cannot be rendered trivial merely by a small change in coordinates since $\Phi$, $\Psi$, $V_i$ and $D_{ij}$ will remain invariant.
When dealing with electromagnetism, we will set to zero the perturbations $\chi_{\mu\nu}$ in eq.~\eqref{PerturbedFLRW} and proceed to solve Maxwell's equations
\begin{align}
\nabla_\nu F^{\mu\nu} = J^\mu,
\qquad\qquad
F_{\mu\nu} \equiv 2\partial_{[\mu} A_{\nu]} .
\label{Maxwell_equations}
\end{align}
Under the gauge transformation
\begin{align}
\label{Maxwell_GaugeTransformation}
A_\mu \to A_\mu - \partial_\mu \mathcal{C} ,
\end{align}
the vector potential $A_\mu$ leaves the Faraday tensor $F_{\mu\nu}$ invariant
for an arbitrary function $\mathcal{C}$. If we perform a scalar-vector decomposition of the vector potential
\begin{align}
A_i \equiv \alpha_i + \partial_i \alpha, \qquad\qquad \partial_i \alpha_i = 0 ,
\end{align}
and that of the electric current
\begin{align}
J_0 & \equiv -\rho , \\
J_i & \equiv \Gamma_i + \partial_i \Gamma, \qquad\qquad \partial_i \Gamma_i = 0 ;
\end{align}
we may proceed to identify the following gauge-invariant scalar $\Phi$ and transverse (helicity$-1$) photon $\alpha_i$:
\begin{align}
\label{Maxwell_GaugeInvariant_ScalarVector}
\Phi \equiv \dot{\alpha} - A_0
\qquad \text{and} \qquad
\alpha_i \equiv A_i^\text{T} .
\end{align}
\footnote{We highlight that, in the electromagnetic case here, some conventions are defined differently relative to \cite{Chu:2016ngc}. For instance, when switching to the gauge-invariant variables used in \cite{Chu:2016ngc}, we may implement the following conversions: $\alpha_i[\text{here}] \to -\alpha_i[\text{\cite{Chu:2016ngc}}]$, $\Phi[\text{here}] \to \Phi[\text{\cite{Chu:2016ngc}}]$, $\rho[\text{here}] \to a^2 \rho[\text{\cite{Chu:2016ngc}}]$, and $\Gamma_i[\text{here}] \to - a^2 \Gamma_i[\text{\cite{Chu:2016ngc}}]$.}In terms of these variables, the Faraday tensor reads
\begin{align}
\label{Maxwell_FaradayTensor}
F_{0i} = \dot{\alpha}_i + \partial_i \Phi,
\qquad\qquad
F_{ij} = 2\partial_{[i} \alpha_{j]} .
\end{align}
We refer the reader to \S V of \cite{Chu:2016ngc} for further details; again, we will cite the relevant results below.
{\bf Electromagnetic Observables} \qquad In classical electromagnetism, it is the electric $F_{0i} = -F_{i0}$ and magnetic $F_{ij}$ fields in eq.~\eqref{Maxwell_FaradayTensor} that are regarded as observables, because they provide the forces on electrically charged systems. We believe the situation for gravity is more subtle, however.
{\bf Gravitational Observables: (Simplified) Weber bar} \qquad Let us begin with a small lump of freely falling material acting as a Weber-bar detector of GWs. In what follows, the assumption of freely falling detectors makes it technically advantageous to describe their trajectories using the synchronous gauge metric -- where the perturbations are purely spatial:
\begin{align}
\label{SynchronousGauge}
g_{\mu\nu} = a^2 \left( \eta_{\mu\nu} \text{d} x^\mu \text{d} x^\nu + \chi_{ij}^{\text{(Synch)}} \text{d} x^i \text{d} x^j \right) .
\end{align}
For, if the particles comprising the detectors experience negligible inter-particle forces, then the synchronous gauge in eq.~\eqref{SynchronousGauge} can be chosen such that each of them would in fact have constant spatial trajectories; specifically, for the $i$th particle its timelike geodesic reads
\begin{align}
Z_{(i)}^\mu = \left( \eta, \vec{Z}_{(i)} \right),
\qquad\qquad
\text{$\vec{Z}_{(i)}$ constant} .
\end{align}
The tidal forces due to a passing GW acting between an infinitesimally nearby pair of particles, whose worldlines are joined by $\ell^\mu$, is given by the geodesic deviation equation
\begin{align}
\label{GeodesicDeviation}
U^\alpha U^\beta \nabla_\alpha \nabla_\beta \ell^\mu
&= -R^\mu_{\phantom{\mu}\nu \alpha\beta} U^\nu \ell^\alpha U^\beta .
\end{align}
The $U^\mu = a^{-1} \delta^\mu_0$ in eq.~\eqref{GeodesicDeviation} is the timelike geodesic vector tangent to either one of the two worldlines. For the freely falling scenario at hand, it is in fact consistent to choose $\ell^\mu$ to be purely spatial, i.e., $\ell^0 = 0$, so that eq.~\eqref{GeodesicDeviation} becomes
\begin{align}
\label{GeodesicDeviation_FreeFall}
\nabla_0 \nabla_0 \ell^i
&= -R^i_{\phantom{\mu}0 j0} \ell^j .
\end{align}
{\it Minkowski analysis} \qquad In a flat background, $a=1$, we witness from eq.~\eqref{GeodesicDeviation_FreeFall} that the $\delta_1 R_{0i0j}$ components of the linearized Riemann tensor must therefore describe the first-order tidal forces between infinitesimally nearby particles within our idealized Weber bar. Any inter-particle (electromagnetic) forces that are responsible for holding the lump of material together must therefore counter the $\delta_1 R_{0i0j}$ on the right hand side of eq.~\eqref{GeodesicDeviation_FreeFall}. Moreover, as long as our Weber bar's proper size is very small compared to that of the GW wavelength, the physical pattern of rarefaction and compression of the material asserted by the GW's passage must also be encoded entirely within $\delta_1 R_{0i0j}$. Now, not only do these components carry physical meaning, the entire linearized Riemann tensor is in fact gauge invariant because the full Riemann tensor is zero when evaluated on the background $g_{\mu\nu} = \eta_{\mu\nu}$. These reasons explain why we will, in the coming sections, compute $\delta_1 R_{0i0j}$ within the gauge invariant formalism:
\begin{align}
\label{LinearizedRiemann_Minkowski_GaugeInvariant}
\delta_1 R_{0i 0j}
= \delta_{ij} \ddot{\Psi} + \partial_i \partial_j \Phi + \partial_{(i} \dot{V}_{j)} - \frac12 \ddot{D}_{ij} ,
\qquad (a=1).
\end{align}
{\it Cosmological background} \qquad If our freely falling particles were in a cosmological background, the geodesic deviation equation \eqref{GeodesicDeviation_FreeFall} continues to be applicable. However, the linearized Riemann tensor is no longer gauge invariant because its background value, when $g_{\mu\nu}=a^2\eta_{\mu\nu}$, is no longer zero. This renders its physical interpretation more subtle. On the other hand, the traceless portion of the Riemann tensor, i.e., the Weyl tensor $C^\mu{}_{\nu \alpha\beta}$, is conformally invariant. This means $C^\mu{}_{\nu \alpha\beta}[g_{\mu\nu}=a^2\eta_{\mu\nu}]=0$ and the $\delta_1 C^i{}_{0j0}$ components of the linearized Weyl tensor is gauge invariant. It may be possible to argue that $\delta_1 C^i{}_{0j0}$ provides the dominant contribution to tidal forces in cosmology -- for, in flat spacetime, it is exactly equivalent to the Riemann tensor whenever the zero cosmological constant form of Einstein's equations holds and the Weber bar is in a vacuum region\footnote{See, for e.g., eq.~(14) of \cite{Chu:2016ngc} or eq.~\eqref{RiemannWeyl}.} -- but we shall leave the detailed analysis of this cosmological case to future work \cite{ChuLiuInPrep}.
{\bf Gravitational observable: (Simplified) Laser Interferometer} \qquad We move on to consider a toy model of a freely falling laser interferometer. If we assume the proper size of the interferometer is small compared to the GW wavelength, it is reasonable to then state the observed interference pattern will be proportional to the differences in its arm-lengths. As argued in \S C of \cite{Chu:2016ngc}, we may again employ the synchronous gauge in eq.~\eqref{SynchronousGauge} to compute the time dependent proper distance between two ends $(\eta,\vec{Y}_0)$ and $(\eta,\vec{Z}_0)$ of a single arm. (Remember the $\vec{Y}_0$ and $\vec{Z}_0$ here are constants.) Using Synge's world function, and assuming the interferometer is turned on at $\eta'$ but does not operate over cosmological timescales, the fractional distortion of this $\vec{Y} \leftrightarrow \vec{Z}$ arm is (cf.~eq.~(6) of \cite{Chu:2016ngc})
\begin{align}
\label{FractionalDistortion_AtConstantCosmicTime_FirstOrderHuman}
\left(\frac{\delta L}{L_0}\right)[\eta > \eta']
= \frac{\widehat{n}^i \widehat{n}^j}{2} \int_{0}^{1} \Delta\chi_{ij}^{\text{(Synch)}} \text{d} \lambda
+ \mathcal{O}\left[ \left(\chi_{mn}^{\text{(Synch)}}\right)^2,(\eta-\eta') \dot{a}[\eta']/a[\eta'] \right],
\end{align}
where $\widehat{n} \equiv (\vec{Z}_0 - \vec{Y}_0)/|\vec{Z}_0 - \vec{Y}_0|$ is the unit radial vector pointing from one mass to the other; and the $\lambda$-integral involves a Euclidean straight line between the two end points:
\begin{align}
\label{FractionalDistortion_AtConstantCosmicTime_Deltachiij}
\Delta \chi_{ij}^{\text{(Synch)}}
\equiv
\chi_{ij}^{\text{(Synch)}}\left[ \eta, \vec{Y}_0 + \lambda \left(\vec{Z}_0-\vec{Y}_0\right) \right]
- \chi_{ij}^{\text{(Synch)}}\left[ \eta', \vec{Y}_0 + \lambda \left(\vec{Z}_0-\vec{Y}_0\right) \right] .
\end{align}
{\it Minkowski analysis} \qquad Now, in the synchronous gauge of eq.~\eqref{SynchronousGauge}, the linearized Riemann tensor reads
\begin{align}
\label{LinearizedRiemann_Minkowski_SynchGauge}
\delta_1 R_{0i 0j}
= -\frac{1}{2} \ddot{\chi}_{ij}^{\text{(Synch)}} .
\end{align}
We may thus solve for the synchronous gauge perturbation needed in eq.~\eqref{FractionalDistortion_AtConstantCosmicTime_FirstOrderHuman} by first connecting it to the gauge-invariant variables. This, in turn, is achieved by exploiting the gauge-invariance of the linearized Riemann tensor in a Minkowski background. In other words, since equations \eqref{LinearizedRiemann_Minkowski_GaugeInvariant} and \eqref{LinearizedRiemann_Minkowski_SynchGauge} refer to the same object, we have
\begin{align}
\label{SynchronousGauge_to_GaugeInvariant}
\ddot{\chi}_{ij}^{\text{(Synch)}}
&= \ddot{D}_{ij} - 2\partial_{(i} \dot{V}_{j)} - 2\delta_{ij} \ddot{\Psi} - 2 \partial_i \partial_j \Phi,
\qquad (a=1) .
\end{align}
Equations \eqref{LinearizedRiemann_Minkowski_GaugeInvariant}, \eqref{LinearizedRiemann_Minkowski_SynchGauge} and \eqref{SynchronousGauge_to_GaugeInvariant} inform us that the fractional distortion formula in eq.~\eqref{FractionalDistortion_AtConstantCosmicTime_FirstOrderHuman} is therefore -- at least in principle -- related to the double time integral of the linearized Riemann tensor itself.\footnote{See, for example, equations (27.22) and (27.24) of \cite{ThorneBlandfordBook}.} In any event, as long as the GW detector is sufficiently far enough from the astrophysical source, we may take the far zone limit of the right hand side of eq.~\eqref{SynchronousGauge_to_GaugeInvariant}. Below, we will use methods different from those in \cite{Ashtekar:2017wgq,Ashtekar:2017ydh} to argue that, this far zone limit yields
\begin{align}
\label{SynchronousGauge_to_GaugeInvariant_StepI}
\left(\ddot{D}_{ij} - 2\partial_{(i} \dot{V}_{j)} - 2\delta_{ij} \ddot{\Psi} - 2 \partial_i \partial_j \Phi\right)_{\text{far zone}}
= \ddot{D}_{ij}[\text{far zone}]
= \ddot{\chi}_{ij}^{\text{tt}}[\text{far zone}] ,
\qquad
(a=1);
\end{align}
where $\chi_{ij}^{\text{tt}}$ is the tt projection of the de Donder gauge solution as $r \equiv |\vec{x}| \to \infty$. Comparing equations \eqref{SynchronousGauge_to_GaugeInvariant} and \eqref{SynchronousGauge_to_GaugeInvariant_StepI} allows us to deduce:
\begin{align}
\label{SynchronousGauge_to_GaugeInvariant_FullAnswer}
\chi_{ij}^{(\text{Synch})}[\eta,\vec{x}]
&= \chi_{ij}^{\text{tt}}[\eta,\vec{x}]
+ (\eta-\eta') \mathcal V_{ij}[\eta',\vec{x}]
+ \mathcal W_{ij}[\eta',\vec{x}],
\end{align}
where $\mathcal V_{ij}$ and $\mathcal W_{ij}$ are the two undetermined initial conditions at $\eta'$, and, on both sides, the far zone limit has been taken. One issue that is often not addressed is, why the initial conditions -- the last two terms on the right hand side of eq.~\eqref{SynchronousGauge_to_GaugeInvariant_FullAnswer} -- are usually neglected. We will take the perspective that realistic GW detectors are sensitive to waves within a limited bandwidth, and since the second and third terms of eq.~\eqref{SynchronousGauge_to_GaugeInvariant_FullAnswer} are zero-frequency `waves' one may ignore their contributions. More explicitly, the ($\omega-$)frequency transform of eq.~\eqref{SynchronousGauge_to_GaugeInvariant_FullAnswer} becomes
\begin{align}
\label{SynchronousGauge_to_GaugeInvariant_FullAnswer_FreqSpace}
\widetilde{\chi}_{ij}^{(\text{Synch})}[\omega,\vec{x}]
&= \widetilde{\chi}_{ij}^{\text{tt}}[\omega,\vec{x}]
- (2\pi i) \delta'[\omega]e^{i\omega\eta'} \mathcal V_{ij}[\eta',\vec{x}]
+ (2\pi) \delta[\omega] \mathcal W_{ij}[\eta',\vec{x}] ,
\end{align}
where $\delta[\omega]$ and $\delta'[\omega]$ are the Dirac delta function and its derivative.
We close this section by providing an expedited method to obtain the synchronous gauge metric perturbation from the gauge-invariance of the linearized Riemann tensor, at least in $(3+1)$ dimensional Minkowski spacetime. The key is that the de Donder gauge $\partial^\mu \chi_{\mu\nu} = (1/2) \partial_\nu \chi$, with $\chi \equiv \eta^{\rho\sigma} \chi_{\rho\sigma}$, allows us to solve $\chi_{\mu\nu}$ rather easily with the massless scalar Green's function. In terms of the `trace-reversed' variable
\begin{align}
\overline{\chi}_{\mu\nu}
&\equiv \chi_{\mu\nu} - \frac{1}{2} \eta_{\mu\nu} \chi,
\qquad
\overline{\chi} = \eta^{\rho\sigma} \overline{\chi}_{\rho\sigma} ;
\end{align}
the linearized Einstein's equations lead us to the following far zone solution:
\begin{align}
\label{deDonder_4D_FarZone}
\overline{\chi}_{\mu\nu}[\eta,\vec{x}]
&= \frac{4G_{\rm N}}{r} \int_{\mathbb{R}^3} \text{d}^3 \vec{x}'
T_{\mu\nu}[\eta-r+\vec{x}'\cdot\widehat{r},\vec{x}'] ,
\qquad
\widehat{r} \equiv \frac{\vec{x}}{|\vec{x}|} .
\end{align}
(Note: we have chosen $\vec{0}$ to lie within the astrophysical system.) We also have, in de Donder gauge, the following components of the linearized Riemann tensor
\begin{align}
\label{LinearizedR0i0j_deDonder}
\delta_1 R_{0i0j}
&= \frac{1}{2} \left( 2\partial_l \partial_{(i} \overline{\chi}_{j)l} - \partial_0^2 \left( \overline{\chi}_{ij} - \frac{1}{2} \delta_{ij} \overline{\chi}_{ll} \right) - \frac{1}{2} \delta_{ij} \partial_l \partial_m \overline{\chi}_{lm} - \frac{1}{2} \partial_i \partial_j \overline{\chi}_{ll} - \frac{1}{2} \partial_i \partial_j \overline{\chi}_{00} \right) .
\end{align}
Upon inserting eq.~\eqref{deDonder_4D_FarZone} into eq.~\eqref{LinearizedR0i0j_deDonder}, one may recognize that every spatial derivative acting on $\overline{\chi}_{\mu\nu}$ may be replaced with a time derivative, via
\begin{align}
\partial_i \to -\widehat{r}^i \partial_0 .
\end{align}
In the far zone limit we are working in, the error incurred by this replacement scales as (time-scale of source)/(observer-source distance) or (characteristic size of source)/(observer-source distance), both of which are small by assumption. At this point, one would find that eq.~\eqref{LinearizedR0i0j_deDonder} has been massaged into
\begin{align}
\label{LinearizedR0i0j_deDonder_FarZone}
\delta_1 R_{0i0j}
&= -\frac{1}{2} \ddot{\chi}_{ij}^{\text{tt}} ,
\end{align}
where the tt perturbation is the following projection (cf.~eq.~\eqref{tt_Projector}) of the de Donder gauge solution
\begin{align}
\chi_{ij}^{\text{tt}} = P_{ij ab} \overline\chi_{ab}[\text{de Donder}] .
\end{align}
Comparing equations \eqref{LinearizedRiemann_Minkowski_SynchGauge} and \eqref{LinearizedR0i0j_deDonder_FarZone} now hands us, for finite frequency $\omega$,
\begin{align}
\widetilde{\chi}_{ij}^{(\text{Synch})}[\omega,\vec{x}]
= \widetilde{\chi}_{ij}^{\text{tt}}[\omega,\vec{x}] \qquad
(\text{Far zone}) .
\end{align}
Finally, we once again leave to future work \cite{ChuLiuInPrep} the connection between the cosmological synchronous gauge metric perturbation to its gauge-invariant counterparts.
\section{Why are the massless spin-1 photon and spin-2 graviton acausal?}
\label{WhyAcausal}
Before we proceed to tackle the computations of the massless spin-1 and spin-2 Green's functions, let us first explain why acausality is to be expected. Since this section is meant to be heuristic, we shall be content to work strictly in a Minkowski background.
{\bf Spin-1 Photons} \qquad We begin by recalling the fact that the Lorenz gauge vector potential, which obeys $\partial^\mu A_\mu = 0$ and $\partial^2 A_\mu = - J_\mu$, is causally dependent on the electric current $J_\mu$. If $G^+_d$ denotes the retarded Green's function of the massless scalar, we have
\begin{align}
A_\mu[x]
= - \int_{\mathbb{R}^{d-1,1}} \text{d}^d x' G^+_d[x-x'] J_\mu[x'].
\end{align}
In even dimensions, where $G^+_d$ propagates signals strictly on the null cone, $A_\mu$ is the field due to the electric current lying on the past light cone of the observer. In odd dimensions, where $G^+_d$ propagates signals strictly inside the null cone (at least for timelike sources \cite{Chu:2016ngc}), $A_\mu$ is the field due to the electric current lying within the past light cone of the observer. On the other hand, the transverse spin-1 photon can be constructed from the Lorenz gauge photon via the following Fourier-space projection involving the spatial Fourier transform of the spatial components of the vector potential $\widetilde{A}_j[\eta,\vec{k}]$:
\begin{align}
\label{Spin1Photon_Minkowski}
\alpha_i[\eta,\vec{x}]
&= \int_{\mathbb{R}^{d-1}} \frac{\text{d}^{d-1} \vec{k}}{(2\pi)^{d-1}} \widehat{P}_{ij}[\vec{k}] \widetilde{A}_j[\eta,\vec{k}] e^{i\vec{k}\cdot\vec{x}} ,
\end{align}
where the transverse projector $\widehat{P}_{ij}[\vec k]$ is defined in eq.~\eqref{TransverseProjector}. Now, in Fourier space, $-1/\vec{k}^2$ is simply the Euclidean Green's function
\begin{align}
\label{EuclideanGreenFunction_Fourier}
G^{(\text{E})}_{d}[\vec{x}-\vec{x}']
= \frac{1}{\partial_i\partial_i}[\vec{x}-\vec{x}']
= \int_{\mathbb{R}^{d-1}} \frac{\text{d}^{d-1}\vec{k}}{(2\pi)^{d-1}} \frac{e^{i\vec{k}\cdot(\vec{x}-\vec{x}')}}{-\vec{k}^2} ,
\end{align}
with the concrete expressions
{\allowdisplaybreaks
\begin{align}
G^{(\text{E,reg})}_{3+2\epsilon}[\vec{x}-\vec{x}'] & = -\frac1{4\pi} \left( \frac1\epsilon - \gamma - \ln[\pi] - 2\ln [\mu R] \right),
\label{EuclideanGreenFunction_3D}\\
G^{(\text{E})}_{d\geq4}[\vec{x}-\vec{x}']
& = - \frac{\Gamma[\frac{d-3}{2}]}{4\pi^{\frac{d-1}{2}} |\vec{x}-\vec{x}'|^{d-3}} ;
\label{EuclideanGreenFunction}
\end{align}}%
where $G^{(\text{E})}_{3}$ has been dimensional-regularized in eq.~\eqref{EuclideanGreenFunction_3D} with an arbitrary mass scale $\mu$ introduced and $\gamma$ being the Euler-Mascheroni constant, and $-\partial_i \partial_j$ is replaced with $k_i k_j$. Utilizing eqs. \eqref{EuclideanGreenFunction_Fourier}, \eqref{EuclideanGreenFunction_3D} and \eqref{EuclideanGreenFunction} in eq.~\eqref{Spin1Photon_Minkowski} informs us that the transverse photon itself must therefore be related to its Lorenz gauge counterpart through the subtraction of the latter's longitudinal piece:
\begin{align}
\label{TransversePhoton_Decomposed}
\alpha_i[\eta,\vec{x}]
&= A_i[\eta,\vec{x}] + A^\parallel_i[\eta,\vec{x}] , \\
\label{TransversePhoton_AcausalPart_3D}
A^\parallel_i[\eta,\vec{x}]
&\equiv -\partial_i \partial_j \int_{\mathbb{R}^{2}} \text{d}^{2}\vec{x}' \, \frac{\ln \big[|\vec{x}-\vec{x}'|\big]}{2\pi} A_j[\eta,\vec{x}'] \qquad\qquad (d = 3) \\
&\equiv \partial_i \partial_j \int_{\mathbb{R}^{d-1}} \text{d}^{d-1}\vec{x}' \frac{\Gamma\left[\frac{d-3}{2}\right]}{4\pi^{\frac{d-1}{2}} |\vec{x}-\vec{x}'|^{d-3}} A_j[\eta,\vec{x}'] \qquad\qquad (d \geq 4).
\label{TransversePhoton_AcausalPart}
\end{align}
We will now explain how the second term $A^\parallel_i[\eta,\vec{x}]$ in eq.~\eqref{TransversePhoton_Decomposed} is most likely acausal, because it is essentially the causal Lorenz gauge vector potential but smeared over all space, weighted by the Euclidean Green's function in eqs.~\eqref{EuclideanGreenFunction_3D} and \eqref{EuclideanGreenFunction}. Referring to Fig.~\eqref{WhySpin1IsAcausal}, we see that eqs.~\eqref{TransversePhoton_AcausalPart_3D} and \eqref{TransversePhoton_AcausalPart} are the weighted superposition of $A_i[\eta,\vec{x}']$ over all $\vec{x}'$, which -- for a fixed $\vec{x}'$ -- receives signals from the electric current from the past light cone of $(\eta,\vec{x}')$ (for even dimensions) or within it (for odd dimensions). But from the perspective of the observer at $(\eta,\vec{x})$, this means $A^\parallel_i$ is getting a signal from the portion of the source residing within the shaded (blue) region, which lies outside its past null cone.
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=5.5in]{ProjectionProcess.pdf}
\caption{The $A_i[\eta,\vec{x}]$ is the Lorenz gauge photon vector potential at the observer's location $(\eta,\vec{x})$. The $A_i[\eta,\vec{x}']$ is the Lorenz gauge potential at some other spatial location but at the same time $\eta$. The solid cone is the past null cone of the observer at $(\eta,\vec{x})$; while the dotted one is that of $(\eta,\vec{x}')$. The gray thick line denotes the worldtube of the electric current. (See Brill and Goodman \cite{BrillGoodman} for a closely related discussion, but from the Coulomb gauge perspective.)}
\label{WhySpin1IsAcausal}
\end{center}
\end{figure}
{\bf Spin-2 Gravitons} \qquad The de Donder gauge $\partial^\mu \chi_{\mu\nu} = (1/2) \partial_\nu \chi$, with $\chi \equiv \eta^{\rho\sigma} \chi_{\rho\sigma}$, like the Lorenz gauge for photons, yields gravitational perturbations that are causally sourced by their matter $T_{\mu\nu}$. Equation (A39) of \cite{Chu:2016ngc} tells us
\begin{align}
\overline{\chi}_{\mu\nu}[x]
&= -16\pi G_{\rm N} \int_{\mathbb{R}^{d-1,1}} \text{d}^d x'
G^+_d[x-x'] T_{\mu\nu}[x'].
\end{align}
The transverse-traceless spin-2 graviton is gotten from its de Donder gauge counterpart via the Fourier space projection
\begin{align}
\chi_{ij}^{\text{TT}}[\eta,\vec{x}]
&= \int_{\mathbb{R}^{d-1}} \frac{\text{d}^{d-1}\vec{k}}{(2\pi)^{d-1}} \widehat P_{ij mn}[\vec{k}] \, \widetilde{\overline\chi}_{mn}[\eta,\vec{k}] e^{i\vec{k}\cdot\vec{x}},
\end{align}
where the projection tensor $\widehat P_{ijmn}[\vec{k}]$ is given in eq.~\eqref{GravitonProjector}, which can also be expressed explicitly in terms of $\vec k$,
\begin{align}
\widehat P_{ij mn}[\vec{k}]
&= \widehat P_{m(i} \widehat P_{j)n} - \frac{1}{d-2} \widehat P_{ij} \widehat P_{mn} , \notag\\
&=
\delta_{m(i} \delta_{j)n}
- \frac{\delta_{ij} \delta_{mn}}{d-2} - \frac{\left( \delta_{m(i} k_{j)} k_n + \delta_{n(i} k_{j)} k_m \right)}{\vec k^2} \notag\\
&\qquad\qquad
+ \frac{\delta_{ij} k_m k_n + \delta_{mn} k_i k_j}{(d-2) \vec{k}^2}
+ \left(\frac{d-3}{d-2}\right) \frac{k_i k_j k_m k_n}{\vec{k}^4} .
\label{GravitonProjector_k}
\end{align}
The same sort of arguments made for the spin-1 photon would apply here to tell us the spin-2 graviton receives signals from $T_{\mu\nu}$ from outside the observer's past light cone. For instance, the third and fourth group of terms in the second equality in eq.~\eqref{GravitonProjector_k} involves two spatial derivatives acting on the weighted superposition of the de Donder GW over all space but at the same observer time $\eta$, namely
\begin{align}
\sim \partial_a \partial_b \int_{\mathbb{R}^{d-1}} \text{d}^{d-1}\vec{x}'
\frac{\overline\chi_{ij}[\eta,\vec{x}']}{|\vec{x}-\vec{x}'|^{d-3}} ;
\end{align}
whereas the last group of terms in eq.~\eqref{GravitonProjector_k} involves four spatial derivatives acting on a different weighted superposition of the same:
\begin{align}
\sim \partial_a \partial_b \partial_c \partial_e \int_{\mathbb{R}^{d-1}}
\text{d}^{d-1}\vec{x}' \frac{\overline\chi_{ij}[\eta,\vec{x}']}{|\vec{x}-\vec{x}'|^{d-5}} .
\end{align}
\section{Minkowski Spacetime} \label{MinkowskiSpacetime}
As demonstrated in the previous section, both massless spin-1 and spin-2 fields are expected to contain the acausal information from their isolated sources. To quantify this acausality, we will in this section perform a detailed analysis of the effective Minkowski spacetime Green's functions of the electromagnetic and gravitational gauge-invariant variables.
\subsection{Electromagnetism}
We will begin with electromagnetism in all spacetime dimensions equal to or higher than three, $d\geq3$; for $d=3$ corresponds to the lowest physical dimension for spin-1 photons to exist. In addition, the discussion for the electromagnetic field here will provide a useful guide for us to tackle the more complicated and subtle case of linearized gravitation.
{\bf Field Equations} \qquad In terms of the gauge-invariant variables \eqref{Maxwell_GaugeInvariant_ScalarVector} and provided the electric current $J^\mu$ is conserved, the non-redundant portions of Maxwell's equations \eqref{Maxwell_equations} -- see \cite{Chu:2016ngc} for a discussion -- are the dynamical wave equation for the transverse spin-1 photon $\alpha_i$,
\begin{align}
\partial^2\alpha_i&=-\Gamma_i,
\label{Spin1_WaveEq_Minkowski}
\end{align}
with $\partial^2\equiv\eta^{\mu\nu}\partial_\mu\partial_\nu$, and a Poisson's equation obeyed by the gauge-invariant scalar $\Phi$,
\begin{align}
\vec\nabla^2\Phi=-\rho.
\label{EMPoisson_Minkowski}
\end{align}
Notice both equations involve non-locality. The source $\Gamma_i$ on the right hand side of eq.~\eqref{Spin1_WaveEq_Minkowski} is the transverse component $J_i$, which (recalling arguments from the previous section) is thus a non-local functional of $J_i$. Whereas the Poisson equation obeyed by $\Phi[\eta,\vec{x}]$ means it is sensitive to the charge density $\rho[\eta,\vec{x}]$ on the right hand side of eq.~\eqref{EMPoisson_Minkowski} at the same instant $\eta$. As we will show later in this section, only when $\alpha_i$ and $\Phi$ are both involved, do the physical observables -- i.e., the field strength $F_{\mu\nu}$ -- become causally dependent on the electromagnetic current $J_\mu$.
{\bf Spin-1 Photons } \qquad To solve for $\alpha_i$ in eq.~\eqref{Spin1_WaveEq_Minkowski} through its effective Green's function convolved against its localized sources, it is convenient to first go to the Fourier space, where the transverse property is implemented through a projection of the Fourier transform of the current $\widetilde{J}_i$. For all $\vec k\neq\vec0$, the $\alpha_i$ can be written as the superposition
\begin{align}
\alpha_i[\eta,\vec x]=-\int_{\mathbb{R}} \text{d}\eta'\int_{\mathbb{R}^{d-1}} \frac{\text{d}^{d-1}\vec{k}}{(2\pi)^{d-1}}\,\widetilde G^{+}_d [\eta,\eta';\vec k]\left(\delta_{ij}-\frac{k_ik_j}{\vec k^2}\right)
\widetilde{J}_j[\eta',\vec{k}]\, e^{i\vec{k}\cdot\vec{x}},
\label{Spin1_Fourier}
\end{align}
where $\widetilde G^{+}_d$ denotes the Fourier transform of the retarded Green's function of the massless scalar. In flat spacetime, it enjoys time-translation symmetry and reads
\begin{align}
\widetilde G^{+}_d [\eta,\eta';\vec k]=-\Theta[T]\frac{\sin |\vec k|T}{|\vec k|},
\label{FourierG+_Minkowski}
\end{align}
with $T\equiv\eta-\eta'$. As we have seen before, $k_ik_j$ in momentum space can be pulled out with the replacement $k_ik_j\to-\partial_i\partial_j$ acting on the Fourier integral. Therefore, the expression \eqref{Spin1_Fourier} can be re-cast into the convolution of the spin-1 effective Green's function $G^+_{ij}$ against the local electromagnetic current,
\begin{align}
\alpha_i[x]=-\int_{\mathbb{R}^{d-1,1}} \text{d}^dx'\,G^+_{ij}[T,\vec R]\,J_j[x'],
\label{Spin1_Convolution_Minkowski}
\end{align}
where $\vec R\equiv\vec x-\vec x'$ and the $G^+_{ij}$ takes the form
\begin{align}
G^+_{ij}[T,\vec R]
&\equiv -\Theta[T]C_{ij}[T,\vec R], \notag \\
C_{ij}[T,\vec R]
&=\delta_{ij}C_{1,d}[T,R]+\partial_i\partial_jC_{2,d}[T,R].
\label{G+ij_Minkowski}
\end{align}
The $C_{1,d}$ and $C_{2,d}$ are respectively defined to be two scalar Fourier integrals
\begin{align}
C_{1,d}[T,R]&\equiv\int_{\mathbb{R}^{d-1}}\frac{\text{d}^{d-1}\vec{k}}{(2\pi)^{d-1}}\frac{\sin |\vec k|T}{|\vec k|}e^{i\vec{k}\cdot\vec{R}},\label{C1_Minkowski} \\
C_{2,d}[T,R]&\equiv\int_{\mathbb{R}^{d-1}}\frac{\text{d}^{d-1}\vec{k}}{(2\pi)^{d-1}}\frac{\sin |\vec k|T}{|\vec k|^3}e^{i\vec{k}\cdot\vec{R}},
\label{C2_Minkowski}
\end{align}
with the observer-source spatial distance denoted as $R\equiv|\vec x-\vec x'|$. Hence, the effective Green's function $G^+_{ij}$ can be gotten explicitly through eq.~\eqref{G+ij_Minkowski} once $C_{1,d}$ and $C_{2,d}$ are known. One of the advantages of using $C_{1,d}$ and $C_{2,d}$ to compute $G^+_{ij}$ is that our calculations can be simplified by exploiting the fact that they both obey the homogeneous wave equation $\partial^2C_{1,d}=0$ and $\partial^2C_{2,d}=0$ with the initial conditions
\begin{align}
\label{C12_Minkowski_T=0}
C_{1,d}[T=0,R]=0 \qquad \text{ and } \qquad C_{2,d}[T=0,R]=0 ;
\end{align}
as well as the initial velocities
\begin{align}
\label{C12dot_Minkowski_T=0}
\dot{C}_{1,d}[T=0,R] = \delta^{(d-1)}[\vec{x}-\vec{x}']
\qquad \text{ and } \qquad
\dot{C}_{2,d}[T=0,R] = -G^{(\text{E})}_d[R] ,
\end{align}
where the overdot denotes the time derivative with respect to $\eta$. In addition, $C_{1,d}$ and $C_{2,d}$ are connected via the spatial Laplacian operator or double-time derivatives:
\begin{align}
\label{C1ddot_Minkowski}
\ddot{C}_{2,d}[T,R] &= -C_{1,d}[T,R], \\
\label{C1Laplacian_Minkowski}
\vec{\nabla}^2 C_{2,d}[T,R] &= -C_{1,d}[T,R].
\end{align}
(Equations \eqref{C12_Minkowski_T=0} through \eqref{C1Laplacian_Minkowski} conditions follow readily from the Fourier representations in equations \eqref{EuclideanGreenFunction_Fourier}, \eqref{C1_Minkowski} and \eqref{C2_Minkowski}.) Moreover, note that
\begin{align}
G^\pm_d [x-x'] = \mp \Theta[\pm T] C_{1,d}[T,R]
\end{align}
are the retarded $(+)$/advanced $(-)$ massless scalar Green's functions; obeying
\begin{align}
\label{MasslessScalarG_Minkowski}
\partial^2 G^\pm_d = \delta^{(d)}[x-x'] .
\end{align}
In other words, $-C_{1,d}$ itself is the retarded minus advanced Green's function:
\begin{align}
\label{C1_Minkowski_RetardedMinusAdvanced}
-C_{1,d} = G^+_d[x-x'] - G^-_d[x-x'] .
\end{align}
Likewise, for the spin-1 case, the $-C_{ij}$ in eq.~\eqref{G+ij_Minkowski} is the difference between the retarded Green's function in eq.~\eqref{G+ij_Minkowski} and that of its advanced counterpart:
\begin{align}
\label{Cij_Minkowski_RetardedMinusAdvanced}
-C_{ij} = G_{ij}^+[x-x'] - G_{ij}^-[x-x'] .
\end{align}
In Quantum Field Theory, the $C_{1,d}$ in eq.~\eqref{C1_Minkowski_RetardedMinusAdvanced} is proportional to the commutator of massless scalar fields. In turn, $C_{ij}$ is proportional to the commutator of (spin-1) photon fields. Therefore, the elucidation of the (classical) causal structure of $C_{ij}$ will also lead to insights regarding the quantization of the associated spin-1 photons.
Before moving on to the analytic solutions, let us show that the source of $G^+_{ij}$ in eq.~\eqref{G+ij_Minkowski} is an extended one, as opposed to the usual spacetime point source of, say, the massless scalar Green's function. Applying the wave operator to the expression \eqref{G+ij_Minkowski} for $G^+_{ij}$ hands us
\begin{align}
\label{Gij_WaveEquation_Minkowski}
\partial^2G^+_{ij}[T,\vec R]
&=\delta_{ij}\delta^{(d)}[x-x']-\delta[T]\partial_i\partial_jG^{\mathrm{(E)}}_d[R] \\
&=\delta[T] \left( \delta_{ij} \delta^{(d-1)}[\vec{x}-\vec{x}'] - \partial_i\partial_j G^{\mathrm{(E)}}_d[R] \right) ;
\end{align}
where the $G^{(\text{E})}_d$ is the Euclidean Green's function of eq.~\eqref{EuclideanGreenFunction_Fourier} and the relations in eq.~\eqref{C12dot_Minkowski_T=0} were employed. We may view eq.~\eqref{Gij_WaveEquation_Minkowski} as a $(d-1) \times (d-1)$ matrix of massless scalar wave equations. That $-\partial_i \partial_j G^{(\text{E})}_d[|\vec{x}-\vec{x}'|]$ is non-zero everywhere in space at $T=0$ tells us, for a fixed pair of indices $ij$, the (retarded) signal it generates likely fills all of spacetime to the future of $\eta'$. This is to be contrasted against the massless scalar Green's function equation itself in eq.~\eqref{MasslessScalarG_Minkowski}; where, because the source at $x'$ is point-like, the signal it generates propagates only on and/or within its future light cone. If the observer at $x$ lies outside the light cone of $x'$, the signal $G^+_d[x-x']$ will be zero and causality respected. Returning to eq.~\eqref{Gij_WaveEquation_Minkowski}, if one continues to insist on viewing $G^+_{ij}[x-x']$ as the signal at $x$ generated at $x'$, since it is non-zero throughout all $\vec{x}$ whenever $\eta>\eta'$, once $x$ lies outside the light cone of $x'$ the observer at $x$ would be led to conclude the signal is acausal.
{\it Recursion relations} \qquad In both Minkowski and spatially flat cosmologies, we are aided by the spatial-translation and spatial-parity invariance of the underlying spacetimes. In particular, these symmetries allow us to solve for $C_{1,d}$ and $C_{2,d}$ for all dimensions once we know their $3-$ and $4-$dimensional solutions. This is because the higher-dimensional ones can be generated through the ``dimension-raising operator''
\begin{align}
\mathcal D_R\equiv - \frac1{2\pi R} \frac{\partial}{\partial R}.
\end{align}
(See appendix (E) of \cite{Chu:2016ngc} for a detailed discussion.) In brief, any bi-scalar function $f_d$ that depends on space solely through $R \equiv |\vec{x}-\vec{x}'|$ and takes the same Fourier integral form
\begin{align}
f_d\left[R\right]
&= \int \frac{\text{d}^{d-1}\vec{k}}{(2\pi)^{d-1}} \widetilde{f}_d\big[|\vec{k}|\big]
e^{i\vec{k}\cdot(\vec{x}-\vec{x}')}
\end{align}
for all relevant spacetime dimensions $d$, obeys the recursion relation
\begin{align}
\label{RecursionRelation}
f_{d+2}[R] = - \frac{1}{2 \pi R} \frac{\partial f_d[R]}{\partial R} .
\end{align}
This remark applies to both $C_{1,d}$ and $C_{2,d}$; specifically, we only need $C_{1,4}$ and $C_{2,4}$ to determine their counterparts in all even $d \geq 4$:
\begin{align}
\label{C1_EvenMinkowski_Recursion}
C_{1,\text{even $d$} \geq 4} &= \mathcal{D}_R^{\frac{d-4}{2}} C_{1,4} , \\
\label{C2_EvenMinkowski_Recursion}
C_{2,\text{even $d$} \geq 4} &= \mathcal{D}_R^{\frac{d-4}{2}} C_{2,4} .
\end{align}
Likewise, we only need $C_{1,3}$ and $C_{2,3}$ to obtain their counterparts in all odd $d \geq 3$:
\begin{align}
\label{C1_OddMinkowski_Recursion}
C_{1,\text{odd $d$} \geq 3} &= \mathcal{D}_R^{\frac{d-3}{2}} C_{1,3} , \\
\label{C2_OddMinkowski_Recursion}
C_{2,\text{odd $d$} \geq 3} &= \mathcal{D}_R^{\frac{d-3}{2}} C_{2,3} .
\end{align}
Also notice that, by counting the powers $|\vec k|$ in the integrals \eqref{C1_Minkowski} and \eqref{C2_Minkowski} as $|\vec k|\to0$, $C_{1,d}$ is finite for all $d$, while $C_{2,d}$ is expected to diverge when $d\leq3$. However, on physical grounds, the full effective Green's function $G^+_{ij}$ should converge for all spacetime dimensions $d \geq 3$. This suggests that, for $d=3$, the two spatial derivatives acting on $C_{2,3}$ in eq.~\eqref{G+ij_Minkowski} will eliminate the divergence completely.
{\it Time integral method} \qquad According to eq.~\eqref{C1_Minkowski_RetardedMinusAdvanced}, $-\Theta[T] C_{1,d}$ is the retarded Green's function $G^+_d$ of the massless scalar. Because eq.~\eqref{C1_Minkowski_RetardedMinusAdvanced} will continue to hold even in cosmology and because the analytic position spacetime solutions to $C_{1,d}$ and $G^+_d$ are known in all Minkowski and constant equation-of-state universes \cite{Chu:2016ngc}, we shall introduce a `time-integral' method here that will allow us to solve the (retarded part of) $C_{2,d}$ in terms of time integrals of $C_{1,d}$. We first recall that eq.~\eqref{C1ddot_Minkowski} provides us a ordinary differential equation (ODE) relating $C_{2,d}$ to $C_{1,d}$. Integrating it twice with respect to time, and taking into account the initial conditions in equations \eqref{C12_Minkowski_T=0} and \eqref{C12dot_Minkowski_T=0},
\begin{align}
C_{2,d}[T,R]
&= -\int_{0}^{T} \text{d} \tau_2 \int_{0}^{\tau_2} \text{d} \tau_1 C_{1,d}[\tau_1,R]
+ T \dot{C}_{2,d}[T=0,R] + C_{2,d}[T=0,R] \\
\label{TimeIntegralMethod_C2d}
&= -\int_{0}^{T} \text{d} \tau_2 \int_{0}^{\tau_2} \text{d} \tau_1 C_{1,d}[\tau_1,R]
- T G^{(\text{E})}_d[R] .
\end{align}
Now, any casual quantity $\mathfrak{Q}^+[\eta,\vec{x};\eta',\vec{x}']$ -- which we define as one that is non-zero only when $T \geq R \geq 0$ -- may be multiplied by $\Theta[T-R^-]$. While any anti-causal expression $\mathfrak{Q}^-[\eta,\vec{x};\eta',\vec{x}']$ -- which we define as one that is non-zero only when $T \leq -R \leq 0$ -- may be multiplied by $\Theta[-T-R^-]$.\footnote{The $R^-$ guarantees that signals on the null cone proportional to $\delta[T-R]$ and its derivatives are included.} If we then consider
\begin{align}
\int_{\eta'}^{\eta' + \tau} \text{d} \eta \, \mathfrak{Q}^+[\eta,\vec{x};\eta',\vec{x}']
= \int_{\eta'}^{\eta' + \tau} \text{d} \eta \,
\Theta[\eta-\eta'-R^-] \mathfrak{Q}^+
= \int_{0}^{\tau} \text{d} \tau' \Theta[\tau'-R^-] \mathfrak{Q}^+ .
\end{align}
We see that, whenever $\tau > R^-$, we may set $\Theta[\tau'-R^-]$ to one for $\tau'\in (R^-, \tau)$; whereas whenever $\tau < R^-$, the latter is to be set to zero.
\begin{align}
\int_{0}^{\tau} \text{d} \tau' \Theta[\tau'-R^-] \mathfrak{Q}^+
= \Theta[\tau-R^-] \left. \int_{R^-}^{\tau} \text{d} \tau' \mathfrak{Q}^+ \right\vert_{\tau' = \eta-\eta'}
\end{align}
Iterating this reasoning, we may deduce that one or multiple nested integrals of a causal quantity would return another causal one:
\begin{align}
\int_{0}^{T} \text{d} \tau_n \dots \int_{0}^{\tau_3} \text{d} \tau_2 \int_{0}^{\tau_2} \text{d} \tau_1 \Theta[\tau_1-R^-] \mathfrak{Q}^+
&= \Theta[T-R^-] \left. \int_{R^-}^{T} \text{d} \tau_n \dots \int_{R^-}^{\tau_3} \text{d} \tau_2 \int_{R^-}^{\tau_2} \text{d} \tau_1 \mathfrak{Q}^+ \right\vert_{\tau_1 = \eta-\eta'} .
\end{align}
Similarly, if we instead consider
\begin{align}
\int_{\eta' + \tau}^{\eta'} \text{d} \eta \, \mathfrak{Q}^-[\eta,\vec{x};\eta',\vec{x}']
= \int_{\eta' + \tau}^{\eta'} \text{d} \eta \,
\Theta[-(\eta-\eta')-R^-] \mathfrak{Q}^-
= \int_{\tau}^0 \text{d} \tau' \Theta[-\tau'-R^-] \mathfrak{Q}^- .
\end{align}
We see that, whenever $\tau < -R^-$, we may set $\Theta[-\tau'-R^-]$ to one for $\tau'\in (\tau, -R^-)$; whereas whenever $\tau \geq -R^-$, the latter is to be set to zero.
\begin{align}
\int_{0}^{\tau} \text{d} \tau' \Theta[-\tau'-R^-] \mathfrak{Q}^-
= \Theta[-\tau-R^-] \left. \int_{-R^-}^{\tau} \text{d} \tau' \mathfrak{Q}^- \right\vert_{\tau' = \eta-\eta'}
\end{align}
Iterating this reasoning, we may deduce that one or multiple nested integrals of an anti-causal quantity would return another anti-causal one:
\begin{align}
\int_{0}^{T} \text{d} \tau_n \dots \int_{0}^{\tau_3} \text{d} \tau_2 \int_{0}^{\tau_2} \text{d} \tau_1 \Theta[-\tau_1-R^-] \mathfrak{Q}^-
&= \Theta[-T-R^-] \left. \int_{-R^-}^{T} \text{d} \tau_n \dots \int_{-R^-}^{\tau_3} \text{d} \tau_2 \int_{-R^-}^{\tau_2} \text{d} \tau_1 \mathfrak{Q}^- \right\vert_{\tau_1 = \eta-\eta'} .
\end{align}
This discussion implies the integral of the difference between the causal Green's function and its anti-causal counterpart -- recall eq.~\eqref{C1_Minkowski_RetardedMinusAdvanced} -- namely $-C_{1,d}$, returns a causal minus anti-causal object:
\begin{align}
\int_{0}^{T} \text{d} \tau' & \big( G^+_d[\tau',R] - G^-_d[\tau',R] \big) \notag\\
& \qquad = \Theta[T-R^-] \int_{R^-}^{T} \text{d} \tau' \, G^+_d[\tau',R] - \Theta[-T-R^-] \int_{-R^-}^{T} \text{d} \tau' \, G^-_d[\tau',R] .
\end{align}
Furthermore, referring to equations \eqref{C1_Minkowski_RetardedMinusAdvanced} and \eqref{TimeIntegralMethod_C2d}, we see that the retarded portion of $C_{2,d}$ -- which is what we need -- is gotten by integrating the retarded portion of $C_{1,d}$:
\begin{align}
\label{TimeIntegralMethod_C2dRet}
C_{2,d}^{+}[T,R]
&= \Theta[T-R^-] \int_{R^-}^{T} \text{d} \tau_2 \int_{R^-}^{\tau_2} \text{d} \tau_1 \, G_d^+[\tau_1,R]
- \Theta[T] \cdot T G^{(\text{E})}_d[R] ,
\end{align}
where $C_{2,d}^{+}\equiv \Theta[T] C_{2,d}$. Observe that the first term on the right hand side is strictly causal, whereas the second term arising from the initial condition is retarded but acausal because it contributes a non-zero signal outside the past null cone. As additional Minkowski and cosmological examples below will further corroborate, the `time-integral' method not only allows us to compute (up to quadrature) the retarded part of $C_{2,d}$ from the known solutions of the massless scalar causal Green's functions, it provides a clean separation between the strictly causal versus the retarded-but-acausal terms arising from the initial conditions -- even if the time-integrals themselves cannot be performed analytically.
{\it Exact solutions in even dimensions $d\geq4$} \qquad The retarded and advanced Green's functions of a massless scalar in 4D Minkowski are
\begin{align}
G^\pm_4 [x-x'] = - \frac{\delta[T \mp R]}{4\pi R} .
\end{align}
The $\delta$-function teaches us that $G^+_4$ propagates signals strictly on the forward null cone; and the $G^-_4$ strictly on the backward null cone. From eq.~\eqref{C1_Minkowski_RetardedMinusAdvanced}, $C_{1,4}$ thus reads:
\begin{align}
C_{1,4}[T,R]
=\frac1{4\pi R}\left(\delta[T-R] - \delta[T+R]\vphantom{\dot T}\right) .
\label{C14_Minkowski}
\end{align}
Of course, $C_{1,4}$ can be worked out straightforwardly from eq.~\eqref{C1_Minkowski} by setting $d=4$.
To compute $C_{2,4}$, on the other hand, we insert eq.~\eqref{C14_Minkowski} into eq.~\eqref{TimeIntegralMethod_C2d} and obtain
\begin{align}
C_{2,4}[T,R]&=\frac1{4\pi}\left(\Theta[T-R]-\Theta[-T-R]\vphantom{\dot T}\right)+\frac T{4\pi R}\left(\Theta[T]\Theta[-T+R]+\Theta[-T]\Theta[T+R]\vphantom{\dot T}\right),
\label{C24_Minkowski}
\end{align}
We may check this result by tackling eq.~\eqref{C2_Minkowski} directly. After integrating over the angular coordinates in $\vec{k}-$space,
\begin{align}
C_{2,4}[T,R]
&= \int_{-\infty}^{+\infty} \frac{\text{d} k}{(2\pi)^{2}}
\frac{\sin kT }{k} \frac{\sin kR }{kR} .
\end{align}
The sines can be converted into exponentials; and because there are no singularities the contour on the real line may be displaced slightly toward the positive or negative imaginary $k-$axis near $k=0$. The resulting expression would consist of $4$ terms, each of which would now be amendable to the residue theorem by closing the contour appropriately in the lower or upper half complex $k$ plane.
From the retarded portion of eq.~\eqref{C24_Minkowski}, we find that the contribution of $C_{2,4}$ comes from both inside and outside the past light cone of the observer; however, the signal that resides within the light cone -- its ``tail" -- is a spacetime constant and will therefore be removed by the spatial derivatives in eq.~\eqref{G+ij_Minkowski}. In contrast, the acausal one with $T<R$ still remains and does contribute to the 4D effective Green's function $G_{ij}^+$, along with some additional light-cone contributions from differentiating the step functions in $C_{2,4}$.
\begin{align}
G_{ij}^{(+,\text{4D})}[T, \vec R]
&= - \left( \delta_{ij} - \widehat R_i \widehat R_j \right) \frac{\delta[T-R]}{4\pi R}
- \Theta[T] \Theta[-T+R] \partial_i \partial_j \left( \frac{T}{4\pi R} \right) .
\end{align}
To sum: the 4D effective Green's function $G^{(+,\text{4D})}_{ij}[x-x']$ propagates signals on and outside the forward light cone of the source at $x'$ -- namely, it is acausal.
With the 4D solutions in eqs.~\eqref{C14_Minkowski} and \eqref{C24_Minkowski} at hand, we may employ eqs. \eqref{C1_EvenMinkowski_Recursion} and \eqref{C2_EvenMinkowski_Recursion} to state:
\begin{align}
C_{1, \text{even }d\geq4}^+[T,R]&=\mathcal D^{\frac{d-4}2}_R\left(\frac{\delta[T-R]}{4\pi R}\right) \label{C1_Even_Minkowski} ,
\end{align}
where $C_{1,d}^+ \equiv \Theta[T] C_{1,d}$, and
\begin{align}
C_{2,\text{even }d\geq4}^+[T,R]&=\mathcal D^{\frac{d-4}2}_R \left( \Theta[T-R]\,\frac1{4\pi} + \Theta[T]\Theta[-T+R]\,\frac T{4\pi R} \right) ;
\label{C2_Even_Minkowski}
\end{align}
where only the retarded contributions are shown, and we highlight that the tail portion of eq.~\eqref{C24_Minkowski} in higher dimensions $d\geq6$ partially cancels the acausal part of it upon differentiation. Like the 4D case, the $T \mathcal{D}_R^{\frac{d-4}{2}} (4\pi R)^{-1}$ part of $C_{2,\text{even }d\geq4}^+$ indicates the latter continues to receive acausal contribution from outside the light cone.
{\it Exact solutions in odd dimensions $d\geq3$} \qquad Odd-dimensional solutions differ from even ones due to the presence of inside-the-null-cone propagation -- ``tail" signals. In $(2+1)-$dimensions, the retarded Green's function of the massless scalar is
\begin{align}
C^+_{1,3}[T,R]=-G^+_{3}[T,R]=\frac{\Theta[T-R]}{2\pi\sqrt{T^2-R^2}},
\label{C13_Minkowski}
\end{align}
which, unlike the 4D case, is pure tail. We now turn to solving the integral $C_{2,3}$, which, as we have reasoned earlier, is expected to blow up when considered alone, but its divergent piece does not really enter the physical spin-1 Green's function $G^+_{ij}$, as it will be eliminated by the two spatial derivatives $\partial_i \partial_j$ in eq.~\eqref{G+ij_Minkowski}. Despite being divergent, $C_{2,3}$ can nonetheless be regularized to a finite expression in the time-integral method, where the divergence only takes place on the initial condition. Within dimensional regularization, the resulting regularized form of it, $C^{(+,\text{reg})}_{2,3+2\epsilon}$, is given by
\begin{align}
C_{2,3+2\epsilon}^{(+,\text{reg})}[T,R]= \Theta[T-R]\, & \frac{-T\ln\left[\mu \left(T+\sqrt{T^2-R^2}\right)\right]+\sqrt{T^2-R^2}}{2\pi} \notag\\
& \qquad \qquad +\Theta[T]\Theta[-T+R]\,\frac{-T\ln [\mu R]}{2\pi} + \Theta[T] \, \frac T{4\pi} \left( \frac1\epsilon - \gamma - \ln[\pi] \right).
\label{C23_Minkowski}
\end{align}
By referring to eq.~\eqref{G+ij_Minkowski}, we see that both tail and the acausal parts of $C_{2,3+2\epsilon}^{\mathrm{(reg)}}$ contribute to the three-dimensional $G^+_{ij}$, with no pure light-cone signals involved.
To further justify the validity of eq.~\eqref{C23_Minkowski}, we independently computed finite $C_{2,5}$ using its Fourier representation in eq.~\eqref{C2_Minkowski}:
\begin{align}
C^+_{2,5}[T,R]&=\Theta[T-R]\,\frac{T-\sqrt{T^2-R^2}}{4\pi^2 R^2} + \Theta[T]\Theta[-T+R]\,\frac T{4\pi^2R^2}.
\label{C25_Minkowski}
\end{align}
This then allows us to verify $C^+_{2,5}=\mathcal D_RC_{2,3+2\epsilon}^{(+,\text{reg})}$ in eq.~\eqref{C2_OddMinkowski_Recursion}. Higher odd-dimensional results follow from eqs.~\eqref{C1_OddMinkowski_Recursion}, \eqref{C2_OddMinkowski_Recursion}, \eqref{C13_Minkowski}, and \eqref{C23_Minkowski}, where we can simply drop the last term of eq.~\eqref{C23_Minkowski} and set the mass scale $\mu$ to one, since they will be removed in $G^+_{ij}$ by the spatial derivatives in eq.~\eqref{G+ij_Minkowski},
{\allowdisplaybreaks
\begin{align}
C^+_{1,\text{odd } d\geq3}[T,R]&=\mathcal D^{\frac{d-3}2}_R\left(\frac{\Theta[T-R]}{2\pi\sqrt{T^2-R^2}}\right),
\label{C1_Odd_Minkowski}\\
C^+_{2,\text{odd } d\geq3}[T,R]&=\mathcal D^{\frac{d-3}2}_R \Bigg(\Theta[T-R]\, \frac{-T\ln\left[ \left(T+\sqrt{T^2-R^2}\right)\right]+\sqrt{T^2-R^2}}{2\pi} +\Theta[T]\Theta[-T+R]\,\frac{-T\ln R}{2\pi} \Bigg) .
\label{C2_Odd_Minkowski}
\end{align}}%
With eqs.~\eqref{C1_Odd_Minkowski} and \eqref{C2_Odd_Minkowski} plugged into eq.~\eqref{G+ij_Minkowski}, we now have the explicit spin-1 Green's function $G^+_{ij}$ for all odd dimensions $d\geq3$. These analytic solutions reveal that, in odd dimensions, the spin-1 photon receives not only the causal tail signals from both $C_{1,d}$ and $C_{2,d}$, with no strictly $\delta$-function light-cone counterpart, but also the acausal contribution from $C_{2,d}[0\leq T<R]$. As a result, we have explicitly shown that, in the presence of the local electromagnetic source, the spin-1 photon being acausal turns out to be a generic feature in any spacetime dimensions $d\geq3$.
{\bf Scalar } \qquad The scalar solution for the gauge-invariant $\Phi$, which obeys the Poisson's equation \eqref{EMPoisson_Minkowski}, is given by a Coulomb-type form,
\begin{align}
\Phi[\eta,\vec x]=\int_{\mathbb{R}^{d-1}} \text{d}^{d-1}\vec{x}'\,G^{(\mathrm{E})}_{d}[R]\, J_0[\eta,\vec x'],
\label{EMPhi_Minkowski}
\end{align}
where we recall that $G^{(\mathrm{E})}_{d}$ takes the form of eq.~\eqref{EuclideanGreenFunction_3D} for $d=3$ and eq.~\eqref{EuclideanGreenFunction} for $d\geq4$. Clearly, this solution manifestly violates causality, in the sense that the scalar $\Phi$ is instantaneously sourced by the local charge density $\rho$. Therefore, neither the spin-1 photon $\alpha_i$ nor the scalar $\Phi$ can be a standalone observable in classical electromagnetism, which then leads us to pose the question: how do these gauge-invariant variables enter the key observable -- the Faraday tensor -- such that the result is causally dependent on their corresponding sources?
{\bf Faraday Tensor} \qquad The causal nature of $F_{\mu\nu}$ can be seen from its own wave equation, derived by taking the divergence of the identity $\partial_{[\mu}F_{\nu\sigma]}=0$ and imposing the Maxwell eqautions \eqref{Maxwell_equations},
\begin{align}
\Box F_{\mu\nu} + R_{\rho\sigma\mu\nu}F^{\rho\sigma}+2R^{\sigma}{}_{[\mu}F_{\nu]\sigma}=-2\nabla_{[\mu}J_{\nu]}, \qquad\qquad \Box \equiv \nabla_\sigma\nabla^\sigma .
\label{WaveEq_FaradayTensor}
\end{align}
In Minkowski spacetime, the geometric tensors vanish and the electromagnetic fields encoded within $F_{\mu\nu}$ are thus given by the massless scalar Green's function convoluted against the first derivatives of the electromagnetic sources,
\begin{align}
F_{\mu\nu}[x]=-2\int_{\mathbb{R}^{d-1,1}} \text{d}^dx'\,\partial_{[\mu}G_d^+J_{\nu]}[x'],
\label{FieldStrength_Sol_Minkowski}
\end{align}
Here, we have dropped the surface terms at infinity when integrating by parts, which can be justified by the causal structures of eqs.~\eqref{C1_Even_Minkowski} and \eqref{C1_Odd_Minkowski} as well as the fact that those at past infinity ($T\to\infty$) in odd dimensions are negligible (see eq.~\eqref{C1_Odd_Minkowski}).
Let us now recover eq.~\eqref{FieldStrength_Sol_Minkowski} within the gauge-invariant formalism. We first make use of the conservation law for the electromagnetic current, $\partial_{j}J_{j}=\dot{J}_0$, to re-write the spin-1 expression \eqref{Spin1_Convolution_Minkowski},
\begin{align}
\alpha_i[x]=-\int_{\mathbb{R}^{d-1,1}} \text{d}^dx'\,\left(G^+_{d}J_i[x']-\Theta[T]\,\partial_i\dot{C}_{2,d}J_0[x']\right),
\label{Spin1_Convolution2_Minkowski}
\end{align}
where the second term is now the convolution with the charge density $J_0$. The surface terms from integration by parts -- namely, $\int \text{d}\eta'\,\text{d}^{d-2}\vec x'\,\Theta[T]\partial_iC_{2,d}J_{j}$ evaluated at spatial infinity and $\int \text{d}^{d-1}\vec{x}'\,\Theta[T]\partial_iC_{2,d}J_0$ at past infinity -- have been neglected, as the former falls off as $R\to \infty$ in both even and odd dimensions, and the electric current is assumed to be isolated; whereas the latter, when $T\to\infty$, has zero contribution in even dimensions and becomes negligible in odd dimensions (see eqs.~\eqref{C2_Even_Minkowski} and \eqref{C2_Odd_Minkowski}). The magnetic field, according to eq.~\eqref{Maxwell_FaradayTensor}, is therefore consistent with eq.~\eqref{FieldStrength_Sol_Minkowski}:
\begin{align}
F_{ij}[x] = 2\partial_{[i} \alpha_{j]}=-2\int_{\mathbb{R}^{d-1,1}} \text{d}^dx'\,\partial_{[i}G^+_d\,J_{j]}[x'].
\label{Fij_Minkowski}
\end{align}
This calculation shows that, despite $\alpha_i$ being acausal, taking the curl of the spin-1 field ends up removing its acausal information encoded in the second term of eq.~\eqref{Spin1_Convolution2_Minkowski}.
According to eq.~\eqref{Maxwell_FaradayTensor}, the electric field $F_{0i}$ is the sum of $\dot{\alpha}_i$ and $\partial_i \Phi$. Employing $\dot C_{2,d}\big|_{T=0}=-G^{\mathrm{(E)}}_d$ and $\ddot{C}_{2,d}=-C_{1,d}$ in eqs.~\eqref{C12dot_Minkowski_T=0} and \eqref{C1ddot_Minkowski}, the time derivative of eq.~\eqref{Spin1_Convolution2_Minkowski} is
\begin{align}
\dot\alpha_i[x]=-\int_{\mathbb{R}^{d-1,1}} \text{d}^dx'\left(\dot G^+_{d}J_i[x']-\partial_iG^+_{d}J_0[x']\right)-\int_{\mathbb{R}^{d-1}} \text{d}^{d-1}\vec{x}'\,\partial_i G^{(\mathrm{E})}_{d}J_0[\eta,\vec x'] .
\label{DotSpin1_Minkowski}
\end{align}
We see that the acausal term containing $G^{(\mathrm{E})}_{d}$ in eq.~\eqref{DotSpin1_Minkowski} is canceled by adding to $\dot{\alpha}_i$ the spatial gradient of eq.~\eqref{EMPhi_Minkowski}. The result is
\begin{align}
F_{0i}[x]=\dot\alpha_i[x]+\partial_i\Phi[x]=-\int_{\mathbb{R}^{d-1,1}} \text{d}^dx'\,\left(\dot G^+_{d}J_i[x']-\partial_iG^+_{d}J_0[x']\right) .
\label{F0i_Minkowski}
\end{align}
which, again, agrees exactly with eq.~\eqref{FieldStrength_Sol_Minkowski}. To sum: the spin-1 photon $\alpha_i$ contains all relevant electromagnetic information, but is acausal. On the other hand, the primary role of $\Phi$ is to cancel the acausal part of $\dot\alpha_i$, rendering the electric field $F_{0i}$ strictly causal. In other words, the electric field turns out to be determined by the causal portion of the velocity of the transverse spin-1 field,
\begin{align}
F_{0i}=\big(\dot{\alpha}_i\big)_{\text{causal}}=\big(\dot A_i^\text{T}\big)_{\text{causal}} .
\label{F0i_CausalSpin1_Minkowski}
\end{align}
Next, we move on to investigate how the spin-1 field and the Faraday tensor behave under certain physically interesting limits.
{\bf Stationary Limit and $\Phi$} \qquad That the electric field in eq.~\eqref{F0i_CausalSpin1_Minkowski} is the causal piece of $\dot{\alpha}_i$ reveals a subtlety in the stationary limit, where the electric current is time independent. For, the first term containing $\dot{G}_d^+$ in eq.~\eqref{DotSpin1_Minkowski} integrates to zero, which then informs us that
\begin{align}
\dot\alpha_i[\vec{x}]
=\int_{\mathbb{R}^{d-1}} \text{d}^{d-1}\vec{x}'\partial_i G^{(\mathrm{E})}_{d}[\vec{x}-\vec{x}'] J_0[\vec{x}']-\int_{\mathbb{R}^{d-1}} \text{d}^{d-1}\vec{x}'\,\partial_i G^{(\mathrm{E})}_{d}[\vec{x}-\vec{x}']J_0[\vec x'] = 0 ;
\label{alphadot_GJ-GJ}
\end{align}
because in the second term of eq.~\eqref{DotSpin1_Minkowski},
\begin{align}
\int_{\mathbb{R}} \text{d} \eta' G^+_{d}[\eta-\eta',\vec{x}-\vec{x}']
= G^{(\mathrm{E})}_{d}[\vec{x}-\vec{x}'] .
\end{align}
In words: within the stationary limit, the causal structure of $\dot{\alpha}_i$ itself becomes degenerate -- the otherwise causal and acausal terms in eq.~\eqref{DotSpin1_Minkowski} cancel one another.
At first sight, eq.~\eqref{EMPoisson_Minkowski} appears to tell us $\Phi$ is the Coulomb potential of a static charge distribution. This seems to be further reinforced by the fact that $\partial_i \Phi$ from eq.~\eqref{EMPhi_Minkowski} is the sole contribution to the electric field $F_{0i}$ in eq.~\eqref{F0i_Minkowski}, since $\dot{\alpha}_i = 0$. But the interpretation that $\partial_i \Phi$ is (the dominant piece of) the electric force becomes erroneous once there is the mildest non-trivial time dependence in the electric current -- as already pointed out -- because $\Phi$ is purely acausal and hence cannot be a standalone physical observable. Instead, the gradient of eq.~\eqref{EMPhi_Minkowski} cancels the (normally acausal) second term in the first equality of eq.~\eqref{alphadot_GJ-GJ} and thus eq.~\eqref{F0i_CausalSpin1_Minkowski} continues to hold:
\begin{align}
F_{0i}[\vec x] = \big(\dot{\alpha}_i \big)_\text{causal}
= \int_{\mathbb{R}^{d-1}} \text{d}^{d-1}\vec{x}'\partial_i G^{(\mathrm{E})}_{d}[\vec{x}-\vec{x}'] J_0[\vec{x}'] .
\end{align}
{\bf Far-Zone Limit} \qquad Provided that the observer is very far from the isolated sources, the leading-order term of the field, which scales as $1/r^{\frac d2-1}$, corresponds to the radiative piece that is capable of carrying energy-momentum to infinity. To extract the leading contribution of the spin-1 field in such a regime, firstly, we re-express the spin-1 commutator $C_{ij}$, by explicitly carrying out the spatial derivatives assuming $R\neq0$, in the following form\footnote{At $R=0$, when two spatial derivatives $\partial_i\partial_j$ act on $1/R^{d-3}$, terms involving $\delta^{(d-1)}[\vec x-\vec x']$ could arise. However, if the observer is away from the source, then those local terms will not contribute to the effective Green's function, and therefore, we can simply ignore them in the calculation.}
\begin{align}
C_{ij}[T,\vec R]= P_{ij}[\vec R]\,C_{1,d}[T,R]+\Pi_{ij}[\vec R]\,2\pi C_{2,d+2}[T,R],
\label{Cij_FarZone_Minkowski}
\end{align}
where the projection tensor $P_{ij}$ and $\Pi_{ij}$, respectively, are defined as
\begin{align}
P_{ij}[\vec R]&\equiv\delta_{ij}-\widehat{R}_i\widehat{R}_j,
\label{TransverseProjector_Position} \\
\Pi_{ij}[\vec R]&\equiv -\delta_{ij}+(d-1)\widehat{R}_i\widehat{R}_j ;
\end{align}
the $\widehat R \equiv (\vec x-\vec x')/|\vec x-\vec x'|$ is pointing from the source location $\vec x'$ to the observer $\vec x$; and $C_{2,d+2}$ in the second term of eqs.~\eqref{Cij_FarZone_Minkowski} is the $(d+2)$-dimensional form of eqs.~\eqref{C2_EvenMinkowski_Recursion} and \eqref{C2_OddMinkowski_Recursion} but its $R$ is the one in $d-1$ spatial dimensions. Note also that, to reach eq.~\eqref{Cij_FarZone_Minkowski}, we have employed the homogeneous wave equation, $\partial^2C_{2,d}=0$, with $R\neq0$, as well as the conversion $\ddot C_{2,d}=-C_{1,d}$, to relate its second spatial derivatives to $C_{1,d}$. Also, it can be checked directly that the expression \eqref{Cij_FarZone_Minkowski} for $R\neq0$ is indeed divergenceless. Altogether, as long as the observer is away from the source, we have an alternative expression for the spin-1 effective Green's function by inserting eq.~\eqref{Cij_FarZone_Minkowski} into $G^+_{ij}=-\Theta[T]C_{ij}$. The purpose of putting $C_{ij}$ in this form is that, the dominant far-zone contribution of the field can be extracted simply by comparing $C_{1,d}$ and $C_{2,d+2}$. Furthermore, each term in eq.~\eqref{Cij_FarZone_Minkowski} is manifestly finite for all spacetime dimensions in which photons exist, since there is no divergence incurred in $C_{1,d}$ and $C_{2,d+2}$ for $d\geq3$.
If $\tau_c$ and $r_c$ are respectively the characteristic time scale and proper size of the source, and $r$ is the observer-source distance, the far zone is defined as the limits $\tau_c/r \ll 1$ and $r_c/r \ll 1$. To perform this limit on the Green's function, we will work in frequency $\omega-$space. Specifically, the far zone then translates into $|\omega| r\gg1$. We shall be content in extracting the leading expressions in the limit $|\omega| R\gg1$.
In terms of the superposition of individual frequencies, the spin-1 field $\alpha_i$ can be written as
\begin{align}
\label{FrequencyTransformOfGij}
\alpha_i[\eta,\vec x]=-\int_{\mathbb{R}^{d-1}} \text{d}^{d-1}\vec{x}'\int_{\mathbb{R}} \frac{\text{d}\omega}{2\pi}\,\widetilde{G}^+_{ij}[\omega,\vec R]e^{-i\omega \eta}\widetilde{J}_j[\omega,\vec{x}'],
\end{align}
with $\widetilde{G}^+_{ij}[\omega,R]$ being the frequency transform of the spin-1 effective Green's function,
\begin{align}
\widetilde{G}^+_{ij}[\omega,\vec R]&=\int_{\mathbb{R}} \text{d} T\,G^+_{ij}[T,\vec R]\,e^{i\omega T} \notag\\
&= P_{ij}[\vec R]\,\widetilde{G}^+_{d}[\omega,R] - \Pi_{ij}[\vec R]\,2\pi\widetilde{C}^+_{2,d+2}[\omega,R],
\label{Gij_FrequencyTransform}
\end{align}
where we have assumed $R\neq0$ and used the expression \eqref{Cij_FarZone_Minkowski}. The $\widetilde{G}^+_{d}[\omega,R]$ and $\widetilde{C}^+_{2,d+2}[\omega,R]$, respectively, denote the frequency transforms of the massless scalar Green's function $G^+_d$ and $C^+_{2,d+2}$.\footnote{The $\widetilde{G}_d^+[\omega,R]$ in eq.~\eqref{Gij_FrequencyTransform} is the frequency transform of $G_{d}^+[\eta-\eta';R]$; this is to be distinguished from $\widetilde{G}_d^+[\eta,\eta';\vec{k}]$ in eq.~\eqref{Spin1_Fourier}, which is the spatial-Fourier transform of the same $G_{d}^+[\eta-\eta';R]$.}
{\it Spin-1 photons in even dimensions $d\geq4$} \qquad A direct calculation starting from eqs.~\eqref{C1_Even_Minkowski} and \eqref{C2_Even_Minkowski} tells us, in all even $d = 4 + 2n \geq 4$,
\begin{align}
\widetilde{G}^+_{4+2n}[\omega,R]& = -\mathcal{D}_R^n \left( \frac{e^{i\omega R}}{4\pi R} \right) = -\frac{i\omega^{2n+1}}{2(2\pi)^{n+1}(\omega R)^n}h^{(1)}_n[\omega R],
\label{C1_Even_FrequencySpace} \\
\widetilde{C}^+_{2,6+2n}[\omega,R]&=\mathcal D^n_R\left(\frac{e^{i\omega R}}{8\pi^2 R^2(i\omega)}-\frac{e^{i\omega R}}{8\pi^2 R^3(i\omega)^2}\right) - \frac{(2n+1)!!\,\omega^{2n+1}}{2(2\pi)^{n+2}(\omega R)^{2n+3}},
\label{C2_Even_FrequencySpace}
\end{align}
where $h^{(1)}_n$ is the spherical Hankel function of the first kind. Notice that the last term in eq.~\eqref{C2_Even_FrequencySpace} does not contain the factor $e^{i\omega R}$; it describes the non-propagating portion of the signal in frequency space, which in turn arises from the acausal effect found in position space. In the limit $|\omega|R\gg1$, eqs.~\eqref{C1_Even_FrequencySpace} and \eqref{C2_Even_FrequencySpace} behave asymptotically as
\begin{align}
\widetilde{G}^+_{4+2n}[\omega,R]&= \frac{(-1)^{n+1}i^n\omega^{2n+1}}{2(2\pi\omega R)^{n+1}}\,e^{i\omega R}\left(1+\mathcal{O}\bigg[\frac1{\omega R}\bigg]\right),
\label{C1_Even_FarZone_FrequencySpace}\\
\widetilde{C}^+_{2,6+2n}[\omega,R]&=\frac{(-1)^{n+1}i^n\omega^{2n+1}}{2(2\pi\omega R)^{n+1}}\,e^{i\omega R}\cdot\frac{i}{2\pi \omega R}\left(1+\mathcal{O}\left[\frac1{\omega R}\right]\right),
\label{C2_Even_FarZone_FrequencySpace}
\end{align}
which reveals that, for any fixed dimension $d=4+2n$ and at leading order, the acausal $\widetilde{C}^+_{2,6+2n}[\omega,R]$ term is suppressed as $1/(\omega R)$ relative to $\widetilde{G}^+_{4+2n}$.\footnote{Strictly speaking, eq.~\eqref{C1_Even_FarZone_FrequencySpace} applies only for $n > 0$. There are no $1/(\omega R)$ corrections in (3+1)-dimensions, because eq.~\eqref{C1_Even_FrequencySpace} informs us that $\widetilde{G}^+_{4}[\omega,R] = - e^{i\omega R}/(4\pi R)$.} Therefore, at leading $1/(\omega R)$ order, the effective Green's function $\widetilde{G}^+_{ij}[\omega,\vec R]$, in frequency space, is exclusively dependent on $\widetilde{G}^+_{d}[\omega,R]$; moreover, with the assumption $r_c/r\ll1$, its far-zone leading contribution can be extracted from the first term of eq.~\eqref{Gij_FrequencyTransform}, which is given by
\begin{align}
\widetilde{G}^+_{ij}[\omega,\vec R]&=P_{ij}\widetilde{G}^{(+,\,\text{fz})}_{4+2n}[\omega;\vec x,\vec x']\left(1+\mathcal{O}\left[\frac1{\omega r},\frac{r_c}{r}\right]\right),
\end{align}
where $P_{ij}$ is the far-zone spatial projector defined in eq.~\eqref{t_Projector} and
\begin{align}
\widetilde{G}^{(+,\,\text{fz})}_{4+2n}[\omega;\vec x,\vec x']\equiv\frac{(-1)^{n+1}(i\omega)^n}{2(2\pi)^{n+1}}\frac{e^{i\omega (r-\vec{x}'\cdot\widehat r)}}{r^{n+1}} .
\label{G+_FrequencySpace_Even_FarZone}
\end{align}
By performing the inverse frequency transform, in the far-zone radiative limit, the transverse spin-1 photon $\alpha_i=A^{\text T}_i$ reduces to a transverse projection in space:
\begin{align}
\lim_{r\to\infty} \alpha_i \to A_i^\text t, \qquad A_i^\text t[x]\equiv P_{ij}\left(-\int_{\mathbb{R}^{d-1,1}} \text{d}^{d}x'\,G^{(+,\,\text{fz})}_{d}[T;\vec x,\vec x']J_j[x']\right),
\label{At_FarZone}
\end{align}
where $G^{(+,\,\text{fz})}_{d}[T;\vec x,\vec x']$ is the far-zone contribution of the massless scalar Green's function,
\begin{align}
G^{(+,\,\text{fz})}_{d}[T;\vec x,\vec x']&=\int_{\mathbb{R}} \frac{\text{d}\omega}{2\pi}\,\widetilde{G}^{(+,\,\text{fz})}_{d}[\omega;\vec x,\vec x']e^{-i\omega T} \notag\\
&=-\frac{1}{2(2\pi r)^{\frac{d-2}2}}\left(\frac{\partial}{\partial \eta}\right)^\frac{d-4}2\!\delta\big[T-r+\vec{x}'\cdot\widehat r\big]. \quad(\text{even }d)
\label{G+_Even_FarZone}
\end{align}
{\it Spin-1 photons in odd dimensions $d\geq3$} \qquad For $d=3+2n$, we can frequency transform the retarded position-space solutions \eqref{C1_Odd_Minkowski} and \eqref{C2_Odd_Minkowski} to obtain
\begin{align}
\widetilde{G}^+_{3+2n}[\omega>0,R]&=-\frac{i\omega^{2n}}{4(2\pi\omega R)^n}H^{(1)}_n[\omega R],
\label{C1_Odd_FrequencySpace} \\
\widetilde{C}^+_{2,5+2n}[\omega>0,R]&= \frac{i\omega^{2n}}{4(2\pi\omega R)^{n+1}}H^{(1)}_{n+1}[\omega R] - \frac{2^nn!\,\omega^{2n}}{(2\pi)^{n+2}(\omega R)^{2n+2}},
\label{C2_Odd_FrequencySpace}
\end{align}
where $H^{(1)}_n$ is the Hankel function of the first kind and its differential recursion relation has been employed, and these expressions are only valid for positive frequencies $\omega>0$; however, since $G^+_{d}$ and $C^+_{2,d+2}$ are real, the negative-frequency modes can be expressed in terms of the complex conjugates of eqs.~\eqref{C1_Odd_FrequencySpace} and \eqref{C2_Odd_FrequencySpace}, $\widetilde{G}^+_{d}[-\omega,R]=\widetilde{G}^{+*}_{d}[\omega,R]$ and $\widetilde{C}^+_{2,d+2}[-\omega,R]=\widetilde{C}^{+*}_{2,d+2}[\omega,R]$, where the asterisk ``$*$'' denotes complex conjugation. As in the even-dimensional case, the non-propagating piece of signals also shows up in the second term of eq.~\eqref{C2_Odd_FrequencySpace}. At leading $1/(\omega R)$ order, the Hankel function goes asymptotically to $H^{(1)}_n[\omega R]=\sqrt{2/\pi\omega R}\,e^{i(\omega R-n\pi/2-\pi/4)}+\mathcal{O}\left[1/(\omega R)^{3/2}\right]$, so
we can read off the leading-order pieces of eqs.~\eqref{C1_Odd_FrequencySpace} and \eqref{C2_Odd_FrequencySpace} accordingly,
\begin{align}
\widetilde{G}^+_{3+2n}[\omega>0,R]&=-\frac{i\omega^{2n}}{2(2\pi\omega R)^{n+\frac12}}e^{i\left(\omega R-\frac{n\pi}2-\frac{\pi}4\right)}\left(1+\mathcal{O}\bigg[\frac1{\omega R}\bigg]\right),
\label{G+_Odd}\\
\widetilde{C}^+_{2,5+2n}[\omega>0,R]&=-\frac{i\omega^{2n}}{2(2\pi\omega R)^{n+\frac12}}e^{i\left(\omega R-\frac{n\pi}2-\frac{\pi}4\right)}\cdot\frac i{2\pi \omega R}\Bigg(1+\mathcal{O}\left[\frac1{\omega R}\right]+\mathcal O\left[\frac1{(\omega R)^{n+\frac12}}\right]\Bigg),
\label{C2+_Odd}
\end{align}
from which we infer that, unlike the even-dimensional results, amplitudes of these tail signals contain fractional powers of frequencies. And, these asymptotic behaviors tell us that, in the far-zone regime $|\omega| R\gg1$, the massless scalar Green's function $\widetilde{G}^+_{d}[\omega,R]$ still dominates over $\widetilde{C}^+_{2,d+2}[\omega,R]$ here. Hence, we can extract the far-zone leading order in $1/r$ piece of $\widetilde{G}^+_{ij}[\omega,\vec R]$ in the same way,
\begin{align}
\label{Gij+_FrequencySpace_Odd_FarZone}
\widetilde{G}^+_{ij}[\omega>0,\vec R]=P_{ij}\widetilde{G}^{(+,\,\text{fz})}_{3+2n}[\omega;\vec x,\vec x']\left(1+\mathcal{O}\left[\frac1{\omega r},\frac{r_c}{r}\right]\right),
\end{align}
with $\widetilde{G}^{(+,\,\text{fz})}_{3+2n}$ defined by
\begin{align}
\widetilde{G}^{(+,\,\text{fz})}_{3+2n}[\omega;\vec x,\vec x']\equiv-\frac{i\omega^{n-\frac12}}{2(2\pi)^{n+\frac12}}\frac{e^{i\left(\omega (r-\vec{x}'\cdot\widehat r)-\frac{n\pi}2-\frac{\pi}4\right)}}{r^{n+\frac12}}.
\label{G+_FrequencySpace_Odd_FarZone}
\end{align}
Consequently, in the radiative limit $r\to\infty$, we reach the same conclusion stated in eq.~\eqref{At_FarZone} for odd dimensions $d=3+2n$ as well, where $G^{(+,\,\text{fz})}_{d}$ is given instead by
\begin{align}
G^{(+,\,\text{fz})}_{d}[T;\vec x,\vec x']=-\frac{1}{(2\pi r)^{\frac{d-2}2}}\,\mathrm{Re}\left[\int^{\infty}_{0} \frac{\text{d}\omega}{2\pi}\,i\omega^{\frac{d-4}2}e^{i\left(\omega (r-\vec{x}'\cdot\widehat r-T)-\frac{(d-2)\pi}4\right)}\right]. \quad (\text{odd }d)
\label{G+_Odd_FarZone}
\end{align}
Thus, based on the frequency-space analysis, the fact that $\alpha_i\equiv A_i^\text T\to A_i^\text t$ as $r\to\infty$ holds generically in any spacetime dimensions $d\geq3$, clearly demonstrating that the acausal portion of the spin-1 field actually contributes negligibly to the far-zone signals.
{\it Summary: Far Zone Transverse Green's Functions} \qquad To sum, the massless spin-1 transverse photon $\alpha_i$ in the radiative regime will coincide with another notion of the ``transverse'' vector potential $A^\text t_i$. While $\alpha_i$ in eq.~\eqref{Spin1Photon_Minkowski} involves a transverse projection in Fourier space, the $A_i^\text{t}$ in eq.~\eqref{At_FarZone} is a local-in-space transverse projection of the far-zone Lorenz-gauge causal solution for the vector potential, i.e., $\widehat{r}^i A_i^\text{t} = 0$, which consists solely of the light-cone signals in even dimensions. In 4D Minkowski spacetime, that the two different notions of the transverse vector potential overlap in the far zone has already been pointed out in Refs.~\cite{Ashtekar:2017wgq,Ashtekar:2017ydh}. The method used here allows us to generalize the conclusion to all dimensions.
{\it Faraday tensor} \qquad Since we have already shown the spin-1 photons $\alpha_i$, for all $d\geq3$, reduce asymptotically to causal $A_i^\text t$ in the far zone, we then expect in this regime the magnetic and electric fields, eqs.~\eqref{Fij_Minkowski} and \eqref{F0i_CausalSpin1_Minkowski}, to become
\begin{align}
F_{ij}&\approx -2\,\widehat r_{[i}\dot A^{\text t}_{j]}, \label{Fij_FarZone}\\
F_{0i}&\approx \dot A^{\text t}_i .
\label{F0i_FarZone}
\end{align}
Here, the far-zone limit has been taken on both sides of the equations, and we have also used the results in eqs. \eqref{C1_Even_FarZone_FrequencySpace}, \eqref{C2_Even_FarZone_FrequencySpace}, \eqref{G+_Odd} and \eqref{C2+_Odd} to deduce -- as far as the leading-order contribution is concerned -- the replacement rule $\partial_i=-\partial_{i'}=-\widehat R_i\partial_0$ holds at the leading $1/(\omega R)$ level, and after which the dominant far-zone contribution in terms of $r$ can be extracted readily. In addition, the far-zone expressions \eqref{Fij_FarZone} and \eqref{F0i_FarZone} can also be checked for consistency through eq.~\eqref{FieldStrength_Sol_Minkowski}, by using the replacement rule as well as the conservation law for the charge current.
{\bf Commutator of Spin-1 Photons} \qquad As already alluded to, the results for the retarded Green's function of the massless spin-1 $\alpha_i$ are intimately related to the commutator of these photon operators in Quantum Field Theory. Let us first consider a free scalar field $\phi$ as a simple example. Its commutator is related to $C_{1,d}$ in the following manner:
\begin{align}
\big[\phi[x],\phi[x']\big]=-iC_{1,d}[T,R].
\end{align}
According to eq.~\eqref{C1_Minkowski_RetardedMinusAdvanced}, since the retarded/advanced Green's functions on the right hand side are strictly zero outside the null cone, this $C_{1,d}$ consists of only causal information -- i.e., it too is zero whenever the two spacetime points are spacelike: $(x-x')^2>0$. In contrast, because the spin-1 Green's functions are non-zero outside the light cone, according to eq.~\eqref{Cij_Minkowski_RetardedMinusAdvanced}, the non-interacting spin-1 commutator is therefore acausal:
\begin{align}
\big[\alpha_i[x],\alpha_j[x']\big]=-iC_{ij}[T,\vec R].
\label{Spin1_Commutator}
\end{align}
In Quantum Field Theory, operators that commute outside the light cone are said to obey micro-causality. Free spin-1 photons are therefore seen to violate micro-causality. It is likely that this acausal character of their commutator is a manifestation of the known tension between Lorentz covariance and gauge invariance when constructing massless helicity-1 theories in flat spacetime.
\subsection{Linearized Gravitation}
We now turn to the linearized theory of General Relativity in a Minkowski background, as described in \S\eqref{Chapter_Observables}. The relevant Green's functions will be computed analytically for all spacetime dimensions $d\geq4$; i.e., excluding those without spin-2 degrees of freedom.
{\bf Field Equations} \qquad The gauge-invariant form of the linearized Einstein's equations can be expressed in terms of the variables defined in eqs.~\eqref{Bardeen_Psi}, \eqref{Bardeen_Phi}, and \eqref{Bardeen_VandDij}, where, as a constrained system, the full set of gauge-invariant field equations can be reduced to four fundamental ones, i.e., eq.~\eqref{PhiPsiRelationship} and the following three \cite{Chu:2016ngc}:
{\allowdisplaybreaks
\begin{align}
(d-2)\vec\nabla^2\Psi&=8\pi G_\text N\rho, \label{Psi_Poisson_Minkowski}\\
\vec\nabla^2 V_i&=-16\pi G_\text N \Sigma_i, \label{Vi_Poisson_Minkowski}\\
\partial^2D_{ij}&=-16\pi G_\text N\sigma_{ij}; \label{Spin-2_WaveEquation_Minkowski}
\end{align}}%
where the source terms $\rho$, $\Sigma_i$, and $\sigma_{ij}$ refer to different parts of the scalar-vector-tensor decomposition of the astrophysical stress-energy tensor $^{(\text{a})}T_{\mu\nu}$ (cf.~eqs.~\eqref{Astro_SVT_IofII} and \eqref{Astro_SVT_IIofII}). These four independent equations, along with the law of energy-momentum conservation, $\partial^\mu\, ^{(\text{a})}T_{\mu\nu}=0$, already imply the other three remaining ones in the linearized Einstein's equations. We see that only the spin-2 graviton field, $D_{ij}\equiv \chi^{\text{TT}}_{ij}$, admits dynamical wave solutions, sourced by the TT portion of $^{(\text{a})}T_{\mu\nu}$, while the Bardeen scalar potential $\Psi$, as well as the vector mode $V_i$, obey Poisson-type ones. This set of equations appear to be similar to their electromagnetic counterparts, and thus, by the same arguments used earlier, we already expect these gauge-invariant variables to be acausal in nature once the GW sources are taken into account. In this sense, none of these gauge-invariant variables -- including the spin-2 $D_{ij}$ -- may be regarded as a standalone observable. Indeed, as we will see in the subsequent discussion, the linearized Riemann tensor $\delta_1 R_{0i 0j}$, discussed in \S\eqref{Chapter_Observables}, in close analogy to the field strength $F_{\mu\nu}$ for electromagnetism, does require all their contributions to become a causal object.
{\bf Spin-2 Gravitons} \qquad The analytic solutions for the effective Green's functions are crucially important for capturing the propagation of wave signals in this linearized system. Here, we start with the massless spin-2 field $D_{ij}$, obeying the wave equation \eqref{Spin-2_WaveEquation_Minkowski}. Since the TT projection of the source takes place locally in Fourier space, as long as $\vec k\neq\vec0$, we can firstly express $D_{ij}$ as
\begin{align}
D_{ij}[\eta,\vec x]=-16\pi G_\text N\int_{\mathbb{R}} \text{d}\eta'\int_{\mathbb{R}^{d-1}} \frac{\text{d}^{d-1}\vec{k}}{(2\pi)^{d-1}}\,\widetilde G^+_{d}[\eta,\eta';\vec k]\,\widehat P_{ijmn}[\vec k]\,
^{(\text{a})}\widetilde{T}_{mn}[\eta',\vec{k}]\, e^{i\vec{k}\cdot\vec{x}},
\label{Spin2_Fourier}
\end{align}
where $\widetilde G^+_d [\eta,\eta';\vec k ]$ is given in eq.~\eqref{FourierG+_Minkowski}, and the spin-2 TT projector $\widehat P_{ijmn}[\vec k]$ is defined in eq.~\eqref{GravitonProjector}. Then, the expression \eqref{Spin2_Fourier}, with each $k_j$ in eq.~\eqref{GravitonProjector_k} replaced by a spatial derivative via $\partial_j \to i k_j$, can be re-written as the spin-2 effective Green's function $G^+_{ijmn}$ convolved against the local stress-energy tensor of the source,
\begin{align}
D_{ij}[x]=-16\pi G_\text N\int_{\mathbb{R}^{d-1,1}} \text{d}^dx'G^+_{ijmn}[T,\vec R]\,^{(\text{a})}T_{mn}[x'],
\label{Spin2_Convolution_Minkowski}
\end{align}
where the spin-2 effective Green's function $G^+_{ijmn}$ is given by the following tensor structure,
\begin{align}
G^+_{ijmn}[T,\vec R]=\,&-\Theta[T]C_{ijmn}[T,\vec R], \notag\\
C_{ijmn}[T,\vec R]=\,&\bigg(\delta_{m(i}\delta_{j)n}-\frac{\delta_{ij}\delta_{mn}}{d-2}\bigg)C_{1,d}[T,R]+\bigg(\delta_{m(i}\partial_{j)}\partial_n+
\delta_{n(i}\partial_{j)}\partial_m \notag\\
& \qquad \qquad -\frac{\delta_{ij}\partial_m\partial_n-\delta_{mn}\partial_i\partial_j}{d-2}
\bigg)C_{2,d}[T,R] +\left(\frac{d-3}{d-2}\right)\partial_i\partial_j\partial_m\partial_nC_{3,d}[T,R],
\label{Cijmn_Minkowski}
\end{align}
with $C_{1,d}$ and $C_{2,d}$ defined previously in eqs.~\eqref{C1_Minkowski} and \eqref{C2_Minkowski}, and $C_{3,d}$ defined by
\begin{align}
C_{3,d}[T,R]\equiv\int_{\mathbb{R}^{d-1}}\frac{\text{d}^{d-1}\vec{k}}{(2\pi)^{d-1}}\frac{\sin |\vec k|T}{|\vec k|^5}e^{i\vec{k}\cdot\vec{R}}.
\label{C3_Minkowski}
\end{align}
Compared with the spin-1 photon case, even though the whole tensor structure of $C_{ijmn}$ here is very different than that of $C_{ij}$ (cf.~eq.~\eqref{G+ij_Minkowski}), the first two terms have structural similarity to $C_{ij}$, and the scalar Fourier integrals $C_{1,d}$ and $C_{2,d}$ have already been dealt with analytically; the only new term that remains to be computed is $C_{3,d}$. Moreover, it can be checked readily that, by employing the relations $\vec\nabla^2 C_{2,d}=-C_{1,d}$ and $\vec\nabla^2 C_{3,d}=-C_{2,d}$ (cf.~eqs.~\eqref{C1_Minkowski}, \eqref{C2_Minkowski}, and \eqref{C3_Minkowski}), the expression \eqref{Cijmn_Minkowski} indeed satisfies the TT conditions $\delta^{ij}G^+_{ijmn}=0$ and $\partial_iG^+_{ijmn}=\partial_jG^+_{ijmn}=0$. We shall soon deploy the time-integral method, which amounts to solving
\begin{align}
\ddot{C}_{3,d} = -C_{2,d} .
\label{C3ddot_Minkowski}
\end{align}
By integrating eq.~\eqref{C3ddot_Minkowski} twice, followed by recalling eq.~\eqref{TimeIntegralMethod_C2d}, we have
{\allowdisplaybreaks
\begin{align}
C_{3,d}[T,R] &= -\int_{0}^{T} \text{d} \tau_2 \int_{0}^{\tau_2} \text{d} \tau_1 \, C_{2,d}[\tau_1,R]
+ T \dot{C}_{3,d}[T=0,R]
+ C_{3,d}[T=0,R], \\
&=\int^T_0 \text{d}\tau_4\int^{\tau_4}_0 \text{d}\tau_3\,\int^{\tau_3}_0 \text{d}\tau_2\int^{\tau_2}_0 \text{d}\tau_1\,C_{1,d}[\tau_1,R]+\frac{T^3}6G^{\mathrm{(E)}}_d[R] + TD_d[R],
\label{TimeIntegralMethod_C3d}
\end{align}}%
where the initial conditions $C_{3,d}[T=0,R]=0$ and $\dot C_{3,d}[T=0,R]=D_d[R]$ have been employed (cf.~eq.~\eqref{C3_Minkowski}), with $D_{d}$ defined by
\begin{align}
D_{d}[R] & \equiv \int_{\mathbb{R}^{d-1}}\frac{\text{d}^{d-1}\vec{k}}{(2\pi)^{d-1}}\frac{e^{i\vec{k}\cdot\vec{R}}}{\vec k^4},
\label{Dd}
\end{align}
whose concrete position-space expressions read
{\allowdisplaybreaks
\begin{align}
D^{(\text{reg})}_{3+2\epsilon}[R] & = -\frac{R^2}{16\pi} \left( \frac1\epsilon - \big(\gamma + \ln[\pi] -1\big) - 2 \ln[\mu R] \right) ,
\label{D3} \\
D^{(\text{reg})}_{4}[R] & = -\frac{R}{8\pi} ,
\label{D4} \\
D^{(\text{reg})}_{5+2\epsilon}[R] & = \frac1{16\pi^2} \left( \frac1\epsilon -\gamma - \ln[\pi] - 2\ln[\mu R] \right) ,
\label{D5} \\
D_{d\geq6}[R] & = \frac{\Gamma\left[\frac{d-5}2\right]}{16\pi^{\frac{d-1}{2}}R^{d-5}} .
\label{Dd_Expression}
\end{align}}%
Note that $D_3$, $D_4$, and $D_5$ are in the dimensional-regularized forms. Finally, parallel to eq.~\eqref{Gij_WaveEquation_Minkowski} in the photon case, acausality encoded in $G^+_{ijmn}$ can be seen at the level of its wave equation,
\begin{align}
\partial^2G^+_{ijmn}[T,\vec R]=\,& \delta[T] \Bigg\{ \bigg(\delta_{m(i}\delta_{j)n}-\frac{\delta_{ij}\delta_{mn}}{d-2}\bigg)\delta^{(d-1)}[\vec x-\vec x']-
\bigg(\delta_{m(i}\partial_{j)}\partial_n+\delta_{n(i}\partial_{j)}\partial_m\notag\\
& \qquad \qquad \qquad \quad -\frac{\delta_{ij}
\partial_m\partial_n-\delta_{mn}\partial_i\partial_j}{d-2}\bigg)G^{\text{(E)}}_{d}[R] +\left(\frac{d-3}{d-2}\right)\partial_i\partial_j\partial_m\partial_nD_{d}[R] \Bigg\}.
\label{Gijmn_WaveEquation_Minkowski}
\end{align}
The last two terms in eq.~\eqref{Gijmn_WaveEquation_Minkowski} correspond to the acausal contributions to the signal attributed to $x' \equiv (\eta',\vec{x}')$, but arising from a non-zero source smeared over the rest of the equal-time spatial hypersurface at $\eta = \eta'$.
To compute $C_{3,d}$ in eq.~\eqref{C3_Minkowski}, we first notice that the integral itself diverges when $d\leq5$, inferred from the power of $|\vec k|$ in the limit $|\vec k|\to0$. However, in the physical spacetime dimensions where spin-2 gravitons exist, namely $d\geq4$, those divergences in $C_{3,4}$ and $C_{3,5}$ are expected to be removed by the multiple spatial derivatives in the spin-2 effective Green's function \eqref{Cijmn_Minkowski}. Furthermore, since the dimension-raising operator still applies to $C_{3,d}$, in either even or odd spacetime dimensions, the lowest-dimensional form is already adequate for us to generate all the higher-dimensional ones.
{\it Exact solutions in even dimensions $d\geq4$} \qquad Since the analytic forms of $C_{1,d}$ and $C_{2,d}$, for all even $d\geq4$, have been obtained in eqs.~\eqref{C1_Even_Minkowski} and \eqref{C2_Even_Minkowski}, we now focus on the $C_{3,d}$ term needed in eq.~\eqref{Cijmn_Minkowski}. Similar methods used to compute $C_{2,d}$ will be performed to tackle this new integral. Here, we start with $C_{3,4}$, the lowest even dimension for the spin-2 graviton to exist. Even though $C_{3,4}$ itself is a divergent integral, we can still extract its regularized finite contribution through the time-integral method, as we did for $C_{2,3}$ in the spin-1 calculation. Utilizing dimensional-regularization, the result turns out to be finite:
\begin{align}
C_{3,4}^{(+,\text{reg})}[T,R]&=\Theta[T-R]\,\frac{-R^2-3T^2}{24\pi}+ \Theta[T] \Theta[-T+R]\,\frac{-3R^2T-T^3}{24\pi R}.
\label{C34_Minkowski}
\end{align}
Similar to $C_{2,4}$, the tail function in eq.~\eqref{C34_Minkowski}, when plugged into eq.~\eqref{Cijmn_Minkowski}, will be eliminated by the spatial derivatives in $G^+_{ijmn}$. Whereas the acausal portion of the signals, from the second term of eq.~\eqref{C34_Minkowski}, will still remain. In addition, this regularized form can be justified by checking whether this expression, with one dimension-rasing operator acting on it, coincides with $C^+_{3,6}$ obtained by a direct contour-integral calculation:
\begin{align}
C^+_{3,6}[T,R]&=\Theta[T-R]\,\frac1{24\pi^2}+\Theta[T]\Theta[-T+R]\,\frac{3R^2T-T^3}{48\pi^2R^3}.
\label{C36_Minkowski}
\end{align}
(One may check, indeed, that $ C_{3,6} = \mathcal{D}_R C^{(+,\text{reg})}_{3,4}$.) With this solution at hand, applying dimension-raising operators to it then produces all the higher even-dimensional results,
\begin{align}
C^+_{3,\text{even }d\geq4}[T,R]&= \mathcal D^{\frac{d-4}2}_R \left( \Theta[T-R]\,\frac{-R^2-3T^2}{24\pi}+ \Theta[T] \Theta[-T+R]\,\frac{-3R^2T-T^3}{24\pi R} \right) .
\label{C3_Even_Minkowski}
\end{align}
Now, plugging eq.~\eqref{C3_Even_Minkowski} for $C_{3,d}$, along with known $C_{1,d}$ and $C_{2,d}$, into the spin-2 effective Green's function \eqref{Cijmn_Minkowski}, we find that the spin-2 causal structure is analogous to that of the spin-1 field. More precisely, for all even dimensions $d\geq4$, no tail signals exist in $G^+_{ijmn}$, but the spin-2 graviton receives not only the light-cone signals but also the acausal ones from both $C_{2,d}$ and $C_{3,d}$.
{\it Exact solutions in odd dimensions $d\geq5$} \qquad For odd spacetime dimensions, we begin with $d=5$, since TT gravitons are non-existent in lower odd dimensions. To calculate $C_{3,5}$, we can make use of the time-integral method to first extract the regularized $C_{3,3}$, namely inserting eq.~\eqref{C13_Minkowski} into eq.~\eqref{TimeIntegralMethod_C3d} along with the dimensional-regularized initial conditions:
\begin{align}
C^{(+,\text{reg})}_{3,3+2\epsilon}[T,R] =\,&\Theta[T-R]\,\frac1{8\pi} \bigg(-\big(T^2-R^2\big)^{\frac32}-\frac1{9}\big(13R^2+2T^2\big)
\sqrt{T^2-R^2} + T\bigg(R^2+\frac23T^2\bigg) \notag\\
& \times \ln\left[\mu\left(T+\sqrt{T^2-R^2}\right)\right]\bigg)+ \Theta[T]\Theta[-T+R] \, \frac{T \ln[\mu R]}{8\pi} \left( R^2 + \frac{2}3 T^2 \right) \notag\\
& - \Theta[T]\frac T{16\pi} \bigg( \bigg( R^2 + \frac23 T^2 \bigg) \left(\frac1\epsilon - \gamma - \ln[\pi] \right) + R^2 \bigg).
\end{align}
By applying the raising operator $\mathcal D_R$ once,
\begin{align}
C^{(+,\text{reg})}_{3,5+2\epsilon}[T,R]=\,&\Theta[T-R] \, \frac1{24\pi^2} \left( \left(2+\frac{T^2}{R^2}\right)\sqrt{T^2-R^2} - 3T\ln\left[\mu\left(
T+\sqrt{T^2-R^2} \right) \right]-\frac{T^3}{R^2}\right) \notag\\
&+\Theta[T]\Theta[-T+R] \, \frac1{24\pi^2} \left(-\frac{T^3}{R^2} - 3T\ln[\mu R]\right) + \Theta[T] \, \frac{T}{16\pi^2}\left( \frac1\epsilon -\gamma - \ln[\pi] \right),
\label{C35_Minkowski}
\end{align}
which is still a regularized expression of the divergent integral $C_{3,5}$, and again, whose validity can be justified in the same way. Through a direct computation of the finite integral $C_{3,7}$, as we did for $C_{2,5}$, we can show that the resulting expression of $C_{3,7}$, namely
\begin{align}
C^+_{3,7}[T,R]=\Theta[T-R]\,\frac{2\left(T^2-R^2\right)^{\frac32}+3R^2T-2T^3}{48\pi^3 R^4}+\Theta[T]\Theta[-T+R]\,\frac{3R^2T-2T^3}{48\pi^3 R^4},
\label{C37_Minkowski}
\end{align}
is simply $\mathcal{D}_R C^{(+,\text{reg})}_{3,5+2\epsilon}$. Following through the same procedures, we can extend the analytic solution \eqref{C35_Minkowski} to all odd dimensions $d\geq5$ via dimension-rasing operators, where, similar to the odd-dimensional photon case, we may discard the last term of eq.~\eqref{C35_Minkowski} and set $\mu=1$ as they will be eliminated in the physical $G^+_{ijmn}$ (cf.~eq.~\eqref{Cijmn_Minkowski}),
\begin{align}
C^+_{3,\text{odd }d\geq5}[T,R]=\,&\mathcal D_R^{\frac{d-5}2}\Bigg( \Theta[T-R] \, \frac1{24\pi^2} \left( \left(2+\frac{T^2}{R^2}\right)\sqrt{T^2-R^2} - 3T\ln\left[ \left(
T+\sqrt{T^2-R^2} \right) \right]-\frac{T^3}{R^2}\right) \notag\\
&+\Theta[T]\Theta[-T+R] \, \frac1{24\pi^2} \left(-\frac{T^3}{R^2} - 3T\ln R\right) \Bigg).
\label{C3_Odd_Minkowski}
\end{align}
As with spin-1 photons in odd dimensions, the spin-2 effective Green's function \eqref{Cijmn_Minkowski} in this case explicitly reveals that, besides pure tails from $C_{1,d}$, the spin-2 graviton receives extra tail and acausal contributions from both $C_{2,d}$ and $C_{3,d}$ for all odd $d\geq5$; and moreover, no signals traveling strictly on its past light cone -- namely, no $\delta$-function light-cone contributions.
Therefore, the acausal nature of the TT spin-2 field in all relevant spacetime dimensions is explicitly confirmed by the analytic solutions of $G^+_{ijmn}$ obtained in this section. Now, we proceed to solve for other gauge-invariant variables involved in this system.
{\bf Bardeen Scalars} \qquad In a flat background, one of the Bardeen scalar potentials $\Psi$ obeys the Poisson's equation \eqref{Psi_Poisson_Minkowski}, which then leads to a Coulomb-type solution,
\begin{align}
\Psi[\eta,\vec x]=\frac{8\pi G_\text N}{d-2} \int_{\mathbb{R}^{d-1}} \text{d}^{d-1}\vec{x}'\, G^{(\mathrm{E})}_{d}[R] {}^{(\text{a})}T_{00}[\eta,\vec x'].
\label{BardeenPsi_Minkowski}
\end{align}
This transparently shows the acausal character of $\Psi$, since it is instantaneously sourced by the local matter energy density. The other Bardeen potential $\Phi$ is related to $\Psi$ via eq.~\eqref{PhiPsiRelationship}; recall that $\Upsilon$ is the nonlocal scalar portion of decomposed $^{(\text{a})}T_{ij}$ (cf.~eq.~\eqref{Astro_SVT_IIofII}), which, in Fourier space with $\vec k\neq\vec0$, is given by the following local projection \cite{Chu:2016ngc},
\begin{align}
\widetilde{\Upsilon}[\eta,\vec k]&=-\left(\frac{d-1}{d-2}\right)\frac1{\vec k^4}\bigg(k_ik_j-\frac{\vec k^2}{d-1}\delta_{ij}\bigg){}^{(\text{a})}\widetilde{T}_{ij}[\eta,\vec k].
\label{Upsilon_Fourier_Minkowski}
\end{align}
By virtue of this local decomposition, we can inverse Fourier transform eq.~\eqref{Upsilon_Fourier_Minkowski} to yield
\begin{align}
\Phi[\eta,\vec x]=\,&(d-3)\Psi[\eta,\vec x] \notag\\
&\quad+ \frac{8\pi G_\text N}{d-2} \int_{\mathbb{R}^{d-1}} \text{d}^{d-1}\vec{x}' \bigg(
G^{\mathrm{(E)}}_{d}[R]{}^{(\text{a})}T_{ll}[\eta,\vec{x}']
-(d-1)\partial_i\partial_jD_d[R]{}^{(\text{a})}T_{ij}[\eta,\vec{x}']\bigg),
\label{PhiPsiRelationship_Convolution}
\end{align}
where $^{(\text{a})}T_{ll}$ denotes the spatial trace of the $(ij)$ components of the matter stress-energy tensor, $^{(\text{a})}T_{ll}\equiv\delta^{ij}\,^{(\text{a})}T_{ij}$, and $D_d[R]$ is defined in eq.~\eqref{Dd}. Inserting eq.~\eqref{BardeenPsi_Minkowski} into eq.~\eqref{PhiPsiRelationship_Convolution}, we can express $\Phi$ itself in terms of the following convolution,
\begin{align}
\Phi[\eta,\vec x]=\,&\frac{8\pi G_\text N}{d-2}\int_{\mathbb{R}^{d-1}} \text{d}^{d-1}\vec{x}'\,\bigg((d-3)G^{(\mathrm{E})}_d[R]{}^{(\text{a})}T_{00}[\eta,\vec{x}']+G^{\mathrm{(E)}}_{d}[R]{}^{(\text{a})}T_{ll}[\eta,\vec{x}']
\notag\\
&-(d-1)\partial_i\partial_jD_d[R] {}^{(\text{a})}T_{ij}[\eta,\vec{x}'] \bigg),
\label{BardeenPhi_Minkowski}
\end{align}
where we see that $\Phi$ is effectively dependent on different components of the matter stress-energy tensor, weighted either by $G^{\mathrm{(E)}}_{d}$ or $D_d$ on the instantaneous $\eta=\eta'$ surface.
{\bf Vector Potential} \qquad The gauge-invariant vector mode $V_i$, in linearized gravity, also obeys the Poisson's equation \eqref{Vi_Poisson_Minkowski}, but is instead sourced by the nonlocal transverse part of $^{(\text{a})}T_{0i}$ (cf.~eq.\eqref{Astro_SVT_IofII}). As before, since the decomposition is local in momentum space,
\begin{align}
\widetilde\Sigma_i[\eta,\vec k]&=\left(\delta_{ij}-\frac{k_ik_j}{\vec k^2}\right)\,^{(\text{a})}\widetilde T_{0j}[\eta,\vec k],
\label{Sigmai_Fourier_Minkowski}
\end{align}
the solution of eq.~\eqref{Vi_Poisson_Minkowski} can first be cast into a Fourier form using eq.~\eqref{Sigmai_Fourier_Minkowski}, and then translated back to the position space,
\begin{align}
V_i[\eta,\vec x]=16\pi G_\text N\int_{\mathbb{R}^{d-1}} \text{d}^{d-1}\vec{x}'\left(\partial_i\partial_jD_d[R]
\,^{(\text{a})}T_{0j}[\eta,\vec{x}']-G^{\mathrm{(E)}}_d[R]{}^{(\text{a})}T_{0i}[\eta,\vec{x}']\right),
\label{Vi_Minkowski}
\end{align}
which is yet again an instantaneous acausal signal. Therefore, in Minkowski background, other than the spin-2 graviton field $D_{ij}$, the rest of the gauge-invariant variables depend exclusively on the weighted superposition of the matter sources evaluated at the instantaneous observer time $\eta$.
{\bf Linearized Riemann Tensor} \qquad In our discussion of gravitational observables in \S\eqref{Chapter_Observables}, we have argued that, in a free-falling synchronous-gauge setup, the $\delta_1R_{0i0j}$ components of the linearized Riemann tensor encode the gravitational tidal forces exerted upon the neighboring test particles in the geodesic deviation equation (cf.~eq.~\eqref{GeodesicDeviation_FreeFall}). And, being also gauge-invariant in Minkowski spacetime, it would reasonably be regarded as a classical physical observable and expected to be strictly causal as well.
As with the Faraday tensor in electromagnetism, it can be directly shown via its second-order wave equation that the linearized Riemann tensor is causal with respect to the flat background. Firstly, by taking the divergence of the Bianchi identity obeyed by the exact Riemann tensor, followed by imposing Einstein's equations, one may obtain
\begin{align}
\label{Riemann_WaveEq}
\Box R^{\rho\sigma}{}_{\mu\nu} + \big[\nabla^\lambda,\nabla^{[\rho}\big]R^{\sigma]}{}_{\lambda\mu\nu} = 32\pi G_\text N\nabla^{[\rho}\nabla_{[\mu} \left(T_{\nu]}{}^{\sigma]}-\delta_{\nu]}^{\sigma]}\frac{T}{(d-2)}\right) .
\end{align}
The linearized version in Minkowski background is thus
\begin{align}
\partial^2\delta_1R^{\rho\sigma}{}_{\mu\nu}&=32\pi G_\text N\,\partial^{[\rho}\partial_{[\mu}\bigg({}^{(\text{a})}T_{\nu]}{}^{\sigma]}-\delta_{\nu]}^{\sigma]}\frac{^{(\text{a})}T}{(d-2)}\bigg),
\label{LinearizedRiemann_WaveEq_Minkowski}
\end{align}
where $^{(\text{a})}T$ is now the trace of the matter stress tensor in flat spacetime, namely $^{(\text{a})}T\equiv\eta^{\mu\nu}\,^{(\text{a})}T_{\mu\nu}$.\footnote{Strictly speaking, the matter stress tensor in eq.~\eqref{Riemann_WaveEq}, when perturbed around a background Minkowski spacetime $g_{\mu\nu} = \eta_{\mu\nu} + \chi_{\mu\nu}$, would typically admit an infinite series in $\chi_{\mu\nu}$. Whereas the matter stress tensor in eq.~\eqref{LinearizedRiemann_WaveEq_Minkowski} does not contain $\chi_{\mu\nu}$. The matter stress tensor appearing everywhere else in this paper denotes the latter.} The wave equation \eqref{LinearizedRiemann_WaveEq_Minkowski} then leads immediately to the fact that the linearized Riemann tensor is causally sourced by the second derivatives of the astrophysical stress-energy tensor, from which its $\delta_1R_{0i0j}$ components can be expressed explicitly as
\begin{align}
\delta_1R_{0i0j}[x]=\,&8\pi G_\text N\int_{\mathbb{R}^{d-1,1}} \text{d}^dx'\Bigg\{\ddot{G}^+_{d}\bigg({}^{(\text{a})}T_{ij}[x']+\frac{\delta_{ij}}{(d-2)}\left({}^{(\text{a})}T_{00}[x']-{}^{(\text{a})}T_{ll}[x']
\right)\bigg)-2\partial_{(i}\dot G_{d}^+\,^{(\text{a})}T_{j)0}[x']\notag\\
&+\frac1{(d-2)}\partial_i\partial_jG^+_{d}\left((d-3){}^{(\text{a})}T_{00}[x']+{}^{(\text{a})}T_{ll}[x']\right)\Bigg\}.
\label{LinearizedRiemann_Sol_Minkowski}
\end{align}
To arrive at eq.~\eqref{LinearizedRiemann_Sol_Minkowski}, we have integrated by parts and dropped the boundary contributions evaluated at the spatial and past infinity. Not only does eq.~\eqref{LinearizedRiemann_Sol_Minkowski} show $\delta_1R_{0i0j}$ is completely causal for all $d\geq4$, it also provides a check for our calculations in the gauge-invariant approach.
As we have shown earlier, all the gauge-invariant variables -- the two scalars $\Psi$ and $\Phi$, one vector $V_i$, and one tensor $D_{ij}$ -- are acausal. From the similar issue encountered in the electromagnetic case, we would expect that, in describing the physical GW observables, a mutual cancellation of the acausal contributions must occur among these variables. Let us now check this statement more carefully using their analytic solutions. Given in eq.~\eqref{LinearizedRiemann_Minkowski_GaugeInvariant} is the linearized Riemann tensor $\delta_1R_{0i0j}$ expressed in terms of four gauge-invariant variables in Minkowski spacetime, all of which are non-trivial whenever matter sources are present. Notice that the spin-2 graviton field $D_{ij}$ enters $\delta_1R_{0i0j}$ through its acceleration; before taking time derivatives, the spin-2 field given in eq.~\eqref{Spin2_Convolution_Minkowski} can firstly be re-cast into another convolution,
{\allowdisplaybreaks
\begin{align}
D_{ij}[\eta,\vec x]=\,&-16\pi G_\text N\int_{\mathbb{R}^{d-1,1}} \text{d}^dx'\,\Bigg\{G^+_{d}\left({}^{(\text{a})}T_{ij}[x']+\frac{\delta_{ij}}{d-2}\left({}^{(\text{a})}T_{00}[x']-{}^{(\text{a})}T_{ll}[x']
\right)\right)
-2\,\Theta[T]\partial_{(i}\dot C_{2,d}{}^{(\text{a})}T_{j)0}[x']\notag\\
&+\frac{1}{d-2}\Theta[T]\partial_i\partial_jC_{2,d}\left((d-3){}^{(\text{a})}T_{00}[x']+{}^{(\text{a})}T_{ll}[x']\right)\Bigg\} \notag\\
&+ \frac{16\pi G_\text N}{d-2}\int_{\mathbb{R}^{d-1}} \text{d}^{d-1}\vec x' \bigg(\delta_{ij}G^{\mathrm{(E)}}_d{} ^{(\text{a})}T_{00}[\eta,\vec x']+(d-3)\partial_i\partial_jD_d{}^{(\text{a})}T_{00}[\eta,\vec x']\bigg),
\label{Spin2_Convolution2_Minkowski}
\end{align}}%
where we have utilized the conservation of the matter stress-energy tensor, ${}^{(\text{a})}\dot{T}_{0j}=\partial_{i}{}^{(\text{a})}T_{ij}$ and ${}^{(\text{a})}\ddot{T}_{00}=\partial_{i}\partial_{j}{}^{(\text{a})}T_{ij}$, the conversion properties $\ddot C_{2,d}=-C_{1,d}$ and $\ddot C_{3,d}=-C_{2,d}$, and the initial conditions $\dot C_{2,d}\big|_{T=0}=-G^{\mathrm{(E)}}_d$ and $\dot C_{3,d}\big|_{T=0}=D_d$. Also, as in the spin-1 case, the boundary terms from integrations by parts all vanish, which can be justified using the analytic expressions of $C_{2,d}$ and $C_{3,d}$ obtained above. Note also that, in eq.~\eqref{Spin2_Convolution2_Minkowski}, the second integral is performed on the equal-time surface, which is clearly acausal, but the whole expression for $D_{ij}$ is not yet a clean separation based on causality, since $C_{2,d}$ in the first integral still contains both causal and acausal pieces. Furthermore, we highlight that the energy-momentum conservation law is always assumed throughout this paper. However, this is no longer true in a self-gravitating system, such as the in-spiraling pairs of black holes/neutron stars whose GWs LIGO have detected to date. Conceptually speaking, to make the theory self-consistent, non-linear corrections of gravity must be incorporated into the right-hand side of the linearized wave equation, so that the conservation of total stress tensor remains valid at linear level. We hope to address this subtlety more systematically in future work.
Now, we proceed to take double-time derivative of the spin-2 field in eq.~\eqref{Spin2_Convolution2_Minkowski},
{\allowdisplaybreaks
\begin{align}
\ddot D_{ij}[\eta,\vec x]&=-16\pi G_\text N\int_{\mathbb{R}^{d-1,1}} \text{d}^dx'\,\Bigg\{\ddot G^+_{d}\left(^{(\text{a})}T_{ij}[x']+\frac{\delta_{ij}}{d-2}\left({}^{(\text{a})}T_{00}[x']-{}^{(\text{a})}T_{ll}[x']\right)\right)
-2\partial_{(i}\dot{G}^+_{d}{}^{(\text{a})}T_{j)0}[x']\notag\\
&+\frac{1}{d-2}\partial_i\partial_jG^+_{d}\left((d-3)\,{}^{(\text{a})}T_{00}[x']+{}^{(\text{a})}T_{ll}[x']\right)\Bigg\} \notag\\
&+16\pi G_\text N\int_{\mathbb{R}^{d-1}} \text{d}^{d-1}\vec{x}'\,\Bigg\{\! -2\partial_m\partial_{(i}G^{\mathrm{(E)}}_d
{}^{(\text{a})}T_{j)m}[\eta,\vec{x}']+\frac1{d-2}\bigg(\delta_{ij}\partial_m\partial_nG^{\mathrm{(E)}}_d{}^{(\text{a})}T_{mn}[\eta,\vec{x}'] \notag\\
&+\partial_i\partial_jG^{\mathrm{(E)}}_d\left((d-3){}^{(\text{a})}T_{00}[\eta,\vec{x}']+{}^{(\text{a})}T_{ll}[\eta,\vec{x}']\right)\bigg)
+\left(\frac{d-3}{d-2}\right)\partial_i\partial_j\partial_m\partial_nD_d{}^{(\text{a})}T_{mn}[\eta,\vec{x}']\Bigg\},
\label{DDotSpin2_Minkowski}
\end{align}}%
where the same properties used in the previous calculation have been employed to carry out the differentiation\footnote{Whenever a second time derivative acts on the expression involving a step function $\Theta[T]$, we make use of the following simplification for any function $F[\eta,\eta']$,
\begin{align}
\partial_\eta^2\left(\Theta[T]F[\eta,\eta']\vphantom{\dot A}\right)=\Theta[T]\ddot F[\eta,\eta']+\dot\delta[T]F[\eta',\eta']+\delta[T]\dot F[\eta,\eta']\big|_{\eta=\eta'}, \notag
\end{align}
where we have made the replacement $T\dot\delta[T]\to-\delta[T]$, which results from differentiating the identity $T\delta[T]=0$ with respect to $\eta$. Notice that the last two terms only contribute at $\eta=\eta'$, and this property will also be utilized later on in the cosmological case.
}, and we observe that the first integral in eq.~\eqref{DDotSpin2_Minkowski} is completely causal and is exactly $-2$ times the expression in eq.~\eqref{LinearizedRiemann_Sol_Minkowski}, whereas the second one is acausally performed over the equal-time hypersurface, which is therefore expected to connect to the other gauge-invariant variables. According to eq.~\eqref{LinearizedRiemann_Minkowski_GaugeInvariant}, the scalar and vector contributions to $\delta_1R_{0i0j}$ come from $\ddot\Psi$, $\partial_i\partial_j\Phi$, and $\partial_{(i} \dot{V}_{j)}$, the explicit forms of which can be readily deduced from eqs.~\eqref{BardeenPsi_Minkowski}, \eqref{BardeenPhi_Minkowski}, and \eqref{Vi_Minkowski},
{\allowdisplaybreaks
\begin{align}
\ddot \Psi[\eta,\vec x]=\,& \frac{8\pi G_\text N}{d-2} \int_{\mathbb{R}^{d-1}} \text{d}^{d-1}\vec{x}'\, \partial_m \partial_n G^{(\mathrm{E})}_{d} {}^{(\text{a})}T_{mn}[\eta,\vec x'], \\
\partial_i\partial_j\Phi[\eta,\vec x]=\,&\frac{8\pi G_\text N}{d-2}\int_{\mathbb{R}^{d-1}} \text{d}^{d-1}\vec{x}'\,\bigg((d-3)\partial_i\partial_jG^{(\mathrm{E})}_d{}^{(\text{a})}T_{00}[\eta,\vec{x}']+
\partial_i\partial_jG^{\mathrm{(E)}}_{d}{}^{(\text{a})}T_{ll}[\eta,\vec{x}']
\notag\\
&-(d-1)\partial_i\partial_j\partial_m\partial_nD_d {}^{(\text{a})}T_{mn}[\eta,\vec{x}'] \bigg), \\
\partial_{(i}\dot{V}_{j)}[\eta,\vec x] =\,& -16\pi G_\text N \int_{\mathbb{R}^{d-1}} \text{d}^{d-1}\vec{x}'\left( \partial_m\partial_{(i}G^{\mathrm{(E)}}_d {}^{(\text{a})}T_{j)m}[\eta,\vec{x}']
-\partial_i\partial_j\partial_m\partial_n
D_d {}^{(\text{a})}T_{mn}[\eta,\vec{x}']\right),
\end{align}}%
where the conservation law of $^{(\text{a})}T_{\mu\nu}$ allows us to switch between different components of the stress-energy tensor. As it turns out, the scalar and vector contributions, added together in accord with eq.~\eqref{LinearizedRiemann_Minkowski_GaugeInvariant}, do conspire to cancel the acausal portion of the acceleration of the spin-2 TT graviton completely, i.e., the second integral of eq.~\eqref{DDotSpin2_Minkowski}. As a result, the remaining part of eq.~\eqref{LinearizedRiemann_Minkowski_GaugeInvariant} is then strictly causal and exactly consistent with eq.~\eqref{LinearizedRiemann_Sol_Minkowski},
\begin{align}
\delta_1 R_{0i 0j}=\delta_{ij}\ddot{\Psi}+ \partial_i\partial_j\Phi+\partial_{(i}\dot{V}_{j)}-\frac12\ddot{D}_{ij}=-\frac12\left(\ddot D_{ij}\right)_{\text{causal}}=-\frac12\left(\vphantom{\dot D}\ddot \chi^{\text{TT}}_{ij}\right)_{\text{causal}},
\label{LinearizedRiemann_CausalSpin2_Minkowski}
\end{align}
which is valid in general weak-field situations. The physical insight gained from the gauge-invariant formalism is that the information regarding gravitational tidal forces is exclusively encoded within the causal part of the acceleration of the spin-2 field; whereas the acausal portion of $D_{ij}$ is completely canceled by the gauge-invariant Bardeen scalars and vector mode. This situation is very similar to that of the spin-1 field describing the electric field in electromagnetism.
{\bf Stationary Limit and $\Phi$} \qquad Like the photon case, the limit where the stress tensor ${}^{(\text a)}T_{\mu\nu}$ becomes time-independent leads to a degenerate causal structure for the acceleration of the spin-2 graviton; namely, its otherwise causal and acausal pieces cancel. In such a situation, one may further verify from eqs. \eqref{BardeenPsi_Minkowski}, \eqref{Vi_Minkowski}, and \eqref{DDotSpin2_Minkowski} that $\ddot{\Psi} = \dot{V}_i = \ddot{D}_{ij} = 0$, leaving the tidal forces to depend only on $\Phi$:
\begin{align}
\delta_1 R_{0i0j} = \partial_i \partial_j \Phi .
\end{align}
Since eq.~\eqref{LinearizedRiemann_CausalSpin2_Minkowski} holds in general, we may maintain that, despite appearances, it is really the acausal pieces of $\ddot{D}_{ij}$ -- which are equal in magnitude but opposite in sign to the causal ones -- that are canceling the $\partial_i \partial_j \Phi$. This interpretation ensures that causality is respected once there is the slightest time-dependence in the ${}^{(\text a)}T_{\mu\nu}$.
\begin{align}
\delta_1 R_{0i 0j}[\vec x]=-\frac12\left(\ddot D_{ij}\right)_{\text{causal}}=\frac{8\pi G_\text N}{d-2}\int_{\mathbb{R}^{d-1}} \text{d}^{d-1}\vec x'\,\partial_i\partial_jG^{(\mathrm{E})}_{d}\left((d-3){}^{(\text{a})}T_{00}[\vec x']+{}^{(\text{a})}T_{ll}[\vec x']\right).
\end{align}
{\bf Far-Zone Limit} \qquad To extract the far-zone GW signals generated by the isolated astrophysical systems, we perform the same frequency space analysis for the spin-2 effective Green's function here as for its spin-1 counterpart. Before taking the far-zone limit, we first re-cast the spin-2 effective Green's function \eqref{Cijmn_Minkowski} into the one analogous to eq.~\eqref{Cij_FarZone_Minkowski} for spin-1 photons, by carrying out all the spatial derivatives involved in $C_{ijmn}$ while avoiding the point $R=0$,
\begin{align}
C_{ijmn}[T,\vec R]&= P_{ijmn}[\vec R]\,C_{1,d}[T,R]+\Pi_{ijmn}[\vec R]\,2\pi C_{2,d+2}[T,R]+\Xi_{ijmn}[\vec R]\,4\pi^2C_{3,d+4}[T,R],
\label{Cijmn_FarZone_Minkowski}
\end{align}
where $ P_{ijmn}[\vec R]$ denotes the TT spatial projector based on the unit vector $\widehat R$,
\begin{align}
P_{ijmn}[\vec R] &\equiv P_{m(i}[\vec R] \, P_{j)n}[\vec R]-\frac{1}{d-2} P_{ij}[\vec R] \, P_{mn}[\vec R],
\end{align}
with $P_{ij}[\vec R]$ given in eq.~\eqref{TransverseProjector_Position}, and the other symmetric tensor structures $\Pi_{ijmn}[\vec R]$ and $\Xi_{ijmn}[\vec R]$, respectively, are defined as
{\allowdisplaybreaks
\begin{align}
\Pi_{ijmn}[\vec R]=\,&-2 P_{ijmn}[\vec R]+\frac{d(d-3)}{d-2}\left(\delta_{m(i}\widehat R_{j)}\widehat R_{n}+\delta_{n(i}\widehat R_{j)}\widehat R_{m}
-2\widehat R_i\widehat R_j\widehat R_m\widehat R_n \right)\\
\Xi_{ijmn}[\vec R]=\,&\left(\frac{d-3}{d-2}\right)\bigg(\delta_{ij}\delta_{mn}+2\delta_{m(i}\delta_{j)n}-(d+1)\left(\delta_{ij}\widehat R_m\widehat R_n+\delta_{mn}\widehat R_i\widehat R_j+2\delta_{m(i}\widehat R_{j)}\widehat R_n\right. \notag\\
&\left.+2\delta_{n(i}\widehat R_{j)}\widehat R_m\right)+(d+1)(d+3)\widehat R_i\widehat R_j\widehat R_m\widehat R_n\bigg) .
\end{align}}%
We have taken advantage of the homogeneous wave equation obeyed by both $C_{2,d}$ and $C_{3,d}$, along with the properties $\ddot C_{2,d}=-C_{1,d}$ and $\ddot C_{3,d}=-C_{2,d}$, to relate different scalar functions. Here, we highlight again that, for a fixed $d$, the notations $C_{2,d+2}$ and $C_{3,d+4}$ used in eq.~\eqref{Cijmn_FarZone_Minkowski} represent their corresponding $(d+2)$ and $(d+4)$-dimensional functional forms, but the $R$ is the one in $d-1$ spatial dimensions. Essentially, as long as $R\neq0$, eq.~\eqref{Cijmn_FarZone_Minkowski} is equivalent to its original expression \eqref{Cijmn_Minkowski}, and, as explained in the similar spin-1 situation, this form is useful for the far-zone analysis and is manifestly finite in all relevant spacetime dimensions, because $C_{1,d}$, $C_{2,d+2}$, and $C_{3,d+4}$ all converge for $d\geq4$. As a consistency check of eq.~\eqref{Cijmn_FarZone_Minkowski}, its TT properties can be shown explicitly by a direct calculation.
The leading contribution of GWs, responsible for the far-zone tidal forces, can be extracted from the spin-2 effective Green's function using eq.~\eqref{Cijmn_FarZone_Minkowski} in frequency space. The relative amplitudes of the three scalar functions can in turn be directly compared in the limit $|\omega|R\gg1$. To begin, we express the spin-2 field $D_{ij}$ in terms of the superposition of monochromatic modes,
\begin{align}
D_{ij}[\eta,\vec x]=-16\pi G_\text N \int_{\mathbb{R}^{d-1}} \text{d}^{d-1}\vec{x}' \int_{\mathbb{R}} \frac{\text{d}\omega}{2\pi}\,\widetilde{G}^+_{ijmn}[\omega,\vec R]\,e^{-i\omega\eta}\,{}^{(\text{a})}\widetilde{T}_{mn}[\omega,\vec{x}'],
\end{align}
where $\widetilde{G}^+_{ijmn}[\omega,\vec R]$ is the frequency transform of the spin-2 effective Green's function assuming $R\neq0$,
\begin{align}
\widetilde{G}^+_{ijmn}[\omega,\vec R]&=\int_{\mathbb{R}} \text{d} T\,G^+_{ijmn}[T,\vec R]\,e^{i\omega T} \notag\\
&= P_{ijmn}[\vec R]\,\widetilde{G}^+_{d}[\omega,R] - \Pi_{ijmn}[\vec R]\,2\pi\widetilde{C}^+_{2,d+2}[\omega,R] - \Xi_{ijmn}[\vec R]\,4\pi^2\widetilde{C}^+_{3,d+4}[\omega,R],
\label{Gijmn_FrequencyTransform}
\end{align}
with $\widetilde{G}^+_{d}[\omega,R]$, $\widetilde{C}^+_{2,d+2}[\omega,R]$, and $\widetilde{C}^+_{3,d+4}[\omega,R]$, respectively, defined to be the frequency transforms of their real-space counterparts, $G^+_d$, $C^+_{2,d+2}$, and $C^+_{3,d+4}$. As we did for the far-zone spin-1 waves, we now take the limit of eq.~\eqref{Gijmn_FrequencyTransform} as $|\omega|R\to \infty$, from which to extract the dominant spin-2 GWs in the radiative regime. Since we have calculated $\widetilde{G}^+_{d}[\omega,R]$ and $\widetilde{C}^+_{2,d+2}[\omega,R]$ earlier in the photon case, $\widetilde{C}^+_{3,d+4}[\omega,R]$ is the only term left to evaluate here.
{\it Spin-2 gravitons in even dimensions $d\geq4$} \qquad For even-dimensional spacetimes, $\widetilde{G}^+_{d}[\omega,R]$ and $\widetilde{C}^+_{2,d+2}[\omega,R]$, for $d=4+2n$, have been obtained in eqs.~\eqref{C1_Even_FrequencySpace} and \eqref{C2_Even_FrequencySpace}. In the same way, with the analytic expression of $C_{3,d}$ given in eq.~\eqref{C3_Even_Minkowski}, its frequency transform $\widetilde{C}^+_{3,d+4}[\omega,R]$ can be computed straightforwardly,
\begin{align}
\widetilde{C}^+_{3,8+2n}[\omega,R]=\,& \mathcal D^n_R\left(\frac{e^{i\omega R}}{16\pi^3 R^3(i\omega)^2}-\frac{3e^{i\omega R}}{16\pi^3 R^4(i\omega)^3}+\frac{3e^{i\omega R}}{16\pi^3 R^5(i\omega)^4}\right) - \frac{(2n+1)!!\,\omega^{2n+1}}{4(2\pi)^{n+3}(\omega R)^{2n+3}} \notag\\
&- \frac{(2n+3)!!\,\omega^{2n+1}}{2(2\pi)^{n+3}(\omega R)^{2n+5}},
\label{C3_Even_FrequencySpace}
\end{align}
which, analogous to $\widetilde{C}^+_{2,d+2}[\omega,R]$, comprises the non-propagating modes associated with the acausal effect, as well as the propagating ones with the $e^{i\omega R}$ factor. Moreover, $\widetilde{C}^+_{3,8+2n}[\omega,R]$ turns out to be suppressed relative to $\widetilde{G}^+_{d}[\omega,R]$ and $\widetilde{C}^+_{2,d+2}[\omega,R]$ when $|\omega|R\to\infty$ (cf.~eqs.~\eqref{C1_Even_FrequencySpace} and \eqref{C2_Even_FrequencySpace}). More explicitly, at leading $1/(\omega R)$ order, it behaves like
\begin{align}
\widetilde{C}^+_{3,8+2n}[\omega,R]=\frac{(-1)^{n+1}i^n\omega^{2n+1}}{2(2\pi\omega R)^{n+1}}\,e^{i\omega R} \cdot \frac{1}{(2\pi \omega R)^2}\left(1+\mathcal{O}\left[\frac1{\omega R}\right]\right).
\label{C3_Even_FarZone_FrequencySpace}
\end{align}
Hence, as inferred from the asymptotic behaviors of three scalar functions \eqref{C1_Even_FarZone_FrequencySpace}, \eqref{C2_Even_FarZone_FrequencySpace}, and \eqref{C3_Even_FarZone_FrequencySpace}, the pure causal one, $\widetilde{G}^+_{d}[\omega,R]$, is still the dominant contribution to the spin-2 effective Green's function in the limit $|\omega|R\gg1$. In close analogy with the spin-1 case, the spin-2 GWs in the radiative zone is dominated by the first term of eq.~\eqref{Gijmn_FrequencyTransform}. That is, under the far-zone assumptions $|\omega|r\gg1$ and $r_c/r\ll1$, the leading $1/r$ piece of $\widetilde{G}^+_{ijmn}$ is given by
\begin{align}
\widetilde{G}^+_{ijmn}[\omega,\vec R]&=P_{ijmn}\widetilde{G}^{(+,\,\text{fz})}_{4+2n}[\omega;\vec x,\vec x']\left(1+\mathcal{O}\left[\frac1{\omega r},\frac{r_c}{r}\right]\right),
\label{Gijmn_Even_FrequencyTransform_FarZone}
\end{align}
where $P_{ijmn}$, the far-zone ``tt'' projector, is defined in eq.~\eqref{tt_Projector} and $\widetilde{G}^{(+,\,\text{fz})}_{4+2n}$ given in eq.~\eqref{G+_FrequencySpace_Even_FarZone}. Accordingly, as already alluded to in \S\eqref{Chapter_Observables}, the spin-2 TT graviton $D_{ij}$, in the far-zone radiative regime ($r\to\infty$), reduces to the causal ``tt'' GWs,
\begin{align}
\lim_{r\to\infty} D_{ij}\to \chi_{ij}^\text{tt}, \qquad \chi_{ij}^\text{tt}[x]\equiv P_{ijmn}\left(-16\pi G_\text N\int_{\mathbb{R}^{d-1,1}} \text{d}^{d}x'\,G^{(+,\,\text{fz})}_{d}[T;\vec x,\vec x']{}^{(\text{a})}T_{mn}[x']\right),
\label{htt_FarZone}
\end{align}
where $G^{(+,\,\text{fz})}_{d}[T;\vec x,\vec x']$, as before, denotes the far-zone version of the massless scalar Green's function, which, for even $d\geq4$, is given in eq.~\eqref{G+_Even_FarZone} consisting of pure light-cone signals. This $\chi_{ij}^\text{tt}$ is thus the tt projection of the far-zone de Donder-gauge solution of the metric perturbations, $\chi_{ij}^\text{tt}=P_{ijmn}\overline\chi_{mn}[\text{de Donder}]$. In other words, like the consequence of the far-zone spin-1 field \eqref{At_FarZone}, the two distinct notions of ``transverse-traceless'' metric perturbations, $D_{ij}=\chi_{ij}^\text{TT}$ and $\chi_{ij}^\text{tt}$, are shown to coincide as $r\to\infty$, where the acausal effect in $D_{ij}$ becomes sufficiently insignificant.
{\it Spin-2 gravitons in odd dimensions $d\geq5$} \qquad Following the similar procedures, we are able to extract the far-zone portion of the spin-2 GWs for odd dimensions as well. Odd-dimensional $\widetilde{G}^+_{d}[\omega,R]$ and $\widetilde{C}^+_{2,d+2}[\omega,R]$ for $d=5+2n$ and positive frequencies can be obtained simply by replacing $n\to n+1$ in eqs.~\eqref{C1_Odd_FrequencySpace} and \eqref{C2_Odd_FrequencySpace}. And, given eq.~\eqref{C3_Odd_Minkowski}, $\widetilde{C}^+_{3,d+4}[\omega,R]$ can be tackled similarly to $\widetilde{C}^+_{2,d+2}[\omega,R]$,
\begin{align}
\widetilde{C}^+_{3,9+2n}[\omega>0,R]=\frac{i\omega^{2n+2}}{4(2\pi\omega R)^{n+3}}H^{(1)}_{n+3}[\omega R] - \frac{2^{n}(n+1)!\,\omega^{2n+2}}{(2\pi)^{n+4}(\omega R)^{2n+4}} - \frac{2^{n+2}(n+2)!\,\omega^{2n+2}}{(2\pi)^{n+4}(\omega R)^{2n+6}},
\end{align}
which, as in the even-dimensional case, resembles the structure of $\widetilde{C}^+_{2,d+2}[\omega,R]$ in eq.~\eqref{C2_Odd_FrequencySpace}, and tends to be more suppressed than both $\widetilde{G}^+_{d}[\omega,R]$ and $\widetilde{C}^+_{2,d+2}[\omega,R]$ at leading $1/(\omega R)$ order. That is, as $|\omega| R\gg1$, the asymptotic behavior of $\widetilde{C}^+_{3,d+4}[\omega,R]$ reads
\begin{align}
\widetilde{C}^+_{3,9+2n}[\omega>0,R]=\frac{-i\omega^{2n+2}}{2(2\pi\omega R)^{n+\frac32}}e^{i\left(\omega R-\frac{(n+1)\pi}2-\frac{\pi}4\right)} \cdot \frac{1}{(2\pi\omega R)^2}\Bigg(1+\mathcal{O}\left[\frac1{\omega R}\right]+\mathcal O\left[\frac1{(\omega R)^{n+\frac12}}\right]\Bigg),
\end{align}
where the expression has been factorized into the leading $1/(\omega R)$ piece of $\widetilde G^+_{5+2n}[\omega,R]$ times the suppression factor. Likewise, among the three scalar functions in eq.~\eqref{Gijmn_FrequencyTransform}, $\widetilde G^+_{d}[\omega,R]$ continues to be the dominant contribution in the limit $|\omega| R\to\infty$. As a result, the far-zone behavior of $\widetilde{G}^+_{ijmn}[\omega,R]$ here admits the same ``tt'' structure as eq.~\eqref{Gijmn_Even_FrequencyTransform_FarZone} for even $d$,
\begin{align}
\widetilde{G}^+_{ijmn}[\omega>0,\vec R]=P_{ijmn}\widetilde{G}^{(+,\,\text{fz})}_{5+2n}[\omega;\vec x,\vec x']\left(1+\mathcal{O}\left[\frac1{\omega r},\frac{r_c}{r}\right]\right),
\end{align}
where $\widetilde{G}^{(+,\,\text{fz})}_{5+2n}$ is given in eq.~\eqref{G+_FrequencySpace_Odd_FarZone} with $n$ replaced by $n+1$; and $P_{ijmn}$ in eq.~\eqref{tt_Projector}. A similar line of arguments then reveals that the spin-2 TT graviton $D_{ij}$, in odd dimensions $d\geq5$, also reduces to $\chi_{ij}^\text{tt}$ as $r\to \infty$, where the acausal nature of $D_{ij}$ becomes trivial; namely, the feature \eqref{htt_FarZone} still holds here, with odd-dimensional $G^{(+,\,\text{fz})}_{d}$ given in eq.~\eqref{G+_Odd_FarZone}.
{\it Linearized Riemann tensor} \qquad Through the analysis of the spin-2 effective Green's function, we have just shown that, in the radiative limit, the spin-2 TT GWs in fact coincide with the tt ones, $D_{ij}\to \chi_{ij}^\text{tt}$, for all spacetime dimensions $d\geq4$. For this reason, the far-zone version of the tidal forces \eqref{LinearizedR0i0j_deDonder_FarZone} for all $d\geq4$, as well as the statement \eqref{SynchronousGauge_to_GaugeInvariant_StepI}, follows immediately from eq.~\eqref{LinearizedRiemann_CausalSpin2_Minkowski} and the fact that $\chi_{ij}^\text{tt}$ is completely causal. This result can alternatively be derived from the expression \eqref{LinearizedRiemann_Sol_Minkowski} for $\delta_1R_{0i0j}$, by repeatedly employing the replacement rule $\partial_i=-\partial_{i'}=-\widehat R_i\partial_0$ and the conservation of the matter stress tensor in the intermediate steps before reaching the final far-zone expression. Furthermore, the far-zone connection between $\chi_{ij}^\text{tt}$ and the synchronous-gauge metric $\chi_{ij}^{(\text{Synch})}$ can be made via eq.~\eqref{SynchronousGauge_to_GaugeInvariant_FullAnswer}, where as explained in \S\eqref{Chapter_Observables} the initial conditions could be dropped for GW detectors sensitive to only finite frequencies. In particular, the fractional distortion spin-2 pattern of the laser interferometer, described in eq.~\eqref{FractionalDistortion_AtConstantCosmicTime_FirstOrderHuman} are exclusively attributed to the causal $\chi_{ij}^\text{tt}$. Such a characterization of the GW observables in terms of $\chi_{ij}^{\text{tt}}$, however, is legitimate only when the GW detector is sufficiently far away from the matter sources.
{\bf Commutator of Spin-2 Gravitons} \qquad Earlier in this section, we have shown micro-causality is violated for the massless spin-1 photons (cf.~eq.~\eqref{Spin1_Commutator}). A similar line of reasoning then reveals that the massless spin-2 graviton field violates micro-causality too. For, the tensor structure $C_{ijmn}$ in eq.~\eqref{Cijmn_Minkowski} is related to the commutator of the corresponding quantum operators via the relationship
\begin{align}
\big[D_{ij}[x],D_{mn}[x']\big]=-iC_{ijmn}[T,\vec R].
\label{Spin2_Commutator}
\end{align}
The acausal nature of $C_{ijmn}$ immediately tells us that the massless spin-2 gravitons do not commute at spacelike separations. Once again, it is likely that this violation of micro-causality is linked to the tension between gauge invariance and Lorentz covariance when constructing massless helicity-2 quantum Fock states.
\section{Spatially Flat Cosmologies with Constant $w$}
\label{SpatiallyFlatw}
We now move on to the spatially-flat cosmological background, driven by a perfect fluid with a constant equation-of-state $w$. Again, we will consider both the electromagnetism and linearized gravitation cases, where the dynamics of the linearized gravitational system, unlike that of electromagnetism, has non-trivial dependence on $w$. In cosmology, there is no longer a time-translation symmetry, and the full analytic expressions for spin-1 and spin-2 effective Green's functions may generally be difficult to attain. On the other hand, the background space-translation symmetry is still preserved, so the similar Fourier-space analysis exploited in Minkowski spacetime continues to apply in the cosmological setup. At a more technical level, the translation symmetry in space would still allow us to utilize the time-integral method to express the spin-1 and spin-2 effective Green's functions in terms of the analytic solutions found in \cite{Chu:2016ngc}, so that the corresponding causal structures can be analyzed.
\subsection{Electromagnetism}
We start with the electromagnetic field in the cosmological background spacetime, described by the metric \eqref{PerturbedFLRW} with $\chi_{\mu\nu}$ set to zero, and our focus here will be the causal structure of the theory in the gauge-invariant content for all $d\geq3$.
{\bf Field Equations} \qquad In spatially-flat cosmologies, the Maxwell's equations \eqref{Maxwell_equations}, in terms of the gauge-invariant variables \eqref{Maxwell_GaugeInvariant_ScalarVector}, are translated into a set of two independent field equations,
\begin{align}
-\frac{\partial_0\left(a^{d-4}\,\dot\alpha_i\right)}{a^{d-4}}+\vec\nabla^2 \alpha_i&=-a^2\Gamma_i,
\label{Spin1_WaveEq_Cosmology} \\
\vec\nabla^2\Phi&=-a^2\rho.
\label{EMPoisson_Cosmology}
\end{align}
Notice that the spatial components of eq.~\eqref{Maxwell_equations} encode not only eq.~\eqref{Spin1_WaveEq_Cosmology} but also the equation $\partial_{0}\left(a^{d-4}\,\Phi\right)/a^{d-4}=a^2\Gamma$, which is in fact redundant, as is already implied by the Poisson's equation \eqref{EMPoisson_Cosmology} inserted into the conservation law of the charge current $\nabla_{\mu}J^{\mu}=0$,
\begin{align}
\frac{\partial_0\left(a^{d-2}J_0\right)}{a^{d-2}}=\partial_iJ_i.
\label{ChargeConservaion_Cosmology}
\end{align}
Moreover, eqs.~\eqref{Spin1_WaveEq_Cosmology} and \eqref{EMPoisson_Cosmology} together show that, except the re-scaling $J_{\mu}\to a^2J_\mu$, the theory is conformally invariant when $d=4$, and, like its Minkowski counterpart, only the spin-1 photon field $\alpha_i$ admits dynamical wave solutions, with the scalar $\Phi$ still obeying a Poisson-type equation.
{\bf Spin-1 Photons} \qquad To solve for the transverse spin-1 field $\alpha_i$ in the cosmological system, we first re-write its wave equation \eqref{Spin1_WaveEq_Cosmology} as
\begin{align}
\left\{\partial^2+\frac{(d-4)}{4}\left((d-4)\mathcal H^2+2\dot{\mathcal H}\right)\right\}
\left(a^{\frac{d-4}2}\alpha_i\right)=-a^{\frac d2}\Gamma_i,
\label{Spin1_WaveEq2_Cosmology}
\end{align}
where $\mathcal H\equiv\dot a/a$, denoting the conformal Hubble parameter. Then, following the same manipulations in Fourier space performed in Minkowski spacetime, we are able to express the spin-1 field $\alpha_i$ in terms of the following convolution based on eq.~\eqref{Spin1_WaveEq2_Cosmology},
\begin{align}
a[\eta]^{\frac{d-4}2}\alpha_i[\eta,\vec x]&=-\int_{\mathbb{R}^{d-1}} \text{d}^{d-1}\vec x'\int_{\eta_{\text p}}^{\eta_{\text f}} \text{d}\eta' \,a[\eta']^{\frac d2}G^{(\gamma,+)}_{ij}[\eta,\eta';R]\,J_j[\eta',\vec x'],
\label{Spin1_Convolution_Cosmology}
\end{align}
where the time interval of integration $(\eta_{\text p},\eta_{\text f})$ covers all the possible values of $\eta$ for an expanding universe, and the spin-1 effective Green's function $G^{(\gamma,+)}_{ij}$ is given by
\begin{align}
G^{(\gamma,+)}_{ij}[\eta,\eta';\vec R]&=-\Theta[T]C^{(\gamma)}_{ij}[\eta,\eta';\vec R], \notag \\
C^{(\gamma)}_{ij}[\eta,\eta';\vec R]&=\delta_{ij}C^{(\gamma)}_{1,d}[\eta,\eta';R]+\partial_i\partial_jC^{(\gamma)}_{2,d}[\eta,\eta';R],
\label{Cij_Cosmology}
\end{align}
with $C^{(\gamma)}_{1,d}$ and $C^{(\gamma)}_{2,d}$, respectively, defined by
{\allowdisplaybreaks
\begin{align}
&C^{(\gamma)}_{1,d}[\eta,\eta';R]\equiv\int_{\mathbb{R}^{d-1}}\frac{\text{d}^{d-1}\vec{k}}{(2\pi)^{d-1}}\,\widetilde C^{(\gamma)}_{1,d}\left[\eta,\eta';|\vec k|\right]\,e^{i\vec{k}\cdot\vec{R}},
\label{C1_Photon_Cosmology}\\
&C^{(\gamma)}_{2,d}[\eta,\eta';R]\equiv\int_{\mathbb{R}^{d-1}}\frac{\text{d}^{d-1}\vec{k}}{(2\pi)^{d-1}}\,\widetilde C^{(\gamma)}_{1,d}\left[\eta,\eta';|\vec k|\right]\,\frac{e^{i\vec{k}\cdot\vec{R}}}{\vec k^2};
\label{C2_Photon_Cosmology}
\end{align}}%
the Fourier transform of $C^{(\gamma)}_{1,d}$ can be equivalently expressed in terms of the following decomposition,
\begin{align}
\widetilde C^{(\gamma)}_{1,d}\left[\eta,\eta';|\vec k|\right]=i\left(v^{(\gamma)}_{|\vec k|}[\eta]\,v^{(\gamma)*}_{|\vec k|}[\eta']-v^{(\gamma)*}_{|\vec k|}[\eta]\,v^{(\gamma)}_{|\vec k|}[\eta']\vphantom{\dot A}\right),
\label{C1_Photon_Cosmology_ModeDecomposition}
\end{align}
where $v^{(\gamma)}_{|\vec k|}$ are in fact the mode functions of the massless scalar field that satisfies the homogeneous (i.e., $\Gamma_i=0$) form of the wave equation \eqref{Spin1_WaveEq2_Cosmology}, the Fourier-space version of which is therefore obeyed by $v^{(\gamma)}_{|\vec k|}$ itself and $\widetilde C^{(\gamma)}_{1,d}$; moreover, $v^{(\gamma)}_{|\vec k|}$ have been normalized so that the initial condition imposed on the time derivative of $\widetilde C^{(\gamma)}_{1,d}$, namely $\dot{\widetilde C}^{(\gamma)}_{1,d}\big|_{\eta=\eta'}=1$, coincides with the Wronskian condition for the mode functions, $v^{(\gamma)}_{|\vec k|}\dot{v}^{(\gamma)*}_{|\vec k|}-v^{(\gamma)*}_{|\vec k|}\dot{v}^{(\gamma)}_{|\vec k|}=i$. In this language, the properties of $C^{(\gamma)}_{1,d}$ and $C^{(\gamma)}_{2,d}$ become more transparent. Specifically, both of the scalar functions obey the homogeneous wave equation associated with the wave operator in eq.~\eqref{Spin1_WaveEq2_Cosmology}, and the equal-time initial conditions for $C^{(\gamma)}_{1,d}$, $C^{(\gamma)}_{2,d}$, and their velocities, can be immediately read off,
{\allowdisplaybreaks
\begin{align}
&C^{(\gamma)}_{1,d}\big|_{\eta=\eta'}=C^{(\gamma)}_{2,d}\big|_{\eta=\eta'}=0, \label{C123_Photon_InitialCondition}\\
&\dot C^{(\gamma)}_{1,d}\big|_{\eta=\eta'}=-\partial_{\eta'} C^{(\gamma)}_{1,d}\big|_{\eta=\eta'}=\delta^{(d-1)}[\vec x-\vec x'], \label{dotC1_Photon_InitialCondition} \\
&\dot C^{(\gamma)}_{2,d}\big|_{\eta=\eta'}=-\partial_{\eta'} C^{(\gamma)}_{2,d}\big|_{\eta=\eta'}=-G^{\mathrm{(E)}}_d[R].
\label{dotC2_Photon_InitialCondition}
\end{align}}%
It turns out that $C^{(\gamma)}_{1,d}$ and $C^{(\gamma)}_{2,d}$ are the cosmological generalization of their Minkowski counterparts \eqref{C1_Minkowski} and \eqref{C2_Minkowski}, and $C^{(\gamma)}_{ij}$ is connected to the commutator of the massless spin-1 photons in the cosmological background,
\begin{align}
\big[\alpha_i[x], \alpha_j[x']\big] = -i \big(a[\eta]a[\eta']\big)^{-\frac{d-4}2} C^{(\gamma)}_{ij}[\eta,\eta';\vec R].
\label{Spin1_Commutator_Cosmology}
\end{align}
When specializing to the constant-$w$ cosmologies considered in this section, where $\mathcal H=2/(q_w\eta)$ with $(\eta_{\text p},\eta_{\text f})=(-\infty,0)$ for $w<-(d-3)/(d-1)$ and $(\eta_{\text p},\eta_{\text f})=(0,\infty)$ for $w>-(d-3)/(d-1)$, the analytic solution of the massless scalar Green's function $G^{(\gamma,+)}_d=-\Theta[T]C^{(\gamma)}_{1,d}$ for $d\geq3$ has been derived in \cite{Chu:2016ngc} via Nariai's ansatz (see eqs.~(205) and (206) of \cite{Chu:2016ngc}), instead of computing eq.~\eqref{C1_Photon_Cosmology}; the result shows that $G^{(\gamma,+)}_d$ contains pure causal signals propagating either on or inside the light cone. Since $C^{(\gamma)}_{1,d}$ is known, we may employ the time integral method to compute $C^{(\gamma)}_{2,d}$ without resorting to tackling the integral in eq.~\eqref{C2_Photon_Cosmology} directly. The homogeneous counterpart of eqs.~\eqref{Spin1_WaveEq2_Cosmology} together with the relation $-\vec{\nabla}^2 C_{2,d}^{(\gamma)} = C_{1,d}^{(\gamma)}$ leads us to
{\allowdisplaybreaks
\begin{align}
\frac{-\partial_\eta\bigg(a^{2\alpha}\,\partial_\eta\left(a^{-\alpha}\, C_{2,d}^{(\gamma)} \vphantom{\dot A}\right)\bigg)}{a^\alpha}&=C_{1,d}^{(\gamma)},
\qquad\qquad
\alpha=\frac{d-4}2 .
\end{align}}%
With the initial conditions \eqref{C123_Photon_InitialCondition} and \eqref{dotC2_Photon_InitialCondition}, we are able to write down
{\allowdisplaybreaks
\begin{align}
C^{(\gamma)}_{2,d}[\eta,\eta';R]=\,&-a[\eta]^{\frac{d-4}2}\int^\eta_{\eta'}\text{d}\eta_2\,a[\eta_2]^{-(d-4)}
\int^{\eta_2}_{\eta'}\text{d}\eta_1\,a[\eta_1]^{\frac{d-4}2}C^{(\gamma)}_{1,d}[\eta_1,\eta';R] \notag\\
&-G^{\mathrm{(E)}}_{d}[R]\left(\frac{a[\eta]}{a[\eta']}\right)^{\frac{d-4}2}\int^\eta_{\eta'}\text{d}\eta_1\,\left(\frac{a[\eta']}{a[\eta_1]}
\right)^{d-4},
\label{C2_Photon_TimeIntegral_Cosmology}
\end{align}}%
If we only consider the retarded piece ($\eta>\eta'$) of $C^{(\gamma)}_{2,d}$, then as explained in \S \eqref{MinkowskiSpacetime} since $C^{(\gamma)}_{1,d}[\eta,\eta';R]$ is strictly causal (i.e., its retarded part is proportional to $\Theta[T-R^-]$), the first term of eq.~\eqref{C2_Photon_TimeIntegral_Cosmology} would in turn yield a strictly causal contribution. Whereas the second term is a smooth function of spacetime, consisting of the signals that pervade all physical spacetime points with $\eta>\eta'$, including the region outside the light cone. Hence, the time-integral method has cleanly elucidated the causal structure of $C^{(\gamma)}_{2,d}$, even if the integrals cannot be performed in closed form: acausality is present for all $d\geq3$ and is encoded only in the second term of eq.~\eqref{C2_Photon_TimeIntegral_Cosmology}. Moreover, it also implies the spin-1 quantum operator violates micro-causality in constant-$w$ cosmologies (cf.~eq.~\eqref{Spin1_Commutator_Cosmology}).
{\bf Scalar } \qquad The Poisson's equation \eqref{EMPoisson_Cosmology} for scalar $\Phi$ can be solved immediately by utilizing the Euclidean Green's function,
\begin{align}
\Phi[\eta,\vec x]=\int_{\mathbb{R}^{d-1}} \text{d}^{d-1}\vec{x}'\,a[\eta]^2G^{(\mathrm{E})}_{d}[R]\, J_0[\eta,\vec x'],
\label{EMPhi_Cosmology}
\end{align}
which, except for the factor $a[\eta]^2$, is precisely the same as its Minkowski counterpart \eqref{EMPhi_Minkowski}. This in turn implies that, despite the distinct waveforms of the spin-1 field in cosmology, the acausal portion of its velocity must take such a simple form to ensure a causal electric field, as we will demonstrate below.
{\bf Faraday Tensor} \qquad Let us now turn to the causal structure of the Faraday tensor $F_{\mu\nu}$ within the cosmological context, where $F_{\mu\nu}$ in the gauge-invariant formalism still takes the form of eq.~\eqref{Maxwell_FaradayTensor}. To make the causality analysis more transparent, we first re-cast the spin-1 photon field \eqref{Spin1_Convolution_Cosmology} in the following convolution that involves both time and spatial components of the electric current $J_\mu$,
\begin{align}
\alpha_i[\eta,\vec x]=-\int_{\mathbb{R}^{d-1}} \text{d}^{d-1}\vec x'\int_{\eta_{\text p}}^{\eta_{\text f}} \text{d} &\eta' \,a[\eta]^{-\frac{d-4}2}a[\eta']^{\frac d2}\bigg(G^{(\gamma,+)}_{d}J_i[\eta',\vec x'] \notag\\
& \qquad +\Theta[T]\,a[\eta']^{\frac{d-4}2}\partial_{\eta'}\left(a[\eta']^{-\frac {d-4}2}\partial_iC^{(\gamma)}_{2,d}\right)J_0[\eta',\vec x']\bigg),
\label{Spin1_Convolution2_Cosmology}
\end{align}
where we have employed the conservation law \eqref{ChargeConservaion_Cosmology} and removed the boundary terms that arise from integration by parts.\footnote{With the assumption that the electric current is sufficiently localized, the boundary contributions evaluated at spatial infinity are zero, whereas the ones at past infinity, namely at $\eta'=-\infty$ for $w<-(d-3)/(d-1)$ and $\eta' = 0$ for $w>-(d-3)/(d-1)$, still require further justification. However, those boundary terms at past infinity are in fact the surface integrals of $C^{(\gamma)}_{2,d}$ at $\eta'= - \infty$ or $\eta' = 0$, indicating the fact that they satisfy the homogeneous wave equation and do not alter the exact inhomogeneous solution to the spin-1 wave equation. We hope to clarify this issue further in our later work.
} The resulting expression is in fact the cosmological generalization of eq.~\eqref{Spin1_Convolution2_Minkowski}, and notice that, due to the lack of time-translation symmetry, the time derivative of $C^{(\gamma)}_{1,d}$ or $C^{(\gamma)}_{2,d}$ with respect to $\eta'$ cannot simply be replaced with the negative of that with respect to $\eta$.
Start with the $(ij)$ components of $F_{\mu\nu}$. Taking the spatial curl of either eq.~\eqref{Spin1_Convolution_Cosmology} or eq.~\eqref{Spin1_Convolution2_Cosmology} gives rise to the magnetic field in the cosmological background (cf.~eq.~\eqref{Maxwell_FaradayTensor}),
\begin{align}
F_{ij}[\eta,\vec x]&=2\partial_{[i}\alpha_{j]}=-2\int_{\mathbb{R}^{d-1}}\text{d}^{d-1}\vec x'\int_{\eta_{\text p}}^{\eta_{\text f}} \text{d} \eta' \,a[\eta]^{-\frac{d-4}2}a[\eta']^{\frac d2}\partial_{[i}G^{(\gamma,+)}_{d}J_{j]}[\eta',\vec x'],
\end{align}
which is completely causal for $d\geq3$, since the acausal effect of the spin-1 photons, encapsulated in the second term of eq.~\eqref{Spin1_Convolution_Cosmology} or eq.~\eqref{Spin1_Convolution2_Cosmology}, is eliminated by the curl operation here -- as was the case in the Minkowski background. Next, to obtain the electric field $F_{0i}$, we first compute the velocity of the spin-1 field by taking the time derivative of eq.~\eqref{Spin1_Convolution2_Cosmology}, with the expression \eqref{C2_Photon_TimeIntegral_Cosmology} substituted for $C^{(\gamma)}_{2,d}$,
\begin{align}
\dot\alpha_i[\eta,\vec x]=\,&-\int_{\mathbb{R}^{d-1}}\text{d}^{d-1}\vec x'\int_{\eta_{\text p}}^{\eta_{\text f}} \text{d} \eta' \,a[\eta]^{-\frac{d-4}2}a[\eta']^{\frac d2}\Bigg\{\left(\dot G^{(\gamma,+)}_d[\eta,\eta';R]-\frac{(d-4)}{q_w\eta} G^{(\gamma,+)}_d[\eta,\eta';R]\right)J_i[\eta',\vec x'] \notag\\
& +\left(\frac{a[\eta']}{a[\eta]}\right)^{\frac{d-4}2}\partial_{\eta'}\bigg(a[\eta']^{-\frac {d-4}2}\int^{\eta}_{\eta'}\text{d}\eta_1\,a[\eta_1]^{\frac{d-4}2}
\partial_iG^{(\gamma,+)}_d[\eta_1,\eta';R]\bigg)J_0[\eta',\vec x']\Bigg\} \notag\\
&-\int_{\mathbb{R}^{d-1}} \text{d}^{d-1}\vec x'\,a[\eta]^{2}\partial_iG^{\mathrm{(E)}}_d[R]J_0[\eta,\vec x'],
\label{DotSpin1_Cosmology}
\end{align}
where the spacetime integral involving $G^{(\gamma,+)}_d$ is purely causal, whereas the last term is an acausal instant-time-surface integral. Then, summing the expression \eqref{DotSpin1_Cosmology} for $\dot\alpha_i$ and the gradient of $\Phi$ in eq.~\eqref{EMPhi_Cosmology} amounts to canceling the acausal term in the last line of eq.~\eqref{DotSpin1_Cosmology}, and yields the purely causal electric field $F_{0i}$,
{\allowdisplaybreaks
\begin{align}
F_{0i}[\eta,\vec x]=\,&\dot\alpha_i[\eta,\vec x]+\partial_i\Phi[\eta,\vec x] \notag\\
=\,&-\int_{\mathbb{R}^{d-1}}\text{d}^{d-1}\vec x'\int_{\eta_{\text p}}^{\eta_{\text f}} \text{d} \eta' \,a[\eta]^{-\frac{d-4}2}a[\eta']^{\frac d2}\Bigg\{\left(\dot G^{(\gamma,+)}_d[\eta,\eta';R]-\frac{(d-4)}{q_w \eta} G^{(\gamma,+)}_d[\eta,\eta';R]\right)J_i[\eta',\vec x'] \notag\\
&+\left(\frac{a[\eta']}{a[\eta]}\right)^{\frac{d-4}2}\partial_{\eta'}\bigg(a[\eta']^{-\frac {d-4}2}\int^{\eta}_{\eta'}\text{d}\eta_1\,a[\eta_1]^{\frac{d-4}2}
\partial_iG^{(\gamma,+)}_d[\eta_1,\eta';R]\bigg)J_0[\eta',\vec x']\Bigg\}.
\label{F0i_Cosmology}
\end{align}}%
This result extends eq.~\eqref{F0i_CausalSpin1_Minkowski} to the cosmological context; i.e., the electric field is still the causal piece of the velocity of the spin-1 photon $\alpha_i$. Furthermore, as a simple check of consistency, one can show eq.~\eqref{F0i_Cosmology} does reduce to its Minkowski counterpart \eqref{F0i_Minkowski}, by setting $a\to1$ and assuming $G^{(\gamma,+)}_d$ takes its Minkowski form $G^+_d$ with time-translation symmetry.
Although we have shown explicitly that causality is preserved for the electromagnetic observables in constant-$w$ spatially-flat cosmologies, the second line of eq.~\eqref{F0i_Cosmology} still involves a time integral from some initial time $\eta'$ to the present $\eta$. However, by introducing a new Green's function $G^{(\gamma|\text{time},+)}_d$ obeying
\begin{align}
\bigg\{\partial^2 + \frac{(d-4) \left(q_w+d-4 \right)}{q^2_w \eta^2} \bigg\}G^{(\gamma|\text{time},+)}_d[\eta,\eta';R]=\delta^{(d)}[x-x'],
\label{Gt+_ElectricField_Cosmology}
\end{align}
the effective Green's function of $F_{0i}$ in eq.~\eqref{F0i_Cosmology} can be further reduced to a localized form,
\begin{align}
F_{0i}[\eta,\vec x]=\,-\int_{\mathbb{R}^{d-1}}\text{d}^{d-1}\vec x' & \int_{\eta_{\text p}}^{\eta_{\text f}} \text{d} \eta' \,a[\eta]^{-\frac{d-4}2}a[\eta']^{\frac d2}\Bigg\{\bigg(\dot G^{(\gamma,+)}_d[\eta,\eta';R] \notag\\
&-\frac{(d-4)}{q_w \eta} G^{(\gamma,+)}_d[\eta,\eta';R]\bigg)J_i[\eta',\vec x'] -\partial_iG^{(\gamma|\text{time},+)}_d[\eta,\eta';R] J_0[\eta',\vec x']\Bigg\}.
\label{F0i_Cosmology_Localized}
\end{align}
We will leave the detailed derivation of eq.~\eqref{F0i_Cosmology_Localized} to our future work \cite{ChuLiuInPrep}. It turns out that the result \eqref{F0i_Cosmology_Localized} is exactly consistent with $F_{0i}$ computed directly using the solution of the generalized Lorenz-gauge vector potential $A_\mu$ in \cite{Chu:2016ngc}; the analytic solutions of $G^{(\gamma,+)}_d = -\Theta[T]\mathcal G^{(\gamma|\text{space})}$ and $G^{(\gamma|\text{time},+)}_d = -\Theta[T]\mathcal G^{(\gamma|\text{time})}$ can be found in eqs.~(B38), (B39), (B40), and (B41) of \cite{Chu:2016ngc}.
\subsection{Linearized Gravitation}
Linearized gravity coupled to the isolated astrophysical sources in cosmology is, in fact, a more sophisticated system, due to the additional first-order perturbations of the very fluid driving cosmic expansion. A detailed analysis of this linearized gravitational system, described in terms of the perturbed metric \eqref{PerturbedFLRW}, has been performed in \cite{Chu:2016ngc} within a constant equation-of-state universe, where the background perfect fluid is modeled through an effective field theory description. As explicitly demonstrated in \cite{Chu:2016ngc}, the field equations for the gauge-invariant metric perturbations, deduced from the full linearized Einstein's equations, can be put in a decoupled form with no perturbed fluid quantities involved. The set of resulting equations then reveals that the dynamics of Bardeen scalar potentials varies in different ranges of the equation-of-state $w$. Here, we will focus on the causal structure of the system in the de Sitter ($w=-1$) and relativistic-fluid ($0<w\leq1$) cases for all $d\geq4$.
{\bf Field Equations} \qquad In terms of the gauge-invariant variables formed in eqs.~\eqref{Bardeen_Psi}, \eqref{Bardeen_Phi}, and \eqref{Bardeen_VandDij}, the relevant equations are derived from Einstein's equations of this cosmological system, linearized about the spatially-flat background with a constant equation-of-state $w$. Based on the results obtained in \cite{Chu:2016ngc}, the character of the decoupled field equation of the Bardeen scalar $\Psi$ for $w=-1$ is distinct from the $0<w\leq1$ case. Hence, in what follows, we will consider these two cases separately.
{\it Field equations for $w=-1$} \qquad When $w=-1$, there is no fluid and the background geometry is de Sitter spacetime. The corresponding gauge-invariant equations are given by eqs.~\eqref{PhiPsiRelationship} and \eqref{Vi_Poisson_Minkowski}, both of which remain unchanged, and
{\allowdisplaybreaks
\begin{align}
(d-2)\vec\nabla^2\Psi&=8\pi G_\text N\left(\rho+(d-1)\mathcal H\Sigma \vphantom{\dot h} \right), \label{Psi_Poisson_Cosmology} \\
-\ddot D_{ij}-(d-2)\mathcal H\dot D_{ij}+\vec\nabla^2 D_{ij}
&= a^2 \, \overline\Box^{(\text S)} D_{ij} \notag\\
&\equiv \frac{a^2\,\partial_\mu \left( \sqrt{|\overline g|} \overline g^{\mu\nu} \partial_\nu D_{ij} \right)}{\sqrt{|\overline g|}} =-16\pi G_\text N\sigma_{ij}
\label{Spin-2_WaveEquation_Cosmology},
\end{align}}%
where $\overline g$ denotes the determinant of the background metric $\overline g_{\mu\nu}=a^2\eta_{\mu\nu}$, the scale factor is $a[\eta]=-1/(H\eta)$, with $H$ denoting the constant Hubble parameter, and the conformal Hubble parameter reads $\mathcal H=-1/\eta$. Compared with their Minkowski counterparts \eqref{Psi_Poisson_Minkowski} and \eqref{Spin-2_WaveEquation_Minkowski}, both $\Psi$ and $D_{ij}$ retain similar dynamics in the de Sitter case here; the Bardeen scalar $\Psi$ still obeys a Poisson-type equation, sourced not only by the local energy density $\rho$, but also by the non-local longitudinal part $\Sigma$ of $^{(\text{a})}T_{0i}$, while the spin-2 field $D_{ij}$ obeys a tensor wave equation in de Sitter background.
{\it Field equations for $0<w\leq1$} \qquad For a physical relativistic equation-of-state within the range $0<w\leq1$, the field equations \eqref{PhiPsiRelationship}, \eqref{Vi_Poisson_Minkowski}, and \eqref{Spin-2_WaveEquation_Cosmology} still hold with $\mathcal H=2/(q_w\eta)$ -- recall eq.~\eqref{ScaleFactor} -- but the Bardeen scalar $\Psi$ now obeys a dynamical wave-like equation \cite{Chu:2016ngc}, instead of being governed by a Poisson-type one,
\begin{align}
-\ddot\Psi-\big(q_w+d-2\big)\mathcal H\dot\Psi+w\vec\nabla^2\Psi
=-8\pi G_\text N\bigg(\frac{\partial_0\left(a^{d-2}\Sigma\right)}{(d-2)a^{d-2}}-\frac{w\rho}{(d-2)}+\mathcal H\dot{\Upsilon}\bigg),
\label{Psi_WaveEq_Cosmology}
\end{align}
\footnote{The left-hand side of eq.~\eqref{Psi_WaveEq_Cosmology} can also be re-expressed in terms of the d'Alembertian ${}^{(\Psi)}\Box$ associated with
\begin{align}
{}^{(\Psi)}g_{\mu\nu} \text{d} x^\mu \text{d} x^\nu =\left(\frac{\eta}{\eta_0}\right)^{\frac{4\left(q_w+d-2\right)}{(d-2)q_w}}\left(-\text{d}\eta^2 + w^{-1} \text{d}\vec{x}\cdot\text{d}\vec{x} \right) . \notag
\end{align}
See eq.~(128) of \cite{Chu:2016ngc}.}which implies the existence of the acoustic cone, $\sqrt w\,T=R$, on which these scalar gravitational signals propagate at speed $\sqrt w$. It is worth noting that there exists no counterpart of this phenomenon in Minkowski and de Sitter spacetimes; this change in the character of the scalar equation for $\Psi$ is presumably tied to the dynamics of the background fluid.
In addition, as we have already noticed from our previous calculations, the energy-momentum conservation of the astrophysical sources, $\overline\nabla^\mu\,^{(\text{a})}T_{\mu\nu}=0$, where $\overline\nabla_\mu$ is the covariant derivative associated with the background metric, will be crucial in extracting the relevant effective Green's functions and their causal structures. For later convenience, in spatially-flat cosmologies, the conservation law can be re-expressed as
\begin{align}
\partial_i{}^{(\text{a})}T_{ij}&=\frac{\partial_0\left(a^{d-2}\,^{(\text{a})}T_{0j}\right)}{a^{d-2}}, \label{EnergyMomentumConservaion_Time_Cosmology}\\
\partial_j{}^{(\text{a})}T_{0j}&=\frac{\partial_0\left(a^{d-2}\,^{(\text{a})}T_{00}\right)}{a^{d-2}}-\mathcal H\left({}^{(\text{a})}T_{00}-{}^{(\text{a})}T_{ll}\right).
\label{EnergyMomentumConservaion_Spatial_Cosmology}
\end{align}
{\bf Spin-2 Gravitons} \qquad The spin-2 wave equation for either $w=-1$ or $0<w\leq1$ takes the same form as eq.~\eqref{Spin-2_WaveEquation_Cosmology} with $\mathcal H=2/(q_w\eta)$; therefore, for both cases, the method used in the previous photon computation can be applied directly to solving eq.~\eqref{Spin-2_WaveEquation_Cosmology} via the spin-2 effective Green's function convolved against the local matter sources.
The first step is to re-cast the tensor wave equation \eqref{Spin-2_WaveEquation_Cosmology} into a conformal re-scaled form,
\begin{align}
\left\{\partial^2 + \frac{(d-2) \left( d-2-q_w \right)}{q^2_w \eta^2} \right\}\left(a^{\frac{d-2}2}D_{ij}\right)=-16\pi G_\text Na^{\frac{d-2}2}\sigma_{ij},
\label{Spin-2_WaveEquation2_Cosmology}
\end{align}
and then, a similar procedure of implementing the local Fourier-space projection of eq.~\eqref{Spin-2_WaveEquation2_Cosmology} leads us to the following convolution for the spin-2 gravitons,
\begin{align}
a[\eta]^{\frac{d-2}2}D_{ij}[\eta,\vec x]&=-16\pi G_\text N\int_{\mathbb{R}^{d-1}} \text{d}^{d-1}\vec x'\int_{\eta_{\text p}}^{\eta_{\text f}} \text{d}\eta' \,a[\eta']^{\frac{d-2}2}G^{(g,+)}_{ijmn}[\eta,\eta';R]{}^{(\text{a})}T_{mn}[\eta',\vec x'],
\label{Spin2_Convolution_Cosmology}
\end{align}
where the time interval $(\eta_{\text p},\eta_{\text f})$ corresponds to $(-\infty,0)$ for $w=-1$ and $(0,\infty)$ for $0<w\leq1$, and $G^{(g,+)}_{ijmn}$ refers to the spin-2 effective Green's function,
{\allowdisplaybreaks
\begin{align}
G^{(g,+)}_{ijmn}[\eta,\eta';\vec R]&=-\Theta[T]C^{(g)}_{ijmn}[\eta,\eta';\vec R], \notag\\
C^{(g)}_{ijmn}[\eta,\eta';\vec R]&=\bigg(\delta_{m(i}\delta_{j)n}-\frac{\delta_{ij}\delta_{mn}}{d-2}\bigg)C^{(g)}_{1,d}[\eta,\eta';R]
+\bigg(\delta_{m(i}\partial_{j)}\partial_n+\delta_{n(i}\partial_{j)}\partial_m \notag\\
&\qquad \qquad -\frac{\delta_{ij}\partial_m\partial_n-\delta_{mn}\partial_i\partial_j}{d-2}
\bigg)C^{(g)}_{2,d}[\eta,\eta';R] +\left(\frac{d-3}{d-2}\right)\partial_i\partial_j\partial_m\partial_nC^{(g)}_{3,d}[\eta,\eta';R],
\label{Cijmn_Cosmology}
\end{align}}%
which has the same tensor structure as eq.~\eqref{Cijmn_Minkowski}, with the scalar functions $C^{(g)}_{1,d}$, $C^{(g)}_{2,d}$, and $C^{(g)}_{3,d}$ generalized to their cosmological versions,
\begin{align}
&C^{(g)}_{1,d}[\eta,\eta';R]\equiv\int_{\mathbb{R}^{d-1}}\frac{\text{d}^{d-1}\vec{k}}{(2\pi)^{d-1}}\,\widetilde C^{(g)}_{1,d}\left[\eta,\eta';|\vec k|\right]\,e^{i\vec{k}\cdot\vec{R}},
\label{C1_Graviton_Cosmology}\\
&C^{(g)}_{2,d}[\eta,\eta';R]\equiv\int_{\mathbb{R}^{d-1}}\frac{\text{d}^{d-1}\vec{k}}{(2\pi)^{d-1}}\,\widetilde C^{(g)}_{1,d}\left[\eta,\eta';|\vec k|\right]\,\frac{e^{i\vec{k}\cdot\vec{R}}}{\vec k^2},
\label{C2_Graviton_Cosmology}\\
&C^{(g)}_{3,d}[\eta,\eta';R]\equiv\int_{\mathbb{R}^{d-1}}\frac{\text{d}^{d-1}\vec{k}}{(2\pi)^{d-1}}\,\widetilde C^{(g)}_{1,d}\left[\eta,\eta';|\vec k|\right]\,\frac{e^{i\vec{k}\cdot\vec{R}}}{\vec k^4};
\label{C3_Graviton_Cosmology}
\end{align}
the Fourier transform of $C^{(g)}_{1,d}$ is denoted by $\widetilde C^{(g)}_{1,d}$. For $G_{ij mn}^{(g,+)}$ to be a Green's function, the $C^{(g)}$s in eqs.~\eqref{Cijmn_Cosmology} must obey the homogeneous version of eq.~\eqref{Spin-2_WaveEquation2_Cosmology}. This implies, for instance,
\begin{align}
\widetilde C^{(g)}_{1,d}\left[\eta,\eta';|\vec k|\right]=i\left(v^{(g)}_{|\vec k|}[\eta]\,v^{(g)*}_{|\vec k|}[\eta']-v^{(g)*}_{|\vec k|}[\eta]\,v^{(g)}_{|\vec k|}[\eta']\vphantom{\dot A}\right),
\label{C1_Graviton_Cosmology_ModeDecomposition}
\end{align}
where $v^{(g)}_{|\vec k|}$ are the mode functions obeying the same homogeneous wave equation as $\widetilde C^{(g)}_{1,d}$, and are normalized to match the Wronskian condition $v^{(g)}_{|\vec k|}\dot{v}^{(g)*}_{|\vec k|}-v^{(g)*}_{|\vec k|}\dot{v}^{(g)}_{|\vec k|}=i$, or the initial condition $\dot{\widetilde C}^{(g)}_{1,d}\big|_{\eta=\eta'}=1$. Furthermore, because their Fourier transforms indicate $-\vec{\nabla}^2 C_{3,d}^{(g)} = C_{2,d}^{(g)}$ and $-\vec{\nabla}^2 C_{2,d}^{(g)} = C_{1,d}^{(g)}$, the homogeneous equations for $C_{1,d}^{(g)}$, $C_{2,d}^{(g)}$, and $C_{3,d}^{(g)}$ translate to the relations:
\begin{align}
\label{Cg12_Cosmology}
\frac{-\partial_\eta\bigg(a^{2\alpha}\,\partial_\eta\left(a^{-\alpha}\, C_{2,d}^{(g)} \vphantom{\dot A}\right)\bigg)}{a^\alpha}&= C_{1,d}^{(g)},
\\
\label{Cg13_Cosmology}
\frac{-\partial_\eta\bigg(a^{2\alpha}\,\partial_\eta\left(a^{-\alpha}\, C_{3,d}^{(g)} \vphantom{\dot A}\right)\bigg)}{a^\alpha}&= C_{2,d}^{(g)},
\qquad \alpha=\frac{d-2}2 .
\end{align}
We may now apply the time-integral method here to relate $C^{(g)}_{2,d}$ and $C^{(g)}_{3,d}$ to $C^{(g)}_{1,d}$, without evaluating their Fourier transform integrals, since $C_{1,d}^{(g)}$ itself has already been derived in \cite{Chu:2016ngc} (cf.~eqs.~(112) and (113) therein). As we shall witness, this will also yield a clean elucidation of their causal structure. To this end, note that the initial conditions for the $C^{(g)}$s at $\eta=\eta'$ may be identified from their Fourier transforms and the anti-symmetric nature of the mode decomposition in eq.~\eqref{C1_Graviton_Cosmology_ModeDecomposition}:
{\allowdisplaybreaks
\begin{align}
&C^{(g)}_{1,d}\big|_{\eta=\eta'}=C^{(g)}_{2,d}\big|_{\eta=\eta'}=C^{(g)}_{3,d}\big|_{\eta=\eta'}=0, \label{C123_Graviton_InitialCondition}\\
&\dot C^{(g)}_{1,d}\big|_{\eta=\eta'}=-\partial_{\eta'} C^{(g)}_{1,d}\big|_{\eta=\eta'}=\delta^{(d-1)}[\vec x-\vec x'], \label{dotC1_Graviton_InitialCondition} \\
&\dot C^{(g)}_{2,d}\big|_{\eta=\eta'}=-\partial_{\eta'} C^{(g)}_{2,d}\big|_{\eta=\eta'}=-G^{\mathrm{(E)}}_d[R],
\label{dotC2_Graviton_InitialCondition} \\
&\dot C^{(g)}_{3,d}\big|_{\eta=\eta'}=-\partial_{\eta'} C^{(g)}_{3,d}\big|_{\eta=\eta'}=D_d[R].
\label{dotC3_Graviton_InitialCondition}
\end{align}}%
With eqs.~\eqref{C123_Graviton_InitialCondition}, \eqref{dotC2_Graviton_InitialCondition}, and \eqref{dotC3_Graviton_InitialCondition} imposed, $C^{(g)}_{2,d}$ and $C^{(g)}_{3,d}$ can both be expressed in terms of (the known) $C^{(g)}_{1,d}$ by integrating eqs.~\eqref{Cg12_Cosmology} and \eqref{Cg13_Cosmology},
{\allowdisplaybreaks
\begin{align}
C^{(g)}_{2,d}[\eta,\eta';R]=\,&-a[\eta]^{\frac {d-2}2}\int^\eta_{\eta'}\text{d}\eta_2\,a[\eta_2]^{-(d-2)}\int^{\eta_2}_{\eta'}\text{d}\eta_1\,a[\eta_1]^{\frac {d-2}2}C^{(g)}_{1,d}[\eta_1,\eta';R] \notag\\
&-G^{(\mathrm E)}_d[R]\left(\frac{a[\eta]}{a[\eta']}\right)^{\frac {d-2}2}\int^\eta_{\eta'}\text{d}\eta_1\left(\frac{a[\eta']}{a[\eta_1]}\right)^{d-2},
\label{C2_Graviton_TimeIntegral_Cosmology} \\
C^{(g)}_{3,d}[\eta,\eta';R]=\,&a[\eta]^{\frac {d-2}2}\int^\eta_{\eta'}\text{d}\eta_4\,a[\eta_4]^{-(d-2)}\int^{\eta_4}_{\eta'}\text{d}\eta_3\,a[\eta_3]^{d-2}\int^{\eta_3}_{\eta'}\text{d}\eta_2
\,a[\eta_2]^{-(d-2)}\int^{\eta_2}_{\eta'}\text{d}\eta_1\,a[\eta_1]^{\frac {d-2}2}C^{(g)}_{1,d}[\eta_1,\eta';R] \notag\\
&+G^{(\mathrm E)}_d[R]\left(\frac{a[\eta]}{a[\eta']}\right)^{\frac{d-2}2}\int^\eta_{\eta'}\text{d}\eta_3\,a[\eta_3]^{-(d-2)}\int^{\eta_3}_{\eta'}\text{d}\eta_2
\,a[\eta_2]^{d-2}\int^{\eta_2}_{\eta'}\text{d}\eta_1\left(\frac{a[\eta']}{a[\eta_1]}\right)^{d-2} \notag\\
&+D_d[R]\left(\frac{a[\eta]}{a[\eta']}\right)^{\frac {d-2}2}\int^\eta_{\eta'}\text{d}\eta_1\left(\frac{a[\eta']}{a[\eta_1]}\right)^{d-2}.
\label{C3_Graviton_TimeIntegral_Cosmology}
\end{align}}%
Although the integrals in the first lines of eqs.~\eqref{C2_Graviton_TimeIntegral_Cosmology} and \eqref{C3_Graviton_TimeIntegral_Cosmology} can be difficult to carry out, just like eq.~\eqref{C2_Photon_TimeIntegral_Cosmology} for $C^{(\gamma)}_{2,d}$, the causal structures of these expressions can still be readily identified. Firstly, the retarded portion of $C^{(g)}_{1,d}$, or equivalently $G^{(g,+)}_d = - \Theta[T]C^{(g)}_{1,d}$, is composed only of the causal signals \cite{Chu:2016ngc}. Then, as already discussed in \S \eqref{MinkowskiSpacetime}, the first terms of both eqs.~\eqref{C2_Graviton_TimeIntegral_Cosmology} and \eqref{C3_Graviton_TimeIntegral_Cosmology} are causal as well when $\eta>\eta'$; whereas the remaining terms of $C^{(g)}_{2,d}$ and $C^{(g)}_{3,d}$, associated with their initial conditions, being non-zero for all $\eta > \eta'$, admit contributions from outside the light cone. For this reason, we see that, after plugging eqs.~\eqref{C2_Graviton_TimeIntegral_Cosmology} and \eqref{C3_Graviton_TimeIntegral_Cosmology} into eq.~\eqref{Cijmn_Cosmology}, the spin-2 effective Green's function $G^{(g,+)}_{ijmn}$ is acausal for all $d\geq4$.
At the quantum level, therefore, the free massless spin-2 operator $D_{ij}$ necessarily violates micro-causality in spatially flat cosmologies:
\begin{align}
\big[D_{ij}[x], D_{mn}[x']\big] = -i \big(a[\eta]a[\eta']\big)^{-\frac{d-2}2} C_{ijmn}^{(g)}[\eta,\eta';\vec R] .
\end{align}
For later convenience, the expression \eqref{Spin2_Convolution_Cosmology} for spin-2 gravitons can be re-cast into another form analogous to their Minkowski counterpart \eqref{Spin2_Convolution2_Minkowski},
{\allowdisplaybreaks
\begin{align}
D_{ij}[\eta,\vec x]&=-16\pi G_\text N\int_{\mathbb{R}^{d-1}} \text{d}^{d-1}\vec x'\int_{\eta_{\text p}}^{\eta_{\text f}} \text{d}\eta'\,\left(\frac{a[\eta']}{a[\eta]}\right)^{\frac {d-2}2}\Bigg\{G^{(g,+)}_{d}\left({}^{(\text{a})}T_{ij}[\eta',\vec x']-\frac{\delta_{ij}}{d-2}{}^{(\text{a})}T_{ll}[\eta',\vec x']\right) \notag\\
&+2\Theta[T]a[\eta']^{\frac{d-2}2}\partial_{\eta'}\left(a[\eta']^{-\frac{d-2}2}\partial_{(i}C^{(g)}_{2,d}\right){}^{(\text{a})}T_{j)0}[\eta',\vec x'] + \frac{\delta_{ij}}{d-2}\Theta[T]a[\eta']^{\frac{d-2}2} \notag\\
&\times \bigg(\partial^2_{\eta'}\left(a[\eta']^{-\frac{d-2}2}C^{(g)}_{2,d}\right)
{}^{(\text{a})}T_{00}[\eta',\vec x'] +\mathcal H[\eta']\,\partial_{\eta'}\left(a[\eta']^{-\frac{d-2}2}C^{(g)}_{2,d}\right)\left({}^{(\text{a})}T_{00}[\eta',\vec x']-{}^{(\text{a})}T_{ll}[\eta',\vec x']\right)\bigg)
\notag\\
&+\frac1{d-2}\Theta[T]\partial_i\partial_jC^{(g)}_{2,d}{}^{(\text{a})}T_{ll}[\eta',\vec x']-\left(\frac{d-3}{d-2}\right)\Theta[T]a[\eta']^{\frac{d-2}2}
\bigg(\partial^2_{\eta'}\left(a[\eta']^{-\frac{d-2}2}\partial_i\partial_jC^{(g)}_{3,d}\right){}^{(\text{a})}T_{00}[\eta',\vec x']\notag\\
&+\mathcal H[\eta']\,\partial_{\eta'}\left(a[\eta']^{-\frac{d-2}2}\partial_i\partial_jC^{(g)}_{3,d}\right)
\left({}^{(\text{a})}T_{00}[\eta',\vec x']-{}^{(\text{a})}T_{ll}[\eta',\vec x']\right) \bigg) \Bigg\} \notag\\
&+\frac{16\pi G_\text N}{d-2}\int_{\mathbb{R}^{d-1}} \text{d}^{d-1}\vec x' \bigg(\delta_{ij}G^{\mathrm{(E)}}_d{} ^{(\text{a})}T_{00}[\eta,\vec x']+(d-3)\partial_i\partial_jD_d{}^{(\text{a})}T_{00}[\eta,\vec x']\bigg),
\label{Spin2_Convolution2_Cosmology}
\end{align}}%
where $^{(\text{a})}T_{ll}\equiv\delta^{ij}\,^{(\text{a})}T_{ij}$, and we have used the conservation laws given in eqs.~\eqref{EnergyMomentumConservaion_Time_Cosmology} and \eqref{EnergyMomentumConservaion_Spatial_Cosmology} as well as the initial conditions \eqref{C123_Graviton_InitialCondition}, \eqref{dotC2_Graviton_InitialCondition}, and \eqref{dotC3_Graviton_InitialCondition}, and removed all the boundary contributions that result from integrations by parts.\footnote{Similar to the spin-1 case, the surface integrals upon integration by parts always involve $C^{(g)}_{2,d}$ and $C^{(g)}_{3,d}$; therefore,
when evaluated at past infinity, i.e., $\eta' = - \infty$ for $w=-1$ or $\eta'=0$ for $0 < w \leq1$, those terms obey the spin-2 homogeneous wave equation and will not change the inhomogeneous solution obtained here.
} The convolution in eq.~\eqref{Spin2_Convolution2_Cosmology} now involves different components of ${}^{(\text{a})}T_{\mu\nu}$ from that in eq.~\eqref{Spin2_Convolution_Cosmology}. Below, the former would help us identify how the acausal portions of the spin-2 contribution to the Weyl tensor are canceled by those from other gauge-invariant variables.
{\bf Bardeen Scalars} \qquad Unlike the spin-2 gravitons, the field equation for the Bardeen scalar $\Psi$ no longer takes an universal form for both $w=-1$ and $0<w\leq1$, which will be solved separately for each case in the following. Once the solution of $\Psi$ is obtained, the other Bardeen scalar potential $\Phi$, related to $\Psi$ via the formula in eq.~\eqref{PhiPsiRelationship}, is given immediately by eq.~\eqref{PhiPsiRelationship_Convolution}.
{\it Solutions for $w=-1$} \qquad In de Sitter background ($w=-1$), the Poisson-type equation \eqref{Psi_Poisson_Cosmology} obeyed by $\Psi$ involves a non-local function $\Sigma$ of the matter source (cf.~eq.~\eqref{Astro_SVT_IofII}), whose momentum-space counterpart for $\vec k\neq \vec0$ reads
\begin{align}
\widetilde\Sigma[\eta,\vec k]&=\frac{k_j}{i\vec k^2}{}^{(\text{a})}\widetilde T_{0j}[\eta,\vec k].
\label{Sigma_Fourier}
\end{align}
Once again, the solution to eq.~\eqref{Psi_Poisson_Cosmology} can be readily derived by implementing the Fourier transform of eq.~\eqref{Psi_Poisson_Cosmology} with eq.~\eqref{Sigma_Fourier},
\begin{align}
\Psi[\eta,\vec x]=\frac{8\pi G_\text N}{d-2} \int_{\mathbb{R}^{d-1}} \text{d}^{d-1}\vec x' \bigg(G^{\mathrm{(E)}}_d[R] {}^{(\text{a})}T_{00}[\eta,\vec x']+(d-1)\mathcal H[\eta]\, \partial_jD_d[R] {}^{(\text{a})}T_{0j}[\eta,\vec x']\bigg),
\label{BardeenPsi_deSitter}
\end{align}
which is again the weighted superposition of local source terms over the equal-time hypersurface.
{\it Solutions for $0<w\leq1$} \qquad When $0<w\leq1$, the Bardeen scalar $\Psi$ becomes dynamical in the sense of being governed by the wave equation \eqref{Psi_WaveEq_Cosmology}, from which we see that the propagation of scalar GWs is in general different than that of the spin-2 ones. However, the strategy of solving the spin-2 wave equation in light of causality still applies here for $\Psi$-waves.
In the same vein, the scalar wave equation \eqref{Psi_WaveEq_Cosmology} can firstly be re-written as a re-scaling form,
\begin{align}
\bigg\{\partial^2_{(w)} + & \frac{ (d-2) \left(q_w + d -2 \right) }{q^2_w\eta^2} \bigg\}\left(a^{\frac12\left(q_w+d-2\right)}\Psi\right) \notag\\
&\qquad =-8\pi G_\text N\,a^{\frac12\left(q_w+d-2\right)}\bigg(\frac{\partial_0\left(a^{d-2}\Sigma\right)}{(d-2)a^{d-2}}-\frac{w\rho}{(d-2)}+\mathcal H\dot{\Upsilon}\bigg),
\label{Psi_WaveEq2_Cosmology}
\end{align}
where $\partial^2_{(w)}\equiv-\partial_\eta^2+w\vec\nabla^2$. Then, using eqs.~\eqref{Upsilon_Fourier_Minkowski} and \eqref{Sigma_Fourier} through the same procedure employed for the spin-2 wave equation, we have
{\allowdisplaybreaks
\begin{align}
a[\eta]&^{\frac12\left(q_w+d-2\right)}\Psi[\eta,\vec x]= \frac{8\pi G_\text N}{d-2} \int_{\mathbb{R}^{d-1}} \text{d}^{d-1}\vec x'\int_0^\infty \text{d}\eta' \, \Theta[T]a[\eta']^{\frac12\left(q_w+d-2\right)} \notag\\
&\times w^{-\frac{d-3}2} \Bigg\{-\partial_jC^{(w)}_{2,d}\left[\eta,\eta';\textstyle{\frac {R}{\sqrt w}}\right] a[\eta']^{-(d-2)} \partial_{\eta'}\left(a[\eta']^{d-2} \,^{(\text{a})}T_{0j}[\eta',\vec x']\right)
-C^{(w)}_{1,d}\left[\eta,\eta';\textstyle{\frac {R}{\sqrt w}}\right] {}^{(\text{a})}T_{00}[\eta',\vec x'] \notag\\
& +\mathcal H[\eta'] \bigg((d-1)w\,\partial_i\partial_jC^{(w)}_{3,d}\left[\eta,\eta';\textstyle{\frac {R}{\sqrt w}}\right]{}^{(\text{a})}\dot{T}_{ij}[\eta',\vec x']
+C^{(w)}_{2,d}\left[\eta,\eta';\textstyle{\frac {R}{\sqrt w}}\right]{}^{(\text{a})}\dot{T}_{ll}[\eta',\vec x']\bigg)\Bigg\},
\label{Psi_Convolution_Cosmology}
\end{align}}%
where the scalar functions $C^{(w)}_{1,d}$, $C^{(w)}_{2,d}$, and $C^{(w)}_{3,d}$, respectively, are defined in a manner similar to eqs.~\eqref{C1_Graviton_Cosmology}, \eqref{C2_Graviton_Cosmology}, and \eqref{C3_Graviton_Cosmology},
{\allowdisplaybreaks
\begin{align}
&C^{(w)}_{1,d}\left[\eta,\eta';\textstyle{\frac {R}{\sqrt w}}\right]\equiv w^{\frac{d-1}2}\int_{\mathbb{R}^{d-1}}\frac{\text{d}^{d-1}\vec{k}}{(2\pi)^{d-1}}\,\widetilde C^{(w)}_{1,d}\left[\eta,\eta';\sqrt{w}|\vec k|\right]\,e^{i\vec{k}\cdot\vec{R}},
\label{C1_Psi_Cosmology}\\
&C^{(w)}_{2,d}\left[\eta,\eta';\textstyle{\frac {R}{\sqrt w}}\right]\equiv w^{\frac{d-3}2}\int_{\mathbb{R}^{d-1}}\frac{\text{d}^{d-1}\vec{k}}{(2\pi)^{d-1}}\,\widetilde C^{(w)}_{1,d}\left[\eta,\eta';\sqrt{w}|\vec k|\right]\,\frac{e^{i\vec{k}\cdot\vec{R}}}{\vec k^2},
\label{C2_Psi_Cosmology}\\
&C^{(w)}_{3,d}\left[\eta,\eta';\textstyle{\frac {R}{\sqrt w}}\right]\equiv w^{\frac{d-5}2}\int_{\mathbb{R}^{d-1}}\frac{\text{d}^{d-1}\vec{k}}{(2\pi)^{d-1}}\,\widetilde C^{(w)}_{1,d}\left[\eta,\eta';\sqrt{w}|\vec k|\right]\,\frac{e^{i\vec{k}\cdot\vec{R}}}{\vec k^4},
\label{C3_Psi_Cosmology}
\end{align}}%
in which $\widetilde C^{(w)}_{1,d}$ represents the Fourier transform of $C^{(w)}_{1,d}$ with respect to $\vec R/\sqrt{w}$, and it obeys the Fourier-transformed homogeneous wave equation of eq.~\eqref{Psi_WaveEq2_Cosmology} with initial conditions specified by $\widetilde C^{(w)}_{1,d}\big|_{\eta=\eta'}=0$ and $\dot{\widetilde C}^{(w)}_{1,d}\big|_{\eta=\eta'}=-\partial_{\eta'}\widetilde C^{(w)}_{1,d}\big|_{\eta=\eta'}=1$.\footnote{The factor of $w$ appearing in eq.~\eqref{C1_Psi_Cosmology} has been arranged such that the corresponding massless scalar Green's function, $G^{(w,+)}_{d}=-\Theta[T]C^{(w)}_{1,d}$ with re-scaled coordinates $(\eta,\vec y) \equiv (\eta,\vec x/\sqrt{w})$, obeys the wave equation
\begin{align}
\bigg\{-\partial^2_\eta+\frac{ (d-2) \left(q_w + d -2 \right) }{q^2_w\eta^2} +\vec\nabla^2_{\vec y}\bigg\}\,G^{(w,+)}_d\left[\eta,\eta';|\vec y-\vec y'|\right]
= \delta[\eta-\eta']\delta^{(d-1)}[\vec y-\vec y'], \notag
\end{align}
where $\vec\nabla^2_{\vec y}$ is the spatial Laplacian with respect to $\vec y$. The analytic solution of $G^{(w,+)}_d$ for $0<w\leq1$ has been obtained in \cite{Chu:2016ngc}.} Analogous to eqs.~\eqref{C1_Photon_Cosmology_ModeDecomposition} and \eqref{C1_Graviton_Cosmology_ModeDecomposition}, $\widetilde C^{(w)}_{1,d}$ admits the following decomposition in terms of the mode functions $v^{(w)}_{\sqrt{w}|\vec k|}$ that obey the same homogeneous wave equation in momentum space,
\begin{align}
\widetilde C^{(w)}_{1,d}\left[\eta,\eta';\sqrt{w}|\vec k|\right]=i\left(v^{(w)}_{\sqrt{w}|\vec k|}[\eta]\,v^{(w)*}_{\sqrt{w}|\vec k|}[\eta']-v^{(w)*}_{\sqrt{w}|\vec k|}[\eta]\,v^{(w)}_{\sqrt{w}|\vec k|}[\eta']\vphantom{\dot A}\right),
\label{C1w_Cosmology_ModeDecomposition}
\end{align}
where the Wronskian condition, $v^{(w)}_{\sqrt{w}|\vec k|}\dot{v}^{(w)*}_{\sqrt{w}|\vec k|}-v^{(w)*}_{\sqrt{w}|\vec k|}\dot{v}^{(w)}_{\sqrt{w}|\vec k|}=i$, is fulfilled to be consistent with the initial condition $\dot{\widetilde C}^{(w)}_{1,d}\big|_{\eta=\eta'}=1$. By construction, these $C^{(w)}$s are solutions to the homogeneous version of eq.~\eqref{Psi_WaveEq2_Cosmology}. Moreover, their equal-time initial conditions may be readily identified,
{\allowdisplaybreaks
\begin{align}
&C^{(w)}_{1,d}\big|_{\eta=\eta'}=C^{(w)}_{2,d}\big|_{\eta=\eta'}=C^{(w)}_{3,d}\big|_{\eta=\eta'}=0,
\label{C123_Psi_InitialCondition}\\
&\dot C^{(w)}_{1,d}\big|_{\eta=\eta'}=-\partial_{\eta'} C^{(w)}_{1,d}\big|_{\eta=\eta'}=w^{\frac{d-1}2}\delta^{(d-1)}[\vec x-\vec x'], \label{dotC1_Psi_InitialCondition} \\
&\dot C^{(w)}_{2,d}\big|_{\eta=\eta'}=-\partial_{\eta'} C^{(w)}_{2,d}\big|_{\eta=\eta'}=-w^{\frac{d-3}2} G^{\mathrm{(E)}}_d[R],
\label{dotC2_Psi_InitialCondition} \\
&\dot C^{(w)}_{3,d}\big|_{\eta=\eta'}=-\partial_{\eta'} C^{(w)}_{3,d}\big|_{\eta=\eta'}=w^{\frac{d-5}2}D_d[R].
\label{dotC3_Psi_InitialCondition}
\end{align}}%
Exploiting their Fourier representations to observe that $-w\vec{\nabla}^2 C_{2,d}^{(w)} = C_{1,d}^{(w)}$ and $-w\vec{\nabla}^2 C_{3,d}^{(w)} = C_{2,d}^{(w)}$, we see that the homogeneous cousins of eq.~\eqref{Psi_WaveEq2_Cosmology} are
{\allowdisplaybreaks
\begin{align}
\frac{-\partial_\eta\bigg(a^{2\alpha}\,\partial_\eta\left(a^{-\alpha}\, C_{2,d}^{(w)} \vphantom{\dot A}\right)\bigg)}{a^\alpha}&= C_{1,d}^{(w)},\\
\frac{-\partial_\eta\bigg(a^{2\alpha}\,\partial_\eta\left(a^{-\alpha}\, C_{3,d}^{(w)} \vphantom{\dot A}\right)\bigg)}{a^\alpha}&= C_{2,d}^{(w)},
\qquad\qquad
\alpha=-\frac{d-2}2 .
\end{align}}%
At this point we may integrate these equations to express $C^{(w)}_{2,d}$ and $C^{(w)}_{3,d}$ in terms of $C^{(w)}_{1,d}$, which had been derived analytically in \cite{Chu:2016ngc} (see eqs.~(131) and (132) of \cite{Chu:2016ngc}). The resulting expressions are
{\allowdisplaybreaks
\begin{align}
C_{2,d}^{(w)}\left[\eta,\eta';\textstyle{\frac {R}{\sqrt w}}\right]=\,&-a[\eta]^{-\frac {d-2}2}\int^\eta_{\eta'} \text{d} \eta_2\,a[\eta_2]^{d-2}\int^{\eta_2}_{\eta'} \text{d} \eta_1\,a[\eta_1]^{-\frac {d-2}2}C^{(w)}_{1,d}\left[\eta_1,\eta';\textstyle{\frac {R}{\sqrt w}}\right] \notag\\
&-w^{\frac{d-3}2}G^{(\mathrm E)}_d[R]\left(\frac{a[\eta']}{a[\eta]}\right)^{\frac {d-2}2}\int^\eta_{\eta'}\text{d}\eta_1\left(\frac{a[\eta_1]}{a[\eta']}\right)^{d-2},
\label{C2_Psi_TimeIntegral_Cosmology} \\
C_{3,d}^{(w)}\left[\eta,\eta';\textstyle{\frac {R}{\sqrt w}}\right]=\,& a[\eta]^{-\frac {d-2}2}\int^\eta_{\eta'}\text{d}\eta_4\,a[\eta_4]^{(d-2)}\int^{\eta_4}_{\eta'}\text{d}\eta_3\,a[\eta_3]^{-(d-2)} \notag\\
& \qquad \qquad \times\int^{\eta_3}_{\eta'}\text{d}\eta_2\,a[\eta_2]^{(d-2)}
\int^{\eta_2}_{\eta'}\text{d} \eta_1\,a[\eta_1]^{-\frac {d-2}2}C^{(w)}_{1,d}\left[\eta_1,\eta';\textstyle{\frac {R}{\sqrt w}}\right]\notag\\
&+w^{\frac{d-3}2}G^{(\mathrm E)}_d[R]\left(\frac{a[\eta']}{a[\eta]}\right)^{\frac{d-2}2}\int^\eta_{\eta'}\text{d}\eta_3\,a[\eta_3]^{d-2}\int^{\eta_3}_{\eta'}\text{d}\eta_2
\,a[\eta_2]^{-(d-2)}\int^{\eta_2}_{\eta'}\text{d}\eta_1\left(\frac{a[\eta_1]}{a[\eta']}\right)^{d-2} \notag\\
&+w^{\frac{d-5}2}D_d[R]\left(\frac{a[\eta']}{a[\eta]}\right)^{\frac {d-2}2}\int^\eta_{\eta'}\text{d}\eta_1\left(\frac{a[\eta_1]}{a[\eta']}\right)^{d-2},
\label{C3_Psi_TimeIntegral_Cosmology}
\end{align}}%
which bear close resemblance to eqs.~\eqref{C2_Graviton_TimeIntegral_Cosmology} and \eqref{C3_Graviton_TimeIntegral_Cosmology}. The retarded part of $C^{(w)}_{1,d}$, or the massless scalar Green's function $G^{(w,+)}_d=-\Theta[T]C^{(w)}_{1,d}$, has been shown to contain only the causal scalar GW signals propagating either on or within the acoustic cone \cite{Chu:2016ngc}. Applying the arguments in \S \eqref{MinkowskiSpacetime} -- the first terms of eqs.~\eqref{C2_Psi_TimeIntegral_Cosmology} and \eqref{C3_Psi_TimeIntegral_Cosmology} are causal when $\eta>\eta'$ because $C_{1,d}^{(w)}$ is; while the rest of the terms are non-zero both inside and outside the light cone of $(\eta',\vec{x}')$. The Bardeen scalar $\Psi$ is therefore acausal for all relevant spacetime dimensions (cf.~eq.~\eqref{Psi_Convolution_Cosmology}).
Alternatively, we can perform integration-by-parts and employ the energy-momentum conservation laws \eqref{EnergyMomentumConservaion_Time_Cosmology} and \eqref{EnergyMomentumConservaion_Spatial_Cosmology}, as well as the properties of $C^{(w)}_{2,d}$ and $C^{(w)}_{3,d}$, to re-express the effective Green's function of $\Psi$ in eq.~\eqref{Psi_Convolution_Cosmology} as
{\allowdisplaybreaks
\begin{align}
\Psi[\eta,\vec x &] = \frac{8\pi G_\text N}{d-2} \int_{\mathbb{R}^{d-1}} \text{d}^{d-1}\vec x'\int_0^\infty \text{d}\eta' \, \left(\frac{a[\eta']}{a[\eta]}\right)^{\frac12\left(q_w+d-2\right)} w^{-\frac{d-3}2}\,\Bigg\{G^{(w,+)}_d
{}^{(\text{a})}T_{00}[\eta',\vec x']\notag\\
&-\Theta[T]a[\eta']^{-\frac12\left(q_w+d-2\right)}\partial_{\eta'}\left(a[\eta']^{\frac12\left(q_w+d-2\right)}\mathcal H[\eta']\,C^{(w)}_{2,d}\right){}^{(\text{a})}T_{ll}[\eta',\vec x'] + (d-2)\Theta[T]a[\eta']^{-\frac12\left(q_w-d+2\right)} \notag\\
&\times \partial_{\eta'}\left(a[\eta']^{\frac12\left(q_w-d+2\right)}\mathcal H[\eta']\,C^{(w)}_{2,d}\right){}^{(\text{a})}T_{00}[\eta',\vec x'] + (d-2)\Theta[T]\mathcal H[\eta']^2\,C^{(w)}_{2,d}\,\left({}^{(\text{a})}T_{00}[\eta',\vec x']-{}^{(\text{a})}T_{ll}[\eta',\vec x'] \right) \notag\\
&-\Theta[T]a[\eta']^{-\frac12\left(q_w-d+2\right)}\partial_{\eta'}\bigg(a[\eta']^{-(d-2)}\partial_{\eta'}
\left(a[\eta']^{\frac12\left(q_w+d-2\right)}\,C^{(w)}_{2,d}\right)\bigg){}^{(\text{a})}T_{00}[\eta',\vec x']\notag\\
&-\Theta[T]a[\eta']^{-\frac12\left(q_w+d-2\right)}\mathcal H[\eta']\partial_{\eta'}\left(a[\eta']^{\frac12\left(q_w+d-2\right)}\,C^{(w)}_{2,d}\right)\left({}^{(\text{a})}T_{00}[\eta',\vec x']
-{}^{(\text{a})}T_{ll}[\eta',\vec x'] \right) \notag\\
&-w(d-1)\Theta[T]a[\eta']^{-\frac12\left(q_w-d+2\right)}\partial_{\eta'}^2\bigg(a[\eta']^{-(d-2)}\partial_{\eta'}
\left(a[\eta']^{\frac12\left(q_w+d-2\right)}\mathcal H[\eta']\,C^{(w)}_{3,d}\right)\bigg){}^{(\text{a})}T_{00}[\eta',\vec x'] \notag\\
&-w(d-1)\Theta[T]a[\eta']^{-\frac12\left(q_w-d+2\right)}\mathcal H[\eta']\partial_{\eta'}\bigg(a[\eta']^{-(d-2)}\partial_{\eta'}
\left(a[\eta']^{\frac12\left(q_w+d-2\right)}\mathcal H[\eta']\,C^{(w)}_{3,d}\right)\bigg)\left({}^{(\text{a})}T_{00}[\eta',\vec x'] \right.\notag\\
& \left.-{}^{(\text{a})}T_{ll}[\eta',\vec x']\right) \Bigg\} + \frac{8\pi G_\text N}{d-2} \int_{\mathbb{R}^{d-1}} \text{d}^{d-1}\vec x'\,\bigg( G^{(\mathrm E)}_d {}^{(\text{a})}T_{00}[\eta,\vec x']+(d-1)\mathcal H[\eta]\partial_jD_d {}^{(\text{a})}T_{0j}[\eta,\vec x']\bigg),
\label{Psi_Convolution2_Cosmology}
\end{align}}%
where all the boundary terms that arise from the integrations by parts have been discarded.\footnote{As previously reasoned in the spin-1 and spin-2 cases, discarding the boundary contributions at past infinity does not affect the inhomogeneous solutions of the Bardeen scalars, since the corresponding homogeneous wave equation is obeyed by those surface terms, which correspond to evaluating the surface integrals of $C^{(w)}_{2,d}$ and $C^{(w)}_{3,d}$ at $\eta'=-\infty$ (for $w = -1$) or $\eta' = 0$ (for $0 < w \leq 1$).
} This form shows more transparently the convolution with the local matter stress-energy tensor, and will be used for our later analysis of the physical observables.
{\bf Vector Potential} \qquad According to \cite{Chu:2016ngc}, the vector mode $V_i$, in de Sitter space ($w=-1$), obeys the Poisson-type equation \eqref{Vi_Poisson_Minkowski}, while for $0<w\leq1$, if perturbations are assumed to be negligible in the far past, then the same vector equation, i.e., eq.~\eqref{Vi_Poisson_Minkowski}, is satisfied as well. Therefore, in both cases, the solution of $V_i$ is that in eq.~\eqref{Vi_Minkowski}.
{\bf Linearized Weyl Tensor} \qquad As we have discussed in \S\eqref{Chapter_Observables}, the linearized Riemann tensor in the cosmological background is no longer gauge invariant due to its non-zero background value. In cosmological spacetimes, which are conformally flat, the causal and gauge-invariant counterpart to the linearized Riemann in flat spacetimes is the linearized Weyl tensor $\delta_1 C^\mu{}_{\nu \alpha\beta}$. More specifically, since the Weyl tensor $C^\mu_{\phantom{\mu}\nu \alpha\beta}$ is conformally invariant, it is zero when evaluated on the unperturbed cosmological geometry $g_{\mu\nu} = a^2 \eta_{\mu\nu}$ and must therefore be gauge-invariant at first order in $\chi_{\mu\nu}$. Furthermore, its exact wave equation is simply the traceless part of eq.~\eqref{Riemann_WaveEq}; but since it is zero at zeroth order, the first order Weyl tensor $\delta_1 C^\mu_{\phantom{\mu}\nu \alpha\beta}$ must therefore obey an equation involving the wave operator with respect to the background FLRW metric.
Motivated by these considerations, we shall proceed to calculate
\begin{align}
\delta_1 C^i{}_{0j0}=\,&\left(\frac{d-3}{d-2}\right)\Bigg\{\left(\partial_i\partial_j-\frac{\delta_{ij}}{d-1}\vec\nabla^2\right)
\left(\Phi+\Psi\right)+\partial_{(i}\dot{V}_{j)}-\frac12\bigg(\ddot{D}_{ij}+\frac1{d-3}\vec\nabla^2D_{ij}\bigg)\Bigg\}.
\label{LinearizedWeyl_GaugeInvariant}
\end{align}
It is likely that $\delta_1 C^i{}_{0j0}$ encodes the dominant contributions to the first-order tidal forces described in eq.~\eqref{GeodesicDeviation_FreeFall}; but we shall leave this analysis to future work \cite{ChuLiuInPrep}. Here, we will instead focus on the causal structure of this quantity with respect to the background spacetime.
{\it Linearized Weyl tensor for $w=-1$} \qquad Within the de Sitter case, plugging into eq.~\eqref{LinearizedWeyl_GaugeInvariant} the solution of $D_{ij}$ in eq.~\eqref{Spin2_Convolution2_Cosmology}, those of $\Psi$ and $\Phi$ in eqs.~\eqref{BardeenPsi_deSitter} and \eqref{PhiPsiRelationship_Convolution}, and that of $V_i$ in eq.~\eqref{Vi_Minkowski}, with $\mathcal H[\eta]=-1/\eta$ and $(\eta_{\text p},\eta_{\text f})=(-\infty,0)$, we find that, after employing the conservation conditions \eqref{EnergyMomentumConservaion_Time_Cosmology} and \eqref{EnergyMomentumConservaion_Spatial_Cosmology}, the scalars and vector act to cancel the acausal signals from the tensor contributions to Weyl. In more detail,
\begin{align}
\delta_1 C^i{}_{0j0}=-\frac12\left(\frac{d-3}{d-2}\right)&\left(\ddot{D}_{ij}+\frac1{d-3}\vec\nabla^2D_{ij}\right)_{\mathrm{causal}}\notag\\
&+ \frac{8\pi G_\text N}{d-2} \left({}^{(\text{a})}T_{ij}-\frac{\delta_{ij}}{d-1}\left((d-3){}^{(\text{a})}T_{00}
+2{}^{(\text{a})}T_{ll}\right)\right) ;
\label{LinearizedWeyl_deSitter}
\end{align}
where the first line of eq.~\eqref{LinearizedWeyl_deSitter} denotes the causal part of the spin-2 contributions that depend exclusively on the retarded Green's function $G^{(g,+)}_d[\eta,\eta';R]$,
{\allowdisplaybreaks
\begin{align}
&\bigg(\ddot{D}_{ij}[\eta,\vec x]+\frac1{d-3}\vec\nabla^2D_{ij}[\eta,\vec x]\bigg)_{\mathrm{causal}}\notag\\
=\,&-16\pi G_\text N\left(\frac{d-2}{d-3}\right)\int_{\mathbb{R}^{d-1}} \text{d}^{d-1}\vec x'\int_{\eta_{\text p}}^{\eta_{\text f}} \text{d}\eta' \,\left(\frac{a[\eta']}{a[\eta]}\right)^{\frac {d-2}2} \Bigg\{\bigg(\ddot G^{(g,+)}_{d}- \frac{2(d-3)}{q_w\eta} \dot G^{(g,+)}_{d} \notag\\
&+ \frac{(d-2)\left(q_w+d-4\right)}{q_w^2 \eta^2} \, G^{(g,+)}_{d}\bigg)\left({}^{(\text{a})}T_{ij}[\eta',\vec x']-\frac{\delta_{ij}}{d-2}{}^{(\text{a})}T_{ll}[\eta',\vec x']\right)\notag\\
&+2a[\eta']^{\frac{d-2}2}\partial_{\eta'}\Bigg(a[\eta']^{-\frac{d-2}2}\bigg(\partial_{(i}G^{(g,+)}_{d} - \frac{2(d-3)}{q_w\eta} a[\eta]^{-\frac{d-2}2}
\int^\eta_{\eta'}\text{d}\eta_1\,a[\eta_1]^{\frac{d-2}2}\partial_{(i}G^{(g,+)}_{d}\bigg)\Bigg){}^{(\text{a})}T_{j)0}[\eta',\vec x'] \notag\\
&+\frac{\delta_{ij}}{d-2}a[\eta']^{\frac{d-2}2}\Bigg(\partial^2_{\eta'}\bigg(a[\eta']^{-\frac{d-2}2}
\bigg(G^{(g,+)}_{d}- \frac{2(d-3)}{q_w\eta} a[\eta]^{-\frac{d-2}2} \int^\eta_{\eta'}\text{d}\eta_1\,a[\eta_1]^{\frac{d-2}2}G^{(g,+)}_{d}\bigg)\bigg){}^{(\text{a})}T_{00}[\eta',\vec x'] \notag\\
&+\frac{2}{q_w\eta'}\partial_{\eta'}\bigg(a[\eta']^{-\frac{d-2}2}\bigg(G^{(g,+)}_{d}-\frac{2(d-3)}{q_w\eta}a[\eta]^{-\frac{d-2}2}
\int^\eta_{\eta'}\text{d}\eta_1\,a[\eta_1]^{\frac{d-2}2}G^{(g,+)}_{d}\bigg)\bigg)\left({}^{(\text{a})}T_{00}[\eta',\vec x'] - {}^{(\text{a})}T_{ll}[\eta',\vec x']\right)\Bigg) \notag\\
&+\frac1{d-2}\bigg(\partial_i\partial_jG^{(g,+)}_{d}-\frac{2(d-3)}{q_w\eta}a[\eta]^{-\frac{d-2}2}
\int^\eta_{\eta'}\text{d}\eta_1\,a[\eta_1]^{\frac{d-2}2}\partial_i\partial_jG^{(g,+)}_{d}\bigg){}^{(\text{a})}T_{ll}[\eta',\vec x'] \notag\\
&+\left(\frac{d-3}{d-2}\right)a[\eta']^{\frac{d-2}2}a[\eta]^{\frac{d-2}2}\Bigg(\partial_{\eta'}^2\bigg(a[\eta']^{-\frac{d-2}2}
\bigg(\int^\eta_{\eta'}\text{d}\eta_2\,a[\eta_2]^{-(d-2)}
\int^{\eta_2}_{\eta'}\text{d}\eta_1\,a[\eta_1]^{\frac{d-2}2}\partial_i\partial_jG^{(g,+)}_{d}\notag\\
&-\frac{2(d-3)}{q_w\eta}a[\eta]^{-(d-2)}\int^\eta_{\eta'}\text{d}\eta_3\,a[\eta_3]^{d-2}\int_{\eta'}^{\eta_3}\text{d}\eta_2\,a[\eta_2]^{-(d-2)}
\int^{\eta_2}_{\eta'}\text{d}\eta_1\,a[\eta_1]^{\frac{d-2}2}\partial_i\partial_jG^{(g,+)}_{d}\bigg)\bigg){}^{(\text{a})}T_{00}[\eta',\vec x'] \notag\\
&+\frac{2}{q_w\eta'}\partial_{\eta'}\bigg(a[\eta']^{-\frac{d-2}2}\bigg(\int^\eta_{\eta'}\text{d}\eta_2\,a[\eta_2]^{-(d-2)}
\int^{\eta_2}_{\eta'}\text{d}\eta_1\,a[\eta_1]^{\frac{d-2}2}\partial_i\partial_jG^{(g,+)}_{d}\notag\\
&-\frac{2(d-3)}{q_w\eta}a[\eta]^{-(d-2)}\int^\eta_{\eta'}\text{d}\eta_3\,a[\eta_3]^{d-2}\int_{\eta'}^{\eta_3}\text{d}\eta_2\,a[\eta_2]^{-(d-2)}
\int^{\eta_2}_{\eta'}\text{d}\eta_1\,a[\eta_1]^{\frac{d-2}2}\partial_i\partial_jG^{(g,+)}_{d}\bigg)\bigg) \notag\\
&\times \left({}^{(\text{a})}T_{00}[\eta',\vec x']-{}^{(\text{a})}T_{ll}[\eta',\vec x']\right)\Bigg)\Bigg\},
\label{WeylSpin2Sector_Retarded}
\end{align}}%
while the second line of eq.~\eqref{LinearizedWeyl_deSitter} consists solely of the stress-energy tensor of the GW source evaluated at the observer location. (Recall that $a[\eta]=-1/(H\eta)$ and $q_w=-2$ in de Sitter spacetime.) As long as the observer at $(\eta,\vec{x})$ is not located at the source, these ${}^{(\text{a})}T_{\mu\nu}[\eta,\vec{x}]$ terms in eq.~\eqref{LinearizedWeyl_deSitter} are zero.\footnote{This calculation is greatly simplified by first using the commutator $C^{(g)}_{1,d}$, and only re-expressing the final result in terms of the massless scalar Green's function via $G^{(g,+)}_d=-\Theta[T]C^{(g)}_{1,d}$ at the very end. In particular, we notice that a local term will show up in the conversion involving a second time derivative, namely $\ddot G^{(g,+)}_{d}=-\delta^{(d)}[x-x']-\Theta[T]\ddot C^{(g)}_{1,d}$ or $\partial_{\eta'}^2 G^{(g,+)}_{d}=-\delta^{(d)}[x-x']-\Theta[T]\partial^2_{\eta'} C^{(g)}_{1,d}$. \newline A simple check of eq.~\eqref{LinearizedWeyl_deSitter} can be made by taking the limit of eq.~\eqref{WeylSpin2Sector_Retarded} as $a\to1$ and assuming $G^{(g,+)}_d$ takes the form of $G^+_d$ in Minkowski spacetime. We may then show explicitly that the first line of eq.~\eqref{LinearizedWeyl_deSitter} reduces to $\delta_1 R_{0i0j}$ given in eq.~\eqref{LinearizedRiemann_Sol_Minkowski}, and the resulting $\delta_1 C^i{}_{0j0}$ is consistent with its Minkowski counterpart obtained from the solutions derived in the last section. Moreover, the Minkowski form of $\delta_1 C^i{}_{0j0}$ also agrees with the relationship between the Riemann and the Weyl tensor,
\begin{align}
C^{\rho}{}_{\sigma\mu\nu}=\,& R^{\rho}{}_{\sigma\mu\nu} - \frac{16\pi G_\text N}{d-2}\left( \delta^\rho_{[\mu}T_{\nu]\sigma}-g_{\sigma[\mu}T_{\nu]}{}^\rho - \delta^\rho_{[\mu}g_{\nu]\sigma}
\frac{2g^{\alpha\beta}T_{\alpha\beta}}{d-1}\right),
\label{RiemannWeyl}
\end{align}
linearized about the Minkowski background; where $T_{\mu\nu}$ is the total energy-momentum tensor of matter and Einstein's equation has been imposed on the trace parts of the Riemann tensor.}
To sum: the result in eq.~\eqref{LinearizedWeyl_deSitter} reveals that $\delta_1 C^i{}_{0j0}$ on a de Sitter background receives only signals from the spin-2 sector, as long as the observer is away from the isolated matter source(s) of GWs.
Analogous to the localization of the effective Green's function of $F_{0i}$ shown in eq.~\eqref{F0i_Cosmology_Localized}, the expression of eq.~\eqref{WeylSpin2Sector_Retarded} in de Sitter spacetime can be further simplified in a localized manner by introducing two additional massless scalar Green's functions $G^{(\text V,+)}_d$ and $G^{(\text{Tr},+)}_d$ that, respectively, obey the following wave equations,
\begin{align}
&\left\{\partial^2+\frac{(d-4)(d-2)}{4\eta^2}\right\}G^{(\text V,+)}_d[\eta,\eta';R]=\delta^{(d)}[x-x'], \label{GV+_deSitter} \\
&\left\{\partial^2+\frac{(d-6)(d-4)}{4\eta^2}\right\}G^{(\text{Tr},+)}_d[\eta,\eta';R]=\delta^{(d)}[x-x'].
\label{GS+_deSitter}
\end{align}
The effective Green's function of $\delta_1 C^i{}_{0j0}$ in eq.~\eqref{LinearizedWeyl_deSitter} can then be localized accordingly in terms of $G^{(g,+)}_d$, $G^{(\text V,+)}_d$, and $G^{(\text{Tr},+)}_d$,
{\allowdisplaybreaks
\begin{align}
\delta_1 C^i{}_{0j0}&[\eta,\vec x]=8\pi G_\text N\int_{\mathbb{R}^{d-1}} \text{d}^{d-1}\vec x'\int_{-\infty}^{0} \text{d}\eta' \,
\left(\frac\eta{\eta'}\right)^{\frac {d-2}2}\Bigg\{ \bigg(\ddot G^{(g,+)}_{d}+\frac{(d-3)}{\eta}\dot G^{(g,+)}_{d}+\frac{(d-2)(d-6)}{4\eta^2}G^{(g,+)}_{d}\bigg) \notag\\
&\times \left({}^{(\text{a})}T_{ij}[\eta',\vec x']-\frac{\delta_{ij}}{d-2}{}^{(\text{a})}T_{ll}[\eta',\vec x']\right)-2\eta^{-\frac{d-4}2}\partial_\eta\left(\eta^{\frac{d-4}2}\partial_{(i}G^{(\text V,+)}_d\right){}^{(\text{a})}T_{j)0}[\eta',\vec x'] \notag\\
&-\frac{\delta_{ij}}{d-2}\Bigg(\left(\eta\eta'\right)^{-\frac{d-4}2}\partial_\eta\partial_{\eta'}
\left(\left(\eta\eta'\right)^{\frac{d-4}2}G^{(\text V,+)}_d\right)
{}^{(\text{a})}T_{00}[\eta',\vec x']+\eta'^{-1}\eta^{-\frac{d-4}2}\partial_\eta\left(\eta^{\frac{d-4}2}G^{(\text V,+)}_d\right){}^{(\text{a})}T_{ll}[\eta',\vec x']\Bigg) \notag\\ &+\frac1{d-2}\partial_i\partial_jG^{(\text{Tr},+)}_d\left((d-3){}^{(\text{a})}T_{00}[\eta',\vec x']+{}^{(\text{a})}T_{ll}[\eta',\vec x']\right)\Bigg\} \notag\\
&+ \frac{8\pi G_\text N}{d-2} \left({}^{(\text{a})}T_{ij}[\eta,\vec x]-\frac{\delta_{ij}}{d-1}\left((d-3){}^{(\text{a})}T_{00}[\eta,\vec x]
+2{}^{(\text{a})}T_{ll}[\eta,\vec x]\right)\right),
\label{LinearizedWeyl_deSitter_Localized}
\end{align}}%
which will be explained in more detail in \cite{ChuLiuInPrep}. Eq.~\eqref{LinearizedWeyl_deSitter_Localized} turns out to be consistent with $\delta_1 C^i{}_{0j0}$ computed from the generalized de Donder gauge $\overline \chi_{\mu\nu}$ solution obtained in \cite{Chu:2016qxp}; the analytic solutions of $G^{(g,+)}_d = -\Theta[T] \mathcal G^{(\text T)}$, $G^{(\text V,+)}_d = -\Theta[T] \mathcal G^{(\text V)}$, and $G^{(\text{Tr},+)}_d = -\Theta[T] \mathcal G^{(\text{Tr})}$ can be found in eqs.~(28), (29), (33), (34), (38), and (39) of \cite{Chu:2016qxp}.
{\it Linearized Weyl tensor for $0<w\leq1$} \qquad To obtain $\delta_1 C^i{}_{0j0}$ for a relativistic equation-of-state $w$ within $0<w\leq1$, we insert into eq.~\eqref{LinearizedWeyl_GaugeInvariant} eq.~\eqref{Spin2_Convolution2_Cosmology} for $D_{ij}$, eqs.~\eqref{Psi_Convolution2_Cosmology} and \eqref{PhiPsiRelationship_Convolution} for $\Psi$ and $\Phi$, and eq.~\eqref{Vi_Minkowski} for $V_i$; recalling that $\mathcal H[\eta]=2/(q_w\eta)$ and $(\eta_{\text p},\eta_{\text f})=(0,\infty)$. A direct computation then reveals that an exact cancellation of the acausal signals takes place again in eq.~\eqref{LinearizedWeyl_GaugeInvariant}, so that
\begin{align}
\delta_1 C^i{}_{0j0}=\,&\left(\frac{d-3}{d-2}\right)\Bigg\{\Bigg(\left(\partial_i\partial_j-\frac{\delta_{ij}}{d-1}\vec\nabla^2\right)
\left(\Phi+\Psi\right)\Bigg)_{\mathrm{causal}}-\frac12\bigg(\ddot{D}_{ij}+\frac1{d-3}\vec\nabla^2D_{ij}\bigg)_{\mathrm{causal}}\Bigg\} \notag\\
&+ \frac{8\pi G_\text N}{d-2} \left({}^{(\text{a})}T_{ij}-\frac{\delta_{ij}}{d-1}\left((d-3){}^{(\text{a})}T_{00}
+2{}^{(\text{a})}T_{ll}\right)\right) .
\label{LinearizedWeyl_RelativisticEoS}
\end{align}
The causal portion of the spin-2 sector takes precisely the same form as eq.~\eqref{WeylSpin2Sector_Retarded} but with scale factor given in eq.~\eqref{ScaleFactor} and $(\eta_{\text p},\eta_{\text f})=(0,\infty)$. On the other hand, the causal contributions of the Bardeen scalar potentials in the first line are given by
{\allowdisplaybreaks
\begin{align}
&\Bigg(\left(\partial_i\partial_j-\frac{\delta_{ij}}{d-1}\vec\nabla^2\right)\big(\Phi[\eta,\vec x]+\Psi[\eta,\vec x]\big)\Bigg)_{\mathrm{causal}} \notag\\
&=-8\pi G_\text N \int_{\mathbb{R}^{d-1}} \text{d}^{d-1}\vec x'\int_0^\infty \text{d}\eta' \, \left(\frac{a[\eta']}{a[\eta]}\right)^{\frac12\left(q_w+d-2\right)}\!\!w^{-\frac{d-1}2}\Bigg\{\frac{\delta_{ij}}{d-1}
\Bigg( \bigg( \ddot G^{(w,+)}_d -\frac{ (d-2) \left(q_w + d -2 \right) }{q^2_w\eta^2} \, G^{(w,+)}_d \notag\\
&-a[\eta']^{-\frac12\left(q_w-3d+8\right)}\partial_{\eta'}\bigg(a[\eta']^{-(2d-5)}\partial_{\eta'}
\left(a[\eta']^{\frac12\left(q_w+d-2\right)}\,G^{(w,+)}_d\right)\bigg) -(d-2) \frac{2q_w + 4(d-3)}{q^2_w\eta'^2} \, G^{(w,+)}_d \bigg){}^{(\text{a})}T_{00}[\eta',\vec x'] \notag\\
& + \frac{2q_w-4(d-2)}{q^2_w \eta'^2} \, G^{(w,+)}_d{}^{(\text{a})}T_{ll}[\eta',\vec x'] \Bigg)
+\delta_{ij}w\,a[\eta']^{-\frac12\left(q_w-d+2\right)}\Bigg(\partial_{\eta'}^2\bigg(a[\eta']^{-(d-2)}\partial_{\eta'}
\bigg(\frac{2}{q_w \eta'} a[\eta']^{\frac12\left(q_w+d-2\right)} a[\eta]^{-\frac{d-2}2} \notag\\
& \times \int^\eta_{\eta'}\text{d}\eta_2\,a[\eta_2]^{(d-2)} \int^{\eta_2}_{\eta'}\text{d}\eta_1\,a[\eta_1]^{-\frac {d-2}2}G^{(w,+)}_d\bigg)\bigg){}^{(\text{a})}T_{00}[\eta',\vec x'] + \frac{2}{q_w\eta'}\partial_{\eta'}\bigg(a[\eta']^{-(d-2)}\partial_{\eta'}
\bigg( \frac{2}{q_w\eta'} a[\eta']^{\frac12\left(q_w+d-2\right)} \notag\\
& \times a[\eta]^{-\frac{d-2}2} \int^\eta_{\eta'}\text{d}\eta_2\,a[\eta_2]^{(d-2)} \int^{\eta_2}_{\eta'}\text{d}\eta_1\,a[\eta_1]^{-\frac {d-2}2}G^{(w,+)}_d\bigg)\bigg)\left({}^{(\text{a})}T_{00}[\eta',\vec x']-{}^{(\text{a})}T_{ll}[\eta',\vec x']\right)\Bigg) \notag\\
&+w \Bigg( \! \bigg(a[\eta']^{-\frac12\left(q_w-3d+8\right)}\partial_{\eta'}\bigg(a[\eta']^{-(2d-5)}\partial_{\eta'}
\bigg(a[\eta']^{\frac12\left(q_w+d-2\right)}
a[\eta]^{-\frac{d-2}2} \!\! \int^\eta_{\eta'}\text{d}\eta_2\,a[\eta_2]^{(d-2)} \!\! \int^{\eta_2}_{\eta'}\text{d}\eta_1\,a[\eta_1]^{-\frac {d-2}2}\partial_i\partial_jG^{(w,+)}_d \! \bigg) \! \bigg) \notag\\
& +(d-2) \frac{2q_w + 4(d-3)}{q^2_w\eta'^2} a[\eta]^{-\frac {d-2}2}\int^\eta_{\eta'}\text{d}\eta_2\,a[\eta_2]^{(d-2)}\int^{\eta_2}_{\eta'}\text{d}\eta_1\,a[\eta_1]^{-\frac {d-2}2}\partial_i\partial_jG^{(w,+)}_d\bigg){}^{(\text{a})}T_{00}[\eta',\vec x'] \notag\\
&- \frac{2q_w-4(d-2)}{q^2_w \eta'^2} a[\eta]^{-\frac {d-2}2}\int^\eta_{\eta'}\text{d}\eta_2\,a[\eta_2]^{(d-2)}\int^{\eta_2}_{\eta'}\text{d}\eta_1\,a[\eta_1]^{-\frac {d-2}2}\partial_i\partial_jG^{(w,+)}_d {}^{(\text{a})}T_{ll}[\eta',\vec x']\Bigg) \notag\\
&-w\,\partial_i\partial_jG^{(w,+)}_d {}^{(\text{a})}T_{00}[\eta',\vec x']
-(d-1)w^2a[\eta']^{-\frac12\left(q_w-d+2\right)}\Bigg(\partial_{\eta'}^2\bigg(a[\eta']^{-(d-2)}\partial_{\eta'}
\bigg(\frac{2}{q_w\eta'}\,a[\eta']^{\frac12\left(q_w+d-2\right)} a[\eta]^{-\frac{d-2}2}\notag\\
&\times \int^\eta_{\eta'}\text{d}\eta_4\,a[\eta_4]^{(d-2)}\int^{\eta_4}_{\eta'}\text{d}\eta_3\,a[\eta_3]^{-(d-2)}\int^{\eta_3}_{\eta'}\text{d}\eta_2\,a[\eta_2]^{(d-2)}
\int^{\eta_2}_{\eta'}\text{d}\eta_1\,a[\eta_1]^{-\frac {d-2}2}\partial_i\partial_jG^{(w,+)}_d\bigg)\bigg) {}^{(\text{a})}T_{00}[\eta',\vec x'] \notag\\
&+\frac{2}{q_w\eta'}\partial_{\eta'}\bigg(a[\eta']^{-(d-2)}\partial_{\eta'}
\bigg( \frac{2}{q_w\eta'}\,a[\eta']^{\frac12\left(q_w+d-2\right)} a[\eta]^{-\frac{d-2}2}\int^\eta_{\eta'}\text{d}\eta_4\,a[\eta_4]^{(d-2)}\int^{\eta_4}_{\eta'}\text{d}\eta_3\,a[\eta_3]^{-(d-2)}
\int^{\eta_3}_{\eta'}\text{d}\eta_2\,a[\eta_2]^{(d-2)}\notag\\
&\times
\int^{\eta_2}_{\eta'}\text{d}\eta_1\,a[\eta_1]^{-\frac {d-2}2}\partial_i\partial_jG^{(w,+)}_d\bigg)\bigg)\left( {}^{(\text{a})}T_{00}[\eta',\vec x']- {}^{(\text{a})}T_{ll}[\eta',\vec x']\right) \Bigg) \Bigg\} .
\label{WeylScalarSector_Retarded}
\end{align}}%
Observe that eq.~\eqref{WeylScalarSector_Retarded} is fully determined by the massless scalar Green's function $G^{(w,+)}_d\big[\eta,\eta';\textstyle{\frac {R}{\sqrt w}}\big]$, and whose contributions to $\delta_1 C^i{}_{0j0}$ are therefore restricted either on or inside the acoustic cone.\footnote{Once again, notice that in eq.~\eqref{WeylScalarSector_Retarded}, we have switched from $C^{(w)}_{1,d}$ to $G^{(w,+)}_d$, where the local terms are incurred in the conversion between their second time derivatives, $\ddot G^{(w,+)}_{d}=-w^{\frac{d-1}2}\delta^{(d)}[x-x']-\Theta[T]\ddot C^{(w)}_{1,d}$ and $\partial_{\eta'}^2G^{(w,+)}_{d}=-w^{\frac{d-1}2}\delta^{(d)}[x-x']-\Theta[T]\partial_{\eta'}^2C^{(w)}_{1,d}$. Similarly, the Minkowski counterpart of $\delta_1 C^i{}_{0j0}$ is again recovered by letting $a\to1$, replacing $G^{(g,+)}_d$ with $G^+_d$ in eq.~\eqref{WeylSpin2Sector_Retarded}, and assuming no scalar contributions of eq.~\eqref{WeylScalarSector_Retarded} to eq.~\eqref{LinearizedWeyl_RelativisticEoS}.}
The physically intriguing feature of the relativistic $w$ result in eq.~\eqref{LinearizedWeyl_RelativisticEoS} is that, not only do spin-2 gravitons contribute to $\delta_1 C^i{}_{0j0}$, it appears the Bardeen scalars do so as well. To be sure, however, it would be prudent to obtain a more explicit expression for eq.~\eqref{WeylScalarSector_Retarded}. In an upcoming work, we hope to tackle this important step towards a more comprehensive understanding of gravitational tidal forces within a cosmological setting.
\section{Summary, Discussions, and Future Directions}
\label{Section_Summary}
In this paper, we have sought to clarify the physical roles played by the TT and tt gravitational perturbations; as well as the analogous issues for the spin-1 photon. Even though the TT GW is gauge-invariant -- it remains un-altered under an infinitesimal change in coordinates -- it is acausal. Since the bulk of the paper involves heavy mathematical analysis for arbitrary dimensions and cosmological equations-of-state, we summarize here the 4D Minkowski case for the reader's convenience.
Let us begin with the electromagnetic sector. The gauge-invariant 4D transverse photon, which obeys $\partial_i \alpha_i = 0$, cannot be a standalone observable because its solution
\begin{align}
\alpha_i[\eta,\vec{x}]
= - \int_{\mathbb{R}^{3,1}} \text{d}^4 x' G_{ij}^+[T,\vec{R}] J_j[\eta',\vec{x}']
\qquad\qquad (T \equiv \eta - \eta' \text{ and } \vec{R} \equiv \vec{x}-\vec{x}')
\end{align}
receives contributions from portions of the electric current $J_j[\eta',\vec{x}']$ lying outside the past lightcone of the observer at $(\eta,\vec{x})$. This is because, the photon retarded Green's function
\begin{align}
G_{ij}^+[T,\vec{R}]
&= - \delta_{ij} \frac{\delta[T-R]}{4\pi R} - \frac{1}{4\pi} \partial_{i} \partial_{j} \left( \Theta[T-R] + \frac{T}{R} \Theta[T] \Theta[R-T] \right)
\end{align}
contains an acausal portion: $G_{ij}^{(+,\text{acausal})} = - (4\pi)^{-1} \Theta[T] \Theta[R-T] T \partial_{i} \partial_{j} R^{-1}$. However, since this acausal term of the photon Green's function is part of a pure gradient, namely $-(4\pi)^{-1} \partial_i \partial_j \big(\Theta[T] \Theta[R-T] T/R\big)$, the magnetic field $F_{ij} = \partial_i \alpha_j - \partial_j \alpha_i$ -- which involves its curl -- is therefore entirely causal. The electric field, on the other hand, is the sum of the photon velocity $\dot{\alpha}_i$ and the gradient of the gauge-invariant scalar potential,
\begin{align}
\label{4D_F0i}
F_{0i} = \dot{\alpha}_i + \partial_i \Phi .
\end{align}
In detail, integration-by-parts (IBPs) and the conservation of the electric current yield
\begin{align}
\dot{\alpha}_i[\eta,\vec x] &= \big(\dot{\alpha}_i\big)_\text{causal} + \int_{\mathbb{R}^{3}} \text{d}^3 \vec{x}' \partial_i \frac{J_0[\eta,\vec{x}']}{4\pi R} ;
\label{4D_PhotonVelocity}
\end{align}
where we have denoted the causal part of the photon velocity as
\begin{align}
\label{4D_PhotonVelocity_Causal}
\big(\dot{\alpha}_i\big)_\text{causal}
&\equiv \int_{\mathbb{R}^{3,1}} \text{d}^4 x' \left( \partial_\eta \frac{\delta[T-R]}{4\pi R} J_i[\eta',\vec x'] - \partial_i \frac{\delta[T-R]}{4\pi R} J_0[\eta',\vec x'] \right) .
\end{align}
Whereas, the gradient of the scalar potential is
\begin{align}
\label{4D_GradScalar}
\partial_i \Phi &= -\int_{\mathbb{R}^{3}} \text{d}^3 \vec{x}' \partial_i \frac{J_0[\eta,\vec{x}']}{4\pi R} .
\end{align}
Adding equations \eqref{4D_PhotonVelocity} and \eqref{4D_GradScalar} to obtain eq.~\eqref{4D_F0i}, we see that the sole purpose of $\Phi$ -- as far as electromagnetic fields are concerned -- is to cancel the acausal part of the photon velocity. This in turn ensures the electric field of eq.~\eqref{4D_F0i}, in a given inertial frame, is the causal part of the latter; namely, $F_{0i} = (\dot{\alpha}_i)_\text{causal}$.
We have also pointed out: upon quantization, these transverse massless spin-1 photon operators violate microcausality, because their Green's functions do not vanish at spacelike intervals.
The transverse-traceless graviton, which obeys $\partial_i D_{ij} = 0 = \delta^{ij} D_{ij}$, also cannot be a standalone observable -- for very similar reasons as its acausal transverse photon counterpart. Its solution
\begin{align}
D_{ij}[\eta,\vec{x}]
= -16\pi G_{\rm N} \int_{\mathbb{R}^{3,1}} \text{d}^4 x' G^+_{ij mn}[T,\vec{R}] T_{mn}[\eta',\vec{x}']
\end{align}
receives signals from regions of the stress tensor $T_{mn}[\eta',\vec{x}']$ outside the past null cone of the observer at $(\eta,\vec{x})$. For, this TT graviton Green's function reads
{\allowdisplaybreaks\begin{align}
G^+_{ijmn}[T,\vec R]=\,& -\bigg(\delta_{m(i}\delta_{j)n}-\frac{\delta_{ij}\delta_{mn}}{2}\bigg)\frac{\delta[T-R]}{4\pi R} \nonumber\\
&\qquad
- \frac1{4\pi}\bigg(\delta_{m(i}\partial_{j)}\partial_n+
\delta_{n(i}\partial_{j)}\partial_m -\frac{\delta_{ij}\partial_m\partial_n-\delta_{mn}\partial_i\partial_j}{2}
\bigg) \left(\Theta[T-R] + \frac{T}{R} \Theta[T]\Theta[R-T] \right) \nonumber\\ &\qquad
+\frac1{48\pi} \partial_i\partial_j\partial_m\partial_n \left( \Theta[T-R]\left(R^2 + 3T^2\right)+ \Theta[T] \Theta[R-T]\,\frac{3R^2T + T^3}{ R} \right) ;
\end{align}}%
which contains the acausal terms $ - \Theta[T]\Theta[R-T] T \big( \delta_{m(i}\partial_{j)}\partial_n + \delta_{n(i}\partial_{j)}\partial_m -(1/2) ( \delta_{ij}\partial_m\partial_n-\delta_{mn}\partial_i\partial_j ) \big) (4\pi R)^{-1}$ and $\Theta[T]\Theta[R-T] \partial_i\partial_j\partial_m\partial_n \big\{ (3R^2T + T^3) (48 \pi R)^{-1} \big\}$.
On the other hand, the dominant physical tidal forces ought to be encoded within the linearized Riemann tensor, which in turn involves all the gauge-invariant variables, not just the spin-2 graviton. In particular, its $0i0j$ components -- which are usually associated with the spatial tidal forces in a given nearly-Lorentzian inertial frame -- are
\begin{align}
\delta_1 R_{0i 0j}
= \delta_{ij} \ddot{\Psi} + \partial_i\partial_j\Phi + \partial_{(i}\dot{V}_{j)} - \frac12\ddot{D}_{ij} .
\end{align}
As one may expect from the preceding discussion for the spin-1 photon, the linearized Riemann $\delta_1 R_{0i0j}$ really only depends on the causal part of the spin-2 graviton acceleration:
\begin{align}
\delta_1 R_{0i 0j} = - \frac12 \big(\ddot{D}_{ij}\big)_\text{causal} ;
\end{align}
where, upon IBPs and invoking the conservation of the energy-momentum-shear-stress tensor, we have
{\allowdisplaybreaks
\begin{align}
\big(\ddot D_{ij}\big)_\text{causal} = \,& 4 G_\text N\int_{\mathbb{R}^{3,1}} \text{d}^4 x'\,\Bigg\{ \partial_\eta^2 \frac{\delta[T-R]}{ R} \left(^{(\text{a})}T_{ij}[\eta',\vec x']+\frac{\delta_{ij}}{2}\left({}^{(\text{a})}T_{00}[\eta',\vec x']-{}^{(\text{a})}T_{ll}[\eta',\vec x']\right)\right) \nonumber\\
&
-2\partial_\eta \partial_{(i} \frac{\delta[T-R]}{ R} {}^{(\text{a})}T_{j)0}[\eta',\vec x']
+\frac{1}{2}\partial_i\partial_j \frac{\delta[T-R]}{ R} \left(\,{}^{(\text{a})}T_{00}[\eta',\vec x']+{}^{(\text{a})}T_{ll}[\eta',\vec x']\right)\Bigg\} .
\end{align}}%
The sole purpose of the rest of the gauge-invariant variables $(\Psi,\Phi,V_i)$, as far as the $\delta_1 R_{0i0j}$ components are concerned, is to cancel the acausal part of the graviton acceleration. Moreover, all of them are needed to ensure causality. We may verify these claims by simply comparing the following expressions.
{\allowdisplaybreaks
\begin{align}
\ddot D_{ij}[\eta,\vec x]&= \big(\ddot D_{ij}\big)_\text{causal} \nonumber\\
&- 4 G_\text N\int_{\mathbb{R}^{3,1}} \text{d}^{3}\vec{x}'\,\Bigg\{\! -2\partial_m\partial_{(i} \frac{1}{ R}
{}^{(\text{a})}T_{j)m}[\eta,\vec{x}']+\frac1{2}\bigg(\delta_{ij}\partial_m\partial_n \frac{1}{ R}{}^{(\text{a})}T_{mn}[\eta,\vec{x}'] \notag\\
&+\partial_i\partial_j \frac{1}{ R} \left({}^{(\text{a})}T_{00}[\eta,\vec{x}']+{}^{(\text{a})}T_{ll}[\eta,\vec{x}']\right)\bigg) + \frac{1}{4}\partial_i\partial_j\partial_m\partial_n R {}^{(\text{a})}T_{mn}[\eta,\vec{x}']\Bigg\}
\end{align}}%
{\allowdisplaybreaks
\begin{align}
\ddot \Psi[\eta,\vec x]=\,& - 4 G_\text N \int_{\mathbb{R}^{3}} \text{d}^{3}\vec{x}'\, \frac{1}{4} \partial_m \partial_n \frac{1}{ R} {}^{(\text{a})}T_{mn}[\eta,\vec x'], \nonumber \\
\partial_i\partial_j\Phi[\eta,\vec x]=\,& -4 G_\text N \int_{\mathbb{R}^{3}} \text{d}^{3}\vec{x}'\, \frac{1}{4} \bigg(\partial_i\partial_j \frac{1}{ R} {}^{(\text{a})}T_{00}[\eta,\vec{x}'] +
\partial_i\partial_j \frac{1}{ R} {}^{(\text{a})}T_{ll}[\eta,\vec{x}']
- \frac32 \partial_i\partial_j\partial_m\partial_n R {}^{(\text{a})}T_{mn}[\eta,\vec{x}'] \bigg), \nonumber\\
\partial_{(i}\dot{V}_{j)}[\eta,\vec x] =\,& 4 G_\text N \int_{\mathbb{R}^{3}} \text{d}^{3}\vec{x}'\left( \partial_m\partial_{(i} \frac{1}{ R} {}^{(\text{a})}T_{j)m}[\eta,\vec{x}']
- \frac12\partial_i\partial_j\partial_m\partial_n
R {}^{(\text{a})}T_{mn}[\eta,\vec{x}']\right) .
\end{align}}%
These massless spin-2 graviton fields, upon quantization, would violate microcausality, because their Green's functions do not vanish at spacelike intervals.
It is worth highlighting, we are not asserting that relativists are computing gravitational wave-forms wrongly. In the far zone, $|\omega| r \gg 1$, we have shown that the distortion of space due to GWs (at finite frequencies) do reduce to the tt ones gotten by performing a local-in-space projection of the (de Donder gauge) spatial perturbations $\chi_{ij}$. These tt GWs, as opposed to their TT counterparts, are in fact the ones computed in the gravitational literature. On the other hand, within this far zone, these tt GWs in fact coincide with the TT ones, because the acausal parts of the latter begin at higher orders in $1/(\omega r)$.
In a cosmology driven by a relativistic fluid, we have uncovered tentative evidence that the Bardeen scalar potentials contribute to gravitational tidal forces, and their wave-like solutions could therefore be legitimately dubbed `scalar gravitational waves' in this sense. More work would be required to confirm or deny this \cite{ChuLiuInPrep}. Nonetheless, if the Bardeen $\Psi$ and $\Phi$ are indeed an integral part of cosmological GWs, we hope this work constitutes the first step towards illuminating not only their associated GW patterns but also potential scalar GW memory effects.
Let us end on a more speculative note. Even though the TT graviton is acausal and cannot be a standalone observable within classical physics, it may be produced quantum mechanically -- and independently of other gauge-invariant perturbations -- during a (still hypothetical) exponentially expanding phase of the early universe.\footnote{We note in passing: when constructing the quantum field theory of photons and gravitons in 4D flat spacetime, Weinberg \cite{Weinberg:1965rz} had to add non-local terms to the Hamiltonian of these massless spin-1 and spin-2 gauge theories in order to preserve the Lorentz invariance of the $S-$matrix.} On the other hand, we have also pointed out that the quantum operators associated with both the free massless spin-1 and spin-2 particles violate micro-causality. Is it possible to exploit this violation to ascertain whether $B-$modes in the Cosmic Microwave Background, if we ever detect them, were truly engendered by {\it quantum} fluctuations of spacetime itself? Or, for the photon case, are there laboratory experiments involving quantum generation of photons that could not only serve as analogs to the inflationary scenario, but also allow the quantum nature of their production mechanism to be probed directly?
\section{Acknowledgments}
YZC is supported by the Ministry of Science and Technology of the R.O.C. under the grant 106-2112-M-008-024-MY3. He also thanks the organizers of the 2018 TGWG Conference (at Tamkang University, Taipei), where he gave a talk about this work; as well as Lior Burko and Feng-Li Lin for questions and Jan Steinhoff for constructive comments. YWL wishes to gratefully acknowledge the support of Shih-Chang Lee and the Department of Physics at National Central University during the completion of the most part of this work. He is supported by the Ministry of Science and Technology of the R.O.C. under Project No.~MOST 108-2811-M-008-503. They would also like to acknowledge Bonga, R\'{a}cz, Flanagan and Hughes for e-mail exchanges.
|
1,116,691,499,271 | arxiv | \section{Introduction}
In a recent paper \cite{DLR3} the first and the third authors introduced the notion of {\em horizontal $\alpha$ harmonic map}, $\alpha\ge 1/2$ with respect to a given $C^1$ planes distribution.
Precisely we consider $P_T\in C^1({\mathbb{R}}^m,M_m({\mathbb{R}}))$ and $P_N\in C^1({\mathbb{R}}^m,M_m({\mathbb{R}}))$ such that
\begin{equation}
\label{I.1}\left\{
\begin{array}{l}
P_T\circ P_T=P_T\quad P_N\circ P_N=P_N\\[3mm]
P_T+P_N=I_m\\[3mm]
\forall\, z\in {\mathbb{R}}^m\quad\forall\, U,V\in T_z{\mathbb{R}}^m\quad <P_TU,P_NV>=0\\[3mm]
\|\partial_z P_T\|_{L^\infty({\mathbb{R}}^m)}<+\infty
\end{array}
\right.
\end{equation}
where $<\cdot,\cdot>$ denotes the standard scalar product in ${\mathbb{R}}^m$. In other words $P_T$ is a $C^1$ map into the orthogonal projections of ${\mathbb{R}}^m$. For such a distribution of projections $P_T$
we denote by
\[
n:=\mbox{rank}(P_T).
\]
Such a distribution identifies naturally with the distribution of $n-$planes given by the images of $P_T$ (or the Kernel of $P_T$) and conversely,
any $C^1$ distribution of $n-$dimensional planes defines uniquely $P_T$ satisfying (\ref{I.1}).
For any $\alpha\ge 1/2$ and for $k\ge 1$ we define the space of {\bf $H^\alpha$-Sobolev horizontal maps }
\[
{\frak{H}}^{\alpha} ({\mathbb{R}}^k):=\left\{u\in H^{\alpha}({\mathbb{R}}^k,{\mathbb{R}}^m)\quad;\quad P_N(u)\nabla u=0\quad\mbox{ in }{\mathcal D}'({\mathbb{R}}^k)\right\}
\]
Observe that this definition makes sense since we have respectively $P_N\circ u\in H^{\alpha}(\mathbb{R}^k,M_m({\mathbb{R}}))$ and ${\nabla u}\in H^{\alpha-1}({\mathbb{R}}^k,{\mathbb{R}}^m)$.
\begin{Definition}
\label{df-legend-harm}
Given a $C^1$ plane distribution $P_T$ in ${\mathbb{R}}^m$ satisfying (\ref{I.1}), a map $u$ in the space ${\frak H}^{\alpha}({\mathbb{R}}^k)$ is called {\bf {horizontal $\alpha$-harmonic} }with respect to $P_T$
if
\begin{equation}
\label{int-I.6}
\forall\, i=1\cdots m\quad\quad\sum_{j=1}^mP_T^{ij}(u)(-\Delta)^\alpha u_j=0\quad\quad\mbox{in }{\mathcal D}'({\mathbb{R}}^k)
\end{equation}
and we shall use the following notation
\[
P_T(u)\,(-\Delta)^\alpha u=0\quad\quad\mbox{in }{\mathcal D}'({\mathbb{R}}^k).\hfill \Box
\]
\end{Definition}
When the plane distribution $P_T$ is {\it integrable} that is to say when
\begin{equation}
\label{I.2}
\forall \ X,Y\in C^1({\mathbb{R}}^m,{\mathbb{R}}^m)\quad P_N [P_T\, X,P_T\,Y]\equiv 0
\end{equation}
where $[\cdot,\cdot]$ denotes the Lie Bracket of vector-fields, by using Fr\"obenius theorem the planes distribution corresponds to
the tangent plane distribution of a $n-$dimensional {\it foliation } ${\mathcal F}$, (see e.g \cite{Lang}). A smooth map $u$ in ${\frak H}^{\alpha}({\mathbb{R}}^m)$ takes values everywhere
into a {\it leaf } of ${\mathcal F}$ that we denote $N^n$ and we are back to the classical theory of $\alpha$ harmonic maps into
manifolds. We recall that the notion of weak $1/2$ harmonic maps into a $n$-dimensional closed manifolds ${N}^n\subset \mathbb{R}^m$ has been introduced by the first and third author in
\cite{DLR1,DLR2}.
These maps
are critical points of the fractional energy on ${\mathbb{R}}^k$
\begin{equation}\label{fracenergy}
E^{1/2}(u):=\int_{{\mathbb{R}}^k}|(-\Delta)^{1/4}u|^2\ dx^k
\end{equation}
within
\[
H^{1/2}({\mathbb{R}}^k,N^n):=\left\{ u\in H^{1/2}({\mathbb{R}}^k,{\mathbb{R}}^m)\ ;\ u(x)\in N^n\ \mbox{ for a. e. }x\in {\mathbb{R}}^k\right\}.
\]
The corresponding Euler-Lagrange equation is given by
\begin{equation}
\label{int-I.3}
\nu(u)\wedge(-\Delta)^{1/2} u=0\quad \quad\mbox{in }{\mathcal D}'({\mathbb{R}}^k),
\end{equation}
where $\nu(z)$ is the Gauss Maps at $z\in\cal{N}$ taking values into the grassmannian $\tilde{G}r_{m-n}({\mathbb{R}}^m)$ of oriented $m-n$ planes in ${\mathbb{R}}^m$ which
is given by the oriented normal $m-n-$plane to $T_z\cal{N}\,.$
One of the main results obtained in \cite{DLR3} is the following
\begin{Theorem}
\label{th-I.1}
Let $P_T$ be a $C^1$ distribution of planes (or projections) satisfying (\ref{I.1}). Any map $u\in {\frak H}^{1/2}({\mathbb{R}})$ (resp. $u\in {\frak H}^{1}({\mathbb{R}^2})$) satisfying
\begin{equation}
\label{I.4}
P_T(u)\,(-\Delta)^{1/2}u=0\quad\mbox{ in }{\mathcal D}'({\mathbb{R}})
\end{equation}
(resp.
\begin{equation}
\label{I.5}
P_T(u)\,\Delta u=0 \quad\mbox{ in }{\mathcal D}'({\mathbb{R}^2}))
\end{equation}
is in $\cap_{\delta<1}C^{0,\delta}({\mathbb{R}})$, (resp. $\cap_{\delta<1}C^{0,\delta}({\mathbb{R}^2})$). \hfill $\Box$
\end{Theorem}
In order to prove Theorem \ref{th-I.1} the authors in \cite{DLR3} use the following two key properties satisfied by respectively horizontal harmonic and horizontal $1/2$-harmonic maps.\par
Horizontal harmonic maps satisfy an elliptic Schr\"odinger type system with an antisymmetric potential $\Omega\in L^{2}({\mathbb{R}}^k,{\mathbb{R}}^k\otimes so(m))$ of the form
\begin{equation}
\label{int-I.7}
-\Delta u=\Omega(P_T)\cdot\nabla u.
\end{equation}
Hence, following the analysis in \cite{Riv} the authors deduced in two dimension the local existence on a disc $D^2$ of $A(P_T)\in L^\infty\cap W^{1,2}(D^2,Gl_m({\mathbb{R}}))$ and $B(P_T)\in W^{1,2}(D^2,M_m({\mathbb{R}}))$ such that
\begin{equation}
\label{int-I.8}
\mbox{div}\left(A(P_T)\,\nabla u\right)=\nabla^\perp B(P_T)\cdot\nabla u
\end{equation}
from which the regularity of $u$ can be deduced using Wente's {\it Integrability by compensation} which can be summarized in the following estimate
\begin{equation}
\label{int-I.9}
\|\nabla^\perp B\cdot\nabla u\|_{H^{-1}(D^2)}\le \, C\, \|\nabla B\|_{L^2(D^2)}\ \|\nabla u\|_{L^2(D^2)}.
\end{equation}
A similar property is satisfied by horizontal $1/2$-harmonic maps. Precisely in \cite{DLR3} conservation laws corresponding to (\ref{int-I.8}) but for general {\it horizontal $1/2-$harmonic maps} have been discovered: locally, modulo some smoother terms
coming from the application of non-local operators on cut-off functions, the authors construct $A(P_T)\in L^{\infty}\cap H^{1/2}(\mathbb{R},Gl_m({\mathbb{R}}))$ and $B(P_T)\in H^{1/2}(\mathbb{R},M_m({\mathbb{R}}))$ such that
\begin{equation}
\label{int-I.9}
(-\Delta)^{1/4}(A(P_T)\, v)=\mathcal{J}(B(P_T),v) +\mbox{cut-off},
\end{equation}
where $v:=(P_T\,(-\Delta)^{1/4}v,{\mathcal R}\, (P_N(-\Delta)^{1/4}v))$ and ${\mathcal R}$ denotes the Riesz operator and ${\mathcal J}$ is a bilinear pseudo-differential operator satisfying
\begin{equation}
\label{int-I.10}
\|\mathcal{J}(B,v)\|_{H^{-1/2}({\mathbb{R}})}\le C\, \|(-\Delta)^{1/4} B\|_{L^2({\mathbb{R}})}\, \|v\|_{L^2({\mathbb{R}})}.
\end{equation}
Moreover by assuming that $P_T\in C^2(\mathbb{R}^m)$and $\|\partial_zP_T\|_{ L^{\infty}(\mathbb{R})}<+\infty$ and by bootstrapping arguments one gets that every horizontal $1/2$ harmonic map $u\in {\frak H}^{1/2}({\mathbb{R}})$ is $C^{1,\alpha}_{loc}(\mathbb{R})$, for every $\alpha<1$, (see \cite{DL3}).\par
We also remark that if $\Pi\colon S^1\setminus \{-i\}\to \mathbb{R}$, $\Pi(\cos(\theta)+i \sin(\theta))=\frac{\cos(\theta)}{1+\sin(\theta)}$ is the classical stereographic projection whose inverse is given by
\begin{equation}\label{formulaPi}
\Pi^{-1}(x)=\frac{2x}{1+x^2}+i\left(-1+\frac{2}{1+x^2}\right).
\end{equation}
then the following relation between the $1/2$ Laplacian in $\mathbb{R}$ and in $S^1$ holds:
\begin{Proposition}[Proposition 4.1, \cite{DLMR}] \label{proppull} Given $u:\mathbb{R}\to\mathbb{R}^m$ set $v:=u\circ \Pi:S^1\to\mathbb{R}^m$. Then $u\in L_{\frac{1}{2}}(\mathbb{R})$\footnote{We recall that
$L_\frac{1}{2}(\mathbb{R}):=\left\{u\in L^1_{loc}(\mathbb{R}):\int_{\mathbb{R}}\frac{|u(x)|}{1+x^2}dx<\infty \right\}$} if and only if $v\in L^1(S^1)$. In this case
\begin{equation}\label{eqlapv1}
\lapfr_{S^1} v(e^{i\theta})=\frac{((-\Delta)_{\mathbb{R}}^\frac12u)(\Pi(e^{i\theta}))}{1+\sin\theta} ~~\mbox{in $\mathcal{D}'(S^1\setminus\{-i\})$},
\end{equation}
Observe that $1+\sin(\theta)=|\Pi^{\prime}(\theta)|,$ and hence we have
$$\int_{S^1} \lapfr v(e^{i\theta})\,\varphi(e^{i\theta})\ d\theta=\int_{\mathbb{R}}\lapfr u(x)\ \varphi\circ\Pi^{-1}(x)\ dx\quad \text{for every }\varphi\in C^\infty_0(S^1\setminus\{-i\}).$$\end{Proposition}
From Proposition \ref{proppull} it follows that $u\in {\frak H}^{1/2}({\mathbb{R}})$ is a horizontal $1/2$ harmonic map in $\mathbb{R}$ if and only if $v:=u\circ \Pi\in {\frak H}^{1/2}({S^1})$ is a horizontal $1/2$ harmonic map in $S^1$.
\par
\medskip
The goal of this paper is to investigate compactness and quantization properties of sequences of horizontal $1/2$ harmonic maps $u_k\in {\frak{H}}^{1/2} ({\mathbb{R}}).$ Our main result is the following:
\begin{Theorem}
\label{th-I.2}
Let $u_k\in {\frak{H}}^{1/2} ({\mathbb{R}})$ be a sequence of horizontal $1/2$-harmonic maps such that
\begin{equation}\label{hypcom}\|u_k\|_{\dot H^{1/2}}\le C,~~~\|(-\Delta)^{1/2}u_k\|_{L^1}\le C\,.\end{equation}
Then it holds:\par
\begin{enumerate}
\item There exist $u_{\infty}\in {\frak{H}}^{1/2} ({\mathbb{R}})$ and a possibly empty set $\{a_1,\ldots, a_\ell\}$, $\ell\ge 1\,,$ such that up to subsequence
\begin{equation}\label{conv}
u_{k }\to u_\infty \quad \mbox{in $\dot W^{1/2,p}_{loc}(\mathbb{R}\setminus\{a_1,\ldots, a_{\ell}\}),~~p\ge 2$ as $k \to +\infty$}
\end{equation}
and
\begin{equation}\label{provv1}
P_T(u_{\infty})(-\Delta)^{1/2} u_{\infty}=0,\quad \mbox{in ${\cal{D}}^{\prime}(\mathbb{R} )$\,.}
\end{equation}
\item There is a family $ \tilde u_{\infty}^{i,j}\in \dot {\frak{H}}^{1/2} ({\mathbb{R}})$ of horizontal $1/2$-harmonic maps $ ( i\in\{1,\ldots,\ell\}, j\in\{1,\ldots ,N_i\}), $ such that up to subsequence
\begin{equation}\label{finalquantintr}
\left\| (-\Delta)^{1/4}\left(u_k-u_{\infty}-\sum_{i,j} \tilde u_{\infty}^{i,j}((x-x^k_{i,j})/r^k_{i,j})\right)\right\|_{L^2_{loc}(\mathbb{R})}\to 0,~~\mbox{as $k \to +\infty$}\,.
\end{equation}
for some sequences $r^k_{i,j}\rightarrow 0$ and $x^k_{i,j}\in {\mathbb{R}}$.
\end{enumerate}
\end{Theorem}
We would like to make some comments and remarks on Theorem \ref{th-I.2}.\par
{\bf 1.} We first mention that the condition $\|(-\Delta)^{1/2}u_k\|_{L^1}\le C$ is always satisfied in the case the maps $u_k$ take values into a closed manifold of $\mathbb{R}^m$ (case of sequences of $1/2$ harmonic maps) as soon as $\|u_k\|_{\dot H^{1/2}}\le C.$ This follows from the fact that if $u$ is a $1/2$-harmonic maps with values into a closed manifold of ${\cal{N}}^n$ of $\mathbb{R}^m$ then
the following inequality holds (see {Proposition} \ref{prL1})
\begin{equation}
\label{L^1}
\|(-\Delta)^{1/2}u\|_{L^1(\mathbb{R})}\le C\|(-\Delta)^{1/4}u\|^2_{L^2(\mathbb{R})}\quad.
\end{equation}
Hence we have the following corollary
\begin{Corollary}
Let ${\mathcal N}^n$ be a closed $C^2$ submanifold of ${\mathbb{R}}^m$ and let Let $u_k\in {{H}}^{1/2} ({\mathbb{R}},{\mathcal N}^n)$ be a sequence of l $1/2$-harmonic maps such that
\begin{equation}\label{hypcom-1}\|u_k\|_{\dot H^{1/2}}\le C\end{equation}
then the conclusions of theorem~\ref{th-I.2} hold. In particular modulo extraction of a subsequence we have the following energy identity: \begin{equation}
\lim_{k\rightarrow +\infty}\int_{\mathbb{R}} |(-\Delta)^{1/4}u_k|^2\ dx=\int_{\mathbb{R}} |(-\Delta)^{1/4}u_\infty|^2\ dx+\sum_{i,j}\int_{\mathbb{R}} |(-\Delta)^{1/4}\tilde u_\infty^{i,j}|^2\ dx
\end{equation}
where $\tilde u_\infty^{i,j}$ are the {\bf bubbles} associated to the weak convergence.
\end{Corollary}
For the moment it remains open to know whether the bound (\ref{L^1}) holds or not in the general case of horizontal $1/2$-harmonic maps.\par
{\bf 2.} The compactness issue (first part of Theorem \ref{th-I.2}) is quite standard. The most delicate part is the quantization analysis consisting in verifying that there is no dissipation of the energy in the region between $u_{\infty}$ and
the {\em bubbles} $\tilde u_{\infty}^{i,j}$ and between the bubbles themselves (the so-called {\em neck-regions}). The strategy of the proof of theorem~\ref{th-I.2} is presented in the next section. One important tool we are using for proving theorem~\ref{th-I.2} is a new {\bf Pohozaev identity} for the half Laplacian in 1 dimension.
\begin{Theorem}\label{th-I.4}{\bf [Pohozaev Identity in $\mathbb{R}$]}
Let $u\in W^{1,2}(\mathbb{R},\mathbb{R}^m)$ be such that
\begin{equation}\label{Iharmequation1}
\frac{du}{dx}\cdot (-\Delta)^{1/2} u=0~~\mbox{a.e in $\mathbb{R}$.}
\end{equation}
Assume that
\begin{equation}\label{IcondPoh}
\int_{\mathbb{R}}|u-u_0 |dx <+\infty,~~\int_{\mathbb{R}}\left\vert \frac{du}{dx}(x)\right\vert\, dx <+\infty
\end{equation}
Then the following identity holds
\begin{equation}\label{I-identityPR}
\left|\int_{x\in\mathbb{R}} \frac{x^2-t^2}{(x^2+t^2)^2}u(x)dx\right|^2=\left|\int_{x\in\mathbb{R}} \frac{2xt}{(x^2+t^2)^2} u(x)dx\right|^2.\end{equation}
\end{Theorem}
We observe that the conditions \rec{IcondPoh} are satisfied by the $1/2$-harmonic maps with valued into a closed $C^2 $ sub-manifold.
By means of the stereographic projection we get an analogous formula in $S^1$.
\begin{Theorem}
\label{Phohozaev-S^1}
{\bf [Pohozaev Identity on $S^1$]}
Let $u$ be a $W^{1,2}$ map from $S^1$ into ${\mathbb{R}}^m$ satisfying
\begin{equation}
\label{pert}
\frac{d u}{d\theta}\cdot (-\Delta)^{1/2} u=0\quad \mbox{ a. e. on }S^1
\end{equation}
then the following identity holds
\begin{equation}
\label{poho-s1}
\left|\int_0^{2\pi}u(\theta)\, \cos\theta\ d\theta\right|^2=\left|\int_0^{2\pi}u(\theta)\, \sin\theta\ d\theta\right|^2
\end{equation}
\hfill $\Box$
\end{Theorem}
We have now to give some explanations why these identities belong to the {\it Pohozaev identities} family. These identities are produced by the conformal invariance of the highest
order derivative term in the Lagrangian from which the Euler Lagrange is issued : in $2$D for instance
\[
E(u)=\int_{\mathbb{R}^2} |\nabla u|^2 dx^2
\]
is conformal invariant, whereas
\[
E^{1/2}(u)=\int_{\mathbb{R}} |(-\Delta)^{1/4}u|^2\ dx
\]
is conformal invariant in $1$D. The infinitesimal perturbations issued from the dilations produce the following infinitesimal variations of these highest order terms respectively
\[
\sum_{i=1}^2 x_i\,\frac{\partial u}{\partial x_i}\cdot\Delta u\quad\mbox{ in $2$D}\quad\mbox{ and }\quad x\frac{d u}{dx}\cdot(-\Delta)^{1/2}u\quad\mbox{ in $1$D}
\]
Being a critical point respectively of $E$ in 2 D and $E^{1/2}$ in $1D$ and assuming enough regularity gives respectively
\[
\sum_{i=1}^2 x_i\,\frac{\partial u}{\partial x_i}\cdot\Delta u=0\quad\mbox{ in $2$D}\quad\mbox{ and }\quad x\frac{d u}{dx}\cdot(-\Delta)^{1/2}u=0\quad\mbox{ in $1$D}
\]
In two dimensions, integrating this identity on a ball $B(x_0,r)$ gives the following {\bf balancing law} between the {\bf radial} part and the {\bf angular} part of the energy
classically known as {\bf Pohozaev identity}.
\begin{Theorem}\label{th-I.6}
Let $u\in W^{2,2} ({\mathbb{R}}^2,\mathbb{R}^m)$ such that
\begin{equation}\label{IharmequationR2}
\sum_{i=1}^2 x_i\,\frac{\partial u}{\partial x_i }\cdot \Delta u=0~~\mbox{a.e in $B(0,1)$.}
\end{equation}
Then it holds
\begin{equation}
\label{PI}
\int_{\partial B(x_0,r)} \left\vert \frac{1}{r}\frac{\partial u}{\partial\theta}\right\vert^2 d\theta=\int_{\partial B(x_0,r)}\left\vert\frac{\partial u}{\partial r}\right\vert^2 d\theta
\end{equation}
for all $r\in [0,1].$
\end{Theorem}
In 1 dimension one might wonder what corresponds to the $2$ dimensional dichotomy between {\bf radial} and {\bf angular} parts. We illustrate below the correspondence
of dichotomies respectively in $1$ and $2$ dimensions.
\[
\begin{array}{ccc}
\mbox{$2$D}& \longleftrightarrow & \mbox{ $1$D}\\[3mm]
\displaystyle\mbox{ {\bf radial} : } \quad\frac{\partial u}{\partial r}& \longleftrightarrow & \mbox{ {\bf symmetric} part of }u\quad:\quad u^+(x):= \frac{u(x)+u(-x)}{2}\\[3mm]
\displaystyle\mbox{{\bf angular} : }\quad \frac{\partial u}{\partial \theta}& \longleftrightarrow & \mbox{ {\bf antisymmetric} part of }u\quad:\quad u^-(x):= \frac{u(x)-u(-x)}{2}
\end{array}
\]
Observe moreover that our {\bf Pohozaev identity in $1$D} (\ref{poho-s1}) can be rewritten as a {\bf balancing law} between the {\bf symmetric} part and the {\bf antisymmetric} part
of $u$.
\begin{equation}
\label{poho-s2}
\left|\int_0^{2\pi}u^+(\theta)\, \cos\theta\ d\theta\right|^2=\left|\int_0^{2\pi}u^-(\theta)\, \sin\theta\ d\theta\right|^2
\end{equation}
This law {\underbar is not invariant} under the action of the M\"obius group but the condition (\ref{pert}) is. Applying for instance rotations by an arbitrary angle $\alpha\in{\mathbb{R}}$, the identity (\ref{poho-s1})
implies
\begin{equation}
\label{poho-s2}
\left\{
\begin{array}{l}
\displaystyle |u_1|=|u_{-1}|\\[3mm]
\displaystyle u_1\cdot u_{-1}=0
\end{array}
\right.
\end{equation}
where
\[
\left\{\begin{array}{l}
\displaystyle u_1:=\frac{1}{2\pi}\int_0^{2\pi}u(\theta)\, \cos\theta\ d\theta\\[3mm]
\displaystyle u_{-1}=\frac{1}{2\pi}\int_0^{2\pi}u(\theta)\, \sin\theta\ d\theta
\end{array}
\right.
\]
The previous implies that there are ``as many'' Pohozaev identities
as elements in this group minus the action of rotations, that is there are a 3-1=2=D+1 dimensional family of identities exactly as in the 2-D case where there are
exactly as many {\bf Pohozaev identities} (\ref{PI}) as choices of center $x_0\in {\mathbb{R}}^2$ and radius $r>0$ (which is again a $D+1=3$ dimensional space).
\subsection{The strategy of the proof of theorem~\ref{th-I.2}.}
\label{strategy}
We first recall the definitions of a bubble and a neck region.
\begin{Definition}[Bubble]
A {\bf Bubble} is a {\bf non-constant} horizontal $1/2$-harmonic map $u\in {\frak{H}}^{1/2} ({\mathbb{R}})$\,. \end{Definition}
\begin{Definition}[Neck region]
A neck region for a sequence $f_k\in L^2(\mathbb{R})$ is the union of finite degenerate annuli of the type $A_k(x)=B(x,R_k)\setminus B(x,r_k)$ with $r_k\to 0$ and $\frac{R_k}{r_k}\to +\infty$ as $k \to +\infty$ such that
\begin{equation}
\label{neck}
\lim_{\Lambda\to \infty}\lim_{k\to \infty}\left( \sup_{\rho\in[\Lambda r_k,(2\Lambda)^{-1}R_k]}\int_{B(x,2\rho)\setminus B(x,\rho)}|f_k|^2 dx\right)^{1/2}=0\,.
\end{equation}
\end{Definition}
\medskip
The main achievement of the present work is to show, under the assumptions of theorem~\ref{th-I.2}, that (\ref{neck}) for $f_k:=(-\Delta)^{1/4}u_k$ can be improved to
\begin{equation}\label{limitneck}
\lim_{\Lambda\to \infty}\lim_{k\to \infty}\left(\int_{B(x,\frac{R_k}{\Lambda})\setminus B(x,\Lambda r_r)}|f_k|^2 dx\right)^{1/2}=0.
\end{equation}\par
The proof of estimate of \rec{limitneck} will be the aim of Section \ref{cq}.\par
{\bf 3.} Theorem \ref{th-I.2} has been proved by the first author in \cite{DL2} in the case of sequences of $1/2$-harmonic maps $u_k$ with values into the ${\cal{S}}^{m-1}$ sphere, see also \cite{LP}. In this case in order to prove
\rec{limitneck} we use the duality of the Lorentz spaces $L^{2,1}-L^{2,\infty}.$\footnote{ see section \ref{secdef} for a definition.}
We first show that the $L^{2,\infty}$ norm of the $u_k$ is arbitrary small in the {\em neck region}
and then we use the fact that $ L^{2,1}$ norm of $1/2$-harmonic maps with values into a sphere is uniformly globally bounded. This last global estimate follows directly by
the formulation of the $1/2$-harmonic maps equation that the first and third author discovered in \cite{DLR1} in terms of special algebraic quantities (three-commutators) satisfying particular integrability compensation properties. \par
The fact that the
$L^{2,\infty}$ norm of a sequence of horizontal $1/2$-harmonic maps $u_k$ is arbitrary small in {\em neck regions} still holds in the general case, precisely we have:
\begin{Lemma}[$L^{2,\infty}$ estimates]\label{neck2}
There exists $ \delta >0$ such that for any sequence of horizontal $1/2$-harmonic maps $u_k\in{\frak{H}}^{1/2} ({\mathbb{R}})$ satisfying
$$ \sup_{\rho\in[\Lambda r_k,{(2\Lambda)}^{-1}R_k]}\left( \int_{B(0,2\rho)\setminus B(0,\rho)}|
(-\Delta)^{1/4} u_k|^2dx\right)^{1/2}<\delta $$
with $r_k\to 0$ and $\frac{R_k}{r_k}\to +\infty$ as $k \to +\infty$ and for all $\Lambda>1$ with $\Lambda r_k<(2\Lambda)^{-1}R_k$
then
\begin{equation}\label{linftyest}
\limsup_{\Lambda\to\infty}\limsup_{k\to\infty } \| (-\Delta)^{1/4} u\|_{L^{2,\infty}(B(0,(2\Lambda)^{-1}R_k)\setminus B(0,\Lambda r_k)} =0.\end{equation}
\end{Lemma}
Unfortunately we do not know if such a a global uniform $ L^{2,1}$ estimate exists even in the case
of $1/2$-harmonic maps with values into a closed submanifold of $\mathbb{R}^m.$
To overcome of this lack of a global $L^{2,1}$ estimate, we show a precise asymptotic development of $(-\Delta)^{1/4}u_k$ in annuli
$A_{r_k,R_k}=B(x,R_k)\setminus B(x,r_k)$ where the $L^2$ norm of $(-\Delta)^{1/4}u_k$ is small.
To get such an asymptotic development we argue as follows. We first prove the following result (for simplicity we will consider an annulus centered at $x=0$).
\begin{Theorem}\label{th-I.3}
Let $u_k\in{\frak{H}}^{1/2} ({\mathbb{R}})$ be a sequence of horizontal $1/2$-harmonic maps satisfying the assumptions of Theorem \ref{th-I.1}. There exists $\delta>0$ such that if
\begin{eqnarray}\label{quant0}
\|(-\Delta)^{1/4}u_k\|_{L^2(B(0,R_k)\setminus B(0,r_k)}&<&\delta,
\end{eqnarray}
with $r_k\to 0$ and $\frac{R_k}{r_k}\to +\infty$ as $k \to +\infty$ , then for all $\Lambda>1$ with $\Lambda r_k<(2\Lambda)^{-1}R_k$ and $x\in B(0,(2\Lambda)^{-1}R_k)\setminus B(0,\Lambda r_k)$ we have
\begin{equation}\label{devel}
(-\Delta)^{1/4}u_k(x)= \frac{a^+_k(x)\overrightarrow{c_{r_k}}}{|x|^{1/2}} + h_k+g_k,
\end{equation}
where $g_k\in L^2(\mathbb{R})$ with
$\displaystyle \limsup_{k\to\infty}\|g_k\|_{L^2} =0,$
\begin{equation}\label{esckI}
\displaystyle \overrightarrow{c_{r_k}}=O\left(\left(\log\left(\frac{R_k}{2\Lambda^2r_k}\right)\right)^{-1/2}\right),~~\mbox{as}~~ k\to +\infty,\Lambda\to +\infty\end{equation} $a^+_k\in L^{\infty}\cap \dot H^{1/2}(\mathbb{R},Gl_m({\mathbb{R}}))$, $h_k\in L^{2,1}(A_{\Lambda r_k,(2\Lambda)^{-1}R_k})$,
\begin{equation}\limsup_{\Lambda\to\infty}\limsup_{k\to+\infty} \|h_k\|_{L^{2,1}(A_{\Lambda r_k,(2\Lambda)^{-1}R_k})}<\infty\end{equation}
and
\begin{equation}\|a_k\|_{\dot H^{1/2}}+\|a_k\|_{L^{\infty}}\le C \|(-\Delta)^{1/4}u_k\|_{L^2(\mathbb{R})}.\end{equation}
\end{Theorem}
From Theorem \ref{th-I.3} we will deduce that in a neck region (centered for simplicity of notation in $x=0$) we have
\ \begin{equation}\label{limitneck2}
\lim_{\Lambda\to \infty}\lim_{k\to \infty}\left(\int_{B(0,\frac{R_k}{2\Lambda_k})\setminus B(0,\Lambda_k r_r)}|((-\Delta)^{1/4}u_k)^{-}|^2 dx\right)^{1/2}=0.
\end{equation}
where for a given function $f\colon\mathbb{R}^n\to\mathbb{R}^m$ we denote by
$f^+$ and $f^-$ respectively the symmetric and antisymmetric part of $f$ with respect to the origin, i.e.
$f^+(x)=\frac{f(x)+f(-x)}{2}$ and $f^-(x)=\frac{f(x)-f(-x)}{2}$. It remains to find a link between the symmetric and the antisymmetric part of $(-\Delta)^{1/4}u_k$.
To this purpose we
make use of the {\bf Pohozaev type formula} theorem~\ref{th-I.4}, which is one of the main result of this paper.
We explain below in some steps how formula \rec{I-identityPR} permits to get an information of the $L^2$ norm of $((-\Delta)^{1/4}u_k)^-$.\par
{$\bullet$} We observe that we can rewrite the l.h.s and r.h.s of \rec{I-identityPR} respectively as follows
\begin{eqnarray*}
\left| \int_{x\in\mathbb{R}} \frac{x^2-t^2}{(x^2+t^2)^2}u^+(x)dx\right|^2
&=&t^{-2}\left\vert \int_{x\in\mathbb{R}} (-\Delta)^{-1/4}\left[\frac{x^2-1}{(x^2+1)^2}\right]((-\Delta)^{1/4}u)^{+}(xt)dx\right|^2.
\end{eqnarray*}
\begin{eqnarray*}\label{I-identityPR3}
\left| \int_{x\in\mathbb{R}} \frac{2xt}{(x^2+t^2)^2} u(x)dx\right|^2
&=&t^{-2}\left\vert \int_{x\in\mathbb{R}} (-\Delta)^{-1/4}\left[\frac{2x}{(x^2+1)^2}\right]((-\Delta)^{1/4}u)^{-}(xt)dx\right|^2.
\end{eqnarray*}
Therefore we can rewrite the formula \rec{I-identityPR} as
\begin{eqnarray}\label{I-identityPR4}
&& \left| \int_{x\in\mathbb{R}} (-\Delta)^{-1/4}\left[\frac{x^2-1}{(x^2+1)^2}\right]((-\Delta)^{1/4}u)^{+}(xt)\right\vert^2\\
&&~~~~~=\left|\int_{x\in\mathbb{R}} (-\Delta)^{-1/4}\left[\frac{2x}{(x^2+1)^2}\right]((-\Delta)^{1/4}u)^{-}(xt)dx\right|^2.\nonumber
\end{eqnarray}\par
{$\bullet$} We prove that
$$M^{+}[v](t):=\int_{x\in\mathbb{R}} (-\Delta)^{-1/4}\left[\frac{x^2-1}{(x^2+1)^2}\right]v(xt)dx$$ and $$M^{-}[v](t):=\int_{x\in\mathbb{R}} (-\Delta)^{-1/4}\left[\frac{2x}{(x^2+1)^2}\right]v(xt)dx$$ are
isomorphisms from $L^{p}_+(\mathbb{R})$ onto $L^{p}_+(\mathbb{R})$ (resp . $L^{p}_-(\mathbb{R})$ onto $L^{p}_-(\mathbb{R})$) for every $p>1$ and by interpolation from $L^{2,1}_+(\mathbb{R})$ onto $L^{2,1}_+(\mathbb{R})$(resp. $L^{2,1}_-(\mathbb{R})$ onto $L^{2,1}_-(\mathbb{R})$). Here $L^p_+$ is the space of even $L^p$ functions and $L^p_-$ the space of odd $L^p$ functions. \par
{$\bullet$} By
plugging the development \rec{devel} into \rec {I-identityPR4} we obtain an information also of the asymptotic behaviour of the $L^2$ norm of $((-\Delta)^{1/4}u_k)^+$ in
an annular region, namely for all $\Lambda>1$ with $\Lambda r_k<(2\Lambda)^{-1}R_k$ and $x\in B(0,(2\Lambda)^{-1}R_k)\setminus B(0,\Lambda r_k)$
\begin{equation}\label{devel2}
\int_{B\left(0,\frac{R_k}{2\Lambda_k}\right)\setminus B(0,\Lambda_k r_r)}|((-\Delta)^{1/4}u_k)^{-}|^2 dx=\tilde h_k+\tilde g_k,\end{equation}
where $\tilde g_k\in L^2(\mathbb{R})$ with
$\displaystyle \limsup_{k\to\infty}\|\tilde g_k\|_{L^2} =0$ and $$\limsup_{\Lambda\to\infty}\limsup_{k\to+\infty} \|\tilde h_k\|_{L^{2,1}(A_{\Lambda r_k,(2\Lambda)^{-1}R_k})}<\infty.$$
{$\bullet$} By combining \rec{devel} and \rec{devel2} we get
\begin{Theorem}\label{th-I.5}
Under the assumptions of Theorem \ref{th-I.3}, we have
\begin{equation}
\limsup_{\Lambda\to\infty}\limsup_{k\to+\infty}\|(-\Delta)^{1/4}u_k\|_{L^2(B(0,(2\Lambda)^{-1}R_k)\setminus B(0,\Lambda_k r_r))}=0.\end{equation}
\end{Theorem}
We would like to establish a link between the above results and the result obtained by the second and third author \cite{LR} in the framework of harmonic maps in $2D$ with values into a closed manifold and that can easily be extended to
horizontal harmonic maps.
Also in this case, the strategy has been to prove an $L^{2,1}$-estimate on the angular part of the gradient (which play the role of the antisymmetric part $(-\Delta)^{1/4}u$). Precisely
suppose we have a sequence $u_k\in{\frak{H}}^{1} (D^2)$ of horizontal harmonic maps such that $\|\nabla u_k\|_{L^2(D^2)}\le C.$ Then they satisfy
\begin{equation}\label{I-Schr}-\Delta u_k=\Omega_k(P_T)\nabla u~~\mbox{in ${\cal{D}}^{\prime}(D^2)$},\end{equation}
where $\Omega\in L^{2}(D^2,{\mathbb{R}}^2\otimes so(m))$ is an antisymmetric potential. In \cite{LR} the authors found the following asymptotic development for $\nabla u_k$ in a annular domain $B(0,R_k)\setminus B(0,r_k)$ where
$\|\Omega_k\|_{L^2(B(0,R_k)\setminus B(0,r_k)}$ is {\em small}:
\begin{equation}\label{develharm}
\nabla u_k(x)= \frac{a_k(|x|)\overrightarrow{c_{r_k}}}{|x|^{1/2}} + h_k,
\end{equation}
where $a_k$ is radial, $a_k\in L^{\infty}(B(0,R_k)\setminus B(0,r_k)),$ $h_k\in L^{2,1}(A_{\Lambda r_k,(2\Lambda)^{-1}R_k})$ (for every $\Lambda>1$ such that $\Lambda r_k<(2\Lambda)^{-1}R_k$),
\begin{equation}\limsup_{\Lambda\to\infty}\limsup_{k\to+\infty} \|h_k\|_{L^{2,1}(A_{\Lambda r_k,(2\Lambda)^{-1}R_k})}<\infty,~~\|a_k\|_{L^{\infty}}\le C \|\nabla u_k\|_{L^2}.\end{equation}
From \rec{develharm} they deduce that
\begin{equation}\label{l21harm}
\limsup_{\Lambda\to\infty}\limsup_{k\to+\infty} \left\Vert \frac{1}{\rho}\frac{\partial u_k}{\partial\theta}\right\Vert_{L^{2,1}(A_{\Lambda r_k,(2\Lambda)^{-1}R_k})}<\infty.
\end{equation}
They also prove that the $L^{2,\infty}$ of $\nabla u_k$ is arbitrary small in degenerating annuli, namely
\begin{equation}\label{linftyharm}
\limsup_{\Lambda\to\infty}\limsup_{k\to+\infty} \|\nabla u_k\|_{L^{2,\infty}(A_{\Lambda r_k,(2\Lambda)^{-1}R_k})}=0.\end{equation}
By combining \rec{l21harm} and \rec{linftyharm} they obtain
\begin{equation}
\limsup_{\Lambda\to\infty}\limsup_{k\to+\infty} \left\Vert \frac{1}{\rho}\frac{\partial u_k}{\partial\theta}\right\Vert_{L^{2}(A_{\Lambda r_k,(2\Lambda)^{-1}R_k})}=0.
\end{equation}
The Pohozaev identity in $2D$ (\ref{PI}) obtained from \rec{IharmequationR2} implies that there is no loss of energy in the neck region:
\begin{equation}
\limsup_{\Lambda\to\infty}\limsup_{k\to+\infty} \|\nabla u_k\|_{L^{2}(A_{\Lambda r_k,(2\Lambda)^{-1}R_k})}=0.
\end{equation}
In a last part, we give a counter example to quantization for general sequence of Schr\"odinger equation with antisymmetric potential. For the Laplacian it has been donne in \cite{LR}. Hence here we gives an example of sequence of $u_k$ satisfying
$$(-\Delta)^{\frac{1}{2} }u_k =\Omega_k u_k +{\Omega_1}_k u_k$$
with $\Omega_k$ antisymmetric whose $L^2$-norm goes to zero in a neck region, and the $L^{2,1}$-norm ${\Omega_1}_k$ goes to zero in the neck region, despite the $L^2$-norm of $u_k$ remains bounded from below. As in the case of the Laplacian in \cite{LR}, we take a modification of the Green function. Since our functions will be even, this insures that that no Pohozaev identity can occur.
\subsection{Definitions and Notations} \label{secdef}
In this section we introduce some definitions and notations that will be used in the paper.\par
\begin{enumerate}
\item Given $\varphi\colon\mathbb{R}\to\mathbb{C}$ we denote by $\hat \varphi$ or ${\cal{F}}[\varphi]$ the Fourier transform of $\varphi$ and
by $\check\varphi$ or ${\cal{F}}^{-1}[\varphi]$ the inverse Fourier transform of $\varphi.$
\item Given $u,v\in\mathbb{R}^n$ we denote by $\langle u, v\rangle $ the standard scalar product of $u$ and $v$.
\item Given $z\in \mathbb{C}$ we denote by $\Re{z}$ and $\Im{z}$ the real and imaginary part of $z$.
\end{enumerate}
Next we introduce some functional spaces.\\
First, we will note $L^p_+$ the space of even $L^p$ functions and $L^p_-$ the space of odd $L^p$ functions.\\
We denote by $L^{2,\infty}(\mathbb{R}^n)$ the space of measurable functions $f$ such that
$$
\sup_{\lambda>0}\lambda |\{x\in\mathbb{R}^n~: |f(x)|\ge\lambda\}|^{1/2}<+\infty\,,
$$
and $L^{2,1}(\mathbb{R}^n)$ is the space of measurable functions satisfying
$$\int_{0}^{+\infty}|\{x\in\mathbb{R}~: |f(x)|\ge\lambda\}|^{1/2} d\lambda<+\infty \,\,.$$
We can check that $L^{2,\infty}$ and $L^{2,1}$ forms a duality pair. \\
We denote by ${\mathcal H}^1({\mathbb{R}}^n)$ the Hardy space which is the space of $L^1$
functions $f$ on ${\mathbb{R}}^n$satisfying
\[
\int_{{\mathbb{R}}^n}\sup_{t>0}|\phi_t\ast f|(x)\ dx<+\infty\quad ,
\]
where $\phi_t(x):=t^{-n}\ \phi(t^{-1}x)$ and $\phi$ is some function in the Schwartz space ${\mathcal S}({\mathbb{R}}^n)$ satisfying $\int_{{\mathbb{R}}^n}\phi(x)\ dx=1$.\\
Finally for every $s\in\mathbb{R}$ and $q>1$ we denote by
$\dot W^{s,q}(\mathbb{R}^n)$ the fractional Sobolev space
$$\{f\in{\cal{S}}^{\prime}:~{\cal{F}}^{-1}[|\xi|^s|{\cal{F}}[f]]\in L^q(\mathbb{R}^n)\}\,.$$
For more properties on the Lorentz spaces, Hardy space ${\mathcal H}^1$ and fractional Sobolev spaces we refer to \cite{Gra1} and \cite{Gra2}.
The paper is organized as follows.
In Section \ref{nonlocalSchr} we analyze the asymptotic behaviour of the solutions of special nonlocal Schr\"odinger systems with a $L^2$ antisymmetric potential in degenerate annular domain.
In Section \ref{PoId} we describe two new Pohozaev type formulae, one on $\mathbb{R}$ and one in $S^1$, which can be obtained one from the other by means of the stereographic projections and which are satisfied in particular by $1/2$-harmonic maps respectively on $\mathbb{R}$ and on $S^1.$ We also compare such formulae with a Pohozev formula in $2D$ which is satisfied by harmoinc maps. In Section 4 we deduce from the results of Section 2 and 3 the quantization of horizontal $1/2$-harmonic maps. Finally in Section 5 we describe an example showing that in general we cannot have quantization for solutions of a general nonlocal Schr\"odinger systems with a $L^2$ antisymmetric potential.
\section{Preliminary results on nonlocal Schr\"odinger system}\label{nonlocalSchr}
As we will see later horizontal $1/2$-harmonic maps in $1$-D satisfy special
nonlocal system of the form
\begin{equation}\label{modelsystem}
(-\Delta)^{1/4} v=\Omega v+\Omega_1 v+{\cal{Z}}(Q,v)+g \end{equation}
where $v\in L^2(\mathbb{R})$, $Q\in H^{1/2}(\mathbb{R})$, $\Omega\in L^2(\mathbb{R},so(m))$, $\Omega_1\in L^{2,1}(\mathbb{R})$, $g\in L^1(\mathbb{R})$ and ${\cal{Z}}\colon H^{1/2}(\mathbb{R})\times L^2(\mathbb{R})\to {\cal{H}}^1(\mathbb{R})$ satisfies the following stability property: if $Q,P\in{ \dot {H}}^{1/2}(\mathbb{R})\cap L^{\infty}(\mathbb{R})$, $v\in L^{2}$ then
\begin{equation}\label{decZ}
P{\cal{Z}}(Q,v)={{{\cal{A}}_{{\cal{Z}}}}}(P,Q)v+J_{{\cal{Z}}}(P,Q,v),
\end{equation}
where
$$
\|{\cal{A}}_{{\cal{Z}}}(P,Q)\|_{L^{2,1}(\mathbb{R})}\leq C\|(-\Delta)^{1/4}[P] \|_{L^2(\mathbb{R})}\|(-\Delta)^{1/4}[Q] \|_{L^2(\mathbb{R})},$$
and
$$
\|{{J}}_{{\cal{Z}}}(P,Q,v)\|_{{\cal{H}}^1(\mathbb{R})}\le C \left(\|(-\Delta)^{1/4}[P] \|_{L^2(\mathbb{R})}+\|(-\Delta)^{1/4}[Q] \|_{L^2(\mathbb{R}) }\right)\|v\|_{L^2(\mathbb{R})}.$$
Actually ${\cal{Z}}$ is a linear combination of the following {\em pseudo-differential operators}:
\begin{equation}\label{opT}
T(Q,v):=(- \Delta)^{1/4}(Qv)-Q(- \Delta)^{1/4} v+ (- \Delta)^{1/4} Q v\,\end{equation}
and
\begin{equation}
\label{opS}
S(Q,v):=(-\Delta)^{1/4}[Qv]-{\cal{R}} (Q{\cal{R}}(-\Delta)^{1/4} v)+{\cal{R}}((-\Delta)^{1/4} Q{\cal{R}} v )
\end{equation}
\begin{equation}\label{opF}
F(Q,v):={\mathcal{R}}[Q]{\mathcal{R}}[v]-Qv.
\end{equation}
\begin{equation}\label{opOm}
\Lambda(Q,v):=Qv+{\mathcal{R}}[Q{\mathcal{R}}[v]]\,.
\end{equation}
In \cite{DL2} the first author impove the estimates on the operators $T,S$ obtained in \cite{DLR1}:
\begin{Theorem}\label{comm2}
Let $v\in L^2(\mathbb{R}),$ $Q\in \dot{H}^{1/2}( \mathbb{R})$. Then $T(Q,v),S(Q,v)\in {\cal{H}}^{1}( \mathbb{R})$ and
\begin{equation}\label{commest2}
\|T(Q,v)\|_{{\cal{H}}^{1}( \mathbb{R})}\le C\|Q\|_{\dot{H}^{1/2}( \mathbb{R})}\|v\|_{L^2( \mathbb{R})}\,.\end{equation}
\begin{equation}\label{commest3}
\|S(Q,v)\|_{{\cal{H}}^{1}( \mathbb{R})}\le C\|Q\|_{\dot{H}^{1/2}( \mathbb{R})}\|v\|_{L^2( \mathbb{R})}\,.~~~\hfill \Box
\end{equation}
\end{Theorem}
\medskip
As a consequence of the Coifman-Rochberg-Weiss estimate \cite{CRW} we also have
\begin{equation}\label{estFhardy}
\|F(f,v)\|_{ {\cal{H}}^1(\mathbb{R})}\le C \|f\|_{L^2(\mathbb{R})}\|v\|_{L^{2}(\mathbb{R})}\,.
\end{equation}
for $f,v\in L^2$.\par
In \cite{DLR3} the first and the third author show that the stability property \rec{decZ} holds for the operators \rec{opT}, \rec{opS},\rec{opF}.
Moreover the authors have shown that if the $L^2$ norm of $\Omega$ is ``small" then the system \rec{modelsystem} is equivalent to
a conservation law:
\begin{Theorem}[Theorem 3.11 in \cite{DLR3}]\label{conservation}
Let $v\in L^{2}(\mathbb{R},\mathbb{R}^m)$ be a solution of \rec{modelsystem},
where $\Omega\in L^2(\mathbb{R},so(m))$, $\Omega_1\in L^{2,1}(\mathbb{R})$, ${\cal{Z}}(Q,v)\in {\cal{H}}^1$ for every $Q\in{H^{1/2}},$ $v\in L^2$ with
\begin{eqnarray*}
\|{\cal{Z}}(Q,v)\|_{{\cal{H}}^1}&\le& C \| Q\|_{{H^{1/2}}}\|v\|_{L^2},
\end{eqnarray*}
and ${\cal{Z}}(Q,v)$ satisfies \rec{decZ}. There exists $\gamma>0$ such that if $\|\Omega\|_{L^2}<\gamma$, then
there exist $A=A(\Omega,\Omega_1,Q)\in L^{\infty}\cap H^{1/2}(\mathbb{R},M_m({\mathbb{R}})))$ and an operator $ B=B(\Omega,\Omega_1,Q)\in H^{1/2}(\mathbb{R})$ such that
\begin{eqnarray}
\|A\|_{ H^{1/2}}+\|B\|_{ H^{1/2}}&\le& C(\|\Omega\|_{L^2}+\|\Omega_1\|_{L^{2,1}} +\| Q\|_{{H^{1/2}}})\\
dist(\{A,A^{-1} \},SO(m))&\le& C(\|\Omega\|_{L^2}+\|\Omega_1\|_{L^{2,1}} +\| Q\|_{{H^{1/2}}})
\end{eqnarray}
and
\begin{equation}\label{conslaw}
(-\Delta)^{1/4}[Av]={\cal{J}}(B,v)+Ag,
\end{equation}
where ${\cal{J}}$ is a linear operator in $B,v$, ${\cal{J}}(B,v)\in {\cal{H}}^1(\mathbb{R})$ and
\begin{equation}\label{J}
\|{\cal{J}}(B,v)\|_{{\cal{H}}^1(\mathbb{R})}\le C \| B \|_{H^{1/2}}\|v\|_{L^2}\,.
\end{equation}
\end{Theorem}
In the following result we show that if the $L^2$-norm of the antisymmetric potential $\Omega$ in \rec{modelsystem} is {\em small} in an annular domain
$A_{r,R}:=B(0,R)\setminus B(0,r)$ then a precise asymptotic development inside $A_{r,R}$ holds for the solution $v$ to \rec{modelsystem}.\par
We assume for simplicity that $g\equiv 0.$
\begin{Proposition}\label{decomposition}
Let $v\in L^{2}(\mathbb{R},\mathbb{R}^m)$ be a solution of
\begin{equation}\label{modelsystem2}
(-\Delta)^{1/4} v=\Omega v+\Omega_1 v+{\cal{Z}}(Q,v) \end{equation}
where $\Omega\in L^2(\mathbb{R},so(m))$, with $\|\Omega\|_{L^2(B(0,R)\setminus B(0,r))}<\gamma$, $\Omega_1\in L^{2,1}(\mathbb{R})$,
${\cal{Z}}(Q,v)\in {\cal{H}}^1$ for every $Q\in{H^{1/2}},$ $v\in L^2$ with
\begin{eqnarray*}
\|{\cal{Z}}(Q,v))\|_{{\cal{H}}^1}&\le& C \| Q\|_{{H^{1/2}}}\|v\|_{L^2},
\end{eqnarray*}
and ${\cal{Z}}(Q,v)$ satisfies \rec{decZ}
Then
there exists $A=A(\Omega,\Omega_1,Q)\in H^{1/2}(\mathbb{R},M_m({\mathbb{R}})))$ with
\begin{eqnarray}
\|A\|_{ H^{1/2}} +\|A^{-1}\|_{ H^{1/2}} &\le& C(\|\Omega\|_{L^2}+\|\Omega_1\|_{L^{2,1}} +\| Q\|_{{H^{1/2}}})\\
dist(\{A,A^{-1} \},SO(m))&\le& C(\|\Omega\|_{L^2}+\|\Omega_1\|_{L^{2,1}} +\| Q\|_{{H^{1/2}}}),\nonumber
\end{eqnarray}
and $\overrightarrow{c_{r}}\in\mathbb{R}^m$ such that for every $x\in B(0,R)\setminus B(0,r)$ and for every $\Lambda>2$ with $\Lambda r< (2\Lambda)^{-1}R$ it holds
\begin{equation}\label{decompv}
v(x)=(A^{-1}(x))^{+}\overrightarrow{c_{r}}\frac{1}{|x|^{1/2}}+h(x)+g(x)\end{equation}
with $h\in L^{2,1}(A_{\Lambda r,(2\Lambda)^{-1}R})$,
$\|h\|_{L^{2,1}(A_{\Lambda r, (2\Lambda)^{-1}R})}\le C\|v\|_{L^2}$ where $C$ is a positive constant independent of $r,R,\Lambda$ and $g\in L^2(\mathbb{R})$ with
$\|g\|_{L^2(\mathbb{R})}\le C |\overrightarrow{c_{r}}|.$
\end{Proposition}
{\bf Proof of Proposition \ref{decomposition}.}
We split $\Omega$ as follows:
$$
\Omega=\Omega^{r}+\Omega^{R}+ \Omega^{R,r}$$
where
\begin{eqnarray*}
\Omega^{R,r} &=&\11_{B(0,R)\setminus B(0,r)}\Omega\\
\Omega^{r}&=&\11_{ B(0,r)}\Omega\\
\Omega^{R}&=&\11_{ B^c(0,R)}\Omega.
\end{eqnarray*}
We write the system \rec{modelsystem2} in the following form
\begin{eqnarray}\label{modelsystem3}
(-\Delta)^{1/4} v&=&\Omega^{R,r} v+\Omega_1 v+{\cal{Z}}(Q,v)+h_{r}+h_R\end{eqnarray}
where
$$
h_{r}=\Omega^{r}\, v,~~\mbox{and} ~~h_{R}=\Omega^{R}\, v .$$
Observe that $ h_{r}, h_{R}\in L^1(\mathbb{R})$ with $supp(h_R)\subseteq B^c(0,R),$ $supp(h_{r})\subseteq B(0,r)$, and $\|h_{r}\|_{L^1}+\|h_{R}\|_{L^1}\le C(\|v\|_{L^2}).$
From Theorem \ref{conservation} there exist $A=A(\Omega,\Omega_1,Q)\in \dot H^{1/2}(\mathbb{R},M_m({\mathbb{R}}))$ and an operator $ B=B(\Omega,\Omega_1,Q)\in \dot H^{1/2}(\mathbb{R})$ such that
\begin{eqnarray}
\|A\|_{ \dot H^{1/2}} +\|A^{-1}\|_{ \dot H^{1/2}}+\|B\|_{ \dot H^{1/2}}&\le& C(\|\Omega\|_{L^2}+\|\Omega_1\|_{L^{2,1}} +\| Q\|_{{\dot H^{1/2}}})\\
dist(\{A,A^{-1} \}, SO(m))&\le& C(\|\Omega\|_{L^2}+\|\Omega_1\|_{L^{2,1}} +\| Q\|_{{\dot H^{1/2}}})
\end{eqnarray}
and
\begin{equation}\label{conslaw2}
(-\Delta)^{1/4}[Av]={\cal{J}}(B,v)+Ah_{r}+Ah_{R},
\end{equation}
with ${\cal{J}}(B,v)$ satisfying \rec{J}.
We write
$v=v_1+v_2+v_3+v_4$ where
\begin{eqnarray}
(-\Delta)^{1/4}[Av_1]&=&{\cal{J}}(B,v);\label{v1}\\
(-\Delta)^{1/4}[Av_2]&=& Ah_{r}-\left(\int_{B(0,r)}Ah_{r} dx\right)\delta_0;\label{v2}\\
(-\Delta)^{1/4}[Av_3]&= &Ah_{R},\label{v3}\\
(-\Delta)^{1/4}[Av_4]&=& \left(\int_{B(0,r)}Ah_{r} dx\right)\delta_0;\label{v4}.
\end{eqnarray}
Let $\Lambda>2$ be such that $\Lambda r< (2\Lambda)^{-1}R.$
We are going to estimate the $L^{2,1}(B(0,(2\Lambda)^{-1}R)\setminus B(0, \Lambda r))$ norm of $v_1,v_2,v_3$.\par
{\bf 1. Estimate of $v_1$\,.}\par
By the properties of $\cal{J}$ we have
\begin{eqnarray}\label{estv1}
\|Av_1\|_{L^{2,1}(\mathbb{R})} &\leq& C\|(-\Delta)^{-1/4}[J(B,v)]\|_{L^{2,1}(\mathbb{R})}\\
&\leq& C \| B \|_{\dot H^{1/2}}\|v\|_{L^2}.
\end{eqnarray}
{\bf 2. Estimate of $v_2$\,.}\par
We set $\phi_{r}:=Ah_{r}-\left(\int_{B(0,r)}Ah_{r} dx\right)\delta_0$ and take $x\in B(0,(2\Lambda)^{-1}R)\setminus B(0, \Lambda r)\,.$
By using the fact that $\int_{B(0,r)} \phi_{r}(y) dy=0$, we get
\begin{eqnarray*}
Av_2(x)&=&\int_{B(0,r)}\phi_{r}(y)\frac{1}{|x-y|^{1/2}} dy= \frac{1}{|x|^{1/2}}\sum_{k=0}^{+\infty} \frac{c_{k}}{|x|^k}\int_{B(0,r)}\phi_{r}(y)y^k dy\\
&=& \sum_{k=1}^{+\infty} \frac{c_k}{|x|^{k+1/2}}\int_{B(0,r)}\phi_{r}(y)y^k dy\,,
\end{eqnarray*}
where the $c_k=\frac{1\times 3\times\dots (2k-1)}{2\times 4\times \dots \times 2k}$.\\
We observe that for every $k\ge 1$, $\frac{1}{|x|^{k+1/2}}\in L^{2,1}(B(0,(2\Lambda)^{-1}R)\setminus B(0, \Lambda r))$ for all $\Lambda>1$ such that $\Lambda r< (2\Lambda)^{-1}R$, with
$$\left\|\frac{1}{|x|^{k+1/2}}\right\|_{ L^{2,1}( B(0,(2\Lambda)^{-1}R)\setminus B(0, \Lambda r))}\precsim \left(\frac{1}{ \Lambda r}\right)^{k}\,.$$
Therefore
\begin{eqnarray*}
\|v_2\|_{ L^{2,1}( B(0,(2\Lambda)^{-1}R)\setminus B(0, \Lambda r))}&\leq&C \|\phi_{r}\|_{L^1}
\sum_{k=1}^{+\infty} \frac{c_k}{( \Lambda r)^k}r^k\\
& \leq &C \|\phi_{r}\|_{L^1}
\sum_{k=1}^{+\infty} \frac{c_k}{\Lambda ^k}<+\infty\\
&\leq& C \|\Omega\|_{L^{2}}\|v\|_{L^{2}}\left( \sum_{k=1}^{+\infty} \frac{c_k}{\Lambda ^k}<+\infty\right).
\end{eqnarray*}
{\bf 3. Estimate of $v_3$\,.}\par
Since $Ah_{R}$ has support in $B^c(0,R)$ then by interpolation one gets that
$A v_3\in L^{2,1}(B(0,(2\Lambda)^{-1}R)\setminus B(0, \Lambda r)))$ and
$$\|A v_3\|_{ L^{2,1}(B(0,(2\Lambda)^{-1}R)\setminus B(0, \Lambda r))}\le C \|h_R\|_{L^1}\le C\|\Omega\|_{L^{2}}\|v\|_{L^{2}})\,.$$
{\bf 3. Estimate of $v_4$\,.}\par
We have
$$v_4(x)=A^{-1}(x)\overrightarrow{c_{r}}\frac{1}{|x|^{1/2}},$$
where
$$\overrightarrow{c_{r}}=\int_{B(0,r)}Ah_{r} dx.$$
Since $A^{-1} \in \dot H^{1/2}(\mathbb{R})$, it verifies the following estimate (see lemma 33.1 of \cite{Tartar}):
\begin{eqnarray}\label{estA}
\left\|\frac{A^{-1}(x)-A^{-1}(-x)}{|x|^{1/2}}\right\|_{L^2(\mathbb{R})}&\le& C\|A^{-1}\|_{\dot H^{1/2}}\end{eqnarray}
Therefore
by combining the $L^{2,1}$ estimates on $ B(0,(2\Lambda)^{-1}R)\setminus B(0, \Lambda r)$ of $v_1,v_2,v_3$ we obtain
\begin{eqnarray}\label{estl21v}
v(x)&=&(A^{-1}(x))^{+}\overrightarrow{c_{r}}\frac{1}{|x|^{1/2}} +(A^{-1}(x))^{-}\overrightarrow{c_{r}}\frac{1}{|x|^{1/2}} +h(x)
\end{eqnarray}
with $h(x)=v_1+v_2+v_3$ and $\|h\|_{L^{2,1}(B(0,(2\Lambda)^{-1}R)\setminus B(0, \Lambda r))}\le C\|v\|_{L^2(\mathbb{R})},$ $\overrightarrow{c_{r}}=\int_{B(0,r)}A(\Omega^{r}\, v)dx.$
Finally from \rec{estA} it follows that $g(x)={\rm asymm}{(A^{-1}(x))}\overrightarrow{c_{r}}\frac{1}{|x|^{1/2}}$ satisfies
$g\in L^2(\mathbb{R})$ with $\|g\|_{L^2(\mathbb{R})}\leq C |\overrightarrow{c_{r}}|.$
We can conclude the proof.\hfill$\Box$\par
\begin{Lemma}[$\varepsilon$-regularity ]\label{epsreg}
Let $v\in L^2(\mathbb{R},\mathbb{R}^m)$ be a solution of
\begin{equation}
(-\Delta)^{1/4} v=\Omega v+\Omega^1 v+{\cal{Z}}(Q ,v ).\end{equation}
Then there exists $\varepsilon_0>0$ such that if
\begin{equation}\label{epsregcond}\sum_{j\ge 0} 2^{-j/2}(\|\Omega\|_{L^2(B(x,2^{j}r)) } +\|\Omega^1\|_{L^{2,1}(B(x,2^{j}r)}+\|(-\Delta)^{1/4}Q\|_{L^2(B(x,2^{j}r)})\le \varepsilon_0\,,\end{equation}
then there is $p>2$ ( independent of $u$) such that for every $y\in B(x,r/2)$ we have
\begin{equation}
\left(r^{p/2-1}\int_{B(y,r/2)}|v|^p dx\right)^{1/p}\le C\sum_{j\ge 0} 2^{-j/2}\left(\|\Omega\|_{L^2(B(x,2^{j}r)) } +\|\Omega^1\|_{L^{2,1}(B(x,2^{j}r)}+\|(-\Delta)^{1/4}Q\|_{L^2(B(x,2^{j}r)}\right),
\end{equation}
where $C>0 $ depends on $\|v\|_{L^2}$\,.
\end{Lemma}
Finally we have
\begin{Lemma}[$L^{2,\infty}$ estimates]\label{neck2}
There exists $ \delta >0$ such that for any $ v\in L^2(\mathbb{R})$ solution of \rec{modelsystem} and $\lambda,\Lambda>0$ with $\lambda<(2\Lambda)^{-1} $ satisfying
$$ \sup_{\rho\in[\lambda,{(2\Lambda)}^{-1}]} \left[\|\Omega\|_{L^{2}(B(0,2\rho)\setminus B(0,\rho))}+\|(-\Delta)^{1/4})Q\|_{L^{2}(B(0,2\rho)\setminus B(0,\rho))}+\|\Omega^1\|_{L^{2,1}(B(0,2\rho)\setminus B(0,\rho))}\right]\le \delta $$
then
\begin{eqnarray}\label{linftyest}
&& \| v\|_{L^{2,\infty}(B(0,(2\Lambda)^{-1})\setminus B(0,\lambda )}\le C\sup_{\rho\in[\lambda,{(2\Lambda)}^{-1}]} \left[\|\Omega\|_{L^{2}(B(0,2\rho)\setminus B(0,\rho))}\right.\\
&&\left.+\|(-\Delta)^{1/4}Q\|_{L^{2}(B(0,2\rho)\setminus B(0,\rho))}+\|\Omega^1\|_{L^{2,1}(B(0,2\rho)\setminus B(0,\rho))}\right] \,.\nonumber
\end{eqnarray}
where $C$ is independent of $\lambda,\Lambda\,.$
\end{Lemma}
The proof of Lemmae \ref{epsreg} and \ref{neck2} are the same of Lemma 2.1 and Lemma 3.2 in \cite{DL2} and therefore we omit them.
We show now a point removability type result for solution of \rec{modelsystem}.
\begin{Proposition}\label{removsing}[ Point removability]
Let $v \in L^2(\mathbb{R})$ be a solution of \rec{modelsystem} in ${\cal{D}}^{\prime} (\mathbb{R}\setminus\{a_1,\ldots,a_{\ell}\})$. Then it is a solution
of \rec{modelsystem} ${\cal{D}}^{\prime} (\mathbb{R})$.
\end{Proposition}
{\bf Proof of Proposition \ref{removsing}\,.}
The fact that
$$
(-\Delta)^{1/4} v=\Omega v+\Omega^1 v+{\cal{Z}}(Q ,v ),
\quad \mbox{in ${\cal{D}}^{\prime} (\mathbb{R}\setminus\{a_1,\ldots,a_{\ell}\})$}$$
implies that
the distribution
$\phi:= (-\Delta)^{1/4} v-\Omega v-\Omega^1 v-{\cal{Z}}(Q ,v )$ is of order $p=1$ and supported in $\{a_1,\ldots,a_{\ell}\}$.
Therefore by Schwartz Theorem (see 6.1.5.of \cite{Bony}) one has
$$\phi =\sum_{|\alpha|\le 1} c_{\alpha}\partial^{\alpha} \delta_{a_i}\,.
$$
Since $\phi\in \dot L^1(\mathbb{R}) $, then the above implies that $c_{\alpha}=0$ and thus
$v$ is a solution of \rec{modelsystem} in { ${\cal{D}}^{\prime} (\mathbb{R} )$}\,.
We conclude the proof of Proposition \ref{removsing}\,.~\hfill$\Box$
\bigskip
\section{Pohozaev Identities}\label{PoId}
In this section we provide some new Pohozaev identities in $\mathbb{R}$ and $S^1$ which are in particular satisfied by horizontal $1/2$ harmonic in ${\frak{H}}^{1/2} ({\mathbb{R}})$.
\subsection{Pohozaev Identities for $(-\Delta)^{1/2}$ in $\mathbb{R}$}\label{Poh}
We first determine the solution of
\begin{equation}\label{solfund1}
\left\{\begin{array}{cc}
\partial_t G+(-\Delta)^{1/2} G=0 & t>0\\
G(0,x)=\delta & t=0\,.
\end{array}\right.
\end{equation}
In order to solve \rec{solfund1} we consider the Fourier Transform of $G$ with respect to $x$
$$\hat G(t,\xi)=\int_{\mathbb{R}} G(t,x) e^{-ix\xi} dx$$
The function $\hat G$ satisfies
the problem
\begin{equation}\label{solfund2}
\left\{\begin{array}{cc}
\partial_t \hat G+|\xi|\hat G=0 & x\in\mathbb{R},\, t>0\\
\hat G(0,x)=\hat\delta=1 & x\in\mathbb{R},\,t=0\,.
\end{array}\right.
\end{equation}
The solution of \rec{solfund2} is
$\hat G(t,\xi)=e^{-t|\xi|}$.
Then $$G(t,x)=\frac{1}{2\pi}\int_{\mathbb{R}} \hat G(t,\xi)e^{ix\xi} d\xi=\frac{1}{\pi} \frac{t}{x^2+t^2}.$$
The following equalities hold
$$\partial_t G=\frac{1}{\pi}\frac{x^2-t^2}{(t^2+x^2)^2},~~\partial_x G=-\frac{1}{\pi}\frac{2xt}{(t^2+x^2)^2}.$$
\begin{Theorem}{[Case on $\mathbb{R}$]}\label{Poho1}
Let $u\in \dot H^{1/2}_{loc}(\mathbb{R},\mathbb{R}^m)$ such that
\begin{equation}\label{harmequation1}
\left\langle \frac{du}{dx}, (-\Delta)^{1/2} u\right\rangle=0~~\mbox{a.e in $\mathbb{R}$.}
\end{equation}
Assume that
\begin{equation}\label{condR}
\int_{\mathbb{R}}|u-u_0|dx <+\infty, ~~\int_{\mathbb{R}}\left|\frac{du}{dx}(x)\right|dx<+\infty..
\end{equation}
Then the following identity holds
\begin{equation}\label{idpohozaevR}
\left|\int_{x\in\mathbb{R}} \frac{x^2-t^2}{(x^2+t^2)^2}u(x)dx\right|^2=\left| \int_{x\in\mathbb{R}} \frac{2xt}{(x^2+t^2)^2} u(x)dx\right|^2.
\end{equation}
\end{Theorem}
{\bf Proof.}
We assume for simplicity that $u_0=0$.
Set $G_t(x):=G(t,x),$ multiply the equation \rec{harmequation1} by $xG_t(x)$ and we integrate. For every $j\in\{1,\ldots,m\}$ we get by Plancherel Theorem
\begin{eqnarray}\label{est1}
0&=&\int_{\mathbb{R}} x G_t \frac{du^j}{dx} \,\overline{(-\Delta)^{1/2} u^j} dx =\int_{\mathbb{R}}{\mathcal{F}}\left[xG_t \frac{du^j}{dx}\right]\overline{{\mathcal{F}}[(-\Delta)^{1/2} u^j]}dx\nonumber
\\&=&
\int_{\mathbb{R}}{\mathcal{F}}[G_t]\ast {\mathcal{F}}\left[x \frac{du^j}{dx}\right] \, |\xi| \overline{{\mathcal{F}}[u^j] }d\xi \\ &
= &\int_{\mathbb{R}}{\mathcal{F}}[G_t]\ast i\frac{d}{d\xi}{\mathcal{F}}\left[ \frac{du^j}{dx}\right] \, |\xi| \overline{{\mathcal{F}}[u^j] }d\xi\nonumber \\ &
= &\int_{\mathbb{R}}{\mathcal{F}}[G_t]\ast i\frac{d}{d\xi}\left(\xi {\mathcal{F}}[u^j]\right)\, |\xi|\overline{{\mathcal{F}}[u^j] }d\xi \nonumber\\ &
= &
-\iint_{\mathbb{R}^2}\hat {G_t}(\xi-\eta)\frac{\partial}{\partial \eta}(\eta \hat{u}^j(\eta))|\xi|
\overline{\hat{u}^j}(\xi) d\xi d\eta \nonumber\\
&=& \iint_{\mathbb{R}^2}\frac{\partial}{\partial \eta}\left[\hat {G_t}(\xi-\eta)\right] \eta \, |\xi|\hat{u}^j(\eta))
\overline{\hat{u}^j}(\xi) d\xi d\eta\nonumber \\
&=&\iint_{\mathbb{R}^2} te^{-t|\xi-\eta|}\frac{\xi-\eta}{|\xi-\eta|}\eta
|\xi| \hat{u}^j(\eta))
\overline{\hat{u}^j}(\xi) d\xi d\eta \nonumber
\end{eqnarray}
We symmetrize \rec{est1}
\begin{eqnarray}\label{sym}
0&=&
\iint_{\mathbb{R}^2} te^{-t|-\xi+\eta|}\frac{-\xi+\eta}{|-\xi+\eta|} \xi
(|\eta|) \hat{u}^j(\xi))
\overline{\hat{u}^j}(\eta) d\xi d\eta
\end{eqnarray}
Then we sum \rec{est1} and \rec{sym} and we take the real part:
\begin{eqnarray}\label{real}
0&=&
\iint_{\mathbb{R}^2} te^{-t|\xi-\eta|}\frac{\xi-\eta}{|\xi-\eta|}[\eta |\xi|-\xi
|\eta|]\Re\left[ \hat{u}^j(\xi)\overline{\hat{u}^j}
\right] d\xi d\eta\nonumber\\
&&=\iint_{\mathbb{R}^2} te^{-t|\xi-\eta|}\frac{\xi-\eta}{|\xi-\eta|}[\eta |\xi|-\xi
|\eta|]\left[ a^j(\xi)a^j(\eta)+b^j(\xi)b^j(\eta)
\right] d\xi d\eta
\end{eqnarray}
where for every $j$ we set
$\hat{u}^j(\eta))=a^j(\eta)+i b^j(\eta).$ \par
We write
\begin{eqnarray*}
a^{j,\pm}(\eta)&=&\frac{a^j(\eta)\pm a^j(-\eta)}{2}\\
b^{j,\pm}(\eta)&=&\frac{b^j(\eta)\pm b^j(-\eta)}{2}.
\end{eqnarray*}
We observe that $a^{j,+}, b^{j,+}$ are even and $a^{j,-}, b^{j,-}$ are odd. Moreover since $u^j$ is real we also have
$b^{j,+}=0, a^{j,-}=0$, $b^{j,-}=b, a^{j,-}=a$.
We can also write
\begin{eqnarray}\label{estaplus}
a^{j,+}(\xi)&=&\int_{\mathbb{R}}\frac{u^j(x)+u^j(-x)}{2}\cos(x\xi)\,dx-i\int_{\mathbb{R}}\frac{u^j(x)+u^j(-x)}{2}\sin(x\xi)\,dx\nonumber\\ &&={\mathcal{F}}(u^{j,+})(\xi);
\end{eqnarray}
\begin{eqnarray}\label{estbminus}
b^{j,-}(\xi)&=&i\left\{\int_{\mathbb{R}}\frac{u^j(x)-u^j(-x)}{2}\cos(x\xi)\,dx-i\int_{\mathbb{R}}\frac{u^j(x)-u^j(-x)}{2}\sin(x\xi)\,dx\right\}\nonumber\\ &&=i{\mathcal{F}}(u^{j,-})(\xi)
\end{eqnarray}
We set
$$
Q(\alpha,\beta):=\iint_{\mathbb{R}^2}\frac{\xi-\eta}{|\xi-\eta|}te^{-t(|\xi-\eta|)}[\eta \, |\xi|-\xi \, |\eta|]\alpha(\eta)\beta(\xi) d\xi\, d\eta.$$
It follows that
\begin{eqnarray}\label{q}
0&=&Q(a^{j,+} ,a^{j,+} )+Q(b^{j,-} ,b^{j,-} ).
\end{eqnarray}
{\bf Estimate of $Q(a^{j,+} ,a^{j,+} )$}
\begin{eqnarray}\label{aplus}
Q(a^{j,+} ,a^{j,+} )&=&\iint_{\mathbb{R}^2} \frac{\xi-\eta}{|\xi-\eta|}te^{-t(|\xi-\eta|)}[\eta \, |\xi|-\xi \, |\eta|] a^{j,+} (\eta)a^{j,+} (\xi)d\xi\,d\eta\\
&=&
2 \int_{\xi>0} \int_{\eta<0} \eta\xi \frac{\xi-\eta}{|\xi-\eta|}te^{-t(|\xi-\eta|)} a^{j,+} (\eta)a^{j,+} (\xi)d\xi\,d\eta\nonumber\\
&-& 2\int_{\xi<0} \int_{\eta>0} \eta\xi \frac{\xi-\eta}{|\xi-\eta|}te^{-t(|\xi-\eta|)} a^{j,+} (\eta)a^{j,+} (\xi)d\xi\,d\eta\nonumber\\
&=& 4\int_{\xi>0} \int_{\eta<0} \eta\xi te^{-t(|\xi-\eta|)} a^{j,+}(\eta) a^{j,+} (\xi)d\xi\,d\eta\nonumber
\\
&=&-4t\left(\int_{\xi>0} \xi e^{-t\xi} a^{j,+} (\xi)d\xi\right)^2\nonumber\\
&=&-4t \left(\int_{\xi>0} \xi e^{-t\xi} {\mathcal{F}}(u^{j,+})(\xi)d\xi\right)^2=-4t\left(\frac{1}{2}\int_{x \in\mathbb{R}}| \xi | e^{-t|\xi |} {\mathcal{F}}(u^{j,+})(\xi)d\xi\right)^2\nonumber\\
&=&- t \left(\int_{\xi\in\mathbb{R}} \mathcal{F}{[(-\Delta)^{1/2} G](\xi)}{\mathcal{F}}(u^{j,+})(\xi)d\xi\right)^2\nonumber\\
&=&- t \left(\int_{x\in\mathbb{R}} \partial_{t} G(x,t) (u^{j,+})(x)dx\right)^2\nonumber\\
&=&t\frac{1}{\pi^2} \left(\int_{x\in\mathbb{R}} \frac{x^2-t^2}{(x^2+t^2)^2} (u^{j,+})(x)dx\right)^2\nonumber
\end{eqnarray}
{\bf Estimate of $Q(b^{j,-} ,b^{j,-} )$}
\begin{eqnarray}\label{bminus}
Q(b^{j,-} ,b^{j,-} )&=&\iint_{\mathbb{R}^2} \frac{\xi-\eta}{|\xi-\eta|}te^{-t(|\xi-\eta|)}[\eta \, |\xi|-\xi \, |\eta|] b^{j,-} (\eta)b^{j,-} (\xi)d\xi\,d\eta\\
&=&
2 \int_{\xi>0} \int_{\eta<0} \eta\xi \frac{\xi-\eta}{|\xi-\eta|}te^{-t(|\xi-\eta|)} b^{j,-} (\eta)b^{j,-} (\xi)d\xi\,d\eta\nonumber\\
&-& 2\int_{\xi<0} \int_{\eta>0} \eta\xi \frac{\xi-\eta}{|\xi-\eta|}te^{-t(|\xi-\eta|)} b^{j,-} (\eta)b^{j,-} (\xi)d\xi\,d\eta\nonumber\\
&=& 4\int_{\xi>0} \int_{\eta<0} \eta\xi te^{-t(|\xi-\eta|)} b^{j,-} (\eta)b^{j,-} (\xi)d\xi\,d\eta\nonumber
\\
&=&4t\left(\int_{\xi>0} \xi e^{-t\xi} b^{j,-} (\xi)d\xi\right)^2\nonumber\\
&=&4t \left(\int_{\xi>0} \xi e^{-t\xi} i {\mathcal{F}}(u^{j,-})(\xi)d\xi\right)^2=4t\left(\frac{1}{2}\int_{\xi\in\mathbb{R}} i\xi e^{-t|\xi|} {\mathcal{F}}(u^{j,-})(\xi)d\xi\right)^2\nonumber\\
&=& t \left(\int_{\xi\in\mathbb{R}} \mathcal{F}[\partial_x G](\xi)\mathcal{F}(u^{j,-})(\xi)d\xi\right)^2\nonumber\\
&=& t \left(\int_{x\in\mathbb{R}} {\partial_{x} G(x,t)} (u^{j,-})(x)dx\right)^2\nonumber\\
&=&t\frac{1}{\pi^2} \left(\int_{x\in\mathbb{R}} \frac{2xt}{(x^2+t^2)^2} (u^{j,-})(x)dx\right)^2\nonumber
\end{eqnarray}
From \rec{q}, \rec{aplus} and \rec{bminus} it follows that
\begin{equation}\label{Po1}
\left|\int_{x\in\mathbb{R}} \frac{x^2-t^2}{(x^2+t^2)^2} (u^+)(x)dx\right|^2= \left|\int_{x\in\mathbb{R}} \frac{2xt}{(x^2+t^2)^2} (u^-)(x)dx\right|^2
\end{equation}
In particular we get
\begin{equation}\label{Po2}
\left|\int_{x\in\mathbb{R}} \frac{x^2-t^2}{(x^2+t^2)^2} u(x)dx\right| = \left|\int_{x\in\mathbb{R}} \frac{2xt}{(x^2+t^2)^2} u(x)dx\right|,
\end{equation}
which achieves the proof of theorem \ref{Poho1}.\hfill$\square$
\subsection*{Preliminary computations}
We observe that
$$
\frac{x^2-t^2}{(x^2+t^2)^2}=\frac{d}{dx}\left(\frac{-x}{x^2+t^2}\right),~\frac{2xt}{(x^2+t^2)^2} =\frac{d}{dx}\left(\frac{-t}{x^2+t^2}\right).$$
{\bf Computation of ${\mathcal{F}}\left[\frac{-1}{x^2+1}\right]$ and ${\mathcal{F}}\left[\frac{-x}{x^2+1}\right].$}\par
\begin{eqnarray*}
{\mathcal{F}}\left[\frac{-1}{x^2+1}\right](\xi)&=&\int_{\mathbb{R}} \frac{-1}{x^2+1} e^{-ix\xi} d\xi\\
&=& -\pi e^{-|\xi|}.
\end{eqnarray*}
\begin{eqnarray*}
{\mathcal{F}}\left[\frac{-x}{x^2+1}\right](\xi)&=&i\frac{d}{d\xi}\left({\mathcal{F}}\left[\frac{-1}{x^2+1}\right]\right)=-i\pi\frac{\xi}{|\xi|}e^{-|\xi|}.\end{eqnarray*}
{\bf Computation of $(-\Delta)^{-1/4}[\frac{x^2-1}{(1+x^2)^2}]$ and $(-\Delta)^{-1/4}[\frac{2x}{(1+x^2)^2}]$}
\begin{eqnarray}
\label{FM+}
(-\Delta)^{-1/4}\left[\frac{x^2-1}{(1+x^2)^2}\right]&=&{\mathcal{F}}^{-1}\left[|\xi|^{-1/2}{\mathcal{F}}\left[\frac{d}{dx}\left(\frac{-x}{x^2+1}\right)\right]\right]\nonumber\\
&=&{\mathcal{F}}^{-1}[|\xi|^{-1/2}i\xi {\mathcal{F}}\left[\frac{-x}{x^2+1}\right]\nonumber\\
&=&\pi {\mathcal{F}}^{-1}\left[|\xi|^{-1/2}\xi \frac{\xi}{|\xi|}e^{-|\xi|}\right]=\pi {\mathcal{F}}^{-1}\left[|\xi|^{1/2}e^{-|\xi|}\right]\\
&=&\pi \frac{1}{2\pi}\int_{\mathbb{R}}|\xi|^{1/2}e^{-|\xi|}e^{-ix\xi} d\xi=\sqrt{\pi} \Re\left(\frac{1}{(1+ix)^{3/2}}\right)\nonumber\\
&=&\sqrt{\pi} \left(\frac{\cos(\arctan(-x))}{(1+x^2)^{3/4}}\right).\nonumber
\end{eqnarray}
\begin{eqnarray}
\label{FM-}
(-\Delta)^{-1/4}\left[\frac{2x}{(1+x^2)^2}\right]&=&{\mathcal{F}}^{-1}\left[|\xi|^{-1/2}{\mathcal{F}}\left[\frac{d}{dx}\left(\frac{-1}{x^2+1}\right)\right]\right]\nonumber\\
&=&{\mathcal{F}}^{-1}[|\xi|^{-1/2}i\xi {\mathcal{F}}\left[\frac{-1}{x^2+1}\right]\nonumber\\
&=&\pi i{\mathcal{F}}^{-1}\left[|\xi|^{-1/2}\xi e^{-|\xi|}\right]=\pi i\frac{1}{2\pi}\int_{\mathbb{R}}|\xi|^{-1/2}\xi e^{-|\xi|}e^{-ix\xi} d\xi\\
&=&\sqrt{\pi}i \Im\left(\frac{1}{(1+ix)^{3/2}}\right)=\sqrt{\pi} \left(\frac{\sin(\arctan(-x))}{(1+x^2)^{3/4}}\right).\nonumber
\end{eqnarray}
Next we define the following operators:
\begin{eqnarray}
M^-[w](t)&:=&\int_{\mathbb{R}} \sqrt{\pi} \left(\frac{\sin(\arctan(-x))}{(1+x^2)^{3/4}}\right)w(tx) dx,\label{Phi}\\
M^+[w](t)&:=&\int_{\mathbb{R}} \sqrt{\pi} \left(\frac{\cos(\arctan(-x))}{(1+x^2)^{3/4}}\right)w(tx) dx,\label{Psi}.
\end{eqnarray}
\begin{Proposition}\label{M+M-}
The operators $M^{+}$ (resp. $M^-$) is an isomorphim from $L^p_+$ to $L^p_{+}$ (resp. $L^p_-$ to $L^p_{-}$) for every $p\in (1,+\infty).$
\end{Proposition}
{\bf Proof.}
We prove the proposition for $M^+$, it is exactly the same for $M^-$.\\
{\bf Claim 1.} $M^+\colon L^p_+(\mathbb{R})\to L^p_+(\mathbb{R})$, for every $p\in (1,+\infty).$\\
{\bf Proof of the claim 1.}
Let $p\in (1,\infty)$ and $p'=\frac{p}{p-1}$ be the conjugate of $p$.
Let us introduce $g(x):=\frac{(1+x^2)^{\beta}}{|x|^{\alpha}}$ where $\beta=\alpha=\frac{1}{4p'},$ and let $w\in L^p(\mathbb{R})$.\par
\begin{eqnarray*}
\|M^+[w](t)\|^p_{L^p(\mathbb{R}^+)}&=& (\pi)^{p/2}\int_\mathbb{R}\left(\int_{\mathbb{R}} \left(\frac{\cos(\arctan(-x))}{(1+x^2)^{3/4}}\right)w(tx) dx\right)^p dt\\
&\le & (\pi)^{p/2}\left(\int_{\mathbb{R}}|w(y)|^p dy\right)\left(\int_{\mathbb{R}}\left(\frac{g(x)}{(1+x^2)^{3/4}}\right)^{p'}dx\right)^{p/p'}\left(\int_{\mathbb{R}}\frac{1}{|x|g^{p}(x)}\,dx\right)\\
&=& (\pi)^{p/2}\left(\int_{\mathbb{R}}|w(y)|^p dy\right)\\
&&\left(\int_{\mathbb{R}} |x|^{-1/4}(1+x^2)^{-\frac{3}{4} p'+1/4}dx\right)^{p/p'}\left(\int_{\mathbb{R}} |x|^{-1+\frac{p}{4p'}}(1+x^2)^{-\frac{p}{4p'}}\,dx\right)\\
&\le& C_p\|w\|^p_{L^p(\mathbb{R})}.~~\hfill\Box
\end{eqnarray*}
{\bf Claim 2: The adjoint of $M^+$ in $L_+^2$ is $\mathcal{F}^{-1}\circ M^+ \circ \mathcal{F}^{-1} $}\\
{\bf Proof of claim 2.}
Thanks to (\ref{FM+}), we have
$$ \pi |\xi|^{-1/2}\xi \frac{\xi}{\vert \xi\vert}e^{-|\xi|} =\mathcal{F}\left(\sqrt{\pi} \left(\frac{\cos(\arctan(-x))}{(1+x^2)^{3/4}}\right) \right).$$
Then let $w\in L^2(\mathbb{R})$, for $t\not=0$,
$$M^+[w](t)=\pi \int_{\mathbb{R}}|\xi|^{1/2} e^{-|\xi|} {\cal{F}}[w(tx)](\xi) d\xi=\pi \int_{\mathbb{R}}|\xi|^{1/2} \vert t\vert^{-1}e^{-|\xi|}\hat{w}\left(\frac{\xi}{t}\right) d\xi .$$
Here we used the fact that $\hat{w}$ is real. Then, let $w,v \in L_+^2$, we have
{\begin{eqnarray*}
\langle v,M^+[w]\rangle&=&\pi \int_\mathbb{R} v(t)\left(\int_{\mathbb{R}}|\xi|^{1/2} e^{-|\xi|}\vert t\vert^{-1}\hat{w}\left(\frac{\xi}{t}\right) d\xi \right) dt\\
&=&\pi \int_\mathbb{R} \int_{\mathbb{R}} |\xi|^{1/2} e^{-|\xi|} v(t)\vert t\vert^{-1}\hat{w}\left(\frac{\xi}{t}\right) dt d\xi \\
&=&\pi \int_\mathbb{R} \int_{\mathbb{R}} |y|^{1/2} e^{-|ty|} v(t)\vert t\vert^{1/2}\hat{w}(y) dt dy\\
&=&\pi \int_\mathbb{R} \int_{\mathbb{R}}\vert x\vert^\frac{1}{2} e^{-|x|} |y|^{-1} v\left(\frac{x}{y}\right)\hat{w}(y) dx dy\\
&=& \langle M^+ \left[\mathcal{F}^{-1}(v)\right], \mathcal{F}(w)\rangle\\
&=& \left\langle \mathcal{F}^{-1}\left( M^+ \left[\mathcal{F}^{-1}(v)\right]\right), w\right\rangle
\end{eqnarray*}}
{\bf Claim 3: $Ker(M^+)=Ker(M^+)^*=\{0\}$}\\
{\bf Proof of claim 3.}
We observe that $M^+[w]=0$ if and only if the Laplace transform of $|\xi|^{-1/2}\xi \hat{w}({\xi})$ is zero. Here we use once more that $w$ is even. Since the Laplace transfom is injective, it implies that $|\xi|^{-1/2}\xi \hat{w}({\xi})=0$ a.e, namely $w= 0$ a.e. The same hold for the adjoint.\\
By combining Claim 1, 2, 3 we deduce that $M^+$ is bijective bounded linear operators from $L^2_+$ to $L^2_+$. The Open Mapping Theorem
implies that it is an isomorphism as well.
\par
By density arguments we get that $M^+$ is also an isomorphism from $L^p_+$ to $L^p_+$ for every $p\in (1,\infty)$ and therefore from
$L^{2,1}_+$ to $L^{2,1}_+$ too.
We conclude the proof of Proposition \ref{M+M-}.~\hfill $\Box$.
\subsection{Pohozaev Identities for $(-\Delta)^{1/2}$ in $S^1$}
In this section we will derive a Pohozaev formula for $C^1$, stationary $1/2$-harmonic on $S^1$, namely satisfying
\begin{equation}
\frac{d}{da}\int_{S^1}| (-\Delta)^{1/4}(u\circ\phi_a)|^2|_{a=0}=0
\end{equation}
where $\phi_a\colon S^1\to S^1$ is a family of diffeomorphisms such that
$\phi_0=Id.$ \par Stationary $1/2$-harmonic maps satisfy
\begin{eqnarray}
0&=&(-\Delta)^{1/2}(u\circ\phi_a)\cdot \frac{d}{d\theta}(u\circ\phi_a)|_{a=0}\\
&=&(-\Delta)^{1/2}(u\circ\phi_a)\cdot \frac{du}{d\phi}\circ\phi_a\frac{d\phi_a}{d\theta}|_{a=0}~~\mbox{in}~{\cal{D}}^{\prime}\nonumber
\end{eqnarray}
If we choose for every $a\in [0,\frac{1}{2})$, $\phi_a=\frac{e^{i\theta}-a}{1-ae^{i\theta}}$ and we set
$u_a(\theta)=u(\frac{e^{i\theta}-a}{1-ae^{i\theta}})$ (namely we consider the composition of $u$ with a
family of M\"obius transformations of the disk).\par
In this case we get
\begin{equation}
\frac{du}{d\phi}\circ\phi_a|_{a=0}=-ie^{-i\theta}\partial_{\theta}u_0(\theta),~~\frac{d\phi_a}{d\theta}|_{a=0}=-1+e^{2i\theta}.\end{equation}
Moreover if $\phi_a$ is conform we have
\begin{equation}
(-\Delta)^{1/2}(u\circ\phi_a)=(-\Delta)^{1/2}u_a= e^{\lambda_a} ((-\Delta)^{1/2}u)\circ \phi_a,
\end{equation}
where $\lambda_a=\log(|\frac{\partial\phi_a}{\partial\theta}(\theta)|).$
Therefore
\begin{eqnarray}\label{harms1}
0&=&(-\Delta)^{1/2}u\circ \phi_a\cdot \frac{du}{d\phi}\circ\phi_a\frac{d\phi_a}{d\theta}|_{t=0}~~\mbox{in}~{\cal{D}}^{\prime}\nonumber\\
&=&(-\Delta)^{1/2}u (e^{i\theta})\cdot (-ie^{-i\theta}\partial_{\theta}u_0(\theta)(-1+e^{2i\theta}))\\
&=&(-\Delta)^{1/2}u_0 \cdot (2\sin(\theta)\partial_{\theta}u_0(\theta) ).\nonumber
\end{eqnarray}
We also observe that
the energy $\int_{S^1} |(-\Delta)^{1/4}u|^2\, d\theta $ is invariant with respect to the trace of M\"obius transformations of the disk and therefore for every function $w\in \dot H^{1/2}(S^1)$ we have
$$\frac{d}{da}\int_{S^1} |(-\Delta)^{1/4}w_a|^2\, d\theta=\frac{d}{da}\int_{S^1} |(-\Delta)^{1/4}w|^2\, d\theta=0.$$
In particular we get
$$2\int_{S^1}\frac{dw_a}{da}(\theta)|_{a=0}(-\Delta)^{1/2}w\, d\theta=4\int_{S^1}\sin(\theta)\partial_{\theta}w (-\Delta)^{1/2}w\, d\theta=0.$$
In the sequel we identify $S^1$ with $[-\pi,\pi]$. We consider the following problem
\begin{equation}\label{solfund3}
\left\{\begin{array}{cc}
\partial_t F+(-\Delta)^{1/2} F=0 & ~~\theta\in[-\pi,\pi],\; t>0\\
F(0,\theta)=\delta_0(x) & \theta\in [-\pi,\pi].
\end{array}\right.
\end{equation}
{\bf Claim:} The solution of \rec{solfund3} is given by
$$
F(\theta,t)=\frac{1}{2\pi}\sum_{n=-\infty}^{+\infty} e^{-t|n|}e^{in\theta}=\frac{e^{2t}-1}{e^{2t}-2e^{t}\cos(\theta)+1}.$$
{\bf Proof of the Claim.}
For every $t>0$ we write
\begin{equation}\label{Fn}
F(\theta,t)=\sum_{n=-\infty}^{+\infty} F_n(t)e^{in\theta}.\end{equation}
By plugging the formula into the equation we get for every $n\in \mathbb{Z}$:
$$
\frac{d}{dt}F_n(t)+|n|F_n(t)=0.$$
Therefore
$F_n(t)=C_ne^{-t|n|}$ and $F(\theta,t)=\sum_{n=-\infty}^{+\infty} C_ne^{-t|n|}e^{in\theta}.$ Since
$F(\theta,0)=\delta_0=\frac{1}{2\pi}\sum_{n=-\infty}^{+\infty} e^{in\theta}$ , namely
$ F(\theta,0)\ast f(\theta)=f(\theta)$ for every distribution $f$, we get that $C_n=\int_{S^1} F(\theta,0)e^{-in\theta} d\theta=\frac{1}{2\pi}$ for every $n\in\mathbb{Z}$.
Hence
$F$ is as in \rec{Fn} and the claim is proved.~$\Box$
\medskip
\begin{Theorem}{[Case on $S^1$]}\label{thms1}
Let $u\in \dot H^{1/2}_{loc}(S^1,\mathbb{R}^m)$ such that
\begin{equation}\label{harmequation2}
\frac{\partial u}{\partial \theta}\cdot (-\Delta)^{1/2} u=0~~\mbox{a.e in $S^1$.}\end{equation}
Then the following identity holds
\begin{equation}\label{pohos1}
\left|\int_{S^1}u(z)\partial_t F(z) \, d\theta\right|^2= \left|\int_{S^1}u(z)\partial_{\theta}F(z) \, d\theta\right|^2
\end{equation}
\end{Theorem}
{\bf Proof of Theorem \ref{thms1}}
Now we write
$u(\theta)=\sum_{n\in\mathbb{Z}} u_ne^{in\theta}$. Then the following equalities hold:
\begin{eqnarray}\label{harms2}
\sin(\theta)\partial_{\theta}u &=&\frac{1}{2}\sum_{n\in\mathbb{Z}}\left(e^{i\theta}-e^{-i\theta}\right) nu_ne^{in\theta}\nonumber\\
&=& \frac{1}{2}\sum_{n\in\mathbb{Z}}e^{in\theta}((n-1)u_{n-1}-(n+1)u_{n+1})\nonumber\\
\overline{(-\Delta)^{1/2}u}&=&\sum_{n\in\mathbb{Z}} |n|\bar{u}_ne^{-in\theta}\nonumber\\
\sin(\theta)\partial_{\theta}u\overline{(-\Delta)^{1/2}u}&=&\sum_{n,m\in\mathbb{Z}}e^{i(n-m)\theta}|m|[(n-1)u_{n-1}-(n+1)u_{n+1}]\bar{u}_m.
\end{eqnarray}
By combining \rec{harms1} and \rec{harms2} we get
\begin{eqnarray}\label{harms3}
0&=&\sum_{n,m\in\mathbb{Z}}|m| e^{-t|n-m|}\left[(n-1)u_{n-1}u_{n-1}-(n+1)u_{n=1}\right]\bar {u}_{m}\nonumber\\
&=&\sum_{n,m\in\mathbb{Z}}|m|nu_n\bar{u}_{m}\left[e^{t|n-m+1|}-e^{-t|n-m-1|}\right].\end{eqnarray}
We first symmetrize \rec{harms3}. We get first
\begin{equation} \label{harms4}
0=- \sum_{n,m\in\mathbb{Z}}|n|mu_m\bar{u}_{n}\left[e^{t|n-m+1|}-e^{-t|n-m-1|}\right].
\end{equation}
We sum \rec{harms3} and \rec{harms4}
\begin{equation} \label{harms5}
0=\sum_{n,m\in\mathbb{Z}}| \left[e^{t|n-m+1|}-e^{-t|n-m-1|}\right] [|m|n u_n\bar{u}_{m}-|n|m\bar {u}_nu_m].
\end{equation}
For every $n\in\mathbb{Z}$ we set
\begin{eqnarray*}
u_n&=&a_n+ib_n\\
a_n^+&=&\frac{a_n+a_{-n}}{2},~~~a_n^-=\frac{a_n-a_{-n}}{2}\\
b_n^+&=&\frac{b_n+b_{-n}}{2},~~b_n^-=\frac{b_n-b_{-n}}{2}.
\end{eqnarray*}
We observe that since the components of $u$ are real we have $a^-_n=0$ and $b^+_n=0$, $a_n^+=u^+_n$, $b_n^-=i u_n^-.$
We take the real part of \rec{harms5} and get
\begin{eqnarray}
0&=&
\sum_{n,m\in\mathbb{Z}}(|m|n-|n|m) \left[e^{-t|n-m+1|}-e^{-t|n-m-1|}\right] [a_ma_n+b_nb_m]\nonumber\\
&=& \sum_{n,m\in\mathbb{Z}}(|m|n-|n|m) \left[e^{-t|n-m+1|}-e^{-t|n-m-1|}\right] [a^+_ma^+_n] \label{harms6}\\
&&+ \sum_{n,m\in\mathbb{Z}}(|m|n-|n|m) \left[e^{-t|n-m+1|}-e^{-t|n-m-1|}\right] [b^-_mb^-_n].\label{harms7}
\end{eqnarray}
{\bf Estimate of \rec{harms6}}
\begin{eqnarray} \label{harms8}
&& \sum_{n,m\in\mathbb{Z}}(|m|n-|n|m) \left[e^{-t|n-m+1|}-e^{-t|n-m-1|}\right] [a^+_ma^+_n] \nonumber\\&=&
\sum_{n>0,m<0}(-2mn) \left[e^{-t|n-m+1|}-e^{-t|n-m-1|}\right] [a^+_ma^+_n] \nonumber\\
&&~~+\sum_{n<0,m>0}(2mn) \left[e^{-t|n-m+1|}-e^{-t|n-m-1|}\right] [a^+_ma^+_n] \nonumber\\
&=& \sum_{n>0,m<0}(-2mn) \left[e^{-t|n-m+1|}-e^{-t|n-m-1|}\right] [a^+_ma^+_n] \nonumber\\
&&~~+\sum_{n>0,m<0}(2(-m)(-n)) \left[e^{-t|-n+m+1|}-e^{-t|-n+m-1|}\right] [a^+_{-m}a^+_{-n}n]\nonumber\\
&=& -4\sum_{n>0,m<0}(mn) \left[e^{-t|n-m+1|}-e^{-t|n-m-1|}\right] [a^+_ma^+_n] \nonumber\\
&=& 4\sum_{n>0,m>0}(-m)n \left[e^{-t|n+m+1|}-e^{-t|n+m-1|}\right] [a^+_{-m}a^+_n]
\nonumber\\
&=& -4\sum_{n>0,m>0}mn \left[e^{-t(n+m+1)}-e^{-t(n+m-1)|}\right] [a^+_{m}a^+_n]
\nonumber\\
&=& -4\left(e^{-t}-e^{t}\right)\left(\sum_{n>0}ne^{-tn} a^+_n\right)^2=8\sinh(t)\left(\sum_{n\in\mathbb{Z}}|n|e^{-t|n|} u^+_n\right)^2\nonumber\\
&=&8\sinh(t)\left(\sum_{n\in\mathbb{Z}}(-\Delta)^{1/2}F_n u^+_n\right)^2\nonumber\\
&=&\frac{1}{2\pi}8\sinh(t)\left(\int_0^{2\pi}(-\Delta)^{1/2}F u^+ d\theta\right)^2=\frac{1}{2\pi}8\sinh(t)\left(\int_0^{2\pi}{\partial_t F} u^+ d\theta\right)^2.
\end{eqnarray}
{\bf Estimate of \rec{harms7}}
\begin{eqnarray} \label{harms9}
&& \sum_{n,m\in\mathbb{Z}}(|m|n-|n|m) \left[e^{-t|n-m+1|}-e^{-t|n-m-1|}\right] [b^-_mb^-_n] \nonumber\\&=&
\sum_{n>0,m<0}(-2mn) \left[e^{-t|n-m+1|}-e^{-t|n-m-1|}\right] [b^-_mb^-_n] \nonumber\\
&&~~+\sum_{n<0,m>0}(2mn) \left[e^{-t|n-m+1|}-e^{-t|n-m-1|}\right] [b^-_mb^-_n] \nonumber\\
&=& \sum_{n>0,m<0}(-2mn) \left[e^{-t|n-m+1|}-e^{-t|n-m-1|}\right] [b^-_mb^-_n] \nonumber\\
&&~~+\sum_{n>0,m<0}(2(-m)(-n)) \left[e^{-t|-n+m+1|}-e^{-t|-n+m-1|}\right] [b^-_{-m}b^-_{-n}n]\nonumber\\
&=& -4\sum_{n>0,m<0}(mn) \left[e^{-t|n-m+1|}-e^{-t|n-m-1|}\right] [b^-_mb^-_n] \nonumber\\
&=& 4\sum_{n>0,m>0}(-m)n \left[e^{-t|n+m+1|}-e^{-t|n+m-1|}\right] [b^-_{-m}b^-_n]
\nonumber\\
&=& 4\sum_{n>0,m>0}mn \left[e^{-t(n+m+1)}-e^{-t(n+m-1)|}\right] [b^-_{m}b^-_n]
\nonumber\\
&=& 4\left(e^{-t}-e^{t}\right)\left|\sum_{n>0}ne^{-tn} b^-_n\right|^2=-8\sinh(t)\left|\sum_{n\in\mathbb{Z}}ine^{-t|n|} u^-_n\right|^2\nonumber\\
&=&-8\sinh(t)\left|\sum_{n\in\mathbb{Z}}(\partial_{\theta} F_n ) u^-_n\right|^2\nonumber\\
&=&-\frac{1}{2\pi}8\sinh(t)\left|\int_0^{2\pi}(\partial_{\theta} F) u^- d\theta\right|^2.
\end{eqnarray}
From \rec{harms6} and \rec{harms8} and \rec{harms9} it follows
\begin{equation}\label{pohozaev2}
\left|{ \int_0^{2\pi}{\partial_t F}} (t,\theta) u(\theta) d\theta\right| =\left| \int_0^{2\pi}{{\partial_{\theta} F}} (t,\theta) u(\theta) d\theta\right|
\end{equation}
where
$${\partial_t F} (t,\theta)=-2e^t\frac{e^{2t}\cos(\theta)-2e^{t}+\cos(\theta)}{(e^{2t}-2e^{t}\cos(\theta)+1)^2}$$
and
$$
{\partial_{\theta} F} (t,\theta) =-2e^{t}\frac{ \sin(\theta)(e^{2t}-1)}{(e^{2t}-2e^{t}\cos(\theta)+1)^2}.$$
\subsection{Pohozaev Identities for the Laplacian in $\mathbb{R}^2$}
In this section we derive a Pohozaev identity in $2D$ which analogous to that found in $1D$. Instead of integrating on balls, the strategy
is to multiply by the fundamental solution of the heat equation. A similar idea has been preformed in \cite{Stru} to study the heat flow.\par
The solution of
\begin{equation}\label{solfundlap}
\left\{\begin{array}{cc}
\partial_t G+(-\Delta) G=0 & t>0\\
G(0,x)=\delta_{x_0} & t=0\,.
\end{array}\right.
\end{equation}
is given by $G(x,t)=(4\pi t)^{-1/2}e^{-\frac{|x-x_0|^2}{4t}}.$
\begin{Theorem}{[Case on $\mathbb{R}^2$]}\label{Poho0}
Let $u\in \mathbb{C}^2(\mathbb{R}^2,\mathbb{R}^m)$ such that
\begin{equation}\label{harmequation0}
\left\langle \frac{\partial u}{\partial x_i},\Delta u\right\rangle=0~~\mbox{a.e in $\mathbb{R}^2$}
\end{equation}
$i=1,2.$
Assume that
\begin{equation}\label{cond1}
\int_{\mathbb{R}}|u-u_0|dx <+\infty, ~~\int_{\mathbb{R}}|\nabla u(x)|dx<+\infty.
\end{equation}
Then for all $x_0\in\mathbb{R}^2$ and $t>0$ the following identity holds
\begin{equation}\label{idpohozaevR2}
\iint_{R^2}e^{-{\frac{|x-x_0|^2}{4t}}}|x-x_0|^2\left|\frac{\partial u}{\partial \nu} \right|^2dx= \iint_{R^2}e^{-{\frac{|x-x_0|^2}{4t}}}\left|\frac{\partial u}{\partial \theta}\right|^2 dx.
\end{equation}
\end{Theorem}
{\bf Proof.}
We multiply the equation \rec{harmequation0} by $x_i e^{-{\frac{|x-x_0|^2}{4t}}}$ and we integrate:
\begin{eqnarray}\label{estpoh0}
0&=&\sum_{k,i=1}^2\iint_{\mathbb{R}^2} e^{-{\frac{|x-x_0|^2}{4t}}}(x_i -{x_0}_i)\frac {\partial u}{\partial x_i}\frac{\partial^2 u}{\partial x_k^2}dx\nonumber\\
&=&-\frac{1}{2t}\iint_{\mathbb{R}^2} e^{-{\frac{|x-x_0|^2}{4t}}}|x -x_0|^2\left|\frac{\partial u}{\partial\nu}\right|^2 dx]-\iint_{\mathbb{R}^2}e^{-{\frac{|x-x_0|^2}{4t}}}|\nabla u|^2 dx\nonumber\\
&-&\frac{1}{2}\sum_{i=1}^2\iint_{\mathbb{R}^2} e^{-{\frac{|x-x_0|^2}{4t}}}(x -x_0)_i\frac{\partial}{\partial x_i} |\nabla u|^2 dx\nonumber\\
&=&-\frac{1}{2t}\iint_{\mathbb{R}^2} e^{-{\frac{|x-x_0|^2}{4t}}}|x -x_0|^2\left|\frac{\partial u}{\partial\nu}\right|^2 dx +\frac{1}{4t}\iint_{\mathbb{R}^2} e^{-{\frac{|x-x_0|^2}{4t}}}|x -x_0|^2\left|\nabla u\right|^2 dx.
\end{eqnarray}
Since $\nabla u=(\frac{\partial u}{\partial\nu},|x-x_0|^{-1}\frac{\partial u}{\partial\theta})$, from \rec{estpoh0} we get the identity
\begin{equation}
\iint_{R^2}e^{-{\frac{|x-x_0|^2}{4t}}}|x-x_0|^2\left|\frac{\partial u}{\partial \nu} \right|^2dx= \iint_{R^2}e^{-{\frac{|x-x_0|^2}{4t}}}\left|\frac{\partial u}{\partial \theta}\right|^2 dx
\end{equation}
and we conclude.~\hfill$\Box$\par
\medskip
\begin{Remark}
We obtain an analogous identity to \rec{idpohozaevR2} if we multiply the equation \rec{harmequation0} by $X_i e^{-{\frac{|x-x_0|^2}{4t}}}$ where
$X_1+i X_2\colon \mathbb{C}\to \mathbb{C}$ is a holomorpic function.
By using the Cauchy-Riemman differential equations we get
\begin{equation}
2 \iint_{R^2}e^{-{\frac{|x-x_0|^2}{4t}}}|x-x_0| \left\langle \frac{\partial u}{\partial \nu}, \frac{\partial u}{\partial X}\right\rangle dx= \iint_{R^2}e^{-{\frac{|x-x_0|^2}{4t}}}\left\langle x-x_0, X\right\rangle |\nabla u|^2 dx.
\end{equation}
\end{Remark}
\section{ Compactness and Quantization of horizontal $1/2$ harmonic maps}\label{cq}
The proof of the first part of Theorem \ref{th-I.2} is exactly the same of that of Lemma 2.3 in \cite{DL2} and we omit it. \par
As far as the quantization issue is concerned the proof goes as that of Theorem 1.1 in \cite{DL2} (namely the decomposition of the $\mathbb{R}$ into converging regions, bubbles domains and neck-regions) once we perform the analysis
of the neck-region. As we have already mentioned in the Introduction global uniform $L^{2,1}$ estimates in degenerating annuli are not anymore available as in the case of $1/2$-harmonic maps with values into a sphere.
Therefore we have to perform a subtle analysis of ${(-\Delta)}^{1/4}u$ where $u\in {\frak{H}}^{1/2} ({\mathbb{R}})$ is a horizontal $1/2$-harmonic map in an annular domain. To this purpose we will make use of the
Pohozaev-type formulae we have discovered in Section \ref{Poh}.
We first show that if $u\in {\frak{H}}^{1/2} ({\mathbb{R}})$ is a horizontal $1/2$-harmonic map with ${(-\Delta)}^{1/2}u\in L^{1}(\mathbb{R})$ then ${\cal{R}}[ ( P^N (-\Delta)^{1/4}u)\in L^{2,1}(\mathbb{R}).$
Next we show that if $u\in \dot H^{1/2}(\mathbb{R},{\cal{N}})$ is a weak harmonic map, then ${(-\Delta)}^{1/2}u\in L^{1}(\mathbb{R})$. \par
\begin{Proposition}\label{prL1}
Let $u\in \dot H^{1/2}(\mathbb{R},{\cal{N}})$ be a weak harmonic maps. Then ${(-\Delta)}^{1/2}u\in L^{1}(\mathbb{R})$.
\end{Proposition}
{\bf Proof of Proposition \ref{prL1}.}
{\bf Step 1.} We prove that ${(-\Delta)}^{1/2}u\in L^{1}(\mathbb{R})$.\par
This follows from the fact that for all $\xi,\eta\in {\cal{N}}$ we have
$$P^N(\xi)\cdot (\xi-\eta)=O(|\xi-\eta|^2).$$
If $u\in \dot H^{1/2}(\mathbb{R},{\cal{N}})$ is a weak harmonic map, then
$P^N(u){(-\Delta)}^{1/2}u ={(-\Delta)}^{1/2}u$ in ${\cal{D}}^\prime(\mathbb{R})$. Therefore
\begin{eqnarray}\label{estL1}
\int_{\mathbb{R}}|{(-\Delta)}^{1/2}u(x)| dx&=& \int_{\mathbb{R}}|P_N(u(x)){(-\Delta)}^{1/2}u(x)| dx\\
&\le & C\int_{\mathbb{R}}\int_{\mathbb{R}}\frac{|u(x)-u(y)|^2}{|x-y|^2} dx dy \nonumber\\
&=& C\|u\|^2_{\dot{H}^{1/2}(\mathbb{R})}<+\infty.\nonumber~\Box
\end{eqnarray}
\begin{Proposition}\label{prL2}
If $u\in {\frak{H}}^{1/2} ({\mathbb{R}})$ is a horizontal $1/2$-harmonic map with ${(-\Delta)}^{1/2}u\in L^{1}(\mathbb{R})$ then ${\cal{R}}[ ( P^N (-\Delta)^{1/4}u)]\in L^{2,1}(\mathbb{R}).$
\end{Proposition}
{\bf Proof of Proposition \ref{prL2}.}
Since $u\in {\frak{H}}^{1/2} ({\mathbb{R}})$, we have $P_N(u)\nabla u=P_N(u){\cal{R}}(-\Delta)^{1/2} u=0.$
The result follows from the fact that $P_N(u)(-\Delta)^{1/4} u$ satisfies the following structure equation
\begin{equation}\label{eqstruct}
(-\Delta)^{1/4} ( P^N (-\Delta)^{1/4}u)= S(P^N,u)-{\cal{R}} [((-\Delta)^{1/4} P^N) ({\cal{R}}(-\Delta)^{1/4} u)]\,.
\end{equation}
We deduce in particular that ${\cal{R}} [((-\Delta)^{1/4} P^N) ({\cal{R}}(-\Delta)^{1/4} u)]\in L^{1}(\mathbb{R})$.\par Since $((-\Delta)^{1/4} P^N) ({\cal{R}}(-\Delta)^{1/4} u)\in L^1(\mathbb{R})$ it follows that
$((-\Delta)^{1/4} P^N) ({\cal{R}}(-\Delta)^{1/4} u)\in {\mathcal{H}}^1$.\footnote{See section \ref{secdef} for the definition.}The Hardy Space can be also characterized as the space of function in $f\in L^1$ such that
${\cal{R}}[f]\in L^1({\mathbb{R}}) $.\par
Hence \rec{eqstruct} implies that
$ (-\Delta)^{1/4}{\cal{R}}[ ( P^N (-\Delta)^{1/4}u)]\in {\mathcal{H}}^1(\mathbb{R})$ and hence ${\cal{R}}[ ( P^N (-\Delta)^{1/4}u)] \in L^{2,1}(\mathbb{R}).$ We conclude the proof.~\hfill$\Box$\par
\bigskip
{\bf Proof of Theorem \ref{th-I.3}}.
We have already remarked that the sequence $$v_k=(P^k_T(-\Delta)^{1/4}u_k,{\cal{R}}[P^k_N(-\Delta)^{1/4} u_k])$$ satisfies a system of the form
\begin{equation}\label{modelsystemk}
(-\Delta)^{1/4} v_k=\Omega_k v_k+\Omega^1_k v+{\cal{Z}}(Q_k,v_k) \end{equation}
where $v_k\in L^2(\mathbb{R})$, $Q_k\in \dot H^{1/2}(\mathbb{R})$, $\Omega_k\in L^2(\mathbb{R},so(m))$, $\Omega_k^1\in L^{2,1}(\mathbb{R})$, $g_k\in L^1(\mathbb{R})$ and ${\cal{Z}}\colon \dot H^{1/2}(\mathbb{R})\times L^2(\mathbb{R})\to {\cal{H}}^1(\mathbb{R})$ with
\begin{equation}
\|\Omega_k\|_{L^2(\mathbb{R})}+\|\Omega^1_k\|_{L^{2,1}(\mathbb{R})}+\|Q_k\|_{\dot H^{1/2}(\mathbb{R})} )\le C\|v_k\|_{L^2}
\end{equation}
If $\delta>0$ is small enough we can apply Proposition \ref{decomposition}.
\begin{equation}\label{estdeltauring}
(P^k_T+ {\mathcal R} {P^k_N} ) (-\Delta)^{1/4} u_k(x)=\11_{A_{r_k,R_k}}[A^{-1}(x)\overrightarrow{c_{r_k}}\frac{1}{|x|^{1/2}}]+h_k(x) +g_k(x)\end{equation}
with
$A^{-1}_k\in L^{\infty}\cap \dot H^{1/2}(\mathbb{R},M_m({\mathbb{R}}))),$ $h_k\in L^{2,1}(A_{\Lambda r_k, (2\Lambda)^{-1}R_k}),$ (for every $\Lambda>2$ such that $\Lambda r_k<(2\Lambda)^{-1}R_k$), $g_k\in L^2(\mathbb{R})$ with ${\rm supp} g_k\subset (A_{r,R})^c.$ Moreover we have
\begin{equation}\label{estck}
\displaystyle \overrightarrow{c_{r_k}}=O\left(\left(\log(\frac{R_k}{2\Lambda^2 r_k})\right)^{-1/2}\right), ~~\mbox{as}~k\to +\infty, \Lambda\to +\infty. \end{equation}
Therefore
\begin{eqnarray}
P^k_T (-\Delta)^{1/4} u_k(x)&=&P^k_T\left(\frac{ \11_{A_{r_k,R_k}{A_k^{-1}(x)}\overrightarrow{c_{r_k}}}}{|x|^{1/2}}\right)+P^k_T h(x)+P^k_T g(x)\label{PTu}\\
P^k_N (-\Delta)^{1/4} u_k(x)&=&P^k_N\left({\mathcal{R}}\left[\frac{ \11_{A_{r_k,R_k}}{A^{-1}(x)}\overrightarrow{c_{r_k}}}{|x|^{1/2}}\right]\right)-P^k_N{\mathcal{R}}[ h(x)]-P^k_N{\mathcal{R}}[ g(x)].\label{PNu}
\end{eqnarray}
Observe that $P^k_N{\mathcal{R}} g_k(x)\in L^{2,1}(A_{\Lambda r_k, (2\Lambda)^{-1}R_k})$ since ${\rm supp}( g_k)\subset (A_{r_k,R_k})^c.$
By combining \rec{PTu} and \rec{PNu} we get
\begin{equation}
(-\Delta)^{1/4}u_k(x)=P^k_T\left(\frac{ \11_{A_{r_k,R_k}}{A_k^{-1}(x)}\overrightarrow{c_{r_k}}}{|x|^{1/2}}\right) +\bar h_k \end{equation}
with $\bar h_k=P^k_T h_k(x)+P^k_T(x)+P^k_N (-\Delta)^{1/4} u_k(x) $ which is in $ L^{2,1}(A_{\Lambda r_k, (2\Lambda)^{-1}R_k}).$
Next
we set
$a_k(x):= P^k_T{A_k^{-1}}$ and we denote for simplicity by $a_k^+$ and $a_k^-$ respectively the symmetric and antisymmetric parts of $a_k$.
Since $A_k^{-1}$ and $P^k_T$ are in $\dot H^{1/2}(\mathbb{R})$, they verify the following estimate (see \cite{Tartar}):
\begin{eqnarray}
\left\|\frac{A_k^{-1}(x)-A_k^{-1}(-x)}{|x|^{1/2}}\right\|_{L^2}&\le& C\|A_k^{-1}\|_{\dot H^{1/2}}\\
\left\|\frac{P^k_T(x)-P^k_T(-x)}{|x|^{1/2}}\right\|_{L^2}&\le& C\|P^k_T\|_{\dot H^{1/2}}\label{tarpt}\\
\end{eqnarray}
By using the fact that $A_k^{-1}$ and $P^k_N,P^k_T$ are also in $L^{\infty}$ we get that
$$\left\| \frac{a_k^{-}\overrightarrow{c_{r_k}}}{|x|^{1/2}}\right\|_{L^2(\mathbb{R})}\le C|\overrightarrow{c_{r_k}}|.$$
Therefore we can write
\begin{equation}
(-\Delta)^{1/4}u_k(x)=\11_{A_{r_k,R_k}} \frac{a_k^{+}\overrightarrow{c_{r_k}}}{|x|^{1/2}} +\bar h_k+\tilde g_k
\end{equation}
where $\bar h_k\in L^{2,1}(A_{\Lambda r_k, (2\Lambda)^{-1}R_k}),$ with
$$\limsup_{\Lambda\to\infty}\limsup_{k\to\infty } \| \bar h_k\|_{L^{2,1}(B(0,(2\Lambda)^{-1}R_k)\setminus B(0,\Lambda r_k)} <+\infty,$$ and
$\tilde g_k=\11_{A_{r_k,R_k}} \frac{a_k^{-}\overrightarrow{c_{r_k}}}{|x|^{1/2}}\in L^{2}(\mathbb{R})$, with $\|\tilde g_k\|_{L^2}\le |\overrightarrow{c_{r_k}}|.$
We can conclude.~~\hfill$\Box$
\par
\bigskip
{\bf Proof of Theorem \ref{th-I.5}}\par
Under the current hypotheses Theorem \ref{th-I.3} and Lemma \ref{neck2} it follows :\par
\begin{equation}\label{linftyestbis}
\limsup_{\Lambda\to\infty}\limsup_{k\to\infty } \| (-\Delta)^{1/4} u_k\|_{L^{2,\infty}(B(0,(2\Lambda)^{-1}R_k)\setminus B(0,\Lambda r_k)} =0\end{equation}
and
\begin{equation}\label{devel3}
(-\Delta)^{1/4}u_k(x)=\11_{A_{r_k,R_k}} \frac{a_k^{+}\overrightarrow{c_{r_k}}}{|x|^{1/2}} +\bar h_k+\tilde g_k,
\end{equation}
where $\bar h_k\in L^{2,1}(A_{\Lambda r_k, (2\Lambda)^{-1}R_k}),$ with
$$\limsup_{\Lambda\to\infty}\limsup_{k\to\infty } \| \bar h_k\|_{L^{2,1}(B(0,(2\Lambda)^{-1}R_k)\setminus B(0,\Lambda r_k)} <+\infty,$$
$\tilde g_k\in L^{2}(\mathbb{R})$ and $\|\tilde g_k\|_{L^2}\le C |\overrightarrow{c_{r}}|.$
By combining \rec{linftyestbis} and \rec{devel3} we get
\begin{equation}\label{estasymi}
\limsup_{\Lambda\to\infty}\limsup_{k\to\infty } \| ((-\Delta)^{1/4} u_k)^-\|_{L^{2}(B(0,(2\Lambda)^{-1}R_k)\setminus B(0,\Lambda r_k)} =0\end{equation}
In order to establish a link between the symmetric and antisymmetric part of $(-\Delta)^{1/4} u_k$ we make use of the formula \ref{idpohozaevR}
than can be rewritten as
\begin{equation}\label{refPoh}
\left(\int_{\mathbb{R}} \sqrt{\pi} \left(\frac{\sin(\arctan(-x))}{(1+x^2)^{3/4}}\right)w(tx) dx\right)^2=\left(\int_{\mathbb{R}} \sqrt{\pi} \left(\frac{\cos(\arctan(-x))}{(1+x^2)^{3/4}}\right)w(tx) dx\right)^2
\end{equation}
Now we plug into \rec{refPoh} the function $w_k(x)=\11_{A_{\Lambda r_k, (2\Lambda)^{-1}R_k}} (-\Delta)^{1/4}u_k(x)$. We observe that
$$\int_{\mathbb{R}} \sqrt{\pi} \left(\frac{\sin(\arctan(-x))}{(1+x^2)^{3/4}}\right)\11_{A_{r_k,R_k}}(tx) \frac{a_k^{+}\overrightarrow{c_{r_k}}}{|tx|^{1/2}} dx=0.$$
Therefore we have that
\begin{eqnarray*}
&&\int_{\mathbb{R}} \sqrt{\pi} \left(\frac{\cos(\arctan(-x))}{(1+x^2)^{3/4}}\right)\11_{A_{r_k,R_k}}(tx) \frac{a_k^{+}\overrightarrow{c_{r_k}}}{|tx|^{1/2}} dx\\
&=&
\int_{\mathbb{R}} \sqrt{\pi} \left(\frac{\sin(\arctan(-x))}{(1+x^2)^{3/4}}\right)\tilde{w}_k(tx) dx - \int_{\mathbb{R}} \sqrt{\pi} \left(\frac{\cos(\arctan(-x))}{(1+x^2)^{3/4}}\right)\tilde{w}_k(tx) dx.
\end{eqnarray*}
where
$$\tilde w_k(x)=w_k(x)-\11_{A_{r_k,R_k}}(x) \frac{a_k^{+}\overrightarrow{c_{r_k}}}{|x|^{1/2}}=\bar h_k+\tilde g_k.$$
Next we use the fact that the operators
\begin{eqnarray}
M^-[w](t)&:=&\int_{\mathbb{R}} \sqrt{\pi} \left(\frac{\sin(\arctan(-x))}{(1+x^2)^{3/4}}\right)w(tx) dx,\label{M+}\\
M^+[w](t)&:=&\int_{\mathbb{R}} \sqrt{\pi} \left(\frac{\cos(\arctan(-x))}{(1+x^2)^{3/4}}\right)w(tx) dx,\label{M-}.
\end{eqnarray}
are isomorphism from $L^{p}_+$ to $ L^{p}_+$ (resp. $L^{p}_-$ to $ L^{p}_-$) for every $p>1$ and from $L^{2,1}_+$ to $ L^{2,1}_+$ (resp. $L^{2,1}_-$ to $ L^{2,1}_-$ ) and we deduce that
$$
\11_{A_{r_k,R_k}}(x) \frac{a_k^{+}\overrightarrow{c_{r_k}}}{|x|^{1/2}}=\varphi_k+\psi_k.$$
with $\varphi_k\in L^{2,1}(A_{\Lambda r_k, (2\Lambda)^{-1}R_k}),~\psi_k\in L^{2}(\mathbb{R})$ and $\|\psi_k\|_{L^2}\le C|\overrightarrow{c_{r}}|.$
\par
Hence
\begin{eqnarray}
&&\limsup_{\Lambda\to\infty}\limsup_{k\to+\infty}\|(-\Delta)^{1/4}u_k-\psi_k\|_{L^2(B(0,(2\Lambda)^{-1}R_k)\setminus B(0,\Lambda_k r_r))}
\\&&\le \limsup_{\Lambda\to\infty}\limsup_{k\to+\infty}\|(-\Delta)^{1/4}u_k-\psi_k\|_{L^{2,\infty}(B(0,(2\Lambda)^{-1}R_k)\setminus B(0,\Lambda_k r_r))}
\nonumber \\
&&\cdot \limsup_{\Lambda\to\infty}\limsup_{k\to+\infty}\|(-\Delta)^{1/4}u_k-\psi_k\|_{L^{2,1}(B(0,(2\Lambda)^{-1}R_k)\setminus B(0,\Lambda_k r_r))}=0.
\end{eqnarray}
Therefore since $\limsup_{k\to+\infty}\|\psi_k\|_{L^2}=0$ we deduce that
$$
\limsup_{\Lambda\to\infty}\limsup_{k\to+\infty}\|(-\Delta)^{1/4}u_k\|_{L^2(B(0,(2\Lambda)^{-1}R_k)\setminus B(0,\Lambda_k r_r))}.$$
We can conclude the proof of Theorem \ref{th-I.5}.~~\hfill $\Box$
\section{Counter-example}
The aim of this section is to construct a sequence of solutions of a Schr\"odinger type equation with antisymmetric potential whose energy is not quantized. In particular, we are going to build a sequence of solutions whose energy of the potential goes to zero in the neck but not the energy of the solutions.\\
We consider
$u(t)= 1 $ on $[-1,1]$ and $\frac{1}{\vert t\vert^{\frac{1}{2}}}$ elsewhere and $v(t)= \frac{1}{(1+t^2)^{\frac{3}{8}}}$.
\begin{Lemma} As $t\rightarrow +\infty$, we have
$$(-\Delta)^\frac{1}{4} u(t) =O\left( \frac{1}{t^\frac{3}{2}}\right)$$
and
$$(-\Delta)^\frac{1}{4} v(t) =O\left( \frac{1}{t^\frac{5}{4}}\right).$$
\end{Lemma}
{\it Proof :}\\
\begin{equation}
\begin{split}
(-\Delta)^\frac{1}{4} u(t) &=\int_{\mathbb{R}} \frac{u(t)-u(s)}{\vert t-s\vert^\frac{3}{2}} \, ds \\
&=\int_{-\infty}^{-1} \frac{u(t)-u(s)}{\vert t-s\vert^\frac{3}{2}} \, ds + \int_{-1}^{1} \frac{u(t)-u(s)}{\vert t-s\vert^\frac{3}{2}} \, ds+ \int_{1}^{+\infty} \frac{u(t)-u(s)}{\vert t-s\vert^\frac{3}{2}} \, ds\\
& =I_1 + I_2+I_3.
\end{split}
\end{equation}
Let $ t >1$
\begin{equation}
\begin{split}
I_2 &= \int_{-1}^1 \frac{t^{-\frac{1}{2}}-1}{\vert t-s\vert^\frac{3}{2}} \, ds \\
&=\frac{t^{-\frac{1}{2}} -1}{ t^\frac{3}{2}}\int_{-1}^{1} \frac{1}{\vert1 -\frac{s}{t}\vert^\frac{3}{2}} \, ds\\
&=\frac{t^{-\frac{1}{2}} -1}{ t^\frac{1}{2}}\int_{-\frac{1}{t}}^{\frac{1}{t}} \frac{1}{\vert1 - u\vert^\frac{3}{2}} \, du\\
&=\frac{t^{-\frac{1}{2}} -1}{ t^\frac{1}{2}} \left[ 2(1-u)^{-\frac{1}{2}} \right]^{\frac{1}{t}}_{-\frac{1}{t}}
=O\left( \frac{1}{t^\frac{3}{2}} \right)
\end{split}
\end{equation}
\begin{equation}
\begin{split}
I_1 &= \int_{-\infty}^{-1} \frac{t^{-\frac{1}{2}}-(-s)^{-\frac{1}{2}}}{\vert t-s\vert^\frac{3}{2}} \, ds \\
&=\frac{1}{t^2}\int_{-\infty}^{-1} \frac{1-(-\frac{s}{t})^{-\frac{1}{2}}}{\vert1 -\frac{s}{t}\vert^\frac{3}{2}} \, ds\\
&=\frac{1}{t} \int^{+\infty}_{\frac{1}{t}} \frac{1-u^{-\frac{1}{2}}}{(1+u)^\frac{3}{2}} \, du
\end{split}
\end{equation}
\begin{equation}
\begin{split}I_3 &= \int_{1}^{+\infty} \frac{t^{-\frac{1}{2}}-s^{-\frac{1}{2}}}{\vert t-s\vert^\frac{3}{2}} \, ds \\
&=\frac{1}{t^2}\int_{1}^{+\infty} \frac{1-(\frac{s}{t})^{-\frac{1}{2}}}{\vert1 -\frac{s}{t}\vert^\frac{3}{2}} \, ds\\
&=\frac{1}{t} \int^{+\infty}_{\frac{1}{t}} \frac{1-u^{-\frac{1}{2}}}{\vert 1-u\vert^\frac{3}{2}} \, du
\end{split}
\end{equation}
We easily check that $\displaystyle \frac{1-u^{-\frac{1}{2}}}{\vert 1-u\vert^\frac{3}{2}} \underset{u\rightarrow 1}{\sim} (1-u)^{-\frac{1}{2}}$ and then the last integral is well defined.\\
But changing the variable $u$ into $\frac{1}{v}$ into $I_1$ and $I_2$ we observe that
$$ \int^{+\infty}_{0} \frac{1-u^{-\frac{1}{2}}}{(1+u)^\frac{3}{2}} \, du = \int_{0}^{+\infty} \frac{1-u^{-\frac{1}{2}}}{\vert 1-u\vert^\frac{3}{2}} \, du=0$$
Which implies that
$$I_1=\frac{-1}{t} \int_0^{\frac{1}{t}} \frac{1-u^{-\frac{1}{2}}}{(1+u)^\frac{3}{2}} \, du = \underset{t\rightarrow +\infty}{O} \left(\frac{1}{t^\frac{3}{2}}\right)$$
and
$$I_3=\frac{-1}{t} \int_0^{\frac{1}{t}} \frac{1-u^{-\frac{1}{2}}}{\vert 1-u\vert^\frac{3}{2}} \, du = \underset{t\rightarrow +\infty}{O} \left(\frac{1}{t^\frac{3}{2}}\right).$$
Which proves the lemma for $u$ when $t>1$. Of course we have the same result for $t<-1$ by symmetry.\\
\begin{equation}
\begin{split}
(-\Delta)^\frac{1}{4} v(t) &=\int_{\mathbb{R}} \frac{u(t)-u(s)}{\vert t-s\vert^\frac{3}{2}} \, ds= \frac{1}{(1+t^2)^{\frac{3}{8}}} \int_{\mathbb{R}} \frac{(1+s^2)^{\frac{3}{8}} -(1+t^2)^{\frac{3}{8} }}{ (1+s^2)^{\frac{3}{8}}\vert t-s\vert^\frac{3}{2}} \, ds\\
&= \frac{1}{(1+t^2)^{\frac{3}{8}}} \int_{-\infty}^{-1} \frac{(1+s^2)^{\frac{3}{8}} -(1+t^2)^{\frac{3}{8} }}{ (1+s^2)^{\frac{3}{8}}\vert t-s\vert^\frac{3}{2}} \, ds + \frac{1}{(1+t^2)^{\frac{3}{8}}} \int_{-1}^1 \frac{(1+s^2)^{\frac{3}{8}} -(1+t^2)^{\frac{3}{8} }}{ (1+s^2)^{\frac{3}{8}}\vert t-s\vert^\frac{3}{2}} \, ds \\
&+ \frac{1}{(1+t^2)^{\frac{3}{8}}} \int_{1}^{+\infty} \frac{(1+s^2)^{\frac{3}{8}} -(1+t^2)^{\frac{3}{8} }}{ (1+s^2)^{\frac{3}{8}}\vert t-s\vert^\frac{3}{2}} \, ds
\\& =I_1 + I_2+I_3.
\end{split}
\end{equation}
We remark that
$$ \frac{(1+s^2)^{\frac{3}{8}} -(1+t^2)^{\frac{3}{8} }}{\vert t-s\vert^\frac{3}{4}} \hbox{ is uniformly bounded.}$$
Then, let $t>1$,
\begin{equation}
\begin{split}
I_2 &= O\left( \frac{1}{\vert t \vert^{\frac{3}{4}}} \int_{-1}^1 \frac{1}{ (1+s^2)^{\frac{3}{8}}\vert t-s\vert^\frac{3}{4}} \, ds\right)\\
&= O\left( \frac{1}{\vert t \vert^{\frac{3}{4}}} \int_{-1}^1 \frac{1}{ \vert t-s\vert^\frac{3}{4}} \, ds\right)\\
&= O\left( \frac{1}{\vert t \vert^{\frac{5}{4}}} \right)
\end{split}
\end{equation}
And
\begin{equation}
\begin{split}
I_1 &= O\left( \frac{1}{\vert t \vert^{\frac{3}{4}}} \int_{-\infty}^{-1} \frac{1}{ (1+s^2)^{\frac{3}{8}}\vert t-s\vert^\frac{3}{4}} \, ds\right)\\
&= O\left( \frac{1}{\vert t \vert^{\frac{3}{4}}} \int_{-\infty}^{-1} \frac{1}{ (-s)^\frac{3}{4} \vert t-s\vert^\frac{3}{4}} \, ds\right)\\
&= O\left( \frac{1}{\vert t \vert^{\frac{3}{4}}} \int_{-\infty}^{-1} \frac{1}{ (-s)^\frac{3}{4} \vert t-s\vert^\frac{3}{4}} \, ds\right)\\
&= O\left( \frac{1}{\vert t \vert^\frac{5}{4} } \int_{-\infty}^{-1/t} \frac{1}{ (-u)^\frac{3}{4} \vert 1-u\vert^\frac{3}{4}} \, ds\right)\\
&= O\left( \frac{1}{\vert t \vert^{\frac{5}{4}}} \right)
\end{split}
\end{equation}
The same estimate work for $I3$ and $t<-1$, which achieved the proof.\hfill$\square$
Then we set $\omega =\frac{(-\Delta)^\frac{1}{4} u}v$ and $\omega_1= \frac{(-\Delta)^\frac{1}{4} v +\omega u}{u}$, which gives
$$(-\Delta)^\frac{1}{4} U=\Omega U +\Omega_1 U$$
with
$$U=\left(\begin{array}{c}u \\v\end{array}\right),$$
$$\Omega = \left(\begin{array}{cc}0 & \omega \\-\omega & 0\end{array}\right)$$
and
$$\Omega_1=\left(\begin{array}{cc}0 & 0 \\ \omega_1 & 0\end{array}\right).$$
Finally, we set $U_n(t)=c_n U(nt)$, $\Omega_n= \sqrt{n} \Omega(nt)$, and ${\Omega_1}_n=\sqrt{n} \Omega_1(nt)$. Where
$$c_n =\frac{1}{\Vert u(nt)\Vert_2}= \left( \frac{n}{\ln\left(\frac{n+\sqrt{1+n^2}}{-n+\sqrt{1+n^2}}\right)}\right)^\frac{1}{2} \sim \left( \frac{n}{\ln(n)}\right)^\frac{1}{2} $$
We easily check that
$$\Vert U_n \Vert_2 \sim1$$
$$\lim_{R\rightarrow +\infty} \lim_{n\rightarrow +\infty} \Vert \Omega_n \Vert_{L^2(B(0,1/R) \setminus B(0,R/n))} + \Vert {\Omega_1}_n \Vert_{L^{2,1} (B(0,1/R) \setminus B(0,R/n))} =0$$
and
$$(-\Delta)^\frac{1}{4} U_n = \Omega_n U_n +{\Omega_1}_n U_n$$
which prove that there is no quantification of the energy, since $\Omega_n$ and ${\Omega_1}_n$ have no energy in the neck region despite $U_n$ get some. Of course such an example doesn't satisfies a Pohozaev identity, since it is symmetric. It is a "local" example since $U_n$ bounded in $L^2$ only on $[-1,1]$, it would be interesting to construct a example define on the whole $\mathbb{R}$. It may be possible by starting on the circle and then projecting on to $\mathbb{R}$. Indeed the main idea here is to take a modification of the Green function of the $(-\Delta)^{\frac{1}{4}}$. Hence playing this game on $S^1$ and projecting then $\mathbb{R}$ should provide an example with good decreasing at infinity.
|
1,116,691,499,272 | arxiv |
\section{Introduction}
\label{sec:intro}
\begin{quote}
\begin{small}
{\em ``In a machine such as this {\em [a quantum computer]} there are
very many other problems due to imperfections. For example, in the
registers for holding the data, there will be problems of cross-talk,
interactions between one atom and another in that register, or
interaction of the atoms in that register directly with things that
are happening along the program line that we didn't exactly bargain
for. In other words, there may be small terms in the Hamiltonian
besides the ones we've written. Until we propose a complete
implementation of this, it is very difficult to analyze. At least some
of these problems can be remedied in the usual way by techniques such
as error correcting codes and so forth, that have been studied in
normal computers. But until we find a specific implementation for this
computer, I do not know how to proceed to analyze these effects.
However, it appears that they would be very important in
practice. This computer seems to be very delicate and these
imperfections may produce considerable havoc.''}
-- Richard P. Feynman,
``Quantum Mechanical Computers,''
{\em Optics News}, February 1985.
\end{small}
\end{quote}
\subsection{Alleviating the ``havoc''}
\label{sec:intro havoc}
The concerns heralded by Feynman remain of concern today in all the
implementations envisioned for quantum information systems. In a
quantum computer it is highly likely that in order to achieve the
desired system objectives, these systems will have to be tuned, or
even entirely determined, using estimated quantities obtained from
data from the actual system rather than solely relying on an initial
design from a theoretical model. The problem addressed here is to
design the experiment in order to yield the optimum information for
the intended purpose. This goal is not just limited to quantum
information systems. It is an essential step in the engineering
practice of system identification \cite[Ch.14]{Ljung:87}. That is,
the design of the experiment which gives the best performance against
a given set of criteria, subject to constraints reflecting the
underlying properties of the system dynamics and/or costs associated
with the implementation of certain operations or controls.
Clearly each application has a specific threshold of performance.
For example, the requirements in quantum chemistry are generally not
as severe as in quantum information systems. The objective of a
measurement, therefore, depends on the way in which information is
encoded into the system to begin with, and this is in turn, depends on
the application. In this paper we are concerned with estimating
quantum system properties: the state, the process which transforms the
state, and parameters in a Hamiltonian model.
The estimation of the state of a quantum system from available
measurements is generally referred to as {\em quantum state
tomography} about which there is extensive literature on both
theoretical and experimental aspects, \emph{e.g.}, see
\cite[Ch.8]{NielsenC:00}, \cite{GriceW:96} and the references therein.
The more encompassing procedure of {\em quantum system identification}
is not so easily categorized as the nomenclature (and methodology)
seems to depend on the type and/or intended use of the identified
model. For example, {\em quantum process tomography} (QPT) refers to
determining the Kraus {\em operator-sum-representation} (OSR) of the
input state to output state (completely positive) map, \emph{e.g.},
\cite[\S 8.4.2]{NielsenC:00}, \cite{ChuangN:97}. {\em Hamiltonian
parameter estimation} refers to determining parameters in a model of
the system Hamiltonian, \emph{e.g.}, \cite{Mabuchi:96}, \cite{ChildsPR:00},
\cite{GeremiaR:02}, \cite{VDM:01}. Somewhere in between quantum
process tomography and Hamiltonian parameter estimation is {\em
mechanism identification} which seeks an estimate of population
transfer between states as the system evolves, \emph{e.g.}, \cite{MitraR:03}.
{\em Maximum likelihood estimation} (MLE), a well established method
of parameter estimation which is used extensively in current
engineering applications, \emph{e.g.}, \cite{Ljung:87}, was proposed in
\cite{Banaszek:98,ParisETAL:01} and \cite{Sacchi:01} for quantum state
tomography of a quantum system with non-continuing measurements, \emph{i.e.},
data is taken from repeated identical experiments. Also, as observed
in \cite{ParisETAL:01,Sacchi:01}, the MLE of the density matrix is a
convex optimization problem.
In this paper we address the related problem of optimal experiment
design (OED) so as to secure an estimate of the best quality. The
approach presented relies on minimizing the Cramer-Rao lower bound
\cite{Cramer:46} where the design parameters are the number of
experiments to be performed while the system is in a specified
configuration. En route we also show that many related problems in
state and process tomography can also be solved using MLE, and
moreover, they are all {\em formally} the same type of convex
optimization problem, namely, a determinant maximization problem,
referred to as a {\em maxdet problem}
\cite{BoydV:04,VBW:98}. Similarly, the OED problem posed here is also
of a single general type of convex optimization problem, namely, a
{\em semidefinite program} (SDP).
Convexity arises in many ways in quantum mechanics and this is briefly
discussed in \refsec{cvx}. The great advantage of convex optimization
is a globally optimal solution can be found efficiently and reliably,
and perhaps most importantly, can be computed to within any desired
accuracy. Achieving these advantages, however, requires the use of
specialized numerical solvers. As described in \refsec{software}, the
appropriate convex solvers have been embedded in some software tools
we have composed which can solve the MLE and OED problems presented
here.
In the remainder of the paper we present both MLE and the
corresponding OED as applied to: quantum state tomography (MLE in
\refsec{mle} and OED in \refsec{expt des}), estimating the
distribution of known input states (MLE in \refsec{dist est} and OED
in \refsec{expdes qs dist}), quantum process tomography using the
Kraus operator sum representation (MLE in \refsec{qpt osr} and
OED in \refsec{expdes osr}), estimating the distribution of a known
OSR set (MLE in \refsec{osr dist} and OED in \refsec{expdes osr
dist}), and to Hamiltonian parameter estimation (MLE in \refsec{mle
hampar} and OED in \refsec{ham oed}). A summary in table form is
presented in \refsec{summary} followed by a discussion in \refsec{iter
adapt} of the relation of MLE and OED to iterative adaptive control of
quantum systems.
\subsection{Convexity and quantum mechanics}
\label{sec:cvx}
Many quantum operations form convex sets or functions. Consider, for
example, the following convex sets which arise from some of the basic
aspects of quantum mechanics:
\begin{center}
\begin{tabular}{lll}
\begin{tabular}{l}
probability outcomes
\end{tabular}
&
$\seq{p_\alpha\in{\mathbf R}}$
&
$
\sum_\alpha p_\alpha=1,
\;\;\;
p_\alpha\geq 0
$
\\&&\\
\begin{tabular}{l}
density matrix
\end{tabular}
&
$\seq{\rho\in{\mathbf C}^{n\times n}}$
&
$
{\rm Tr}\ \rho=1,
\;\;\;
\rho \geq 0
$
\\&&\\
\begin{tabular}{l}
positive operator
\\
valued measure (POVM)
\end{tabular}
&
$\seq{O_\alpha\in{\mathbf C}^{n\times n}}$
&
$
\sum_\alpha\ O_\alpha = I_n,
\;\;\;
O_\alpha \geq 0
$
\\&&\\
\begin{tabular}{l}
operator sum
\\
representation (OSR)
\\
in fixed basis
\\
$\set{B_i\in{\mathbf C}^{n\times n}}{i=1,\ldots,n^2}$
\end{tabular}
&
$\seq{X\in{\mathbf C}^{n^2\times n^2}}$
&
$
\sum_{ij}\ X_{ij}\ B_i^*B_j = I_n,
\;\;\;
X \geq 0
$
\end{tabular}
\end{center}
An example of a convex function relevant to quantum information is
{\em worst-case gate fidelity}, a measure of the ``distance'' between
two unitary operations on the same input. As pointed out in
\cite{GilchristETAL:04}, there are many ways to define this
measure. Consider, for example,
\[[eq:wc fidelity]
f^{\rm wc}(U_{\rm des},\ U_{\rm act})
=
\min_{\norm{\psi}=1}
\left|
\left(U_{\rm des} \psi\right)^*
\left( U_{\rm act}\psi \right)
\right|^2
\]
where $U_{\rm des}\in{\mathbf C}^{n\times n}$ is the desired unitary and $U_{\rm
act}\in{\mathbf C}^{n\times n}$ is the actual unitary. In this case the worst-case
fidelity can be interpreted as the minimum probability of obtaining
the desired output state $U_{\rm des}\psi$ over all possible pure input
states $\psi$ which produce the actual output state $U_{\rm act}\psi$. If
$U_{\rm des}$ and $U_{\rm act}$ differ by a scalar phase then the worst-case
fidelity is clearly unity; which is consistent with the fact that a
scalar phase cannot be measured. This is not the case for the error
norm $\norm{U_{\rm des}-U_{\rm act}}$.
As shown in Appendix \refsec{fidelity}, obtaining the worst-case
fidelity requires solving the following (convex) {\em quadratic
programming} (QP) problem:
\[[eq:fidelity qp]
\begin{array}{ll}
\mbox{minimize}
&
z^T(aa^T+bb^T)z
\\
\mbox{subject to}
&
\sum_{k=1}^n\ z_k = 1,
\;\;\;
z_k \geq 0
\end{array}
\]
with the vectors $a,b$ in ${\mathbf R}^n$ the real and imaginary parts,
respectively, of the eigenvalues of the unitary matrix $U_{\rm des}^*U_{\rm act}$,
that is,
$a={\rm Re}\ {\bf eig}(U_{\rm des}^*U_{\rm act}),\
b={\rm Im}\ {\bf eig}(U_{\rm des}^*U_{\rm act})$.
In some cases it is possible to compute the worst-case fidelity
directly, \emph{e.g.}, in the example in Section \refsec{ex hampar} and in
some examples in \cite[\S 9.3]{NielsenC:00}. Although the optimal
objective value $f^{\rm wc}(U_{\rm des},\ U_{\rm act})$ is global, the optimal
worst-case state which achieves this value is not unique.
In addition to these examples, convex optimization has been exploited
in \cite{AudD:03} and \cite{Sacchi:01} in an attempt to realize
quantum devices with certain properties. In \cite{EldarMV:03} and
\cite{KosutWER:04}, convex optimization is used to design optimal
state detectors which have the maximum efficiency.
In general, convex optimization problems enjoy many useful properties.
From the introduction in \cite{BoydV:04}, and as already stated, the
solution to a convex optimization problem can be obtained to within
any desired accuracy. In addition, computation time does not explode
with problem size, stopping criteria always produce a lower bound on
the solution, and if no solution can be found a proof of infeasibility
is provided. There is also a complete duality theory which can yield
more efficient computation as well as optimality conditions. This is
explored briefly in Section \refsec{dual}.
\subsection{Software for tomography \& experiment design}
\label{sec:software}
We have composed some MATLAB m-files which can be used to solve a
subset of the QPT and OED convex optimization problems presented
here. The examples shown here were generated using this software. The
software, available upon request from the first author, requires the
convex solvers YALMIP \cite{Yalmip} and SDPT3 \cite{Sdpt3} which can
be downloaded from the internet. These solvers make use of {\em
interior-point methods} for solving convex optimization problems, \emph{e.g.},
\cite[Ch.11]{BoydV:04}, \cite{NN94}.
\section{Quantum State Tomography}
\label{state est}
Consider a quantum system which has ${n_{\rm out}}$ distinct {\em outcomes},
labeled by the index $\alpha,\ \alpha=1,\ldots,{n_{\rm out}}$, and which can be
externally manipulated into ${n_{\rm cfg}}$ distinct {\em configurations},
labeled by the index $\gamma,\ \gamma=1,\ldots,{n_{\rm cfg}}$. Configurations
can include wave-plate angles for photon counting, sample times at
which measurements are made, and settings of any experimental
``knobs'' such as external control variables, \emph{e.g.}, laser wave shape
parameters, magnetic field strengths, and so on. For quantum process
tomography (\refsec{qpt osr}) and Hamiltonian parameter estimation
(\refsec{mle hampar}), configurations can also include distinctly
prepared initial states.
The problem addressed in this section is to determine the minimum
number of experiments per configuration in order to obtain a state
estimate of a specified quality, \emph{i.e.}, what is the tradeoff between
number of experiments per configuration and estimation quality. The
method used to solve this problem is based on minimizing the size of
the Cram\'{e}r-Rao lower bound on the estimation error
\cite{Cramer:46}.
\subsection{Data collection}
\label{sec:data collect}
The data is collected using a procedure referred to here as {\em
non-continuing measurements}. Measurements are recorded from
identical experiments in each configuration $\gamma$ repeated
$\ell_\gamma$ times. The set-up for data collection is shown
schematically in Figure \ref{fig:rho} for configuration $\gamma$.
\begin{figure}[h]
\[
\begin{array}{rr}
\rho^{\rm true}\in{\mathbf C}^{n\times n} & \longrightarrow
\\
\mbox{} & \mbox{}
\\
\gamma & \longrightarrow
\end{array}
\mathbox{
\begin{array}{c}
\mbox{System}
\\
Q
\\
\mbox{}
\end{array}
}
\begin{array}{r}
\sigma^{\rm true}_\gamma\in{\mathbf C}^{n\times n}
\\
\mbox{----------}\!\!\!\longrightarrow
\\
\gamma \longrightarrow
\end{array}
\mathbox{
\begin{array}{c}
\mbox{POVM}
\\
M_{\alpha\gamma} \in{\mathbf C}^{n\times n}
\\
\alpha=1,\ldots,{n_{\rm out}}
\end{array}
}
\longrightarrow
\begin{array}{c}
\mbox{Outcome counts}
\\
\seq{
\begin{array}{c} n_{\alpha\gamma},\ \ell_\gamma~\mbox{trials}
\\ \alpha=1,\ldots,{n_{\rm out}} \end{array}
}
\end{array}
\]
\caption{System/POVM.}
\label{fig:rho}
\end{figure}
\noindent
Here $\rho^{\rm true}\in{\mathbf C}^{n\times n}$ is the true, unknown state to be estimated,
$\sigma^{\rm true}_\gamma\in{\mathbf C}^{n\times n}$ is the {\em reduced density matrix} which
captures all the statistical behavior of the $Q$-system under the
action of the measurement apparatus, and $n_{\alpha\gamma}$ is the number
of times outcome $\alpha$ is obtained from the $\ell_\gamma$ experiments.
Thus,
\[
{\sum_{\alpha=1}^m} \ n_{\alpha\gamma} = \ell_\gamma,
\;\;
{\ell_{{\rm expt}}}=\sum_{\gamma=1}^{{n_{\rm cfg}}}\ \ell_\gamma
\]
where ${\ell_{{\rm expt}}}$ is the total number of experiments. The {\em data set}
consists of all the outcome counts,
\[[eq:data]
D
=
\set{n_{\alpha\gamma}}
{\alpha=1,\ldots,{n_{\rm out}},\gamma=1,\ldots,{n_{\rm cfg}}}
\]
The design variables used to optimize the experiment are the
non-negative integers $\{\ell_\gamma\}$ represented by the vector,
\[[eq:ell]
\ell=[\ell_1 \cdots \ell_{{n_{\rm cfg}}}]^T
\]
Let $p^{\rm true}_{\alpha\gamma}$ denote the true probability of obtaining
outcome $\alpha$ when the system is in configuration $\gamma$ with state
input $\rho^{\rm true}$. Thus,
\[[eq:av nalfgam]
{\bf E} \ n_{\alpha\gamma} = \ell_\gamma p^{\rm true}_{\alpha\gamma}
\]
where the expectation $ {\bf E} (\cdot)$ taken with repect to the
underlying quantum probability distributions.
We pose the following {\em model} of the system,
\[[eq:palfgam]
p_{\alpha\gamma}(\rho) = {\rm Tr}\ M_{\alpha\gamma} \sigma_\gamma(\rho)
\]
where $p_{\alpha\gamma}(\rho)$ is the outcome probability of measuring
$\alpha$ when the system is in configuration $\gamma$ with input state
$\rho$ belonging to the set of density matrices,
\[[eq:rhoset]
\set{\rho\in{\mathbf C}^{n\times n}}
{\rho \geq 0,\; {\rm Tr}\ \rho =1}
\]
$\seq{M_{\alpha\gamma}}$ are the POVM elements of the measurement
apparatus, and thus, for $\gamma=1,\ldots,{n_{\rm cfg}}$,
\[[eq:povm]
{\sum_{\alpha=1}^m} \ M_{\alpha\gamma}=I_n,
\;\;
M_{\alpha\gamma} \geq 0,\
\alpha=1,\ldots,{n_{\rm out}}
\]
and $\sigma_\gamma(\rho)$ is the reduced density output state of the
$Q$-system model. A general (model) representation of the $Q$
system is the {\em Kraus operator-sum-representation} (OSR) which can
account for many forms of error sources as well as decoherence
\cite{NielsenC:00}. Specifically, in configuration $\gamma$, the
$Q$-system model can be parametrized by the set of {\em Kraus
matrices}, $K_\gamma=\set{K_{\gamma k}\in{\mathbf C}^{n\times n}}{k=1,\ldots,{\kappa_\gamma}}$ as
follows:
\[[eq:osr]
\sigma_\gamma(\rho)
=Q(\rho,K_\gamma)
=
\sum_{k=1}^{\kappa_\gamma}\ K_{\gamma k} \rho K_{\gamma k}^*,
\;\;\;
\sum_{k=1}^{\kappa_\gamma}\ K_{\gamma k}^* K_{\gamma k} = I_n
\]
with ${\kappa_\gamma} \leq n^2$. Implicit in this OSR is the assumption that
the $Q$-system is trace preserving. Combining this with the
measurement model \refeq{povm} gives the model probability outcomes,
\[[eq:pout osr]
p_{\alpha\gamma}(\rho)
=
{\rm Tr}\ O_{\alpha\gamma}\rho,
\;\;
O_{\alpha\gamma}
=
\sum_{k=1}^{\kappa_\gamma}\ K_{\gamma k}^* M_{\alpha\gamma} K_{\gamma k}
\]
In this model, the outcome probabilities are {\em linear} in the input
state density matrix. Moreover, the set
$O_\gamma=\set{O_{\alpha\gamma}}{\alpha=1,\ldots,{n_{\rm out}}}$, satisfies
\refeq{povm}, and hence, is a POVM. \footnote{
In a more general OSR the $Q$-system need not be trace preserving,
hence the Kraus matrices in \refeq{osr} need not sum to identity as
shown, but rather, their sum is bounded by identity. Then the set
$O_\gamma$ is not a POVM, however, satisfies,
$
{\sum_{\alpha=1}^m} \ O_{\alpha\gamma} \leq I_n,
\;\;
O_{\alpha\gamma} \geq 0,\
\alpha=1,\ldots,{n_{\rm out}}
$
}
If the $Q$-system is modeled as a unitary system, then,
\[[eq:povm u]
\sigma_\gamma(\rho) = U_\gamma \rho U_\gamma^*,
\;\;
U_\gamma^* U_\gamma = I_n
\Longrightarrow
O_{\alpha\gamma}
=
U_\gamma^* M_{\alpha\gamma} U_\gamma
\]
The set $O_\gamma$ is still a POVM; in effect the OSR has a single
element, namely, $K_\gamma=U_\gamma$.
\subsubsection*{System in the model set}
We make the following assumption throughout: {\em the true system is
in the model set}. This means that,
\[[eq:s in m]
p^{\rm true}_{\alpha\gamma} = p_{\alpha\gamma}(\rho^{\rm true})
={\rm Tr}\ O_{\alpha\gamma}\rho^{\rm true}
\]
This is always a questionable assumption and in most engineering
practice is never true. Relaxing this assumption is an active research
topic particularly when identification (state or process) is to be
used for control design, \emph{e.g.}, see \cite{Kosut:01} and
\cite{SafonovT:97}. The case when the system is {\em not} in the model
set will not be explored any further here except for the effect of
measurement noise which is discussed next. It is important to
emphasize that in order to produce an accurate unbiased estimate of
the true density it is necessary to know the noise elements (as
described next) which is a consequence of assumption \refeq{s in m}.
\subsubsection*{Noisy measurements}
\label{sec:noisy meas}
Sensor noise can engender more noisy outcomes than noise-free
outcomes. Consider, for example, a photon detection device with two
photon-counting detectors. If both are noise-free, meaning, perfect
efficiency and no dark count probability, then, provided one photon is
always present at the input of the device, there are only two possible
outcomes: $\seq{10,\ 01}$. If, however, each detector is noisy, then
either or both detectors can misfire or fire even with a photon always
present at the input. Thus in the noisy case there are {\em four}
possible outcomes: $\seq{10,\ 01,\ 11,\ 00}$.
Let $\set{M_{\alpha\gamma}}{\alpha=1,\ldots,{n_{\rm out}}}$ denote the noisy POVM and
let $\set{\bar{M}_{\alpha\gamma}}{\alpha=1,\ldots,{\bar{n}_{\rm out}}}$ denote the noise-free
POVM with ${n_{\rm out}} \geq {\bar{n}_{\rm out}}$ where,
\[[eq:oalfmn]
M_{\alpha\gamma} = \sum_{\beta=1}^{\bar{n}_{\rm out}} \
\nu_{\alpha\beta\gamma}\
\bar{M}_{\beta\gamma},\
\alpha=1,\ldots,{n_{\rm out}},\
\gamma=1,\ldots,{n_{\rm cfg}}
\]
The $\seq{ \nu_{\alpha\beta\gamma} }$ represents the noise in the
measurement, specifically, the conditional probability that $\alpha$ is
measured given the noise-free outcome $\beta$ with the system in
configuration $\gamma$. Since $ {\sum_{\alpha=1}^m} \
\nu_{\alpha\beta\gamma}=1,\ \forall \beta,\gamma$, it follows that if the
noise-free set is a POVM then so is the noisy set.
\subsection{Maximum likelihood state estimation}
\label{sec:mle}
The Maximum Likelihood (ML) approach to quantum state estimation
presented in this section, as well as observing that the estimation is
convex, can be found in \cite{ParisETAL:01}, \cite{VDM:01} and the
references therein. Using convex programming methods, such as an
interior-point algorithm for computation, was not exploited in these
references.
If the experiments are independent, then the probability of obtaining
the data \refeq{data} is a product of the individual model
probabilities \refeq{palfgam}. Consequently, for an {\em assumed}
initial state $\rho$, the model predicts that the probability of
obtaining the data set \refeq{data} is given by,
\[[eq:proballdata]
\prob{D,\rho}
=
\prod_{\alpha,\gamma}
p_{\alpha\gamma}(\rho)^{n_\alpha\gamma}
\]
The data is thus captured in the outcome counts $\seq{n_{\alpha\gamma}}$
whereas the model terms have a $\rho$-dependence. The function
$\prob{D,\rho}$ is called the {\em likelihood} function and since it
is positive, the {\em maximum likelihood estimate} (MLE) of $\rho$ is
obtained by finding a $\rho$ in the set \refeq{rhoset} which maximizes
the {\em log-likelihood function}, or equivalently, minimizes the {\em
negative log-likelihood function},
\[[eq:loglike1]
\begin{array}{rcl}
L(D,\rho)
&=&
-\log\ \prob{D,\rho}
\\
&=&
\displaystyle
-\sum_{\alpha,\gamma}
n_{\alpha\gamma}
\log\ p_{\alpha\gamma}(\rho)
\\
&=&
\displaystyle
-\sum_{\alpha,\gamma}
n_{\alpha\gamma}
\log{\rm Tr}\ O_{\alpha\gamma}\rho
\end{array}
\]
These expressions are obtained by combining \refeq{proballdata},
\refeq{loglike1} and \refeq{pout osr}. The Maximum Likelihood state
estimate, $\rho^{\rm ML}$, is obtained as the solution to the optimization
problem:
\[[eq:maxlike1]
\begin{array}{ll}
\mbox{minimize}
&
L(D,\rho)
=
-\sum_{\alpha,\gamma}
n_{\alpha\gamma}
\log{\rm Tr}\ O_{\alpha\gamma}\rho
\\
\mbox{subject to}
&
\rho\geq 0,\; {\rm Tr}\ \rho=1
\end{array}
\]
$L(D,\rho)$ is a positively weighted sum of log-convex functions of
$\rho$, and hence, is a log-convex function of $\rho$. The constraint
that $\rho$ is a density matrix forms a convex set in $\rho$. Hence,
\refeq{maxlike1} is in a category of a class of well studied
log-convex optimization problems, \emph{e.g.}, \cite{BoydV:04}.
\subsubsection*{Pure state estimation}
\label{eq:pure est}
Suppose it is known the the true state is pure, that is, $\rho^{\rm true} =
\psi_{\rm true} \psi_{\rm true}^*$ with $\psi_{\rm true}\in{\mathbf C}^n$ and
$\psi_{\rm true}^*\psi_{\rm true}=1$. In practice we have found that solving
\refeq{maxlike1} when the true state is pure gives solutions which are
easily approximated by pure states, that is, the estimated state has
one singular value near one and all the rest are very small and
positive.
To deal directly with pure state estimation we first need to
characterize the set of all density matrices which are pure. This is
given by the set $\set{\rho\in{\mathbf C}^{n\times n}} {\rho \geq 0,\; {\bf rank}\ \rho
=1}$, which is equivalent to,
\[[eq:rho pure]
\set{\rho\in{\mathbf C}^{n\times n}}
{\rho \geq 0,\; {\rm Tr}\ \rho =1,\; {\rm Tr}\ \rho^2=1}
\]
The corresponding ML estimate is then the solution of,
\[[eq:maxlike pure]
\begin{array}{ll}
\mbox{minimize}
&
L(\rho)=
-\sum_{\alpha,\gamma}
n_{\alpha\gamma}
\log{\rm Tr}\ O_{\alpha\gamma}\rho
\\
\mbox{subject to}
&
\rho\geq 0,\; {\rm Tr}\ \rho=1,\; {\rm Tr}\ \rho^2=1
\end{array}
\]
This is not a convex optimization problem because the equality
constraint, ${\rm Tr}\ \rho^2=1$, is not convex. However, relaxing this
constraint to the convex {\em inequality} constraint, ${\rm Tr}\ \rho^2
\leq 1$, results in the convex optimization problem:
\[[eq:maxlike pure rlx]
\begin{array}{ll}
\mbox{minimize}
&
L(\rho)=
-\sum_{\alpha,\gamma}
n_{\alpha\gamma}
\log{\rm Tr}\ O_{\alpha\gamma}\rho
\\
\mbox{subject to}
&
\rho\geq 0,\; {\rm Tr}\ \rho=1,\; {\rm Tr}\ \rho^2 \leq 1
\end{array}
\]
If the solution is on the boundary of the set ${\rm Tr}\ \rho^2 \leq 1$,
then a pure state has been found. There is however, no guaranty that
this will occur.
\subsubsection*{Least-squares (LS) state estimation}
\label{sec:lsapp}
In a typical application the number of trials per configuration,
$\ell_\gamma$, is sufficiently large so that the {\em empirical
estimate} of the outcome probability,
\[[eq:pemp]
p^{\rm emp}_{\alpha\gamma} = \frac{n_{\alpha\gamma}}{\ell_\gamma}
\]
is a good estimate of the true outcome probability
$p^{\rm true}_{\alpha\gamma}$. The empirical probability estimate also
provides the smallest possible value of the negative log-likelihood
function, that is, $p^{\rm emp}_{\alpha\gamma}$ is the solution to,
\[[eq:pemp opt]
\begin{array}{ll}
\mbox{minimize}
&
L(p)=
-\sum_{\alpha,\gamma}\
n_{\alpha\gamma} \log p_{\alpha\gamma}
\\
\mbox{subject to}
&
{\sum_{\alpha=1}^m} \ p_{\alpha\gamma}=1,\forall \gamma,
\;
p_{\alpha\gamma}\geq 0,\forall\alpha,\gamma
\end{array}
\]
with optimization variables $p_{\alpha\gamma},\forall\alpha,\gamma$. Thus,
for any value of $\rho$ we have the lower bound,
\[[eq:pemp lb]
-\sum_{\alpha,\gamma}\
n_{\alpha\gamma} \log \frac{n_{\alpha\gamma}}{\ell_\gamma}
\leq
-\sum_{\alpha,\gamma}\
n_{\alpha\gamma} \log {\rm Tr}\ O_{\alpha\gamma} \rho
\]
In particular, assuming \refeq{av nalfgam} holds, and the $\ell_\gamma$
trials are independent, then the variance of the empirical estimate is
known to be \cite{Papoulis:65},
\[[eq:var pemp]
{\bf var}\ p^{\rm emp}_{\alpha\gamma}
=
\frac{1}{\ell_\gamma}
p^{\rm true}_{\alpha\gamma}
\left( 1-p^{\rm true}_{\alpha\gamma} \right)
\]
It therefore follows that for large $\ell_\gamma$, $p^{\rm emp}_{\alpha\gamma}
\approx p^{\rm true}_{\alpha\gamma}$, and if as assumed \refeq{s in m}, the
system is in the model set, then $p^{\rm true}_{\alpha\gamma} = {\rm Tr}\
O_{\alpha\gamma} \rho^{\rm true}$. These two conditions lead to taking the state
estimate as the solution to the constrained weighted least-squares
problem:
\[[eq:lsopt cvx]
\begin{array}{ll}
\mbox{minimize}
&
\sum_{\alpha,\gamma}\
w_\gamma
\left( p^{\rm emp}_{\alpha\gamma} - {\rm Tr}\ O_{\alpha\gamma} \rho \right)^2
\\
\mbox{subject to}
&
\rho\geq 0,\; {\rm Tr}\ \rho=1
\end{array}
\]
The weights, $w_\gamma$, are chosen by the user to emphasize different
configurations. A typical choice is the distribution of experiments
per configuration, hence, $w_\gamma\geq 0,\ \sum_\gamma\
w_\gamma=1$. Because of the semi-definite constraint, this
weighted-least-squares problem is a convex optimization in the
variable $\rho$. For large $\ell_\gamma$, the solution ought to be a
good estimate of the true state. There is, however, little numerical
benefit in solving \refeq{lsopt cvx} as compared to \refeq{maxlike1}
-- they are both convex optimization problems and the numerical
complexity is similar provided \refeq{maxlike1} is solved using an
interior-point method \cite{BoydV:04}. Some advantage is obtained by
dropping the semidefinite constraint $\rho\geq 0$ in \refeq{lsopt cvx}
resulting in,
\[[eq:lsopt]
\begin{array}{ll}
\mbox{minimize}
&
\sum_{\alpha,\gamma}\
w_\gamma
\left( p^{\rm emp}_{\alpha\gamma} - {\rm Tr}\ O_{\alpha\gamma} \rho \right)^2
\\
\mbox{subject to}
&
{\rm Tr}\ \rho=1
\end{array}
\]
This is a standard least-squares problem with a linear equality
constraint which can be solved very efficiently using a singular value
decomposition to eliminate the equality constraint
\cite{GolubV:83}. For sufficiently large $\ell_\gamma$ the resulting
estimate {\em may} satisfy the positivity constraint $\rho\geq 0$. If
not, it is usually the case that some of the small eigenvalues of the
state estimate or estimated outcome probabilities are slightly
negative which can be manually set to zero. Solving \refeq{lsopt} {\em
is} numerically faster than solving \refeq{maxlike1}, but not by
much. Even with a large amount of data the solution to \refeq{lsopt}
can produce estimates which are not positive if the data is not
sufficiently rich. In this case the estimates from any procedure which
eliminates the positivity constraint can be very misleading.
It thus appears that even for large $\ell_\gamma$, there is no
significant benefit accrued, either because of numerical precision or
speed, to using the empirical estimate followed by standard
least-squares. If, however, the $\ell_\gamma$ are not sufficiently large
and/or the data is not sufficiently rich, then it is unlikely that the
estimate from \refeq{lsopt} will be accurate.
One possible advantage does come about because the solution to
\refeq{lsopt} can be expressed analytically, and thus it is possible
to gain an understanding of how to select the POVM. For example, in
\cite{NielsenC:00} special POVM elements are selected to essentially
diagonalize the problem, thereby making the least-squares problem
\refeq{lsopt} simpler, \emph{i.e.}, the elements of the density matrix can be
estimated one at a time. However, implementing the requisite POVM set
may be very difficult depending on the physical apparatus involved.
\subsection{Experiment design for state estimation}
\label{sec:expt des}
In this section we describe the experiment design problem for quantum
state estimation. The objective is to select the number of experiments
per configuration, the elements of the vector $\ell=[\ell_1 \cdots
\ell_{n_{\rm cfg}}]^T\in{\mathbf R}^{n_{\rm cfg}}$, so as to minimize the error between the
state estimate, $\hat{\rho}(\ell)$, and the true state
$\rho^{\rm true}$. Specifically, we would like to solve for $\ell$ from:
\[[eq:expdes 0]
\begin{array}{ll}
\mbox{minimize}
&
{\bf E} \ \norm{\hat{\rho}(\ell)-\rho^{\rm true}}_{\rm frob}^2
\\
\mbox{subject to}
&
\sum_{\gamma=1}^{{n_{\rm cfg}}}\ \ell_\gamma = {\ell_{{\rm expt}}}
\\
&
\mbox{integer}\ \ell_\gamma \geq 0,\;
\gamma=1,\ldots,{n_{\rm cfg}}
\end{array}
\]
where ${\ell_{{\rm expt}}}$ is the desired number of total experiments. This is a
difficult, if not insoluble problem for several reasons. First, the
solution depends on the estimation method which produces
$\hat{\rho}(\ell)$. Secondly, the problem is integer combinatorial because
$\ell$ is a vector of integers. And finally, the solution depends on
$\rho^{\rm true}$, the very state to be estimated. Fortunately all these
issues can be circumvented.
We first eliminate the dependence on the estimation method. The
following result can be established using the {\em Cram\'{e}r-Rao\ Inequality}
\cite{Cramer:46}. The derivation is in Appendix \refsec{var rho}.
\begin{quote}
{\bf State estimation variance lower bound} \footnote{
Cram\'{e}r-Rao\ bounds previously reported in the literature are not quite
correct as they do not include the linear constraint ${\rm Tr}\ \rho=1$
as is done here.
}
\\
Suppose the system generating the data is in the model set used for
estimation, \emph{i.e.}, \refeq{s in m} holds. For $\ell=[\ell_1 \cdots
\ell_{{n_{\rm cfg}}}]$ experiments per configuration, suppose $\hat{\rho}(\ell)$
is a density matrix {\em and} an unbiased estimate of $\rho^{\rm true}$, \emph{i.e.},
$\hat{\rho}(\ell) \geq 0$, ${\rm Tr}\ \hat{\rho}(\ell)=1$, and $ {\bf E} \ \hat{\rho}(\ell)
= \rho^{\rm true}$. Under these conditions, the estimation error variance
satisfies,
\[[eq:var rho]
{\bf E} \ \norm{\hat{\rho}(\ell)-\rho^{\rm true}}_{\rm frob}^2
\geq
V(\ell,\rho^{\rm true})
=
{\rm Tr}\ G(\ell,\rho^{\rm true})^{-1}
\]
where\footnote{
The $\vec$ operation takes the rows of a matrix and stacks them one
row at a time on top of each other. Two useful expressions are
$\vec(AXB)=(B^T\otimes A)\vec\ X$ and ${\rm Tr}\ AX=(\vec\ A^T)^T \vec\
X$.
}
\[[eq:var rho 1]
\begin{array}{rcl}
\displaystyle
G(\ell,\rho^{\rm true})
&=&
\displaystyle
\sum_{gam=1}^{n_{\rm cfg}}\ \ell_\gamma G_\gamma(\rho^{\rm true})
\in{\mathbf R}^{n^2-1 \times n^2-1}
\\&&\\
\displaystyle
G_\gamma(\rho^{\rm true})
&=&
\displaystyle
C_{\rm eq}^T
\left(
{\sum_{\alpha=1}^m} \
\frac{a_{\alpha\gamma} a_{\alpha\gamma}^*}{p_{\alpha\gamma}(\rho^{\rm true})}
\right)
C_{\rm eq}
\in{\mathbf R}^{n^2 \times n^2}
\\&&\\
a_{\alpha\gamma} &=& \vec\ O_{\alpha\gamma} \in {\mathbf C}^{n^2}
\end{array}
\]
and $C_{\rm eq}\in{\mathbf R}^{n^2 \times n^2-1}$ is part of the unitary matrix in
the singular value decomposition:
$ \vec\ I_n = USW^T\in{\mathbf R}^{n^2},\ W=[c\ C_{\rm eq}]\in{\mathbf R}^{n^2\times n^2}$.
\end{quote}
This theorem states the for any unbiased estimate of $\rho^{\rm true}$, the
variance of the estimate satisfies the inequality \refeq{var rho}. The
power of the result is that it is independent of {\em how} the
estimate is obtained, \emph{i.e.}, no estimation algorithm which produces an
unbiased estimate can have an estimation error variance smaller than
that in \refeq{var rho}. There is a generalization for biased
estimators but we will not pursue that here.
In general it is difficult to determine if any estimate will achieve
the lower bound. However, under the conditions stated in the above
result, the ML estimate, $\rho^{\rm ML}(\ell)$, the solution to
\refeq{maxlike1}, approaches $\rho^{\rm true}$ with probability one,
asymptotically as ${\ell_{{\rm expt}}}$ increases, and the asymptotic distribution
becomes Gaussian with covariance given by the Cram\'{e}r-Rao\ bound (see
\refsec{crao} for the covariance expression and \cite{Ljung:87} for a
derivation).
The one qualifier to the Cram\'{e}r-Rao\ bound as presented is that the
indicated inverse exists. This condition, however, is necessary and
sufficient to insure that the state is identifiable. More precisely,
the state is {\em identifiable} if and only if,
\[[eq:identifiable]
G(\ell=1_{n_{\rm cfg}},\ \rho^{\rm true})
=
C_{\rm eq}^*
\left(
\sum_{\gamma=1}^{{n_{\rm cfg}}} {\sum_{\alpha=1}^m} \
\frac{a_{\alpha\gamma} a_{\alpha\gamma}^*}{p_{\alpha\gamma}(\rho^{\rm true})}
\right)
C_{\rm eq}
\;\;\;
\mbox{is invertible}
\]
Under the condition of identifiability, the experiment design problem
can be expressed by the following optimization problem in the vector
of integers $\ell$:
\[[eq:expdes]
\begin{array}{ll}
\mbox{minimize}
&
V(\ell,\rho^{\rm true}) = {\rm Tr}\ G(\ell,\rho^{\rm true})^{-1}
\\
\mbox{subject to}
&
\sum_{\gamma=1}^{{n_{\rm cfg}}}\ \ell_\gamma = {\ell_{{\rm expt}}}
\\
&
\mbox{integer}\ \ell_\gamma \geq 0,\;
\gamma=1,\ldots,{n_{\rm cfg}}
\end{array}
\]
where ${\ell_{{\rm expt}}}$ is the desired number of total experiments. The good
news is that the objective, $V(\ell,\rho^{\rm true})$, is convex in $\ell$
\cite[\S 7.5]{BoydV:04}. Unfortunately, there are still two
impediments: (i) restricting $\ell$ to a vector of integers makes the
problem combinatorial; (ii) the lower-bound function
$V(\ell,\rho^{\rm true})$ depends on the true value, $\rho^{\rm true}$. These
difficulties can be alleviated to some extent. For (i) we can use the
convex relaxation described in \cite[\S 7,5]{BoydV:04}. For (ii) we
can solve the relaxed experiment design problem with either a set of
``what-if'' estimates as surrogates for $\rho^{\rm true}$, or use nominal
values to start and then ``bootstrap'' to more precise values by
iterating between state estimation and experiment design. We now
explain how to perform these steps.
\subsubsection*{Relaxed experiment design for state estimation}
Following the procedure in \cite[\S 7.5]{BoydV:04}, introduce the
variables $\lambda_\gamma=\ell_\gamma/{\ell_{{\rm expt}}}$, each of which is the fraction of
the total number of experiments performed in configuration
$\gamma$. Since all the $\ell_\gamma$ and ${\ell_{{\rm expt}}}$ are non-negative
integers, each $\lambda_\gamma$ is non-negative and {\em rational},
specifically an integer multiple of $1/{\ell_{{\rm expt}}}$, and in addition,
$\sum_\gamma \lambda_\gamma=1$. Let $\rho^\surr$ denote a surrogate for
$\rho^{\rm true}$, \emph{e.g.}, an estimate or candidate value of $\rho^{\rm true}$. Using
\refeq{var rho}-\refeq{var rho 1} gives,
\[[eq:nexscale]
V(\ell={\ell_{{\rm expt}}}\lambda,\rho^\surr) = \frac{1}{{\ell_{{\rm expt}}}}V(\lambda,\rho^\surr)
\]
Using \refeq{var rho 1},
\[[eq:vlamrho]
\begin{array}{rcl}
V(\lambda,\rho^\surr)
&=&
{\rm Tr}\
G(\lambda,\rho^\surr)^{-1}
\\
\displaystyle
G(\lambda,\rho^\surr)
&=&
\displaystyle
\sum_{\gamma=1}^{{n_{\rm cfg}}}\ \lambda_\gamma G_\gamma(\rho^\surr)
\end{array}
\]
Hence, the objective function $V(\ell,\rho^\surr)$ can be replaced with
$V(\lambda,\rho^\surr)$ and the experiment design problem \refeq{expdes} is
equivalent to.
\[[eq:expdes rat]
\begin{array}{ll}
\mbox{minimize}
&
V(\lambda,\rho^\surr) = {\rm Tr}\ G(\lambda,\rho^\surr)^{-1}
\\
\mbox{subject to}
&
\sum_{\gamma=1}^{{n_{\rm cfg}}}\ \lambda_\gamma = 1
\\
&
\lambda_\gamma \geq 0,\;
\mbox{integer multiple of $1/{\ell_{{\rm expt}}}$},\;
\gamma=1,\ldots,{n_{\rm cfg}}
\end{array}
\]
The objective is now a convex function of the $\lambda_\gamma$, but it is
still a combinatorial problem because the $\lambda_\gamma$ are constrained
to each be an integer multiple of $1/{\ell_{{\rm expt}}}$. If $\lambda_\gamma$ is only
otherwise constrained to the non-negative reals, then this has the
effect of relaxing the constraint that the $\ell_\gamma$ are integers.
As phrased in \cite{BoydV:04}, the {\em relaxed} experiment design
problem is:
\[[eq:rel expdes]
\begin{array}{ll}
\mbox{minimize}
&
V(\lambda,\rho^\surr)
=
{\rm Tr}\left(
\sum_{\gamma=1}^{{n_{\rm cfg}}}\ \lambda_\gamma G_\gamma(\rho^\surr)
\right)^{-1}
\\
\mbox{subject to}
&
\sum_{\gamma=1}^{{n_{\rm cfg}}}\ \lambda_\gamma = 1
\\
&
\lambda_\gamma\geq 0,\;
\gamma=1,\ldots,{n_{\rm cfg}}
\end{array}
\]
The objective is convex, the equality constraint is linear, and the
inequality constrains are convex, hence, this is a convex optimization
problem in $\lambda\in{\mathbf R}^{n_{\rm cfg}}$. Let $\lambda^{\rm opt}$ denote the optimal
solution to \refeq{rel expdes}. Since the problem no longer depends on
${\ell_{{\rm expt}}}$, $\lambda^{\rm opt}$ can be viewed as a distribution of experiments per
configuration.\footnote{
{\bf Caveat emptor}: The relaxed optimal experiment design
distribution, $\lambda^{\rm opt}$, is optimal with respect to the initial state
$\rho^\surr$, a surrogate for $\rho^{\rm true}$. Thus, $\lambda^{\rm opt}$ is {\em not}
optimal with respect to $\rho^{\rm true}$. This should be no surprise because
the underlying goal is to find a good estimate of $\rho^{\rm true}$.
}
Clearly there is no guaranty that ${\ell_{{\rm expt}}}\lambda^{\rm opt}$ is a vector of integer
multiples of $1/{\ell_{{\rm expt}}}$. A practical choice for obtaining a vector of
integer multiples of $1/{\ell_{{\rm expt}}}$ is,
\[[eq:ellrnd]
\ell^{\rm round}_{\rm expt} = \round{{\ell_{{\rm expt}}} \lambda^{\rm opt}}
\]
If $\ell^{\rm opt}$ is the (unknown) integer vector solution to \refeq{expdes},
then we have the relations:
\[[eq:bnds]
V(\ell^{\rm round}_{\rm expt},\rho^\surr)
\geq
V(\ell^{\rm opt},\rho^\surr)
\geq
V({\ell_{{\rm expt}}}\lambda^{\rm opt},\rho^\surr)
\]
The optimal objective is thus bounded above and below by known values
obtained from the relaxed optimization. The gap within which falls
the optimal solution can be no worse than the difference between
$V(\ell^{\rm round}_{\rm expt},\rho^\surr)$ and $V({\ell_{{\rm expt}}}\lambda^{\rm opt},\rho^\surr)$, which can be computed
solely from $\lambda^{\rm opt}$. If the gap is sufficiently small then for all
practical purposes the ``optimal'' solution is $\lambda^{\rm opt}$. From now on
we will refer to $\lambda^{\rm opt}$ as the optimal solution rather than the
relaxed optimal.
\subsubsection*{Performance tradeoff}
The optimal distribution $\lambda^{\rm opt}$ can be used to guide the
elimination of small values of $\lambda^{\rm opt}$. For example, consider the
{\em suboptimal} distribution, $\lambda^{\rm sub}$, obtained by selecting the
largest $n_{\rm sub}$ out of ${n_{\rm cfg}}$ non-zero values of $\lambda^{\rm opt}$. Let
$\ell^{\rm sub}_{\rm expt}$ denote the integer vector of configurations,
\[[eq:nexsub]
\ell^{\rm sub}_{\rm expt}=\round{{\ell_{{\rm expt}}}\lambda^{\rm sub}}
\]
Using \refeq{nexscale}, the minimum number of experiments so that
$V(\ell^{\rm sub}_{\rm expt},\rho^\surr)\leq V_0$ is given by,
\[[eq:nexmin]
\ell^{\rm sub}_{\rm expt} =\round{V(\lambda^{\rm sub},\rho^\surr)/V_0}
\]
As $n_{\rm sub}$ is varied, the graph $\{n_{\rm sub},\ \ell^{\rm sub}_{\rm expt}\}$ establishes a
tradeoff between the number of configurations per experiment versus
the total number of experiments such that the lower bound on the
estimation variance does not exceed the desired value $V_0$. When
$n_{\rm sub}={n_{\rm cfg}}$, \refeq{nexmin} is identical with \refeq{ellrnd}.
The condition number of the matrix $G(\lambda^{\rm opt},\rho^\surr)$ gives an
indication of the identifiability of the density matrix $\rho^\surr$. A
very large condition number means that the linear combination of
elements of the density matrix associated with the small eigenvalue
will be more difficult to obtain then those combinations associated
with a large eigenvalue. The condition number of $G(\lambda^{\rm opt},\rho^\surr)$ is
not only affected by the number of experiments per configuration,
$\lambda^{\rm opt}$, but by the configurations themselves. Examining
$G(\lambda^{\rm opt},\rho^\surr)$ for different $\rho^\surr$ (surrogates of $\rho^{\rm true}$)
and different configurations can help establish a good experiment
design.
\subsubsection*{Bootstrapping}
A standard approach used to circumvent not knowing the true state
needed to optimize the experiment design is to proceed adaptively, or
by ``bootstrapping.'' The idea is to use the current estimate of the
initial state found from \refeq{maxlike1}, then solve \refeq{rel
expdes}, and then repeat. The algorithm at the $k$-th iteration looks
like this:
\[
\begin{array}{rcl}
\hat{\rho}(k)
&=&
\displaystyle
\arg\min_{\rho}
V(\hat{\ell}(k-1),\rho)
\\&&\\
\lambda^{\rm opt}(k)
&=&
\displaystyle
\arg\min_{\lambda}
V(\lambda,\rho=\hat{\rho}(k))
\\&&\\
\hat{\ell}(k)
&=&
\displaystyle
\round{{\ell_{{\rm expt}}} \lambda^{\rm opt}(k)}
\end{array}
\]
The initial distribution $\hat{\ell}(0)$ could be chosen as uniform, \emph{e.g.},
the same for a not too large number of configurations. The algorithm
could also start by first solving for a distribution from an initial
state surrogate. In each iteration we could also vary ${\ell_{{\rm expt}}}$. Although
each optimization is convex, the joint problem may not be. Conditions
for convergence would need to be investigated as well as establishing
that this method is efficient, \emph{i.e.}, reduces the number of trials. We
will not pursue this any further here.
\subsubsection*{Dual experiment design problem}
\label{sec:dual}
{\em Lagrange Duality Theory} can provide a lower bound on the
objective function in an optimization problem as well as establishing
optimality conditions often leading to insights into the optimal
solution structure \cite[Ch.5]{BoydV:04}. In many cases the largest
lower bound -- the solution of the {\em dual problem} -- is equal to
the optimal objective function. The dual problem associated with the
experiment design problem \refeq{rel expdes} is,
\[[eq:dual expdes]
\begin{array}{ll}
\mbox{maximize}
&
\left( {\rm Tr}\ W^{1/2} \right)^2
\\
\mbox{subject to}
&
{\rm Tr}\ W G_\gamma(\rho^\surr) \leq 1,
\;
\gamma=1,\ldots,{n_{\rm cfg}}
\\
&
W > 0
\end{array}
\]
The optimization variable is $W\in{\mathbf C}^{n^2-1 \times n^2-1}$. The
above form of the dual is given in \cite[\S 7.5.2]{BoydV:04} for a
slightly simpler problem (the $G_\gamma$ are dyads) but is essentially
the same. A key observation arises from the {\em complementary
slackness condition},
\[[eq:cs cond]
\lambda^{\rm opt}_\gamma
\left(
{\rm Tr}\ W^{\rm opt} G_\gamma(\rho^\surr) - 1
\right)
=0,
\;\;
\gamma=1,\ldots,{n_{\rm cfg}}
\]
where $\lambda^{\rm opt}$ is the solution to the {\em primal problem},
\refeq{rel expdes}, and $W^{\rm opt}$ is the solution to the dual problem,
\refeq{dual expdes}. Thus, only when the equality constraint holds,
${\rm Tr}\ W^{\rm opt} G_\gamma(\rho^\surr)=1$, is the associated $\lambda^{\rm opt}_\gamma$ not
necessarily equal to zero. It will therefore be usually the case that
many of the elements of the optimal distribution will be zero.
Strong duality also holds for this problem, thus the optimal primal
and dual objective values are equal,
\[[eq:strdual]
{\rm Tr}\left(
\sum_{\gamma=1}^{{n_{\rm cfg}}}\ \lambda^{\rm opt}_\gamma G_\gamma(\rho^\surr)
\right)^{-1}
=
\left( {\rm Tr}\ (W^{\rm opt})^{1/2} \right)^2
\]
For this problem, a pair $(\lambda,\ W)$ is optimal with respect to
$\rho^\surr$ if and only if:
\[[eq:oc expdes]
\begin{array}{rcl}
\sum_{\gamma=1}^{{n_{\rm cfg}}}\ \lambda_\gamma &=& 1
\\
\lambda_\gamma &\geq& 0,\ \forall\gamma
\\
\lambda^{\rm opt}_\gamma
\left(
{\rm Tr}\ W^{\rm opt} G_\gamma(\rho^\surr) - 1
\right)
&=& 0,\ \forall\gamma
\\
{\rm Tr}\ W G_\gamma(\rho^\surr) &\leq& 1,\ \forall\gamma
\\
{\rm Tr}\left(
\sum_{\gamma=1}^{{n_{\rm cfg}}}\ \lambda_\gamma G_\gamma(\rho^\surr)
\right)^{-1}
&=&
\left( {\rm Tr}\ (W^{\rm opt})^{1/2} \right)^2
\end{array}
\]
\subsection{Example: experiment design for state estimation}
\label{sec:example qs}
A schematic of an apparatus for state tomography of a pair of
entangled photons specified by the quantum state (density matrix)
$\rho$ is shown in figure \reffig{tomag01}.
\begin{figure}[h]
\centering
\epsfxsize=4in
\epsfboxit{tomog_qh02.ps}
\caption{Detection apparatus for two-photon tomography}
\label{fig:tomag01}
\end{figure}
The set up has four photon-counting detectors, A, B, C, D. There are
four continuous variable settings for the quarter-wave plates and
half-wave plates, \emph{i.e.}, $q,h,q',h'$. For any settings of these
parameters one of the detectors in each arm will register a
photon. The objective is to determine the optimal settings of these
parameters and the number of experiments per setting for estimation of
the state $\rho$ of the pair using as data the photon counts from the
four detectors.
Because the photon sources are not completely efficient, the input
quantum state actually consists of either two or zero photons. The
detectors register a 0 or 1 depending on whether a photon is incident
on them or not. The basis states for the upper arm are therefore:
$\ket{0}_e\ket{0}_f,\ \ket{0}_e\ket{1}_f,\ \ket{1}_e\ket{0}_f$. There
is a similar set for the lower arm (modes $g, h$).
The firing patterns for an arbitrary setting of the wave plates,
assuming perfect detection efficiency and no dark counts are given in
the table:
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
A & B & C & D \\
\hline
0 & 1 & 0 & 1 \\
0 & 1 & 1 & 0 \\
1 & 0 & 0 & 1 \\
1 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 \\
\hline
\end{tabular}
\end{center}
The probabilities for these patterns are given by
\[
p_{ijk\ell}
=
{\rm Tr}\ (M_{AB}^{ij} \otimes M_{CD}^{k\ell}) \rho
\]
where $\{i,j,k,\ell\}\in\{0,1\}$, and $M_{AB}^{ij}$ is the projector
for detector A to register count $i$ and simultaneously detector B to
register count $j$. Similarly, $M_{CD}^{k\ell}$ is the projector for
detector C to register count $j$ and simultaneously detector D to
register count $\ell$. The projectors for A and B in the above basis
are:
\[[eq:projmat]
\begin{array}{rcl}
\displaystyle
M_{AB}^{00}
&=&
\left[
\begin{array}{ccc}
1 & 0 & 0\\
0 & 0 & 0\\
0 & 0 & 0
\end{array}
\right]
\\&&\\
\displaystyle
M_{AB}^{10}
&=&
\displaystyle
\left[
\begin{array}{c}
0
\\
\psi_1(h,q)
\end{array}
\right]
\left[
0\ \psi_1(h,q)^*
\right],\
\psi_1(h,q)
=
\frac{1}{\sqrt{2}}
\left[
\begin{array}{c}
\sin 2h+i\sin 2(h-q)
\\
\cos 2h-i\cos 2(h-q)
\end{array}
\right]
\\&&\\
\displaystyle
M_{AB}^{01}
&=&
\displaystyle
\left[
\begin{array}{c}
0
\\
\psi_2(h,q)
\end{array}
\right]
\left[
0\ \psi_2(h,q)^*
\right],\
\psi_2(h,q)
=
\frac{1}{\sqrt{2}}
\left[
\begin{array}{c}
\cos 2h+i\cos 2(h-q)
\\
-\sin 2h+i\sin 2(h-q)
\end{array}
\right]
\end{array}
\]
A similar set of projectors can be written for C and D with the
variables $h,q$ replaced by their primed counterparts
$h',q'$.
The protocol is to measure the probabilities for enough settings of
the variables that the elements of the two-photon density operator can
be estimated. The two-photon density operator is the direct product of
the one-photon density operator, for which the set of 3 states given
above forms a basis. The basis states of the two-photon $(9\times 9)$
density operator, $\rho$, are:
$\ket{ijk\ell}=\ket{i}_e\ket{j}_f\ket{k}_g\ket{\ell}_h$
with $i,j,k,\ell \in \{0,1\}$
Hence,
\[
\begin{array}{ll}
\mbox{zero photon} & \ket{0000} \\
\mbox{one photon} & \ket{0100},\ \ket{1000},\ \ket{0001},\ \ket{0010} \\
\mbox{two photon} & \ket{0101},\ \ket{0110},\ \ket{1001},\ \ket{1010}
\end{array}
\]
\subsubsection*{simulation results: one-arm}
Consider only one arm of the apparatus in figure \reffig{tomag01}, say
the upper arm with detectors (A,B). Suppose the wave plate settings
are,
\[
\set{h_\gamma,\ q_\gamma}{\gamma=1,\ldots,{n_{\rm cfg}}}
\]
Assume also that the incoming state {\em always} is one photon, never
none. Hence, $\rho\in{\mathbf C}^{2\times 2}$ and the projectors are:
\[[eq:povm hqgam]
\begin{array}{rcl}
\displaystyle
M_\gamma^{10}
&=&
\psi_1(h_\gamma,q_\gamma)
\psi_1(h_\gamma,q_\gamma)^*
\\&&\\
\displaystyle
M_\gamma^{01}
&=&
\psi_2(h_\gamma,q_\gamma)
\psi_2(h_\gamma,q_\gamma)^*
\end{array}
\]
with $\psi_1,\psi_2$ from \refeq{projmat}. Assuming each detector has
efficiency $\eta,\ 0\leq\eta\leq 1$ and a non-zero dark count
probability, $\delta,\ 0\leq\delta\leq 1$, then there are four possible
outcomes at detectors A,B given in the following table:
\begin{center}
\begin{tabular}{|c||c|c|}
\hline
$\alpha$ & A & B \\
\hline
10 & 1 & 0 \\
01 & 0 & 1 \\
00 & 0 & 0 \\
11 & 1 & 1 \\
\hline
\end{tabular}
\end{center}
Following \cite{GriceW:96,WalW:98} the probability of a dark count is
denoted by the conditional probability,
\[
\nu_{1|0}=\delta
\]
where $1|0$ means the detector has fired ``1'' given that no photon is present
at the detector ``0.'' As shown in \cite{GriceW:96}, it therefore follows
that the probability that the detector does not fire ``0'' although a
photon is present at the detector ``1'' is given by,
\[
\nu_{0|1}=(1-\eta)(1-\delta)
\]
Here $1-\eta$ is the probability of no detection and $1-\delta$ is the
probability of no dark count. The remaining conditional probabilities
are, by definition, constrained to obey:
\[
\begin{array}{rcl}
\nu_{1|0} + \nu_{0|0} &=& 1
\\
\nu_{1|1} + \nu_{0|1} &=& 1
\end{array}
\]
The probabilities for the firing patterns in the above table are thus
given by \refeq{palfgam} with the following observables $M_{\alpha\gamma}$:
\[
\begin{array}{rcl}
\displaystyle
M_{10,\gamma}
&=&
\displaystyle
\nu_{1|1}\nu_{0|0} M_\gamma^{10}+\nu_{1|0} \nu_{0|1} M_\gamma^{01}
\\&&\\
\displaystyle
M_{01,\gamma}
&=&
\displaystyle
\nu_{0|1} \nu_{1|0} M_\gamma^{10}+\nu_{0|0} \nu_{1|1} M_\gamma^{01}
\\&&\\
\displaystyle
M_{00,\gamma}
&=&
\displaystyle
\nu_{0|1} \nu_{0|0} M_\gamma^{10}+\nu_{0|0} \nu_{0|1} M_\gamma^{01}
\\&&\\
\displaystyle
M_{11,\gamma}
&=&
\displaystyle
\nu_{1|1} \nu_{1|0} M_\gamma^{10}+\nu_{1|0} \nu_{1|1} M_\gamma^{01}
\end{array}
\]
Numerical computer simulations were performed for two input state cases:
\[[eq:rhot]
\begin{array}{ll}
\mbox{\bf pure state:}
&
\displaystyle
\rho_{\rm pure}
=
\frac{1}{2}
\left[
\begin{array}{cc}
1 & 1\\
1 & 1
\end{array}
\right]
=
\psi_0 \psi_\ast,\
\psi_0
=
\frac{1}{\sqrt{2}}
\left[\begin{array}{c} 1 \\ 1 \end{array} \right]
\\&\\
\mbox{\bf mixed state:}
&
\displaystyle
\rho_{\rm mixd}
=
\left[
\begin{array}{cc}
0.6 & -0.2i\\
0.2i & 0.4
\end{array}
\right]
\end{array}
\]
For each input state case we computed $\hat{\lam}$ with and without ``noise:''
\[
\begin{array}{rl}
\mbox{no noise}
&
\left\{
\begin{array}{ll}
\mbox{detector efficiency} & \eta=1
\\
\mbox{dark count probability} & \delta=0
\end{array}
\right.
\\&\\
\mbox{yes noise}
&
\left\{
\begin{array}{ll}
\mbox{detector efficiency} & \eta=0.75
\\
\mbox{dark count probability} & \delta=0.05
\end{array}
\right.
\end{array}
\]
For all cases and noise conditions we used the wave plate settings:
\[[eq:hq1arm]
\begin{array}{l}
h_i=(i-1)(5^\circ),\ i=1,\ldots,10
\\
q_i=(i-1)(5^\circ),\ i=1,\ldots,10
\end{array}
\]
Both angles are set from $0$ to $45^\circ$ in $5^\circ$
increments. This yields a total of ${n_{\rm cfg}}=10^2=100$ configurations
corresponding to all the wave plate combinations.
Figure \reffig{lamopt1arm} shows the optimal distributions $\lambda^{\rm opt}$
versus configurations $\gamma=1,\ldots,100$ for all four test cases: two
input states with and without noise. Observe that the optimal
distributions are {\em not} uniform but are concentrated near
the same particular wave plate settings. These settings are very close
to those established in \cite{JamesETAL:01}.
To check the gap between the relaxed optimum $\hat{\lam}$ and the unknown
integer optimum we appeal to \refeq{ellrnd}-\refeq{bnds}. The
following table shows that these distributions are good approximation
to the unknown optimal integer solution for even not so large ${\ell_{{\rm expt}}}$
for the two state cases with no noise. Similar results were obtained
for the noisy case.
\[
\begin{array}{|c||c|c|}
\hline
&&\\
{\ell_{{\rm expt}}}
&
\fracds{V({\ell_{{\rm expt}}}{\lamopt_{\rm pure}},\rho_{\rm pure})}{V({\ell^{\rm round}_{\rm expt}(\rho_{\rm pure})},\rho_{\rm pure})}
&
\fracds{V({\ell_{{\rm expt}}}{\lamopt_{\rm mixd}},\rho_{\rm mixd})}{V({\ell^{\rm round}_{\rm expt}(\rho_{\rm mixd})},\rho_{\rm mixd})}
\\
&&\\
\hline
100 & .9797 & .7761
\\
1000 & .9950 & .9735
\\
10,000 & .9989 & .9954
\\
\hline
\end{array}
\]
The following table compares the distributions for optimal, suboptimal
with 8 angles, uniform at the 8 suboptimal angles, and uniform at all
100 angles by computing the minimum number of experiments required to
obtain an RMS estimation error of no more than 0.01.
\[
\renewcommand{\arraystretch}{1}
\begin{array}{|c||c|c|c|c|}
\hline
\mbox{input state}
&
\begin{array}{c}
\mbox{optimal}
\\
\round{{\ell_{{\rm expt}}} \lambda^{\rm opt}}
\\
{n_{\rm cfg}}=100
\end{array}
&
\begin{array}{c}
\mbox{suboptimal}
\\
\round{{\ell_{{\rm expt}}}\lambda^{\rm sub}}
\\
{n_{\rm cfg}}=8
\end{array}
&
\begin{array}{c}
\mbox{uniform}
\\
(1/8)\ {\bf 1}(\lambda^{\rm sub})
\\
{n_{\rm cfg}}=8
\end{array}
&
\begin{array}{c}
\mbox{uniform}
\\
(1/100)\ {\bf 1}(\lambda^{\rm opt})
\\
{n_{\rm cfg}}=100
\end{array}
\\
\hline
\mbox{$\rho_{\rm pure}$, no noise} & 20,308 & 20,308 & 20,638 & 29,274\\
\mbox{$\rho_{\rm pure}$, yes noise} & 37,775 & 64,129 & 40,178 & 52,825\\
\mbox{$\rho_{\rm mixd}$, no noise} & 41,890 & 92,471 & 69,750 & 64,780\\
\mbox{$\rho_{\rm mixd}$, yes noise} & 61,049 & 134,918 & 101,425 & 94,385\\
\hline
\end{array}
\]
For these optical experiments there is significant cost (in time)
associated with changing wave plate angles and very little cost (in
time) for an experiment. As a result, although the uniform
distribution at all 100 angles does not require a significant increase
in the number of experiments, and in some cases fewer experiments than
the suboptimal case, it is {\em very} costly in terms of changing wave
plate angles. The following table shows the wave plate settings and
suboptimal distributions from the above table.
\[
\begin{array}{|c|c||c|c|c|c|}
\hline
& & \lambda^{\rm sub}(\rho_{\rm pure}) & \lambda^{\rm sub}(\rho_{\rm pure}) & \lambda^{\rm sub}(\rho_{\rm mixd}) & \lambda^{\rm sub}(\rho_{\rm mixd}) \\
h & q & \mbox{no noise} & \mbox{yes noise}
& \mbox{no noise} & \mbox{yes noise}\\
\hline
0 & 0 & 0.24 & 0.23 & 0.14 & 0.14 \\
5 & 0 & 0 & 0.11 & 0.11 & 0.11 \\
10 & 0 & 0 & 0.04 & 0 & 0 \\
15 & 45 & 0 & 0 & 0.07 & 0.07 \\
20 & 40 & 0.01 & 0 & 0 & 0 \\
20 & 45 & 0.25 & 0.12 & 0.19 & 0.19 \\
25 & 45 & 0.25 & 0.13 & 0.21 & 0.21 \\
30 & 45 & 0 & 0 & 0.07 & 0.07 \\
35 & 0 & 0 & 0.04 & 0 & 0 \\
40 & 0 & 0 & 0.09 & 0.07 & 0.07 \\
45 & 0 & 0.24 & 0.23 & 0.14 & 0.14 \\
45 & 5 & 0 & 0 & 0 & 0 \\
\hline
\end{array}
\]
This table shows that only a few wave plate angle changes are
necessary if the suboptimal distributions are invoked. And from the
previous table, as already observed, the suboptimal settings do not
require a significant increase in the number of experiments required
to achieve the desired estimation accuracy.
\psfrag{gam}{$\gamma$}
\psfrag{r01}{$\lambda^{\rm opt}_{\rm pure}$, no noise}
\psfrag{r01n}{$\lambda^{\rm opt}_{\rm pure}$, yes noise}
\psfrag{r02}{$\lambda^{\rm opt}_{\rm mixd}$, no noise}
\psfrag{r02n}{$\lambda^{\rm opt}_{\rm mixd}$, yes noise}
\begin{figure}[h]
\epsfxsize=6.5in
\epsfboxit{lamopt1arm.ps}
\caption{optimal distributions -- one-arm}
\label{fig:lamopt1arm}
\end{figure}
\clearpage
\subsubsection*{simulation results: both arms}
We explore the two arm case under the assumption that two photons are
always present at the input, thereby excluding the zero photon
case.\footnote{
In an actual laboratory setting it is important to include the zero
photon case; it is often that no photon is actually present at the
input.
}
The table of detector firing patterns is:
\begin{center}
\begin{tabular}{|c||c|c|c|c|}
\hline
$\alpha$ & A & B & C & D \\
\hline
0101 & 0 & 1 & 0 & 1 \\
0110 & 0 & 1 & 1 & 0 \\
1001 & 1 & 0 & 0 & 1 \\
1010 & 1 & 0 & 1 & 0 \\
\hline
\end{tabular}
\end{center}
The following three $4\times 4$ input states are considered:
\[[eq:rhoott]
\begin{array}{l}
\rho_{\rm pure}\otimes\rho_{\rm pure}
\\
\rho_{\rm pure}\otimes\rho_{\rm mixd}
\\
\rho_{\rm mixd}\otimes\rho_{\rm mixd}
\end{array}
\]
with $\rho_{\rm pure},\ \rho_{\rm mixd}$ given by \refeq{rhot}. We use the angles
from the one-arm optimal distribution for the largest 8 values:
\[
\begin{array}{ll}
h\in[0,20,25,45] & q\in[0,20,25,45]
\\
h'\in[0,20,25,45] & q'\in[0,20,25,45]
\end{array}
\]
This yields a total of ${n_{\rm cfg}}=4^4=256$ wave plate settings.
Figure \reffig{lamopt2arm} shows the optimal distributions $\hat{\lam}$
versus configurations $\gamma=1,\ldots,256$ for all three input states.
In this case the optimal distributions $\hat{\lam}$ are nearly uniform in
magnitude. The following table compares the distributions for
optimal, suboptimal with 25 angles, uniform at the 25 suboptimal
angles, and uniform at all 256 angles by examining the minimum number
of experiments required to obtain an RMS estimation error of no more
than 0.01.
\[
\renewcommand{\arraystretch}{1}
\begin{array}{|c||c|c|c|c|}
\hline
\mbox{input state}
&
\begin{array}{c}
\mbox{optimal}
\\
\round{{\ell_{{\rm expt}}} \lambda^{\rm opt}}
\\
{n_{\rm cfg}}=256
\end{array}
&
\begin{array}{c}
\mbox{suboptimal}
\\
\round{{\ell_{{\rm expt}}}\lambda^{\rm sub}}
\\
{n_{\rm cfg}}=25
\end{array}
&
\begin{array}{c}
\mbox{uniform}
\\
(1/25)\ {\bf 1}(\lambda^{\rm sub})
\\
{n_{\rm cfg}}=25
\end{array}
&
\begin{array}{c}
\mbox{uniform}
\\
(1/256)\ {\bf 1}(\lambda^{\rm opt})
\\
{n_{\rm cfg}}=256
\end{array}
\\
\hline
\rho_{\rm pure}\otimes\rho_{\rm pure} & 81,870 & 81,877 & 86,942 & 115,165\\
\rho_{\rm pure}\otimes\rho_{\rm mixd} & 135,158 & 139,151 & 156,219 & 226,234\\
\rho_{\rm mixd}\otimes\rho_{\rm mixd} & 225,739 & 288,163 & 262,833 & 427,292\\
\hline
\end{array}
\]
As might be expected it is easier to estimate all pure states than
mixed states. The table also shows that the optimal solution can be
used effectively to guide the selection of suboptimal distributions.
\psfrag{gam}{$\gamma$}
\psfrag{r0101}{$\lambda^{\rm opt}$ for $\rho_{\rm pure}\otimes\rho_{\rm pure}$}
\psfrag{r0102}{$\lambda^{\rm opt}$ for $\rho_{\rm pure}\otimes\rho_{\rm mixd}$}
\psfrag{r0202}{$\lambda^{\rm opt}$ for $\rho_{\rm mixd}\otimes\rho_{\rm mixd}$}
\begin{figure}[h]
\epsfxsize=6.5in
\epsfboxit{lamopt2arm.ps}
\caption{optimal distributions -- two-arms}
\label{fig:lamopt2arm}
\end{figure}
\clearpage
\subsection{Maximum likelihood state distribution estimation}
\label{sec:dist est}
A variation on the state estimation problem is to estimate the {\em
distribution} of a known set of input states. The set-up for data
collection is shown schematically in Figure \ref{fig:dist} for
configuration $\gamma$.
\begin{figure}[h]
\[
\begin{array}{rr}
\rho^{\rm true}\in\Omega(f^{\rm true}) & \longrightarrow
\\
\mbox{} & \mbox{}
\\
\gamma & \longrightarrow
\end{array}
\mathbox{
\begin{array}{c}
\mbox{System}
\\
Q
\\
\mbox{}
\end{array}
}
\begin{array}{r}
\sigma_\gamma\in{\mathbf C}^{n\times n}
\\
\mbox{----------}\!\!\!\longrightarrow
\\
\gamma \longrightarrow
\end{array}
\mathbox{
\begin{array}{c}
\mbox{POVM}
\\
M_{\alpha\gamma} \in{\mathbf C}^{n\times n}
\\
\alpha=1,\ldots,{n_{\rm out}}
\end{array}
}
\longrightarrow
\begin{array}{c}
\mbox{Outcome counts}
\\
\seq{
\begin{array}{c} n_{\alpha\gamma},\ \ell_\gamma~\mbox{trials}
\\ \alpha=1,\ldots,{n_{\rm out}} \end{array}
}
\end{array}
\]
\caption{System/POVM.}
\label{fig:dist}
\end{figure}
\noindent
In this case the input state $\rho^{\rm true}$ is drawn from,
\[[eq:omin]
\Omega(f^{\rm true})
=
\set{\rho(i),\ f^{\rm true}(i)}
{i=1,\ldots,{n_{\rm in}}}
\]
which consists of a set of known states,
$\seq{\rho(1),\ldots,\rho({n_{\rm in}})}$, with corresponding unknown
occurrence probabilities $f^{\rm true}=\seq{f^{\rm true}(1),\ldots,f^{\rm true}({n_{\rm in}})}$
where $0\leqf^{\rm true}(i)\leq 1,\forall i$ and $\sum_i f^{\rm true}(i)=1$. The
objective is to use the data and knowledge of the input state set to
estimate the vector of occurrence probabilities $f^{\rm true}$. Proceeding
as before, assume that the input state {\em model} is $\rho\in\Omega(f)$
where $\Omega(f)=\set{\rho(i),\ f(i)} {i=1,\ldots,{n_{\rm in}}} $ and where the
vector of distributions $f=[f(1),\ldots,f({n_{\rm in}})]^T\in{\mathbf R}^{n_{\rm in}}$ is to
be estimated. In this case the input state can be represented by the
mixed state,
\[[eq:rhof]
\rho(f) = \sum_{i=1}^{n_{\rm in}}\ f(i) \rho(i)
\]
The ML estimate of $f$ is then the solution of the optimization
problem:
\[[eq:maxlike f]
\begin{array}{ll}
\mbox{minimize}
&
L(f)=
-\sum_{\alpha,\gamma}
n_{\alpha\gamma}
\log{\rm Tr}\ O_{\alpha\gamma}\rho(f)
\\
\mbox{subject to}
&
\sum_{i=1}^{n_{\rm in}} f(i)=1,
\;\;
f(i)\geq 0,\ i=1,\ldots,{n_{\rm in}}
\end{array}
\]
As in the MLE for quantum state estimation \refeq{maxlike1}, the
objective is log-convex in the state $\rho(f)$, the state is linear in
$f\in{\mathbf R}^{n_{\rm in}}$, and the constraints form a convex set in $f$. Hence,
this is a convex optimization problem in the variable $f$. Combining
\refeq{rhof} with \refeq{maxlike f} gives the more explicit form,
\[[eq: maxlike f]
\begin{array}{ll}
\mbox{minimize}
&
L(f)=
-\sum_{\alpha,\gamma}
n_{\alpha\gamma}
\log a_{\alpha\gamma}^T f
\\
\mbox{subject to}
&
\sum_{i=1}^{n_{\rm in}} f(i)=1,
\;\;
f(i)\geq 0,\ i=1,\ldots,{n_{\rm in}}
\\
&
a_{\alpha\gamma}
=
\left[
{\rm Tr}\ O_{\alpha\gamma}\rho(1)
\cdots
{\rm Tr}\ O_{\alpha\gamma}\rho({n_{\rm in}})
\right]^T\in{\mathbf R}^{n_{\rm in}},\;
\forall \alpha,\gamma
\end{array}
\]
Here again as in \refeq{lsopt cvx} we could solve for $f$ using the
empirical estimate of the outcome probabilities as in \refeq{lsopt}:
\[[eq:lsopt f]
\begin{array}{ll}
\mbox{minimize}
&
\sum_{\alpha,\gamma}\
w_\gamma
\left( p^{\rm emp}_{\alpha\gamma} - a_{\alpha\gamma}^T f \right)^2
\\
\mbox{subject to}
&
\sum_{i=1}^{n_{\rm in}} f(i)=1,
\;\;
f(i)\geq 0,\ i=1,\ldots,{n_{\rm in}}
\end{array}
\]
\subsection{Experiment design for state distribution estimation}
\label{sec:expdes qs dist}
Let $f^\surr\in{\mathbf R}^{n_{\rm in}}$ be a surrogate for the true state
distribution, $f^{\rm true}$. Following the derivation of \refeq{var rho}
in Appendix \refsec{var rho}, the associated (relaxed) optimal
experiment design problem is,
\[[eq:expdes qs dist]
\begin{array}{ll}
\mbox{minimize}
&
V(\lambda,f^\surr)=
{\rm Tr}\left(
\sum_{\gamma=1}^{{n_{\rm cfg}}}\ \lambda_\gamma G_\gamma(f^\surr)
\right)^{-1}
\\
\mbox{subject to}
&
\sum_{\gamma=1}^{{n_{\rm cfg}}}\ \lambda_\gamma = 1
\\
&
\lambda_\gamma\geq 0,\;
\gamma=1,\ldots,{n_{\rm cfg}}
\end{array}
\]
where
\[[eq:expdes qs 1]
\begin{array}{rcl}
G_\gamma(f^\surr)
&=&
\displaystyle
C_{\rm eq}^T
\left(
{\sum_{\alpha=1}^m} \
\frac{a_{\alpha\gamma}a_{\alpha\gamma}^T}
{p_{\alpha\gamma}(f^\surr)}
\right)
C_{\rm eq}
\in{\mathbf R}^{{n_{\rm in}}-1 \times {n_{\rm in}}-1}
\end{array}
\]
with $a_{\alpha\gamma}\in{\mathbf R}^{n_{\rm in}}$ from \refeq{maxlike f} and where
$C_{\rm eq}\in{\mathbf R}^{{n_{\rm in}}\times{n_{\rm in}}-1}$ is part of the unitary matrix $W$ in
the singular value decomposition: $1_{{n_{\rm in}}}^T=USW^T,\ W=[c\
C_{\rm eq}]\in{\mathbf R}^{{n_{\rm in}}\times{n_{\rm in}}}$.
\section{Quantum Process Tomography: OSR Estimation}
\label{sec:qptosr}
We explore two methods of identification for determining the
$Q$-system from data: (i) in this section, estimating the OSR in a
fixed basis, and (ii) in Section \refsec{ham est}, estimating
Hamiltonian parameters. In either case, the set-up is now as shown in
Figure \ref{fig:qpt}.
\begin{figure}[h]
\[
\rho_\gamma\in{\mathbf C}^{n\times n} \longrightarrow
\mathbox{
\begin{array}{c}
\mbox{System}
\\
Q
\\
\mbox{}
\end{array}
}
\begin{array}{r}
\sigma_\gamma\in{\mathbf C}^{n\times n}
\\
\mbox{----------}\!\!\!\longrightarrow
\\
\gamma \longrightarrow
\end{array}
\mathbox{
\begin{array}{c}
\mbox{POVM}
\\
M_{\alpha\gamma} \in{\mathbf C}^{n\times n}
\\
\alpha=1,\ldots,{n_{\rm out}}
\end{array}
}
\longrightarrow
\begin{array}{c}
\mbox{Outcome counts}
\\
\seq{
\begin{array}{c} n_{\alpha\gamma},\ \ell_\gamma~\mbox{trials}
\\ \alpha=1,\ldots,{n_{\rm out}} \end{array}
}
\end{array}
\]
\caption{System/POVM.}
\label{fig:qpt}
\end{figure}
\noindent
The main difference between state tomography (Figure \ref{fig:rho})
and process tomography (Figure \ref{fig:qpt}) is that in the latter
case the input state is prepared at specific values, $\rho_\gamma$,
depending on the configuration, whereas the $Q$-system does not
depend on the configuration. If the process varies with every change
in configuration it would be much more difficult to estimate; a model
of configuration dependence would need to be established. This
situation is perhaps amenable with Hamiltonian parameter estimation
but will not be pursued here.
\subsection{Maximum likelihood OSR estimation}
\label{sec:qpt osr}
As already stated, the Krause matrices for modeling the (trace
preserving) $Q$-system in this case are {\em not} dependent on the
configuration $\gamma$, specifically, $K=\set{K_k}{k=1,\ldots,\kappa}$
with $\kappa \leq n^2$. Using \refeq{osr}, the reduced state model in
Figure \ref{fig:qpt} as a function of $K$ is,
\[[eq:osr est]
\sigma_\gamma(K)
=Q(\rho_\gamma,K)
=
\sum_{k=1}^\kappa\ K_k \rho_\gamma K_k^*,
\;\;\;
\sum_{k=1}^\kappa\ K_k^* K_k = I_n
\]
Combining the above with the measurement model \refeq{povm} gives the
probability outcomes model,
\[[eq:pout osr est]
p_{\alpha\gamma}(K)
=
{\rm Tr}\ O_{\alpha\gamma}(K)\rho_\gamma,
\;\;
O_{\alpha\gamma}(K)
=
\sum_{k=1}^\kappa\ K_k^* M_{\alpha\gamma} K_k
\]
The log-likelihood function \refeq{loglike1} is,
\[[eq:loglike osr]
L(D,K)
=
-\sum_{\alpha,\gamma}\
n_{\alpha\gamma} \log {\rm Tr}\ O_{\alpha\gamma}(K)\rho_\gamma
\]
An ML estimate of $K$ is then a solution to,
\[[eq:mlosr k]
\begin{array}{ll}
\mbox{minimize}
&
L(D,K)
=
-\sum_{\alpha,\gamma}\
n_{\alpha\gamma} \log {\rm Tr}\
\sum_{k=1}^\kappa\ K_k^* M_{\alpha\gamma} K_k
\rho_\gamma
\\
\mbox{subject to}
&
\sum_{k=1}^\kappa\ K_k^* K_k = I_n
\end{array}
\]
This is not a convex optimization for two reasons: the equality
constraint is not linear in $K$ and the objective function is not
convex. The problem can be transformed -- more accurately, embedded
-- into a convex optimization problem by expanding the Kraus matrices
in a fixed basis. The procedure, described in \cite[\S
8.4.2]{NielsenC:00}, is as follows: since any matrix in ${\mathbf C}^{n\times n}$ can
be represented by $n^2$ complex numbers, let
\[[eq:basis]
\set{B_i\in{\mathbf C}^{n\times n}}{i=1,\ldots,n^2}
\]
be a basis for matrices in ${\mathbf C}^{n\times n}$. The Kraus matrices can thus be
expressed as,
\[[eq:krausb]
K_k = \sum_{i=1}^{n^2}\ a_{ki} B_i,\;
k=1,\ldots,\kappa
\]
where the $n^2$ coefficients $\seq{a_{ki}}$ are complex scalars.
Introduce the matrix $X\in{\mathbf C}^{n^2\times n^2}$, often referred to as
the {\em superoperator}, with elements,
\[[eq:osr2]
X_{ij} = \sum_{k=1}^\kappa\ a_{ki}^* a_{kj},
\;\;\;
i,j=1,\ldots,n^2
\]
As shown in \cite{NielsenC:00}, from the requirement to preserve
probability, $X$ is restricted to the convex set,
\[[eq:xset]
X \geq 0,
\;\;\;
\sum_{i,j=1}^{n^2}\ X_{ij}\ B_i^* B_j = I_n
\]
The system output state \refeq{osr est} and outcome probabilities
\refeq{pout osr est} now become,
\[[eq:osr1]
\begin{array}{rcl}
\sigma_\gamma(X)
&=&
\displaystyle
Q(\rho_\gamma,X)
=
\sum_{i,j=1}^{n^2}\ X_{ij}\ B_i \rho_\gamma B_j^*
\\&&\\
p_{\alpha\gamma}(X)
&=&
\displaystyle
{\rm Tr}\ O_{\alpha\gamma}\ Q(\rho_\gamma,X)
=
{\rm Tr}\ X R_{\alpha\gamma}
\end{array}
\]
where the matrix $R_{\alpha\gamma}\in{\mathbf C}^{n^2\times n^2}$ has elements,
\[[eq:rmat]
[R_{\alpha\gamma}]_{ij}
=
{\rm Tr}\ B_j \rho_\gamma B_i^* O_{\alpha\gamma},\;
i,j = 1,\ldots,n^2
\]
Quantum process tomography is then estimating $X\in{\mathbf C}^{n^2\times
n^2}$ from the data set $D$ \refeq{data}. An ML estimate is obtained
by solving for $X$ from:
\[[eq:mle x]
\begin{array}{ll}
\mbox{minimize}
&
L(D,X)
=
-\sum_{\alpha,\gamma}\
n_{\alpha\gamma} \log {\rm Tr}\ X R_{\alpha\gamma}
\\
\mbox{subject to}
&
X \geq 0,
\;\;\;\;
\sum_{ij}\ X_{ij}\ B_i^*B_j = I_n
\end{array}
\]
This problem has essentially the same form as \refeq{maxlike1}, and
hence is also a convex optimization problem with the optimization
variables being the elements of the matrix $X$. Since
$X=X^*\in{\mathbf C}^{n^2\times n^2}$, it can be parametrized by $n^4$ real
variables. Accounting for the $n^2$ real linear equality constraints,
the number of free (real) variables in $X$ is thus $n^4-n^2$. This can
be quite large even for a relatively small number of qubits, \emph{e.g.}, for
$q=[1,\ 2,\ 3,\ 4]$ qubits, $n=2^q=[2,\ 4,\ 8,\ 16]$ and
$n^4-n^2=[12,\ 240,\ 4032,\ 65280]$. This exponential (in qubit)
growth is the main drawback to using this approach.
The $X$ (superoperator) matrix can be transformed back to Kraus
operators via the singular value decomposition \cite[\S
8.4.2]{NielsenC:00}. Specifically, let $X=VSV^*$ with unitary $V\in
{\mathbf C}^{n^2\times n^2}$ and $S={\rm diag}(s_1\ \cdots\ s_{n^2})$ with
the singular values ordered so that $s_1 \geq s_2 \geq\ \cdots\ \geq
s_{n^2} \geq 0$. Then the coefficients in the basis representation of
the Kraus matrices \refeq{krausb} are,
\[[eq:aki]
a_{ki} = \sqrt{s_k}\ V_{i k}^*,\
k,i=1,\ldots,n^2
\]
Theoretically there can be fewer then $n^2$ Kraus operators. For
example, if the $Q$ system is unitary, then,
\[[eq:u rho u]
Q(\rho) = U \rho U^*
\]
In effect, there is one Kraus operator, $U$, which is unitary and of
the same dimension as the input state $\rho$. The corresponding $X$
matrix is a dyad, hence ${\bf rank}\ X=1$. A rank constraint is not
convex. However, the $X$ matrix is symmetric and positive
semidefinite, so the heuristic from \cite{FazelHB:01} applies where the
rank constraint is replaced by the trace constraint,
\[[eq:rankx]
{\rm Tr}\ X \leq \eta
\]
From the singular value decomposition of $X$, ${\rm Tr}\ X = \sum_k
s_k$, and hence, adding the constraint \refeq{rankx} to \refeq{mle x}
will force some (or many) of the $s_k$ to be small which can be
eliminated (post-optimization) thereby reducing the rank. The
auxiliary parameter $\eta$ can be used to find a tradeoff between
simpler realizations and performance. The estimation problem is then:
\[[eq:mle x eta]
\begin{array}{ll}
\mbox{minimize}
&
L(D,X)
=
-\sum_{\alpha,\gamma}\
n_{\alpha\gamma} \log {\rm Tr}\ X R_{\alpha\gamma}
\\
\mbox{subject to}
&
X \geq 0,
\;\;\;\;
\sum_{ij}\ X_{ij}\ B_i^*B_j = I_n
\\
&
{\rm Tr}\ X \leq \eta
\end{array}
\]
\subsection{Experiment design for OSR estimation}
\label{sec:expdes osr}
Let $X^\surr\in{\mathbf C}^{n^2\times n^2}$ be a surrogate for the true OSR, $X^{\rm true}$. As
derived in Appendix \refsec{expdes osr der}, the associated (relaxed)
optimal experiment design problem is,
\[[eq:expdes osr]
\begin{array}{ll}
\mbox{minimize}
&
V(\lambda,X^\surr)=
{\rm Tr}\left(
\sum_{\gamma=1}^{{n_{\rm cfg}}}\ \lambda_\gamma G_\gamma(X^\surr)
\right)^{-1}
\\
\mbox{subject to}
&
\sum_{\gamma=1}^{{n_{\rm cfg}}}\ \lambda_\gamma = 1
\\
&
\lambda_\gamma\geq 0,\;
\gamma=1,\ldots,{n_{\rm cfg}}
\end{array}
\]
where
\[[eq:expdes osr 1]
\begin{array}{rcl}
G_\gamma(X^\surr)
&=&
\displaystyle
C_{\rm eq}^*
\left(
{\sum_{\alpha=1}^m} \
\frac{a_{\alpha\gamma} a_{\alpha\gamma}^* }{p_{\alpha\gamma}(X^\surr)}
\right)
C_{\rm eq}
\\&&\\
a_{\alpha\gamma}
&=&
\displaystyle
\vec\ R_{\alpha\gamma} \in {\mathbf C}^{n^4}
\end{array}
\]
and $C_{\rm eq}\in{\mathbf C}^{n^4\times n^4-n^2}$ is part of the unitary matrix
$W=[C\ C_{\rm eq}]\in{\mathbf C}^{n^4\times n^4}$ in the singular value
decomposition of the $n^2\times n^4$ matrix,
\[[eq:expdes osr 2]
\left[a_1\ \cdots\ a_{n^4}\right]
=
U \left[ \sqrt{n}I_{n^2}\;\; 0_{n^2\times n^4-n^2} \right] W^*
\]
with with $a_k=\vec(B_i^*B_j)\in{\mathbf C}^{n^2}$ for $k=i+(j-1)n^2,\
i,j=1,\ldots,n^2$. The columns of $C_{\rm eq}$, \emph{i.e.}, the last $n^4-n^2$
columns of $W$, are a basis for the nullspace of $\left[a_1\ \cdots\
a_{n^4}\right]$.
\subsection{Example: experiment design for OSR estimation}
\label{sec:ex expdes osr}
Consider the POVM set from the one-arm photon detector
(\refsec{example qs}) using all combinations of the following set
wave-plate angles,
\[
h=[0\ 30\ 45],\; q=[0\ 30\ 45]
\]
Assume detector efficiency $\eta=0.75$ and dark count probability
$\delta=0.05$. The set of inputs (state configurations) is
\[
\ket{0},\; \ket{1},\; \ket{{\scriptstyle +}}=(\ket{0}+\ket{1})/\sqrt{2},\;
\ket{{\scriptstyle -}}=(\ket{0}+i\ket{1})/\sqrt{2}
\]
The 9 combinations of angles together with the 4 combinations of input
states gives a total of 36 configurations, $\gamma=1,\ldots,{n_{\rm cfg}}=36$.
\iffalse
\begin{verbatim}
h=[0 30 45]*pi/180;q=h;
del=0.05;eta=0.75;
ket0=[1;0];
ket1=[0;1];
ketp=(ket0+ket1)/sqrt(2);
ketm=(ket0+jay*ket1)/sqrt(2);
objuoed= 130.46 85.668
ceil(objuoed/(.05^2)) = 52183 34268
ceil(objuoed/(.01^2)) = 1304561 856676
\end{verbatim}
\fi
Figure \ref{fig:lamopt_qp} shows the optimal distribution of
experiments for the 36 configurations using the true OSR corresponding
to the Pauli basis set
$
\seq{
I_2/\sqrt{2},\
\sigma_x/\sqrt{2},\
\sigma_y/\sqrt{2},\
\sigma_z/\sqrt{2}
}
$.
Since the system is simply the identity, $Q(\rho)=\rho$, with this
basis choice, $X^{\rm true}={\rm diag}(2\ 0\ 0\ 0)$. (No knowledge of the
system being identity is used, hence, all elements of $X^{\rm true}$ are
estimated, not just the single element in the ``11'' location.)
The following table displays the minimum number of experiments
required to meet estimation accuracies of 0.05 and 0.01 for both
uniform and optimal distributions.
\[
\begin{array}{|c||c|c|}
\hline
\mbox{accuracy} & \lambda^{\rm opt} & \lambda^{\rm unif}
\\
\hline
0.01 & 856,676 & 1,304,561\\
0.05 & 34,268 & 52,183\\
\hline
\end{array}
\]
Approximately 35\% fewer experiments are needed using the optimal
distribution. Although not dramatic, as in the photon estimation
example \refsec{example qs}, there is a large penalty, in terms of
time, for changing wave-plate angles.
\psfrag{gam}{$\gamma$}
\psfrag{lam}{$\lambda^{\rm opt}$}
\begin{figure}[h]
\centering
\epsfysize=3.5in
\epsfxsize=6.5in
\epsfboxit{lamopt_qp.eps}
\caption{optimal distributions for OSR estimation}
\label{fig:lamopt_qp}
\end{figure}
The following Table shows the 22 out of 36 configurations for
$\lambda^{\rm opt}_\gamma > 0.01$.
\[
\begin{array}{|c|c|c|c|c|}
\hline
\gamma & \lambda^{\rm opt}_\gamma & h_\gamma & q_\gamma & \rho_\gamma
\\
\hline
1 & 0.011 & 0 & 0 & \ket{0} \\
2 & 0.011 & 0 & 0 & \ket{1} \\
3 & 0.073 & 0 & 0 & \ket{{\scriptstyle +}} \\
4 & 0.068 & 0 & 0 & \ket{{\scriptstyle -}} \\
5 & 0.033 & 0 & 30 & \ket{0} \\
6 & 0.033 & 0 & 30 & \ket{1} \\
7 & 0.071 & 0 & 30 & \ket{{\scriptstyle +}} \\
8 & 0.012 & 0 & 30 & \ket{{\scriptstyle -}} \\
12 & 0.018 & 0 & 45 & \ket{{\scriptstyle -}} \\
21 & 0.068 & 30 & 45 & \ket{0} \\
22 & 0.068 & 30 & 45 & \ket{1} \\
23 & 0.063 & 30 & 45 & \ket{{\scriptstyle +}} \\
24 & 0.141 & 30 & 45 & \ket{{\scriptstyle -}} \\
25 & 0.011 & 45 & 0 & \ket{0} \\
26 & 0.011 & 45 & 0 & \ket{1} \\
27 & 0.073 & 45 & 0 & \ket{{\scriptstyle +}} \\
28 & 0.068 & 45 & 0 & \ket{{\scriptstyle -}} \\
29 & 0.033 & 45 & 30 & \ket{0} \\
30 & 0.033 & 45 & 30 & \ket{1} \\
31 & 0.071 & 45 & 30 & \ket{{\scriptstyle +}} \\
32 & 0.012 & 45 & 30 & \ket{{\scriptstyle -}} \\
36 & 0.018 & 45 & 45 & \ket{{\scriptstyle -}} \\
\hline
\end{array}
\]
\subsection{Maximum likelihood OSR distribution estimation}
\label{sec:osr dist}
Suppose the Kraus matrices are known up to a scale factor which is
related to its probability of occurrence, that is,
\[[eq:osr dist]
\begin{array}{lll}
K_k = \sqrt{q_k}\ \bar{K}_k
&
\sum_{k=1}^\kappa q_k \bar{K}_k^* \bar{K}_k=I_n
&
{}
\\
\sum_{k=1}^\kappa q_k=1
&
q_k \geq 0
&
k=1,\ldots,\kappa
\end{array}
\]
One interpretation of this system model is that one of the matrices,
say $\bar{K}_1$, is the nominal (unperturbed) system, and the others,
$\bar{K}_k,k=2,\ldots,\kappa$, are perturbations, each of them occurring
with probability $q_k$. Examples of perturbations include the typical
errors which can be handled by quantum error correction codes, \emph{e.g.},
depolarization, phase damping, phase and bit flip; see, \emph{e.g.},
\cite[Ch.8]{NielsenC:00}.
The goal is to use the data to estimate the unknown vector of
probabilities, $q=[q_1 \cdots q_\kappa]^T \in{\mathbf R}^\kappa$. Using the
system model \refeq{osr dist}, the model probability outcomes are,
\[[eq:pout osr dist]
\begin{array}{rcl}
p_{\alpha\gamma}
&=&
{\rm Tr}\ M_{\alpha\gamma} \sum_{k=1}^\kappa
q_k\ \bar{K}_k \rho_\gamma \bar{K}_k^*
=
a_{\alpha\gamma}^T q
\\
a_{\alpha\gamma}
&=&
\left[
{\rm Tr}\ M_{\alpha\gamma} \bar{K}_1 \rho_\gamma \bar{K}_1^*
\cdots
{\rm Tr}\ M_{\alpha\gamma} \bar{K}_\kappa \rho_\gamma \bar{K}_\kappa^*
\right]^T
\in{\mathbf R}^\kappa
\end{array}
\]
The ML estimate of $q\in{\mathbf R}^\kappa$ is the solution of the
optimization problem,
\[[eq:mle osr dist]
\begin{array}{ll}
\mbox{minimize}
&
L(q)=
-\sum_{\alpha,\gamma}
n_{\alpha\gamma}
\log a_{\alpha\gamma}^T q
\\
\mbox{subject to}
&
\sum_{k=1}^\kappa q_k=1,
\;\;
q_k\geq 0,\ k=1,\ldots,\kappa
\end{array}
\]
This is a convex optimization problem and is essentially in the same
form as problem \refeq{maxlike f} which seeks the ML estimate of the
input state distribution.
\subsection{Experiment design for OSR distribution estimation}
\label{sec:expdes osr dist}
The formulation here is directly analogous to that of experiment
design for state distribution estimation \refsec{expdes qs dist}.
Let $q^\surr\in{\mathbf R}^{n_{\rm in}}$ be a surrogate for the true OSR distribution,
$q^{\rm true}$. Following the lines of the derivation in Appendix
\refsec{expdes osr der}, the associated (relaxed) optimal
experiment design problem is,
\[[eq:expdes osr dist]
\begin{array}{ll}
\mbox{minimize}
&
V(\lambda,q^\surr)=
{\rm Tr}\left(
\sum_{\gamma=1}^{{n_{\rm cfg}}}\ \lambda_\gamma G_\gamma(q^\surr)
\right)^{-1}
\\
\mbox{subject to}
&
\sum_{\gamma=1}^{{n_{\rm cfg}}}\ \lambda_\gamma = 1
\\
&
\lambda_\gamma\geq 0,\;
\gamma=1,\ldots,{n_{\rm cfg}}
\end{array}
\]
where
\[[eq:expdes qs 2]
\begin{array}{rcl}
G_\gamma(q^\surr)
&=&
\displaystyle
C_{\rm eq}^T
\left(
{\sum_{\alpha=1}^m} \
\frac{a_{\alpha\gamma}a_{\alpha\gamma}^T}
{p_{\alpha\gamma}(q^\surr)}
\right)
C_{\rm eq}
\in{\mathbf R}^{{n_{\rm in}}-1 \times {n_{\rm in}}-1}
\end{array}
\]
with $a_{\alpha\gamma}\in{\mathbf R}^{n_{\rm in}}$ from \refeq{mle osr dist} and where
$C_{\rm eq}\in{\mathbf R}^{{n_{\rm in}}\times{n_{\rm in}}-1}$ is part of the unitary matrix $W$ in
the singular value decomposition: $1_{{n_{\rm in}}}^T=USW^T,\ W=[c\
C_{\rm eq}]\in{\mathbf R}^{{n_{\rm in}}\times{n_{\rm in}}}$.
\subsection{Example: experiment design for OSR distribution estimation}
\label{sec:ex osr dist}
Consider a quantum process, or channel, where a single qubit state,
$\rho\in{\mathbf C}^{2\times 2}$, is corrupted by a {\em bit-flip error} with
occurrence probability $q_B$ and a {\em depolarizing error} with
occurrence probability $q_D$. The process is described by the quantum
operation,\footnote{
$X$ is one of the three $2\times 2$ Pauli spin matrices:
$
X = \mattwo{0}{1}{1}{0},\
Y = \mattwo{0}{-i}{i}{0},\
Z = \mattwo{1}{0}{0}{-1}
$
}
\[[eq:bf dep]
\begin{array}{l}
Q(\rho,q)
=
\displaystyle
q_I\ \rho + q_B\ X\rho X
+
q_D\ I/2
\\
\\
q_I+q_B+q_D = 1
\end{array}
\]
where $q_I = 1 -(q_B+q_D)$ is the probability of no error occurring.
The probability of observing outcome $\alpha$ with the system in
configuration $\gamma$ is,\footnote{
As shown in \cite[\S 8.3]{NielsenC:00}, an equivalent set of OSR
elements which describe \refeq{bf dep} are,
$
\sqrt{q_I}\ I,\
\sqrt{q_B}\ X,\
\sqrt{q_D}\ I/2,\
\sqrt{q_D}\ X/2,\
\sqrt{q_D}\ Y/2,\
\sqrt{q_D}\ Z/2
$.
Forming the probability outcomes in terms of this expansion
results in an overparamtrization.
}
\[[eq:palfgam q]
\begin{array}{rcl}
p_{\alpha\gamma}(q)
&=&
{\rm Tr}\ M_{\alpha\gamma} Q(\rho_\gamma,q)
\\
&=&
\left[
\begin{array}{ccc}
{\rm Tr}\ M_{\alpha\gamma}\rho_\gamma
&
{\rm Tr}\ M_{\alpha\gamma} X\rho_\gamma X
&
{\rm Tr}\ M_{\alpha\gamma}/2
\end{array}
\right]
\
\left[
\begin{array}{c} q_I \\ q_B \\ q_D \end{array}
\right]
\\
&=&
a_{\alpha\gamma}^T\ q
\end{array}
\]
An interesting aspect of this problem is that not all input states
$\rho_\gamma$ lead to identifiability of the occurrence probabilities. And
this is independent of the choice of POVM $M_{\alpha\gamma}$. To see this
consider the single pure input state,
\[
\rho_\gamma=\psi\psi^*,
\;
\psi=\left[\begin{array}{c} a \\ b \end{array}\right],
\;
|a|^2 + |b|^2 =1
\]
The output of the channel \refeq{bf dep} is then,
\[
Q(\psi\psi^*,q)
=
\left[
\begin{array}{cc}
q_I |a|^2 + q_B |b|^2 + q_D/2
&
q_I ab^* + q_B a^*b
\\
q_I a^*b + q_B ab^*
&
q_I |b|^2 + q_B |a|^2 + q_D/2
\end{array}
\right]
\]
Suppose we knew the elements of $Q(\psi\psi^*,q)$ perfectly; call them
$Q_{11},Q_{12},Q_{22}$. Then in principal we could solve for the three
occurrence probabilities from the linear system of equations,
\[
\underbrace{
\left[
\begin{array}{ccc}
|a|^2 & |b|^2 & 1/2
\\
|b|^2 & |a|^2 & 1/2
\\
ab^* & a^*b & 0
\end{array}
\right]
}_{R}
\
\left[
\begin{array}{c} q_I \\ q_B \\ q_D \end{array}
\right]
=
\left[
\begin{array}{c} Q_{11} \\ Q_{22} \\ Q_{12} \end{array}
\right]
\]
If $\det\ R=0$ then no unique solution exists; the occurrence
probabilities are not {\em identifiable}. Specifically, $\det\ R=0$
for all $a,b\in{\mathbf C}$ such that,
\[
({\rm Re}\ ab^*)(|b|^2-|a|^2) = 0,
\;
|a|^2+|b|^2=1
\]
Equivalently, $\det\ R=0$ for the following sets of $a,b\in{\mathbf C}$:
\[[eq:bad ab]
(a=0,\ |b|=1),
\;
(|a|=1,\ b=0),
\;
(|a| = |b| = 1/\sqrt{2})
\]
Let the input state be a single pure state of the form
\[
\psi(\th)=\left[\begin{array}{c} \cos\th \\ \sin\th \end{array}\right],
\]
Suppose the angle $\th$ is restricted to the range $0 \leq \th \leq
90^\circ$. Using \refeq{bad ab}, the occurrence probabilities are not
identifiable for the angles $\th$. and respectively, the states
$\psi(\th)$, in the sets,
\[[eq:bad th]
\th\in\seq{0,\ 45^\circ,\ 90^\circ},
\;
\psi(\th)\in
\seq{
\left[\begin{array}{c} 1 \\ 0 \end{array}\right],\
\left[\begin{array}{c} 1/\sqrt{2} \\ 1/\sqrt{2} \end{array}\right],\
\left[\begin{array}{c} 0 \\ 1 \end{array}\right]
}
\]
Unfortunately, this excludes inputs identical to the computational
basis states $\ket{0}$ or $\ket{1}$, respectively, $\psi(\th)$ with
$\th=0$ or $\th=90^\circ$.
We now solve the (relaxed) experiment design problem \refeq{expdes osr
dist} for occurrence probabilities $(q_I,\ q_B,\ q_D)=(0.6,\ 0.2,\
0.2)$ and with the POVM set given by \refeq{povm hqgam}. For
illustrative purposes, we use only 16 of the 100 configurations
represented by the wave plate angles \refeq{hq1arm}. Specifically, the
wave-plate angles are:
$
\seq{h=0,\ 15,\ 30,\ 45} \times
\seq{q=0,\ 15,\ 30,\ 45}
$.
The optimization results are presented in the following table which
shows the number of experiments per configuration (each of the 16
angle pairs) required to achieve an accuracy of 0.01 for the input
states corresponding to the angles $\th\in\seq{2,\ 10,\ 25,\ 35,\
44}$.
\begin{center}
\begin{tabular}{|c|c||r|r|r|r|r|}
\hline
\mcol{2}{|c||}{configurations} & \mcol{5}{|c|}{experiments per configuration}\\
\hline\
$h$ & $q$ & $\th=2$ & $\th=10$ & $\th=25$ & $\th=35$ & $\th=44$ \\
\hline\hline
0 & 0 & 62,244 & 15,453 & 13,386 & 31,275 & 2,136,560 \\
0 & 15 & 1 & 1 & 1 & 1 & 1 \\
0 & 30 & 1 & 0 & 0 & 0 & 1 \\
0 & 45 & 1 & 0 & 0 & 0 & 1 \\
15 & 0 & 1 & 0 & 0 & 0 & 1 \\
15 & 15 & 1 & 1 & 1 & 1 & 1 \\
15 & 30 & 73,096 & 11,277 & 4,765 & 9,006 & 107,371 \\
15 & 45 & 2,080,984 & 89,588 & 18,598 & 14,573 & 62,187 \\
30 & 0 & 1 & 0 & 0 & 0 & 1 \\
30 & 15 & 1 & 0 & 0 & 0 & 1 \\
30 & 30 & 1 & 0 & 0 & 0 & 1 \\
30 & 45 & 2,080,984 & 89,588 & 18,598 & 14,573 & 62,187 \\
45 & 0 & 62,244 & 15,453 & 13,386 & 31,275 & 2,136,560 \\
45 & 15 & 1 & 1 & 1 & 1 & 1 \\
45 & 30 & 1 & 0 & 0 & 0 & 1 \\
45 & 45 & 1 & 0 & 0 & 0 & 1 \\
\hline\hline
\mcol{2}{|c||}{${\ell_{{\rm expt}}}\ =$} & 4,359,563 & 221,362 & 68,736 & 100,705 & 4,504,876 \\
\hline
\end{tabular}
\end{center}
The numerical example shows that for input states close to those
states which make the problem not identifiable \refeq{bad th}, the
number of experiments required to achieve the specified accuracy grows
very large. In this case, $\th=2$ and $\th=44$ are close to the bad
angles 0 and 45, and the number of experiments is quite large.
\section{Hamiltonian Parameter Estimation}
\label{sec:ham est}
The process of modeling a quantum system in this case begins with the
construction of a Hamiltonian {\em operator} on an infinite
dimensional Hilbert space. {\em Eventually}, a finite dimensional
approximation is invoked in order to calculate anything. (In some
cases a finite dimensional model is immediately appropriate, \emph{e.g.}, spin
systems, \cite[Ch.12-9]{FeynmanLS:65}.) The finite dimensional model
is the starting point here.
\subsection{Maximum likelihood Hamiltonian parameter estimation}
\label{sec:mle hampar}
The quantum system is modeled by a finite dimensional Hamiltonian
matrix $H(t,\th)\in{\mathbf C}^{n\times n}$, having a known dependence on time $t,\
0\leq t\leq t_f$, and on an unknown parameter vector
$\th\in{\mathbf R}^{{n_\th}}$. The model density matrix will depend on $\th$ and
the initial (prepared and known) state drawn from the set of states
$\set{\rho^{\rm init}_\beta\in{\mathbf C}^{n\times n}}{\beta=1,\ldots,{n_{\rm in}}}$. Thus, the
density matrix associated with initial state $\rho^{\rm init}_\beta$ is
$\rho_\beta(t,\th)\in{\mathbf C}^{n\times n}$ which evolves according to,
\[[eq:rhobet]
i\hbar\dot\rho_\beta = [H(t,\th),\rho_\beta],\
\rho_\beta(0,\th)=\rho^{\rm init}_\beta
\]
Equivalently,
\[[eq:rho1]
\rho_\beta(t,\th) = U(t,\th) \rho^{\rm init}_\beta U(t,\th)^*
\]
where $U(t,\th)\in{\mathbf C}^{n\times n}$ is the unitary propagator associated with
$H(t,\th)$ which satisfies,
\[[eq:prop]
i\hbar\dot U = H(t,\th) U,\ U(0,\th)=I_n
\]
At each of ${n_{\rm sa}}$ sample times in a time interval of duration ${t_{\rm f}}$,
measurements are recorded from identical repeated
experiments. Specifically, let $\set{t_\tau}{\tau=1,\ldots,{n_{\rm sa}}}$
denote the sample times relative to the start of each experiment. Let
$n_{\alpha\beta\tau}$ be the number of times the outcome $\alpha$ is
recorded at $t_\tau$ with initial state $\rho^{\rm init}_\beta$ from
$\ell_{\beta\tau}$ experiments. The data set thus consists of all the
outcome counts,
\[[eq:data th]
D = \set{n_{\alpha\beta\tau}}
{\alpha=1,\ldots,{n_{\rm out}},\
\beta=1,\ldots,{n_{\rm in}},\
\tau=1,\ldots,{n_{\rm sa}}}
\]
The {\em configurations} previously enumerated and labeled by
$\gamma=1,\ldots,{n_{\rm cfg}}$ are in this case all the combinations of input
states $\rho^{\rm init}_\beta$ and sample times $\tau$, thus ${n_{\rm cfg}}={n_{\rm in}}{n_{\rm sa}}$.
For the POVM $M_\alpha$, the model outcome probability per configuration
pair $(\rho^{\rm init}_\beta,\ t_\tau)$ is,
\[[eq:palfbettau]
\begin{array}{rcl}
p_{\alpha\beta\tau}(\th)
&=&
{\rm Tr}\ M_\alpha \rho_\beta(t_\tau,\th)
=
{\rm Tr}\ O_{\alpha\tau}(\th) \rho^{\rm init}_\beta
\\
O_{\alpha\tau}(\th)
&=&
U(t_\tau,\th)^* M_\alpha U(t_\tau,\th)
\end{array}
\]
The Maximum Likelihood estimate, $\th^{\rm ML}\in{\mathbf R}^{{n_\th}}$, is obtained
as the solution to the optimization problem:
\[[eq:maxlike th]
\begin{array}{ll}
\mbox{minimize}
&
L(D,\th)
=
-\sum_{\alpha,\beta,\tau}
n_{\alpha\beta\tau}
\log{\rm Tr}\ O_{\alpha\tau}(\th)\rho^{\rm init}_\beta
\\
\mbox{subject to}
&
\th \in \Theta
\end{array}
\]
where $\Theta$ is a set of constraints on $\th$. For example, it may be
known that $\th$ is restricted to a region near a nominal value, \emph{e.g.},
$\Theta=\set{\th}{\norm{\th-\th_{\rm nom}} \leq \delta}$. Although this
latter set is convex, unfortunately, the likelihood function,
$L(D,\th)$, is not guaranteed to be convex in $\th$. It is possible,
however, that it is convex in the restricted region $\Theta$, for
example, if $\delta$ is sufficiently small.
\subsection{Experiment design for Hamiltonian parameter estimation}
\label{sec:ham oed}
Despite the fact that Hamiltonian parameter estimation is not convex,
the (relaxed) experiment design problem is convex. A direct
application of the Cram\'{e}r-Rao\ bound to the likelihood function in
\refeq{maxlike th} results in the following.
\begin{quote}
{\bf Hamiltonian parameter estimation variance lower bound}
Suppose the system generating the data is in the model set used for
estimation, \emph{i.e.}, \refeq{s in m} holds. For $\ell=[\ell_1 \cdots
\ell_{{n_{\rm cfg}}}]$ experiments per configuration $(\rho^{\rm init}_\beta,t_\tau)$,
suppose ${\hat{\theta}}(\ell)\in{\mathbf R}^{n_\th}$ is an unbiased estimate of
$\th^{\rm true}\in{\mathbf R}^{n_\th}$. Under these conditions, the estimation error
variance satisfies,
\[[eq:var th]
{\bf E} \ \norm{{\hat{\theta}}(\ell)-\th^{\rm true}}^2
\geq
V(\ell,\th^{\rm true})
=
{\rm Tr}\
G(\ell,\th^{\rm true})^{-1}
\]
where
\[[eq:var th 1]
\begin{array}{rcl}
\displaystyle
G(\ell,\th^{\rm true})
&=&
\displaystyle
\sum_{\beta,\tau}\ \ell_{\beta\tau} G_{\beta\tau}(\th^{\rm true})
\in{\mathbf R}^{{n_\th} \times {n_\th}}
\\&&\\
\displaystyle
G_{\beta\tau}(\th^{\rm true})
&=&
\displaystyle
\left.
{\sum_{\alpha=1}^m} \
\left(
\frac{
\left(\nabla_\th\ p_{\alpha\beta\tau}(\th) \right)
\left(\nabla_\th\ p_{\alpha\beta\tau}(\th) \right)^T
}
{p_{\alpha\beta\tau}(\th)}
-
\nabla_{\th\th}\ p_{\alpha\beta\tau}(\th)
\right)
\right|_{\th=\th^{\rm true}}
\in{\mathbf R}^{{n_\th}\times{n_\th}}
\end{array}
\]
\end{quote}
The relaxed experiment design problem with respect to the surrogate
${\hat{\theta}}$ for $\th^{\rm true}$ is,
\[[eq:expdes th]
\begin{array}{ll}
\mbox{minimize}
&
V(\lambda,{\hat{\theta}})
=
{\rm Tr}\left(
\sum_{\beta,\tau}\ \lambda_{\beta\tau} G_{\beta\tau}({\hat{\theta}})
\right)^{-1}
\\
\mbox{subject to}
&
\sum_{\beta,\tau}\ \lambda_{\beta\tau} = 1
\\
&
\lambda_{\beta\tau} \geq 0,\;
\forall\ {\beta,\ \tau}
\end{array}
\]
with optimization variables $\lambda_{\beta\tau}$, the distribution of
experiments per configuration $(\rho^{\rm init}_\beta,\ t_\tau)$. The
difference between this and the previous formulation is that there are
no equality constraints on the parameters. The gradient $\nabla_\th\
p_{\alpha\beta\tau}(\th)$ and Jacobian $\nabla_{\th\th}\
p_{\alpha\beta\tau}(\th)$ are dependent on the parametric structure of
the Hamiltonian $H(t,\th)$.
\subsection{Example: experiment design for Hamiltonian parameter estimation}
\label{sec:ex hampar}
Consider the system Hamiltonian,
\[
H = \th^{\rm true} {\varepsilon}\
\left(X + Z \right)/\sqrt{2},
\]
with {\em constant} control $\varepsilon$. The goal is to select the control
to make the Hadamard logic gate, $U_{\rm had}=(X+Z)/\sqrt{2}$. If $\th^{\rm true}$
were known, then the control $\varepsilon=1/\th^{\rm true}$ would produce the
Hadamard (to within a scalar phase) at time $t=\pi/2$, that is,
\[
U(t=\pi/2)
=
\exp\seq{-i(\pi/2)H(\varepsilon=1/\th^{\rm true})}
=
-i\ U_{\rm had}
\]
We assume that only the estimate ${\hat{\theta}}$ of $\th^{\rm true}$ is available.
Using the estimate and knowledge of the Hamiltonian model structure,
the control is $\varepsilon=1/{\hat{\theta}}$. This yields the {\em actual} gate at
$t=\pi/2$,
\[
\begin{array}{rcl}
U_{\rm act}
&=&
\exp\seq{-i(\pi/2)H(\varepsilon=1/{\hat{\theta}})}
=
-i\ U_{\rm had}\ \exp\seq{-i \delta\ (\pi/2)\ U_{\rm had}}
\\
\delta &=& \th^{\rm true}/{\hat{\theta}}-1
\end{array}
\]
Since the parameter estimate, ${\hat{\theta}}$, is a random variable, so is the
normalized parameter error $\delta$. Assuming the estimate is unbiased,
the expected value of the worst-case gate fidelity \refeq{wc fidelity}
is given explicitly by,
\[[eq:wcfid]
{\bf E} \
\min_{\norm{\psi}=1}\
\left|
\left(U_{\rm had} \psi\right)^*
\left(U_{\rm act}\psi\right)
\right|^2
=
{\bf E} \ \cos^2\left(\frac{\pi}{2}\ \delta\right)
\approx
1-\left(\frac{\pi}{2}\right)^2\ {\bf E} (\delta^2)
\]
Consider the case where the system is in the model set, the POVMs are
projectors in the computational basis $(\ket{0},\ \ket{1})$, and the
configurations consist of combinations of input states and sample
times. Specifically, the example problem is as follows:
\[
\begin{array}{ll}
\mbox{{model Hamiltonian}}
&
H(\varepsilon,\ \th) =
\th\ \varepsilon
\left( X + Z \right)/\sqrt{2}
\\&\\
\mbox{{true Hamiltonian}}
&
H^{\rm true}=\th^{\rm true}\varepsilon\ \left( X + Z \right)/\sqrt{2}
\\&\\
\mbox{{POVM}}
&
M_1 = \ket{0}\bra{0},
\;\;
M_2 = \ket{1}\bra{1}
\\&\\
\mbox{{configurations}}
&
\left\{
\begin{array}{c}
\mbox{sample times}
\\
\\
t_k=\delta(k-1),\ k=1,\ldots,100,\ \delta=(\pi/2)/99
\\
\\
\mbox{with pure input state}
\\
\\
\psi^{\rm init}=\ket{0}\;
\mbox{or}\;
\psi^{\rm init}=U_{\rm had}\ket{0}
\end{array}
\right.
\end{array}
\]
In this example, with a single parameter and a single input state, the
optimal experiment design problem \refeq{expdes th} becomes:
\[[eq:expdes single th]
\begin{array}{ll}
\mbox{minimize}
&
V(\lambda,\th^{\rm surr})
=
\left(
\sum_{\tau}\ \lambda_{\tau} g_{\tau}(\th^{\rm surr})
\right)^{-1}
\\
\mbox{subject to}
&
\sum_{\tau}\ \lambda_{\tau} = 1
\\
&
\lambda_{\tau} \geq 0,\;
\forall\ {\tau}
\end{array}
\]
The trace operation in \refeq{expdes th} is eliminated because the
matrix $G_{\beta\tau}(\th^{\rm surr})$ is now the scalar,
\[[eq:g sinlge th]
g_{\tau}(\th^{\rm surr})
=
\left.
{\sum_{\alpha=1}^m} \
\left(
\frac{
\left(\nabla_\th\ p_{\alpha\tau}(\th) \right)
\left(\nabla_\th\ p_{\alpha\tau}(\th) \right)^T
}
{p_{\alpha\tau}(\th)}
-
\nabla_{\th\th}\ p_{\alpha\tau}(\th)
\right)
\right|_{\th=\th^{\rm surr}}
\in{\mathbf R}
\]
The solution can be determined directly: concentrate all the
experiments at the recording time $t_\tau$ where $g_{\tau}(\th^{\rm surr})$ is a
maximum, specifically,
\[[eq:topt]
t^{\rm opt} = \set{t_s}{g_{s}(\th^{\rm surr}) \geq g_{\tau}(\th^{\rm surr}),\ \forall s,\tau}
\]
The following tables show the minimum number of experiments at the
optimal recording time, $t^{\rm opt}$, in order to achieve 0.01 accuracy
(deviation) in the parameter estimate with $\th^{\rm surr}=\th^{\rm true}$. The
two cases shown are for the two input states with the control set to
unity.
\[[eq:tables e1]
\begin{array}{cc}
\begin{array}{c}
\varepsilon = 1
\\
\psi^{\rm init}=\ket{0}
\\
\\
\renewcommand{\arraystretch}{1}
\begin{array}{|c||c|c|}
\hline
\th^{\rm surr}=\th^{\rm true} & t^{\rm opt}/(\pi/2) & {\ell_{{\rm expt}}}\\
\hline\hline
0.9 & 0.68 & 8,876\\
1.0 & 0.61 & 10,957\\
1.1 & 0.56 & 13,262\\
\hline
\end{array}
\end{array}
&
\begin{array}{c}
\varepsilon = 1
\\
\psi^{\rm init}=U_{\rm had}\ket{0}
\\
\\
\renewcommand{\arraystretch}{1}
\begin{array}{|c||c|c|}
\hline
\th^{\rm surr}=\th^{\rm true} & t^{\rm opt}/(\pi/2) & {\ell_{{\rm expt}}}\\
\hline\hline
0.9 & 1.0 & 2052\\
1.0 & 1.0 & 2069\\
1.1 & 1.0 & 2052\\
\hline
\end{array}
\end{array}
\end{array}
\]
To make the Hadamard the control update is $\varepsilon=1/{\hat{\theta}}$ which follows
from the (risky) assumption that the estimate, ${\hat{\theta}}$, is perfectly
correct. For any of the true values from the above table
\refeq{tables e1}, all the estimates have the same accuracy. Hence,
the average value of the worst-case gate fidelity after the update is
approximately,
\[
{\bf E} \
\min_{\norm{\psi}=1}
\left|
\left(U_{\rm des} \psi\right)^*
\left( U_{\rm act}\psi \right)
\right|^2
\approx
1-\left(\frac{\pi}{2}\right)^2\ (0.01^2)
=
1-0.00024672
=
0.999753
\]
The ease of obtaining the estimate by minimizing the negative
log-likelihood function can be determined by examining the {\em
average likelihood function},
\[[eq:avlike]
{\bf E} \ L(\th)
=
\sum_{\alpha,\gamma}
\ell_\gamma\
p_{\alpha\gamma}(\th^{\rm true})\
\log p_{\alpha\gamma}(\th)
\]
which is obtained from \refeq{maxlike th} and \refeq{av nalfgam}.
Figure \ref{fig:avlike_e1r1} shows plots of normalized\footnote{
The plots show $ {\bf E} \ L(\th)$ divided by its minimum value, thus
{\em normalized} to have a minimum of unity.
}
$ {\bf E} \ L(\th),\ 0.8 \leq \th \leq 1.2$ for the three true parameter
values and corresponding optimal recording times from the table in
\refeq{tables e1} with control $\varepsilon=1$ and initial state $\ket{0}
$. In all three cases, over the $\th$ range shown, $ {\bf E} \ L(\th)$ is
convex.
Figure \ref{fig:avlike_e1rh1} shows plots of normalized
$ {\bf E} \ L(\th),\ 0.8 \leq \th \leq 1.2$ for the three true parameter
values and corresponding optimal recording times from the table in
\refeq{tables e1} with control $\varepsilon=1$ and initial state
$U_{\rm had}\ket{0}$. For initial state $\ket{0}$, a range of 8876 to 13262
experiments are needed at the optimal recording time to achieve 0.01
deviation in the estimate. With the initial states $U_{\rm had}\ket{0}$ the
same accuracy only requires about 2000 experiments. This difference
can be inferred partly by comparing the curvature in the plots in
Figure \ref{fig:avlike_e1r1} with \ref{fig:avlike_e1rh1}; as they are
plotted on the same normalized scale. Note the increased curvature of
$ {\bf E} \ L(\th)$ in Figure \ref{fig:avlike_e1rh1} in the neighborhood of
the true value.
If we further increase the control effort, say to $\varepsilon=5$,
the number of experiments required to achieve 0.01 accuracy
is significantly reduced as seen in the following table.
\[[eq:table e5]
\begin{array}{c}
\varepsilon = 5
\\
\psi^{\rm init}=\ket{0}
\\
\\
\renewcommand{\arraystretch}{1}
\begin{array}{|c||c|c|}
\hline
\th^{\rm surr}=\th^{\rm true} & t^{\rm opt}/(\pi/2) & {\ell_{{\rm expt}}}\\
\hline\hline
0.9 & 0.93 & 98\\
1.0 & 0.84 & 121\\
1.1 & 1.00 & 122\\
\hline
\end{array}
\end{array}
\]
However, Figure \ref{fig:avlike_e5r1} shows clearly that the average
likelihood function is now significantly more oscillatory, and
certainly not convex over the range shown. It is of course convex in a
much smaller neighborhood of the true value. This would require,
therefore, very precise prior knowledge about the true system. Thus
we see a clear tradeoff between the number of experiments to achieve a
desired estimation accuracy and the ease of obtaining the estimate as
seen by the convexity, or lack thereof, with respect to minimizing the
likelihood function, which is the optimization objective.
\psfrag{th}{$\begin{array}{c}{}\\ \th\end{array}$}
\psfrag{th0t0=0.9,0.68}{$\th^{\rm true}=0.9,\ t^{\rm opt}/(\pi/2)=0.68$}
\psfrag{th0t0=1,0.61}{$\th^{\rm true}=1,\ t^{\rm opt}/(\pi/2)=0.61$}
\psfrag{th0t0=1.1,0.56}{$\th^{\rm true}=1.1,\ t^{\rm opt}/(\pi/2)=0.56$}
\begin{figure}[h]
\epsfysize=3.5in
\epsfxsize=6in
\epsfboxit{avlike_e1r1.ps}
\caption{
Normalized average likelihood function $ {\bf E} \ L(\th)$ with control
$\varepsilon=1$ and input state $\psi^{\rm init}=\ket{0}$ for the
true parameter values and associated optimal recording times as
indicated above and given in \refeq{tables e1}.
}
\label{fig:avlike_e1r1}
\end{figure}
\psfrag{th}{$\begin{array}{c}{}\\ \th\end{array}$}
\psfrag{th0t0=0.9,1}{$\th^{\rm true}=0.9,\ t^{\rm opt}/(\pi/2)=1$}
\psfrag{th0t0=1,1}{$\th^{\rm true}=1,\ t^{\rm opt}/(\pi/2)=1$}
\psfrag{th0t0=1.1,1}{$\th^{\rm true}=1.1,\ t^{\rm opt}/(\pi/2)=1$}
\begin{figure}[h]
\epsfysize=3.5in
\epsfxsize=6in
\epsfboxit{avlike_e1rh1.ps}
\caption{
Normalized average likelihood function $ {\bf E} \ L(\th)$ with control
$\varepsilon=1$ and initial state $\psi^{\rm init}=U_{\rm had}\ket{0}$
for the true parameter values and associated optimal recording times
as indicated above and given in \refeq{tables e1}.
}
\label{fig:avlike_e1rh1}
\end{figure}
\psfrag{th}{$\begin{array}{c}{}\\ \th\end{array}$}
\psfrag{th0t0=0.9,0.93}{$\th^{\rm true}=0.9,\ t^{\rm opt}/(\pi/2)=0.93$}
\psfrag{th0t0=1,0.84}{$\th^{\rm true}=1,\ t^{\rm opt}/(\pi/2)=0.84$}
\psfrag{th0t0=1.1,1}{$\th^{\rm true}=1.1,\ t^{\rm opt}/(\pi/2)=1$}
\begin{figure}[h]
\epsfysize=3.5in
\epsfxsize=6in
\epsfboxit{avlike_e5r1.ps}
\caption{
Normalized average likelihood function $ {\bf E} \ L(\th)$ with control
$\varepsilon=5$ and initial state $\psi^{\rm init}=\ket{0}$
for the true parameter values and associated optimal recording times
as indicated above and given in \refeq{table e5}.
}
\label{fig:avlike_e5r1}
\end{figure}
\clearpage
\section{Summarizing Maximum Likelihood Estimation
\& Optimal Experiment Design}
\label{sec:summary}
The results presented show that an efficient numerical method based on
convex programming is possible for optimizing the experiment for state
and process tomography. In addition, the estimation of the state
and/or process using data from non-continuing measurements is
copacetic with Maximum Likelihood Estimation. Both the experiment
design and estimation work naturally together and both can be solved
using convex optimization methods.
\subsection*{Maximum likelihood estimation}
The general form for estimating the parameter $\pi^{\rm true}$ is obtained as
the solution to the optimization problem,
\[[eq:mle gen]
\mathbox{
\begin{array}{ll}
\mbox{minimize}
&
L({\pi}) = -\sum_{\alpha,\gamma}\
n_{\alpha\gamma}\ \log\ p_{\alpha\gamma}({\pi})
\\&\\
\mbox{subject to}
&
{\pi} \in \Pi
\end{array}
}
\]
Under the assumption that the system generating the data is in the
model set used for estimation, the estimate, $\pi^{\rm ML}$, the
solution to \refeq{mle gen}, is unbiased and has the asymptotic
variance,
\[[eq:varmle]
\begin{array}{c}
{\bf E} \ \norm{\pi^{\rm ML}-\pi^{\rm true}}^2
\to
\frac{1}{{\ell_{{\rm expt}}}}\ V(\lambda,\pi^{\rm true})
\;\;\;
\mbox{as ${\ell_{{\rm expt}}}\to\infty$}
\end{array}
\]
where $\lambda$ is the vector of fraction of experiments per
configuration, and $V(\lambda,\pi^{\rm true})$ is obtained from the Cram\'{e}r-Rao\
inequality.
\subsection*{Optimal experiment design}
The general form for estimating the configuration distribution $\lambda$
is obtained as the solution to the optimization problem,
\[
\mathbox{
\begin{array}{ll}
\mbox{minimize}
&
V({\lambda},\hat{\pi}) =
{\rm Tr}\
\left(
\sum_{\gamma=1}^{{n_{\rm cfg}}}\ {\lambda}_\gamma\ G_\gamma(\hat{\pi})
\right)^{-1}
\\&\\
\mbox{subject to}
&
\sum_{\gamma=1}^{{n_{\rm cfg}}}\ {\lambda}_\gamma=1,
\;\;\;
{\lambda}_\gamma \geq 0
\end{array}
}
\]
where $\hat{\pi}$ is a surrogate for $\pi^{\rm true}$.
Tables \ref{tab:mle} and \ref{tab:oed} summarize the class of
estimation and experiment design problems, respectively.
\newpage
\begin{small}
\begin{table}[t]
\begin{tabular}{|c||c|c|c|}
\hline
&&&
\\
\blue{objective} & \blue{$p_{\alpha\gamma}(\pi)$} & \blue{$\pi\in\Pi$}
& \blue{comment}
\\
&&&
\\
\hline\hline
\black{\begin{tabular}{l}
Hamiltonian\\
parameter\\
estimation
\end{tabular}}
& $p_{\alpha\gamma}(\red{\th})={\rm Tr}\ O_{\alpha\gamma}(\red{\th})\rho_\gamma$
& $\norm{\red{\th}-\th_{\rm nom}} \leq \delta$
&
\begin{tabular}{l}
not convex\\
in $\red{\th}$\\
many local\\
minima
\end{tabular}
\\
\hline
\black{\begin{tabular}{l}
state\\
estimation
\end{tabular}}
&
$p_{\alpha\gamma}(\red{\rho}) = {\rm Tr}\ O_{\alpha\gamma}\ \red{\rho}$
&
${\rm Tr}\ \red{\rho}=1,\; \red{\rho} \geq 0$
& convex in $\red{\rho}$
\\
\hline
\black{\begin{tabular}{l}
state\\
distribution\\
estimation
\end{tabular}}
&
$
\begin{array}{l}
p_{\alpha\gamma}(\red{f})=a_{\alpha\gamma}^T \red{f}
\\
(a_{\alpha\gamma})_i = {\rm Tr}\ O_{\alpha\gamma}\ \rho_i
\end{array}
$
& $\sum_i \red{f_i}=1, \;\;\; \red{f_i}\geq 0$
& convex in $\red{f}$
\\
\hline
\black{\begin{tabular}{l}
OSR
\\
fixed basis
\\
($B_i$)
\end{tabular}}
&
$
\begin{array}{l}
p_{\alpha\gamma}(\red{X}) = {\rm Tr}\ R_{\alpha\gamma}\ \red{X}
\\
\left[ R_{\alpha\gamma} \right]_{ij} = {\rm Tr}\ B_j^* M_{\alpha\gamma} B_i\ \rho_\gamma
\end{array}
$
&
$
\begin{array}{l}
\red{X} \geq 0
\\
\sum_{i,j}\ \red{X_{ij}}\ B_i^*\ B_j = I
\end{array}
$
& convex in $\red{X}$
\\
\hline
\black{\begin{tabular}{l}
OSR
\\distribution
\\
($\bar{K}_i$ basis)
\end{tabular}}
&
$
\begin{array}{l}
p_{\alpha\gamma}(\red{q})
=
a_{\alpha\gamma}^T \red{q}
\\
(a_{\alpha\gamma})_i = {\rm Tr}\ M_{\alpha\gamma} \bar{K}_i \rho_\gamma \bar{K}_i^*
\end{array}
$
&
$\sum_i \red{q_i}=1, \;\;\; \red{q_i} \geq 0$
& convex in $\red{q}$
\\
\hline
\end{tabular}
\caption{
Summary of maximum likelihood estimation.
Except for Hamiltonian parameter estimation,
all other cases are convex optimization problems.
}
\label{tab:mle}
\end{table}
\end{small}
\begin{small}
\begin{table}[b]
\begin{tabular}{|c||c|c|}
\hline
&&
\\
\blue{objective} & \blue{$p_{\alpha\gamma}(\pi^{\rm surr})$}
&
\blue{$G_\gamma(\pi^{\rm surr})$}
\\
&&
\\
\hline\hline
\black{\begin{tabular}{l}
Hamiltonian\\
parameter\\
estimation
\end{tabular}}
& $p_{\alpha\gamma}(\blues{\th})={\rm Tr}\ O_{\alpha\gamma}(\blues{\th})\rho_\gamma$
&
$
{\sum_{\alpha=1}^m} \left(
\fracds{1}{p_{\alpha\gamma}}
(\nabla_\th\ p_{\alpha\gamma})(\nabla_\th\ p_{\alpha\gamma})^T
-\nabla_{\th\th}\ p_{\alpha\gamma}
\right)
$
\\
\hline
\black{\begin{tabular}{l}
state\\
estimation
\end{tabular}}
&
$p_{\alpha\gamma}(\blues{\rho}) = {\rm Tr}\ O_{\alpha\gamma}\ \blues{\rho}$
&
$
C_{\rm eq}^*\left(
{\sum_{\alpha=1}^m} \fracds{1}{p_{\alpha\gamma}}
(\vec\ O_{\alpha\gamma}) (\vec\ O_{\alpha\gamma})^*
\right)C_{\rm eq}
$
\\
\hline
\black{\begin{tabular}{l}
state\\
distribution\\
estimation
\end{tabular}}
&
$
\begin{array}{l}
p_{\alpha\gamma}(\blues{f}) = a_{\alpha\gamma}^T \blues{f}
\\
(a_{\alpha\gamma})_i = {\rm Tr}\ O_{\alpha\gamma} \rho_i
\end{array}
$
&
$
C_{\rm eq}^*\left(
{\sum_{\alpha=1}^m} \fracds{1}{p_{\alpha\gamma}} a_{\alpha\gamma} a_{\alpha\gamma}^T
\right)C_{\rm eq}
$
\\
\hline
\black{\begin{tabular}{l}
OSR
\\
fixed basis
\\
($B_i$)
\end{tabular}}
&
$
\begin{array}{l}
p_{\alpha\gamma}(\blues{X}) = {\rm Tr}\ R_{\alpha\gamma}\ \blues{X}
\\
\left[ R_{\alpha\gamma} \right]_{ij} = {\rm Tr}\ B_j^* M_{\alpha\gamma} B_i\ \rho_\gamma
\end{array}
$
&
$
C_{\rm eq}^*\left(
{\sum_{\alpha=1}^m} \fracds{1}{p_{\alpha\gamma}}
(\vec\ R_{\alpha\gamma}) (\vec\ R_{\alpha\gamma})^*
\right)C_{\rm eq}
$
\\
\hline
\black{\begin{tabular}{l}
OSR
\\distribution
\\
($\bar{K}_i$ basis)
\end{tabular}}
&
$
\begin{array}{l}
p_{\alpha\gamma}(\blues{q})
=
a_{\alpha\gamma}^T \blues{q}
\\
(a_{\alpha\gamma})_i = {\rm Tr}\ M_{\alpha\gamma} \bar{K}_i \rho_\gamma \bar{K}_i^*
\end{array}
$
&
$
C_{\rm eq}^*\left(
{\sum_{\alpha=1}^m} \fracds{1}{p_{\alpha\gamma}} a_{\alpha\gamma} a_{\alpha\gamma}^T
\right)C_{\rm eq}
$
\\
\hline
\end{tabular}
\caption[oed]{ Summary of optimal experiment designs. These are convex
optimization problems in all cases. The matrix $C_{\rm eq}$ comes from the
associated parameter equality constraints, \emph{e.g.},
${\rm Tr}\ \blues{\rho} =1,\
\sum_i \blues{f_i}=1$, $\sum_{i,j}\ \blues{X_{ij}}\ B_i^*\ B_j = I$, \emph{etc.}. }
\label{tab:oed}
\end{table}
\end{small}
\clearpage
\section{Iterative Adaptive Control}
\label{sec:iter adapt}
Quantum process tomography can be used for adaptive control design as
described in \cite{KosutRW:03}. Adaptive control systems are in
general one of two types \cite{AstromW:95,AndersonETAL:86}: (1) {\em
indirect adaptive control} -- use the data to first determine
parameters in a system model, then based on the model determine the
control parameters, and (2) {\em direct adaptive control} -- use the
data to directly select control parameters by comparing actual
performance to an ideal. Applications of direct adaptive control (and
learning principles) to quantum systems can be found in
\cite{PhanR:97,ZhuR:98,PhanR:99}. It could be argued that the direct
approach is simpler in that a system model is not needed.
However, this is a little deceptive because in effect the {\em
closed-loop system} is being modeled indirectly via the ideal
performance. This is made more explicit in the {\em
unfalsification/invalidation} approach to adaptive control,
\cite{SafonovT:97}, \cite{Kosut:01}.
\subsection{Indirect adaptive control}
\label{sec:indirect adapt}
Hamiltonian parameter estimation and associated optimal experiment
design can be combined in an iterative indirect adaptive control
approach as depicted in Figure \ref{fig:adapt}.
\begin{small}
\begin{figure}[h]
\unitlength 1.2mm
\thicklines
\centering
\begin{picture}(104,75)(-10,-15)
\put(0,0){\framebox(30,15){\footnotesize Controller}}
\put(30,7.5){\vector(1,0){20}}
\put(50,0){\framebox(30,15){\footnotesize System}}
\put(80,7.5){\line(1,0){10}}
\put(0,25){\framebox(30,15)
{\shortstack{\footnotesize Control\\ \footnotesize Design}}}
\put(50,25){\framebox(30,15)
{\shortstack{\footnotesize Model\\ \footnotesize Parameter\\ \footnotesize Estimator}}}
\put(-20,11){\vector(1,0){20}}
\put(-23,12){$r$}
\put(-10,4){\vector(1,0){10}}
\put(-10,4){\line(0,-1){14}}
\put(-10,-10){\line(1,0){100}}
\put(90,-10){\line(0,1){42.5}}
\put(90,32.5){\vector(-1,0){10}}
\put(40,7.5){\line(0,1){25}}
\put(40,32.5){\vector(1,0){10}}
\put(65,40){\line(0,1){10}}
\put(15,50){\vector(0,-1){10}}
\put(15,25){\vector(0,-1){10}}
\put(15,50){\line(1,0){10}}
\put(25,42.5){\framebox(30,15)
{wait}}
\put(65,50){\vector(-1,0){10}}
\put(70,50){$\th_ {(i+1)} $}
\put(8,50){${\hat{\theta}}_ {(i)} $}
\put(40,3){$\varepsilon_ {(i)} $}
\put(93,7.5){$n_ {(i)} $}
\end{picture}
\caption{Iterative adaptive control}
\label{fig:adapt}
\end{figure}
\end{small}
Typical steps in the iteration are:
\[[eq:indirect adapt]
\begin{array}{ll}
\mbox{control design}
&
\displaystyle
\varepsilon^ {(i)}
=
\arg{\rm opt}_{\varepsilon}\ J(\varepsilon,\th^ {(i)} )
\\&\\
\mbox{experiment design}
&
\displaystyle
\ell^ {(i)} = {\bf round}({\ell_{{\rm expt}}}\ \lambda^ {(i)} )
\\
&
\displaystyle
\lambda^ {(i)}
=
\arg\min
\sett{
V(\lambda,\varepsilon^ {(i)} ,\th^ {(i)} )
}{
\lambda\geq 0,\ \sum_\gamma\ \lambda_\gamma=1
}
\\&\\
\mbox{collect data}
&
\displaystyle
D^ {(i)} = \sett{ n_{\alpha\gamma}^ {(i)} }{
\alpha=1,\ldots,{n_{\rm out}},\ \gamma=1,\ldots,{n_{\rm cfg}}
}
\\&\\
\mbox{estimate parameters}
&
\displaystyle
\th^ {(i+1)}
=
\arg\min
\sett{ L(D^ {(i)} ,\th) }{ \norm{\th-\th_{\rm nom}} \leq \delta }
\end{array}
\]
The ``opt'' in the control design step could be to maximize worst-case
gate fidelity \refeq{wc fidelity},
\[[eq:max wc fid]
\varepsilon^ {(i)}
=
\max_\varepsilon\
\underbrace{
\min_{\norm{\psi}=1}
\left|
\left(U_{\rm des} \psi\right)^*
\left( U({t_{\rm f}},\varepsilon,\th^ {(i)} )\psi \right)
\right|^2
}_{\displaystyle J(\varepsilon,\th^ {(i)} )}
\]
where $U(t,\varepsilon,\th)$ is the propagator arising from the parametric
Hamiltonian model. The control design step is not necessarily convex.
Even if it were, it is not yet known under what conditions the
complete iterative procedure will converge to the optimal control, or
converge at all \cite{KosutRW:03}. For example, in the simulations to
follow convergence to the optimum is dependent on the initial
parametrization. The properties of this type of iteration remain an
area for further study.
\subsubsection*{Example}
The spin-coherent photon transmitter/receiver system proposed in
\cite{VriETAL:00,YabETAL:03} creates quantum logic gates by
manipulating electron spin via external potentials (gate voltages) to
effect the {\em g-factors} in the semiconductor material in the
presence of an external (rotating) magnetic field (\cite{Awsc:03}
explores {\em g-tensor} control without the rotating field). Following
\cite[III,Ch.12-9]{FeynmanLS:65} on models of spin systems, an
idealized model of the normalized Hamiltonian in the rotating frame of
a two-qubit gate under ``linear g-factor control'' is given by,
\[
\begin{array}{rcl}
H &=& H_1 + H_2 +H_{12}
\\&&\\
H_1
&=&
\frac{1}{2}
\left[
{\varepsilon_{1z}}
\omega_0(Z\otimes I_2)
+
{\varepsilon_{1x}}
\omega_1(X\otimes I_2)
\right]
\\&{}&\\
H_2
&=&
\frac{1}{2}
\left[
{\varepsilon_{2z}}
\omega_0(I_2\otimes Z)
+
{\varepsilon_{2x}}
\omega_1(I_2\otimes X)
\right]
\\&&\\
H_{12}
&=&
\displaystyle
{\varepsilon_c}
\omega_c
\left(
X^{\otimes 2}+Y^{\otimes 2}+Z^{\otimes 2}
\right)
\end{array}
\]
The design goal is to use the 5 controls
$
({
\varepsilon_{1z},\
\varepsilon_{1x},\
\varepsilon_{2z},\
\varepsilon_{2x},\
\varepsilon_c
}
)
$
to make the Bell transform,
\[
U_{\rm bell}
=
\frac{1}{\sqrt{2}}
\left[
\begin{array}{cccc}
1 & 0 & 1 & 0 \\
0 & 1 & 0 & 1 \\
0 & 1 & 0 & -1 \\
1 & 0 & -1 & 0
\end{array}
\right]
\]
One of the many possible decompositions of the Bell transform is the
following:
\[
U_{\rm bell}
=
(U_{\rm had} \otimes I_2)\
\sqrt{\Uswap}\
( X^{-1/2} \otimes X^{1/2} )
\sqrt{\Uswap}\
( I_2 \otimes X)
\]
Each operation in this sequence uses only the single qubit and swap
``gates'' produced by simultaneously pulsing the 5 controls as shown
in the following table.
\begin{figure}[h]
{
\renewcommand{\arraystretch}{2}
\[
\begin{array}{|cc|cc|c||c||c|}
\hline
\varepsilon_{1z} & \varepsilon_{1x} & \varepsilon_{2z} & \varepsilon_{2x} & \varepsilon_c & \Delta t & \mbox{gate}\\
\hline\hline
0&0&0&1&0&\frac{\pi}{{\omega_1}}&
-iI_2\otimes X \\
\hline
0&0&0&0&1&\frac{\pi}{8\omega_c}&
e^{-i\frac{\pi}{8}} \sqrt{\Uswap} \\
\hline
0&0&0&1&0&\frac{\pi}{2{\omega_1}}&
e^{-i\frac{\pi}{4}} I_2\otimes X^{1/2} \\
\hline
0&1&0&0&0&\frac{3\pi}{2{\omega_1}}&
e^{-i\frac{3\pi}{4}} X^{-1/2}\otimes I_2 \\
\hline
0&0&0&0&1&\frac{\pi}{8\omega_c}&
e^{-i\frac{\pi}{8}} \sqrt{\Uswap} \\
\hline
\frac{\omega_ {\rm had} }{\omega_0\sqrt{2}}&\frac{\omega_ {\rm had} }{{\omega_1}\sqrt{2}}&
0&0&0&
\frac{\pi}{\omega_ {\rm had} }&-iU_{\rm had}\otimes I_2 \\
\hline
\end{array}
\]
}
\caption{Pulse control table}
\end{figure}
The resulting gate at the final time, ${t_{\rm f}}$, is $U_{\rm bell}$ to within a
scalar phase:
\[
U({t_{\rm f}})
=
e^{-i\frac{\pi}{4}}\ U_{\rm bell},
\;\;\;
{t_{\rm f}}
=
\left(
\frac{3}{{\omega_1}}+\frac{1}{4\omega_c}+\frac{1}{\omega_ {\rm had} }
\right)\pi
\]
Suppose the only unknown parameter is $\omega_1$. Consider the following
simplified version of \refeq{indirect adapt}:
\[
\begin{array}{ll}
\mbox{control design}
&
\varepsilon^ {(i)}
=
\bar{\eps}(\hat{\omega}_1^ {(i)} ),
\;\;
{t_{\rm f}}^ {(i)} = {t_{\rm f}}(\hat{\omega}_1^ {(i)} )
\\
\mbox{estimation}
&
\hat{\omega}_1^ {(i+1)}
=
\displaystyle
\arg\min_{\omega_1}\
{\bf E} \ L(\omega_1,\ \varepsilon^ {(i)} )
\end{array}
\]
where the control design function $\bar{\eps}(\hat{\omega}_1^ {(i)} )$ represents the
pulse design from the above table, and where the average likelihood
function follows from the description in Section \refsec{mle hampar}
with the following parameters:
\[
\begin{array}{ll}
\mbox{single initial state}
&
\rho^{\rm init}=\ket{0}\bra{0}
\;\;
(\beta=1)
\\
\mbox{POVM}
&
M_1=\ket{0}\bra{0},
\;\;
M_2=\ket{1}\bra{1}
\;\;
({n_{\rm out}}=2)
\\
\mbox{sample times}
&
\mbox{
either $\seq{{t_{\rm f}}(\hat{\omega}_1),\ ({n_{\rm sa}}=1)}$
or
$\seq{{t_{\rm f}}(\hat{\omega}_1)/2,\ {t_{\rm f}}(\hat{\omega}_1),\ ({n_{\rm sa}}=2)}$
}
\end{array}
\]
Using Hamiltonian parameters $(\omega_0^{\rm true}=1,\ \omega_1^{\rm true}=0.01,\
\omega_c^{\rm true}=0.01)$, Figure \reffig{avlike2cd2eli} shows $ {\bf E} \
L(\hat{\omega}_1,\ {n_{\rm sa}}=1)$ vs. $\hat{\omega}_1/\omega_i^{\rm true}$ for sequences of adaptive
iterations using the estimate $\hat{\omega}_1$ obtained from a local hill
climbing algorithm, \emph{i.e.}, the local maximum of the average likelihood
function is obtained. The estimation is followed by a control using
the estimated value in the pulse control table. In the two cases shown
the algorithm converges to the true value.
\psfrag{avlike}{$ {\bf E} \ L(\hat{\omega}_1,\rho_0)$}
\psfrag{om1}{$\hat{\omega}_1/\omega_1$}
\begin{figure}[h]
\begin{tabular}{cc}
\epsfysize=5.5in
\epsfxsize=3in
\epsfmaxlike{avlike2cd2eli01.ps}
&
\epsfysize=5.5in
\epsfxsize=3in
\epsfmaxlike{avlike2cd2eli02.ps}
\end{tabular}
\caption{\small Iterative adaptation for ${n_{\rm sa}}=1$ at ${t_{\rm f}}(\hat{\omega}_1)$ for
two starting values of $\hat{\omega}_1$.}
\label{fig:avlike2cd2eli}
\end{figure}
Although not shown, the algorithm does not converge from all initial
values of $\hat{\omega}_1$. Figure \reffig{directerr01} shows
$\norm{U({t_{\rm f}}(\hat{\omega}_1),\ \bar{\eps}(\hat{\omega}_1))-U_{\rm des}}_{\rm frob}$ vs. estimate
$\hat{\omega}_1/\omega_1^{\rm true}$ with the control from the table. The function is
clearly not convex. The region of convergence for ${n_{\rm sa}}=1$ and
${n_{\rm sa}}=2$ sample times are shown in blue and green, respectively. The
region of attraction is increased for ${n_{\rm sa}}=2$. These results, of
course, are specific to this example and can not be generalized. To
re-iterate, conditions for convergence, region of attraction, and so
on, are only partially understood, in general, for this type of
iteration \cite{HjalGD:96}.
\psfrag{norm(uf-udes)}{}
\psfrag{om1h}{$
\begin{array}{l} {}\\ \hat{\omega}_1/\omega_1 \end{array}
$}
\psfrag{nsa=1}{\blue{${n_{\rm sa}}=1$ @ ${t_{\rm f}}$}}
\psfrag{nsa=2}{\green{${n_{\rm sa}}=2$ @ $[{t_{\rm f}},\ {t_{\rm f}}/2]$}}
\begin{figure}[h]
\begin{tabular}{c}
$\norm{U({t_{\rm f}}(\hat{\omega}_1),\ \bar{\eps}(\hat{\omega}_1))-U_{\rm des}}_{\rm frob}$
\\
\epsfysize=4in
\epsfxsize=6.25in
\epsfmaxlike{directerr01.ps}
\end{tabular}
\caption{Regions of convergence.}
\label{fig:directerr01}
\end{figure}
\subsection{Direct adaptive control}
\label{sec:direct adapt}
In a direct adaptive control system no model is posed for the system;
ideally only a performance measure is available and the control
parameters are adjusted to improve the performance. The adjustment
``directions'', however, clearly must depend on the shape of the
``landscape'', otherwise, it would not be possible to know how to make
the adjustment. In effect then, a model of the landscape is either
available or is computed intrinsically. Consider the following
bipartite system whose Hamiltonian depends on the two controls
$(\varepsilon_z,\ \varepsilon_x)$.
\[
\begin{array}{rcl}
H(\varepsilon) &=& H_Q(\varepsilon)\otimes I_E+I_Q\otimes H_E + H_{QE}
\\
H_Q(\varepsilon)
&=&
(\varepsilon_z-1)\omega_{Qz}Z/2 + \varepsilon_x\omega_{Qx}X/2
\\
H_E &=& \omega_{Ez}Z/2 + \omega_{Ex}X/2
\\
H_{QE} &=& \omega_{QE}
\left(
X^{\otimes 2}+Y^{\otimes 2}+Z^{\otimes 2}
\right)
\end{array}
\]
The Q-part of the system is assumed to be accessible to the user and
the E-part, the ``environment'', is not. The goal is to select the
controls to make the Q-system behave as a {\em bit-flip} device, \emph{i.e.},
the Pauli $X$ matrix. Suppose the Q-system is prepared in the initial
state $\rho_Q^{\rm init}=\ket{1}\bra{1}$ and a measurement is made at
${t_{\rm f}}=\pi/\omega_{Qx}$ of the state $\ket{0}$. hence, ideally, the outcome
probability, $p(\varepsilon)$, should be unity. Due to the uncontrolled
$E$-system coupling, however, the goal is to select the controls
$\varepsilon$ to make $p(\varepsilon)$ as large as possible. Under these conditions
the outcome probability $p(\varepsilon)$ arises from,
\[
\begin{array}{rcl}
p(\varepsilon) &=& {\rm Tr}\ M U({t_{\rm f}})\rho_0 U({t_{\rm f}})^*
\\
i\dot U(t) &=& H(\varepsilon(t)) U(t),\ U(0) = I,\ 0\leq t\leq{t_{\rm f}}
\\
\rho_0 &=& \rho_Q^{\rm init} \otimes \rho_E^{\rm init}
\end{array}
\]
Of course the system Hamiltonian is not known, only at best the
outcome probability would be known after enough repetitions of the
experiment.
Figure \ref{fig:angel} shows the landscape of the system with
parameters,
$
\omega_{Qz}=1,\ \omega_{Qx}=0.01,\
\omega_{Ez}=1,\ \omega_{Qx}=0,\
\omega_{QE}=0.005,\
\rho_E^{\rm init}=I_2/2
$
and for {\em constant} values of the controls over the ranges $0.96
\leq \varepsilon_z \leq 1.04,\ 0.1 \leq \varepsilon_x \leq 5.2$.
\psfrag{pz}{$p(\varepsilon)$}
\psfrag{ex}{$\varepsilon_z$}
\psfrag{ey}{$\varepsilon_x$}
\begin{figure}[h]
\centering
\epsfysize=3.5in
\epsfxsize=6.5in
\epsfboxit{angela.eps}
\caption{Two parameter landscape; the maximum probability
of 0.96 is achieved
with $\varepsilon_z=0$ and $\varepsilon_x=5.2$.}
\label{fig:angel}
\end{figure}
\iffalse
rho0=kron(ket1*ket1',s0/2);
omq0=1;omq1=.01;
ome0=1;ome1=0.0;
omqe=.005;
meas=kron(ket0*ket0',s0);
tf=pi/omq1;
e0=[.96:.002:1.04]';e1=[.1:.02:5.2]';
hq=(e0(ie0)-1)*omq0*sz/2+e1(ie1)*omq1*sx/2;
he=ome0*sz/2+ome1*sx/2;
hqe=omqe*ss;
h=kron(hq,s0)+kron(s0,he)+hqe;
uf=expm(-jay*tf*h);
rhof=uf*rho0*uf';
pf(ie0,ie1)=real(trace(meas*rhof));
\fi
The landscape clearly has several local maximum values. Thus without
some knowledge of the landscape or an exhaustive search, it might be
difficult to find the global maximum. In this two-parameter case, of
course, an exhaustive search is not too exhausting. Nevertheless, it
is clear that any direct adaptive algorithm will face some
difficulties. It is also clear that prior knowledge can alleviate
many of the difficulties, \emph{e.g.}, knowledge of system parameter ranges or
nominal response which is close to a good outcome, \emph{etc.}.
A more in-depth analysis of the landscape for control of quantum
systems can be found in \cite{RabitzHR:04}. There it is shown that for
unconstrained time-varying controls, if the system is controllable,
then {\em all} the local maximum are global. That is, the outcome
probability at every local maximum is unity and all other extrema give
the minimum probability of zero. The complexity shown here results
principally from the fact that the choice of controls is constrained
to be constant. This points out the importance of constraints either
imposed by the physics or by the designer in inadvertently making the
"wrong" choice of the control structure. In particular, suppose that
the landscape is actually very simple with even possibly one extremum
when viewed with no constraints on the controls. However, when
constraints are imposed, or a new set of controls is defined, the
landscape may then exhibit structure that was not evident in the
freely floating original set. An extreme opposite case could be for a
choice of variables where the new variables actually do hardly
anything at all with regard to control action. In this case one would
conclude that the landscape is totally flat!
Many of the current "working" adaptive feedback control experiments
are operating somewhere between these two extremes, ranging from
having highly constrained knobs to the totally wrong knobs. This
comment comes from the observation that physical effects spanning a
dynamic range in quantum wavelength of about $10^7$ are being
controlled by a single type of laser, \emph{i.e.}, the Ti:Sapphire laser,
working over a range of about 1-10 on that scale. That is, a domain
in laser wavelength of 10 (in some units) is split up into about 100
small pieces as controls, and those very narrow controls manage
everything over a huge dynamic range. The fact that the experiments
work at all is rather amazing. A guess is that an examination of the
landscapes will show considerable detail, much of which is likely
false structures arising from having highly constrained controls.
From a positive perspective, as more bandwidth becomes available the
control landscape will become less complicated and more regular in the
sense that more of the local optimum values will provide performance
close to the global optimum.
\begin{appendix}
\section{Appendix}
\subsection{Worst-case gate fidelity}
\label{sec:fidelity}
From the definition of worst-case fidelity \refeq{wc fidelity},
\[
\begin{array}{rcl}
\left|
\left(U_{\rm des} \psi\right)^*
\left( U_{\rm act}\psi \right)
\right|^2
&=&
\left|
\psi^* (U_{\rm des}^*U_{\rm act})\psi
\right|^2
\\
&=&
\left|
(V^*\psi)^*\Omega(V^*\psi)
\right|^2
\\
&=&
\left|
\sum_{k=1}^n\ \omega_k\ |x_k|^2
\right|^2
\\
&=&
\left|
\sum_{k=1}^n\ \omega_k\ z_k
\right|^2
\end{array}
\]
where the last three lines follow directly from (i) the eigenvalue
decomposition of the unitary: $U_{\rm des}^*U_{\rm act}=V\Omega V^*,\ V^*V=I_n,\
\Omega={\rm diag}(\omega_1\ \cdots\ \omega_n),\ |\omega_k|=1$, (ii) defining
$x=V^*\psi\in{\mathbf C}^n$, and (iii) defining $z_k=|x_k|^2\in{\mathbf R}$. Using
the definitions of the vectors $(a,b)$ in \refeq{fidelity qp} gives,
\[
\left|
\left(U_{\rm des} \psi\right)^*
\left( U_{\rm act}\psi \right)
\right|^2
=
\left|
\sum_{k=1}^n\ (a_k+ib_k)\ z_k
\right|^2
=
z^T(aa^T+bb^T)z
\]
The QP follows from the relations:
\[
\norm{\psi}=1\
\Leftrightarrow\
\norm{x=V^*\psi}=1\
\Leftrightarrow\
\sum_k\ (z_k=|x_k|^2) = 1,\ z_k\geq 0
\]
\subsection{Cram\'{e}r-Rao\ Inequality}
\label{sec:crao}
The following is the classical form of the Cram\'{e}r-Rao\ Inequality.
\begin{quote}
{\bf Cram\'{e}r-Rao Inequality}\cite{Cramer:46}
\\
Let $\th_0\in{\mathbf R}^p$ be the true parameter to be estimated from a data
set $D$. Let $L(D,\th_0)$ be the true negative log-likelihood function
of the system generating the data. Let ${\hat{\theta}}\in{\mathbf R}^p$ be an unbiased
estimate of $\th_0$, \emph{i.e.}, $ {\bf E} \ {\hat{\theta}}=\th_0$. Then, the covariance of
the estimate,
\[[eq:covth]
{\bf cov}{{\hat{\theta}}}
=
{\bf E} {
\left(
{\hat{\theta}}-\th_0
\right)
\left(
{\hat{\theta}}-\th_0
\right)^T
}
\]
satisfies the matrix inequality,
\[[eq:cramer1]
\left[
\begin{array}{cc}
{\bf cov}\ {\hat{\theta}} & I
\\
I & F(\th_0)
\end{array}
\right]
\geq 0
\]
where $F(\th_0)$ is the {\em Fisher information matrix},
\[[eq:fisher]
F(\th_0)
=
{\bf E} \
\nabla_{\th\th}
L(D,\th)
\bigg|_{\th=\th_0}
\]
If $F(\th_0)>0$, then \refeq{cramer1} is equivalent to,
\[[eq:cramer2]
{\bf cov}{{\hat{\theta}}}
\geq
F(\th_0)^{-1}
\]
\end{quote}
This famous theorem states the for any unbiased estimator, the
covariance of the estimate satisfies the inequality \refeq{cramer1},
or equivalently \refeq{cramer2}, provided the Fisher matrix is
invertible. Usually only \refeq{cramer2} is given as the theorem: the
minimum covariance of the estimate is given by the inverse of the
Fisher information matrix. The power of the result \refeq{cramer1} is
that it is independent of {\em how} the estimate is obtained. The
lower bound only depends on the model structure, the experiment
design, and the information in the data.
\subsection{Derivation of \refeq{var rho}}
\label{sec:var rho}
Define the vectors $r,\ a_{\alpha\gamma}\in{\mathbf C}^{n^2}$ as,
\[
r=\vec\ \rho,
\;\;\;
a_{\alpha\gamma}=\vec\ O_{\alpha\gamma}
\]
Then
\[
p_{\alpha\gamma}={\rm Tr}\ O_{\alpha\gamma}\rho = a_{\alpha\gamma}^* r
\]
and
\[
\begin{array}{rcl}
{\rm Tr}\ \rho=1
&\Leftrightarrow&
b^T r=1,
\;\;
b=\vec\ I_n
\end{array}
\]
The next step eliminates the equality constraint $b^Tr=1$ reducing the
$n^2$ unknowns in $r$ to $n^2-1$. Since $b^Tb=n$, the SVD of $b$ is,
\[
b=W\left[\begin{array}{c} \sqrt{n} \\ 0_{n^2-n} \end{array} \right]
\]
with unitary $W\in{\mathbf R}^{n^2\times n^2}$, Partition $W=[c\ C_{\rm eq}]$ with
$C_{\rm eq}\in{\mathbf R}^{n^2\times n^2-1}$. Then all $r$ satisfying $b^Tr=1$ are
given by
\[
r=c/\sqrt{n} + C_{\rm eq} z
\]
for all $z\in{\mathbf C}^{n^2-1}$. The likelihood function with the equality
constraint (${\rm Tr}\ \rho=1$ or $b^Tr=1$) eliminated is then a function
only of $z$,
\[
L(D,z)=-\sum_{\alpha,\gamma} n_{\alpha\gamma} \log (c/\sqrt{n} + C_{\rm eq} z)
\]
To obtain the Cram\'{e}r-Rao\ bound, we first compute,
\[
\nabla_{zz}L(D,z)
=
\sum_{\alpha,\gamma}
\frac{n_{\alpha\gamma}}{p_{\alpha\gamma}(z)^2}
(C_{\rm eq}^T a_{\alpha\gamma})(C_{\rm eq}^T a_{\alpha\gamma})^*
\]
Using \refeq{av nalfgam}, $ {\bf E} \ n_{\alpha\gamma}=\ell_\gamma
p_{\alpha\gamma}(\rho^{\rm true}),\ \rho^{\rm true}=c/\sqrt{n}+C_{\rm eq} z^{\rm true}$, and hence
the Fisher information matrix is, with respect to $z$,
\[
\begin{array}{rcl}
F &=& {\bf E} \ L(D,z=z^{\rm true})
\\
&=&
\displaystyle
\sum_{\alpha,\gamma}
\frac{\ell_\gamma}{p_{\alpha\gamma}(\rho^{\rm true})}
(C_{\rm eq}^T a_{\alpha\gamma})(C_{\rm eq}^T a_{\alpha\gamma})^*
\\
&=&
G(\ell,\rho^{\rm true})
\end{array}
\]
where the last line comes from the definition of $G(\ell,\rho^{\rm true})$ in
\refeq{var rho 1}. Let $\hat{r}=c/\sqrt{n}+C_{\rm eq}\hat{z}$ be an unbiased
estimate of $r^{\rm true}$. Thus,
\[
{\bf cov}\ \hat{r} = C_{\rm eq} {\bf cov}\ \hat{z}\ C_{\rm eq}^T \geq C_{\rm eq} F^{-1} C_{\rm eq}^T
\]
Using ${\bf var}\ \hat{\rho} = {\bf var}\ \hat{r} ={\rm Tr}\ {\bf cov}\ \hat{r}$ and the fact that $W$ is
unitary, and hence, $C_{\rm eq}^TC_{\rm eq}=I_{n^2-1}$, we get,
\[
{\bf var}\ \hat{\rho} \geq {\rm Tr}\ C_{\rm eq} F^{-1} C_{\rm eq}^T = {\rm Tr}\ F^{-1}
\]
which is the final result \refeq{var rho}-\refeq{var rho 1}.
\subsection{Derivation of \refeq{expdes osr}}
\label{sec:expdes osr der}
Define $x,\ r_{\alpha\gamma}\ \in{\mathbf C}^{n^4}$ as,
\[
x = \vec\ X,
\;\;
r_{\alpha\gamma} = \vec\ R_{\alpha\gamma}
\]
The likelihood function in \refeq{mle x} can then be written as,
\[
L(D,x)=-\sum_{\alpha,\gamma}\ n_{\alpha\gamma} \log r_{\alpha\gamma}^* x
\]
and the equality constraint in \refeq{mle x} becomes,
\[
Ax=\vec\ I_n,
\;\;
A=[a_1\ \cdots\ a_{n^4}]\in{\mathbf C}^{n^2 \times n^4}
\]
with $a_k=\vec(B_i^*B_j)\in{\mathbf C}^{n^2}$ for $k=i+(j-1)n^2,\
i,j=1,\ldots,n^2$. Perform the singular value decomposition
$A=U[S\ 0]W^*,\ W=[C\ C_{\rm eq}]$ with $C\in{\mathbf C}^{n^2\times n^2},\,
C_{\rm eq}\in{\mathbf C}^{n^2\times n^4-n^2}$ as given in \refeq{expdes osr 2}.
From the definition of the basis functions \refeq{basis} it follows
that $S=\sqrt{n}\ I_{n^2}$. Observe also that the columns of $C_{\rm eq}$
are a basis for the nullspace of $A$. Hence all $x$ satisfying the
equality constraint are given by,
\[
x=\bar{x}+C_{\rm eq} z,\; \forall z
\;
\mbox{with}
\;
\bar{x}=(1/\sqrt{n}) C U^*\vec\ I_n
\]
The likelihood function with the equality constraint ($Ax=\vec\ I_n$)
eliminated is then a function only of $z$,
\[
L=-\sum_{\alpha,\gamma} n_{\alpha\gamma} \log (\bar{x} + C_{\rm eq} z)
\]
This is exactly the same form of the likelihood function in
\refsec{var rho} after the single equality constraint there is
eliminated. Hence, to obtain the Cram\'{e}r-Rao\ bound \refeq{expdes osr}
repeat, {\em mutatis mutandis}, the procedure in \refsec{var rho}.
\end{appendix}
\bibliographystyle{plain}
|
1,116,691,499,273 | arxiv | \section*{Acknowledgements}
This work was supported by the Robert A. Welch Foundation.
\section*{References}
|
1,116,691,499,274 | arxiv | \section{INTRODUCTION \label{section_introduction}}
The Large Synoptic Survey Telescope (LSST) has been designed to provide a deep six-band (0.3-1.1 $\micron$) astronomical
imaging survey of more than $20,000$ square degrees of the southern sky. Using active optics, the 8.4-meter aperture and 9.6 square degree
field-of-view of the telescope will allow $\sim1000$ visits to each patch of sky in ten years with a final
depth reaching $r\sim27.5$ and a mean delivered PSF better than 0.7$\arcsec$ (LSST Science Collaborations et al. 2009).
The LSST project will deliver fully-calibrated, science quality images,
catalogs, and derived data products to the US public with no proprietary period.
We refer readers to LSST Science Book II (LSST Science Collaboration 2009) for extensive discussions on the science that the survey
will enable.
Among the most critical science applications that the LSST will revolutionize is weak gravitational lensing.
The key observable in weak gravitational lensing is the shape of distant galaxies weakly distorted by foreground masses. Because each galaxy is sheared only by
a small amount, the lensing signal must be extracted from an average over a population of galaxies. Consequently, the total number of resolved galaxies at the
usable surface brightness limit
is one
useful measure to quantify the effectiveness of an astronomical survey for weak-lensing science. Another important factor is the number density of
usable galaxies (shear noise weighted effective number), which determines the lower limit of the angular scale that lensing can probe. The LSST project is designed to maximize
the scientific return balancing both the total number (area) and the number density within the ten-year nominal mission period.
A sky coverage of apptroximately $20,000$ square degrees is required to keep the sample variance below the statistical limit, and the
high-resolution ($<0.7\arcsec$ median delivered seeing), deep imaging ($r\sim27.5$), providing $>40$ high S/N galaxies per square arcmin enabling a reliable
probe of matter structure on galaxy cluster scales. Residual uncorrected PSF shape variation requirements are summarized in the Science Requirement
Documents: http://www.lsst.org/filesdocs/SRD.pdf.
These performance requirements determine the desired level of accuracy in point spread function (PSF) correction. The effect of the PSF on
the galaxy shape measurement cannot be overemphasized. Anisotropic PSFs mimic gravitational lensing signal, generating a coherent alignment
of galaxy shapes. Even without this anisotropy, PSFs circularize shapes of barely resolved galaxies
and dilute the intrinsic lensing signal. For various reasons, the observed
PSF changes in size and anisotropy across the field and also over time. Moreover, we need the PSF extrapolated to the positions of
galaxies, whereas the PSF model is constructed from the high S/N stars that are much less densely distributed.
Therefore, in addition to very low surface brightness limits, a key factor in precision lensing lies in one's ability to carefully
model and remove the complicated effects of PSFs.
Although the seeing at the LSST site is among the best in existing ground-based facilities, the accurate description of the PSF
is highly challenging because of the optical and focal plane design. As described below, the telescope optics and camera are continuously aligned via
wavefront sensing. Optimized for the unprecedented large field of view, the effective
$f$-ratio of LSST is small $f/\sim1.2$, which makes the optical aberration highly sensitive to alignment and focal plane deviation. Because
LSST's focal plane will be tiled with 189 4k $\times$ 4k CCDs, any height fluctuation both across and within the CCDs will translate into a very
complicated PSF pattern, which is characterized by abrupt changes across the CCD gaps and smooth, but possibly high-frequency variations within the CCDs.
Any sub-optimal modeling of these PSFs will leave systematic residuals on the scales of the CCD sizes and the ``potato chip'' effect\footnote{
The substrate bonding and packaging process of the CCD fabrication induces a height variation, which leads to non-negligible focus variation (thus
variation in aberration) within a CCD.}.
These
systematic residuals, mimicking lensing signals, will, of course, hamper the correct interpretation of the lensing analysis.
Unfortunately, with the existing algorithms the ability to precisely describe the PSF alone does not guarantee the success in the extraction
of gravitational shear to the accuracy that future LSST-like weak-lensing surveys require. This obstacle is well noted in the recent
large collaborative shear measurement campaign GREAT08 Challenge (Bridle et al. 2010). The campaign let the participants perform
blind shear measurements on simulated images, where the PSF was known, but the input shear was unknown. Although many algorithms
were shown to provide the accuracy for high S/N images
suitable for the existing survey data, no method came close to the target accuracy $Q\sim1000$ when the noise level of the images matched
the realistic value (see Bridle et al. 2010 for the definition of $Q$). Nevertheless, it is important to note that the main challenge is
purely mathematical/statistical and thus improvable as more mathematical sophistication is incorporated in the algorithms.
For example, Bernstein (2010) claims that when the bias arising
when the true galaxy profiles do not match the models being fit is properly addressed, the resulting algorithm can achieve $Q\sim3000$
for high S/N galaxies of the GREAT08 sample.
We launched the LSST shear calibration project in order to diagnose the key factors in the telescope and camera
engineering specifications affecting the weak-lensing science and to develop
new algorithms for optimal shear extraction in the presence of different combinations of systematics. The current paper is the first in the series of this topic with an emphasis on the realization, characterization, and reconstruction of LSST PSFs.
The paper is organized as follows. In \textsection\ref{section_lsst_optics}, we provide a brief review on the telescope optics.
Our implementation of the atmospheric turbulence is
described in \textsection\ref{section_atmosphere}. In Section \textsection\ref{section_focal_plane}, we discuss the focal plane design of LSST and the resulting behavior of the PSF. \textsection\ref{section_pca} presents our algorithm to measure and reconstruct the PSFs. In \textsection\ref{section_simulation}, we describe our simulation. Finally, the detailed comparison between the observed and modeled PSFs is presented in
\textsection\ref{section_analysis} before the conclusion in \textsection\ref{section_conclusion}.
\section{LSST ENVIRONMENT AND OPTICS}
\label{section_lsst_optics}
Figure~\ref{fig_lsst_optics} shows the optical design of the LSST, which is a modified Paul-Baker three-mirror system.
The active optics telescope consists of an 8.4 m f/1.18 concave primary, a 3.4 m f/1.0 convex secondary, and 5.2 m f/0.83 concave tertiary mirrors.
The final focal ratio f/1.23 (focal length of 10.3 m)
gives a plate scale $0.0197\arcsec/\micron$ over the 64 cm diameter focal plane ($3.5\degr \times 3.5\degr$).
The 27\% obscuration yields a total effective light collecting area of 35 $\mbox{m}^2$, which corresponds to an effective
6.7 m diameter clear circular aperture. Current wide field telescopes were not designed for the stringent PSF demands of LSST weak lensing goals.
This optical design provides a wide field of view while a maintaining uniform image quality across the field (Seppala 2002). The remarkably small variation of the
PSF size over the field is illustrated in Figure~\ref{fig_lsst_ee}, where the diffraction-limited PSF images are generated with ZEMAX at the nominal
flat focal plane.
The peak-to-valley 80\% encircled energy radius is within $\sim7$\% of the
mean value from the center to $\sim1.4\degr$ for simulated $i$ filter point sources. The maximum deviation ($\sim17$\%) is seen
at the edge of the field ($1.75\degr$).
The image brightness is stable across the field, providing a nearly uniform illumination within
a radius of 1.2$\degr$ and about 10\% decrease at the edge (Seppala 2002; LSST Science Collaborations 2010).
Finally, the LSST field distortion is remarkably small: less than 0.1\% over the full field.
LSST will have continuous correction of its optics, using curvature wavefront sensing. Four special purpose rafts, mounted at the corners of the
science array, contain wavefront sensors and guide sensors (right panel of Figure \ref{fig_lsst_optics}). Wavefront measurements are accomplished using curvature sensing,
in which the spatial intensity distribution of de-focused stars is measured at equal distances on either side of focus. Each curvature sensor is composed
of two CCD detectors, with one positioned slightly above the focal plane, the other positioned slightly below the focal plane.
The CCD technology for the curvature sensors is identical to that used for the science detectors in the focal plane, except that the curvature
sensor detectors are half-size so they can be mounted as an in-out defocus pair. Detailed analyses (Manuel et al. 2010) have verified that this configuration can reconstruct the wavefront within the $\lesssim0.2\mu$m accuracy.
These four corner rafts also hold two guide sensors each. The guide sensors monitor the locations of bright stars at a frequency of $\sim10$ Hz to provide feedback for a loop
that controls and maintains precision tracking of the telescope during an exposure.
The fast focal ratio and the rapid pointing changes make any hardware-based atmospheric dispersion correction technically difficult.
Consequently, the effect of the atmospheric dispersion sets the maximum angle away from the zenith. We estimate the
atmospheric differential refraction for the six proposed filters ($u,g,r,i,z,$ and $Y$) using the models summarized in Filippenko (1982).
Figure~\ref{fig_atmospheric_dispersion} displays the results for the input parameters of $f=8$ mm Hg (water vapor pressure), $T=5\degr$C (atmosphere temperature), and
$P=520$ mm Hg (atmospheric pressure), which are the approximate average conditions at Cerro Pach\'{o}n (Claver et al. 2004).
Because the weak-lensing shape measurements will be carried out in $r$ and $i$ band images, it is critical for the $r$ and $i$ band imaging to provide the excellent seeing
while still covering the required sky area of 20,000 square degrees.
From Cerro Pach\'{o}n, the required 20,000 square degrees of survey area are viewable above a zenith angle of 45 degrees.
At this 45$\degr$ limit, the corresponding dispersion in the $r$ and $i$ bands is less than $\sim0.3\arcsec$, which elongates the PSF in the zenith angle direction by $\sim9$\% and gives a coherent PSF ellipticity of $\lesssim0.04$ for $\sim0.7\arcsec$ seeing.
This amount of ellipticity by atmospheric dispersion is small and comparable to the values induced by the optical aberrations (after atmospheric turbulence). Therefore, we conclude
that the LSST design does not require atmospheric dispersion corrector (ADC).
\section{IMPLEMENTATION OF ATMOSPHERIC TURBULENCE}
\label{section_atmosphere}
After the optics and camera are corrected by the wavefront sensing, the largest residuals are due to the atmosphere and the focal plane non-flatness.
The simulation of the atmospherically distorted wavefront is the first step for the generation of the realistic PSF of LSST.
A typical short exposure image of a star through atmospheric turbulence can be viewed as a superposition of a number of
speckles. The spatial extent of each speckle is roughly determined by the diffraction limit of the instrument whereas
the relative centroid shift between these speckles reflects the severity of the turbulence mainly depending
on the amplitude of the low frequencies. To mimic these effects numerically, we follow the well-known classical approach which
regards the atmosphere as consisting of multiple moving layers each with
independent phase screen and velocity.
Although a single phase screen has been shown to provide many characteristics of the real environments qualitatively,
the wind speed must be unrealistically high $\sim100~\mbox{km~s}^{-1}$ in order to correctly represent the time scale of the observed speckle
changes.
In principle, the turbulence affects both the magnitude and phase of the wavefront. However, because
the phase distortion has a much larger effect for non-saturated scintillation (Fried 1994), in this paper we only consider the phase effect on the PSF behavior.
The power spectrum of the phase screen is usually described by the Kolmogorov/von K{\'a}rm{\'a}n model (Kolmogorov 1941; von K{\'a}rm{\'a}n 1930; Sasiela 1994):
\begin{equation}
\Psi (\nu)=0.023 r_0^{-5/3} \left ( \nu^2 + \frac{1}{L_0^2} \right )^{-11/6},
\end{equation}
\noindent
where $\nu$, $r_0$, and $L_0$ are the two-dimensional frequency, the Fried parameter, and the outer scale, respectively.
The outer scale $L_0$ is infinite for Kolmogorov screens. However, for a large
telescope this approximation becomes invalid, leading to an over-prediction of the PSF size (e.g., Tokovinin \& Travouillon 2006).
We set
$L_0$ to 25 m, which is close to the average value at Cerro Pach{\'o}n determined by Ziad et al. (2000) from the measurement
of the angle-of-arrival fluctuations.
The realization of the Kolmogorov/von Karman phase screen in the spatial domain is obtained by taking the inverse Fast Fourier Transform (FFT) of
$\psi (\nu)$, whose modulus is given by $L\sqrt{\Psi(\nu)}$; $L$ refers to the physical length of the screen and
the phase of $\psi (\nu)$ is random.
For the two-dimensional array, where $i$ and $j$ are the indexes
in the spatial domain, and $k$ and $l$ are the indexes in the frequency domain, $\psi(i,j)$ (the phase value at $i$ and $j$) is given by (e.g., Carbillet \& Riccardi 2010)
\begin{eqnarray}\nonumber
\psi (i,j)&=& \sqrt{2} \sqrt{0.023} \left ( \frac{L}{r_0} \right )^{5/6} \times \\
&& \mbox{FFT}_{R/I}^{-1} \left [ \left ( k^2+l^2+\left ( \frac{L}{L_0} \right )^2 \right )^{-11/12}
e^{\phi (k,l) \bvec{i}} \right ]. \label{eqn_phase_screen}
\end{eqnarray}
\noindent
In equation~\ref{eqn_phase_screen}, $\mbox{FFT}_{R/I}^{-1}$ means that either real or imaginary part of the inverse FFT result is
used (thus the additional factor $\sqrt{2}$ is present). $\phi(k,l)$ is the uniformly distributed random number in the range $-\pi\leq\phi<\pi$.
Because we used FFT, one obvious drawback of the phase screen generated in this way is the lack of frequencies lower than the value
set by the physical dimension of the array (i.e., $1/L$). Lane et al. (1992) suggest a compensation of this artifact by
adding subharmonics to the result. However, this problem is not relevant in the current study because the simulation of LSST
requires a large ($\gtrsim$ 1 km) phase screen to account for the large field of view.
For the realization of the specific environment for LSST, we adopt the results of Ellerbroek (2002),
who provides the parameters of the atmospheric turbulence profile based on the measurements at Cerro Pach\'{o}n, Chile, the
site of the current Gemini-South telescope and also the future LSST observatory.
The atmospheric profile consists of six layers
at altitudes of 0, 2.58, 5.16, 7.73, 12.89, and 15.46 km as shown in Figure~\ref{fig_phase_screen} with
relative weights of 0.652, 0.172, 0.055, 0.025, 0.074, and 0.022, respectively (Ellerbroek 2002).
The Fried parameter ($r_0$) is set to 0.16 m, which is reported to be an approximate median condition at the location by Ellerbroek (2002);
for a large telescope with a diameter $D>>r_0$ short-exposure seeing
is inversely proportional to $r_0$.
Each layer is allowed to move independently at a constant velocity during the integration. The maximum wind velocity at the highest altitude
is kept under $\sim20 ~\mbox{m}~\mbox{s}^{-1}$,
which determines the minimum time step 0.005 s (corresponding to $\sim0.5$ pixel shift) to satisfy Nyquist sampling rate.
We choose the dimension of our resulting phase screen array to be sufficiently large ($6\times8192\times8192$ or $6\times 1.3 \mbox{km}\times 1.3 \mbox{km}$) in order to keep the moving phase screens covering the telescope field of view during the 15 s integration time.
The weighted average $\psi(x,y)$ of the six phase screens at a given moment can be converted to a snapshot of the telescope PSF
in the absence of the optical aberration via the following equations:
\begin{equation}
p(x,y)=A(x,y) e^{ i 2\pi \psi (x,y) /\lambda } \label{eqn_p}
\end{equation}
\begin{equation}
PSF=| FFT(p) |^2, \label{eqn_psf}
\end{equation}
\noindent
In equation~\ref{eqn_p}, $A(x,y)$ is the telescope pupil function (a mask showing the obscuration)
and the phase difference $\psi(x,y)$ has the dimension of length.
The pixel scale of the PSF array given by equation~\ref{eqn_psf} is simply $F \lambda/2$, where $F$ is the f-ratio.
Figure~\ref{fig_psf_time_evolution} displays the examples of the LSST PSFs generated in this way at different integration times. The PSF at $t=0$ shows
the typical instantaneous speckle. In this figure we do not include either the optical aberration or the charge diffusion by CCDs, the effects of which
we however later add to generate the simulated LSST images.
As the exposure time increases, more speckles are stacked together, which makes the resulting PSF rounder and the irregular features present in the
individual speckles more smeared (de Vries et al. 2007).
A quantitative study on the impact of the atmospheric turbulence on the ellipticity and its spatial correlation is needed to support the validity of
our simulation hereafter. As is discussed in \textsection\ref{section_focal_plane}, we model LSST PSF variation CCD-by-CCD with polynomials.
If the anisotropic power from the atmosphere within 15 sec exposure turns out to be too strong and changes on a very small scale, any conventional interpolation
scheme with the finite number of stars will over-smooth the inherent PSF variation.
We define the ellipticity of PSF as $(a-b)/(a+b)$, where
$a$ and $b$ are the semi-major and -minor axes of the PSFs, respectively.
In general, the PSF isophotes change with radius. Thus, the measurement somewhat depends on the weighting scheme.
We measure the ellipticity of the PSF using the following quadrupole moments:
\begin{equation}
Q_{ij} = \frac{ \int d^2 \theta W(\symvec{\theta}) I(\symvec{\theta}) (\theta_i - \bar{\theta_i})(\theta_j - \bar{\theta_j}) }
{\int d^2 \theta W(\symvec{\theta}) I(\symvec{\theta}) }, \label{eqn_quadrupole}
\end{equation}
\noindent
where $I(\symvec{\theta})$ is the pixel intensity at $\symvec{\theta}$, $\bar{\theta}_{i(j)}$ is
the center of the star, and $W(\symvec{\theta})$ is the optimal weight function required to suppress the noise in the outskirts.
For the diffraction-limited PSF of LSST (Figure~\ref{fig_ellipticity_focal_plane_no_atmos}), we choose a Gaussian with a FWHM of $\sim$0.3\arcsec for $W(\symvec{\theta})$ whereas for the turbulence-limited PSF the FWHM of the Gaussian weight function used is $\sim0.7\arcsec$.
The quadrupole moments are converted to the ellipticity $\symvec{\epsilon}$ via:
\begin{eqnarray}\nonumber
\symvec{\epsilon} &=& (\epsilon_+,\epsilon_{\times}) \\
\nonumber
& = & \left ( \frac{Q_{11}-Q_{22}} {Q_{11}+Q_{22}+2(Q_{11} Q_{22}-Q^2_{12})^{1/2}} , \right. \\
&& \left. \frac {Q_{12}} {Q_{11}+Q_{22}+2(Q_{11} Q_{22}-Q^2_{12})^{1/2}} \right ) \label{eqn_delta}.
\end{eqnarray}
\noindent
The magnitude of the stick is proportional to $|\symvec{\epsilon}|=\sqrt{\epsilon_+^2 + \epsilon_{\times}^2}$, and the orientation
angle is given by $0.5\tan^{-1}( \epsilon_{\times}/ \epsilon_{+})$.
Figure~\ref{fig_at_psf}a displays the PSF ellipticity variation for the 15 s exposure within a 4k$\times$4k CCD of LSST by atmosphere. Because no optical aberration is introduced yet, this 4000 ``whiskers'' show the purely atmospheric contribution.
The size of the sticks represents the magnitude of the ellipticity whereas the
the orientation of the stick is aligned with the position angle of the elongation. A clear spatial correlation is visible on a scale of $\sim$1$\arcmin$ ($\sim$300 pixels). However, the average magnitude is less than 1\%, and thus the resulting anisotropic power is not strong. Figure~\ref{fig_at_psf}b shows the
ellipticity correlation based on the following equation:
\begin{equation}
\xi_{\pm}(r)=\langle e_{t} (r^{\prime} )~ e_{t} (r^{\prime} + r) \pm e_{\times} (r^{\prime} )~ e_{\times} (r^{\prime} + r) \rangle,
\end{equation}
\noindent
where the product $e (r^{\prime} )~ e (r^{\prime} + r)$ is performed for each pair separated by $r$, and
the $e_{t}$ and $e_{\times}$ are the tangential and the 45$\degr$ components,respectively, with respect to the line
connecting the pair.
The amplitude of the correlation function decreases with exposure time. From the comparison with the contribution due to the telescope and camera aberration
(see \textsection\ref{section_focal_plane}), we conclude that the small scale anisotropic power for the delivered PSF is not dominated by the atmospheric
turbulence. These results have been validated with a series of 15 sec exposures of the globular cluster NGC2419 on the Subaru telescope.
\section{LSST FOCAL PLANE AND IMPACTS ON PSF}
\label{section_focal_plane}
The 64 cm diameter focal plane of LSST will be tiled with 189 4k$\times$4k CCDs. As-built heights might vary up to $\sim10$ microns (peak-to-peak) in a complicated way
relative to the nominal flat surface. Although this flatness deviation is small compared with other cameras in existing facilities, the small f-ratio
of the instrument makes this focal plane flatness variation play a critical role in aberration-induced PSF behavior (depth of focus $\propto$ f-ratio).
In Table 1 we summarize the current specification of the focal plane error budget originally issued to vendors, as well as the values used for the current simulation. The CCD flatness technology and testing is reviewed in Takacs et al. (2006) and Radeka et al. (2009). The focal plane errors that we assume for the simulations in this paper are larger than the current expectation so that the results presented here are conservative. Several vendors are now exceeding these flatness specifications.
A realization of the LSST focal plane based on these error distributions
is displayed in Figure~\ref{fig_focal_plane}.
The resulting behavior of the PSF ellipticity is illustrated in Figure~\ref{fig_ellipticity_focal_plane_no_atmos}. These PSFs are obtained by evaluating optical path difference
functions across the focal plane in the absence of atmospheric turbulence using the ZEMAX software; see also Jarvis, Schechter, \& Jain (2008) for the discussion on the impact of the telescope focus on the PSF ellipticity.
Several features are noteworthy in Figure~\ref{fig_ellipticity_focal_plane_no_atmos}. First, the aberration-induced ellipticity is large. Most of the large sticks
in the plot exceed $\sim10$\% ellipticity, approaching nearly $30$\% at the field edges.
Because the maximum deviation in this realization
is small ($\sim10$ $\mu$m), these large values of ellipticity remind us that the focal error tolerance of LSST is indeed narrow;
however, we note that in the central region ($\lesssim 1$\degr) the sensitivity of PSF elongation to height error is somewhat mitigated.
Second, sharp discontinuities of PSF ellipticity are present where the CCD heights also vary abruptly.
Third, although it may be a bit difficult to recognize this feature in Figure~\ref{fig_ellipticity_focal_plane_no_atmos}, a small-scale variation is
observed even within a CCD mainly due to the
tilt and potato chip effects.
The presence of these smooth, small-scale variation and discontinuous changes across CCD gaps are the most challenging aspects of the PSF description and modeling
for LSST. Any attempt to use a single set of polynomials to characterize the PSF behavior across the entire focal plane fails because the small-scale variation
requires impractical, high-order terms in the polynomials, and in addition no interpolation scheme can satisfactorily
reproduce the sharp discontinuities across the CCD borders. This is the reason that in the current paper we perform interpolation CCD-by-CCD
to model LSST PSFs. Considering the 1-2 cycles of the potato chip effect within a CCD, we estimate that 3-4 order polynomials suffice.
The remaining
question is what features of PSF are interpolated. A simple description of the two components of the PSF ellipticity has been used in the early weak-lensing
analysis (e.g., Valdes, Jarvis, \& Tyson 1983; Kaiser, Squires, \& Broadhurst 1995). However, certainly ellipticity alone does not fully characterize a PSF. One obvious missing piece of information is the size of the PSF.
One can imagine that a 10\% ellipticity PSF with FWHM=$1\arcsec$ more strongly affects the shape of galaxies than a PSF with the same
ellipticity but with FWHM=0.5$\arcsec$. Many authors (e.g., Hoekstra et al 1998, Kaiser 2000; Rhodes, Refreiger, \& Groth 2001)
suggested modifications to this earlier method (Kaiser, Squires, \& Broadhurst 1995) for the treatment of the PSF size.
Nevertheless, this increased sophistication (i.e., ellipticity with size) is not still sufficient to obtain the accuracy that the LSST weak-lensing science requires.
On the other extreme, one might consider interpolating pixel values of observed PSFs. The result of this type of PSF interpolation is
the actual image of the PSF, which in principle contains the most complete information at any point in the focal plane.
However, this scheme faces severe practical challenges in the implementation because
1) except for the pixels near the core, individual PSF images are noisy, and 2) the number of needed polynomials become prohibitively large as one enlarges
the dimension of the PSF. Therefore, one might imagine that a new PSF description scheme should provide sufficient details with both an efficient
noise filtering method and a compact set of polynomials. A natural choice is the description of the PSFs using some basis functions, preferably
an orthogonal set to avoid degeneracy and thus improve compactness. Bernstein \& Jarvis (2002) and Refregier (2003) suggest the use of eigenfunctions of
two-dimensional quantum harmonic oscillator (also known as shapelets) for this purpose. Although the shapelet approach is shown to work
in many applications, Jee et al. (2007) demonstrate that the basis functions are still sub-optimal, truncating the PSF profile much earlier than desired.
Consequently, Jee et al. (2007) proposed to derive the basis functions from the data themselves via principal component analysis (PCA).
Lupton et al. (2001) first mentioned the use of PCA for the modeling of the Sloan Digital Sky Survey (SDSS).
Although the details of the implementation were not given, the method is similar to that of Jee et al. (2007) in that
PCA is employed to derive the optimal basis functions from the data themselves. On the other hand,
the use of PCA by Jarvis \& Jain (2004) and Schrabback et al. (2010) for the description of the PSF variation is somewhat different
from the method of Jee et al. (2007) because they use PCA to characterize the pattern of the variation (e.g., coefficients or ellipticity components),
not to capture the pixelated PSF features.
Our PCA approach has been so far applied to weak-lensing studies only with Hubble Space Telescope images (e.g., Jee et al. 2009).
In the current investigation, we
adopt this PCA scheme to characterize/model the LSST PSFs. As our PCA algorithm is independent of the specification of the instrument,
the implementation in Jee et al. (2007) can be used with little modifications.
The technical details of the PCA method will be reviewed in the following section.
\section{PRINCIPAL COMPONENT ANALYSIS }
\label{section_pca}
PCA is the mathematical procedure for analyzing multivariate data by finding and utilizing a new orthogonal coordinate system, which
substantially reduces the number of axes relative to the original number of variables for the characterization of the data.
Inevitably, the procedure involves information loss. However, in the typical data set where the noise is significant, this
reduction of the information in fact works as filtering, enabling a compact description of the data without significantly harming its integrity.
In the current paper, we explain the procedure
using our specific data set, i.e., series of noisy two-dimensional PSF images. A similar account can be also found in Jee et al. (2007).
Let us assume that we have $N$ observed stars in a 4k$\times$4k CCD image with random positions and brightness.
The two-dimensional
array of each PSF image can be rearranged to form a one-dimensional vector. How we exactly scramble the pixels of the PSF image
is not important in the subsequent analysis as long as we have the ability to re-order the pixels to recover the original two-dimensional image.
However, it is critical that each PSF image is carefully background-subtracted and normalized prior to the rearrangement.
Then, the $N$ one-dimensional vectors are stacked to comprise a matrix. If each PSF image contains $M$ pixels, the result is
a $N\times M$ matrix. The next step is to evaluate mean values along the columns, and subtract this $1\times M$ mean vector
from each row of the matrix. In practice, each row of the $N\times M$ matrix contains star images contaminated by neighboring
objects, cosmic rays, bad pixels, etc. Therefore, the mean values are preferably evaluated iteratively with some outlier rejection.
Also, the rejected pixel values in the $N\times M$ matrix should be replaced with the mean values.
Because we need this mean vector to reconstruct the PSF later, this mean vector (or mean PSF rearranged
in one-dimension) must be saved.
Now, we are ready to find a set of $P$ ($\ll MIN\{N,M\}$) orthogonal vectors that compactly describe the $N\times M$ matrix. One way to
derive such orthogonal vectors is Singular Value Decomposition (SVD; Press et al. 1992). If we call the $N\times M$ matrix \bvec{S},
the matrix \bvec{S} can be decomposed in the following way:
\begin{equation}
\bvec{S}=\bvec{U}\bvec{W}\bvec{V}^T \label{eqn_SVD},
\end{equation}
\noindent
where $\bvec{U}$ is an $N\times M$ column-orthogonal matrix, $\bvec{W}$ is an $M\times M$ diagonal matrix, and
$\bvec{V}$ is an $M\times M$ orthogonal matrix. The diagonal elements of $\bvec{W}$ are called singular values.
Therefore, the matrix component $\bvec{S}_{ij}$ can be expressed as
\begin{equation}
S_{ij} = \sum_{k=1}^{M} w_k U_{ik} V_{jk}. \label{eqn_svd_component},
\end{equation}
\noindent
which implies that, if some singular values $w_k$ are relatively small, we can substitute zeros for those $w_k$'s without greatly
compromising the numerical accuracy of the matrix evaluation. In other words, the reconstruction of $\bvec{S}$ is
possible with fewer columns in both $\bvec{U}$ and $\bvec{V}$, and thus the procedure provides a substantial compression
of the information in the original matrix $\bvec{S}$, where photon and detector noise is a significant part.
The remaining columns (principal components) of $\bvec{V}$ form an orthogonal basis, and in this analysis we refer to these eigenfunctions
as eigenPSFs. We display examples of these eigenPSFs in Figure~\ref{fig_eigenPSF}. These eigenPSFs in this figure are
obtained from the PCA of 4000 simulated stars on the entire focal plane in the absence of telescope aberration and CCD height fluctuation, but
in the presence of the atmospheric turbulence. The eigenPSFs are ranked according to the size of their eigenvalues with $n=0$ representing the
largest. Note the remarkable resemblance of the first several eigenPSFs to the basis functions in the shapelet approach. However, also
note the asymmetry of the eigenPSFs due to the characteristic pattern of the specific data set. For increasing $n$, more high-frequency
features are evident. Above a certain limit $n=n_c$, the resulting eigenPSFs start to reflect the noise features without contributing
to the PSF reconstruction. Consequently, we need only a truncated set of eigenPSFs to describe the behavior of the PSF.
This minimal number of eigenPSFs can be determined by examining the eigenvalues of each eigenPSF. In general, there
is a conspicuous ``kink'' at a certain $n=n_c$, beyond which the decrease of the eigenvalue suddenly slows down
(e.g., Figure 5 of Jee et al. 2007). We find that $\sim20$ eigenPSFs are sufficient to describe the details for both
space- and ground-based telescope PSFs.
After the eigenPSFs are obtained, the $k^{th}$ star used for the derivation is decomposed as
\begin{equation}
C_k (i,j)= \sum_{n=0}^{n_{max}} a_{kn} P_n (i,j) + T(i,j),
\end{equation}
\noindent
where $C_k (i,j)$ is the normalized pixel value of the $k^{th}$ star at the
pixel coordinate $(i,j)$,
$P_n$ is the $n^{th}$ eigenPSF, $a_{kn}$ is
the projection of the $k^{th}$ star in $P_n$, and $T$ is the mean PSF. Because $P_n$'s are orthogonal to
one another, one does not need to determine $a_{kn}$ through some $\chi^2$ minimization here.
Now the PSF at any arbitrary location can be evaluated through polynomial interpolation of $a_{kn}$'s:
\begin{equation}
a_n = \sum_{l,m=0}^{l+m\leq N} d_{nlm} x^l y^m
\end{equation}
\noindent
where $x$ and $y$ are the coordinate of the position, at which one wants to know the PSF, $d_{nlm}$
is the coefficient of the term $x^l y^m$ for the $n^{th}$ eigenPSF, and $N$ is the maximum
order of the polynomial. It is important to determine $d_{nlm}$ without the application
of the S/N weight of the star because we are interested in the spatial variation
of the PSF.
\section{SIMULATION OF LSST IMAGES AND PSF RECONSTRUCTION}
\label{section_simulation}
For the robust simulation of the weak-lensing shear extraction using LSST,
it is crucial that the simulated image contains astronomical objects with
realistic photometric, morphological, and statistical properties, as well as the instrumental artifacts resulting
from the known characteristics of the telescope, camera, and the environment.
Specifically, the requirements include our ability to correctly represent:
\begin{enumerate}
\item the number density and luminosity function of unsaturated high S/N Galactic stars,
\item the number density, luminosity function, size, and morphology of galaxies,
\item position-dependent PSF variations,
\item geometric distortion, and
\item atmospheric turbulence and dispersion.
\end{enumerate}
The first requirement is needed because the number of available high S/N stars
determines how well we can measure the behavior of the PSFs. In particular, we need
to investigate if the nominal 15 s exposure image provides sufficient number of usable stars
within a CCD so as to correctly sample the position-dependent PSF variation.
The next point relates to the issue of both statistical and systematic errors
of the shear measurement from galaxy ellipticity. The number of usable galaxies
for testing lensing analysis affects the statistical accuracy of the estimate of shear and our determination of mass.
In addition, one important source of bias in many ellipticity measurement techniques is
the under-fitting of galaxy shapes (i.e., the model galaxy profile assumed is different
from the real profile). Since many $z\gtrsim1$ galaxies are dominated by
so-called faint blue galaxy (FBG) population incapable of being described by
analytic functions, it is important that the image contain a realistic fraction of
these irregular galaxies.
The extremely high sensitivity of the LSST PSF to the focal plane errors makes the position-dependent
PSF one of the most critical features to be simulated.
We need to generate the realistic PSF and convolve the galaxy image with this position-dependent
PSF in a time-efficient way.
Although expected to be very small, the geometric distortion by the telescope and atmosphere
can correlate ellipticity of galaxies and mimic gravitational lensing signal similar to the way that
uncorrected PSF can cause systematic alignment of galaxy ellipticity. It is essential to
estimate the level of accuracy in rectifying the raw images and the effects on shear measurement.
The atmospheric turbulence and dispersion are dominant sources of time-dependent PSF variation.
Unlike the telescope and camera optical aberration, the atmospheric turbulence is believed to circularize
the PSF rather than to introduce significant anisotropy whereas the atmospheric dispersion obviously
elongates the PSF in the direction of the zenith angle. It goes without saying that these atmospheric
effects should be properly included in the simulation.
Our simulation of the LSST shear test is an on-going project, and the final goal is to carry out an
end-to-end simulation from the realization of the LSST image from an input cosmology to the
full recovery of the galaxy shear signal and the cosmological parameters. Hence, it is our ultimate aim to rely on the
final simulator not only to implement the above requirements but also to address other non-shear measurement
issues such as diffuse components, cosmic rays, hot/warm/bad pixels, etc.
This end-to-end simulation is being done by The LSST Simulations Team (Connolly et al. 2010).
Here, as a pilot study, we focus on generating realistic LSST PSFs and recovering these input
PSFs from sparse distribution of the stars in the presence of background galaxies. Therefore, some simplifications
are made where they do not significantly compromise the integrity of our simulation relevant to the focal plane
flatness issue.
\subsection{Generation of Input PSF Model }
\label{section_inputpsf}
First, we generate the coordinates of the model stars that are densely distributed across the focal plane.
Because we will evaluate the fiducial PSFs at these locations and interpolate the results to describe the PSF
on every pixel, the density of these stars should be high enough to accommodate the expected small scale
variation. We assigned about 800 stars to each CCD and verify that these 800 stars sample with ample resolution the
spatial variation, which is dominantly modulated by potato-chip effects (1-2 cycles per CCD).
Then, we evaluate the diffraction-limited PSF at the location of those stars using ZEMAX. Strictly speaking,
in this step we save phase data rather than PSFs because the optical phase data should be later combined with
the atmospheric phase screen to generate the final PSF. We use the simulated focal plane height fluctuation shown in Figure~\ref{fig_focal_plane}
in order to provide the focus value for each star. In the current investigation, our main
interest is the ability to model/recover the PSF in the $r$ band. Thus, we simulate the broadband effect by obtaining the data at five uniformly spaced wavelengths
between 5450-7030 \AA.
Next, the atmospheric phase screen is evaluated at the location of each star. This atmospheric phase information is
merged with the optical data to construct the next stage PSF. Finally, the result is further convolved twice
to mimic the effects of the charge diffusion ($\sim0.4$ pixel rms) within a CCD and the atmospheric dispersion (close
to zenith).
We derive sets of eigenPSFs separately for each CCD. Alternatively, one can consider finding one set of eigenPSFs utilizing all PSFs on the focal plane.
Although this latter scheme may reduce statistical errors, it appears that the resulting eigenPSFs are suboptimal, somewhat under-predicting
the systematic pattern unique in each CCD. We choose to keep the most signifincant 20 eigenPSFs, which accounts for
more than 99\% of the total variance. Now each PSF is described by 20 coefficients each representing the amplitude in each eigenPSF direction.
We fit 4th order polynomials to the coefficients of all stars to describe the PSF at any arbitrary location.
\subsection{LSST Galaxy Image Generation}
\label{section_lsst_image}
One ideal way to generate galaxy images for a given cosmology is to make a catalog using large numerical simulation results (e.g., Millennium Simulation, Springel
et al. 2005) with
ray-tracing technique, to assign proper spectral energy distribution (SED) and morphology to individual galaxies, and to
convert the catalog to astronomical images. However, this full scale end-to-end simulation (Connolly et al. 2010) is beyond the scope
of the current project, and here we instead utilize Hubble Space Telescope (HST) Ultra Deep Field (UDF) images to sample
galaxies. The HST UDF (Beckwith et al. 2003) is a
412-orbit HST Cycle 12 program to image a single field with the Wide
Field Camera (WFC) of the Advanced Camera for Surveys (ACS) in
four filters: F435W, F606W, F775W, and F850LP. The total exposure time of
the F775W image is $\sim350$ ks, and the 10 $\sigma$ limiting magnitude is 29 ABmag. The depth of the UDF image
is sufficient for the generation of the galaxies that will be imaged by LSST, which reaches $r\sim 27.5$ mag ($\sim5\sigma$) in the final depth.
Hence, virtually, the UDF data provide noiseless, seeing-free galaxies (the size of the ACS PSF is less than that of one LSST pixel) for
a 15 s exposure.
As the field of view of the ACS is $\sim3\arcmin\times3\arcmin$, even smaller than the area covered by one LSST $\sim14\arcmin\times14\arcmin$ CCD,
it is inevitable that the same UDF galaxies must appear multiple times across the focal plane. However, this is not likely to compromise the
accuracy of the current weak-lensing simulation, where we do not attempt to recover any input cosmology.
The high-level science product of the UDF is publicly available\footnote{http://archive.stsci.edu/prepds/udf/}, and we use the F775W image
to sample galaxies. Objects were detected via SExtractor (Bertin \& Arnouts 1996) by looking for 8 continuous pixels above the sky rms.
Postage stamp images were created by cropping a square region centered on the object with the side five times the semimajor axis of the SExtractor
estimate. We used the segmentation map to check if any pixels belonging to other objects were present in the square. If so, we filled these
pixels with the randomized background whose statistics were tuned to match the SExtractor estimate.
Any objects whose half-light radii were less than the upper limit of the HST PSF value were discarded.
The next procedure is to plant these postage stamp UDF galaxy images onto each 4k $\times$ 4k CCD image. We looped over the list of the postage stamp images sequentially but
randomizing the location and the orientation. The loop ended when the desired number density was reached. A large fraction of the pixels were
the result of the superposition of many different postage stamp images. Thus, it is apparent that the noise level of these pixels vary slightly depending
on the number of overlaps. However, because we later introduce much larger additional noise to simulate the 15 s exposure data,
this noise variation is negligible.
A stellar count study based on the SDSS catalog (Juri{\'c} et al. 2008) shows that
a single 4k $\times$ 4k CCD image ($\sim14\arcsec\times14\arcsec$) in 15 s exposure at the Galactic north pole
is expected to contain at least $\sim140$ high S/N stars ($\geq 30\sigma$),
which can be used to model the PSF. However, in the current simulation
we conservatively distributed 40 point sources per CCD in such a way that
the S/N of those stars remain above 30 in 15 s exposure, assuming the following differential star count: $\phi(m_r) \propto 10^{0.2 m_r}$;
we find that PSF reconstruction with different magnitude distribution does not impact the quality of the PSF reconstruction
as long as all the stars used have S/N$>30$.
Prior to the noise generation, the ``perfect'' 4k $\times$ 4k image must be convolved with the position-dependent PSF of the CCD.
Because our PSF is represented by 20 coefficients and 20 basis functions, it provides us with a convenient way to implement the spatially varying convolution.
To begin with, we convolve a 4k$\times$4k seeing-free astronomical image with 20 eigenPSFs via FFT. The results of this operation are 20 4k$\times$4k images.
Then, using the resulting 4th order polynomial fitting we generate another 20 4k$\times$4k images with each pixel containing
the coefficient (amplitude) of the eigenPSF. Next, we multiply each eigenPSF-convolved image with
the matching coefficient image, obtaining 20 new 4k$\times$ 4k images.
These 20 images are simply stacked together to produce one 4k$\times$4k image. This image is not yet the final
image that we desire to generate because the 20 eigenPSFs are derived from the mean-subtracted
PSFs. Therefore, we need to convolve the seeing-free 4k$\times$4k image with the mean PSF, and add the result
to the former 4k$\times$4k image to yield the final 4k$\times$4k LSST image.
Figure~\ref{fig_simulated_lsst} displays a $100\arcsec\times100\arcsec$ cutout from one of the 189 4k $\times$ 4k images.
The noise-free (except for the existing low-level noise in the ACS UDF image) LSST-seeing-convolved image is shown
in the left panel. This ``ultra deep'' image is about 2 mag deeper than the
median depth LSST image (middle) after the completion of the nominal 10 year mission with 200 exposures per sky patch.
The nominal 15 s LSST single exposure simulation is shown in the right panel.
\section{SIMULATED IMAGE ANALYSIS AND RESULTS}
\label{section_analysis}
\subsection{PSF Reconstruction}
\label{section_psf_reconstruction}
The procedure to apply PCA to the simulated LSST image containing galaxies and stars is similar to the one discussed in the case of the input PSF modeling.
The difference is that now we have far fewer stars, realistic noise, and neighboring objects. As a
matter of course, this will provide the reconstructed PSF with fewer degrees of freedom in the description of the
spatial variation. Hence, if the intrinsic PSF variation within a CCD is more complicated than what the reconstructed
model can predict, the weak-lensing science with LSST will be substantially compromised. In this section,
we present the detailed comparison between the reconstructed PSF and the input PSF in order to address this
critical issue.
Astronomical objects in the simulated 189 4k$\times$4k images were detected with SExtractor. Stars were selected
by utilizing half light radius versus magnitude relation as performed in typical lensing analysis. The stellar
locus is well defined for bright stars. In addition, using
ellipticity, comparison of half-light radius with FWHM measurement, and source extraction history (e.g., presence
of close neighbor, edge clipping, etc.), the contamination becomes negligible in the current 10 sq. degree simulation.
The cutout size is chosen
to be $31\mbox{pixel}\times31\mbox{pixel}$ ($6.2\arcsec\times 6.2\arcsec$) after the star is sub-pixel shifted
to place the centroid on a center of pixel. The ellipticity of the selected stars are displayed in Figure~\ref{fig_ellipticity_focal_plane}.
We overlay the ``whiskers'' over the focal plane image that we use for the input PSF generation.
The pattern of the PSF variation resembles the case displayed in Figure~\ref{fig_ellipticity_focal_plane_no_atmos}, where no
atmospheric effect is included. However, the reduction of the magnitude is apparent due to the circularization
effect of the atmospheric turbulence. We observe that no significant anisotropy is introduced by the moving phase screens.
The PSF elongation in the zenith angle direction is very small here because we choose a telescope
pointing to be close to the zenith.
We derive eigenPSFs for each CCD separately and use 3rd order polynomials to describe the spatial variation.
For the visualization of how well the reconstructed model represents the observed pattern, we display the residual
ellipticity distribution across the focal plane in Figure~\ref{fig_residual_ellipticity_focal_plane}.
The residual in Figure~\ref{fig_residual_ellipticity_focal_plane} is the difference in ellipticity ($e_{+}$,$e_{\times}$) of the observed (simulated) stars and
the model PSF evaluated at the location of the stars. Consequently, any correct model should not only reduce
the size of the ``whiskers'' in Figure~\ref{fig_ellipticity_focal_plane},
but also make the correlation of their size and the orientation decrease significantly. Visual comparison
of Figure~\ref{fig_residual_ellipticity_focal_plane} with Figure~\ref{fig_ellipticity_focal_plane}
shows qualitatively that our interpolated PSF at the location of the stars closely mimics the
pattern of the observed PSFs. One method to quantitatively examine the reduction of the whiskers
is to compare the ellipticity component distribution in the two cases.
In Figure~\ref{fig_ellipticity_component} the blue
dots represent the ellipticity components of the observed stars whereas the red dots indicate
those of the residuals. Note that the residual ellipticity distribution is much more compact
and is precisely centered on the origin $(e_{+},e_{\times})=(0,0)$.
Figure~\ref{fig_ellipticity_correlation} displays the ellipticity correlation.
The spatial correlation of the observed ellipticity (before PCA correction) has an amplitude of $3\times10^{-5}$.
For a sanity check, we compared this amplitude with the values that we obtain from real observations, and found that
they agree well. For example, the amplitude of the intrinsic PSF correlation before correction
is $\sim4\times10^{-5}$ and $\sim3\times10^{-5}$ in the Deep Lens Survey (DLS; Wittman et al. 2002) field from
the 4m Kitt Peak telescope and 8m Subaru telescope images, respectively; one should use caution when comparing these values to other published results because of different ellipticity definitions and weight functions.
Also displayed in Figure~\ref{fig_ellipticity_correlation} is the ellipticity correlation function of the purely atmospheric PSF (Figure~\ref{fig_at_psf}),
which is generated by ``turning off" optical aberrations. These PSFs are free from interpolation errors and thus help us to assess
the amount of the atmospheric contribution to the PSF anisotropy. In particular, the comparison enables us to verify that
$\sim800$ stars per CCD is sufficient to model the PSF pattern within a CCD. If the atmospheric PSF possessed a significant
power on a small scale $r<1\arcmin$, those $\sim800$ stars could not adequately sample the PSF variation.
Obviously, the amplitude $3\times10^{-5}$ of the ellipticity correlation for the observed PSF (blue) is
higher than the expected cosmic shear signal for the $\Lambda$ CDM universe at large angles.
This implies that without PSF
correction the galaxy ellipticity correlation will be dominated by the PSF systematics rather than by the
intrinsic lensing signal. It is reassuring that the correlation function of the residual ellipticity
components has an amplitude of $\sim10^{-8}$, which is more than 3 orders of magnitude reduction in correlation and
also more than 2 orders of magnitude below the anticipated cosmic shear lensing signal (see Rowe 2010 for
the discussion and use of the residual ellipticity correlation in detail). We find similar improvement when we apply the PCA
PSF corrections to the DLS data.
It is important to remember that the above PSF correlation is measured on a single 15 s exposure image.
In fact, LSST will visit the same patch of the sky roughly 200 times in each filter used for weak-lensing with two 15 s exposures.
Before each exposure the telescope optics is updated from the wavefront sensing system.
The pupil rotation, sky rotation, and $x-y$ dithers will mix different systematic PSF patterns and reduce the
intrinsic ellipticity of the PSFs in the final stack. In addition, this will also
remove the spatial correlation that we saw within a single CCD even before we apply PSF corrections.
Although one can come up with an optimized operation scheme, which takes full
advantage of hundreds of visits, we observe that even a simple random field rotation
can dramatically reduce the intrinsic PSF ellipticity correlation. Figure~\ref{fig_field_rotation}
schematically displays this simple observing pattern and the resulting PSF ellipticity distribution.
We illustrate an over-simplified observing scenario, where a same patch of the sky is
visited 100 times with randomized roll angles.
In order to create the stack image out of these 100 frames,
we arbitrarily choose one frame as a reference and rotate the rest of the frames for the alignment.
Stacking rotated images are highly non-trivial in the presence of anisotropy in pixel boundaries, atmospheric dispersion,
satellite trails, cosmic rays, detector response variation, internally scattered light/ghosts, etc\footnote{While beyond the scope of this paper,
our ``Multi-Fit'' approach (Tyson et al. 2008) to galaxy photometry and shape measurement fits a galaxy morphological model to all
the exposures, and thus can avoid the propagation of these issues into the LSST weak-lensing science.}
Numerous schemes exist for fractional pixel handing. We use
bi-cubic interpolation (Press et al. 1992), which closely mimics the ideal sinc interpolation or
its variations (e.g., Lanczos kernels), for both stacking images and constructing PSFs;
the merits of using Lanczos kernels are discussed in the literature (e.g., Jee et al. 2007; Bertin 2010).
The atmospheric dispersion is very small in the current simulation because we simulated pointings near the zenith.
Thus, the dispersion correction is omitted in the current simulation. However, we think this atmospheric
dispersion and other aforementioned real world problems must be handled with care because any small bias may
appear as second-order systematics after analysing many billion galaxies in the 40,000 sq. degree area.
These issues will be discussed in our future publications.
We use the single realization of the focal plane (shown in Figure~\ref{fig_focal_plane}) in the above multi-epoch simulation although we randomize the phase of the atmospheric turbulence screen for each epoch.
It is important to realize that this simulation setup for the telescope and camera closely reflects the planned operation scheme of LSST. The LSST camera will
maintain its fixed focal plane configuration throughout the mission, and the mirror deformation and alignment changes will be
corrected with actuators to $\lesssim0.2\mu$m accuracy (Manuel et al. 2010). That is, the difference in aberration for different visits is small
compared to the aberration introduced by the focal plane non-flatness.
In principle, this small wavefront reconstruction error may slightly alter the resulting telescope aberration patterns exposure by exposure.
However, the wavefront reconstruction error in different exposures are random, and stacking many exposures will reduce the correlation, circularizing the final PSFs to our advantage.
In Figure \ref{fig_field_rotation} the observed PSF ellipticity after stacking
(upper right) is much smaller than what is seen in individual single 15 s observations (upper left).
The lower left panel shows the reconstructed PSF by first performing
PCA on each exposure and then stacking all model PSFs after correct application of field rotation;
the rotated PSFs are created by re-sampling with bi-cubic interpolation, which is identical
to the procedure in the image stacking above.
Stacking different PSF patterns reduces the correlation
in observed PSF (Figure~\ref{fig_ellipticity_correlation_rotation}.) However, it does not bring down the amplitude
below $10^{-7}$ over the entire range of angular scales. This is because our toy model observing scheme without $x-y$ dithers does not
mix different PSFs efficiently.
\subsection{Null Test with Galaxies}
The high-fidelity reconstruction of the observed PSF through PCA demonstrated in \textsection\ref{section_psf_reconstruction}
hints at the prospect that the resulting galaxy shear measurement can be carried out with similarly small systematic errors.
However, there remain several issues. First, anisotropic PSFs
change the pixel noise properties also anisotropically, and thus even with the knowledge of the perfect PSF
the resulting galaxy shape measurement can be still biased along the elongation of the PSFs. Related to this point is
the centroid bias (e.g.,Kaiser 2000; Bernstein \& Jarvis 2002), which causes
the object centroid to be more uncertain in the direction of the PSF elongation.
Second, our comparison between the model and the observation is limited to the PSFs at the location of the stars. If the
intrinsic spatial variation of the PSF has higher frequency than the order of the polynomial (3rd order here), it is
possible that galaxy shape measurements would be carried out with suboptimal PSFs.
Third, apart from PSF modeling, the difference in galaxy radial profile between assumption and reality can
cause biases in shear extraction as suggested by Bernstein (2010). Fourth, it is not yet clear how to handle
the pixelization effect optimally. Because photons are collected in finite CCD pixels,
any shape measurement algorithm needs to consider this procedure to avoid bias. Similar to the convolution
by seeing, the pixelization also somewhat destroys the information, especially at the core of object, where
the radial profile is most steep. [However, the low surface brightness limit of each exposure allows shape measurement at
large radius.] Fifth, object detection from real astronomical images involves many subtleties such
as deblending, spurious object identification, etc. The current algorithms need to be
improved substantially to minimize the contamination from these undesired detections or at least to quantify
the fraction more precisely in order to estimate the dilution of the lensing signal.
In terms of shear extraction, the above concerns can be largely divided into two issues: the removal of the anisotropic bias and the calibration
bias. For convenience, many authors (Heymans et al. 2006; Massey et al. 2007; Bridle et al. 2010)
formulate the two issues with the following expression:
\begin{equation}
\gamma^o - \gamma^t = m \gamma^t + c,
\end{equation}
\noindent
where $m$ and $c$ are the calibration and anisotropic biases, respectively, and $\gamma^o$ and $gamma^t$ are
the observed and true shears, respectively.
Despite the difficulties mentioned above, as a last resort the calibration bias or multiplicative factor $m$ can be
determined through extensive image simulations. We prefer a method which
provides self-contained calibration factor and relies on the image simulation result only as a confirmation tool.
Within the scope of this paper, we address the second issue, i.e., the removal of anisotropic bias, by performing ``null'' test
on the simulated LSST images. In particular, we focus on how well a simple field rotation can remove this additive factor.
For the measurement of the galaxy ellipticity, we fit PSF-convolved elliptical Gaussians (e.g., Jee et al. 2009) to galaxies. As mentioned
in \textsection\ref{section_psf_reconstruction}, the PSF on the final stack is constructed from the
superposition of the 100 PSFs from all visits. We discard objects whose half-light radius is less than
1.2 times the median value of the stars or ellipticity measurement errors greater than 0.2.
The resulting number of usable galaxies is $\sim40$ per sq. arcmin for the depth at the completion of the nominal LSST survey, which
meets the weak-lensing science requirement (LSST Science Collaborations et al. 2009). The exact value of the selection
criteria should be determined in conjunction with the study of shear calibration, and thus will be
the subject of our future publications.
As no shear is introduced, the galaxy correlation function should possess amplitudes consistent with the shot noise.
One should remember that since the intrinsic ellipticity of individual galaxies are at least a factor of 10 larger than that of PSFs,
the correlation function within a single LSST field will not approach the level of the residual PSF correlation function.
In Figure~\ref{fig_g_correlation}, we present this ``null'' test result with galaxy ellipticity for the 10 sq. degree field.
If anisotropic PSF effects are corrected, the correlation amplitude should simply reflect the statistical galaxy shot noise as observed.
As the number of exposures increases,
the correlation amplitude goes down because the residual PSFs become more uncorrelated (systematic noise) and also
the increased depth gives higher number of galaxies usable for shear measurements (statistical shot noise).
The number density of galaxies with sufficiently high S/N for ellipticity measurement for a single,
10, and 100 exposures are 10, 24, and 42 galaxies per sq. arcmin,
respectively. We find that the mean amplitudes of the galaxy ellipticity correlation functions
are closely proportional to the inverse of these number densities (Figure~\ref{fig_shear_correlation_error}).
This tight relation implies that the level of the systematic error for the 100 exposure data set
is much less than that of the statistical error ($\lesssim10^{-7}$).
The expected signal for a $\Lambda$CDM cosmology corresponds to a galaxy ellipticity correlation $>10^{-6}$ over
this range of angular scale.
The exact level of the systematic floor can be estimated only by extending
the area of the simulation by a large amount. For the case that the amplitude
of the PSF ellipticity correlation function $\sim10^{-8}$ in Figure~\ref{fig_ellipticity_correlation_rotation} happens to be the
fundamental limit, we estimate the extrapolated correlation function for the 20,000 sq. degree area survey (cyan line in Figure~\ref{fig_g_correlation}).
\section{CONCLUSIONS AND SUMMARY}
\label{section_conclusion}
We have investigated how the atmospheric turbulence and the telescope aberrations will impact the PSF of the active optics LSST in the context
of weak lensing science. The atmospheric turbulence simulated by six moving Kolmogorov phase screens mainly contributes
to the broadening of the diffraction-limited PSF (that would have been obtained in the absence of the atmosphere) without
significantly inducing additional anisotropy. The FWHM $\sim0.7\arcsec$ of the simulated PSF nicely agrees with the mean value determined by the
LSST seeing monitoring program at Cerro Pach\'{o}n.
The instrumental aberrations place potential challenges to precision weak-lensing analysis with LSST because of the fast f-ratio of the optics.
Our simulation shows that the resulting shallow focal depth creates abrupt changes in PSF pattern across CCD borders.
As the $3.5\degr \times 3.5\degr$ focal plane will be tiled with 189 CCDs each having its own surface distortion (i.e., potato-chip effect),
the PSF pattern on the entire focal plane is very complex.
However, we demonstrate that the complexity of the PSF pattern over the entire focal plane can be nicely described by
interpolation with basis functions derived by PCA. In order to account for the discontinuity across the CCD borders with
polynomials, the basis functions and the polynomial coefficients are determined separately for each CCD.
This PSF recovery test is carried out with simulated LSST images containing realistic galaxies, as well as stars in the
presence of photon noise and neighboring objects. The residual PSF ellipticity correlation function has
amplitudes of $10^{-8}$ over the $r=0.2\sim3.0\degr$ range, significantly lower than the amplitudes of the cosmic shear and
the raw PSF ellipticity correlation.
It is critical to remember that in actual LSST operations the same patch of sky will be
visited multiple times with different combinations of pupil rotation, sky rotation, and $x-y$ dither. Hence, this will
substantially reduce our burden in modeling the PSF for individual exposures because the mix of
different PSF pattens will circularize the PSF in the stacked image and also decreases the spatial correlation.
To illustrate how much the stacking procedure can improve the systematics, we simulate
a toy case, where 100 visits are randomly rotated around the center of the field. The reduction in both ellipticity and
correlation prior to any PSF modeling and correction is dramatic even in this simplistic scenario.
The ``null'' test with galaxies in the stacked image indicates that the amplitude of the
galaxy shear correlation is consistent with the level of the shot noise, indicating that the systematic error
is not detectable within a single $\sim10$ sq. degree field with 100 exposures.
The PSF anisotropy correction (additive factor in shear recovery) is one of the most
poorly investigated effects, and the literature disagrees on the best approaches.
Our current simulation shows that the anisotropic PSF effects can be sufficiently corrected
with the current design of LSST. This improvement is enabled by both a new PCA algorithm and the
large \'{e}tendue and active optics of LSST capable of turning systematics into statistics.
In a forthcoming paper, we will extend the current simulation with the addition of
input galaxy shear and discuss the accuracy of the shear recovery with existing algorithms.
M. James Jee acknowledges support for the current research from the TABASGO foundation in the form of
the Large Synoptic Survey Telescope Cosmology Fellowship. We thank Lynne Jones for help with stellar
density estimation at the Galactic pole. We thank Chuck Claver for providing the ZEMAX model of the LSST.
We thank David Wittman, Andrew Connolly, Andrew Becker, and
Andrew Rasmussen for useful discussions on focal plane non-flatness and PSF impact. We thank Perry Gee
for help in the parallelization of our simulation code. Finally, we thank the anonymous referee who helped
us to improve the quality of the paper significantly.
|
1,116,691,499,275 | arxiv | \section{Introduction}
Exchangeable relational data~\cite{nowicki2001estimation,ishiguro2010dynamic,nonpa2013schmidt}, such as tensor data~\cite{tnnls_8497036,pensky2019dynamic} and collaborative filtering data~\cite{tnnls_7112169,tnnls_8525418,Li_transfer_2009}, are commonly observed in many real-world applications. In general, exchangeable relational data describe the relationship between two or more nodes~(e.g.~friendship linkages in social networks; user-item rating matrices in recommendation systems;
and protein-to-protein interactions in computational biology), where exchangeability refers to the phenomenon that the joint distribution over all observed relations remains invariant under node permutations.
Techniques for modelling exchangeable relational data
include node partitioning to form ``homogeneous blocks"~\cite{nowicki2001estimation,kemp2006learning,roy2009mondrian}, graph embedding methods to generate low-dimensional representations~\cite{tnnls_6508899,tnnls_6842607,tnnls_8587135}, and optimization strategies to minimize prediction errors~\cite{tnnls_6208890,tnnls_8440680}.
\textit{Graphon theory}~\cite{orbanz2009construction, orbanz2014bayesian,RandomFunPriorsExchArrays} has recently recently been proposed as a unified theoretical framework for modelling exchangeable relational data. In graphon theory, each relation from a node $i$ to another node $j$ is represented by an \textit{intensity} value generated by a \textit{graphon function}, which maps from the corresponding coordinates of the node pair in a unit square, $(u_{i}^{(1)}, u_{j}^{(2)})$, to an intensity value in a unit interval. Many existing Bayesian methods for modelling exchangeable relational data can be described using graphon theory with various graphon functions~(see Fig.~\ref{fig:graphon_comparison}). Representative models include the Stochastic Block Model (SBM)~\cite{nowicki2001estimation,kemp2006learning}, the Mondrian Process Relational Model (MP-RM)~\cite{roy2009mondrian}, the Rectangular Tiling Process Relational Model (RTP-RM)~\cite{nakano2014rectangular},
the Rectangular Bounding Process Relational Model~(RBP-RM)~\cite{NIPS2018_RBP} and the Gaussian Process Prior Relational Model~(GP-RM)~\cite{orbanz2009construction}.
These existing models can be broadly classified into two categories. The first category, which includes the SBM, MP-RM, RTP-RM and RBP-RM models, uses node-partitioning strategies to construct the relational model.
By partitioning the set of nodes into groups along node co-ordinate margins, blocks can be constructed from these marginal groups that partition the full-dimensional co-ordinate space according to a given construction method (Fig.~\ref{fig:graphon_comparison}).
These models then assume that the relation intensity for node pairs is constant within each block. That is, the graphon function that generates intensity values over node co-ordinate space is constructed in a piecewise-constant manner. However, such piecewise-constant graphons can only provide limited modelling flexibility with a fixed and constant number of intensity values~(e.g.~equivalent to the number of blocks).
As a result, they are restricted in their ability to model the ground-truth well.
The second category of relational models, which includes the GP-RM,
aims to address this limitation as the graphon function can provide continuous intensity values.
However, the computational complexity for estimating this graphon function is proportional to the cube of the number of nodes, which makes it practically non-viable for medium or large sized datasets.
In this paper, we propose to apply a smoothing procedure to piecewise-constant graphons to form \textit{smoothing graphons}, which will naturally permit continuous intensity values for describing relations without impractically increasing computational costs. As the Stochastic Block Model is one of the most popular Bayesian methods for modelling exchangeable relational data, we focus on developing smoothing strategies within the piecewise-constant SBM graphon framework. In particular, we develop two variant smoothing strategies for the SBM: the Integrated Smoothing Graphon~(ISG) and the Latent Feature Smoothing Graphon~(LFSG).
\begin{figure*}[ht]
\centering
\includegraphics[width = 0.9\textwidth]{graphon_comparison_vis.pdf}
\caption{Visualisation of Bayesian graphon-construction methods for modeling exchangeable relational data.
From left to right: the Stochastic Block Model~(SBM); the Mondrian Process Relational Model~(MP-RM); the Rectangular Tiling Process Relational Model~(RTP-RM) and the Gaussian Process Prior Relational Model~(GP-RM). For any pair of node coordinates $(u_{i}^{(1)}, u_{j}^{(2)})$ the relation intensity is mapped from the unit square to a unit interval using a graphon
function~(denoted $g_1,\ldots,g_4$),
where a darker colour observed in the unit square represents a higher mapped intensity in the unit interval.}
\label{fig:graphon_comparison}
\end{figure*}
\begin{itemize}
\item ISG: In contrast to existing piecewise-constant graphons, which determine the intensity value based only on the block within which a node pair resides,
the ISG alternatively calculates a mixture intensity for each pair of nodes by taking into account the intensities of all other blocks. The resulting mixture graphon function is constructed so that its output values are continuous.
\item LFSG: This strategy introduces auxiliary pairwise hidden labels to decompose the calculation of the mixture intensity used in the ISG, in order to enables efficient inference.
In addition, the introduction of these labels allows each node to belong to multiple groups in each dimension (e.g.~a user may interact with different people by playing different roles in a social network), which provides more modelling flexibility compared with the ISG (and existing piecewise-graphons) where each node is assigned to one group only.
\end{itemize}
Note that while we develop the ISG and LFSG for SBM-based graphons, our smoothing approach can easily be applied to other piecewise-constant graphons. The main contributions of our work are summarised as follows:
\begin{itemize}
\item We identify the key limitation of existing piecewise-constant graphons and develop a smoothing strategy to flexibly generate continuous graphon intensity values, which may better reflect the reality of a process.
\item We develop the ISG smoothing strategy for the SBM to demonstrate how piecewise-constant graphons can be converted into smoothing graphons.
\item We improve on the ISG by devising the LFSG, which achieves the same objective of generating continuous intensity values but without sacrificing computation efficiency. Compared with the ISG where each node belongs to only one group, the LFSG allows each node to belong to multiple groups (e.g.~so that the node plays different roles in different relations), and thereby also providing a probabilistic interpretation of node groups.
\item We evaluate the performance of our methods on the task of link prediction by comparing with the SBM and other benchmark methods. The experimental results clearly show that the smoothing graphons can achieve significant performance improvement over piecewise-constant graphons.
\end{itemize}
\section{Preliminaries}
\subsection{Graphon theory}
The Aldous--Hoover theorem~\cite{hoover1979relations,aldous1981representations} provides the theoretical foundation for modelling exchangeable multi-dimensional arrays~(i.e.~exchangeable relational data) conditioned on a stochastic partition model. A random $2$-dimensional array is called {\em separately exchangeable} if its distribution is invariant under separate permutations of rows and columns.
\begin{theorem}~\cite{orbanz2014bayesian, RandomFunPriorsExchArrays}: \label{theoremgraphon}
A random array $(R_{ij})$ is separately exchangeable if and only if it can be represented as follows: there exists a random measurable function $F:[0,1]^3 \mapsto \mathcal{X}$ such that $(R_{ij}) \overset{d}{=} \left(F(u_i^{(1)}, u_j^{(2)}, \nu_{ij})\right)$, where $\{u_i^{(1)}\}_i, \{u_j^{(2)}\}_j$ and $\{\nu_{ij}\}_{i,j}$ are two sequences and an array of i.i.d.~uniform random variables in $[0,1]$, respectively.
\end{theorem}
Many existing Bayesian methods for modelling exchangeable relational data can be represented as in Theorem \ref{theoremgraphon}, using specific forms of the mapping function $F$. For instance, as illustrated in Fig.~\ref{fig:graphon_comparison}, given the uniformly distributed node coordinates $(u_i^{(1)}, u_j^{(2)})$, the SBM corresponds to a regular-grid constant graphon; the MP-RM uses a $k$-d tree structured constant graphon; the RTP-RM adopts an arbitrary rectangle constant graphon; and the GP-RM induces a continuous $2$-dimensional function. While taking different forms, these graphon functions commonly map from pairs of node coordinates in a unit square to intensity values in a unit interval. As shown in Fig.~\ref{fig:graphon_comparison}, the darker colour at the pair of node coordinates the higher intensity in the interval, which corresponds to a larger probability of observing or generating the relation between the pair of nodes.
\subsection{Piecewise-constant graphons and their limitations}
Many alternative piecewise-constant graphons can be implemented to model exchangeable relational data $R$, where $R$ is a binary adjacency matrix which can be either directed~(asymmetric) or undirected~(symmetric). Here we consider the more complicated situation where $R$ is a $n\times n$ asymmetric matrix with $R_{ji}\neq R_{ij}$ (the extension of our method to the symmetric case is straightforward). For any two nodes in $R$, if node $i$ is related to node $j$ then $R_{ij}=1$, otherwise $R_{ij}=0$.
We take the SBM as an illustrative example.
In a two-dimensional SBM, there are two distributions generating the groups, $\pmb{\theta}^{(1)},\pmb{\theta}^{(2)}\sim\text{Dirichlet}(\pmb{\alpha}_{1\times K})$, where $K$ is the number of groups and $\pmb{\alpha}_{1\times K}$ is the $K$-vector concentration parameter.
Each node $i\in \{1, \ldots, n\}$ is associated with two hidden labels $z_i^{(1)}, z_i^{(2)}\in\{1, \ldots, K\}$, and $\{z_i^{(1)}\}_{i}\sim\text{Categorical}(\pmb{\theta}^{(1)}), \{z_i^{(2)}\}_{i}\sim\text{Categorical}(\pmb{\theta}^{(2)})$ for $i=1,\ldots,n$. Hence, $z_i^{(1)}$ and $z_i^{(2)}$ denote the particular groups that node $i$ belongs to in two dimensions, respectively (that is, $z_i^{(1)}$ is the group of node $i$ when $i$ links to other nodes, and $z_i^{(2)}$ is the group of node $i$ when other nodes link to it). The relation $R_{ij}$ from node $i$ to node $j$ is then generated based on the interaction between their respective groups $z_i^{(1)}$ and $z_j^{(2)}$.
Let $\pmb B$ be a $K\times K$ matrix, where each entry $B_{k_1, k_2}\in[0, 1]$ denotes the probability of generating a relation from group $k_1$ in the first dimension to group $k_2$ in the second dimension. For $k_{1},k_{2}=1, \ldots, K, B_{k_{1},k_{2}}\sim\text{Beta}(\alpha_0, \beta_0)$, where $\alpha_0, \beta_0$ are hyper-parameters for $\{B_{k_1,k_2}\}_{k_1, k_2}$. That is, we have $P(R_{ij}=1|z_i^{(1)}, z_j^{(2)}, \pmb B)=B_{z_i^{(1)}, z_j^{(2)}}$.
Now consider the SBM from the graphon perspective (Fig.~\ref{fig:graphon_comparison}; left). Let $\pmb{\theta}^{(1)},\pmb{\theta}^{(2)}$ be group (or segment) distributions of the two dimensions in a unit square respectively. The generation of hidden labels $z_i^{(1)}$ for node $i$ and $z_j^{(2)}$ for node $j$ proceeds as follows: Uniform random variables $u_i^{(1)}$ and $u_j^{(2)}$ are respectively generated in the first and second dimensions. Then, $z_i^{(1)}$ and $z_j^{(2)}$ can be determined by checking in which particular segments of $\pmb{\theta}^{(1)}$ and $\pmb{\theta}^{(2)}$, $u_i^{(1)}$ and $u_j^{(2)}$ are respectively located. Formally, we have:
\begin{align} \label{eq:SBM_partition_geneartion}
& \pmb{\theta}^{(1)}, \pmb{\theta}^{(2)}\sim\text{Dirichlet}(\pmb{\alpha}_{1\times K}),\:\: \:\:u_i^{(1)}, u_j^{(2)}\sim\text{Unif}[0, 1] \nonumber \\
&z_i^{(1)}=(\pmb{\theta}^{(1)})^{-1}(u_i^{(1)}),\:\: \:\: z_j^{(2)}=(\pmb{\theta}^{(2)})^{-1}(u_j^{(2)}),
\end{align}
where $(\pmb{\theta}^{(1)})^{-1}(u_i^{(1)})$ and $(\pmb{\theta}^{(2)})^{-1}(u_j^{(2)})$ respectively map $u_i^{(1)}$ and $u_j^{(2)}$ to particular segments of $\pmb{\theta}^{(1)}$ and $\pmb{\theta}^{(2)}$.
A regular-grid partition~($\boxplus$) can be formed in the unit square by combining the segment distributions $\pmb{\theta}^{(1)}, \pmb{\theta}^{(2)}$ in two dimensions. Each block in this regular-grid partition is presented in a rectangular shape. Let $L_{k}^{(1)}=\sum_{k'=1}^k{{\theta}}_{k'}^{(1)}$ {and $L_{k}^{(2)}=\sum_{k'=1}^k{{\theta}}_{k'}^{(2)}$ be the accumulated sum of the first $k$ elements of $\pmb{\theta}^{(1)}$ and $\pmb{\theta}^{(2)}$ respectively~(w.l.o.g.~$L_{0}^{(1)}=L_{0}^{(2)}=0, L_{K}^{(1)}=L_{K}^{(2)}=1$).} Use $\Box_{k_1,k_2}=[L^{(1)}_{k_1-1}, L^{(1)}_{k_1}]\times [L^{(2)}_{k_2-1}, L^{(2)}_{k_2}]$ to represent the $(k_1, k_2)$-th block in the unit square of $[0,1]^2$, such that
$\bigcup_{k_1, k_2}\Box_{k_1,k_2}=[0,1]^2$. Then, an intensity function defined on the pair $(u_i, u_j)$ can be obtained by the piecewise-constant graphon function
\begin{align} \label{SBM_graphon_intensity_function}
g\left(u_i^{(1)}, u_j^{(2)}\right) = \sum_{k_1, k_2} \pmb{1}{((u_i^{(1)},u_j^{(2)})\in \Box_{k_1,k_2})} \cdot B_{k_1,k_2}
\end{align}
where $\pmb{1}(A)=1$ if $A$ is true and $0$ otherwise, and
where $B_{k_1, k_2}\in[0, 1]$ is the intensity of the $(k_1, k_2)$-th block. We term \eqref{SBM_graphon_intensity_function} the
SBM-graphon. Thus, the generative process of the SBM-graphon can be described as:
\begin{enumerate}
\item For $k_{1},k_{2}=1, \ldots, K$, generate $B_{k_{1},k_{2}}\sim\text{Beta}(\alpha_0, \beta_0)$, where $\alpha_0, \beta_0$ are hyper-parameters;
\item Generate the segment distributions $\pmb{\theta}^{(1)}, \pmb{\theta}^{(2)}$ via Eq.~(\ref{eq:SBM_partition_geneartion}) and form the partition~($\boxplus$) according to combinations of $\pmb{\theta}^{(1)}, \pmb{\theta}^{(2)}$ in the unit square;
\item Uniformly generate the $1^{st}$ dimension coordinates $\{u_i^{(1)}\}_{i=1}^n$ and the $2^{nd}$ dimension coordinates $\{u_i^{(2)}\}_{i=1}^n$ for all nodes;
\item For $i,j=1, \ldots, n$
\begin{enumerate}
\item Calculate the intensity $g(u_i^{(1)}, u_j^{(2)})$ according to Eq.~(\ref{SBM_graphon_intensity_function}) based on the node coordinates $(u_i^{(1)}, u_j^{(2)})$;
\item Generate $R_{ij}\sim\mbox{Bernoulli}(g(u_i^{(1)}, u_j^{(2)}))$.
\end{enumerate}
\end{enumerate}
Alternatively, if considering the latent labels~($z_i^{(1)}, z_j^{(2)}$) for nodes $i$ and $j$, then from Eq.~(\ref{eq:SBM_partition_geneartion}), step 4) can also be written as
\begin{enumerate}
\setcounter{enumi}{3}
\item For $i, j=1, \cdots, n$,
\begin{enumerate}
\item Generate the latent labels~($z_i^{(1)}, z_j^{(2)}$) via Eq.~(\ref{eq:SBM_partition_geneartion});
\item Generate $R_{ij}\sim\text{Bernoulli}\left(B_{z_i^{(1)}, z_j^{(2)}}\right)$.
\end{enumerate}
\end{enumerate}
The SBM-graphon has several limitations.
Firstly, the SBM-graphon function
(Eq.~(\ref{SBM_graphon_intensity_function}))
is piecewise-constant. That is, the generated intensities for node pairs are discrete and the number of different intensity values is limited to the number of blocks in the partition ($\boxplus$). Consequently, this leads to an over-simplified description when modelling real
relational data, which can result in at least two issues. On the one hand, as long as two nodes belong to the same segment in one dimension, their probabilities of generating relations with another node are the same even if the distance between the two nodes in that dimension is quite far. Conversely, given two nodes that are close in one dimension but belong to two adjacent segments, their probabilities of generating relations with another node could be dramatically different, depending on the respective block intensities (e.g., $B_{k_1,k_2}$).
The second limitation of the SBM-graphon is that it determines the intensity value for a pair of nodes by considering only the block ($\Box_{k_1,k_2}$) in which $(u_i, u_j)$ resides. However, the nodes relations with other nodes, especially neighbouring nodes in adjacent blocks, may also be expected to have a certain influence on the generation of the target relation, if one considers the relational data collectively. As a result, it may perhaps be beneficial to consider the interactions that naturally exist among all blocks when generating the relation $R_{ij}$.
The third limitation of the SBM-graphon is that it provides latent information of node clustering as a side-product through the hidden labels $\{z_i^{(1)}, z_i^{(2)}\}_{i=1}^n$. However, the clustering information may not be ideal because each node is assigned to only one cluster in each dimension. That is, when considering the outgoing relations from node $i$, it is assumed that node $i$ consistently plays one single role in any relation with other nodes. In fact, in practice a node may play different roles by participating in different relations with different nodes. As a result, it would be more useful and flexible to allow a node to belong to multiple clusters in each dimension.
To address the limitations of piecewise-constant graphons (and in particular, the SBM-graphon), we propose a smoothing strategy to enable piecewise-constant graphons to produce continuous intensity values. The proposed smoothing graphons naturally consider interactions between the partitions and allow each node to play multiple roles in different relations.
\section{Main Models}
\subsection{The Integrated Smoothing Graphon~(ISG)}
In order to improve on the limitations of the piecewise-constant graphon
we first develop the Integrated Smoothing Graphon~(ISG), based on the SBM-graphon construction.
The piecewise-constant nature of the SBM-graphon is created through the use of an indicator function in \eqref{SBM_graphon_intensity_function} that selects only the particular block accommodating the target node pair.
Accordingly, we replace the indicator function with an alternative that can produce continuous intensity values. Moreover, to capture the interaction between all blocks, we construct the smoothing graphon function to generate the intensity value as a summation over all block intensities, weighted by the importance of each block. Let $\bar{F}_{\Box_{k_1,k_2}}(u_i^{(1)}, u_j^{(2)})$ be the weight of the block $\Box_{k_1,k_2}$ with respect to $(u_i^{(1)}, u_j^{(2)})\in[0, 1]^2$. The mixture intensity $g\left(u_i^{(1)}, u_j^{(2)}\right)$, used to determine the $R_{ij}$, can then be represented as
\begin{align} \label{smooth_graphon_intensity_function}
g\left(u_i^{(1)}, u_j^{(2)}\right) = \sum_{k_1, k_2}\bar{F}_{\Box_{k_1,k_2}}(u^{(1)}_i, u^{(2)}_j)\cdot B_{k_1, k_2},
\end{align}
where { $\sum_{k_1,k_2}\bar{F}_{\Box_{k_1,k_2}}(u^{(1)}_i, u^{(2)}_j)=1$.}
The ISG generative process can be summarised as:
1)$\sim$3) The block intensities~($\pmb B$), graphon partition~($\boxplus$) and $2$-dimensional coordinates ($\{u_{i}^{(1)}, u_{i}^{(2)}\}_{i=1}^n$) are generated as for the SBM-graphon;
\begin{enumerate}
\setcounter{enumi}{3}
\item For $i, j=1, \cdots, n$,
\begin{enumerate}
\item Calculate the mixture intensity $g\left(u_i^{(1)}, u_j^{(2)}\right)$ according to \eqref{smooth_graphon_intensity_function} for the node coordinates $(u_i^{(1)}, u_j^{(2)})$;
\item Generate $R_{ij}\sim\mbox{Bernoulli}\left(g\left(u_i^{(1)}, u_j^{(2)}\right)\right)$.
\end{enumerate}
\end{enumerate}
As a consequence, while the SBM-graphon determines the relation intensity based only on the single block in which $(u_i^{(1)}, u_j^{(2)})$ resides, the ISG computes a mixture intensity as a weighted (and normalised) sum of all block intensities. That is, instead of assigning a weight of $1$ for one particular block and weights of $0$ for all other blocks, the ISG weights the importance of each block with respect to the pair of node coordinates $(u_i^{(1)}, u_j^{(2)})$. As long as the weighting function $\bar{F}_{\Box_{k_1,k_2}}(u_i^{(1)}, u_j^{(2)})$ is continuous, then the mixture intensity \eqref{smooth_graphon_intensity_function} is also continuous. The intensity function \eqref{smooth_graphon_intensity_function} then becomes a smoothing graphon function.
The ISG allows the mixture intensity to take any value between the minimum and maximum of all block intensities. As a result, the ISG provides more modelling flexibility compared to the SBM-graphon, where only limited discrete intensity values~(equivalent to the number of blocks) are available to describe relations.
\subsection{Construction of the mixture intensity}
To ensure that the graphon function \eqref{smooth_graphon_intensity_function} is continuous, we consider an integral-based weighting function of the form
{\small\begin{align}
{F}_{\Box_{k_1,k_2}}(u_i^{(1)}, u_j^{(2)}) \propto
\int_{L^{(1)}_{k_1-1}}^{L^{(1)}_{k_1}}f(x-u_i^{(1)})dx
\cdot {\int_{L^{(2)}_{k_2-1}}^{L^{(2)}_{k_2}}f(x-u_j^{(2)})dx}
\end{align}}
where $f(x-u)$ is a
univariate derivative function. Beyond the continuity requirement, $f(x-u)$ and ${F}_{\Box_{k_1,k_2}}(u_i^{(1)}, u_j^{(2)})$ should satisfy the following three conditions:
\begin{enumerate}
\item $f(x-u)$ is non-negative;
\item $f(x-u)$ increases with decreasing distance between $x$ and the corresponding coordinate $u$ (i.e.~$|x-u_i^{(1)}|$ or $|x-u_j^{(2)}|$).
This condition means that the closer the block $\Box_{k_1,k_2}$ is to the pair of node coordinates $(u_i^{(1)}, u_j^{(2)})$, the larger weight the block will be assigned. The maximum weight value is achieved when $|x-u_i^{(1)}|=0$ and $|x-u_j^{(2)}|=0$;
\item
The total weight of all blocks remains invariant regardless of different partitioning of the unit space. That is, ${F}_{\Box_{k_1,k_2}}(u_i^{(1)}, u_j^{(2)})={F}_{\Box'_{k_1,k_2}}(u_i^{(1)}, u_j^{(2)})+{F}_{\Box''_{k_1,k_2}}(u_i^{(1)}, u_j^{(2)})$, where $\Box'_{k_1,k_2}, \Box''_{k_1,k_2}$ are sub-boxes of $\Box_{k_1,k_2}$ such that $\Box_{k_1,k_2}=\Box'_{k_1,k_2}\cup\Box''_{k_1,k_2}$ and
$\Box'_{k_1,k_2}\cap\Box''_{k_1,k_2}=\emptyset$.
\end{enumerate}
\begin{figure}[h]
\centering
\includegraphics[width = 0.35 \textwidth]{Figure_3_a.pdf}
\includegraphics[width = 0.36 \textwidth]{Figure_3b.pdf}
\caption{The influence of the $\lambda$ parameter on the Laplace probability density function with coordinate located at $u=0.3$~(top), and the corresponding mixture intensities for $\{u_i\}_{i=1}^n$~(bottom). Different colors represent different settings of $\lambda$. The gray dotted lines represent segment division in one dimension with $\pmb{\theta}=(0.15, 0.27, 0.08, 0.5)^\top\sim\text{Dirichlet}(1, 1, 1,1)$. }
\label{fig:weight_function_vis}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width = 0.45 \textwidth]{Figure_4.pdf}
\caption{Visualizations of the Integrated Smoothing Graphon under the Stochastic Block Model for different values of $\lambda$. Darker shading represents higher graphon intensity.}
\label{fig:lambda_related_visulizaton}
\end{figure}
There are many candidate functions satisfying these conditions, such as Gaussian or Laplace probability density functions. For ease of computation, we use the Laplace density (with location parameter $\mu = 0$)
as the derivative function, so that $f_{\lambda}(x-u)=\frac{{{\lambda}}}{2}e^{{-{{\lambda}} |x-u|}}$. Let $G_{\lambda}(x-u)=\left\{\begin{array}{lr}
\frac{1}{2}e^{{{\lambda}} (x-u)}; & (x-u)<0 \\
1-\frac{1}{2}e^{-{{\lambda}} (x-u)}; & (x-u)\ge0
\end{array}\right.$. We then have $\int_{L_{k_1-1}}^{L_{k_1}}f_{\lambda}(x-u)dx=G_{\lambda}(L_{k_1}-u)-G_{\lambda}(L_{k_1-1}-u) $. As a result, for relation $R_{ij}$ and corresponding node coordinates $(u^{(1)}_i, u^{(2)}_j)$, the normalised weight $\bar{F}_{\Box_{k_1,k_2}}(u^{(1)}_i, u^{(2)}_j)$ of the $(k_1, k_2)$-th block $\Box_{k_1,k_2}$ contributing to the mixture intensity of $R_{ij}$ is given by
\begin{align} \label{eq:pi_val}
\bar{F}_{\Box_{k_1,k_2}}(u^{(1)}_i, u^{(2)}_j) =& \frac{G_{\lambda}(L^{(1)}_{k_1}-u^{(1)}_i)-G_{\lambda}(L_{k_1-1}^{(1)}-u^{(1)}_i)}{G_{\lambda}(1-u^{(1)}_i)-G_{\lambda}(-u^{(1)}_i)}\nonumber \\
&\times\frac{G_{\lambda}(L^{(2)}_{k_2}-u^{(2)}_j)-G_{\lambda}(L_{k_2-1}^{(2)}-u^{(2)}_j)}{G_{\lambda}(1-u^{(2)}_j)-G_{\lambda}(-u^{(2)}_j)}.
\end{align}
\begin{proposition}
$\sum_{k_1,k_2}\bar{F}_{\Box_{k_1,k_2}}(u^{(1)}_i, u^{(2)}_j)=1$.
\end{proposition}
\begin{proof}
{\small\begin{align}
&\sum_{k_1,k_2}\bar{F}_{\Box_{k_1,k_2}}(u^{(1)}_i, u^{(2)}_j) \nonumber \\
=&\left[\sum_{k_1}\frac{G_{\lambda}(L^{(1)}_{k_1}-u^{(1)}_i)-G_{\lambda}(L_{k_1-1}^{(1)}-u^{(1)}_i)}{G_{\lambda}(1-u^{(1)}_i)-G_{\lambda}(-u^{(1)}_i)}\right]\nonumber \\
&\cdot\left[\sum_{k_2}\frac{G_{\lambda}(L^{(2)}_{k_2}-u^{(2)}_j)-G_{\lambda}(L_{k_2-1}^{(2)}-u^{(2)}_j)}{G_{\lambda}(1-u^{(2)}_j)-G_{\lambda}(-u^{(2)}_j)}\right]=1.
\end{align}}
\end{proof}
Fig.~\ref{fig:weight_function_vis} (top) illustrates the function curves of $f_{\lambda}(x-u)$ for $u=0.3$ and Fig.~\ref{fig:weight_function_vis} (bottom) shows the resulting one-dimensional mixture intensities
under varying scale parameter values $\lambda=0.25, 25$ and $250$.
It is easily observed that when $\lambda$ is smaller, both the curves of the derivative function and the mixture intensity become flatter and smoother. Conversely, for larger $\lambda$, the mixture intensity values (generated for the coordinate $0.3$) become more discrete. Fig.~\ref{fig:lambda_related_visulizaton} visualizes the mixture intensities obtained by applying the ISG to the SBM under the same three $\lambda$ values.
\begin{proposition}
$\lambda$ controls the smoothness of the graphon, with $\lambda\rightarrow\infty$ recovering the piecewise-constant graphon, and $\lambda\rightarrow 0$ resulting in a globally constant graphon.\end{proposition}
\begin{proof}
Using the L'hospital rule, when $\lambda\rightarrow 0$, we have
\begin{align}
&\lim_{\lambda\rightarrow 0} \frac{G_{\lambda}(L^{(1)}_{k_1}-u^{(1)}_i)-G_{\lambda}(L_{k_1-1}^{(1)}-u^{(1)}_i)}{G_{\lambda}(1-u^{(1)}_i)-G_{\lambda}(-u^{(1)}_i)}\nonumber \\
=&\frac{l_{k_1}^{(1)}-u_i^{(1)}-(l_{{k_1}-1}^{(1)}-u_i^{(1)})}{1-u_i^{(1)}+u_i^{(1)}} = l_{k_1}^{(1)}-l_{{k_1}-1}^{(1)}
\end{align}
Thus, we get $\bar{F}_{\Box_{k_1,k_2}}(u^{(1)}_i, u^{(2)}_j) = (l_{k_1}^{(1)}-l_{k_1-1}^{(1)})(l_{k_2}^{(2)}-l_{k_2-1}^{(2)})$, which is unrelated to the coordinate of $(u^{(1)}_i, u^{(2)}_j)$. The graphon is a globally constants graphon.
We have three different cases when $\lambda\rightarrow \infty$: case (1), $L_{k_1}^{(1)}>L_{k_1-1}^{(1)}>u_i^{(1)}$, we have
\begin{align}
&\lim_{\lambda\rightarrow \infty} \frac{G_{\lambda}(L^{(1)}_{k_1}-u^{(1)}_i)-G_{\lambda}(L_{k_1-1}^{(1)}-u^{(1)}_i)}{G_{\lambda}(1-u^{(1)}_i)-G_{\lambda}(-u^{(1)}_i)}\nonumber \\
= & \lim_{\lambda\rightarrow \infty} \frac{1-\frac{1}{2}e^{-\lambda(L^{(1)}_{k_1}-u^{(1)}_i)}-(1-\frac{1}{2}e^{-\lambda(L^{(1)}_{k_1-1}-u^{(1)}_i)})}{1-\frac{1}{2}e^{-\lambda(1-u^{(1)}_i)}-\frac{1}{2}e^{-\lambda(u^{(1)}_i)}}=0;\nonumber
\end{align}
case (2), $L_{k_1-1}^{(1)}<L_{k_1-1}^{(1)}<u_i^{(1)}$, we have
\begin{align}
&\lim_{\lambda\rightarrow \infty} \frac{G_{\lambda}(L^{(1)}_{k_1}-u^{(1)}_i)-G_{\lambda}(L_{k_1-1}^{(1)}-u^{(1)}_i)}{G_{\lambda}(1-u^{(1)}_i)-G_{\lambda}(-u^{(1)}_i)}\nonumber \\
= & \lim_{\lambda\rightarrow \infty} \frac{\frac{1}{2}e^{\lambda(L^{(1)}_{k_1}-u^{(1)}_i)}-(\frac{1}{2}e^{\lambda(L^{(1)}_{k_1-1}-u^{(1)}_i)})}{1-\frac{1}{2}e^{-\lambda(1-u^{(1)}_i)}-\frac{1}{2}e^{-\lambda(u^{(1)}_i)}}=0;\nonumber
\end{align}
case (3), $L_{k_1-1}^{(1)}<u_i^{(1)}<L_{k_1-1}^{(1)}$, we have
\begin{align}
&\lim_{\lambda\rightarrow \infty} \frac{G_{\lambda}(L^{(1)}_{k_1}-u^{(1)}_i)-G_{\lambda}(L_{k_1-1}^{(1)}-u^{(1)}_i)}{G_{\lambda}(1-u^{(1)}_i)-G_{\lambda}(-u^{(1)}_i)}\nonumber \\
= & \lim_{\lambda\rightarrow \infty} \frac{1-\frac{1}{2}e^{-\lambda(L^{(1)}_{k_1}-u^{(1)}_i)}-(\frac{1}{2}e^{\lambda(L^{(1)}_{k_1-1}-u^{(1)}_i)})}{1-\frac{1}{2}e^{-\lambda(1-u^{(1)}_i)}-\frac{1}{2}e^{-\lambda(u^{(1)}_i)}}=1;\nonumber
\end{align}
That is, $\bar{F}_{\Box_{k_1,k_2}}(u^{(1)}_i, u^{(2)}_j)=1$ if and only if the coordinate $(u^{(1)}_i, u^{(2)}_j)$ locates in the $(k_1, k_2)$-th block. Thus, the graphon would turn into a piecewise-constant one when $\lambda\rightarrow\infty$.
\end{proof}
Accordingly, we refer to $\lambda$ as the smoothing parameter.
\subsection{Latent Feature Smoothing Graphon~(LFSG) with probabilistic assignment}
While the ISG addresses the limitations of the SBM-graphon by generating continuous intensity values, its graphon function \eqref{smooth_graphon_intensity_function} indicates that all blocks are involved in calculating the mixture intensity for generating individual relations. Accordingly, the additive form for evaluating the mixture intensity makes it difficult to form efficient inference schemes for all random variables.
To
improve
inferential
efficiency we introduce auxiliary pairwise latent labels $\{s_{ij}\}_{j=1}^n$ (associated with node $i$) and $\{r_{ij}\}_{i=1}^n$ (associated with node $j$) for individual relations $\{R_{ij}\}_{i,j=1}^n$, where $s_{ij},r_{ij} \in\{1, \ldots, K\}$. The $\{s_{ij}\}_{j=1}^n$ and $\{r_{ij}\}_{i=1}^n$ are sampled from the respective node categorical distributions in their corresponding dimensions using normalised weights as probabilities. In particular
\begin{align} \label{eq:s_ij_r_ij}
\{s_{ij}\}_{j=1}^n&\sim\text{Categorical}(\bar{F}^{(1)}_{1}(u^{(1)}_i), \ldots, \bar{F}^{(1)}_{K}(u^{(1)}_i)) \nonumber \\
\{r_{ij}\}_{i=1}^n&\sim\text{Categorical}(\bar{F}^{(2)}_{1}(u^{(2)}_j), \ldots, \bar{F}^{(2)}_{K}(u^{(2)}_j)),
\end{align}
where $\bar{F}_{k}(u)=\frac{G_{\lambda}(L_{k}-u)-G_{\lambda}(L_{k-1}-u)}{G_{\lambda}(1-u)-G_{\lambda}(-u)}$ is the normalised weight of segment $k$ in the dimension of coordinate $u$.
For each relation from node $i$ to node $j$ ($R_{ij}$), the hidden label $s_{ij}$ denotes the group that node $i$ belongs to (in the $1$st dimension) and $r_{ij}$ denotes the group that node $j$ belongs to (in the $2$nd dimension). Through the introduction of the two labels, the final intensity in determining $R_{ij}$ can be obtained similarly to the Mixed Membership Stochastic Block Model (MMSB)~\cite{airoldi2009mixed}:
\begin{align}
P(R_{ij}=1|s_{ij}, r_{ij}, \pmb B)=B_{s_{ij}, r_{ij}}.
\end{align}
Note that since both $\{s_{ij}\}_{j=1}^n$ and $\{r_{ij}\}_{j=1}^n$ are $n$-element arrays, each node has the potential to belong to multiple segments, rather than the single segment under the SBM-graphon. When participating in different relations, each outgoing node $i$ (incoming node $j$) may fall into different segments, which means that each node may play different roles when taking part in different relations.
Note that taking expectations over the hidden labels $s_{ij}$ and $r_{ij}$, results in the same intensity as for the ISG, so that
\begin{align}
\mathbb{E}_{s_{ij}, r_{ij}}\left[P(R_{ij}=1\vert s_{ij}, r_{ij}, \pmb B)\right]= g\left((u_i^{(1)}, u_j^{(2)})\right).
\end{align}
We term this approach the Latent Feature Smoothing Graphon~(LFSG). Its generative process is described as follows
1)$\sim$3) The block intensities~($\pmb B$), graphon partition~($\boxplus$) and $2$-dimensional coordinates ($\{u_{i}^{(1)}, u_{i}^{(2)}\}_{i=1}^n$) are generated as for the SBM-graphon;
\begin{enumerate}
\setcounter{enumi}{3}
\item For $i=1, \cdots, n$, calculate the hidden label distributions in each dimension, $\bar{\pmb{F}}^{(1)}(u_i^{(1)})$ and $\bar{\pmb{F}}^{(2)}(u_i^{(2)})$, where $\bar{\pmb{F}}^{(1)}(u_i^{(1)})=({\bar{F}}^{(1)}_1(u_i^{(1)}), \ldots, {\bar{F}}^{(1)}_K(u_i^{(1)}))$;
\item For $i, j=1, \cdots, n$,
\begin{enumerate}
\item Generate the hidden labels $s_{ij}\sim\bar{\pmb{F}}^{(1)}(u_i^{(1)}), r_{ij}\sim\bar{\pmb{F}}^{(2)}(u_j^{(2)})$ following \eqref{eq:s_ij_r_ij}
\item Generate $R_{ij}\sim\text{Bernoulli}\left(B_{s_{ij}, r_{ij}}\right)$.
\end{enumerate}
\end{enumerate}
The graphical models for implementing the ISG within the SBM (referred to as the ISG-SBM), and also implementing the LFSG within the SBM (referred to as the LFSG-SBM) are illustrated in Fig.~\ref{fig:graphical_model}.
The main difference between the two models -- the introduction of the pairwise hidden labels $s_{ij}$ and $r_{ij}$ for generating each relation $R_{ij}$ -- allows the LFSG-SBM to enjoy the following advantages over the ISG-SBM:
\begin{itemize}
\item The aggregated counting information of the hidden labels enables efficient Gibbs sampling of the block intensities $\pmb{B}$.
\item Calculation involving $\bar{F}_{\Box}$ is instead reduced to calculation involving $\bar{F}_{k}(u)$, avoiding the inclusion of all blocks when calculating the mixture intensity.
\item
Because each node is allowed to belong to different groups when linking to other nodes, permitting differences in the natures of those links, the group distribution $\bar{F}(u)$ is then easily interpretable as the group membership distribution for that node.
For example, a higher membership degree in group $k$ indicates the node is more important or active in group $k$.
\end{itemize}
\begin{figure}[h]
\centering
\includegraphics[width = 0.45 \textwidth]{graphical_model.pdf}
\caption{The graphical model for (a) the ISG-SBM and (b) the LFSG-SBM. (a) The weights of the blocks $\bar{F}_{\Box}$ are first calculated by using the partition $\boxplus$, node coordinates $U$ and smoothing parameter $\lambda$. Then the $\bar{F}_{\Box}$ and block intensities $\pmb B$ are integrated together to generate the exchangeable relations $R$. (b) The weight $\bar{F}$ for each node is individually generated using the partition $\boxplus$, node coordinates $U$ and the smoothing parameter $\lambda$, based on the auxiliary hidden labels $s, r$ for the node pair relationship. Then $s, r$ are used to generate the exchangeable relational, $R$, together with the block intensities $\pmb B$.}
\label{fig:graphical_model}
\end{figure}
\subsection{Extensions to other piecewise-constant graphons}
The major difference between the construction of the existing piecewise-constant graphons is the generation process of partitions~($\boxplus$; Fig.~\ref{fig:graphon_comparison}). As a result, our smoothing approach, while described for the SBM-graphon, can be straighforwardly applied to other piecewise-constant graphons. For example, to apply the ISG to other piecewise-constant graphons, we can similarly calculate a mixture intensity as a weighted sum of the intensities of all existing blocks.
When the partitioned blocks are rectangular-shaped (as for e.g.~the MP-graphon, RTP-graphon and RBP-graphon), the intensity for each block can be computed by independently integrating the derivative function over two dimensions. If the partitioned blocks are shaped as convex-polygons (as for e.g.~the BSP-graphon~\cite{pmlr-v84-fan18b}), the intensity can be generated via integrating the derivative function over the polygon.
\section{Inference}
\begin{algorithm}[t]
\caption{{MCMC for the ISG}}
\label{detailInference_ISG}
\begin{algorithmic}
\REQUIRE Exchangeable relational data $R\in\{0, 1\}^{n\times n}$, hyperparameters $\alpha_0, \beta_0, \pmb{\alpha}_{1\times K}$, iteration time $T$
\ENSURE $\{u_{i}^{(1)}, u_{i}^{(2)}\}_{{i=1}}^n, \pmb{\theta}^{(1)}, \pmb{\theta}^{(2)}, \pmb{B}, \lambda$
\FOR{$t=1,\cdots, T$}
\FOR{$i=1,\ldots,n$}
\STATE Sample $u_i^{(1)}, u_i^{(2)}$; // according to \eqref{eq:sample_u_i}
\ENDFOR
\STATE Sample $\pmb{\theta}^{(1)}, \pmb{\theta}^{(2)}$; // according to \eqref{eq:sample_theta}
\FOR{$k_1, k_2=1,\ldots,K$}
\STATE Sample $B_{k_1,k_2}$; // according to \eqref{eq:sample_B_k1k2}
\ENDFOR
\STATE Sample $\lambda$; // according to \eqref{eq:sample_lambda}
\ENDFOR
\end{algorithmic}
\end{algorithm}
\begin{algorithm}[t]
\caption{MCMC for the LFSG}
\label{detailInference_SG}
\begin{algorithmic}
\REQUIRE Exchangeable relational data $R\in\{0, 1\}^{n\times n}$, hyperparameters $\alpha_0, \beta_0, \pmb{\alpha}_{1\times K}$, iteration time $T$
\ENSURE $\{u_{i}^{(1)}, u_{i}^{(2)}\}_{{i=1}}^n, \pmb{\theta}^{(1)}, \pmb{\theta}^{(2)}, \pmb{B}, \{s_{ij},r_{ij}\}_{i,j=1}^n, \lambda$
\FOR{$t=1,\cdots, T$}
\FOR{$i=1,\ldots,n$}
\STATE Sample $u_i^{(1)}, u_i^{(2)}$; // according to \eqref{eq:sample_u_i}
\ENDFOR
\STATE Sample $\pmb{\theta}^{(1)}, \pmb{\theta}^{(2)}$; // according to \eqref{eq:sample_theta}
\FOR{$k_1, k_2=1,\ldots,K$}
\STATE Sample $B_{k_1,k_2}$; // according to \eqref{eq:sample_B_k1k2}
\ENDFOR
\FOR{$i,j=1,\ldots,n$}
\STATE Sample $s_{ij},r_{ij}$; // according to \eqref{eq:sample_s_r_ij}
\ENDFOR
\STATE Sample $\lambda$; // according to \eqref{eq:sample_lambda}
\ENDFOR
\end{algorithmic}
\end{algorithm}
We present a Markov Chain Monte Carlo~(MCMC) algorithm for posterior model inference, with detailed steps for the ISG and the LFSG as illustrated in Algorithms \ref{detailInference_ISG} and \ref{detailInference_SG} respectively.
In general, the joint distribution over the hidden labels $\{s_{ij}, r_{ij}\}_{i,j=1}^n$, pairwise node coordinates $\{u_{i}^{(1)}, u_{i}^{(2)}\}_{i=1}^n$, group distributions $\pmb{\theta}^{(1)}, \pmb{\theta}^{(2)}$, the block intensities $\pmb{B}$ and the smoothing parameter $\lambda$ is:
\begin{align}
& P(\{s_{ij}, r_{ij}, R_{ij}\}_{i,j=1}^n, \{u_{i}^{(1)}, u_{i}^{(2)}\}_{{i=1}}^n, \pmb{\theta}^{(1)}, \pmb{\theta}^{(2)}, \pmb{B}, \lambda|\alpha_0, \beta_0)\nonumber \\
\propto & \prod_{i,k}\left[\bar{F}_{k}^{(1)}(u_i^{(1)}|\pmb{\theta}^{(1)}, \lambda)^{m_{ik}^{(1)}}\bar{F}_{k}^{(2)}(u_i^{(2)}|\pmb{\theta}^{(2)}, \lambda)^{m_{ik}^{(2)}}\right]\nonumber \\
&\cdot \prod_{k_1, k_2}\left[B_{k_1, k_2}^{N_{k_1, k_2}^{(1)}+\alpha_0-1}(1-B_{k_1, k_2})^{N_{k_1, k_2}^{(0)}+\beta_0-1}\right]\nonumber \\
&\cdot\prod_{k}\left[(\theta_{k}^{(1)})^{\alpha_{k}-1}(\theta_{k}^{(2)})^{\alpha_{k}-1}\right]\cdot P(\lambda)
\end{align}
where $m_{ik}^{(1)}=\sum_{j=1}^n\pmb{1}(s_{ij}=k), m_{{ik}}^{(2)}=\sum_{j=1}^n\pmb{1}(r_{ji}=k), N_{k_1, k_2}^{(1)}=\sum_{(i,j):s_{ij}=k_1, r_{ij}=k_2}\pmb{1}(R_{ij}=1), N_{k_1, k_2}^{(0)}=\sum_{(i,j):s_{ij}=k_1, r_{ij}=k_2}\pmb{1}(R_{ij}=0)$. In this joint distribution, we have set the following prior distributions for the variables: $s_{ij}\sim\text{Categorical}(\bar{\pmb{F}}(u_i^{(1)}|\pmb{\theta}^{(1)}, \lambda)), B_{k_1,k_2}\sim\text{Beta}(\alpha_0, \beta_0), \pmb{\theta}^{(1)}\sim\text{Dirichlet}(\pmb{\alpha}_{1\times K})$.
The details for updating each parameter in the ISG and LFSG MCMC algorithms are listed below.
\paragraph{Updating $\{u_{i}^{(1)}, u_{i}^{(2)}\}_{i=1}^n$}
Independent Metropolis-Hastings steps can be used to update the variables $u_{i}^{(1)}, u_{i}^{(2)}$. We propose a new sample for $u_{i}^{(1)}$
as $u^*\sim\text{Beta}[\alpha_u, \beta_u]$, and accept this proposal with probability $\min(1, \alpha_{u_i^{(1)}})$ where
\begin{align} \label{eq:sample_u_i}
\alpha_{u_i^{(1)}} = \frac{Be(u_i^{(1)}|\alpha_u,\beta_u)}{Be(u^*|\alpha_u,\beta_u)}
\prod_{k}\frac{\bar{F}_{k}^{(1)}(u^{*}|\pmb{\theta}^{(1)}, \lambda)^{m_{ik}^{(1)}}}{\bar{F}^{(1)}_{k}(u_i^{(1)}|\pmb{\theta}^{(1)}, \lambda)^{m_{ik}^{(1)}}},
\end{align}
where $Be(u|\alpha,\beta)$ denotes the Beta density with parameters $\alpha$ and $\beta$ evaluated at $u$.
The update for $u_i^{(2)}$ proceeds similarly. Note that each of the $2n$ parameters $\{u_i^{(1)},u_i^{(2)}\}_{i=1}^n$ can be updated in parallel. In our simulations we found that $\alpha_u=\beta_u=1$ gave good sampler performance.
\paragraph{Updating $\pmb{\theta}^{(1)}, \pmb{\theta}^{(2)}$}
A Metropolis-Hastings step can be used to update $\pmb{\theta}^{(1)}$, and $\pmb{\theta}^{(2)}$. For $\pmb{\theta}^{(1)}$ or $\pmb{\theta}^{(2)}$ we draw a proposed sample $\pmb{\theta}^*\sim\text{Dirichlet}(\pmb{\alpha}_{1\times K})$ from a Dirichlet distribution with concentration parameters $\pmb{\alpha}_{1\times K}$. We accept the proposal $\pmb{\theta}^*$ for w.l.o.g.~$\pmb{\theta}^{(1)}$ with probability $\min(1, \alpha_{\pmb{\theta}^{(1)}})$, where
\begin{align} \label{eq:sample_theta}
\alpha_{\pmb{\theta}^{(1)}} = \prod_{i,k}\frac{\bar{F}^{(1)}_{k}(u_i^{(1)}|\pmb{\theta}^{*}, \lambda)^{m_{ik}^{(1)}}}{\bar{F}^{(1)}_{k}(u_i^{(1)}|\pmb{\theta}^{(1)}, \lambda)^{m_{ik}^{(1)}}},
\end{align}
with a similar update for $\pmb{\theta}^{(2)}$. Both $\pmb{\theta}^{(1)}$ and $\pmb{\theta}^{(2)}$ can be updated in parallel.
\paragraph{Updating $\pmb B$}
The conjugacy between the prior and the conditional likelihood for $\pmb B$ means that we can update $\pmb B$ via a Gibbs sampling step.
Specifically, each entry $B_{k_1, k_2}$ can be updated in parallel via
\begin{align} \label{eq:sample_B_k1k2}
B_{k_1, k_2}\sim\text{Beta}(\alpha_0+N_{k_1, k_2}^{(1)}, \beta_0+N_{k_1, k_2}^{(0)}), \forall k_1, k_2.
\end{align}
\paragraph{Updating $\{s_{ij}, r_{ij}\}_{i,j=1}^n$}
The posterior distribution of $s_{ij}$ is a categorical distribution, where the probability of $s_{ij}=k$ is
\begin{align} \label{eq:sample_s_r_ij}
P(s_{ij}=k|\theta_{k}^{(i)}, R_{ij}, B_{k,r_{ij}})\propto\: & \bar{F}_{k}^{(1)}(u_i^{(1)}|\pmb{\theta}^{(1)}, \lambda)\\&\times B_{k,r_{ij}}^{R_{ij}}\nonumber
(1-B_{k,r_{ij}})^{1-R_{ij}},
\end{align}
and from which $s_{ij}$ may be straightforwardly updated ($r_{ij}$ may be updated in a similar way).
Each of the $2n$ parameters can be updated in parallel.
\paragraph{Updating $\lambda$}
A Metropolis-Hastings step can be used to update $\lambda$. We draw a proposed value $\lambda^*\sim\Gamma(1, 1)$ and accept it with probability $\min(1, \alpha_{\lambda})$, where
\begin{align} \label{eq:sample_lambda}
\alpha_{\lambda}=\prod_{i,k}\frac{\bar{F}^{(1)}_{k}(u_i^{(1)}|\pmb{\theta}^{(1)}, \lambda^*)^{m_{i{k}}^{(1)}}\bar{F}^{(2)}_{k}(u_i^{(2)}|\pmb{\theta}^{(2)}, \lambda^*)^{m_{i{k}}^{(2)}}}{\bar{F}^{(1)}_{k}(u_i^{(1)}|\pmb{\theta}^{(1)}, \lambda)^{m_{ik}^{(1)}}\bar{F}^{(2)}_{k}(u_i^{(2)}|\pmb{\theta}^{(2)}, \lambda)^{m_{ik}^{(2)}}}.
\end{align}
\section{Related work}
There are many notable Bayesian methods for modelling exchangeable relational data. First, we review related models that can be viewed via graphon theory. We then discuss the Mixed-Membership Stochastic Block Model~(MMSB)~\cite{airoldi2009mixed} and highlight the differences between the MMSB and the LFSG. Finally, we analyse and compare the computational complexities of our model compared to existing methods.
\subsection{Graphons for modelling exchangeable relational data}
The Mondrian process relational model~(MP-RM; Fig.~\ref{fig:graphon_comparison}, centre-left)~\cite{roy2007learning,roy2009mondrian,roy2011thesis} is a representative model which generates $k$-d tree structured piecewise-constant graphons. In general, the Mondrian process recursively generates axis-aligned cuts in the unit square
and partitions the space in a hierarchical fashion known as a $k$-d tree. The tree structure is regulated by attaching an exponentially distributed cost to each axis-aligned cut, so that the tree generation process terminates when the accumulated cost exceeds a budget value.
The Binary Space Partitioning-Tree process relational model~(BSP-RM)~\cite{pmlr-v84-fan18b,pmlr-v89-fan18a} also generates tree structured partitions. The difference between the BSP-RM and the MP-RM is that the BSP-RM uses two dimensions to form oblique cuts and thus generate convex polyhredon-shaped blocks. These oblique cuts concentrate more on describing the inter-dimensional dependency and can produce more efficient space partitions.
The regular-grid piecewise-constant graphon is similar earlier model (Fig.~\ref{fig:graphon_comparison}, left). Generally, it is constructed from $2$ independent partition processes in a $2$-dimensional space. The resulting orthogonal crossover between these dimensions produces regular grids in the space. Typical regular-grid partition models include the SBM~\cite{nowicki2001estimation} and its infinite states variant, the Infinite Relational Model~(IRM)~\cite{kemp2006learning}. The SBM uses a Dirichlet distribution~(or Dirichlet process for the IRM) to independently generate a finite~(or infinite for the IRM) number of segments in each dimension.
The Rectangular Tiling process relational model~(RTP-RM; Fig.~\ref{fig:graphon_comparison}, centre-right)~\cite{nakano2014rectangular} produces a flat partition structure on a two-dimensional array by assigning each entry to an existing block or a new block in sequence, without violating the rectangular restriction of the blocks. By relaxing the restrictions of the hierarchical or regular-grid structure, the RTP-RM aims to provide more flexibility in block generation. However, the process of generating blocks is quite complicated for practical usage. Moreover, the hierarchical and regular-grid partition models can be used for continuous space and multi-dimensional arrays~(after trivial modifications), the RTP-RM is restricted to (discrete) arrays only.
The Rectangular Bounding process relational model~(RBP-RM)~\cite{NIPS2018_RBP} uses a bounding strategy to generate rectangular blocks in the space. In contrast to the previously described cutting strategies, the RBP-RM concentrates more on the important regions of the space and avoids over-modelling sparse and noisy regions. In the RBP-RM, the number of possible intensities
is equivalent to the number of blocks, which follows a Poisson distribution and is finite almost surely.
The Gaussian process relational model~(GP-RM; Fig.~\ref{fig:graphon_comparison}, right)~\cite{RandomFunPriorsExchArrays} utilises a prior over a random function in the unit square to form a continuous graphon. In this way it can potentially generate desired continuous intensity values via the graphon function. However, the computational cost of the GP-RM is the same as that of the Gaussian process, which scales to the cube of the number of nodes ($n$).
\subsection{Comparing the LFSG and the MMSB} The MMSB is another notable Bayesian method for modelling exchangeable relational data. In contrast to other graphon methods,
the MMSB allows each node $i$ to have a group distribution $\pmb{F}_i$, which follows a Dirichlet distribution. To form the relation between any two nodes $i,j$, a latent label pair consisting of a sender and a receiver $(s_{ij}, r_{ij})$ is first generated via $s_{ij}\sim\text{Categorical}(\pmb{F}_i)$, and $r_{ij}\sim\text{Categorical}(\pmb{F}_j)$. The relation $R_{ij}$ may then be generated based on the intensity of the block $\pmb{B}$ formed by group $s_{ij}$ and group $r_{ij}$: $R_{ij}\sim\text{Bernoulli}(B_{s_{ij},r_{ij}})$.
Our proposed LFSG shares similarities with the MMSB, since both of them use group distributions to represent individual nodes and the likelihood generation method is the same. However, there are key differences. These are: (1) The priors for the group distributions are different. In the MMSB, the group distributions of all nodes are generated independently from a Dirichlet distribution, whereas in the LFSG, the group distributions are highly correlated~(determined by the node coordinates and the unified partition structure); (2) The MMSB requires $nK$ parameters to form the group distributions, while the LFSG requires only {$2(n+K)$} parameters; (3) The MMSB cannot be described by graphon theory (because it involves $n$ independent group distribution $\pmb{F}_i$ for generating group distributions), whereas the LFSG naturally fits within the graphon framework.
\begin{table}
\centering
\caption{Model complexity comparison}
\begin{tabular}{l|cc}
\hline
Model & Intensity computation & Label sampling\\
\hline
SBM & $\mathcal{O}(K^2L)$ & $\mathcal{O}(nK)$ \\
ISG & $\mathcal{O}(K^2n^2)$ & -- \\
LFSG & $\mathcal{O}(K^2L)$ & $\mathcal{O}(n^2K)$ \\
MMSB & $\mathcal{O}(K^2L)$ & $\mathcal{O}(n^2K)$ \\
GP-RM & $\mathcal{O}(n^3)$ & -- \\
\hline
\end{tabular}
\label{table:model_complexity}
\end{table}
\subsection{Computational complexities}
Table~\ref{table:model_complexity} compares the computational complexities of the ISG and the LFSG against representative existing models, including the SBM, the MMSB and the GP-RM. In terms of calculating the intensity for generating the relations $\{R_{ij}\}_{i,j=1}^n$, the ISG requires a scale of $\mathcal{O}(n^2K^2)$ since the calculation of the mixture intensity for each relation involves a pair of coordinates~(giving a total of $n^2$) and all of the block intensities~(which is $K^2$). However, the uncoupling strategy applied in the LFSG lowers this cost dramatically to $\mathcal{O}(K^2L)$, where $L$ is the number of positive links~(i.e.~$R_{ij}=1$) observed in the data (Table \ref{rm_dataset} enumerates $L$ for each data set analysed below). Note that the mixture intensity computation cost of the LFSG is the same as
\begin{figure*}[t]
\centering
\includegraphics[width = 0.9 \textwidth]{auc_pr_vis_1.pdf}
\includegraphics[width = 0.9 \textwidth]{auc_pr_vis_2.pdf}
\caption{Average area under the curve receiver operating characteristic~(AUC) and the precision recall~(Precision) under the Stochastic Block Model~(SBM, blue line), Mixed-membership Stochastic Block Model~(MMSB, green line), Mondrian Process-Relational Model~(MP-RM, red line), Latent Feature Smoothing Graphon on the SBM~(LFSG, orange line) and Integrated Smoothing Graphon on the SBM~(ISG, purple line) for each of the Delicious, Digg, Facebook, Flickr, Gplus and Twitter datasets, under different proportions of training data ($x$-axis).
}
\label{fig:auc_precision}
\end{figure*}
that of both the SBM and the MMSB. As a result, the continuous intensities of the LFSG compared to the discrete intensities of the SBM is achieved without sacrificing computation complexity. In contrast, the computational cost of computing the mixture intensity for the GP-RM is $\mathcal{O}(n^3)$, which is the highest among these methods, even though it can also provide continuous intensities. Regarding the complexity of sampling the labels, both the LFSG and the MMSB provide multiple labels for each node and incur the same cost of $\mathcal{O}(n^2K)$. However, while the SBM requires a smaller cost of $\mathcal{O}(nK)$ for label sampling, it only allows a single-label for each node.
\section{Experiments}
We now evaluate the performance of the ISG-SBM and the LFSG-SBM on real-world data sets, comparing them with three state-of-the-art methods: the SBM, the MP-RM and the MMSB. Although the MMSB cannot be explained using graphon theory, it is included as it shares some similarities with with the LFSG-SBM. We implement posterior simulation for the SBM and the MMSB using Gibbs sampling and a conditional Sequential Monte Carlo algorithm\cite{andrieu2010particle,LakOryTeh2015ParticleGibbs,pmlr-v89-fan18a} for the MP-RM.
\begin{table}
\centering
\caption{Dataset summary information}
\begin{tabular}{c|cc||c|cc}
\hline
Dataset & $L$ & $S(\%)$ & Dataset & $L$ & $S(\%)$\\
\hline
Delicious & $10,775$ & \phantom{1}$4.31 $ & Gplus & $76,575$ & $30.63 $ \\
Digg & $25,943$ & $ 10.38$ & Facebook & $54,476$ & $ 21.79$ \\
Flickr & $49,524$ & $ 19.81$ & Twitter & $24,378$ & $ \phantom{1}9.75$ \\
\hline
\end{tabular}
\label{rm_dataset}
\end{table}
\subsection{Data sets}
We examine six real-world exchangeable relational data sets: Delicious~\cite{Zafarani+Liu:2009}, Digg~\cite{Zafarani+Liu:2009}, Flickr~\cite{Zafarani+Liu:2009}, Gplus~\cite{leskovec2012learning}, Facebook~\cite{leskovec2012learning}, and Twitter~\cite{leskovec2012learning}. To construct the exchangeable relational data matrix we extract
the top $1\,000$ active nodes based on node interaction frequencies,
and then randomly sample $500$ nodes from these top $1\,000$ nodes to form the $500\times 500$ interaction binary matrix. Table~\ref{rm_dataset} summarizes the number of positive links~($L$) and the corresponding sparsity ($S\%$), which is defined as the ratio of the number of positive links to the total number of links, for each dataset.
\begin{figure*}[p]
\centering
\includegraphics[width = 0.49 \textwidth]{delicious_0.pdf}\quad
\includegraphics[width = 0.49 \textwidth]{digg_0.pdf}\quad
\includegraphics[width = 0.49 \textwidth]{flickr_0.pdf}\quad
\includegraphics[width = 0.49 \textwidth]{gplus_0.pdf}\quad
\includegraphics[width = 0.49 \textwidth]{facebook_0.pdf}\quad
\includegraphics[width = 0.49 \textwidth]{twitter_0.pdf}\quad
\caption{Visualisation of mixture intensity from one posterior draw, the posterior mean of smoothing parameter $\lambda$ and pairwise hidden labels on the Delicious, Digg, Flickr, Gplus, Facebook and Twitter datasets when implementing the Latent Feature Smoothing Graphon within the Stochastic Block Model. There are three figures for each dataset. Left: the grey level in the unit square illustrates the predicted mixture intensity for each relation (darker = higher intensity), the dotted lines indicate the related partition~($\boxplus=\pmb{\theta}^{(1)}\times\pmb{\theta}^{(2)}$), and the coordinates of all the nodes are visualised by the blue dots.
Right: the different colours
represent the different values of the latent labels $s_{ij}$~(right top) and $r_{ij}$~(right bottom), with the $x$-axis indicating different nodes (sorted by the ratio of the labels) and the $y$-axis showing the proportion of different labels for each node.}
\label{fig:graphon_visualization}
\end{figure*}
\subsection{Experimental setting}
The hyper-parameters for each method are set as follows: for the SBM, LFSG-SBM, ISG-SBM, MMSB and MP-RM, the hyper-parameters $\alpha_0, \beta_0$ used in generating the block intensities are set as {$\alpha_0=S\%, \beta_0=1-S\%$}, where $S\%$ refers to the sparsity shown in Table~\ref{rm_dataset}, such that the block intensity has an expectation equivalent to the sparsity of the exchangeable relational data; for the SBM, LFSG-SBM, ISG-SBM and MMSB, we set the group distribution of $\pmb{\theta}$ as $\text{Dirichlet}(\pmb{1}_{1\times 4})$. Hence, the number of groups in each dimension in these models is set as $4$, with a total of $16$ blocks generated in the unit square; for the MP-RM, the budget parameter is set to $3$, which suggests that approximately $(3+1)\times(3+1)$ blocks would be generated.
\subsection{Link prediction performance}
The performance of each models in the task of link prediction is shown in Fig.~\ref{fig:auc_precision}, which reports both the average area under the curve of the receiver operating characteristic~(AUC) and the precision-recall~(Precision).
The AUC denotes the probability that the model will rank a randomly chosen positive link higher than a randomly chosen zero-valued link. The precision is the average ratio of correctly predicted positive links to the total number of predicted positive links. Higher values of AUC and precision indicate better model performance.
For each dataset, we vary the ratio of training data from $10\%$ to $90\%$ and use the remainder for testing. The training/test data split is created in a row-wise manner. In particular, we take the same ratio of training data from each row of the relational matrix, so that each node shares the same amount of training data.
From Fig.~\ref{fig:auc_precision},
both the AUC and precision of all models improves as the amount of training data increases.
The trend generally becomes steady when the proportion is larger than $0.3$, indicating
the amount of data required to fit a model with a $\sim$16-block complexity.
Except for the Facebook data, we can see that the AUC and precision of both the ISG-SBM and the LFSG-SBM are better than for the piecewise-constant graphon models (i.e.~the SBM and MP-RM). The proposed smoothing graphons typically achieve similar performance to the MMSB, demonstrating that the smoothing graphon strategy
is useful for improving model performance.
For the Facebook dataset, the SBM seems to perform better than the smoothing graphon-based models. This is examined in greater detail in the next section.
\subsection{Graphon and hidden label visualisation}
In addition to the quantitative analysis, we visualise the generated graphons and hidden labels under the LFSG-SBM on all six data sets in Fig.~\ref{fig:graphon_visualization}. For each dataset, we visualise the resulting mixture intensities for one posterior sample, with the learned posterior mean of the smoothing parameter $\lambda$, based on using $90\%$ training data.
We observe that the displayed graphon intensities exhibit smooth transitions between blocks for each dataset, highlighting
that continuous, rather than discrete, mixture intensity values are generated under the smoothing graphon.
The transition speed of the intensity between blocks is influenced by the smoothing parameter $\lambda$ -- a larger value of $\lambda$ leads to a less smooth graphon, and a smaller value of $\lambda$ to a more smooth graphon -- similar to that observed in Fig.~\ref{fig:weight_function_vis}.
In Fig.~\ref{fig:graphon_visualization}, for each dataset, we also display the posterior proportions of the pairwise hidden labels $s_{ij}$ (top right) and $r_{ij}$ (bottom right) for each node. Here the $x$-axis indicates different nodes~(sorted by the label probabilities) and the $y$-axis displays the {posterior mean} of label probabilities~(each label represented by a different colour). For each node $i$ on the $x$-axis, the more colours observed on the $y$-axis indicates a greater diversity of groups associated with that node,
which in turn represents a higher potential for that node to belong to different groups when interacting with other nodes. In other words, the larger the tendency away from vertical line transitions between groups in these plots means
a larger number of nodes belonging to multiple groups.
Compared with the value of the smoothing parameter $\lambda$ learned on the other four data sets, the values of $\lambda$ estimated from the Facebook and Twitter datasets are larger. Further, the visualisations of the hidden labels for these two data sets are partitioned by almost straight vertical lines, which suggests that only one label is a realistic possibility for most of the nodes. This could explain why both the AUC and precision values of the ISG-SBM and the LFSG-SBM are less competitive with those of the SBM on these two datasets (Fig.~\ref{fig:auc_precision}). Here, that the SBM assigns each node to exactly one group only, which aligns well with the ground-truth for these two datasets.
Another explanation for the performance on the datasets of Facebook and Twitter is that we can recover the SBM if and only if $\lambda= \infty$. For any finite value of the smoothing parameter $\lambda$, it is impossible to have any posterior mass on the SBM. To this end, we might use a mapping to map $\lambda$ from $(0, \infty)\rightarrow (0, 1)$ (e.g. $1-e^{-\lambda}$) such that we are able to place posterior mass close to $1$. As the mapped value would be easily to approximate $1$, the models of ISG-SBM and LFSG-SBM could be able to perform at least as well as the SBM, even for the datasets of Facebook and Twitter.
\section{Conclusion}
In this paper, we have introduced a smoothing strategy to modify conventional piecewise-constant graphons in order to increase their continuity. Through the introduction of a single smoothing parameter $\lambda$, we first developed the Integrated Smoothing Graphon~(ISG) that addresses the key limitation of existing piecewise-constant graphons which only generate a limited number of discrete intensity values. To improve the computational efficiency of the ISG and to allow for the possibility of each node to belong to multiple groups, we further developed the Latent Feature Smoothing Graphon~(LFSG) by the introduction of auxiliary hidden labels. Our experimental results verify the effectiveness of this smoothing strategy in terms of greatly improved AUC and precision scores in the task of link prediction. The visualisations of the generated graphons and the posterior hidden label summaries further provide an intuitive understanding of the nature of the smoothing mechanism for the given dataset.
|
1,116,691,499,276 | arxiv | \section{Keywords:} Sun, Corona, Magnetohydrodynamics, Waves, Coronal Seismology}
\end{abstract}
\section{Introduction}
The outermost layer of the solar atmosphere, the corona, consists of a fully ionised and strongly magnetised plasma, which is able to respond periodically or quasi-periodically to any impulsive perturbation. The interest in studying coronal waves and oscillations is connected with their possible role in the enigmatic coronal heating problem \citep{2020SSRv..216..140V} and with a promising and sometimes unique opportunity to probe the coronal plasma parameters with the method of magnetohydrodynamic (MHD) seismology \citep{2020ARA&A..58..441N}. In particular, fast magnetoacoustic wave modes, directly observed in the corona as e.g. kink oscillations of coronal loops or fast-propagating quasi-periodic wave trains, are extensively used for probing the coronal magnetic field strength and twist, density stratification, and cross-field fine structuring \citep[see e.g.][for the most recent comprehensive reviews]{2021SSRv..217...73N, 2020SSRv..216..136L, 2022SoPh..297...20S}. The slow mode of magnetoacoustic waves, which appear in standing \citep[e.g.][]{2021SSRv..217...34W}, propagating \citep[e.g.][]{2021SSRv..217...76B}, and sloshing \citep[e.g.][]{2019ApJ...874L...1N} forms, has in turn been found sensitive to both the magnetic and thermodynamic properties of the coronal plasma, which makes it a powerful seismological tool too.
The seismological applications of slow waves in the corona span from probing the absolute value of the magnetic field in active regions \citep[e.g.][]{2007ApJ...656..598W, 2016NatPh..12..179J} and the magnetic field direction \citep{2009ApJ...697.1674M} to estimating the effective adiabatic index of the coronal plasma \citep[e.g.][]{2011ApJ...727L..32V, 2015ApJ...811L..13W, 2018ApJ...868..149K}, its effective energy transport coefficients \citep{2015ApJ...811L..13W, 2018ApJ...860..107W}, multi-thermal nature of coronal loops \citep[e.g.][]{2003A&A...404L...1K, 2017ApJ...834..103K}, and even properties of the coronal heating function \citep{2019ApJ...884..131R, 2020A&A...644A..33K}. Moreover, a similarity between the properties of the phenomenon of quasi-periodic pulsations, observed in solar and stellar flare lightcurves and attributed to the modulation of the flare emission by slow waves, allowed for revealing new solar-stellar analogies \citep{2016ApJ...830..110C} and stimulated the development of the theory of slow waves in stellar coronal conditions \citep[e.g.][]{2018ApJ...856...51R, 2022ApJ...931...63L}.
The majority of seismological estimations with slow waves have been carried out under the assumption of weak non-adiabaticity of the coronal plasma, i.e. assuming the energy exchange and energy transfer processes (such as thermal conduction, compressive viscosity, optically thin radiation, etc) are weak and slow in comparison with the oscillation period of a slow wave as its characteristic timescale. Under this assumption, the seismological analysis with slow waves gets substantially simplified. However, it cannot properly account for such important observable effects as rapid damping of slow waves, with the damping time being about the oscillation period \citep[see e.g.][for the most recent multi-instrumental statistical survey]{2019ApJ...874L...1N}, apparently linear scaling between the slow wave damping time and oscillation period \citep[see e.g.][]{2016ApJ...830..110C, 2016ApJ...820...13M}, strong modification of a slow wave speed and effective adiabatic index of the corona \citep[see e.g.][who detected the effective adiabatic index to vary from about 5/3 to 1]{2018ApJ...868..149K}, large phase shifts between the plasma temperature and density perturbed by slow waves \citep{2019MNRAS.483.5499K}, and yet undetected effects such as coupling of the slow and entropy wave modes \citep{2021SoPh..296...96Z}. Furthermore, the transport coefficients of those non-adiabatic processes are often considered as free parameters in the corona, and their deviation from the classical values prescribed by \citet{1962pfig.book.....S} and \citet{1965RvPP....1..205B} due to essentially dynamic and turbulent nature of the coronal plasma remains a subject to intensive studies. In particular, the parametric study of the dynamics of slow waves in coronal loops with suppressed field-aligned thermal conduction and of their diagnostic potential was performed recently by \citet{2019ApJ...886....2W}. Likewise, the question of \lq\lq anomalous transport\rq\rq\ remains open in other astrophysical plasma environments too \citep[see e.g.][for the discussion of this topic in the Earth's magnetospheric plasma context]{2017PhPl...24b2104M}.
In this work, we delineate the applicability of a {weak thermal conduction theory} of slow waves to coronal seismology. For this, we numerically model the linear evolution of a standing slow oscillation in a hot coronal loop (alike those observed with the SUMER instrument onboard the SOHO spacecraft or in \lq\lq hot\rq\rq\ channels of SDO/AIA) with {full conductivity}, and compare the model outcomes to those obtained in a {weakly conductive limit}. In particular, we focus on the measurements of the phase shift and relative amplitude ratio between density and temperature perturbations and their use for probing the effective adiabatic index of the coronal plasma. The applicability of a polytropic assumption for estimating the effective adiabatic index is also discussed. The paper is structured as follows. In Sec.~\ref{sec:model}, we describe the numerical model and plasma loop parameters. In Sec.~\ref{sec:analysis}, we present the analysis of oscillatory variations of plasma loop density and temperature, caused by the standing slow wave, and the comparison of those in the numerical solution with {full conductivity} and in an approximate {weakly conductive limit}. The application of the obtained oscillation parameters to probing the effective adiabatic index of the coronal plasma, in the polytropic assumption and as ratio of the effective wave speed to the standard sound speed, is demonstrated in Sec.~\ref{sec:gamma}. The discussion of the obtained results and conclusions are summarised in Sec.~\ref{sec:disc}.
\section{Governing equations and modelling}
\label{sec:model}
We model the dynamics of a standing slow wave in a low-beta coronal plasma in the infinite magnetic field approximation \citep[see Sec. 2.3 of][and references therein]{2021SSRv..217...34W}, using the following set of linearised governing equations,
\begin{align}\label{eq_motion}
&\rho_0 \frac{\partial V_{z}}{\partial t} = -\frac{\partial p}{\partial z},\\
&\frac{\partial \rho}{\partial t} + \rho_0\frac{\partial V_{z}}{\partial z} = 0,\\
&p=\frac{k_\mathrm{B}}{m}\left( \rho_0 T + T_0 \rho\right),\\
&\frac{\partial {T}}{\partial t} - (\gamma-1)\frac{T_0}{\rho_0}\frac{\partial {\rho}}{\partial t}=\frac{\kappa_\parallel}{\rho_0 C_\mathrm{V}}\frac{\partial^2{T}}{\partial z^2}.\label{eq:energy}
\end{align}
In Eqs.~(\ref{eq_motion})--(\ref{eq:energy}), the direction of the wave propagation along the $z$-axis is prescribed by the infinitely stiff (not perturbed by the wave) magnetic field; $V_z$, $p$, $\rho$, and $T$ represent perturbations of the plasma velocity, pressure, density, and temperature, respectively; the subscripts \lq\lq 0\rq\rq\ correspond to the values of those variables at $t=0$; $m$, $\gamma$, and $k_\mathrm{B}$ are the mean particle mass, standard adiabatic index $5/3$, and Boltzmann constant, respectively; $C_\mathrm{V} = (\gamma-1)^{-1}k_\mathrm{B}/m$ is the standard specific heat capacity; and the coefficient of thermal conduction along the field $\kappa_\parallel$ is treated as a free parameter in this study. The effects of other non-ideal processes, such as compressive viscosity, optically thin radiation, heating, and the wave-induced misbalance between them, are omitted, with the field-aligned thermal conduction considered as the main wave damping mechanism \citep[e.g.][]{2002ApJ...580L..85O, 2003A&A...408..755D, 2004A&A...415..705D, 2016ApJ...826L..20R, 2019A&A...628A.133K}.
The presence of a dissipative term on the right-hand side of energy equation (\ref{eq:energy}) makes the model essentially non-adiabatic and {may lead} to the appearance of the phase shift $\Delta \varphi$ between the temperature and density perturbations and modify the ratio of their instantaneous relative amplitudes $A_T$ and $A_\rho$, respectively \citep[see e.g. a series of works by][]{2009A&A...494..339O, 2011ApJ...727L..32V, 2015ApJ...811L..13W, 2018ApJ...868..149K, 2022SoPh..297....5P}. In a {weakly conductive limit}, i.e. assuming the damping time by thermal conduction is much longer than the oscillation period and the wave speed remains equal to the standard sound speed $C_\mathrm{s} = \sqrt{\gamma k_\mathrm{B}T_0/m}$, the parameters $\Delta \varphi$ and $A_T/A_\rho$ have been previously derived as
\begin{align}
&\tan \Delta \varphi \approx \frac{2\pi(\gamma-1)\kappa_\mathrm\parallel m}{k_\mathrm{B}C_\mathrm{s}^2 P_0 \rho_0},\label{eq:phi_weak}\\
&\frac{A_T}{A_\rho} \approx (\gamma-1)\cos \Delta \varphi,\label{eq:amp_weak}
\end{align}
where $P_0=2L/C_\mathrm{s}$ is oscillation period in the ideal adiabatic case, with $L$ being the loop length.
\begin{figure}
\begin{center}
\includegraphics[width=0.39\textwidth]{u0.pdf
\includegraphics[width=0.61\textwidth]{oscillations.pdf}
\end{center}
\caption{Left: The form of the initial perturbation of the plasma velocity, $V_{z0} = 0.1 C_\mathrm{s} \cos(2 \pi z/\lambda)$, applied to the loop model described in Sec.~\ref{sec:model}.
Right: Variations of the slow wave-perturbed plasma velocity, density, and temperature in red, blue, and green, measured at $z=0.1\lambda$ and for $\kappa_\parallel=10\kappa_\mathrm{Sp}$, normalised to $C_\mathrm{s}$, $\rho_0$, and $T_0$, respectively. $C_\mathrm{s} = \sqrt{\gamma k_\mathrm{B}T_0/m}$ is the adiabatic sound speed; $\lambda$ is the wavelength (prescribed by the loop length, $L$, as $\lambda = 2L$); $P_0$ is the adiabatic acoustic oscillation period, $2L/C_s$. {{The grey lines illustrate the numerical error estimate for the velocity perturbation.}}
}
\label{fig:1}
\end{figure}
In this work, we solve Eqs.~(\ref{eq_motion})--(\ref{eq:energy}) numerically in the mathematical environment \emph{Maple 2020.2}, {{using the built-in function \textit{pdsolve}. It implements a second order (in space and time) centred, implicit finite difference scheme\footnote{\url{https://www.maplesoft.com/support/help/Maple/view.aspx?path=pdsolve/numeric}}, with timestep 0.02$P_0$ and spacestep 0.02$\lambda$ ($\lambda=2L$) providing the numerical accuracy up to 0.2\% of the equilibrium plasma parameters during the first five oscillation cycles (estimated with the \textit{errorest} keyword of the \textit{pdsolve} command) for the initial perturbation amplitude of 10\%.}}
We do not apply the assumption of {weak conductivity} and vary the field-aligned thermal conduction coefficient $\kappa_\parallel$ from 0.01 to 10 of the standard \emph{Spitzer} value $\kappa_\mathrm{Sp} = 10^{-11}T_0^{5/2}\,\mathrm{W\,m}^{-1}\,\mathrm{K}^{-1}$. {The considered interval of $\kappa_\parallel$ is motivated by previous observational findings. In particular, \citet{2021SSRv..217...34W} (see Sec. 8.1) demonstrated that to account for the coronal polytropic index measured by \citet{2011ApJ...727L..32V}, the thermal conductivity needs to be enhanced by an order of magnitude.}
The following values of the equilibrium plasma parameters are considered: $\rho_0 = 3 \times 10^{-12}$\,kg\,m$^{-3}$ and $T_0 = 6.3$\,MK (both uniform along the loop), $L=180$\,Mm, $m=0.6\times1.67\times10^{-27}$\,kg, typical for coronal loops hosting SUMER oscillations. We excite the fundamental harmonic of a standing slow wave by perturbing the plasma velocity with a harmonic function with maximum at $z=0$ and the wavelength equal to double the loop length, $\lambda=2L$, and apply rigid-wall boundary conditions at $z=\pm 0.25 \lambda$ mimicking the effective slow wave reflection from the transition region and dense chromosphere \citep[e.g.][]{2004A&A...414L..25N}. The form of the initial perturbation and an example of time variations of the plasma velocity, density, and temperature, obtained numerically for e.g. $\kappa_\parallel=10\kappa_\mathrm{Sp}$, are shown in Fig.~\ref{fig:1}.
{All oscillatory signals used in the further analysis are taken at $z=0.1\lambda$.}
\section{Temperature/density phase shifts and amplitudes}
\label{sec:analysis}
We begin the analysis of the numerically modelled standing slow wave with obtaining the dependence of the oscillation period $P$ on the thermal conduction coefficient $\kappa_\mathrm\parallel$. It is estimated through the fast Fourier transform applied to the plasma velocity perturbations for several different values of $\kappa_\mathrm\parallel$ (see e.g. the red line in Fig.~\ref{fig:1}, right panel) and is presented in the left panel of Fig.~\ref{fig:2}. As one can see, the oscillation period increases by about 30\% with $\kappa_\mathrm\parallel$, from the ideal adiabatic value $P_0$ determined by the standard sound speed $C_\mathrm{s}$, to a new value in the isothermal regime (achieved for high $\kappa_\mathrm\parallel$), determined by the isothermal sound speed $C_\mathrm{s}/\sqrt{\gamma}$. Throughout this work, the loop length $L$ (and therefore the wavelength $\lambda$) is kept constant. In other words, in the {strongly conductive regime}, the effective slow wave speed gets significantly modified, which leads to the corresponding modification in the wave travel time along the loop, i.e. the oscillation period. This empirical result is consistent with previous analytical estimations by e.g. \citet{2021A&A...646A.155D}.
\begin{figure}
\begin{center}
\includegraphics[width=0.4\textwidth]{period.pdf
\includegraphics[width=0.48\textwidth]{omega_im2re_tcond.pdf}
\end{center}
\caption{Left: Slow-wave oscillation period, $P$, estimated empirically via the fast Fourier transform of the numerical solution for the plasma velocity perturbation, vs. the coefficient of the field-aligned thermal conduction normalised to the standard Spitzer value. The horizontal dashed and dot-dashed lines show the period values in the ideal adiabatic and isothermal limits, obtained with the standard adiabatic and isothermal sound speeds, $C_\mathrm{s}$ and $C_\mathrm{s}/\sqrt{\gamma}$, respectively.
Right: Dependence of the ratio of the imaginary part $\omega_i=1/\tau_D$ to the real part $\omega_r = 2\pi/P$ of the slow-wave angular frequency $\omega$ on the field-aligned thermal conduction coefficient (green), with $\tau_D$ being the oscillation exponential damping time estimated empirically from the numerical solution.
{The dark blue line in the right panel shows the dimensionless parameter $\omega_r \tau_\mathrm{cond}$, with $\tau_\mathrm{cond} = \rho_0 C_\mathrm{V}\lambda^2/\kappa_\parallel$ being the characteristic time scale of thermal conduction.}
}
\label{fig:2}
\end{figure}
The phase shifts between density and temperature perturbations are estimated from our numerical solution through the cross-correlation analysis. More specifically, we obtain the time lag $\Delta t$ for which the cross-correlation between density and temperature oscillations is the highest. With this, the phase shift $\Delta \varphi$ is obtained as $\Delta \varphi = (\Delta t/P)\times 360^{\circ}$ for each considered value of $\kappa_\mathrm\parallel$, using the dependence of the oscillation period $P$ on $\kappa_\mathrm\parallel$ obtained above. Thus, the dependence of $\Delta \varphi$ on $\kappa_\mathrm\parallel$, revealed empirically, is shown in the left panel of Fig.~\ref{fig:3} in red. It is seen to vary from $0^{\circ}$ in the ideal adiabatic case to about $80^{\circ}$ in the {strongly conductive regime} with high $\kappa_\mathrm\parallel$ \citep[cf.][]{2022SoPh..297....5P}.
\begin{figure}
\begin{center}
\includegraphics[width=0.38\textwidth]{phase_shifts.pdf
\includegraphics[width=0.43\textwidth]{amp_ratio.pdf}
\end{center}
\caption{The phase shift (left) and relative amplitude ratio (right) between plasma temperature and density perturbations by the standing slow wave, obtained from the analysis of the numerical solution of Eqs.~(\ref{eq_motion})--(\ref{eq:energy}) with {full thermal conductivity} as described in Sec.~\ref{sec:analysis} (in red) and from the approximate analytical solutions (\ref{eq:phi_weak})--(\ref{eq:amp_weak}) in a {weakly conductive limit} (in blue). The grey curve in the right panel shows the relative error between the estimations of the temperature and density relative amplitude ratio, shown in red and blue.
}
\label{fig:3}
\end{figure}
For estimating the ratio between temperature and density perturbation amplitudes, we obtain the instantaneous amplitudes $A_T(t)$ and $A_\rho(t)$ as oscillation envelopes by exponential fitting and with the use of the Hilbert transform \citep[see][for apparently the first use of the Hilbert transform for coronal seismology by slow waves]{2019ApJ...884..131R}. The edge effects of the Hilbert transform are mitigated by mirroring the signals with respect to the vertical axis and smoothing the resulting oscillation envelopes over a half of the oscillation period. The examples of $A_T$ and $A_\rho$ for $\kappa_\parallel = \kappa_\mathrm{Sp}$, obtained with the Hilbert transform and their best-fits by decaying exponential functions, are shown in the left panel of Fig.~\ref{fig:4}. It shows, in particular, that actual temperature and density perturbation amplitudes do not obey the exponential law during the first cycle of oscillation, which we attribute to the simultaneous development and rapid decay of the entropy mode \citep{2011A&A...533A..18M, 2021SoPh..296...96Z}.
Thus, associating this mismatch with a possible signature of the slow wave coupling with entropy waves, the ratio $A_T/A_\rho$ in slow waves is estimated via the exponentially decaying instantaneous amplitudes obtained by fitting (the right panel of Fig.~\ref{fig:4}).
{More specifically, the red dashed lines in the right panel of Fig.~\ref{fig:4} show exponentially decaying $A_T(t)$ vs. exponentially decaying $A_\rho(t)$ for three different values of $\kappa_\parallel$. As the slow wave damping rate is the same in perturbations of all plasma parameters, the ratio of exponentially decaying $A_T(t)$ and $A_\rho(t)$ is independent of time. In other words, it may be characterised by the $y$-intercept of the red dashed lines shown in the right panel of Fig. 4 (indeed, if $A_T = \mathrm{const}\times A_\rho$, then $\log(A_T) = \log(\mathrm{const})+\log(A_\rho)$).}
The dependence of the obtained values of $A_T/A_\rho$ on $\kappa_\parallel$ is shown in the right panel of Fig.~\ref{fig:3} in red. It varies from 2/3 ($\equiv \gamma - 1$) in the ideal adiabatic regime to almost zero in the isothermal regime for high $\kappa_\parallel$, in which temperature gradients are effectively smoothed out by thermal conduction. {We also note that the use of a non-exponential damping envelope may lead to underestimated values of $A_T/A_\rho$ in slow waves.}
In addition, this analysis allows us to estimate the conductive damping rate of standing slow waves {and the characteristic thermal conduction time scale $\tau_\mathrm{cond}=\rho_0 C_\mathrm{V}\lambda^2/\kappa_\parallel$} for various values of the conduction coefficient $\kappa_\parallel$ (the right panel in Fig.~\ref{fig:2}). For this, we consider the ratio of the imaginary part of the angular oscillation frequency $\omega_i$ (estimated as the reciprocal of the exponential damping time) to the real part $\omega_r = 2\pi/P$. The obtained non-monotonic behaviour of $\omega_i/\omega_r$ is consistent with the previous analytical estimations by e.g. \citet{2003A&A...408..755D}, with the highest value of about $0.15$, detected for $\kappa_\parallel\approx 2\kappa_\mathrm{Sp}$, being consistent with the observed rapid damping of SUMER oscillations with quality factors (i.e. ratio of the oscillation damping time to period) of about unity \citep[e.g.][]{2019ApJ...874L...1N, 2021SSRv..217...34W}. {The right panel of Fig.~\ref{fig:2} also shows the dimensionless parameter $\omega_r \tau_\mathrm{cond}$ that can be used for a quantitative discrimination between weak and strong conductive limits. Thus, $\omega_r \tau_\mathrm{cond} \gg 1$ and $\omega_i/\omega_r\propto 1/ \omega_r\tau_\mathrm{cond}$ in the weak limit, and $\omega_r \tau_\mathrm{cond} \ll 1$ and $\omega_i/\omega_r\approx\omega_r\tau_\mathrm{cond}$ in the strong limit \citep[see e.g.][]{2014ApJ...789..118K, 2021A&A...646A.155D}. For $\kappa_\parallel\approx 2\kappa_\mathrm{Sp}$ consistent with observations, $\omega_r \tau_\mathrm{cond}$ is about 1 by an order of magnitude, so that neither of those limits is fully justified.}
\begin{figure}
\begin{center}
\includegraphics[width=0.48\textwidth]{temp-dens-hilbert.pdf}
\includegraphics[width=0.5\textwidth]{temp-dens.pdf
\end{center}
\caption{The time profiles of the temperature and density perturbations by a standing slow wave, obtained numerically from Eqs.~(\ref{eq_motion})--(\ref{eq:energy}) at $z=0.1\lambda$, for $\kappa_\parallel=\kappa_\mathrm{Sp}$, and normalised to $T_0$ and $\rho_0$, respectively (left). The envelopes of the temperature and density perturbations are obtained with the Hilbert transform (grey solid) and by exponential fitting (red dashed). The vertical dotted lines in the left panels indicate the apparent transition time from a non-exponential to exponential damping. The right panel shows examples of the amplitude ratios for several values of $\kappa_\parallel$ (shown in the inlet), estimated with the Hilbert transform (grey diamonds) and by exponential fitting (red dashed). Mind the logarithmic scale in the right panel.
}
\label{fig:4}
\end{figure}
We now compare the dependences of $\Delta \varphi$ and $A_T/A_\rho$ on $\kappa_\parallel$ obtained from the analysis of our numerical solution to those prescribed by approximate solutions (\ref{eq:phi_weak}) and (\ref{eq:amp_weak}), derived in a {weakly conductive limit} (see the red and blue lines in Fig.~\ref{fig:3}). For both $\Delta \varphi$ and $A_T/A_\rho$, the approximate and numerical solutions seem to perfectly agree for low values of $\kappa_\parallel$, i.e. in the {weakly conductive regime}. For higher $\kappa_\parallel$, the phase shifts $\Delta \varphi$, estimated in the {fully conductive and weakly conductive regimes}, remain well consistent with each other, with a relative error being below a few percent which is practically indistinguishable in observations. In contrast, the amplitude ratio $A_T/A_\rho$ is found to differ significantly from its {weakly conductive estimation} for $\kappa_\parallel \gtrsim \kappa_\mathrm{Sp}$. The relative error of this offset is seen to be about 5\% for $\kappa_\parallel = \kappa_\mathrm{Sp}$ and reaches 30--40\% for higher $\kappa_\parallel$.
\section{Effective adiabatic index}
\label{sec:gamma}
In this section, we demonstrate the application of the obtained wave parameters to probing the effective adiabatic index of the coronal plasma, $\gamma_\mathrm{eff}$, and assess the suitability of a commonly used polytropic assumption for it.
Following e.g. \citet{2018ApJ...860..107W} and \citet{2019PhPl...26h2113Z}, we define $\gamma_\mathrm{eff}$ as a measure of the deviation of the observed phase speed $V_\mathrm{p}$ of slow waves affected by non-adiabatic effects (the field-aligned thermal conduction in our model) from the standard sound speed $C_\mathrm{s}$, i.e.
\begin{equation}
\gamma_\mathrm{eff} = \gamma \left(\frac{V_\mathrm{p}}{C_\mathrm{s}} \right)^2.\label{eq:gamm_speeds}
\end{equation}
In the solar corona, the standard sound speed can be estimated as $C_\mathrm{s} \mathrm{[km/s]} \approx 152 \sqrt{T_0\mathrm{[MK]}}$. On the other hand, as the wavelength of the discussed standing slow wave is prescribed by the loop length and thus remains constant, we can use the dependence of the oscillation period $P$ on $\kappa_\mathrm\parallel$ obtained in Sec.~\ref{sec:analysis} (see Fig.~\ref{fig:2}) as a proxy (observable parameter) of the slow wave phase speed $V_\mathrm{p}$. With this, the definition of $\gamma_\mathrm{eff}$ (\ref{eq:gamm_speeds}) can be re-written as
\begin{equation}
\gamma_\mathrm{eff} = \gamma \left(\frac{P_0}{P} \right)^2,\label{eq:gamm_periods}
\end{equation}
with $P_0$ being the slow wave oscillation period in the ideal adiabatic case. The dependence of $\gamma_\mathrm{eff}$ estimated by Eq.~(\ref{eq:gamm_periods}) on the field-aligned thermal conduction coefficient $\kappa_\parallel$ is shown in the left panel of Fig.~\ref{fig:5} in red, using the dependence of the oscillation period $P$ on $\kappa_\parallel$ shown in Fig.~\ref{fig:2}. As expected, the obtained values of $\gamma_\mathrm{eff}$ decrease with $\kappa_\parallel$ from 5/3 to 1 in the ideal adiabatic and isothermal regimes, respectively.
In the polytropic assumption, i.e. assuming the plasma density and pressure perturbations to be connected through a power-law as $p\propto\rho^{\gamma_\mathrm{eff}}$, $\gamma_\mathrm{eff}$ can be estimated through the ratio of the instantaneous relative amplitudes $A_T$ and $A_\rho$ of temperature and density perturbations \citep[][]{2011ApJ...727L..32V} as
\begin{equation}
\gamma_\mathrm{eff} \approx \frac{A_T}{A_\rho} + 1.\label{eq:gamma_poly}
\end{equation}
Despite being not strictly consistent with the observed non-zero phase difference between temperature and density perturbations, this assumption is widely used for probing the effective adiabatic index of the corona with both standing \citep[e.g.][]{2015ApJ...811L..13W, 2019ApJ...884..131R} and propagating \citep[e.g.][]{2011ApJ...727L..32V, 2018ApJ...868..149K} slow waves.
\begin{figure}
\begin{center}
\includegraphics[width=0.4\textwidth]{gamma_eff.pdf
\includegraphics[width=0.4\textwidth]{gamma_eff_err.pdf}
\end{center}
\caption{The dependence of the effective adiabatic index $\gamma_\mathrm{eff}$ of the coronal plasma on the field-aligned thermal conduction coefficient $\kappa_\parallel$ (left), estimated numerically as ratio of the effective slow wave speed to the standard sound speed (red), see Eqs.~(\ref{eq:gamm_speeds}) and (\ref{eq:gamm_periods}), and under the polytropic assumption (blue) with Eq.~(\ref{eq:gamma_poly}). The horizontal dashed and dot-dashed lines indicate the values of $\gamma_\mathrm{eff}$ in the ideal adiabatic case (5/3) for low $\kappa_\parallel$ and in the isothermal regime (1) for high $\kappa_\parallel$, respectively.
The right panel shows the relative error between the estimations of $\gamma_\mathrm{eff}$, shown in red and blue in the left panel.
}
\label{fig:5}
\end{figure}
The dependence of $\gamma_\mathrm{eff}$ (\ref{eq:gamma_poly}) on $\kappa_\parallel$, using $A_T/A_\rho$ estimated empirically in Sec.~\ref{sec:analysis} (the red line in Fig.~\ref{fig:3}, right panel), is shown in the left panel on Fig.~\ref{fig:5} in blue. Its comparison with $\gamma_\mathrm{eff}$ (\ref{eq:gamm_periods}), as ratio of the effective wave speed to the standard sound speed (i.e. {ratio of the slow wave period in adiabatic case to the observed period}), justifies the use of the polytropic assumption in a {weakly conductive regime} (for $\kappa_\parallel \lesssim \kappa_\mathrm{Sp}$ and $\gamma_\mathrm{eff}$ being between approximately 1.5 and 5/3) and reveals the relative errors (the right panel of Fig.~\ref{fig:5}) comparable to those detected in observations for $\kappa_\parallel > \kappa_\mathrm{Sp}$, that reach the maximum of 14\% for $\kappa_\parallel \approx 3\kappa_\mathrm{Sp}$. Even in the isothermal regime with high $\kappa_\parallel$, the mismatch between $\gamma_\mathrm{eff}$ (\ref{eq:gamm_periods}) and its polytropic approximation (\ref{eq:gamma_poly}) remains above 5\% {{(e.g. for $\kappa_\parallel = 6 \kappa_\mathrm{Sp}$, $\gamma_\mathrm{eff}$ is about 1.0 by Eq.~(\ref{eq:gamm_periods}) and is about 1.1 by Eq.~(\ref{eq:gamma_poly})).}}
\section{Discussion and conclusions}
\label{sec:disc}
The applicability of a weakly non-adiabatic because of finite thermal conduction along the field and polytropic assumptions to coronal seismology with slow waves has been studied in this work. We numerically modelled a 1D evolution of the fundamental harmonic of a standing slow wave in a strongly magnetised coronal plasma loop, with the field-aligned thermal conduction as the dominant wave damping mechanism and the conduction coefficient $\kappa_\parallel$ as a free parameter. In the model, no restrictions on the effectiveness of thermal conduction were imposed. The time profiles of the plasma velocity, density, and temperature perturbations have been treated as effective observables to which the standard data analysis techniques, such as the fast Fourier transform and cross-correlation analysis, and more advanced Hilbert transform, were applied. The outcomes of this analysis have been compared to the approximate analytical solutions. The main results of this work can be summarised as:
\begin{itemize}
\item The finite thermal conductivity along the field modifies the effective speed of slow waves, which leads to the modification of the observed oscillation period by up to 30\% from the value estimated in the ideal regime and used in the {weakly conductive limit}. Accounting for additional non-adiabatic effects, such as e.g. the wave-induced misbalance between coronal heating and cooling processes \citep{2021PPCF...63l4008K}, may make this modification even stronger.
\item The dependences of the phase shift $\Delta \varphi$ between the loop's temperature and density perturbations on the thermal conductivity $\kappa_\parallel$, estimated in the {strongly and weakly conductive cases}, are well consistent with each other for both low and high values of $\kappa_\parallel$. The obtained ratio of temperature and density relative amplitudes $A_T/A_\rho$, in contrast, agrees with the {weak conduction theory} for $\kappa_\parallel \lesssim \kappa_\mathrm{Sp}$ only. For higher $\kappa_\parallel$, the mismatch can reach up to 30--40\%, which clearly requires accounting for higher-order non-adiabatic effects.
\item From the practical point of view, the previous finding allows one to reduce the analytical solutions for $\Delta \varphi$ and $A_T/A_\rho$ obtained with {full conductivity} \citep[see e.g. Eqs.~(51) and (52) in][]{2021SSRv..217...34W}, which are essentially coupled through two unknowns $\kappa_\parallel$ and $\gamma_\mathrm{eff}$ and therefore cannot be used independently, to Eq.~(\ref{eq:phi_weak}) for $\Delta \varphi$ and
\begin{equation}
\frac{A_T}{A_\rho}=\frac{(\gamma-1)\cos \Delta \varphi}{1-2\pi\gamma d \chi (\gamma/\gamma_\mathrm{eff})},\label{eq:amp_strong}
\end{equation}
where $d \equiv {(\gamma-1)\kappa_\parallel m}/{\gamma k_\mathrm{B}C_\mathrm{s}^2 P_0 \rho_0}$, $\chi \equiv \omega_i/\omega_r$, and $\gamma_\mathrm{eff}$ is given by Eq.~(\ref{eq:gamm_speeds}). As such, Eqs.~(\ref{eq:phi_weak}) and (\ref{eq:amp_strong}) are de-coupled with respect to $\kappa_\parallel$ and $\gamma_\mathrm{eff}$ (the right-hand side of Eq.~(\ref{eq:phi_weak}) has $\kappa_\parallel$ only), which would significantly simplify their future seismological applications without the loss of accuracy.
\item The polytropic assumption (\ref{eq:gamma_poly}) can be used for probing the effective adiabatic index of the coronal plasma, $\gamma_\mathrm{eff}$, in the {weakly conductive regime} only, i.e. with $\kappa_\parallel \lesssim \kappa_\mathrm{Sp}$ and small deviations of $\gamma_\mathrm{eff}$ from the adiabatic value 5/3. For $\kappa_\parallel > \kappa_\mathrm{Sp}$ or if $\gamma_\mathrm{eff}$ is deemed to differ from 5/3 by more than 10\%, it should be estimated either as a ratio of the observed slow wave oscillation period (phase speed) to the period expected in the ideal adiabatic case (standard sound speed) or via the ratio of relative amplitudes $A_T/A_\rho$ using Eq.~(\ref{eq:amp_strong}). Otherwise, the relative errors may reach up to 14\% \citep[cf. 7\% uncertainty in the estimation of $\gamma_\mathrm{eff}$, detected by][for example]{2018ApJ...868..149K}.
\item As an additional side result of this work, a non-exponential damping of slow waves during approximately the first cycle of oscillation was detected with the use of the Hilbert transform. Similarly to the transition time from a Gaussian to exponential damping of coronal kink oscillations {by mode coupling with torsional Alfv\'en waves} \citep[e.g.][]{2017A&A...600A..78P}, the revealed non-exponential damping of slow waves can be used as an indirect signature of the entropy mode evolution with yet unexploited seismological potential. {In particular, this non-exponential damping of slow waves is seen to be more pronounced in the perturbation of plasma temperature and for lower values of $\kappa_\parallel$ in our analysis, the reason for which is to be understood.}
\end{itemize}
This work establishes an important ground for the application of the method of coronal seismology by slow waves in strongly non-adiabatic conditions. Moreover, the performed analysis can be readily generalised for additional non-adiabatic effects, such as compressive viscosity, optically thin radiation and enigmatic coronal heating, and used for validation of the corresponding theories \citep[e.g.][]{2022SoPh..297....5P} without the need to deploy full-scale viscous 3D MHD simulations \citep[e.g.][]{2022ApJ...926...64O} or dedicated MHD spectral codes \citep[e.g.][]{2020ApJS..251...25C}.
\section*{Conflict of Interest Statement}
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
\section*{Author Contributions}
DYK is the sole author and is responsible for the entire content of this work.
\section*{Funding}
The work was supported by the STFC Consolidated Grant ST/T000252/1.
\section*{Acknowledgments}
The author is grateful to Dr Dmitrii Zavershinskii, Prof Valery Nakariakov, and members of the International Online Team \lq\lq Effects of Coronal Heating/Cooling on MHD Waves\rq\rq\ for inspiring discussions and comments.
\section*{Data Availability Statement}
The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.
\bibliographystyle{Frontiers-Harvard}
|
1,116,691,499,277 | arxiv | \section{Introduction}\label{sec1}
A core part of regression analysis involves the examination and
handling of individual cases (Weis\-berg, \citeyear{weisberg:book}).
Traditionally, cases have been
removed or downweighted as outliers or because they exert
an overly large influence
on the fitted regression surface. The mechanism by which they
are downweighted or removed is through inclusion of case-specific
indicator variables. For a least-squares fit, inclusion of a
case-specific indicator in the model is equivalent to removing
the case from the data set; for a normal-theory,
Bayesian regression analysis,
inclusion of a case-specific indicator with an appropriate prior
distribution is equivalent to inflating the variance of the
case and hence downweighting it. The tradition in robust regression
is to handle the case-specific decisions automatically, most often
by downweighting outliers according to an iterative procedure
(\citep{huber}).
This idea of introducing
case-specific indicators also applies naturally to criterion based
regression procedures.
Model selection criteria such as AIC or BIC take aim at choosing a model
by attaching a~penalty for each additional parameter in the model.
These criteria can be applied
directly to a larger space of models---namely those in which the
covariates are augmented by a set of case indicators, one for
each case in the data set. When considering inclusion of a case
indicator for a large outlier, the criterion will judge the trade-off
between the empirical risk (here, negative log-likelihood) and model complexity
(here, number of parameters) as favoring the more complex model. It will
include the case indicator in the model, and, with a least-squares
fit, effectively remove the case from the data set. A more considered
approach would allow differential penalties for case-specific indicators
and ``real'' covariates.
With adjustment, one can essentially recover the familiar
$t$-tests for outliers (e.g., \citep{weisberg:book}), either
controlling the error
rate at the level of the individual test or controlling the Bonferroni
bound on the familywise error rate.
Case-specific indicators can also be used in conjunction with
regularization methods such as the LASSO (\citep{lasso}).
Again, care must
be taken with details of their inclusion. If these new covariates are
treated in the same fashion as the other covariates in the problem,
one is making an implicit judgment that they should be penalized in
the same fashion. Alternatively, one can allow a second parameter
that governs the severity of the penalty for the indicators. This
penalty can be set with a view of achieving robustness in the analysis,
and it allows one to tap into a large, extant body of knowledge about
robustness (\citep{huber}).
With regression often serving as a motivating\break theme, a host of
regularization methods for model selection and estimation problems
have been developed.
These methods range broadly across the field of statistics. In addition
to traditional normal-theory linear regression,
we find many methods motivated by a~loss which is composed of a negative
log-likeli\-hood and a penalty for model complexity.
Among these regularization methods are
penalized linear regression methods [e.g., ridge regression (Hoerl and Kennard, \citeyear{ridge})
and the LASSO], regression with a~nonparametric mean function,
[e.g., smoothing\break splines\vadjust{\goodbreak} (\citep{wahba90splines}) and generalized
additive
models (\citep{gam})], and extension to regression with nonnormal error
distributions, name\-ly, generalized linear models (\citep{glm}).
In all of these cases, one can add case-specific indicators along
with an appropriate penalty in order to yield an automated, robust analysis.
It should be noted that, in addition to a different severity for the
penalty term, the case-specific indicators sometimes require a different
form for their penalty term.
A second class of procedures open to modification with case-specific
indicators are those motivated by minimization of
an empirical risk function.
The risk function may not be a negative log-likelihood.
Quantile regression (whether linear or nonlinear) falls into this category,
as do modern classification
techniques such as the support vector machine (\citep{vapnik98}) and
the $\psi$-learner (\citep{psi03}). Many of these procedures
are designed with the robustness of the analysis in mind, often
operating on an estimand defined to be the population-level minimizer of
the risk. The procedures are consistent across a wide variety
of data-generating mechanisms and hence are asymptotically robust. They
have little need of further robustification. Instead, scope for bettering
these procedures lies in improving their finite sample properties.
The finite sample performance of many procedures in this
class can be improved by including case-specific indicators in the problem,
along with an appropriate penalty term for them.
This paper investigates the use of case-specific indicators
for improving modeling and prediction procedures in a regularization
framework.
Section~\ref{sec:robustmodel} provides a formal description
of the optimization problem which arises with the introduction
of case-specific indicators. It also describes a computational
algorithm and conditions that ensure the algorithm will obtain the
global solution to the regularized problem.
Section \ref{sec:reg} explains the
methodology for a~selection of regression methods, motivating particular
forms for the penalty terms. Section \ref{sec:class} describes how the
methodology applies to several classification schemes.
Sections \ref{sec:simulate} and \ref{sec:apply} contain simulation
studies and worked examples. We discuss implications
of the work and potential extensions in Section~\ref{sec:disc}.\looseness=1
\section{Robust and Efficient Modeling Procedures}
\label{sec:robustmodel}
Suppose that we have $n$ pairs of observations denoted by
$(x_i,y_i)$, $i=1,\ldots,n$, for statistical modeling and prediction. Here
$x_i = (x_{i1},\ldots,x_{ip})^\top$ with $p$ covariates and the $y_i$'s are
responses. As in the standard setting of regression and
classification, the $y_i$'s are assumed
to be conditionally independent given the $x_i$'s. In this paper, we take
modeling of the data as a procedure of finding a functional
relationship between $x_i$ and $y_i$, $f(x; \beta)$ with
unknown parameters $\beta\in\Bbb{R}^p$ that is consistent with the data.
The discrepancy or lack of fit of $f$ is measured by a loss function
$\mathcal{L}(y, f(x; \beta))$.
Consider a modeling procedure, say, $\mathcal{M}$ of finding $f$
which minimizes ($n$ times) the empirical risk
\[
R_n(f)=\sum_{i=1}^n \mathcal{L}(y_i, f(x_i; \beta))
\]
or its penalized version,
$ R_n(f)+\lambda J(f) =
\sum_{i=1}^n \mathcal{L}(y_i,\break f(x_i; \beta))+ \lambda J(f)$,
where $\lambda$ is a positive penalty parameter for balancing the data fit
and the model complexity of $f$ measured by $J(f)$.
A variety of common modeling procedures are subsumed under
this formulation, including ordinary linear regression,
generalized linear models, nonparametric regression,
and supervised learning techniques.
For\break brevity of exposition, we identify $f$ with $\beta$ through a parametric
form and view $J(f)$ as a functional depending on $\beta$.
Extension of the formulation presented in this paper
to a nonparametric function $f$ is straightforward via a~basis expansion.
\subsection{Modification of Modeling Procedures}
First, we introduce case-specific parameters,
$\underline{\gamma}=(\gamma_1,\ldots,\gamma_n)^\top$, for the $n$ observations
by augmenting the covariates with $n$ case-specific indicators.
For convenience, we use $\gamma$ to refer to a generic element of
$\underline{\gamma}$, dropping the subscript.
Motivated by the beneficial effects of regularization,
we propose a general scheme to modify the modeling procedure
$\mathcal{M}$ using the case-specific parameters $\underline{\gamma}$, to enhance
$\mathcal{M}$ for robustness or efficiency.
Define modification of $\mathcal{M}$ to be the
procedure of finding the original model parameters, $\beta$,
together with the case-specific parameters, $\underline{\gamma}$, that minimize
\begin{eqnarray}
\label{eq:robustM}
L(\beta,\underline{\gamma})
&=& \sum_{i=1}^n \mathcal{L}(y_i, f(x_i; \beta)+\gamma_i)
\nonumber
\\[-8pt]
\\[-8pt]
\nonumber
&&{}+ \lambda_\beta J(f)
+ \lambda_\gamma J_2(\underline{\gamma}).
\end{eqnarray}
If $\lambda_\beta$ is zero, $\mathcal{M}$ involves
empirical risk minimization, otherwise penalized risk minimization.
The adjustment that the added case-specific parameters bring to
the loss function $\mathcal{L}(y, f(x; \beta))$ is the same regardless of
whether $\lambda_\beta$ is zero or not.\vadjust{\goodbreak}
In general, $J_2(\underline{\gamma})$ measures the size of
$\underline{\gamma}$. When concerned
with robustness, we often take $J_2(\underline{\gamma}) =
\|\underline{\gamma}\|_1=\sum_{i=1}^n|\gamma_i|$.
A rationale for this choice is that with added flexibility,
the case-specific parameters can curb the undesirable influence of individual
cases on the fitted model.
To see this effect, consider minimizing $L(\hat{\beta}, \underline
{\gamma})$ for
fixed $\hat{\beta}$, which decouples to a~minimization of
$\mathcal{L}(y_i, f(x_i; \hat{\beta})+\gamma_i)
+ \lambda_\gamma|\gamma_i|$ for each $\gamma_i$. In most cases,
an explicit form of the minimizer $\hat{\underline{\gamma}}$ of
$L(\hat{\beta},\underline{\gamma})$ can be obtained.
Generally $\hat{\gamma}_i$'s
are large for observations with large ``residuals'' from the current
fit, and
the influence of those observations
can be reduced in the next round of fitting $\beta$ with the
$\hat{\underline{\gamma}}$-adjusted data.
Such a case-specific adjustment would be necessary
only for a small number of potential outliers, and
the $\ell_1$ norm which yields sparsity works to that effect.
The adjustment in the process of sequential updating of $\beta$
is equivalent to changing the loss from
$\mathcal{L}(y, f(x;\beta))$ to $\mathcal{L}(y, f(x;\beta)+\hat{\gamma})$,
which we call the \textit{$\gamma$-adjusted loss} of $\mathcal{L}$.
The $\gamma$-adjusted loss is a re-expression of $\mathcal{L}$
in terms of the adjusted residual, used as a conceptual aid to illustrate
the effect of adjustment through the case-specific parameter~$\gamma$ on
$\mathcal{L}$.
Concrete examples of the adjustments will be given in the following sections.
Alternatively, one may view
$\mathcal{L}_{\lambda_\gamma}(y, f(x;\beta))
:=\min_{\gamma\in\Bbb{R}} \{ \mathcal{L}(y, f(x; \beta)+\gamma) + \lambda
_\gamma
|\gamma|\} = \mathcal{L}(y, f(x; \beta)+\hat{\gamma})
+ \lambda_\gamma|\hat{\gamma}|$
as a whole to be the ``effective loss'' in terms of $\beta$ after profiling
out $\hat{\gamma}$. The effective loss replaces
$\mathcal{L}(y, f(x; \beta))$ for the modified $\mathcal{M}$ procedure.
When concerned with efficiency, we often take
$J_2(\underline{\gamma}) = \|\underline{\gamma}\|_2^2=\sum_{i=1}^n\gamma_i^2$.
This choice has the effect of increasing the impact of selected,
nonoutlying cases on
the analysis.
In subsequent sections,
we will take a few standard statistical methods for regression and
classification and illustrate how this general scheme applies.
This framework allows us to see established procedures in a new
light and also generates new procedures.
For each method, particular attention will be paid to the form of
adjustment to
the loss function by the penalized case-specific parameters.
\subsection{General Algorithm for Finding Solutions}
Although the computational details for obtaining the solution to
\eqref{eq:robustM} are specific to each modeling procedure $\mathcal{M}$,
it is feasible to describe a common computational strategy which is
effective for a wide range of procedures that optimize a convex function.
For fixed $\lambda_\beta$ and $\lambda_\gamma$, the solution
pair of $\hat{\beta}$ and $\hat{\underline{\gamma}}$ to the modified
$\mathcal{M}$
can be found with little extra computational cost.
A~generic algorithm below alternates estimation of $\beta$ and
$\underline{\gamma}$.\vadjust{\goodbreak}
Given $\hat{\underline{\gamma}}$, minimization of $L(\beta,
\hat{\underline{\gamma}})$ is done via the original modeling procedure
$\mathcal{M}$.
In most cases we consider, minimization of $L(\hat{\beta}, \underline
{\gamma})$
given $\hat{\beta}$ entails simple adjustment of ``residuals.''
These considerations lead to the following iterative algorithm
for finding $\hat{\beta}$ and $\hat{\underline{\gamma}}$:
\begin{enumerate}
\item Initialize $\hat{\underline{\gamma}}^{(0)}=0$ and
$\hat{\beta}^{(0)}=\operatorname{arg} \min_\beta L(\beta,0)$
(the ordinary $\mathcal{M}$ solution).
\item Iteratively alternate the following two steps, $m=0,1,\ldots$:
\begin{itemize}
\item$\hat{\underline{\gamma}}^{(m+1)}
= \operatorname{arg} \min_{\underline{\gamma}\in\Bbb{R}^n}
L(\hat{\beta}^{(m)},\underline{\gamma})$ modifies ``re\-siduals.''
\item$\hat{\beta}^{(m+1)}=\operatorname{arg} \min_{\beta\in\Bbb{R}^p} L(\beta,
\hat{\underline{\gamma}}^{(m+1)})$.
This step\break amounts to reapplying the $\mathcal{M}$ procedure to
$\hat{\underline{\gamma}}^{(m+1)}$-adjusted data although the nature of
the data adjustment would largely depend on~$\mathcal{L}$.
\end{itemize}
\item Terminate the iteration when
$\|\hat{\beta}^{(m+1)}-\hat{\beta}^{(m)}\|^2 < \varepsilon$,
where $\varepsilon$ is a prespecified convergence tolerance.
\end{enumerate}
In a nutshell, the algorithm attempts to find the joint minimizer
$(\beta,\underline{\gamma})$ by combining the minimizers $\beta$ and
$\underline{\gamma}$ resulting from the projected subspaces.
Convergence of the iterative updates
can be established under appropriate conditions.
Before we state the conditions and results for convergence,
we briefly describe implicit assumptions on the loss function and the
complexity or penalty terms, $J(f)$ and $J_2(\underline{\gamma})$.
$\mathcal{L}(y, f(x;\beta))$ is assumed to be nonnegative.
For simplicity, we assume that $J(f)$ of $f(x;\beta)$ depends on~$\beta$ only,
and that it is of the form $J(f)=\|\beta\|_k^k$ and
$J_2(\underline{\gamma})=\|\underline{\gamma}\|_k^k$ for $k \geq1$.
The LASSO penalty has $k=1$ while a ridge regression type
penalty sets $k=2$. Many other penalties of this form for $J(f)$
can be adopted as well to achieve better model selection
properties or certain desirable performance
of $\mathcal{M}$. Examples include those for the elastic net
(Zou and Hastie. \citeyear{elasticnet}), the grouped LASSO (\citep{glasso})
and the hierarchical LASSO (\citep{hlasso}).
For certain combinations of the loss $\mathcal{L}$ and the penalty functionals,
$J(f)$ and $J_2(\underline{\gamma})$, more efficient computational
algorithms can be devised, as in Has\-tie et~al. (\citeyear{svmpath}), \citet{lars} and
\citet{rosset:zhu}.
However, in an attempt to provide a~general computational recipe applicable
to a variety of modeling procedures which can be implemented with simple
modification of existing routines,
we do not pursue the optimal implementation tailored to a specific
procedure in this paper.
Convexity of the loss and penalty terms plays a~primary role
in characterizing\vadjust{\goodbreak} the solutions of the iterative algorithm.
For a general reference to properties of
convex functions and convex optimization,
see \citet{convex}. Nonconvex problems require different
optimization strategies.
If $L(\beta,\underline{\gamma})$ in \eqref{eq:robustM} is continuous and
strictly convex in $\beta$ and $\underline{\gamma}$ for fixed $\lambda
_\beta$ and
$\lambda_\gamma$, the minimizer pair $(\beta,\underline{\gamma})$ in
each step is
properly defined. That is, given $\underline{\gamma}$, there exists a
unique minimizer
$\beta(\underline{\gamma}):=\operatorname{arg} \min_\beta
L(\beta,\underline{\gamma})$, and vice versa.
The assumption that $L(\beta,\underline{\gamma})$ is strictly convex holds
if the loss $\mathcal{L}(y, f(x;\beta))$ itself is strictly convex.
Also, it is satisfied when a convex $\mathcal{L}(y, f(x;\beta))$ is combined
with $J(f)$ and $J_2(\underline{\gamma})$ strictly convex in $\beta$
and $\underline{\gamma}$, respectively.
Suppose that $L(\beta,\underline{\gamma})$ is strictly convex in
$\beta$ and~$\underline{\gamma}$ with a unique minimizer
$(\beta^*,\underline{\gamma}^*)$ for fixed $\lambda_\beta$ and~$\lambda
_\gamma$.
Then, the iterative algorithm gives a sequence of
$(\hat{\beta}^{(m)},\hat{\underline{\gamma}}^{(m)})$ with strictly decreasing
$L(\hat{\beta}^{(m)},\hat{\underline{\gamma}}^{(m)})$. Moreover,
$(\hat{\beta}^{(m)},\hat{\underline{\gamma}}^{(m)})$ converges to
$(\beta^*,\underline{\gamma}^*)$. This result of convergence of the iterative
algorithm is well known in convex optimization, and
it is stated here without proof.
Interested readers can find a formal proof in \citet{regcaseparam}.
\section{Regression}
\label{sec:reg}
Consider a linear model of the form
$y_i=x_i^\top\beta+ \varepsilon_i$. Without loss of generality,
we assume that each covariate is standardized.
Let $X$ be an
$n \times p$ design matrix with $x_i^\top$ in the $i$th row and let
$Y=(y_1,\ldots,y_n)^\top$.
\begin{figure*}
\centering
\begin{tabular}{@{}cc@{}}
\includegraphics{377f01a.eps}
& \includegraphics{377f01b.eps}\\
\footnotesize{(a)} & \footnotesize{(b)}\\[6pt]
\includegraphics{377f01c.eps}
& \includegraphics{377f01d.eps}\\
\footnotesize{(c)} & \footnotesize{(d)}
\end{tabular}
\caption{Modification of the squared error loss with a case-specific
parameter.
\textup{(a)} $\gamma$ versus the residual $r$,
\textup{(b)} the adjusted residual $r^*$ versus the ordinary residual $r$,
\textup{(c)} a truncated squared error loss as the $\gamma$-adjusted
loss and \textup{(d)} the effective loss.}
\label{fig:winsorize}
\end{figure*}
\subsection{Least Squares Method}
Taking the least squares method as a baseline modeling procedure
$\mathcal{M}$, we make a link between its modification via
case-specific parameters and a classical robust regression procedure.
The least squares estimator of $\beta=(\beta_1,\ldots,\beta_p)^\top$
is the minimizer $\hat{\beta} \in\Bbb{R}^p$ of
$ L(\beta)=\frac{1}{2} (Y-X\beta)^\top(Y-X\beta)$.
To reduce the sensitivity of the estimator to influential
observations, the $p$ covariates are augmented by $n$ case indicators.
Let $z_i$ be the indicator variable taking 1 for the $i$th observation
and~0 otherwise, and let
$\underline{\gamma}=(\gamma_1,\ldots,\gamma_n)^\top$ be the
coefficients of
the case indicators. The additional design matrix $Z$
for $z_i$ is the identity matrix, and $Z\underline{\gamma}$ becomes
$\underline{\gamma}$ itself.
The proposed modification of the least squares method
with $J_2(\underline{\gamma})=\|\underline{\gamma}\|_1=\sum
_{i=1}^n|\gamma_i|$
leads to a~well-known robust regression procedure.
For the robust modification, we find
$\hat{\beta} \in\Bbb{R}^p$ and $\hat{\underline{\gamma}} \in
\Bbb{R}^n$ that minimize
\begin{eqnarray}
\label{eq:robustls}\qquad
L(\beta,\underline{\gamma})
&=&\tfrac{1}{2}\{Y-(X\beta+\underline{\gamma})\}^\top
\{Y-(X\beta+\underline{\gamma})\}
\nonumber
\\[-8pt]
\\[-8pt]
\nonumber
&&{}+ \lambda_{\gamma}\|\underline{\gamma}\|_1,
\end{eqnarray}
where $\lambda_{\gamma}$ is a fixed regularization
parameter constraining $\underline{\gamma}$.
Just as the ordinary LASSO with the~$\ell_1$ norm penalty stabilizes regression
coefficients by\break shrinkage and selection,
the additional penalty in \eqref{eq:robustls} has the same effect
on $\underline{\gamma}$, whose components gauge the extent of case influences.
The minimizer
$\hat{\underline{\gamma}}$ of $L(\hat{\beta},\underline{\gamma})$ for
a fixed $\hat{\beta}$ can be found
by soft-thresholding the residual vector $r=Y-X\hat{\beta}$.
That is, $\hat{\gamma}_i = \operatorname{sgn}(r_i)(|r_i|-\lambda_{\gamma})_+$.
For observations with
small residuals, $|r_i|\le\lambda_{\gamma}$,
$\hat{\gamma}_i$ is set equal to zero with no effect on
the current fit, and for those with large residuals,
$|r_i|> \lambda_{\gamma}$, $\hat{\gamma}_i$ is
set equal to the residual $r_i=y_i-x_i^\top\hat{\beta}$
offset by $\lambda_{\gamma}$ toward zero.
Combining $\hat{\underline{\gamma}}$ with $\hat{\beta}$,
we define the adjusted residuals to be
$r^*_i=y_i-x_i^\top\hat{\beta}-\hat{\gamma}_i$;
that is, $r^*_i=r_i$ if $|r_i|\le\lambda_{\gamma}$, and
$r^*_i=\operatorname{sgn}(r_i)\lambda_{\gamma}$, otherwise.
Thus, introduction of the case-specific parameters along with the
$\ell_1$ penalty on $\underline{\gamma}$ amounts to
winsorizing the ordinary residuals.
The $\gamma$-adjusted loss is equivalent to
truncated squared error loss which is
$(y-x^\top\beta)^2$ if $|y-x^\top\beta|\le
\lambda_{\gamma}$, and is $\lambda_{\gamma}^2$ otherwise.
Figure \ref{fig:winsorize} shows (a) the relationship between
the ordinary residual~$r$ and the corresponding $\gamma$,
(b)~the residual and the adjusted residual $r^*$, (c) the
$\gamma$-adjusted loss as a~function of~$r$, and (d) the effective
loss.\looseness=-1
The effective loss is
$\mathcal{L}_{\lambda_\gamma}(y,x^\top\beta)=(y-x^\top\beta)^2/2$ if
$|y-x^\top\beta|\le\lambda_{\gamma}$, and
$\lambda_{\gamma}^2/2+\lambda_{\gamma}(|y-x^\top\beta|-\lambda_{\gamma})$
otherwise.
This effective loss matches Huber's loss function for robust regression
(\citep{huber}).
As in robust regression, we choose a sufficiently large~$\lambda_{\gamma}$
so that only a modest fraction of the residuals are adjusted.
Similarly, modification of the LASSO as a~penalized regression procedure
yields the Huberized LASSO described by \citet{huberlasso}.\vspace*{2pt}
\subsection{Location Families}\vspace*{2pt}
More generally, a wide class of problems can be cast in the
form of a minimization of
$L(\beta) =\break \sum_{i=1}^n g(y_i - x_i^\top\beta)$
where $g(\cdot)$ is the negative log-likelihood derived from a location
family. The assumption that we have a location family
implies that the negative log-likelihood is a function only
of $r_i=y_i - x_i^\top\beta$.
Dropping the subscript,
common choices for the negative log-likelihood, $g(r)$, include~$r^2$
(least squares, normal distributions) and $|r|$ (least
absolute deviations, Laplace distributions).
Introducing the case-specific parameters $\gamma_i$, we wish to
minimize
\[
L(\beta, \underline{\gamma}) = \sum_{i=1}^n g(y_i - x_i^\top\beta-
\gamma_i) +
\lambda_\gamma\|\underline{\gamma}\|_1.
\]
For minimization with a fixed $\hat{\beta}$,
the next result applies to a broad class of $g(\cdot)$ (but not to
$g(r) = |r|$).
\begin{proposition}
\label{prop:locfam}
Suppose that $g$ is strictly convex with the minimum at 0,\ and
$\lim_{r\rightarrow\pm\infty}g'(r)=\pm\infty$, respectively.
Then,
\begin{eqnarray*}
\hat{\gamma}&=&\operatorname{arg}\min_{\gamma\in\Bbb{R}} ~ g(r - \gamma) + \lambda
_\gamma|\gamma|\\
&=&\cases{
r - g'^{-1}(\lambda_{\gamma}),\vspace*{2pt}\cr
\quad\hspace*{11pt}\mbox{for } r > g'^{-1}(\lambda_{\gamma}), \vspace*{2pt}\cr
0, \quad\mbox{for } g'^{-1}(-\lambda_{\gamma})
\leq r \leq g'^{-1}(\lambda_{\gamma}), \vspace*{2pt}\cr
r - g'^{-1}(-\lambda_{\gamma}), \vspace*{2pt}\cr
\quad\hspace*{11pt}\mbox{for } r < g'^{-1}(-\lambda
_{\gamma}).}
\end{eqnarray*}
\end{proposition}
The proposition follows from straightforward algebra. Set the first
derivative of the decoupled minimization equation equal to $0$ and
solve for
$\gamma$. Inserting these values for $\hat{\gamma}_i$ into the equation for
$L(\beta, \underline{\gamma})$ yields
\[
L(\hat{\beta}, \hat{\underline{\gamma}})
= \sum_{i=1}^n g(r_i - \hat{\gamma_i}) +
\lambda_\gamma\|\hat{\underline{\gamma}}\|_1.
\]
The first term in the summation can be decomposed into three parts. Large
$r_i$ contribute $g(r_i - r_i + g'^{-1}(\lambda_{\gamma})) =
g(g'^{-1}(\lambda_{\gamma}))$. Large, negative $r_i$ contribute
$g(g'^{-1}(-\lambda_{\gamma}))$. Those $r_i$ with intermediate values
have $\hat{\gamma}_i = 0$ and so contribute $g(r_i)$. Thus a~graphical
depiction of the $\gamma$-adjusted loss is much like that in Figure
\ref{fig:winsorize},
panel (c), where the loss is truncated above.
For asymmetric distributions (and hence
asymmetric log-likelihoods), the truncation point may differ for positive
and negative residuals. It should be remembered that when $|r_i|$ is
large, the corresponding $\hat{\gamma}_i$ is large, implying a large
contribution of $\|\underline{\gamma}\|_1$
to the overall minimization problem. The residuals will
tend to be large for vectors $\beta$ that are at odds with the data. Thus,
in a sense, some of the loss which seems to disappear due to the effective
truncation of $g$ is shifted into the penalty term for $\gamma$.
Hence the effective loss
$\mathcal{L}_{\lambda_\gamma}(y,f(x; \beta))=g(y-f(x; \beta)-\hat{\gamma})
+ \lambda_\gamma|\hat{\gamma}|$ is the same as the original loss,
$g(y-f(x; \beta))$ when the residual is in
$[g'^{-1}(-\lambda_{\gamma}),\break g'^{-1}(\lambda_{\gamma})]$ and is
linear beyond the interval. The linearized part of $g$ is joined with
$g$ such that $\mathcal{L}_{\lambda_\gamma}$ is differentiable.
Computationally, the minimization of $L(\beta, \hat{\underline{\gamma}})$
giv\-en~$\hat{\underline{\gamma}}$ entails
application of the same modeling procedure $\mathcal{M}$ with $g$
to winsorized pseudo responses $y^*_i=y_i-\hat{\gamma}_i$,
where $y^*_i=y_i$ for $g'^{-1}(-\lambda_{\gamma})
\leq r_i \leq g'^{-1}(\lambda_{\gamma})$,
$y^*_i = g'^{-1}(\lambda_{\gamma})$ for $r > g'^{-1}(\lambda_{\gamma})$,
and $y^*_i = g'^{-1}(-\lambda_{\gamma})$ for
$r < g'^{-1}(-\lambda_{\gamma})$.
So, the $\hat{\underline{\gamma}}$-adjusted data in Step~2 of the main algorithm
consist of $(x_i,y_i^*)$ pairs in each iteration.
A related idea of subsetting data and model-fitting to the subset
iteratively for robustness can be found in the computer vision
literature, the random sample consensus algorithm (Fischler and Bolles, \citeyear{RSC}) for
instance.
\subsection{Quantile Regression}
Consider median regression with absolute deviation loss
$\mathcal{L}(y,x^\top\beta)=|y-x^\top\beta|$, which is not covered
in the foregoing discussion. It can be verified easily
that the $\ell_1$-adjustment of $\mathcal{L}$ is void due to the
piecewise linearity of the loss, reaffirming the robustness of median
regression.
For an effectual adjustment, the $\ell_2$ norm regularization of
the case-specific parameters is considered.
With the case-specific parameters $\gamma_i$, we
have the following objective function for modified median regression:
\begin{equation}
\label{eq:robustmedian}
L(\beta, \underline{\gamma})
=\sum_{i=1}^n |y_i - x_i^\top\beta- \gamma_i|
+ \frac{\lambda_\gamma}{2}\|\underline{\gamma}\|_2^2.
\end{equation}
For a fixed $\hat{\beta}$ and residual
$r=y-x^\top\hat{\beta}$, the $\hat\gamma$ minimizing
$ |r - \gamma|+ \frac{\lambda_\gamma}{2}\gamma^2$
is given by
\[
\operatorname{sgn}(r)\frac{1}{\lambda_\gamma}I \biggl(|r|> \frac{1}{\lambda_\gamma}\biggr)
+r I\biggl(|r|\le\frac{1}{\lambda_\gamma}\biggr).
\]
The $\gamma$-adjusted loss for median regression is
\[
\mathcal{L}(y, x^\top\beta+\hat{\gamma})
=\biggl|y-x^\top\beta-\frac{1}{\lambda_\gamma}\biggr|
I\biggl(|y-x^\top\beta|> \frac{1}{\lambda_\gamma}\biggr),
\]
as shown in Figure \ref{fig:modloss}(a) below.
Interestingly, this $\ell_2$-adjusted absolute deviation loss
is the same as the so-called ``$\varepsilon$-insensitive linear loss''
for support vector regression (\citep{vapnik98}) with
$\varepsilon=1/\lambda_\gamma$.
With this adjustment, the effective loss is Huberized
squared error loss.
The $\ell_2$ adjustment makes
median regression more efficient by rounding the sharp corner of the loss,
and leads to a hybrid procedure which lies between mean and median regression.
Note that, to achieve the desired effect
for median regression, one chooses quite a different value of
$\lambda_\gamma$ than one would when adjusting squared error loss
for a robust mean regression.
The modified median regression procedure can be also combined with
a penalty on $\beta$ for shrinkage and/or selection.
\citet{Bietal} considered support vector regression
with the $\ell_1$ norm penalty $\|\beta\|_1$ for simultaneous robust
regression and variable selection.
These authors relied on the
$\varepsilon$-insensitive linear loss which comes out as the $\gamma$-adjusted
loss of the absolute deviation. In contrast, we rely on the
effective loss which produces a different solution.
In general, quantile regression (Koenker and Bassett, \citeyear{KB1978}; \citep{qr})
can be used to estimate conditional quantiles of $y$ given $x$.
It is a~useful regression technique when
the assumption of normality on the distribution of the errors~$\varepsilon$
is not appropriate, for instance, when the error distribution is skewed or
heavy-tailed. For the $q$th quantile, the check function $\rho_q$
is employed:
\begin{equation}
\label{eq:qr}
\rho_q(r) = \cases{
q r, & $\mbox{for }r \ge0,$\vspace*{2pt}\cr
-(1-q)r, & $\mbox{for } r<0.$}
\end{equation}
The standard procedure for the $q$th quantile regression finds $\beta$
that minimizes the sum of asymmetrically weighted absolute errors with
weight $q$ on positive errors and weight $(1-q)$ on negative errors:
\[
L (\beta)= \sum_{i=1}^{n} \rho_q(y_i-x_i^\top\beta).
\]
Consider modification of $\rho_q$ with a case-specific parameter $\gamma
$ and
$\ell_2$ norm regularization. Due to the asymmetry in the loss, except for
$q=1/2$, the size of reduction in the loss by
the case-specific parameter $\gamma$ would depend on its sign.
Given $\hat{\beta}$ and residual $r=y-x^\top\hat{\beta}$,
if $r\ge0$, then the positive $\gamma$ would lower $\rho_q$ by $q\gamma
$, while
if $r<0$, the negative $\gamma$ with the same absolute value would lower
the loss by $(q-1)\gamma$. This asymmetric impact on the loss is
undesirable. Instead, we create a penalty that leads to the same
reduction in loss for positive and negative $\gamma$ of the same
magnitude.
In other words, the desired $\ell_2$ norm penalty needs to
put $q\gamma_+$ and $(1-q)\gamma_-$ on an equal footing. This leads to
the following penalty proportional to $q^2\gamma_+^2$ and
$(1-q)^2\gamma_-^2$:
\[
J_2(\gamma):=\{q/(1-q)\}\gamma_+^2 +\{(1-q)/q\}\gamma_-^2.
\]
When $q=1/2$, $J_2(\gamma)$ becomes the symmetric
$\ell_2$ norm of $\gamma$.
With this asymmetric penalty, given $\hat{\beta}$,
$\hat{\gamma}$ is now defined as
\begin{equation}\label{eq:quantile2}
\operatorname{arg} \min_{\gamma\in\Bbb{R}} \mathcal{L}_{\lambda_\gamma}(\hat{\beta},
\gamma)
:=\rho_q(r - \gamma)
+ \frac{\lambda_\gamma}{2} J_2(\gamma),
\end{equation}
and is explicitly given by
\begin{eqnarray*}
&&-\frac{q}{\lambda_\gamma}I \biggl(r<-\frac{q}{\lambda_\gamma}\biggr)
+ r I\biggl(-\frac{q}{\lambda_\gamma}\le r <
\frac{1-q}{\lambda_\gamma}\biggr)
\\
&&\quad{}+\frac{1-q}{\lambda_\gamma}I\biggl(r\ge\frac{1-q}{\lambda_\gamma}\biggr).
\end{eqnarray*}
The effective loss $\rho_q^\gamma$ is then given by
\begin{equation}
\label{eq:modcheck}
\rho_q^\gamma(r) = \cases{
(q-1)r-\displaystyle\frac{q(1-q)}{2\lambda_\gamma},\vspace*{2pt}\cr
\quad\mbox{for } r<-\displaystyle\frac{q}{\lambda_\gamma}, \vspace*{2pt}\cr
\displaystyle\frac{\lambda_\gamma}{2} \frac{1-q}{q}r^2,\vspace*{2pt}\cr
\quad\mbox{for } -\!\displaystyle\frac{q}{\lambda_\gamma}\le r < 0, \vspace*{2pt}\cr
\displaystyle\frac{\lambda_\gamma}{2} \frac{q}{1-q}r^2,\vspace*{2pt}\cr
\quad\mbox{for } 0\le r < \displaystyle\frac{1-q}{\lambda_\gamma}, \vspace*{2pt}\cr
qr-\displaystyle\frac{q(1-q)}{2\lambda_\gamma}, \vspace*{2pt}\cr
\quad\mbox{for } r \ge\displaystyle\frac{1-q}{\lambda_\gamma},}
\end{equation}
and its derivative is
\begin{equation}
\label{eq:modcheckderiv}
\psi_q^\gamma(r) = \cases{
q-1, & $\mbox{for } \displaystyle r<-\frac{q}{\lambda_\gamma},$ \vspace*{2pt}\cr
\displaystyle\lambda_\gamma\frac{1-q}{q}r,
& $\mbox{for } \displaystyle-\!\frac{q}{\lambda_\gamma}\le r < 0,$ \vspace*{2pt}\cr
\displaystyle\lambda_\gamma\frac{q}{1-q}r,
& $\mbox{for } 0\le r < \displaystyle\frac{1-q}{\lambda_\gamma},$ \vspace*{2pt}\cr
q, & $\mbox{for } r \ge\displaystyle\frac{1-q}{\lambda_\gamma}.$}
\end{equation}
We note that, under the assumption that the density is locally
constant in a neighborhood of the quantile, the quantile remains
the $0$ of the effective $\psi_q^\gamma$ function.
\begin{figure*}
\includegraphics{377f02.eps}
\caption{The derivative of the check loss
in the left panel, $\psi_q$, and that of the modified check loss
in the right panel, $\psi_q^\gamma$, for $q = 0.2$, $0.5$ and
$0.7$.}\label{fig:psi_fct}\vspace*{-4pt}
\end{figure*}
Figure \ref{fig:psi_fct} compares the derivative of the check loss with
that of the effective loss in \eqref{eq:modcheck}.
Through penalization of a case-specific parameter, $\rho_q$ is
modified to have a continuous derivative at the origin joined by
two lines with a different slope that depends on~$q$.
The effective loss is reminiscent of the
asymmetric squared error loss ($q(r_+)^2 + (1-q)(r_-)^2$)
considered by \citet{NeweyPowell:1987} and \citet{Efron:1991}
for the so-called expectiles.
The proposed modification
of the check loss produces a hybrid of the check loss and asymmetric
squared error loss, however, with different weights than those for
expectiles, to\vadjust{\goodbreak} estimate quantiles.
The effective loss is formally similar to the rounded-corner
check loss of \citet{NychkaEtAl:1995} who used a vanishingly
small adjustment to speed computation. \citet{KoenkerPortnoy:1997}
thoroughly discussed efficient computation for quantile regression.
Redefining $J_2(\underline{\gamma})$ as the sum of the asymmetric~pen\-alty
for the case-specific parameter $\gamma_i$, $i=1,\ldots, n$, modified
quantile regression is formulated as a~procedure that finds
$\beta$ and $\underline{\gamma}$ by minimizing
\begin{equation}
\label{eq:qrmodobj}\quad
L(\beta, \underline{\gamma})
=\sum_{i=1}^n \rho_q(y_i - x_i^\top\beta-\gamma_i)
+ \frac{\lambda_\gamma}{2} J_2(\underline{\gamma}).
\end{equation}
In extensive simulation studies (Jung, MacEachern and Lee, \citeyear{l2QR}), such
adjustment of the standard quantile regression procedure generally
led to more accurate estimates.
See Section \ref{subsec:qrsim} for a summary of the studies.
This is confirmed in the\break NHANES data analysis in Section \ref{subsec:nhanes}.
For large enough samples, with a fixed $\lambda_{\gamma}$,
the bias of the enhanced estimator will typically
outweigh its benefits. The natural approach is to adjust the penalty attached
to the case-specific covariates as the sample size increases.
This can be accomplished by increasing the parameter $\lambda_\gamma$
as the sample size $n$ grows.
Let $\lambda_\gamma:=cn^\alpha$ for some constant $c$ and $\alpha>0$.
The following theorem shows that if $\alpha$ is sufficiently large,
the modified quantile regression estimator $\hat{\beta}_n^\gamma$,
which minimizes $\sum_{i=1}^n \rho_q^\gamma(y_i - x_i^\top\beta)$
or equivalent\-ly~\eqref{eq:qrmodobj},
is asymptotically equivalent to the standard estimator $\hat{\beta}_n$.
\citet{Knight:1998} proved the asymptotic normality of the regression quantile
estimator $\hat{\beta}_n$ under some mild regularity conditions.
Using the arguments in \citet{Koenker:2005}, we show that
$\hat{\beta}_n^\gamma$ has the same limiting distribution as $\hat{\beta}_n$,
and thus it is $\sqrt{n}$-consistent if $\alpha$ is sufficiently
large.\vadjust{\goodbreak}
Allowing a potentially different error distribution for each observation,
let $Y_1, Y_2, \ldots$ be independent random variables with c.d.f.'s
$F_1, F_2, \ldots$ and suppose that each $F_i$ has
continuous p.d.f. $f_i$.
Assume that the $q$th conditional quantile
function of $Y$ given $x$ is linear in $x$ and given by $x^\top\beta(q)$,
and let $\xi_i(q):=x_i^\top\beta(q)$.
Now consider the following regularity conditions:
\begin{enumerate}[(C-1)]
\item[(C-1)] $f_i(\xi)$, $i=1,2,\ldots,$ are uniformly bound\-ed away
from 0
and $\infty$ at $\xi_i$.
\item[(C-2)] $f_i(\xi)$, $i=1,2,\ldots,$ admit a first-order Taylor expansion
at $\xi_i$, and $f'_i(\xi)$ are uniformly bound\-ed at $\xi_i$.
\item[(C-3)] There exists a positive definite matrix $D_0$ such that
$ \lim_{n\rightarrow\infty}n^{-1}\sum x_ix_i^\top=D_0$.
\item[(C-4)] There exists a positive definite matrix $D_1$ such that
$ \lim_{n\rightarrow\infty}n^{-1}\sum f_i(\xi_i)x_ix_i^\top=D_1$.
\item[(C-5)] $\max_{i=1,\ldots,n}\| x_i \| /\sqrt{n} \rightarrow0$
in probability.
\end{enumerate}
(C-1) and (C-3) through (C-5) are the conditions considered for
the limiting distribution of the standard regression quantile
estimator $\hat\beta_n$ in \citet{Koenker:2005}
while (C-2) is an additional assumption that we make.
\begin{theorem}
\label{thm:reg_qtl} Under the conditions \textup{(C-1)--(C-5)},
if $\alpha>1/3$, then
\[
\sqrt{n}\bigl(\hat{\beta}^\gamma_{n}-\beta(q)\bigr) \stackrel{d} \rightarrow
N\bigl(0,q(1-q)D_1^{-1}D_0D_1^{-1}\bigr).
\]
\end{theorem}
The proof of the theorem is in the \hyperref[app]{Appendix}.
\begin{figure*}
\centering
\begin{tabular}{@{}ccc@{}}
\includegraphics{377f03a.eps}
& \includegraphics{377f03b.eps} & \includegraphics{377f03c.eps}\\
\footnotesize{(a)} & \footnotesize{(b)} & \footnotesize{(c)}
\end{tabular}
\caption{Modification of
\textup{(a)} absolute deviation loss for median regression with $\ell_2$ penalty,
\textup{(b)} negative log-likelihood for logistic regression with $\ell_1$
penalty, and
\textup{(c)} hinge loss for the support vector machine with $\ell_2$ penalty.
The solid lines are for the effective loss, the dashed lines are
for the $\gamma$-adjusted loss, and the dotted lines are
for the original loss in each panel.}
\label{fig:modloss}\vspace*{6pt}
\end{figure*}
\section{Classification}
\label{sec:class}
Now suppose that $y_i$'s indicate binary outcomes.
For modeling and prediction of the binary responses, we mainly consider
margin-based procedures such as
logistic regression, support vector machines
(\citep{vapnik98}), and\vadjust{\goodbreak} boosting (\citep{freund:schapire97}). These procedures
can be modified by the addition of case indicators.\vspace*{1pt}
\subsection{Logistic Regression}\vspace*{1pt}
Although it is customary to label a binary outcome as 0 or 1
in logistic regression, we instead adopt the symmetric labels of $\{
-1,1\}$
for $y_i$'s. The symmetry facilitates comparison of different
classification procedures. Logistic regression takes
the negative log-likelihood as a loss for estimation of logit
$f(x)=\log[p(x)/(1-p(x))]$. The loss,
$\mathcal{L}(y,f(x))=\log[1+\exp(-yf(x))]$, can be viewed as a function of
the so-called margin, $yf(x)$. This functional margin of $yf(x)$ is
a pivotal quantity for defining a family of loss functions in classification
similar to the residual in regression.
As in regression with continuous responses, case indicators can be
used to modify the logit function $f(x)$ in logistic regression to minimize
\begin{eqnarray}\label{eq:penlogistic}
&& L(\beta_0,\beta, \underline{\gamma})
\nonumber\\
&&\quad= \sum_{i=1}^n \log\bigl(1+\exp\bigl(-y_i\{f(x_i; \beta_0,\beta)+\gamma_i\}
\bigr)\bigr)\\
&&\qquad{}+ \lambda_\gamma\|\underline{\gamma}\|_1,\nonumber
\end{eqnarray}
where $f(x; \beta_0,\beta)=\beta_0 + x^\top\beta$.
When it is clear in context, $f(x)$ will be used as abbreviated notation
for $f(x; \beta_0,\beta)$, a discriminant function, and the
subscript $i$ will be dropped.
For fixed $\hat{\beta}_0$ and $\hat{\beta}$, the minimization
decouples, and $\gamma_i$ is determined
by minimizing
\[
\log\bigl(1+\exp\bigl(-y_i\{f(x_i;\hat{\beta}_0,\hat{\beta})+\gamma_i\}
\bigr)\bigr)+
\lambda_\gamma|\gamma_i|.
\]
First note that the minimizer $\gamma_i$
must have the same sign as $y_i$. Letting $\tau=yf(x)$ and assuming that
$0<\lambda_\gamma<1$, we have
$\arg\min_{\gamma\ge0}
\log(1+\exp(-\tau-\gamma))+ \lambda_\gamma|\gamma|
=\log\{(1-\lambda_\gamma)/\lambda_\gamma\} - \tau$
if $\tau\le\log\{(1-\lambda_\gamma)/\lambda_\gamma\}$, and
0 otherwise.
This yields a truncated negative log-likelihood given by
\[
\mathcal{L}(y,f(x))=\cases{
\log\bigl(1+ \lambda_\gamma/(1-\lambda_\gamma)\bigr),\vspace*{2pt}\cr
\quad \mbox{if }yf(x) \le\log\{(1-\lambda_\gamma)/\lambda_\gamma\},
\vspace*{2pt}\cr
\log\bigl(1+\exp(-yf(x))\bigr), \vspace*{2pt}\cr
\quad \mbox{otherwise,}}
\]
as the $\gamma$-adjusted loss.
This adjustment is reminiscent of
\citeauthor{pregibon82}'s (\citeyear{pregibon82}) proposal
tapering the deviance function so as to downweight extreme observations,
thereby producing a robust logistic regression.
See Figure \ref{fig:modloss}(b) for the $\gamma$-adjusted loss
(the dashed line), where
$\eta_\lambda:=\log\{(1-\lambda_\gamma)/\lambda_\gamma\}$ is
a decreasing function of $\lambda_\gamma$.
$\lambda_\gamma$ determines the level of truncation of the loss.
As $\lambda_\gamma$ tends to $1$, there is no truncation.
Figure \ref{fig:modloss}(b) also shows the effective loss (the solid
line) for
the $\ell_1$ adjustment, which linearizes the negative log-likelihood
below $\eta_\lambda$.\vspace*{1pt}
\subsection{Large Margin Classifiers}\vspace*{1pt}
With the symmetric class labels, the foregoing characterization of
the case-specific parameter $\gamma$ in logistic regression can be
easily generalized to various margin-based classification procedures.
In classification, potential outliers are
those cases with large negative margins.
Let $g(\tau)$ be a loss function of the margin $\tau=yf(x)$. The
following proposition, analogous to Proposition \ref{prop:locfam},
holds for a general family of loss functions.
\begin{proposition}
\label{prop:margin}
Suppose that $g$ is convex and monotonically decreasing in $\tau$,
and $g'$ is continuous.
Then, for $\lambda_{\gamma} < -\lim_{\tau\rightarrow-\infty}g'(\tau)$,
\begin{eqnarray*}
\hat{\gamma}
&=&\arg\min_{\gamma\in\Bbb{R}} ~ g(\tau+ \gamma) + \lambda_\gamma
|\gamma|
\\
&=&\cases{
g'^{-1}(-\lambda_{\gamma})-\tau, & $\mbox{for } \tau
\le g'^{-1}(-\lambda_{\gamma}),$\vspace*{2pt}\cr
0, & $\mbox{for } \tau> g'^{-1}(-\lambda_{\gamma}).$}
\end{eqnarray*}
\end{proposition}
The proof is straightforward. Examples of the mar\-gin-based loss $g$
satisfying the assumption include
the exponential loss $g(\tau)=\exp(-\tau)$ in boosting,
the squared hinge loss $g(\tau)=\{(1-\tau)_+\}^2$ in the support vector
machine (\citep{leemangasarian}), and the negative log-likelihood
$g(\tau)=\log(1+\exp(-\tau))$ in logistic regression.
Although their theoretical targets are different,
all of these loss functions are truncated above for large negative margins
when adjusted by~$\gamma$. Thus, the effective loss
$\mathcal{L}_{\lambda_\gamma}(y, f(x))=g(yf(x)+\hat{\gamma})
+ \lambda_\gamma|\hat{\gamma}|$ is obtained by linearizing
$g$ for $yf(x)<g'^{-1}(-\lambda_{\gamma})$.
The effect of $\hat{\gamma}$-adjustment
depends on the form of~$g$, and hence on the classification
method. For boosting, $\hat{\gamma}\! =\! -\log\lambda_{\gamma}-yf(x)$ if
$yf(x)\!\le\!-\log\lambda_{\gamma}$,~and is 0 otherwise.
This gives
$L(\beta_0, \beta,\hat{\underline{\gamma}})
\!=\!\sum_{i=1}^n \exp(-y_i\cdot f(x_i;\beta_0, \beta)-\hat{\gamma}_i)
\!=\!\sum_{i=1}^n \exp({-}\hat{\gamma}_i) \exp({-}y_if(x_i; \beta_0,\beta)\!)$.
So, finding $\beta_0$ and $\beta$ given $\hat{\underline{\gamma}}$
amounts to weighted boosting, where
the positive case-specific parameters $\hat{\gamma}_i$ downweight the
corresponding cases by\break $\exp(-\hat{\gamma}_i)$.
For the squared hinge loss in the support vector machine, $\hat{\gamma}
= 1-yf(x)-\lambda_{\gamma}/2$ if $yf(x) \le1-\lambda_{\gamma}/2$,
and is
$0$ otherwise.
A positive case-specific parameter $\hat{\gamma}_i$
has the effect of relaxing the margin requirement, that is,
lowering the joint of the hinge for that specific case.
This allows the associated slack variable to be smaller in the primal
formulation. Accordingly, the adjustment affects the coefficient of the linear
term in the dual formulation of the quadratic programming problem.
As a related approach to robust classification,
\citet{Wu:Liu} proposed truncation of margin-based loss functions
and studied theoretical properties that ensure classification
consistency. Similarity exists between our proposed adjustment of a loss
function with $\gamma$ and truncation of the loss at some point.
However, it is the linearization of a margin-based loss function
on the negative side that produces its effective loss, and
minimization of the effective loss is quite different from
minimization of the truncated (i.e., adjusted) loss.
Linearization is more conducive to computation
than is truncation. Application of the result
in \citet{bjm-ccrb-05} shows that the linearized loss
functions satisfy sufficient conditions for classification consistency,
namely Fisher consistency, which is the main property investigated by
\citet{Wu:Liu} for truncated loss functions.
\citet{Xuetal:2009} showed that regularization in the standard
support vector machine is equivalent to a robust formulation
under disturbances of $x$ without penalty. In contrast, under our
approach, robustness of classification methods is considered through
the margin, which is analogous to the residual in regression.
This formulation can cover outliers due to perturbation in $x$ as
well as mislabeling of~$y$.
\subsection{Support Vector Machines}
As a special case of a large margin classifier,
the linear support vector machine (SVM) looks for the optimal hyperplane
$f(x;\beta_0,\beta)=\beta_0 + x^\top\beta=0$ minimizing
\begin{equation}\label{eq:svm2}
\hspace*{19pt} L_{\lambda}(\beta_0,\beta)
= \sum_{i=1}^n [ 1 - y_i f(x_i;\beta_0,\beta) ]_+
+ \frac{\lambda}{2}\|\beta\|_2^2,
\end{equation}
where $[t]_+ = \max(t, 0)$ and $\lambda>0$ is a regularization parameter.
Since the hinge loss for the SVM, $g(\tau)=(1-\tau)_+$, is piecewise linear,
its linearization with $\|\underline{\gamma}\|_1$ is void,
indicating that it has little need of further robustification.
Instead, we consider
modification of the hinge loss with $\|\underline{\gamma}\|_2^2$.
This modification is expected to improve efficiency,
as in quantile regression.
Using the case indicators $z_i$ and their coefficients~$\gamma_i$,
we modify \eqref{eq:svm2}, arriving at the problem of minimizing
\begin{eqnarray}\label{eq:modsvm2}
L(\beta_0, \beta, \underline{\gamma})
&=& \sum_{i=1}^n [ 1 - y_i \{f(x_i;\beta_0,\beta)+\gamma_i\} ]_+
\nonumber
\\[-8pt]
\\[-8pt]
\nonumber
&&{}+ \frac{\lambda_\beta}{2}\|\beta\|_2^2
+ \frac{\lambda_\gamma}{2}\|\underline{\gamma}\|_2^2.
\end{eqnarray}
For fixed $\hat{\beta}_0$ and $\hat{\beta}$, the minimizer
$\hat{\underline{\gamma}}$
of $L(\hat{\beta}_0,\hat{\beta},\underline{\gamma})$ is obtained by
solving the decoupled optimization problem of
\[
\min_{\gamma_i \in\Bbb{R}}
[1-y_if(x_i;\hat{\beta}_0,\hat{\beta})-y_i\gamma_i]_+
+ \frac{\lambda_\gamma}{2}\gamma_i^2 \quad\mbox{for each }\gamma_i.
\]
With an argument similar to that for logistic regression,
the minimizer $\hat{\gamma}_i$ should have the same sign as~$y_i$.
Let $\xi\!=\!1-yf$. A simple calculation shows that\looseness=-1
\begin{eqnarray*}
&&\arg\min_{\gamma\ge0} [\xi-\gamma]_+ + \frac{\lambda_\gamma
}{2}\gamma^2
\\
&&\quad=\cases{
0, & $\mbox{if }\xi\le0,$ \vspace*{2pt}\cr
\xi, & $\mbox{if } 0< \xi< 1/\lambda_\gamma,$\vspace*{2pt}\cr
1/\lambda_\gamma, & $\mbox{if } \xi\ge1/\lambda_\gamma.$}
\end{eqnarray*}\looseness=0
Hence, the increase in margin $y_i\hat{\gamma}_i$ due to inclusion of
$\gamma$ is given by
\begin{eqnarray*}
&&\{1-y_if(x_i)\}I\biggl(0< 1-y_if(x_i) < \frac{1}{\lambda_\gamma}\biggr)\\
&&\quad{}+
\frac{1}{\lambda_\gamma} I\biggl(1-y_if(x_i)\ge\frac{1}{\lambda_\gamma}\biggr).
\end{eqnarray*}
The $\gamma$-adjusted hinge loss is
$\mathcal{L}(y,f(x))=[1-1/\lambda_\gamma-yf(x)]_+$ with the hinge lowered
by $1/\lambda_\gamma$ as shown in Figure \ref{fig:modloss}(c)
(the dashed line).
The effective loss (the solid line in the figure)
is then given by a smooth function with
the joint replaced with a quadratic piece between $1-1/\lambda_\gamma$
and 1
and linear beyond the interval.
\section{Simulation Studies}
\label{sec:simulate}
We present results from various numerical experiments
to illustrate the effect of the proposed modification of modeling procedures
by regularization of case-specific parameters.
\subsection{Regression}
\subsubsection{\texorpdfstring{$\ell_2$-adjusted quantile regression}{l2-adjusted quantile regression}}
\label{subsec:qrsim}
The effectiveness of the $\ell_2$-adjusted quantile regression
depends on the penalty parameter $\lambda_\gamma$ in \eqref{eq:modcheck},
which yields $(-q/\lambda_\gamma, (1-q)/\lambda_\gamma)$
as the interval of quadratic adjustment.
We undertook extensive simulation studies (available in \citep{l2QR})
to establish guidelines for selection of the penalty
parameter~$\lambda_\gamma$ in the linear regression model setting. The studies encompassed
a range of sample sizes, from $10^{2}$ to $10^{4}$, a variety of
quantiles, from $0.1$ to $0.9$,
and distributions exhibiting symmetry, varying degrees of asymmetry,
and a variety of tail behaviors.
The modified quantile regression method was directly implemented by specifying
the effective $\psi$-function~$\psi_q^\gamma$,
the derivative of the effective loss, in the
\texttt{rlm} function in the R package.
An empirical rule was established via
a (robust) regression analysis. The analysis considered
$\lambda_\gamma$ of the form
$c_q n^\alpha/\hat{\sigma}$, where
$c_q$ is a constant depending on $q$ and $\hat{\sigma}$ is a robust
estimate of the scale of the error distribution.
The goal of the analysis was to find $\lambda_\gamma$ which, across a
broad range of conditions, resulted in an MSE near the condition-specific
minimum MSE.
Here MSE is defined as mean squared error of estimated regression
quantiles at a new $X$ integrated over the distribution of the covariates.
After initial examination of the MSE with a range of $\alpha$
values, we made a decision to set $\alpha$ to $0.3$ for good finite sample
performance across a wide range of conditions.
With fixed $\alpha$, we varied $c_q$ to obtain the smallest MSE
by grid search for each condition under consideration.
For a quick illustration, Figure \ref{fig:window} shows the intervals
of adjustment with such optimal $c_q$ for various error distributions,
$q$ values, and sample sizes.
Wider optimal intervals indicate that
more quadratic adjustment is preferred to the standard quantile
regression for reduction of MSE.
Clearly, Figure \ref{fig:window} demonstrates the benefit of the proposed
quadratic adjustment of quantile regression in terms of MSE across
a broad range of situations, especially when the sample size is small.
In general, MSE values begin to decrease as the size of adjustment increases
from zero and increase after hitting the minimum, due to an increase
in bias. There is an exception of this typical pattern when estimating
the median with normally distribut\-ed errors. MSE monotonically
decreases in this case as the interval of adjustment widens,
confirming the optimality properties of least
squares regression for normal theory regression.
The comparisons between sample mean and sample median can
be explicitly found under the $t$ error distributions using different
degrees of freedom. The benefit of the median relative to the mean is
greater for thicker tailed distributions.
We observe that this qualitative behavior carries over to the optimal
intervals. Thicker tails lead to shorter optimal intervals, as shown
in Figure \ref{fig:window}.
Modeling the optimal condition-specific $c_q$ as a~function of $q$
through a robust regression analysis led to the rule, with $\alpha
=0.30$, of
$c_q \approx0.5\exp(-2.118-1.097q)$ for $q<0.5$ and
$c_q \approx0.5\exp(-2.118-1.097(1-q))$ for $q \ge0.5$.
The simulation studies show that this choice of penalty parameter
results in an accurate estimator of the quantile surface.
\begin{figure*}
\includegraphics{377f04.eps}
\caption{``Optimal'' intervals of adjustment for different quantiles ($q$),
sample sizes ($n$), and error distributions.
The intervals range from the quantile minus $q/\lambda_{\gamma}$ to
the quantile plus $(1-q)/\lambda_{\gamma}$, with $\lambda_{\gamma}$
minimizing \textit{MSE}.
The vertical lines in each distribution
indicate the true quantiles. The stacked horizontal
lines for each quantile are corresponding optimal intervals.
Five intervals at each quantile are for $n=10^2$, $10^{2.5}$,
$10^3$, $10^{3.5}$ and $10^4$, respectively, from the bottom.}
\label{fig:window}
\end{figure*}
\begin{table*}[b]
\tabcolsep=0pt
\caption{Distribution of difference in the number of selected variables
for the fitted model to contaminated data from that
to clean data}\label{tab:simulation1}
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}l ccccccc c ccccccc@{}}
\hline
& \multicolumn{7}{c}{\textbf{LARS}} && \multicolumn{7}{c@{}}{\textbf{Robust LARS}}\\
\ccline{2-8,10-16
\textbf{Scenario} & $\bolds{-3}$ & $\bolds{-2}$ & $\bolds{-1}$ & \textbf{0} & \textbf{1} & \textbf{2} & \textbf{3} &
& $\bolds{-3}$ & $\bolds{-2}$ & $\bolds{-1}$ & \textbf{0} & \textbf{1} & \textbf{2} & \multicolumn{1}{c@{}}{\textbf{3}}\\
\hline
$\varepsilon$ contamination \\
Sparse & 5* & \phantom{0}6 & 21 & 48 & 13 & 5 & \phantom{0}2* &
& 1* & 4 & 12 & 71 & \phantom{0}7 & \phantom{0}5 & 0\\
Intermediate & 5\phantom{*} & 10 & 14 & 46 & 21 & 3 & 1 &
& 1\phantom{*} & 3 & 14 & 64 & 14 & \phantom{0}4 & 0 \\
Dense & 2\phantom{*} & \phantom{0}1 & 16 & 80 & \phantom{0}1 & 0 & 0 &
& 0\phantom{*} & 0 & \phantom{0}8 & 89 & \phantom{0}3 & \phantom{0}0 & 0 \\[3pt]
$X$ contamination \\
Sparse & 7* & \phantom{0}5 & 15 & 34 & 20 & 7 & 12* &
& 5* & 3 & 16 & 36 & 22 & 12 & 6 \\
Intermediate & 1* & \phantom{0}5 & 13 & 55 & 21 & 3 & 2 &
& 1\phantom{*} & 3 & 18 & 50 & 23 & \phantom{0}4 & 1\\
Dense & 0\phantom{*} & \phantom{0}0 & \phantom{0}5 & 93 & \phantom{0}2 & 0 & 0 &
& 0\phantom{*} & 0 & \phantom{0}4 & 94 &\phantom{0}2 & \phantom{0}0 & 0\\
\hline
\end{tabular*}
\tabnotetext[]{}{\textit{Note}:
The entries with * are the cumulative counts of the specified
case and more extreme cases.}
\end{table*}
\subsubsection{Robust LASSO}
We investigated the sensitivity of the LASSO
(or LARS) and its robust version (obtained by the
proposed $\ell_1$ modification)
to contamination of the data through simulation.
For the robust LASSO, the iterative algorithm in Section
\ref{sec:robustmodel} was implemented
by using LARS (Efron et~al., \citeyear{lars}) as the baseline modeling procedure
and winsorizing the residuals with $\lambda_{\gamma}$ as a bending
constant. The bending constant was taken to be scale invariant, so
that\vadjust{\goodbreak}
$\lambda_{\gamma} = k \hat{\sigma}$, where $k$ is a constant and
$\hat{\sigma}$ is a~robust scale estimate.
The standard robust statistics literature (\citep{huber})
suggests that good choices of $k$ lie in the range from $1$ to $2$.
For brevity, we report only that portion of the results pertaining to accuracy
of the fitted regression surface and inclusion of variates in the model
when $k=2$.
Similar results were obtained for $k$ near $2$. The results differ
for extreme values of $k$.
Throughout the simulation,
the standard linear model $y= x^\top\beta + \varepsilon$ was assumed.
Following the simulation setting in \citet{lasso},
we generated $x=(x_1,\ldots,x_8)^\top$ from a multivariate normal distribution
with mean zero and standard deviation 1.
The correlation between $x_i$ and $x_j$
was set to $\rho^{|i-j|}$ with $\rho=0.5$.
Three scenarios were considered with a varying degree of sparsity
in terms of the number of nonzero true coefficients:
(i)~sparse: $\beta=(5,0,0,0,0,0,0,0)$,
(ii) intermediate: $\beta=(3,1.5,0,0,2,0,0,0)$ and
(iii) dense: $\beta_j=0.85$ for all $j=1,\ldots,8$.
In all cases, the sample size was $100$.
For the base case, $\varepsilon_i$ was assumed to follow
$N(0,\sigma^2)$ with $\sigma=3$.
For potential outliers in~$\varepsilon$,
the first 5\%
of the $\varepsilon_i$'s were tripled, yielding a data set with more outliers.
We also investigated sensitivity to high leverage cases. For this setting,
we tripled the first 5\% of the values of $x_1$. Thus the replicates
were blocked across the three settings.
The $C_p$ criterion was used to select the model.
\begin{figure}
\includegraphics{377f05.eps}
\caption{Mean squared error (\textit{MSE}) of $\hat\beta$ for LARS and
its robust version under three different scenarios in the simulation study.
In each scenario, \textsf{o}, \textsf{e}, and \textsf{x} indicate
clean data, data with contaminated measurement errors, and
data with mismeasured first covariate.
The dotted lines are for LARS while
the solid lines are for robust LARS.
The points are the average \textit{MSE} for 100 replicates.}
\label{fig:mse.beta}\vspace*{3pt}
\end{figure}
Figure \ref{fig:mse.beta} shows mean squared error (MSE) between the
fitted and true regression surfaces, omitting intercepts.
MSE is integrated across the distribution of a
future $X$, taken to be that for the base case of the simulation. Over the
$m=100$ replicates in the simulation,
$\operatorname{MSE} = m^{-1} \sum_{i=1}^m (\hat{\beta}^i-\beta)^\top
\Sigma(\hat{\beta}^i-\beta)$,
where $\hat{\beta}^i$ is the estimate of $\beta$ for the $i$th
replicate, and $\Sigma$ is the covariance matrix of $X$.
LARS and robust LARS perform comparably in the base case, with the MSE
for robust LARS being greater by $1$ to $6$ percent.
For both LARS and robust LARS, MSE in the base case increases as one moves
from the sparse to the dense scenario. MSE increases noticeably when
$\varepsilon$ is contaminated, by a factor of $1.31$ to $1.41$ for LARS.
For robust LARS, the factor for increase over the base case with LARS is
$1.12$ to $1.22$.
For contamination in $X$, results under LARS and robust LARS are
similar in the
intermediate and dense cases, with increases in MSE over the base
case. For
the sparse case, the coefficient of the contaminated covariate, $x_1$,
is large
relative to the other covariates. Here, robust LARS performs noticeably better
than LARS, with a smaller increase in MSE.
Table \ref{tab:simulation1} presents results on the difference in
number of
selected variables for pairs of models. In each pair, a contaminated model
is contrasted with the corresponding uncontaminated model.
The top half of the table presents results for contamination of
$\varepsilon$.
The distribution of the differences in the number of selected variables
for the pairs of fitted models has a mode at $0$ in each scenario for both
LARS and robust LARS. There is, however,
substantial spread around $0$. The fitted models for the data
with contaminated errors tend to have fewer variables than those
for the original data, especially in the dense scenario. This may well be
attributed to inflated estimates of $\sigma^2$ used in $C_p$ for the
contaminated data, favoring relatively smaller models. The effect is
stronger for LARS than for robust LARS, in keeping with the lessened
impact of outliers on the robust estimate of $\sigma^2$.
The bottom half of Table \ref{tab:simulation1} presents results for
contamination of $X$. Again, the distributions of differences in model size
have modes at $0$ in all scenarios.
The distributions have substantial spread around
$0$. Under the sparse scenario in which the contamination has a substantial
impact on MSE, the distribution under robust LARS is more
concentrated than
under LARS.
The simulation demonstrates that the proposed robustification is
successful in dealing\vadjust{\goodbreak} with both contaminated errors and contaminated
covariates. As expected, in contrast to LARS, robust LARS
is effective in identifying observations with
large measurement errors and lessening their influence. It is also
effective at reducing the impact of high leverage cases, especially when
the high leverage arises from a covariate with a large regression coefficient.
The combined benefits of robustness to outliers and high leverage cases render
robust LARS effective at dealing with influential cases in an automated
fashion.
\subsection{Classification}
A three-part simulation study was carried out to examine the effect of the
proposed modification of loss functions for classification.
The primary focus is on (i)~the efficiency of the modified SVM relative
to the SVM with hinge loss
and its smoothed version with quadratically modified hinge loss,
and (ii) the robustness of logistic regression relative to modified
logistic regression (via the linearized deviance). The secondary
focus is on ensuring that robustness does not significantly degrade
as efficiency is improved, and that efficiency does not suffer
too much as robustness is improved.
All three parts of the simulation begin with $n=100$ cases
generated from a pair of five-dimensional multivariate normal
distributions, with identical covariance matrices and equal proportions
for two clas\-ses ($y=\pm1$). Without loss of generality,
the covariance matrices were taken to be the identity.
For the first part of the simulation, the separation between the
two classes is fixed. The separation is determined by the
difference in means of the multivariate normals, which, in turn,
determine the Bayes error rate for the underlying problem. Throughout,
once a~method was fit to the data (i.e., a discriminant function
was obtained), the error rate was calculated analytically. Each
part of the simulation consisted of $400$ replicates.
Six methods were considered in this study: LDA (linear discriminant
analysis) as a baseline method for the normal setting, the standard
SVM, its variant with squared hinge loss
(called Smooth SVM in \citep{leemangasarian}),
another variant with quadratically modified hinge loss
(referred to as Huberized SVM in \citep{rosset:zhu}),
logistic regression, and the method with linearized binomial deviance
(referred to as linearized LR in this study).
The Huberized SVM and linearized LR were implemented through
the fast Newton--Armijo algorithm proposed\vadjust{\goodbreak} for Smooth SVM
in \citet{leemangasarian}.
To focus on the effect of the loss functions on the classification
error rate,
no penalty was imposed on the parameters of discriminant functions.
For the first part of the study, the mean vectors were set with a
difference of
2.7 in the first coordinate and 0 elsewhere, yielding a Bayes
error rate of 8.851\%.
Figure \ref{fig:svmvshuberwithk} compares the SVM and its variants
in terms of the average excess error from the Bayes error rate.
The $k$ on the $x$-axis corresponds to
the bending constant, $1-1/\lambda_\gamma$ in the Huberized SVM.
When $k$ is as small as $-1$, we see that
quadratic modification in the Huberized SVM effectively yields the
same result as Smooth SVM. As $k$ tends to 1, the Huberized SVM
becomes the standard SVM. Clearly, there is a range of $k$ values
for which the mean error rate of the Huberized SVM is lower than that
of the standard SVM, demonstrating improved efficiency in
classification. In fact, the improved efficiency of smooth versions of the
hinge loss in the normal setting can be verified theoretically
for large sample cases, where the relative efficiency is defined as
the ratio of mean excess errors. See \citet{lee:wang} for details.
\begin{figure}
\includegraphics{377f06.eps}
\caption{Mean excess error of the SVM variant with quadratically
modified hinge loss (Huberized SVM) and the method with
linearized deviance loss (linearized LR)
as the bending constant $k$ varies.
The gray band indicates a one standard error bound
around the mean estimate for Huberized SVM from 400 replicates.
The standard error for comparison of the Huberized SVM to another method
varies, but is considerably smaller, due to the simulation design.
The horizontal lines from top to bottom are for SVM, logistic
regression and Smooth SVM, respectively.}\vspace*{-2pt}
\label{fig:svmvshuberwithk}
\end{figure}
\begin{figure*}
\centering
\begin{tabular}{@{}cc@{}}
\includegraphics{377f07a.eps}
& \includegraphics{377f07b.eps}\\
\footnotesize{(a)} & \footnotesize{(b)}
\end{tabular}
\caption{Mean excess errors of \textup{(a)} logistic regression and linearized LR,
and \textup{(b)} SVM and its variants, as the proportion of perturbation varies.
The gray band indicates a one standard error bound
around the mean estimate for \textup{(a)} linearized LR and \textup{(b)} Huberized SVM
from 400 replicates. The standard errors for comparisons are
considerably smaller than indicated by the bands.}
\label{fig:robustclass}
\end{figure*}
Figure \ref{fig:svmvshuberwithk} also displays a comparison
between logistic regression and the linearized LR of
Section \ref{sec:class}, with
bending constant
$k = \log\{(1-\lambda_\gamma)/\lambda_\gamma\}$.
There is no appreciable difference in the excess error between
logistic regression and its linearized version for negative\vadjust{\goodbreak} values of $k$.
Enhancing the robustness of logistic regression (shown in part two of the
study) sacrifices almost none of its efficiency.
The value of the bending constant $k$ leading to the minimum
error rate depends on the underlying problem itself,
and the range of best $k$ values may differ for the
Huberized SVM and linearized LR.
The results in Figure \ref{fig:svmvshuberwithk} suggest that
values of $k$ ranging from $-1$ to $0$ yield excellent
performance for both procedures in this setting.
The second part of the study focuses on robustness. To study this,
we perturbed each sample by flipping the
class labels of a certain proportion of cases selected at random,
and applied the six procedures to the perturbed sample. The
estimated discriminant rules were evaluated in the same
way as in the setting without perturbation.
Figure \ref{fig:robustclass}(a) highlights increased robustness of
linearized LR (with $k=-0.5$) compared to logistic regression when
some fraction of labels are flipped.
As the proportion of mislabeled data increases,
excess error rises for all of the procedures, including
the baseline method of LDA. However, the rate of increase in error
is slower for the modified logistic regression, as the linearized
deviance dampens the influence of mislabeled cases on
the discriminant rule.
Comparison of the SVM and its variants in the same setting
reveals a trade-off between efficiency and robustness.
Figure \ref{fig:robustclass}(b) shows that the squared
hinge loss yields a lower error rate than hinge loss
when the perturbation fraction is less than 6\%. The trend is reversed
when the fraction is higher than 6\%. This trade-off is\vadjust{\goodbreak}
reminiscent of that between the sample mean and median
as location parameter estimators. The Huberized SVM (with $k=-0.5$)
as a hybrid method strikes a balance between the two.
We note that the robustness of the SVM, compared with its variants,
is more visible when two
classes have less overlap (not shown here).
\begin{table*}
\tabcolsep=0pt
\caption{Mean error rates of classification methods under various
settings of mean difference and perturbation fraction.
The lowest error rates are in bold when compared among the
methods of the same type (either SVM or LR) for each scenario}\label{tab:error}
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}lccccccc@{}}
\hline
& & \multicolumn{2}{c}{\textbf{Huberized SVM}} &
& \multicolumn{2}{c}{\textbf{Linearized LR}} & \\
\ccline{3-4,6-7}
\textbf{Scenario} & \textbf{SVM}& $\bolds{k=-0.5}$ & $\bolds{k=-1}$ &\textbf{Smooth SVM} & $\bolds{k=-0.5}$ & $\bolds{k=-1}$ &\textbf{LR}\\
\hline
Easy
& 0.0385 & \textbf{0.0376} & \textbf{0.0376} & \textbf{0.0376}
& \textbf{0.0362} & 0.0363 & 0.0363 \\
Intermediate
&0.1028 & 0.1009 & \textbf{0.1008} & \textbf{0.1008}
& 0.1014 & \textbf{ 0.1013} & \textbf{0.1013} \\
Hard & 0.1753 & 0.1727 & \textbf{0.1726} & \textbf{0.1726}
& 0.1730 & 0.1729 & \textbf{0.1728}\\[3pt]
Easy${} + {}$5\% flip & \textbf{0.0348} & 0.0362 & 0.0371 & 0.0372
& \textbf{0.0383} & 0.0395 & 0.0411 \\
Intermediate${} + {}$5\% flip
& 0.1063 & \textbf{0.1050} & 0.1057 & 0.1059
& \textbf{0.1054} & 0.1061 & 0.1071\\
Hard${} + {}$5\% flip &0.1790 & \textbf{0.1769} & 0.1773 & 0.1774
& \textbf{0.1772} & 0.1773 & 0.1778 \\[3pt]
Easy${} + {}$10\% flip
& \textbf{0.0370} & 0.0415 & 0.0423 & 0.0421
& \textbf{0.0445} & 0.0465 & 0.0481 \\
Intermediate${} + {}$10\% flip
& \textbf{0.1107} & 0.1117 & 0.1127 & 0.1127
& \textbf{0.1125} & 0.1136 & 0.1150 \\
Hard${} + {}$10\% flip
& 0.1846 & \textbf{0.1833} & 0.1839 & 0.1840
& \textbf{0.1836} & 0.1841 & 0.1848 \\
\hline
\end{tabular*}
\end{table*}
The third part of the study provides a comprehensive comparison
of the methods.
Three scenarios with differing degree of difficulty were considered;
``easy,'' ``intermediate'' and ``hard'' settings refer to the multivariate
normal setting with the Bayes error rates of
2.275\%, 8.851\% and 15.866\%, respectively. In addition,
for scenarios with mislabeled cases, 5\% and 10\% of labels were flipped
under each of the three settings. Two values of the bending constant
($k=-0.5$ and $-1$) were used for the Huberized SVM and the linearized LR.
The results of comparison under nine scenarios are
summarized in Table \ref{tab:error}. The tabulated values are the mean
error rates of the discriminant rules under each method.
When there are no mislabeled cases, the smooth variants of the SVM
improve upon the performance of the standard SVM.
As the separation between classes increases, the reduction in
error due to modification of the hinge loss with fixed $k$
diminishes. Linearization of deviance in logistic regression does not
appear to affect the error rate.
In contrast, when there are mislabeled cases, linearization of
the deviance renders logistic regression more robust across all the
scenarios with differing class separations.
Similarly, the standard SVM is less sensitive to mislabeling than
its smooth variants. This makes the SVM more preferable as the
proportion of mislabeled cases increases.
However, in the difficult problem of little class
separation,
the quadratic modification in the Huberized SVM performs better
than the SVM.
\section{Applications}
\label{sec:apply}
\subsection{Analysis of the NHANES Data}
\label{subsec:nhanes}
We numerically compare standard quantile regression with modified
quantile regression for analysis of real data.
The Centers for Disease Control and Prevention conduct the
National Health and Nutrition Examination Survey (NHANES),
a large-scale survey designed to monitor the health and nutrition of
residents of the United States. Many are concerned about the record
levels of obesity in the population, and the survey contains
information on height and weight of individuals, in addition to
a variety of dietary and health-related questions. Obesity is
defined through body mass index (BMI) in $\mathrm{kg}/\mathrm{m}^2$, a measure which adjusts
weight for height. In this analysis,
we describe the relationship between height and BMI
among the $5938$ males over the age of $18$ in the aggregated
NHANES data sets from 1999, 2001, 2003 and 2005. Our analyses
do not adjust for NHANES' complex survey design. In particular,
no adjustment has been made for oversampling of selected groups
or nonresponse.
Since BMI is weight adjusted for height, the null expectation
is that BMI and height are unrelated.
We fit a nonparametric quantile regression model to the data.
The model is a six-knot regression spline using the natural basis expansion.
The knots (held constant across quantiles) were chosen by eye.
The rule for\vadjust{\goodbreak} selection of the penalty parameter $\lambda_\gamma$
described in Section \ref{subsec:qrsim}
was used for the NHANES data analysis.
Figure \ref{fig:NHANES} displays the fits from standard (QR)
and modified (QR.M) quantile regressions for the quantiles between $0.1$
and $0.9$ in steps of $0.05$.
The fitted curves show a slight
upward trend, some curvature overall, and mildly increasing spread
as height increases. There is a noticeable bump upward in the
distribution of BMI for heights near $1.73$ meters.
The differences between the two methods of fitting the quantile
regressions are most apparent in the tails, for example the $0.6$th
and $0.85$th quantiles for large heights.
\begin{figure*}[t]
\includegraphics{377f08.eps}
\caption{Regression spline estimates of conditional BMI quantiles in steps
of $0.05$, from $0.1$ to $0.9$ for the NHANES data. Natural spline
bases and six knots are used in each fitted curve.} \label{fig:NHANES}
\end{figure*}
\begin{figure*}[b]
\includegraphics{377f09.eps}
\caption{Scatterplots of 10-fold CV scores from standard quantile
regression (QR) and modified quantile regression (QR.M) at $0.25$th,
$0.5$th and $0.9$th quantiles.
Regression splines with natural
spline bases and six knots are fitted to the NHANES data. Each of 500
points represents a pair of CV scores as in (\protect\ref{eq:cv}).}
\label{fig:NHANES_MSE_male_df7}
\end{figure*}
The predictive performances of the standard and modified
quantile regressions are compared in
Figure~\ref{fig:NHANES_MSE_male_df7}. To compare the methods,
$10$-fold cross-valida\-tion was repeated $500$ times for different
splits of
the data. Each time, a cross-validation score was computed as
\begin{equation}
\label{eq:cv} \mathrm{CV}=
\frac{1}{n}\sum_{i=1}^{n}\rho_q(y_i-\hat{y_i}),
\end{equation}
where $y_i$ is the observed BMI for an individual in the
hold-out sample, $\hat{y}_i$ is the fitted value under QR or QR.M,
and the sum runs over the hold-out sample. The figure contains
plots of the $500$ $\mathrm{CV}$ scores. The great majority of $\mathrm{CV}$ scores are
to the lower right side of the 45 degree line, indicating that
the modified quantile regression outperforms the standard method---even
when the QR empirical risk function is used to evaluate performance.
Mean and 1000 times standard deviation of the $\mathrm{CV}$ scores for the
methods are
summarized in Table~\ref{tab:NHANES}.
The pattern shown in these panels
is consistent across other quantiles (not shown here). The pattern
becomes a bit stronger when the QR.M empirical risk function is
used to evaluate performance.
Quantile regression has the property that
$100\cdot q$\% of the responses fall at or below the fitted $q$th quantile
surface. This does not have to hold for the modified quantile
regression fit.
However, as the cross-validation shows, QR.M does provide a
better quantile regression surface than QR.
\begin{figure*}[b]
\includegraphics{377f10.eps}
\caption{Differences between fitted median line and the other fitted
quantiles for standard quantile regression (QR), modified quantile
regression (QR.M), and $\ell_1$ penalized quantile regression (QR.L1)
for the NHANES data.
The dashed lines are the minimum and maximum of the observed heights.}
\label{fig:NHANES_diff}
\end{figure*}
Modified quantile regression has an additional advantage which
is apparent for small and large heights. The standard quantile
regression fits show several crossings of estimated quantiles,
while crossing behavior
is reduced considerably with modified quantile regression.
Crossed quantiles correspond to\break a~claim that a lower quantile
lies above a higher quantile, contradicting the laws of probability.
Figure \ref{fig:NHANES_diff} shows this behavior.
Fixes for this behavior have been proposed (e.g., \citep{He:1997}), but
we consider it desirable to lessen crossing without any explicit
fix. The reduction in crossing holds up across other data sets that we have
examined and with regression models that differ in their details.
\begin{table}
\caption{Mean and (1000 times standard deviation) of CV
scores at selected quantiles based on 500 replicates from NHANES data}\label{tab:NHANES}
\begin{tabular*}{\columnwidth}{@{\extracolsep{\fill}}lccc@{}}
\hline
\textbf{Method} & $\bolds{q=0.25}$ & $\bolds{q=0.5}$ & \multicolumn{1}{c@{}}{$\bolds{q=0.9}$} \\
\hline
QR & 1.5040 (0.6105) & 2.0405 (0.7272) & 1.1267 (1.0714)\\
QR.M & 1.5039 (0.5855) & 2.0402 (0.6576) & 1.1263 (1.0030)\\
QR.L1 & 1.5039 (0.8963) & 2.0393 (0.5140) & 1.1289 (0.8569)\\
\hline
\end{tabular*}
\end{table}
In addition, we compare both methods with $\ell_1$ parameter-penalized quantile
regression (QR.L1),\break
where the estimator $\hat{\beta}$ is defined as the minimizer~of
\[
\sum_{i=1}^n\rho_q(y_i-x_i^\top\beta) + \lambda_{\beta}\sum
_{j=1}^p|\beta_j|.
\]
The \texttt{rq.fit.lasso} function in the \texttt{quantreg} R package was
used for QR.L1.\vadjust{\goodbreak}
Keeping the same split of data into 90\% of training and 10\% of
testing for each replicate, we have chosen $\lambda_\beta$
among 100 candidate values by 9-fold cross-validation.
The results are in Table \ref{tab:NHANES}.
The effect of parameter penalization differs from modification of
the loss function. Figure \ref{fig:NHANES_diff} illustrates the difference.
The quantiles estimated under QR.L1 (with $\lambda_\beta$ chosen by
10-fold cross-validation) show less variation across $x$
relative to the fitted median line, due to the shrinkage of each $\beta_j$
toward 0. This effect is more visible for large quantiles.
Such nondifferential penalty can degrade performance, unless
the parameters are of comparable size.
This adverse effect is numerically evidenced in the large $\mathrm{CV}$ score of
QR.L1 for $q=0.9$ in Table \ref{tab:NHANES}.
For $q=0.25$ and $0.5$, QR.L1 yields similar results to the other two
methods in terms of the $\mathrm{CV}$ scores.
\subsection{Analysis of Language Data}
\label{subsec:langdata}
\citet{balota04} conducted an extensive lexical decision experiment
in which subjects were asked to identify whether a string of letters was
an English word or a nonword. The words were monosyllabic, and the
nonwords were constructed to closely resemble words on a number of
linguistic dimensions. Two groups were studied---college students and
older adults. The data consist of response times by word, averaged over
the thirty subjects in each group. For each word, a number of
covariates was recorded. Goals\vadjust{\goodbreak} of the experiment include determining
which features of a word (i.e., covariates) affect response time, and
whether the active features affect response time in the same fashion
for college students and older adults. The authors make a case for the
need to conduct and analyze studies with regression techniques in mind,
rather than simpler ANOVA techniques.
\citet{baayen} conducted an extensive analysis of a slightly modified data
set which is available in his \texttt{languageR} package. In his
analysis, he creates and selects variables to include in a regression
model, addresses issues of nonlinearity, collinearity and interaction,
and removes selected cases as being influential and/or outlying. He
trims a total of $87$ of the $4568$ cases. The resulting model, based
on ``typical'' words, is used to address issues of linguistic importance.
It includes seventeen basic covariates which enter the model as linear
terms, a nonlinear term for the written frequency of a word (fit as a
restricted cubic spline with five knots), and an interaction term between
the age group and the (nonlinear) written frequency of the word.
We consider two sets of potential covariates for the model. The small set
consists of Baayen's $17$ basic covariates and three additional covariates
representing a squared term for written frequency and the interaction
between age group and the linear and squared terms for written frequency.
Age group has been coded as $\pm1$ for the interactions. The large set
augments these covariates with nine additional covariates that were not
included in Baayen's final model. Baayen excluded some of these covariates
for a lack of significance, others because of collinearity.
To investigate the performance of the LASSO and robust LASSO, a simulation
study was conducted on the $4568$ cases in the data set. For a single
replicate in the simulation, the data were partitioned into a training
data set and a test data set. The various methods were fit to the training
data, with evaluation conducted on the test data. The criteria for
evaluation were sum of squared differences between the fitted and observed
responses, either over all cases in the test data or over the test data
with the cases identified by Baayen as outliers removed. We refer to these
criteria as predictive mean squared error ($\mathrm{PMSE}$).
The simulation investigated several factors, including the amount of
training data (10\% of the full data, 20\%, 30\%, etc.), the regularization
parameter $\lambda_{\gamma} = k \hat{\sigma}$, and
the method used to select the model.
Three methods were used to
select the model (i.e., the fraction\vadjust{\goodbreak} of the distance along the solution
path): minimum $C_p$, generalized cross-validation,
and 10-fold cross-validation on the training data.
The results of a $300$ replicate simulation show a~convincing benefit to
use of the robust LASSO. The benefit of the robust LASSO is most apparent
when~$k$ is in the ``sweet spot'' ranging from $1.4$ or so to well
above $2.0$. As
expected, for very small $k$ (near~1), the robust LASSO may not perform
as well as the LASSO. The reduction in $\mathrm{PMSE}$ for moderate values of $k$,
both absolute and percent, is slightly larger when the evaluation is
conducted after outliers (as identified by Baayen---not by the fitted model)
have been dropped from the test data set. The benefit is largest for small
training data sets and decreases as the size of the training data set
increases. For large training data sets (e.g., 90\% of the data), little
test data remains for calculation of $\mathrm{PMSE}$ and the evaluation is less
stable. These patterns were apparent over all three methods of model
selection. Figure~\ref{fig:LangSimPMSE}
shows the results for a training sample size of $1827$
cases (40\% of the data), with model selected by cross-validation, for a
variety of values of $k$. The $\mathrm{PMSE}$ for the robust LASSO dips below the
mean $\mathrm{PMSE}$ for the LASSO for a wide range of $k$. The figure also presents
95\% confidence intervals, based on the $300$ replicates in the simulation,
for the difference between mean $\mathrm{PMSE}$ under the robust LASSO and the LASSO.
The intervals are indicated by the vertical lines, and statistical
significance is indicated where the lines do not overlap the mean
$\mathrm{PMSE}$ under the LASSO. The narrowing of the intervals is a consequence
of the greater similarity of LASSO and robust LASSO fits as the bending
constant increases.
The patterns just described hold for both the small set of
covariates and the large set
of covariates.
\begin{figure*}
\centering
\begin{tabular}{@{}cc@{}}
\includegraphics{377f11a.eps}
& \includegraphics{377f11b.eps}\\
\footnotesize{(a)} & \footnotesize{(b)}
\end{tabular}
\caption{Predictive mean squared error (\textit{PMSE}) for the test data in the
simulation study, after removal of cases identified by Baayen
as outliers.
The horizontal
line is the mean \textit{PMSE} for the LASSO while the points represent
the mean of \textit{PMSE}s for the robust LASSO. The vertical lines have
the width of
approximate 95\% confidence intervals for the difference
in mean \textit{PMSE} under the LASSO and robust LASSO.
Panel \textup{(a)} presents results
for the small set of covariates and panel \textup{(b)} presents results for
the large set of covariates.}
\label{fig:LangSimPMSE}
\end{figure*}
\begin{figure*}[b]
\centering
\begin{tabular}{@{}cc@{}}
\includegraphics{377f12a.eps}
& \includegraphics{377f12b.eps}\\
\footnotesize{(a)} & \footnotesize{(b)}
\end{tabular}
\caption{Sum of squared deviations (\textit{SSD}) from Baayen's fits in the simulation
study. The horizontal
line is the mean \textit{SSD} for the LASSO while the points represent
the mean of SSDs for the robust LASSO. The vertical lines have
the width of
approximate 95\% confidence intervals for the difference in mean \textit{SSD}
under the LASSO and robust LASSO.
Panel \textup{(a)} presents results
for the small set of covariates and panel \textup{(b)} presents results for
the large set of covariates.}
\label{fig:LangSimSSD}
\end{figure*}
In addition to using the test data as a target, we studied how well the
two methods could reproduce Baayen's expert fit. This makes a good target
for inference, as there is evidence that humans can produce a better fit
than automated methods (\citep{Yu:AOAS2011}). Taking a fitted surface as a
target allows us to remove the noise inherent in data-based out-of-sample
evaluations.\break The results from a $5000$ replicate simulation study with a
training sample size of $400$ appear in Figure~\ref{fig:LangSimSSD}.
The criterion is sum
of squared deviations ($\mathrm{SSD}$) between the (robust) LASSO fit and Baayen's
fit, with the sum taken over only those covariate values contained in the
test data set. The results presented here are for models
selected with the minimum $C_p$ criterion.
The robust LASSO outperforms the LASSO over a wide range of values for $k$
for both the small and large sets of covariates.
Figures \ref{fig:LangSimPMSE} and \ref{fig:LangSimSSD} reveal an
interesting difference across targets in the behavior of the
small and large sets of covariates. When the target is an
expert fit, as in the second study, adding covariates not present
in the expert's model to the pool of potential covariates
allows the LASSO and robust LASSO to produce a near-equivalent fit
to the data, but with different coefficients for the regressors.
An examination of the variables present in the fitted models and
their coefficients
uncovers patterns. As an example, the two covariates
``WrittenFrequency'' and ``Familiarity'' appear in nearly all
of the models for both the LASSO and the robust LASSO, while
Baayen includes only ``WrittenFrequency'' in his model, and these
covariate(s) have negative coefficients.
Subjects are able to decide that a familiar
word is a~word more quickly (and more accurately) than an
unfamiliar word. Although there seems to be no debate on
whether this conceptual effect of similarity exists, there
are a variety of viewpoints on how to best capture the effect.
Regularization methods allow one to include a suite of
covariates to address a single conceptual effect, and this
produces a difference between the LASSO and robust LASSO fits
on one hand and a least-squares, variable-selection style fit
on the other hand. The end result is that the regularized
fits with the large set
of covariates show greater departures from Baayen's fit than
do regularized fits with the small set of covariates.
In contrast, under the data-based target
of the first study, the large set of covariates results in a
smaller $\mathrm{PMSE}$.
\section{Discussion}
\label{sec:disc}
In the preceding sections, we have laid out an approach to
modifying modeling procedures. The approach is based
on the creation of case-specific covariates which are then
regularized. With appropriate choices of
penalty terms, the addition of these covariates allows us to
robustify those procedures which lack robustness and also allows
us to improve the efficiency of procedures which are very robust,
but not particularly efficient. The method is fully compatible
with regularized estimation methods. In this case, the case-specific
covariates are merely included as part of the regularization.
The techniques are easy to
implement, as they often require little modification of existing
software. In some cases, there is no need for modification of
software, as one merely feeds a modified data set into
existing routines.
The motivation behind this work is a desire to move relatively
automated modeling procedures
in the direction of traditional data analysis
(e.g., Weisberg, \citeyear{weisberg:lars}).
An important component of this type of analysis
is the ability to take different looks at a~data set. These
different looks may suggest creation of new variates
and differential handling of individual cases or groups of cases.
Robust methods allow us to take such a look, even when data sets
are large. Coupling robust regression techniques with the ability
to examine an entire solution path provides a~sharper view of the
impact of unusual cases on the analysis.
A second motivation for the work is the desire to improve robust,
relatively nonparametric methods. This is accomplished by introducing
case-specific parameters in a controlled fashion whereby the finite sample
performance of estimators is improved.
The perspective provided by this work suggests several directions for
future research. Adaptive penalties, whether used for robustness or
efficiency, can be designed to satisfy specified invariances.
The asymmetric $\ell_2$ penalty for modified quantile regression was
designed to satisfy a specified invariance. For a locally constant
residual density, it keeps the $0$ of the $\psi_q^\gamma$ function
invariant as the width of the interval of adjustment varies.
Specific, alternative forms of invariance for quantile regression are
suggested by\vadjust{\goodbreak} consideration of parametric driving forms for the residual
distribution. A motivating parametric model, coupled with invariance
of the $0$ of the $\psi_q^\gamma$ function to the size of the penalty
term $\lambda_\gamma$, yields a~path of penalties. Increasing the size
of the covari\-ate-specific penalty at an appropriate rate leads to
asymptotic equivalence with the quantile regression estimator.
This allows one to fit the model nonparametrically
while tapping into an approximate parametric form
to enhance finite sample performance. Similarly, when case-specific
penalties are applied to a model such as the generalized linear model,
the asymmetry of the likelihood, coupled with invariance, suggests an
asymmetric form for the $\ell_1$ penalty used to enhance robustness
of inference.
Following development of the technique for quantile regression, one can
apply the adaptive loss para\-digm for model assessment and selection.
For example, in cross-validation, a summary of a model's fit is computed
as an out-of-sample estimate of empirical risk and the evaluation is
used for choosing the model (parameter value) with the smallest estimated
risk. For model averaging, estimated risks are converted into weights which
are then attached to model-specific predictions that are then combined
to yield an overall prediction. The use of modified loss functions for
estimation of risks is expected to improve stability and efficiency in
model evaluation and selection.\vspace*{-1pt}
\begin{appendix}
\section*{Appendix}\label{app}
Proof of Theorem \ref{thm:reg_qtl}.
Let $u_i:=y_i-x_i^\top\beta(q)$ and consider the objective function
\renewcommand{\theequation}{\arabic{equation}}
\setcounter{equation}{12}
\begin{equation}
\quad Z_n^\gamma(\delta):=
\sum^{n}_{i=1}\bigl\{\rho_q^\gamma\bigl(u_i-x_i^\top\delta/\sqrt{n}\bigr)-\rho_q(u_i)\bigr\}.
\end{equation}
Note that $Z_n^\gamma(\delta)$ is minimized at $\hat\delta_n:=
\sqrt{n}(\hat{\beta}^\gamma_{n}-\beta(q))$, and
the limiting distribution of $\hat\delta_n$ is determined by
the limiting behavior of $Z_n^\gamma(\delta)$. To study the limit
of $Z_n^\gamma(\delta)$, decompose $Z_n^\gamma(\delta)$ as
\begin{eqnarray*}
Z_n^\gamma(\delta)& = &
\sum^{n}_{i=1}\bigl\{\rho_q^\gamma\bigl(u_i-x_i^\top\delta/\sqrt{n}\bigr)-
\rho_q\bigl(u_i-x_i^\top\delta/\sqrt{n}\bigr)\bigr\}
\\
&&{}+\sum^{n}_{i=1}\bigl\{\rho_q\bigl(u_i-x_i^\top\delta/\sqrt{n}\bigr)-\rho_q(u_i)\bigr\}\\
&=&\sum^{n}_{i=1}\bigl\{\rho_q^\gamma\bigl(u_i-x_i^\top\delta/\sqrt{n}\bigr)-
\rho_q\bigl(u_i-x_i^\top\delta/\sqrt{n}\bigr)\bigr\}\\
&&{}+Z_n(\delta),
\end{eqnarray*}
where $Z_n(\delta)
:=\sum^{n}_{i=1}\{\rho_q(u_i-x_i^\top\delta/\sqrt{n})-\rho_q(u_i)\}$.
By showing that the first sum converges to zero in\vadjust{\goodbreak} probability
up to a sequence of constants that do not depend on $\delta$,
we will establish the asymptotic equivalence
of $\hat{\beta}^\gamma_{n}$ to $\hat{\beta}_{n}$.
Given $\lambda_\gamma=cn^\alpha$, first observe that
\begin{eqnarray*}
&&\hspace*{-5pt}E \bigl\{ \rho_q^\gamma\bigl(u_i-x_i^\top\delta/\sqrt{n}\bigr)-
\rho_q\bigl(u_i-x_i^\top\delta/\sqrt{n}\bigr)\bigr\}\\
&&\hspace*{-5pt}\qquad{}+ q(1-q)/2\lambda_\gamma\\
&&\hspace*{-5pt}\quad=\int^{{(1-q)}/{\lambda_\gamma}+x_i^\top\delta/\sqrt{n}}_{x_i^\top
\delta/\sqrt{n}}
\biggl(\frac{\lambda_\gamma}{2} \frac{q}{1-q}
\biggl(u-\frac{x_i^\top\delta}{\sqrt{n}}\biggr)^2
\\
&&\hspace*{82pt}\qquad{}-q\biggl(u-\frac{x_i^\top\delta}{\sqrt{n}}\biggr
+ \frac{q(1-q)}{2\lambda_\gamma}\biggr)\\
&&\hspace*{101pt}\cdot f_i(\xi_i+u)\,du\\
&&\hspace*{-5pt}\qquad{}+\int^{x_i^\top\delta/\sqrt{n}}_{-q/{\lambda_\gamma}+x_i^\top\delta
/\sqrt{n}}
\biggl(\frac{\lambda_\gamma}{2} \frac{1-q}{q}
\biggl(u-\frac{x_i^\top\delta}{\sqrt{n}}\biggr)^2
\\
&&\hspace*{79pt}\qquad{}-(q-1)\biggl(u-\frac{x_i^\top\delta}{\sqrt{n}}\biggr)
\\
&&\hspace*{128pt}\qquad{}
+ \frac{q(1-q)}{2\lambda_\gamma}\biggr)\\
&&\hspace*{96pt}{}\cdot f_i(\xi_i+u)\,du\\
&&\quad=\int^{{(1-q)}/{\lambda_\gamma}+x_i^\top\delta/\sqrt{n}}_{x_i^\top
\delta/\sqrt{n}}
\frac{\lambda_\gamma}{2} \frac{q}{1-q}\\
&&\hspace*{79pt}\qquad{}\cdot\biggl(u-\frac{x_i^\top\delta}{\sqrt{n}}-\frac{1-q}{\lambda_\gamma}\biggr)^2\\
&&\hspace*{79pt}\qquad{}\cdot f_i(\xi_i+u)\,du\\
&&\qquad{}+\int^{x_i^\top\delta/\sqrt{n}}_{-q/{\lambda_\gamma}+x_i^\top\delta
/\sqrt{n}}
\frac{\lambda_\gamma}{2} \frac{1-q}{q}
\biggl(u-\frac{x_i^\top\delta}{\sqrt{n}}+\frac{q}{\lambda_\gamma}\biggr)^2
\\
&&\hspace*{79pt}\qquad{}\cdot f_i(\xi_i+u)\,du.
\end{eqnarray*}
Using a first-order Taylor expansion of $f_i$ at $\xi_i$ from the
condition (C-2) and the expression above, we have
\begin{eqnarray*}
&&E \sum^{n}_{i=1}\bigl\{\rho_q^\gamma\bigl(u_i-x_i^\top\delta/\sqrt{n}\bigr)-
\rho_q\bigl(u_i-x_i^\top\delta/\sqrt{n}\bigr)\bigr\} \\
&&\qquad{}+ nq(1-q)/2\lambda_\gamma\\
&&\quad= \frac{q(1-q)}{6c^2n^{2\alpha}}\sum^{n}_{i=1}f_i(\xi_i) +
\frac{q(1-q)}{6c^2n^{2\alpha}}
\sum^{n}_{i=1}\frac{f'_i(\xi_i)x_i^\top\delta}{\sqrt{n}}
\\
&&\qquad{}+o(n^{-2\alpha+1/2}).
\end{eqnarray*}
Note that $\sum^{n}_{i=1}f'_i(\xi_i)x_i^\top\delta/\sqrt{n}=O(\sqrt
{n})$ as
$f'_i(\xi_i)$, $i=1,\ldots,n$, are uniformly bounded from the
condition (C-2), and
$|x_i^\top\delta| \le\|x_i\|_2\|\delta\|_2\le
(\|x_i\|_2^2+\|\delta\|_2^2)/2$ while
$\sum^{n}_{i=1} \|x_i\|_2^2=O(n)$ from\vadjust{\goodbreak} the condition \mbox{(C-3)}.
Taking
$C_n:= -{q(1-q)}/ {(2cn^{\alpha-1})}+
{q(1-q)}/\break {(6c^2n^{2\alpha})}\sum^{n}_{i=1}f_i(\xi_i)$,
we have that
\begin{eqnarray*}
&&E \sum^{n}_{i=1}\bigl\{\rho_q^\gamma\bigl(u_i-x_i^\top\delta/\sqrt{n}\bigr)-
\rho_q\bigl(u_i-x_i^\top\delta/\sqrt{n}\bigr)\bigr\}-C_n \\
&&\quad{}\rightarrow0 \quad\mbox{if }
\alpha> 1/4.
\end{eqnarray*}
Similarly, it can be shown that
\begin{eqnarray*}
&&\operatorname{Var} \sum^{n}_{i=1}\bigl\{\rho_q^\gamma\bigl(u_i-x_i^\top\delta/\sqrt{n}\bigr)-
\rho_q\bigl(u_i-x_i^\top\delta/\sqrt{n}\bigr)\bigr\}\\
&&\quad= \sum^{n}_{i=1}\frac{q^2(1-q)^2f_i(\xi_i)}{20c^3n^{3\alpha}}
+o(n^{-3\alpha+1})\\
&&\qquad\rightarrow0 \quad\mbox{if } \alpha> 1/3.
\end{eqnarray*}
Thus, if $\alpha$ $>1/3$,
\begin{eqnarray*}
&&\sum^{n}_{i=1}\rho_q^\gamma\bigl(u_i-x_i^\top\delta/\sqrt{n}\bigr)-
\rho_q\bigl(u_i-x_i^\top\delta/\sqrt{n}\bigr) - C_n \\
&&\quad\rightarrow0
\quad\mbox{in probability}.
\end{eqnarray*}
This implies that the limiting behavior of $Z_n^\gamma(\delta)$ is
the same as
that of $Z_n(\delta)$. From the proof of Theorem 4.1 in \citet{Koenker:2005},
$Z_n(\delta) \stackrel d \rightarrow-\delta^\top W
+\frac{1}{2}\delta^\top D_1\delta$,
where $W$ $\sim N(0,q(1-q)D_0)$. By the convexity argument in \citet
{Koenker:2005}
(see also \citep{pollard:1991}; \citep{hjort:pollard}; \citep{Knight:1998}),
$\hat\delta_n$,
the minimizer of $Z_n^\gamma(\delta)$,
converges to $\hat\delta_0:=D_1^{-1}W$, the unique minimizer of
$-\delta^\top W+\frac{1}{2}\delta^\top D_1\delta$ in distribution.
This completes the proof.
\end{appendix}
\section*{Acknowledgments}
We thank the Editor, Associate Editor and referees for their
thoughtful comments which helped us improve the presentation of this paper.
This research was supported in part by NSA Grant H98230-10-1-0202.
|
1,116,691,499,278 | arxiv |
\section{Introduction}
The basic structure of an accepting network of evolutionary processors (ANEP for short) is
widely met in distributed, parallel and evolutionary computing: a virtual undirected graph whose
nodes are processors handling some data. All node processors act simultaneously on the local data
in accordance with some predefined rules, and then local data becomes a mobile agent which can
navigate in the network following a given protocol.
All the nodes send simultaneously their data and the receiving nodes handle also simultaneously all the arriving messages. Only the data able to pass a filtering process can be communicated to the other processors. This filtering process may require the data to satisfy some conditions imposed by the sending processor, by the receiving processor or by both of them.
In a series of papers starting with \cite{gnep} (for the generating variants)
and \cite{dna10} (for the accepting variants) this general structure is particularized in a
bio-inspired way: each node may be viewed as a cell having genetic information encoded in DNA sequences which may evolve by very simple evolutionary events, that is point mutations. Each node is specialized in just one of these evolutionary operations. Furthermore, the data in each node is organized in the form of multisets of words, each word appearing in an arbitrarily large number of copies, and all the copies are processed in a massive parallel manner, such that all the possible events that can take place do actually take place.
Furthermore, the filtering process is based on simple ``random-context" conditions, namely the presence/absence of some symbols. Clearly, the biological hints presented above are intended to explain in an informal way how some biological phenomena are {\it sources of inspiration} for the mathematical computing model.
In \cite{dna10} one presents a characterization of the complexity class {\bf NP} based on ANEPs. The work \cite{mscs} discusses how ANEPs can be considered as problem solvers. In \cite{ipl}, one shows that every recursively enumerable language can be accepted by an ANEP with $24$ nodes. Moreover, this construction proves that for every {\bf NP}-language there exists an ANEP of size $24$ deciding that language in polynomial time. While the number of nodes of this ANEP does not depend on the language, the other parameters of the network (rules, symbols, filters) depend on it. This result may be also interpreted as a method for solving every {\bf NP}-problem in polynomial time by ANEPs of constant size.
All the aforementioned results were obtained via simulations of Turing machines by ANEPs.
It is expected that having filters associated with each node, as in the case of ANEPs, allows a strong control of the computation. Indeed, every node
has an associated input and output filter; two nodes can exchange data if it passes the output filter of the sender {\it and} the input filter of the receiver. Moreover, if some data is sent out by a node and not able to enter any other node, then it is lost. In \cite{jucsTOM} the ANEP model considered in \cite{dna10} was simplified by moving the filters from the nodes to the edges. Each edge was viewed as a two-way channel such that the input and output filters, respectively, of the two nodes connected by the edge coincide; now two nodes can exchange data if it passes the filters of the edge existing between the two nodes. Clearly, the possibility of controlling the computation in such networks seemed to be diminished. For instance, there was no possibility to loose data during the communication steps. However, in \cite{jucsTOM}
(a simplified proof can be found in \cite{{ijfcsDM}}) one proves that these newly introduced
devices, called accepting networks of evolutionary processors with filtered connections (ANEPFCs for short), were still computationally complete. Furthermore,
in \cite{ijfcsDM} it is constructed an universal ANEPFC with $18$ nodes, and it is shown that every recursively enumerable language $L$ is recognized by an ANEPFC with $27$ processors, having the property that only $7$ of its nodes depend on the language $L$, while the others remain unchanged regardless the accepted language; moreover, any nondeterministic Turing machine can be simulated efficiently (with respect to time and space complexity) by an ANEPFC with $26$ processors. All these results were also based on
simulations of Turing machines by ANEPFCs.
Here we aim to improve the results reported in \cite{ijfcsDM}. More precisely, we first look for a universal ANEPFC with a smaller number of nodes. To this end, we propose a simulation of 2-tag systems introduced in \cite{minsky}. Second, we are interested in finding a way to design ANEPFCs with less than $26$ nodes that simulate computationally efficient nondeterministic Turing machines.
Note that the universal ANEPFC obtained
from the simulation of a tag system does not solve the second problem as a $2$-tag system can efficiently simulate any deterministic Turing machine but not nondeterministic ones.
Based on a similar idea to that used in the simulation of tag systems, we propose a simulation of nondeterministic Turing machines with ANEPFCs of size $16$ which maintain the working time of the Turing machine. That is, every language accepted by a nondeterministic Turing machine in time $f(n)$ can be accepted by an ANEPFC of size $16$ in time $O(f(n))$. Consequently, the class ${\bf NP}$ equals the class of languages accepted in polynomial time by ANEPFCs of size $16$. This result considerably improves the known bound of $26$ reported in \cite{ijfcsDM}.
\section{Basic definitions}
We start by summarizing the notions used throughout the paper; for all unexplained notions
the reader is referred to \cite{handbook}. An
{\it alphabet} is a finite and nonempty set of symbols. The
cardinality of a finite set $A$ is written $card(A)$. Any sequence
of symbols from an alphabet $V$ is called {\it word (string)} over
$V$. The set of all words over $V$ is denoted by $V^*$ and the
empty word is denoted by $\varepsilon$. The length of a word $x$ is
denoted by $|x|$ while $alph(x)$ denotes the minimal alphabet $W$ such
that $x\in W^*$. For a word $x\in W^*$, $x^r$ denotes the reversal of the word.
We consider here the following definition of $2$-tag systems that appears in~\cite{rogo}.
It is slightly different but equivalent to those from \cite{post,minsky}. A $2$-tag system
\hbox{$T=(V,\phi)$} consists of a finite alphabet of symbols~$V$, containing a special {\it halting symbol} $H$ and a finite set of rules $\phi:V\setminus\{H\} \rightarrow V^+$ such that $|\phi(x)|\geq 2$ or $\phi(x)=H$.
Furthermore, $\phi(x)=H$ for just one $x\in V\setminus\{H\}$.
A halting word for the system $T$ is a word that contains the halting symbol $H$ or whose length is less than $2$; the transformation $t_T$ (called the tag operation) is defined on the set of non-halting words as follows: if $x$ is the leftmost symbol of a non-halting word $w$, then $t_T(w)$ is the result of deleting the leftmost $2$ symbols of $w$ and then appending the word $\phi(x)$ at the right end of the obtained word.
A computation by a $2$-tag system as above is a finite sequence of words produced by iterating the transformation $t$, starting with an initially given non-halting word $w$ and halting when a halting word is produced. Note that a computation is not considered to exist unless a halting word is produced in finitely-many iterations. We recall that such restricted $2$-tag systems are universal \cite{rogo}.
A nondeterministic Turing machine is a construct $M=(Q,$ $V,$ $U,$
$\delta ,$ $q_0,$ $B,$ $F)$, where $Q$ is a finite set of states,
$V$ is the input alphabet, $U$ is the tape alphabet, $V \subset
U$, $q_0$ is the initial state, $B \in U \setminus V$ is the
``blank'' symbol, $F \subseteq Q$ is the set of final states, and
$\delta$ is the transition mapping,\linebreak
$ \delta : (Q\setminus F)\!\times\! U\! \rightarrow\! 2^{Q\times (U\setminus
\{B\})\times\{R,L\}}$. In this paper, we assume without
loss of generality that any Turing machine we consider has a
semi-infinite tape (bounded to the left) and makes no stationary
moves; the computation of such a machine is described in \cite{handbook,hartmanis2,Papa}. An input word is accepted if and only if after a finite number of moves the
Turing machine enters a final state. The language accepted by the
Turing machine is a set of all accepted words. We say a Turing
machine \emph{decides} a language $L$ if it accepts $L$ and
moreover halts on every input.
The reader is referred to \cite{hartmanis2,Papa} for the classical
time and space complexity classes defined for Turing machines.
We say that a rule $a\rightarrow b$, with $a,b\in V\cup\{\varepsilon\}, a\neq b,$ is a {\it substitution rule} if both $a$ and $b$ are not $\varepsilon$; it is a {\it deletion rule} if $a\ne\varepsilon$ and $b=\varepsilon$; it is an {\it insertion rule} if $a=\varepsilon$ and $b\ne\varepsilon$. The set of all substitution, deletion, and insertion rules over an alphabet $V$ are denoted by $Sub_V$, $Del_V$, and $Ins_V$, respectively.
Given a rule as above $\sigma$ and a word $w\in V^*$, we define the following
\emph{actions} of $\sigma$ on $w$:
\begin{itemize}
\item If $\sigma\equiv a\rightarrow b\in Sub_V$, then
$$\sigma^*(w)=\sigma^r(w)=\sigma^l(w)=\left\{
\begin{array}{ll}
\{ubv:\ \exists u,v\in V^*\ (w=uav)\},\\
\{w\},\mbox{ otherwise}.
\end{array}\right.$$
\item If $\sigma\equiv a\to \varepsilon\in Del_V$, then
$
\sigma^*(w)=\left\{
\begin{array}{ll}
\{uv:\ \exists u,v\in V^*\ (w=uav)\},\\
\{w\},\mbox{ otherwise},
\end{array}\right.$
$$
\begin{array}{llc}
\sigma^r(w)=\left\{
\begin{array}{ll}
\{u:\ w=ua\},\\
\{w\},\mbox{ otherwise},
\end{array}\right. & \qquad &
\sigma^l(w)=\left\{
\begin{array}{ll}
\{v:\ w=av\},\\
\{w\},\mbox{ otherwise}.
\end{array}\right.
\end{array}
$$
\item If $\sigma\equiv \varepsilon\to a\in Ins_V$, then
$\sigma^*(w)=\{uav:\ \exists u,v\in V^*\ (w=uv)\},\ \sigma^r(w)=\{wa\},\
\sigma^l(w)=\{aw\}.$
\end{itemize}
\noindent In the following $\alpha\in\{*,l,r\}$ expresses the way of applying a deletion or insertion rule to a word, namely at any position ($\alpha=*$), at the left ($\alpha=l$), or at the right ($\alpha=r$) end of the word, respectively. For every rule $\sigma$, action $\alpha\in \{*,l,r\}$, and $L\subseteq V^*$, we define the \emph{$\alpha$-action of $\sigma$ on $L$} by $\sigma^\alpha(L)=\bigcup_{w\in L} \sigma^\alpha(w)$. Given a finite set of rules $M$, we define the \emph{$\alpha$-action of $M$} on the word $w$ and the language $L$ by
\begin{center}$ M^{\alpha}(w)=\bigcup_{\sigma\in M}
\sigma^{\alpha}(w)\ \mbox{ and } \ M^{\alpha}(L)=\bigcup_{w\in
L}M^{\alpha}(w),$\end{center}
respectively. In what follows, we shall refer to the rewriting operations defined above as {\it evolutionary operations} since they may be viewed as linguistic formulations of local gene mutations.
For two disjoint subsets $P$ and $F$ of an alphabet $V$ and a word $x$ over $V$, we define the predicates
\begin{align*}
\varphi^{s}(x;P,F)&\equiv P\subseteq alph(x) \,\wedge\, F\cap alph(x)=\emptyset,\\
\varphi^{w}(x;P,F)&\equiv alph(x)\cap P \ne \emptyset \,\wedge\, F\cap alph(x)=\emptyset.
\end{align*}
The construction of these predicates is based on {\it random-context conditions} defined by the two sets $P$ ({\it permitting contexts/symbols})
and $F$ ({\it forbidding contexts/symbols}). Informally, the former condition requires ($s$ stands for strong) that all permitting symbols are and no forbidding symbol is present in $x$, while the latter ($w$ stands for weak) is a weaker variant such that at least one permitting symbol appears in $x$ but still no forbidding symbol is present in $x$.
For every language $L\subseteq V^*$, $P$, $F$ as above, and $\beta\in \{s,w\}$, we define:
\begin{center}$\varphi^\beta(L,P,F)=\{x\in L\mid \varphi^\beta(x;P,F)\}.$\end{center}
An \emph{accepting network of evolutionary processors with filtered connections} (abbreviated ANEPFC) is a $9$-tuple
\begin{center}
$\Gamma=(V,U,G,{\cal R},{\cal N},\alpha,\beta,x_I,x_O),$
\end{center}
where:
\begin{itemize}
\item $V$ and $U$ are the \emph{input and network alphabet}, respectively; we have $V\subseteq U$.
\item $G=(X_G,E_G)$ is an undirected graph without loops with the set of nodes
$X_G$ and the set of edges $E_G$. Each edge is given in the form of a
binary set. $G$ is called the \emph{underlying graph} of the network.
\item ${\cal R}:X_G\longrightarrow 2^{Sub_U}\cup 2^{Del_U}\cup 2^{Ins_U}$ is a mapping which associates
with each node \emph{the set of evolutionary rules} that can be applied in that node.
Note that each node is associated only with one type of evolutionary rules, namely for
every $x\in X_G$ either ${\cal R}(x)\subset Sub_U$ or ${\cal R}(x)\subset Del_U$
or\linebreak ${\cal R}(x)\subset Ins_U$ holds.
\item ${\cal N}:E_G\longrightarrow 2^U\times 2^U$ is a mapping which associates with each
edge \hbox{$e\in E_G$} \emph{the permitting and forbidding filters of that edge};
formally, \hbox{${\cal N}(e)=(P_e,F_e)$}, with $P_e\cap F_e=\emptyset$.
\item $\alpha: X_G\longrightarrow \{*,l,r\}$; $\alpha(x)$
gives \emph{the action mode of the rules} of node $x$ on the words existing in that
node.
\item $\beta: E_G\longrightarrow \{s,w\}$ defines \emph{the filter type of an edge}.
\item $x_I, x_O \in X_G$ are \emph{the input and the output node} of $\Gamma$, respectively.
\end{itemize}
We say that $card(X_G)$ is the size of $\Gamma$. Generally, the ANEPs considered in the literature have complete underlying graphs, namely graphs without loops in which every two nodes are connected. Starting from the observation that every ANEPFC can be immediately transformed into an equivalent ANEPFC with a complete underlying graph (the edges that are to be added are associated with filters which make them useless), for the sake of simplicity, we discuss in what follows ANEPFCs whose underlying graphs have useful edges only. Note that this is not always possible for ANEPs.
A \emph{configuration} of an ANEPFC $\Gamma$ as above is a mapping $C:X_G\longrightarrow 2^{U^*}$ which associates a set of words with every node of the graph. A configuration may be understood as the sets of words which are present in any node at a given moment. Given a word $w \in V^*$, \emph{the initial configuration} of $\Gamma$ on $w$ is defined by $C_0^{(w)}(x_I)=\{w\}$ and $C_0^{(w)}(x)=\emptyset$ for all $x\in X_G\setminus\{x_I\}$.
A configuration can change either by an {\it evolutionary step} or by a {\it communication step}. When changing by an evolutionary step, each component $C(x)$ of the configuration $C$ is changed in accordance with the set of evolutionary rules ${\cal R}(x)$ associated with the node $x$ and the way of applying these rules $\alpha(x)$. Formally, we say that the configuration $C'$ is obtained in \emph{one evolutionary step} from the configuration $C$, written as $C\Longrightarrow C'$, iff
$C'(x)=({\cal R}(x))^{\alpha(x)}(C(x))\mbox{ for all } x\in X_G.$
When changing by a communication step, each node processor $x\in X_G$ sends one copy of each word it contains to every node processor $y$ connected to $x$, provided they can pass the filter of the edge between $x$ and $y$. It keeps no copy of these words but receives all the words sent by any node processor $z$ connected with $x$ providing that they can pass the filter of the edge between $x$ and $z$.
Formally, we say that the configuration $C'$ is obtained in \emph{one communication step} from configuration~$C$, written as $C\vdash C'$, iff
\begin{eqnarray*}
C'(x)&=&(C(x)\setminus(\bigcup_{\{x,y\}\in E_G}\varphi^{\beta(\{x,y\})}(C(x),{\cal N}(\{x,y\}))))\cup (\bigcup_{\{x,y\}\in E_G} \varphi^{\beta(\{x,y\})}(C(y),{\cal N}(\{x,y\})))
\end{eqnarray*}
for all $x\in X_G.$
Let $\Gamma$ be an ANEPFC; the computation of $\Gamma$ on the input word $z\in V^*$ is a sequence of configurations $C_0^{(z)},C_1^{(z)},C_2^{(z)},\dots$, where $C_0^{(z)}$ is the initial configuration of $\Gamma$ on $z$, $C_{2i}^{(z)}\Longrightarrow C_{2i+1}^{(z)}$ and
$C_{2i+1}^{(z)}\vdash C_{2i+2}^{(z)}$, for all $i\geq 0$. By the previous definitions, each configuration $C_i^{(z)}$ is uniquely determined by the configuration $C_{i-1}^{(z)}$, thus each computation in an ANEPFC can be seen as deterministic.
A computation {\it halts} (and it is said to be {\it finite}) if one of the following two conditions holds:\\
(i) There exists a configuration in which the set of words existing in the output node $x_O$ is non-empty. In this case, the computation
is said to be an {\it accepting computation}.\\
(ii) There exist two identical configurations obtained either in consecutive evolutionary steps or in consecutive communication steps.
The {\it language accepted} by $\Gamma$ is
$$L_a(\Gamma)=\{z\in V^*\mid \mbox{ the computation of $\Gamma$ on $z$}\mbox{ is an accepting one.}\}$$
We say that an ANEPFC $\Gamma$ decides the language $L\subseteq V^*$, and write $L(\Gamma)=L$ iff $L_a(\Gamma)=L$ and the computation of $\Gamma$ on every $z\in V^*$ halts.
In a similar way to Turing machines, we define two computational complexity measures
using ANEPFC as the computing model. To this aim we consider an ANEPFC $\Gamma$ with the
input alphabet~$V$ that halts on every input. The {\it time complexity} of the finite
computation $C_0^{(x)}, C_1^{(x)}, C_2^{(x)}, \dots, C_m^{(x)}$ of $\Gamma$ on $x\in V^*$ is denoted by $Time_{\Gamma}(x)$ and equals $m$.
The time complexity of $\Gamma$ is the partial function from $\mathbb{N}$ to $\mathbb{N}$:
$Time_{\Gamma}(n)=\mbox{max}\{Time_{\Gamma}(x)\mid x\in
V^*, |x|=n\}.$
We say that $\Gamma$ decides $L$ in time $O(f(n))$ if $Time_{\Gamma}(n)\in O(f(n))$.
For a function $f:{\mathbb{N}}\longrightarrow {\mathbb{N}}$ we define:
\begin{eqnarray*}
{\bf Time}_{\mathit{ANEPFC}_p}(f(n))&=&\{L\mid \mbox{there exists an
ANEPFC $\Gamma$ of size $p$ deciding $L$,}\\
&& \quad\mbox{and $n_0$ such that
$Time_{\Gamma}(n) \le f(n)$ for all $n \geq n_0$}\}.
\end{eqnarray*}
Moreover, we write ${\bf PTime}_{\mathit{ANEPFC}_p}=\displaystyle\bigcup_{k\ge 0} {\bf Time}_{\mathit{ANEPFC}_p}(n^k)$ for all $p\ge 1$ as well as
$${\bf PTime}_{\mathit{ANEPFC}}=\displaystyle\bigcup_{p\ge 1} {\bf PTime}_{\mathit{ANEPFC}_p}.$$
We recall from \cite{ijfcsDM}:
\begin{theorem}\label{result_ipl}
${\bf NP} = {\bf PTime}_{\mathit{ANEPFC}_{26}}$.
\end{theorem}
\section{Decreasing the size of universal ANEPFCs}
In the following we show how a $2$-tag system can be simulated by an ANEPFC of size $10$.
\begin{theorem}\label{2tag}
For every $2$-tag system $T=(V,\phi)$ there exists a complete ANEPFC $\Gamma$
of size $10$ such that $L(\Gamma)=\{w\mid \mbox{ $T$ halts on $w$}\}$.
\end{theorem}
\begin{proof} Let $V=\{a_1,a_2,\ldots,a_n,a_{n+1}\}$ be the alphabet of the tag system $T$
with $a_{n+1}=H$ and\linebreak \hbox{$V'=V\setminus \{H\}$}.
We consider the ANEPFC
$$\Gamma=(V',U,K_{10},{\cal R},{\mathcal N},\alpha,\beta,1, 10)$$
with the $10$ nodes labeled with the numbers from $1$ to $10$.
The working alphabet of the network is defined as follows:\\
\centerline{$U=V\cup\{\$,\#,a_0',a_0'',\prec\!\! a_0\!\!\succ'\}\cup\{a',a'',a^{\circ}\mid a\in V'\}\cup
\{[x],\langle x \rangle ,\prec\!\! x\!\!\succ \mid x\in X\}$,}\\
where $X=\{x\in (V\cup\{a_0\})^*\mid |x|\leq \mbox{max}\{|\phi(a)|\mid a\in V'\}\}, $ and $\$,\# \notin V $.
The processors placed in the nodes of the network are defined as follows (we assume that the output node $10$ has an empty set of rules):
\begin{itemize}\small
\item \underline{The node $1$}:
$M = \{ a\rightarrow [\phi(a)], a\rightarrow a^{\circ}\mid a\in V'\}$, $\alpha = *$.
\item \underline{The node $2$}:
$M = \{\varepsilon\rightarrow a_0'\}$, $\alpha= r$.
\item \underline{The node $3$}:
$M = \{\varepsilon\rightarrow \$\}$, $\alpha= r$.
\item \underline{The node $4$}:
$M = \{a_{k}'\rightarrow a_{k}''\mid 0\le k\le n\}\cup{}$\\
\hspace*{\fill}$\{[a_kx]\rightarrow \langle a_{k-1}x\rangle, \prec\!\! a_kx \!\!\succ\rightarrow \langle a_{k-1}x\rangle,
\prec\!\! a_0a_kx\!\!\succ\rightarrow \langle a_{k-1}x\rangle \mid x\in X, 1\!\le\! k\!\le\! n+1\}$,\\
\hspace*{\fill}$\alpha = *$.
\item \underline{The node $5$}:
$M = \{a_{k-1}''\rightarrow a_{k}'\mid 1\le k\le n\}\cup\{a_{k-1}''\rightarrow a_{k}\mid 1\le k\le n+1\}
\cup \{\langle x\rangle\rightarrow \prec\!\! x\!\!\succ \mid x\in X\}$, $\alpha = *$.
\item \underline{The node $6$}:
$M = \{\$\rightarrow \varepsilon\}$, $\alpha= r$.
\item \underline{The node $7$}:
$M = \{\prec\!\! a_0\!\!\succ\rightarrow\prec\!\! a_0\!\!\succ'\}$,
$\alpha =* $.
\item \underline{The node $8$}:
$M = \{\prec\!\! a_0\!\!\succ'\rightarrow \varepsilon\}$, $\alpha= l$.
\item \underline{The node $9$}:
$M = \{a^{\circ}\rightarrow\varepsilon\mid a\in V' \}$, $\alpha =l $.
\end{itemize}
The edges of the network and their filters are defined as follows:
\begin{itemize}\small
\item \underline{The edge $\{1,2\}$} has $\beta = w$ and
$P = \{a^\circ\mid a\in V\},\
F = \{a'\mid a\in V\}\cup \{a_{n+1},a_0'\}.$
\item \underline{The edge $\{2,3\}$} has $\beta = w$ and
$ P = \{[\phi(a)]\mid a\in V'\},\
F = \{\$, \prec\!\! a_0\!\!\succ\}.$
\item \underline{The edge $\{3,4\}$} has $\beta = w$ and
$ P = \{\$\},\
F = \{\prec\!\! a_0\!\!\succ\}\cup \{\langle x \rangle \mid x\in X\}\cup \{a''\mid a\in V\cup\{a_0\}\}.$
\item \underline{The edge $\{4,5\}$} has $\beta = w$ and
$ P = \{a'' \mid a\in V\cup\{a_0\}\},\
F = \{[x]\mid x\in X\}\cup \{\prec\!\! x\!\!\succ \mid x\in X\} .$
\item \underline{The edge $\{5,6\}$} has $\beta = w$ and
$ P = \{\prec\!\! x\!\!\succ \mid x\in X\},\
F = \{a''\mid a\in V\cup\{a_0\}\}\cup\{[x],\langle x\rangle\mid x\in X\}.$
\item \underline{The edge $\{6,2\}$} has $\beta = w$ and
\begin{align*}
P &= \{\prec\!\! a_0x\!\!\succ\mid x\in X, x\neq \lambda\},\\
F &= \{\$,a_0'\}\cup\{[x],\langle x\rangle\mid x\in X\}\cup \{\prec\!\! x\!\!\succ\mid x\neq a_0y, y\in X\}.
\end{align*}
\item \underline{The edge $\{6,3\}$} has $\beta = w$ and
\begin{align*}
P &= \{\prec\!\! x\!\!\succ\mid x\neq a_0y, y\in X\},\\
F &= \{\$\}\cup\{[x],\langle x\rangle\mid x\in X\}\cup \{\prec\!\! x\!\!\succ\mid x= a_0y, y\in X\}.
\end{align*}
\item \underline{The edge $\{6,7\}$} has $\beta = w$ and
$ P = \{\prec\!\! a_0\!\!\succ \},\
F = \{\$\}\cup\{\prec\!\! x\!\!\succ\mid x\in X, x\neq a_0\}\cup\{[x],\langle x\rangle\mid x\in X\}.$
\item \underline{The edge $\{7,8\}$} has $\beta = w$ and
$ P = \{\prec\!\! a_0\!\!\succ'\},\
F =\{\prec\!\! a_0\!\!\succ \}.$
\item \underline{The edge $\{7,3\}$} has $\beta = w$ and
$ P = \{\prec\!\! a_0\!\!\succ\},\
F =\{\$\}.$
\item \underline{The edge $\{8,9\}$} has $\beta = w$ and
$ P = \{a^\circ\mid a\in V\},\
F =\{\prec\!\! a_0\!\!\succ'\}.$
\item \underline{The edge $\{9,10\}$} has $\beta = w$ and
$ P = \{a_{n+1}\},\
F =U\setminus V.$
\item \underline{The edge $\{9,1\}$} has $\beta = w$ and
$ P =V,\
F =\{a_{n+1}\}\cup (U\setminus V).$
\end{itemize}
We show that $\Gamma$ accepts a word $w$ that does not contain $H$ if an only if $T$ eventually halts on $w$.
Let $w=aby, a,b\in V, y\in V^*$ be a word that does not contain $H$ such that $T$ eventually halts on~$w$.
We show how $w$ can be accepted by $\Gamma$.
At the beginning of the computation $w$ is found in node $1$, where the first symbol $a$ can be replaced by $[\phi(a)]$ but the new string cannot pass the filter of any edge, thus remaining in this node during
the next communication step.
In the next step, we can rewrite $b$ as~$b^{\circ}$, getting the new word $[\phi(a)]b^{\circ}y$ which is sent out to node $2$. Here, the symbol $a_0'$ is inserted to its righthand end obtaining $[\phi(a)]b^{\circ}ya_0'$. This word can only enter node $3$. In this node, the string becomes $[\phi(a)]b^{\circ}ya_0'\$ $, and goes to node $4$.\\
Let $\phi(a)=a_ix$, for some $1\le i\le n+1$ and $x\in X$. In node $4$, $[a_ix]b^{\circ}ya_0'\$ $ is first converted into $[a_{i}x]b^{\circ}ya_0''\$ $, which remains in this node for the next communication step,
and then into $\langle a_{i-1}x\rangle b^{\circ}ya_0''\$ $. This string is sent out to node $5$, where it is transformed into $\prec\!\! a_{i-1}x\!\!\succ b^{\circ}ya_1'\$ $, via $\prec\!\! a_{i-1}x\!\!\succ b^{\circ}ya_0''\$ $. This string goes to node $6$, where $\$ $ is deleted, and the string becomes $\prec\!\! a_{i-1}x\!\!\succ b^{\circ}ya_1'$.
If $i>1$, this string first returns to node $3$, resulting in $\prec\!\! a_{i-1}x\!\!\succ b^{\circ}ya_1'\$ $, and then goes to node $4$; in this node, the string is transformed into $\langle a_{i-2}x\rangle b^{\circ}ya_1''\$ $, and sent to node $5$. This process is repeated until a string of the form $\langle a_0x\rangle b^{\circ}ya_{i-1}''\$ $ arrives in node $5$. Here the string becomes $\prec\!\! a_0x\!\!\succ b^{\circ}ya_{i}\$ $ and goes to node $6$, where the $\$ $ symbol is deleted. \\
Now, if $x\neq \varepsilon$ (namely, $x=a_jy$ with $j\geq 1$), the string $\prec\!\! a_0x\!\!\succ b^{\circ}ya_{i} $ goes to node~$2$, where it becomes $\prec\!\! a_0x\!\!\succ b^{\circ}ya_{i}a_0' $; then, the string enters node $3$, where it is transformed into $\prec\!\! a_0x\!\!\succ b^{\circ}ya_{i}a_0'\$ $. Further, the string enters node $4$, where we obtain the string $\langle a_{j-1}y\rangle b^{\circ}ya_{i}a_0''\$ $, and the process described above is resumed.\\
On the other hand, if $x=\varepsilon$, the string equals $\prec\!\! a_0\!\!\succ b^{\circ}y\phi(a) $, and goes to node $7$, where it becomes $\prec\!\! a_0\!\!\succ' b^{\circ}ya_{i} $. From this node, the string can only go to node $8$ where the $\prec\!\! a_0\!\!\succ'$ symbol is deleted. Then, it enters node $9$ where the $b^\circ$ symbol is deleted, and the string becomes $y\phi(a) $. In this moment, the string can go either to node $1$, provided that it doesn't contain $a_{n+1}$ and the whole procedure we described above is resumed, or to node $10$ and the input string is accepted.
We now argue why the above simulation is the only possible derivation in $T$, so that it
halts on a word $w$ if and only if $w$ is accepted by $\Gamma$.
In many steps, the derivation stated above is the only possible derivation. However, there are a few cases we need to consider more closely.\\
First of all, in node $1$, only one substitution $a\rightarrow a^{\circ}$ can be performed before the string is sent out, but potentially zero or more than one $a\rightarrow [\phi(a)]$ substitutions. If no such substitution is performed, the resulting string enters node $2$ when it remains forever.
If the outgoing word from node $1$
contains more than one symbol $[x]$, $x\in X$, then after several processing steps, either all of them will be transformed into $\prec\!\! a_0\!\!\succ$, and the string will reach node $7$, or only some of them will be transformed into $\prec\!\! a_0\!\!\succ$, and the string will be further blocked in node $6$. On the other hand,
if the current string contains more than one $\prec\!\! a_0\!\!\succ $ symbol, then it enters node $7$, where exactly one of these symbols is transformed into $\prec\!\! a_0\!\!\succ'$. Then the string goes to node $3$, where it is blocked.\\
Thus, an accepting computation is only possible if exactly one of each of the symbols $a^{\circ}$ and $[x]$ are present when leaving node $1$. However, both symbols could be on any position of the string. Assume that they do not occupy the first two positions in the way described above. The simulation would then go on as described,
until a string $y_1\prec\!\! a_0\!\!\succ' y_2b^{\circ}y_3$ or $y_1b^{\circ}y_2\prec\!\! a_0\!\!\succ' y_3$,
\hbox{$y_1,y_2,y_3\in (V\setminus\{H\})^*$} is reached. However, in all of these cases the string will be communicated between nodes $7$, $8$ and $9$ only, thus it will not affect the computation.\\
This covers all possible cases, proving that if $w\in L(\Gamma)$, then $T$ will eventually halt on $w$.
\end{proof
Since $2$-tag systems are universal \cite{rogo,minsky}, the following corollary is immediate:
\begin{cor} There exists a universal ANEPFC with $10$ nodes.
\end{cor}
This result significantly improves the result reported in \cite{mscs} where a universal ANEPFC with $18$ nodes was constructed.
\section{Decreasing the size of ANEPFCs accepting recursively enumerable languages}
Although $2$-tag systems efficiently simulate deterministic Turing machines, via
cyclic tag systems (see, e.\,g., \cite{woods}),
the previous result does not allow us to infer a bound on the size of the networks accepting
in a computationally efficient way all recursively enumerable languages.
We now discuss how an efficient
ANEPFC accepting (deciding) every recursively enumerable (recursive) language can be constructed.
\begin{theorem}\label{TM}
For any recursively enumerable (recursive) language $L$ accepted (decided) by a Turing machine
there exists a complete ANEPFC $\Gamma$ of size $16$ accepting (deciding) $L$.\\
Moreover, if $L\in \mathit{NTIME}(f(n))$, then \hbox{$Time_\Gamma(n)\in {\cal O}(f(n))$}.
\end{theorem}
\begin{proof}
Due to space requirements, we present a sketch of the proof only. However, the principles of this construction rely on the same mechanisms as the ones we used in the proof of Theorem \ref{2tag}, so we believe the reader is able to infer a clear idea of how the construction works.
Let $M=(Q,V,W,q_0,B,F,\delta)$ be a nondeterministic Turing machine; we construct the ANEPFC $\Gamma=(V,U,G,{\cal R},{\mathcal N},\alpha,\beta,1,16)$ with $16$ nodes, labeled with the numbers $1$, $2,\ldots,16$, working as follows.
We stress from the beginning that the edges of the graph, and their filters, are defined such that only the following derivation can take place. Also, nodes $3$, $7$, $8$, $10$ and $14$ do not actually contribute to the simulation of a given derivation, but are used to keep away other derivations from occurring.
In our simulation, a string $v_1[q,a,q',b,X]v_2$ corresponds to a configuration of~$M$ where the current state is $q$, the tape content is $v_2v_1$ and the head of $M$ reads the first symbol of $v_1$.
To obtain a string having this form corresponding to the initial configuration of $M$
from the input string, two nodes of the network are used: the input node $1$ and node $2$, both of them right insertion nodes. Let now $(q',b,X)\in \delta(q,a)$ be the next transition of $M$.
From a word as above, $\Gamma$ simulates the transition of $M$ in a so-called Simulation Phase. As a first step in this simulation, we need to check that the first symbol of $v_1$ is $a$. We regard $W$, the working alphabet of Turing machine $M$, as the ordered alphabet $\{a_1,a_2,\ldots,a_n\}$ such that $V=\{a_{1},\ldots,a_m\}$ for some $m<n$ and $B=a_n$. Now we can perform this check by simultaneously lowering the index of the first symbol of $v_1$ and that of $a$ in the symbol $[q,a,q',b,X]$ in the same way as we did in Theorem \ref{2tag}. Only when this check is successful, we obtain a string of the form $\$ v_1'[q,\$,q',b,X]v_2$, where $v_1=av_1'$. The leftmost symbol $\$$ is then deleted and we move on to the next stage of the simulation. The procedure described above is carried out by eight of the network's nodes, namely the nodes $3$, $4$, $5$, $6$, $7$, $8$, $9$ and $11$. The node $3$ is a right insertion node, nodes $4$, $5$, $6$ and $8$ are substitution nodes, node $7$ is a deletion node and $9$ is a left deletion node. If the end of the tape (i.\,e., the deleted symbol $a$ equals the blank symbol $B$) is reached during the computation and further tape space is needed, then a special insertion node ensures that the blank symbol $B$ is introduced before continuing the simulation. This is done using node $11$, a left insertion node. Also, node $10$ is used during this step of the computation to collect and block strings that should not be processed further.
Now, $\Gamma$ has to write the symbol $b$ and simulate the correct repositioning of the head for the next move. This is done differently depending on whether the head moves left or right.
If $(q',b,R)\in \delta(q,a)$, it suffices to append the symbol $b$ to the right end of the string. This is done by first inserting a symbol $a_0\notin W$ to the right, to obtain $v_1'[q,\$,q',b,R]v_2a_0$. Then, the index of this last symbol is increased, while that of $b$ in the symbol $[q,\$,q',b,R]$ is simultaneously lowered. After this process finishes, we have a string of the form $v_1'[q,\$,q',\#,R]v_2b$. At this point the simulation is complete, and the symbol $[q,\$,q',\#,R]$ can be rewritten as $[q',a',q'',b',X]$ for some $(q'',b',X)\in \delta(q',a')$.
If $(q',b,L)\in \delta(q,a)$, then some more work is necessary. First $b$ is written at the left end of the string, just like explained above. This gives a string of the form $bv_1'[q,\$,q',\#,L]v_2$. Moreover, since now the head reads the last symbol of $v_2$, this symbol has to be moved to the left end of the string. Let $v_2=v_2'c$. Now a symbol $a_0'$ is inserted to the left, giving $a_0'bv_1'[q,\$,q',\#,L]v_2'c$. Again, the index of the first symbol is increased, while that of the last symbol is decreased; moreover, the symbol encoding the move of the Turing machine is updated. Finally, we obtain $c'bv_1'[q',c',q'',d,X]v_2'\bot$, for some $(q'',b',X)\in \delta(q',c')$, from which the symbol $\bot$ is deleted.
To perform the procedure described above five of the network's nodes are used: node $12$, a right insertion node, node $13$, a left insertion node, nodes $3$, $4$, $5$, $6$, $7$ mentioned above, and nodes $14$, a substitution node, and $15$, a right deletion node.
From here a new simulation restarts, unless $q'\in F$, in which case the symbol $[q,\$,q',\#,X]$ was replaced by a special symbol $\Delta$ and the word can enter the output node $16$, ensuring that $\Gamma$ accepts the word if and only if $M$ accepts it.
Moreover, if $M$ stops on the input string $w$, in $f(|w|)$ steps, then $\Gamma$ stops on the input string $w$ after $f(|w|)$ executions of the Simulation Phase, described above.
\end{proof}
We can easily state the following corollary of the previous Theorem:
\begin{cor}${\bf NP} = {\bf PTime}_{\mathit{ANEPFC}_{16}}$.
\end{cor}
This result provides a significantly improvement of the results in \cite{ijfcsDM}, where the same
characterization of {\bf NP} was obtained for ANEPFCs with $26$ nodes.
\bibliographystyle{eptcs}
|
1,116,691,499,279 | arxiv | \section{Introduction}
Let $\mathfrak{S}_n$ denote the symmetric group of order $n$. Recall that the \textbf{Robinson-Schensted correspondence} associates with a permutation $\sigma^{(n)}\in \mathfrak{S}_n$ a pair of standard Young tableaux $(P_n,Q_n)$ whose common shape $\lambda$ is a Young diagram of order $n$.
A fruitful area of study concerns asymptotic properties of the Robinson-Schensted shape $\lambda$ and the tableaux $P_n,Q_n$ associated with a \emph{random} permutation $\sigma^{(n)}$ sampled from the uniform distribution on $\mathfrak{S}_n$. The existing results on this subject are too numerous to list here, but some of the important highlights of the theory are the limit shape result of Logan-Shepp \cite{loganshepp} and Vershik-Kerov \cite{vershikkerov1, vershikkerov2}, which led to the solution of the so-called Ulam-Hammersley problem on the typical length of a longest increasing subsequence in random permutations; and the celebrated Baik-Deift-Johansson theorem \cite{baik-deift-johansson} and its refinements and variants \cite{baik-deift-johansson2, borodin-okounkov-olshanski, johansson} that tied the behavior of longest increasing subsequences in random permutations to the Tracy-Widom distribution and other naturally-occurring stochastic processes from random matrix theory. See the book \cite{romik} for a survey of many of these developments that also touches on diverse connections to random growth processes, interacting particle systems, representation theory and more.
In this paper we continue this line of investigation by studying the \textbf{bumping route} computed during the application of an insertion procedure, which is the fundamental building block of the Robinson-Schen\-sted correspondence.
Let us recall briefly the relevant definitions. A \textbf{Young diagram} $\lambda$ of order~$n$ is an integer partition of $n$, that is, an array of positive integers $\lambda_1\ge\dots\ge\lambda_m\ge 0$ such that $n=\sum_{j=1}^m \lambda_j$, represented graphically as a diagram of left-justified square boxes wherein the $j$th row contains $\lambda_j$ boxes.
If $\lambda$ is a Young diagram of order $n$ and $x_1,\dots,x_n$ are distinct real numbers, an \textbf{increasing tableau} of shape $\lambda$ and entries given by $x_1,\dots, x_n$ is a filling of the boxes of $\lambda$ with the numbers $x_1,\dots,x_n$ that is increasing along rows and columns. A \textbf{standard Young tableau} is such an increasing tableau whose entries are precisely the numbers $1,\dots,n$.
Given an increasing tableau $P$ with entries $x_1,\dots, x_n$ and a number $z$ distinct from $x_1,\dots,x_n$, the \textbf{insertion procedure} applied to $P$ and $z$ produces a new increasing tableau $P\leftarrow z$ with entries $x_1,\dots,x_n,z$ whose shape $\lambda^+$ is obtained from $\lambda$ by the addition of a single box.
The new tableau $P\leftarrow z$ is computed by performing a succession of \textbf{bumping steps} whereby the number $z$ is inserted into the first row of the diagram, displacing an existing entry from the first row;
the displaced entry is bumped onto the second row, and in turn bumps an entry of the second row onto the third row; and so on, until finally the entry being bumped settles down in an unoccupied position outside the diagram $\lambda$. In each row, the position where the bumping (or settling-down, in the last step) occurs is the leftmost position containing an entry bigger than the incoming number, or a new unoccupied position to the right of all existing entries if no such entry exists.
An example is shown in Figure~\ref{fig:bumping-example}.
\newcommand{blue!50}{blue!50}
\begin{figure}[t]
\begin{center}
\begin{tikzpicture}[scale=0.7]
\begin{scope}
\begin{scope} \clip (3,0) rectangle +(1,1); \draw[line width=6pt,blue!50] (3,0) rectangle +(1,1); \end{scope}
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\begin{scope} \clip (1,2) rectangle +(1,1); \draw[line width=6pt,blue!50] (1,2) rectangle +(1,1); \end{scope}
\begin{scope} \clip (1,3) rectangle +(1,1); \draw[line width=6pt,blue!50] (1,3) rectangle +(1,1); \end{scope}
\draw (0.5,0.5) node {1};
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\begin{scope}
\clip (0,0) -- (7,0) -- (7,1) -- (5,1) -- (5,2) -- (4,2) -- (4,3) -- (3,3) -- (3,4) -- (1,4) -- (1,5) -- (0,5) -- cycle;
\draw (0,0) grid (50,50);
\end{scope}
\draw (5.5,-1.5) node {7};
\draw[->,ultra thick] (5,-1.5) .. controls (4,-1.5) and (3.5,-1.5) .. (3.5,-0.5);
\end{scope}
\draw (8,2.5) node {$\longrightarrow$};
\begin{scope}[shift={(10,0)}]
\begin{scope} \clip (3,0) rectangle +(1,1); \draw[line width=6pt,blue!50] (3,0) rectangle +(1,1); \end{scope}
\begin{scope} \clip (2,1) rectangle +(1,1); \draw[line width=6pt,blue!50] (2,1) rectangle +(1,1); \end{scope}
\begin{scope} \clip (1,2) rectangle +(1,1); \draw[line width=6pt,blue!50] (1,2) rectangle +(1,1); \end{scope}
\begin{scope} \clip (1,3) rectangle +(1,1); \draw[line width=6pt,blue!50] (1,3) rectangle +(1,1); \end{scope}
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\draw (1.5,3.5) node {11};
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\draw (1.5,4.5) node {17};
\draw[thick](0,0) -- (7,0) -- (7,1) -- (5,1) -- (5,2) -- (4,2) -- (4,3) -- (3,3) -- (3,4) -- (2,4) -- (2,5) -- (0,5) -- cycle;
\begin{scope}
\clip (0,0) -- (7,0) -- (7,1) -- (5,1) -- (5,2) -- (4,2) -- (4,3) -- (3,3) -- (3,4) -- (2,4) -- (2,5) -- (0,5) -- cycle;
\draw (0,0) grid (50,50);
\end{scope}
\end{scope}
\end{tikzpicture}
\caption{Inserting a number into a tableau results in a cascade of bumping events. The bumping route is the sequence of positions where a bumping occurred.}
\label{fig:bumping-example}
\end{center}
\end{figure}
Define the \textbf{bumping route} $\textnormal{\textsf{B}}_{P,z}$ associated with an insertion procedure performed on the tableau $P$ with a new input~$z$ to be the sequence of positions where a bumping occurred during the insertion, together with the position of the final box added to the shape. The $j$th position in the bumping route is of the form $\big(b_{P,z}(j),j\big)$, so it is convenient to encode the bumping route using only the $x$-coordinates of the positions, which form a monotone nonincreasing sequence of positive integers $b_{P,z}(1) \ge b_{P,z}(2) \ge \dots \ge b_{P,z}(k_{P,z})$ whose length we denote by $k_{P,z}$. For example, the bumping route associated with the insertion step in Figure~\ref{fig:bumping-example} is $(4,3,2,2,2)$.
The \textbf{insertion tableau} corresponding to a sequence $x_1,\dots,x_n$ is defined as
the outcome of the iterative application of the insertion procedure
\[ P(x_1,\dots,x_n):=\Big( \big( (\emptyset \leftarrow x_1) \leftarrow x_2 \big) \leftarrow \dots \Big) \leftarrow x_n,\]
starting with the empty tableau $\emptyset$.
The \textbf{Robinson-Schensted correspondence} associates with a permutation $\sigma^{(n)}\in \mathfrak{S}_n$ a pair of standard Young tableaux $(P_n,Q_n)$ with the first one $P_n=P\big(\sigma^{(n)})=P\big(\sigma^{(n)}(1),\dots,\sigma^{(n)}(n)\big)$ being the insertion tableau corresponding to the permutation; the definition of the second tableau $Q_n$, known as the \textbf{recording tableau}, will not be necessary for our purposes. More details on Young tableaux, the Robinson-Schensted correspondence and their properties can be found in several well-known sources such as \cite{fulton, knuthvol2, stanleyvol2}.
If $\sigma^{(n)}$ is a uniformly random permutation of order $n$, the bumping route computed in the last insertion step
performed while calculating the insertion tableau $P_n$
is equal to $\textnormal{\textsf{B}}_{P_{n-1},\sigma^{(n)}(n)}$, where
$P_{n-1}=P\left(\sigma^{(n)}(1),\dots,\sigma^{(n)}(n-1)\right)$ denotes the insertion tableau computed from the truncated sequence. The question we wish to address is that of understanding the asymptotic behavior of this bumping route.
Since the computation depends only on the relative order of the numbers $\sigma^{(n)}(j), 1\le j\le n$, it will be equivalent, and more convenient, to formulate the result in terms of a sequence $X_1,\dots,X_n$ of independent and identically distributed (i.i.d.) random variables with the uniform distribution $U[0,1]$ on the unit interval $[0,1]$, which gives a canonical way of realizing uniformly random order structures of all orders $n$ simultaneously. If we denote by $T_{n}=P(X_1,\dots,X_n)$ the corresponding insertion tableau, then the bumping route $\textnormal{\textsf{B}}_{T_{n-1},X_n}$ is equal in distribution to $\textnormal{\textsf{B}}_{P_{n-1},\sigma^{(n)}(n)}$.
Our result will pertain to an even more general scenario in which the final input $X_n$ is taken to be an arbitrary (non-random) number $\alpha$ in the unit interval $[0,1]$. Note that, by obvious monotonicity properties of the insertion procedure, as $\alpha$ increases from $0$ to $1$, the bumping route is deformed monotonically between the two extreme cases $\alpha=0$ and $\alpha=1$, where in the case $\alpha=0$ the bumping route will be the first column of the diagram and an additional new box at the top of the first column, and in the case $\alpha=1$ the bumping route consists of a single new box at the end of the first row of the diagram.
Note also that the bumping route (except for the last box)
is contained in the Young diagram of the tableau $T_{n-1}$; this random Young diagram, whose distribution is known as the \textbf{Plancherel measure} of order $n-1$, converges to a well-known limit shape discovered in the celebrated works of Logan-Shepp \cite{loganshepp} and Vershik-Kerov \cite{vershikkerov1, vershikkerov2}.
Figure~\ref{fig:simulation} shows the bumping routes $\textnormal{\textsf{B}}_{T_{n-1},\alpha}$ for various values of $\alpha$ in a numerical simulation with $n=10^4$. Our goal will be to show that the bumping routes converge after scaling to a family of deterministic limiting curves, which are shown in Figure~\ref{fig:theory}. As preparation for the precise formulation of this result, let us first define this family of curves. First, define auxiliary functions $F$, $\Omega$, $u_\alpha$, $v_\alpha$, $x_\alpha$, $y_\alpha$, $\kappa$ by
\begin{figure}[t]
\hfill
\subfloat[][]{\includegraphicsA[width=0.45\textwidth]{figures/bumping-simulation10000} \label{fig:simulation}}
\hfill
\subfloat[][]{\includegraphicsA[width=0.45\textwidth]{figures/bumping-limits} \label{fig:theory}}
\hfill
\caption{\protect\subref{fig:simulation} Bumping routes $\textnormal{\textsf{B}}_{T_{n-1},\alpha}$ in a simulation with $n=10^4$ and the values $\alpha=1/10,2/10,\dots,9/10$;
\protect\subref{fig:theory} the limiting curves $\big(\beta_\alpha(s),s\big)$ for the same values of $\alpha$. The region bounding the limit shapes is the Logan-Shepp-Vershik-Kerov limit shape of Plancherel-random Young diagrams \cite{romik}.}
\label{fig:bumping-routes}
\end{figure}
\begin{align}
\Omega(u) &= \frac{2}{\pi} \left( u \sin^{-1}\left(\frac{u}{2}\right) + \sqrt{4-u^2} \right) & (|u|\le 2), \label{eq:def-omega} \\
F(u) &= \frac12 + \frac{1}{\pi} \left( \frac{u \sqrt{4-u^2}}{4} + \sin^{-1}\left( \frac{u}{2}\right) \right) & (|u|\le 2), \label{eq:def-semicircle-cdf} \\
u_\alpha(t) &= \sqrt{t} \, F^{-1}\left(\frac{\alpha}{t}\right) & (0\le \alpha \le t \le 1), \label{eq:ualpha}
\\
v_\alpha(t) &= \sqrt{t} \,\Omega\!\left( F^{-1}\left(\frac{\alpha}{t}\right) \right) & (0\le \alpha \le t \le 1), \label{eq:valpha} \\
x_\alpha(t) &= \frac{v_\alpha(t)+u_\alpha(t)}{2} & (0\le \alpha \le t \le 1),
\label{eq:xalpha}
\\
y_\alpha(t) &= \frac{v_\alpha(t)-u_\alpha(t)}{2} & (0\le \alpha \le t \le 1),
\label{eq:yalpha} \\
\kappa(\alpha) &= y_\alpha(1) = \frac{\Omega(F^{-1}(\alpha)) - F^{-1}(\alpha)}{2} & (0\le \alpha \le 1).
\end{align}
The \textbf{limiting bumping route curves} are now defined as the one-parameter family $\big(\beta_\alpha(t)\big)_{0\le \alpha< 1}$ of functions, where for each $\alpha\in [0,1)$, $\beta_\alpha(\cdot)$ is given by
\begin{equation}
\beta_\alpha(s) = x_\alpha ( y_\alpha^{-1}(s)) \qquad (0\le s\le \kappa(\alpha)). \label{eq:def-beta}
\end{equation}
Our main result is as follows.
\begin{thm}[Limit shapes of bumping routes]
\label{thm:main-result}
For each $0\le \alpha<1$, the curve $\beta_\alpha(\cdot)$ describes the limiting bumping route $\textnormal{\textsf{B}}_{T_{n-1},\alpha}$, in the following precise sense: for any $\epsilon>0$, we have that
\begin{align}
\mathbb{P} &\left(
\left| \frac{k_{T_{n-1},\alpha}}{\sqrt{n}} - \kappa(\alpha) \right| > \epsilon
\right) \xrightarrow[n\to\infty]{} 0, \textrm{ and}
\label{eq:limit1} \\[3pt]
\mathbb{P} & \left(
\max_{1 \le m\le k_{T_{n-1},\alpha}}
\left| \frac{b_{T_{n-1},\alpha}(m)}{\sqrt{n}} -
\beta_\alpha\left( \frac{m}{\sqrt{n}} \wedge \kappa(\alpha) \right)
\right| > \epsilon
\right) \xrightarrow[n\to\infty]{} 0.
\label{eq:limit2}
\end{align}
\end{thm}
The problem of understanding the limit shapes of the bumping routes, which our result answers, was posted by C.~Moore on his personal web page (along with simulation results similar to our Figure~\ref{fig:simulation}) in 2006 \cite{moore}.
\section{Preliminary remarks}
As a first step towards proving Theorem~\ref{thm:main-result}, let us recall some facts from the theory of Plancherel measure, which will immediately prove \eqref{eq:limit1} and also help elucidate the somewhat involved definition of the family of limiting curves $\beta_\alpha(\cdot)$. First, when discussing Plancherel-random Young diagrams and Young tableaux it is convenient to use the so-called rotated (also known as ``Russian'') coordinate system, related to the standard $x$-$y$ coordinates by the linear change of variables
\begin{equation} \label{eq:russian}
\begin{array}{c} u = x-y, \\[5pt] v=x+y. \end{array}
\end{equation}
In this coordinate system, the curve $v=\Omega(u)$, where $\Omega(u)$ is defined in~\eqref{eq:def-omega}, describes the Logan-Shepp-Vershik-Kerov limit shape of Plancherel-random Young diagrams mentioned in the introduction.
Second, the function $F(u)$ defined in \eqref{eq:def-semicircle-cdf} is the cumulative distribution function of the semicircle distribution on $[-2,2]$; that is, we have
$$ F(u) = \frac{1}{2\pi} \int_{-2}^u \sqrt{4-s^2}\,ds \qquad (|u|\le 2). $$
Its importance for the present discussion is that, according to one of the main results of our previous paper \cite{romik-sniady}, the point
\begin{equation} \label{eq:u-v-alpha-def}
\big(U(\alpha),V(\alpha)\big)=\left(F^{-1}(\alpha), \Omega\big(F^{-1}(\alpha)\big)\right)
\end{equation}
is the limiting scaled position (in rotated coordinates) of the new box added to the Robinson-Schensted shape after applying the insertion procedure with the number $\alpha \in[0,1]$ to the existing insertion tableau $T_{n-1}$. More precisely, when stated using our current terminology, the result \cite[Theorem~5.1]{romik-sniady}, which we dubbed the ``\textbf{asymptotic determinism of RSK insertion},'' says that the last position $\big(b_\alpha(k_{T_{n-1,\alpha}}),k_{T_{n-1,\alpha}}\big)$ of the bumping route $\textnormal{\textsf{B}}_{T_{n-1},\alpha}$ satisfies
\begin{multline}
\label{eq:asym-det}
\frac{1}{\sqrt{n}} \big(b_\alpha(k_{T_{n-1,\alpha}})-k_{T_{n-1,\alpha}}, b_\alpha(k_{T_{n-1,\alpha}})+k_{T_{n-1,\alpha}}\big)
\\ \xrightarrow[n\to\infty]{\mathbb{P}} \big(U(\alpha),V(\alpha)\big).
\end{multline}
After applying the inverse transformation of \eqref{eq:russian} to rewrite the result in $x$-$y$ coordinates, and noting that
$$\kappa(\alpha)=y_\alpha(1)= \frac{V(\alpha)-U(\alpha)}{2},$$
we get the relation \eqref{eq:limit1}, the first claim of Theorem~\ref{thm:main-result}.
Next, turn to the pair of functions $\big(u_\alpha(t), v_\alpha(t)\big)$ defined in \eqref{eq:ualpha}--\eqref{eq:valpha}, with the associated pair $\big(x_\alpha(t),y_\alpha(t)\big)$ from \eqref{eq:xalpha}--\eqref{eq:yalpha} representing the same functions in $x$-$y$ coordinates. Note that for any fixed $\alpha$, the planar curve $\big(x_\alpha(t),y_\alpha(t)\big)_{\alpha\le t\le 1}$ is a reparametrized version of the curve $\big(\beta_\alpha(s),s\big)_{0\le s\le \kappa(\alpha)}$, which according to our claim \eqref{eq:limit2} is the limit shape of the bumping route $\textnormal{\textsf{B}}_{T_{n-1},\alpha}$ (one needs to note that $y_\alpha(\cdot)$ is a strictly decreasing function; see Lemma~\ref{lem:decreasing} below).
It turns out that the parametrization of the curve as $\big(x_\alpha(t),y_\alpha(t)\big)$ is the correct one when trying to prove the limit shape result (although the parametrization $\big(\beta_\alpha(s),s\big)$ is the one that answers the original question). To see why, we need to explain the role of the parameter $t$. Note that for fixed $t$, the points $\big(u_\alpha(t), v_\alpha(t)\big)_{0\leq \alpha\leq 1}$ all lie on the curve $v=\sqrt{t} \, \Omega(u/\sqrt{t})$, which is a copy of the limit shape $v=\Omega(u)$ scaled down by the factor $\sqrt{t}$. The idea is that this scaled-down copy represents the limiting shape of the ``$t$-sublevel tableau'' of $T_{n-1}$---that is, the subset of boxes of $T_{n-1}$ containing an entry $\le t$. We will show below that the point $\big(u_\alpha(t),v_\alpha(t)\big)$ (or $\big(x_\alpha(t),y_\alpha(t)\big)$, in the usual coordinate system) corresponds to the limiting position, after scaling, of the point at which the bumping route $\textnormal{\textsf{B}}_{T_{n-1},\alpha}$ exits this sublevel tableau. The reason for this is that this exit position relates to the sublevel tableau in roughly the same way that the final bumping route position $\big(b_\alpha(k_{T_{n-1,\alpha}}),k_{T_{n-1,\alpha}}\big)$ relates to the entire tableau $T_{n-1}$, except that there is a scaling relation that causes the number being inserted to change from $\alpha$ to $\alpha/t$. A more precise formulation of this statement is discussed in the next section, after which we will see that our main result follows without much difficulty by another appeal to the ``asymptotic determinism of RSK'' theorem. A schematic illustration of the argument described above is shown in Figure~\ref{fig:summary-picture}.
\begin{figure}[t]
\begin{center}
\begin{picture}(320,300)(0,0)
\put(0,0){\scalebox{1.2}{\includegraphicsA[]{figures/summary-picture}}}
\put(207,39){$v=\Omega(u)$}
\put(174,22){$\scriptscriptstyle v=\sqrt{t}\,\Omega(u/\!\sqrt{t})$}
\put(106,26){$\scriptscriptstyle x=\beta_\alpha(y)$}
\put(40,1){$\scriptstyle \beta_\alpha(s)=x_\alpha(t)$}
\put(-19,91){$\scriptstyle s=y_\alpha(t)$}
\put(-7,184){$\scriptstyle \kappa(\alpha)$}
\end{picture}
\caption{The meaning of the parameter $t$: for a fixed value of $\alpha$, the intersection $\big(u_\alpha(t),v_\alpha(t)\big)$ of the asymptotic bumping curve with the scaled-down copy $v=\sqrt{t}\,\Omega(u/\sqrt{t})$ of the Logan-Shepp-Vershik-Kerov limit shape is computed by applying the asymptotic determinism theorem to the $t$-sublevel tableau; when the numbers in the sublevel tableau are scaled to the range $[0,1]$, the number $\alpha$ being inserted is scaled to~$\alpha/t$.}
\label{fig:summary-picture}
\end{center}
\end{figure}
We conclude this section with another small but useful observation.
\begin{lem} \label{lem:decreasing}
The function $y_\alpha(\cdot)$ is strictly decreasing. In particular, the limit shape functions $\beta_\alpha(\cdot)$ defined in \eqref{eq:def-beta} are well-defined.
\end{lem}
\begin{proof}
Denote
$$\big(X(\alpha),Y(\alpha)\big)=\frac12\big(V(\alpha)+U(\alpha), V(\alpha)-U(\alpha)\big),$$
where $U(\alpha),V(\alpha)$ are defined in \eqref{eq:u-v-alpha-def}. First, note that $Y(\cdot)$ is strictly decreasing, since it is the composition of the increasing function $\alpha\mapsto F^{-1}(\alpha)$ with the function $u\mapsto \frac12(\Omega(u)-u)$, the latter being decreasing (this can be seen by interpreting this function geometrically, or by differentiating and using the fact that $\Omega'(u)=\frac{2}{\pi}\sin^{-1}(u/2)$). Now,
if $t<t'$ are numbers in $[\alpha,1]$ then
$$
y_\alpha(t) = \sqrt{t}\, Y(\alpha/t) < \sqrt{t'}\, Y(\alpha/t) < \sqrt{t'}\, Y(\alpha/t') = y_\alpha(t'),
$$
proving the claim.
\end{proof}
\section{Plancherel measure and sublevel tableaux}
Recall that the distribution of the insertion tableau $P_n$ associated via the Robinson-Schensted correspondence with a uniformly random permutation $\sigma^{(n)}$ in $\mathfrak{S}_n$ is also called the Plancherel measure of order $n$ (this refers to the measure on standard Young tableaux; the distribution of the shape of this tableau is the Plancherel measure on Young diagrams of order $n$ that was mentioned in the introduction). By the remarks made in the introduction, a tableau $P_n$ with this distribution arises by taking the tableau $T_n$ and ``standardizing'' it by replacing each entry $X_k$ with the ranking of $X_k$ in the list $X_1,\dots,X_n$ (i.e., the number $\sigma(k)$ such that $X_k = X^{(n)}_{\sigma(k)}$, where $X^{(n)}_1\le \dots \le X^{(n)}_n$ are the order statistics of $X_1,\dots,X_n$).
Note also that the vector
$(X^{(n)}_1, \dots , X^{(n)}_n)$
of order statistics is independent of the tableau $P_n$ and is distributed uniformly on the simplex $$\Delta_n=\{ (x_1,\dots,x_n)\,:\, 0\le x_1\le \dots \le x_n\le 1\}.$$
It follows that, conversely, if we start with a random standard Young tableau $P_n$ distributed according to the Plancherel measure of order $n$ and a random vector $(W_1,\dots,W_n)$ that is independent of $P_n$ and is distributed uniformly on $\Delta_n$, then the tableau $T_n'$ obtained by replacing each entry $p$ of $P_n$ by $W_p$ is equal in distribution to $T_n$.
We will now apply these observations to prove a simple lemma about sublevel tableaux.
As mentioned above, for any $0<t\le1$, the \textbf{$t$-sublevel tableau of $T_{n-1}$}, which we denote by $T_{n-1}^{(t)}$, is the subtableau of $T_{n-1}$ consisting of those boxes with entries $\le t$. An essential fact that makes our main result possible is a self-similarity property that says that $T_{n-1}^{(t)}$ is distributed roughly as a scaled version of $T_k$ for $k \approx tn$. Since the number of entries in $T_{n-1}^{(t)}$ is itself random, we need to condition on its value to make this statement precise. The details are as follows.
\begin{lem}
\label{lem:self-similarity}
Let $0<t\le 1$ and $0\le k\le n-1$.
\begin{itemize}
\item The number $|T_{n-1}^{(t)}|$ of boxes in $T_{n-1}^{(t)}$ satisfies
\begin{equation}
\label{eq:lln}
\frac{|T_{n-1}^{(t)}|}{n-1} \xrightarrow[n\to\infty]{\mathbb{P}} t.
\end{equation}
\item Conditioned on the event $|T_{n-1}^{(t)}|=k$, the rescaled sublevel tableau $\frac{1}{t} T_{n-1}^{(t)}$ (where the rescaling means that each entry of $T_{n-1}^{(t)}$ is divided by $t$) is equal in distribution to $T_k$.
\end{itemize}
\end{lem}
\begin{proof}
Recall that the entries of $T_{n-1}$ are the i.i.d.~uniform random numbers $X_1,\dots,X_{n-1}$, so the entries of the sublevel tableau $T_{n-1}^{(t)}$ consist of the subset of the numbers $X_1,\dots,X_{n-1}$ which are $\le t$. It follows that the random variable $Z_n=|T_{n-1}^{(t)}|$ is given by
$$ Z_n =\sum_{j=1}^{n-1} 1_{\{X_j\le t\}}, $$
so the first claim follows from the law of large numbers.
Denote by $(Y_1,\dots,Y_{Z_n})$ the sequence (of random length $Z_n$) of those $X_j$'s (for $1\le j\le n-1$) for which $X_j\le t$, in the order in which they appear. From elementary probability theory, it is easy to see that, conditioned on the event $\{Z_n=k\}$, the random variables $Y_1,\dots,Y_k$ are independent and uniformly distributed in $[0,t]$. But now observe that (still conditioning on $\{Z_n=k\}$) the sublevel tableau $T_{n-1}^{(t)}$ can be constructed as follows:
\begin{description}
\item[Standardize] \hfill \\Replace $T_{n-1}$ by a standard Young tableau $P_{n-1}$, where each entry of $P_{n-1}$ is the ranking of the corresponding entry of $T_{n-1}$ in the list $X_1,\dots,X_{n-1}$.
\item[Sublevel] \hfill \\ Take the $k$-sublevel tableau $P_{n-1}^{(k)}$ of $P_{n-1}$.
\item[Destandardize] \hfill \\Replace each entry $p$ of $P_{n-1}^{(k)}$ by the $p$th order statistic $Y^{(k)}_p$ of the sequence $Y_1,\dots,Y_k$.
\end{description}
By the remarks made at the beginning of this section, $P_{n-1}$ is a Plancherel-random standard Young tableau of order $n-1$. We now use the elementary fact that the Plancherel measures are a consistent family of probability measures, in the sense that for any $k\le m$, the $k$-sublevel tableau of a Plancherel-random tableau $Q_m$ of order $m$ is a Plancherel-random tableau of order $k$. (The case $k=m-1$ of this claim corresponds to the simple operation of removing the maximal entry of a Plancherel-random tableau; this clearly implies the general case by induction, and the fact that the claim is true in this case is a version of a well-known property of the Plancherel measures, mentioned for example in \cite[Lemma~1.25]{romik}.)
So, the tableau $P^{(k)}_{n-1}$ is distributed according to the Plancherel measure of order $k$. Finally, since (still conditioning on the event $\{Z_n=k\}$ as before) the vector of order statistics
$(Y^{(k)}_1,\dots,Y^{(k)}_k)=(X^{(n)}_1,\dots,X^{(n)}_k)$
is independent of $P_{n-1}$ (and hence also of $P^{(k)}_{n-1}$) and is distributed like $t$ times a random vector distributed uniformly in $\Delta_k$, again by the remarks made above we have that $T^{(t)}_{n-1}$ is (conditionally on $\{Z_n=k\}$) equal in distribution to $t\cdot T_k$.
\end{proof}
\section{Finishing the proof}
To prove \eqref{eq:limit2}, we first reparametrize the bumping route $\textnormal{\textsf{B}}_{T_{n-1},\alpha}$ according to the parameter $t$ associated with the sublevel tableaux. For each $0\le \alpha< 1$, this reparametrized bumping route will now be a random function $\Phi_{n,\alpha}:[\alpha,1]\to\mathbb{N}\times\mathbb{N}$ defined by
\begin{equation} \label{eq:bumping-reparam}
\Phi_{n,\alpha}(t) = \big(b_{T_{n-1},\alpha}(m),m\big)
\end{equation}
where for each $t$ we denote by $m$ the minimal number for which \linebreak $\big(b_{T_{n-1},\alpha}(m),m\big)$ lies outside the sublevel tableau $T_{n-1}^{(t)}$. Note that almost surely we have that
\begin{align*}
\Phi_{n,\alpha}(\alpha)=& \big(b_{T_{n-1},\alpha}(1),1\big),\\
\Phi_{n,\alpha}(1)=& \big(b_{T_{n-1},\alpha}(k_{T_{n-1},\alpha}),k_{T_{n-1},\alpha}\big),
\end{align*}
and the range of $\Phi_{n,\alpha}$ consists of the entire bumping route $\textnormal{\textsf{B}}_{T_{n-1},\alpha}$.
\begin{thm}
For any $0\le \alpha<1$ and $\epsilon>0$ we have
\begin{equation} \label{eq:stronger}
\mathbb{P}\left( \max_{\alpha\le t\le 1} \left\lVert
\frac{\Phi_{n,\alpha}(t)}{\sqrt{n}} - \big(x_\alpha(t),y_\alpha(t)\big)
\right\rVert > \epsilon
\right) \xrightarrow[n\to\infty]{} 0.
\end{equation}
\end{thm}
\begin{proof}
Fix $\alpha\in [0,1)$. First, we prove the weaker statement that for any $\epsilon>0$ and $t\in[\alpha,1]$ we have
\begin{equation} \label{eq:weaker}
\mathbb{P}\left( \left\lVert
\frac{\Phi_{n,\alpha}(t)}{\sqrt{n}} - \big(x_\alpha(t),y_\alpha(t)\big)
\right\rVert > \epsilon
\right) \xrightarrow[n\to\infty]{} 0.
\end{equation}
Denote $Z_n=|T_{n-1}^{(t)}|$ as before, and let $\delta>0$ be some small number (depending on $\epsilon$) whose value will be fixed shortly. We have
\begin{multline}
\mathbb{P}\left( \left\lVert
\frac{\Phi_{n,\alpha}(t)}{\sqrt{n}} - \big(x_\alpha(t),y_\alpha(t)\big)
\right\rVert > \epsilon
\right)
\\
\shoveleft{\leq
\mathbb{P}\left(
\left| \frac{Z_n}{n-1} - t \right| > \delta
\right)}
\\
{ \hspace{42pt} + \hspace{-5pt}
\sum_{\left|\frac{k}{n-1}-t\right| \le \delta} \hspace{-5pt} \mathbb{P}(Z_n=k)
\mathbb{P}\left( \left\lVert
\frac{\Phi_{n,\alpha}(t)}{\sqrt{n}} - \big(x_\alpha(t),y_\alpha(t)\big)
\right\rVert > \epsilon \,\Big|\, Z_n=k
\right)}
\\
\shoveleft{\leq
\mathbb{P}\left(
\left| \frac{Z_n}{n-1} - t \right| > \delta
\right)}
\\
{ +
\max_{\left|\frac{k}{n-1}-t\right| \le \delta}
\mathbb{P}\left( \left\lVert
\frac{\Phi_{n,\alpha}(t)}{\sqrt{n}} - \big(x_\alpha(t),y_\alpha(t)\big)
\right\rVert > \epsilon \,\Big|\, Z_n=k
\right).}
\label{eq:prob-twoparts}
\end{multline}
In the last expression, the first term tends to $0$ as $n\to\infty$, by \eqref{eq:lln}.
Let $k=k(n)$ be the value for which the maximum of the second term is attained.
Note that the second claim of Lemma~\ref{lem:self-similarity} implies that the conditional probability in the second term can be replaced by its unconditional counterpart
\begin{multline}
\label{eq:prob-replaced}
\mathbb{P} \left( \left\lVert
\frac{\Phi_{k+1,\alpha/t}(1)}{\sqrt{n}} - \big(x_\alpha(t),y_\alpha(t)\big)
\right\rVert > \epsilon
\right)
\\ =
\mathbb{P}\left( \left\lVert
\sqrt{\frac{k}{n}}\ \frac{\Phi_{k+1,\alpha/t}(1)}{\sqrt{k}} - \sqrt{t}\ \big(x_{\alpha/t}(1),y_{\alpha/t}(1)\big)
\right\rVert > \epsilon
\right).
\end{multline}
If we had the precise equality $k=tn$, it would immediately follow from \eqref{eq:asym-det} that this probability tends to $0$ as $n$ (and therefore also $k$) tends to $\infty$. As it is, such an equality does not hold; however, we restricted $k$ to a range such that
$$
t-\delta \le \liminf_{n\to\infty} \frac{k}{n} \leq \limsup_{n\to\infty} \frac{k}{n} \le t+\delta.
$$
This is good enough, since it is easy to check that if $\delta$ is taken (as a function of $\epsilon$) to be a small enough positive number, then the right-hand side of \eqref{eq:prob-replaced} can be bounded from above by
\begin{multline}
\mathbb{P}\left( \left\lVert
\frac{\Phi_{k+1,\alpha/t}(1)}{\sqrt{k}} - \big(x_{\alpha/t}(1),y_{\alpha/t}(1)\big)
\right\lVert > \frac{\epsilon}{2}
\right)
\\ +
\mathbb{P}\left( \left\lVert
\frac{\Phi_{k+1,\alpha/t}(1)}{\sqrt{k}} \right\rVert > 3\sqrt{2}
\right).
\label{eq:prob-bound2}
\end{multline}
The first probability tends to $0$ as $n\to\infty$
by \eqref{eq:asym-det}. The second probability is bounded by the probability that a Plancherel-random Young diagram of order $k$ has a row or column of length $\ge 3\sqrt{k}$; it is well-known that this probability decreases to $0$ at a rate that is exponential in $\sqrt{k}$ (see \cite[Lemma~1.5]{romik}). Thus, combining these observations with \eqref{eq:prob-twoparts}, \eqref{eq:prob-replaced} and the bound \eqref{eq:prob-bound2} proves \eqref{eq:weaker}.
Finally, to finish the proof we need to show that \eqref{eq:weaker} implies \eqref{eq:stronger}. This is a standard argument: first, \eqref{eq:weaker} clearly implies a version of \eqref{eq:stronger} in which the maximum is taken over finitely many values $\alpha \le t_1 < \dots < t_p \le 1$ of $t$. Second, since both the functions $t\mapsto \Phi_{n,\alpha}(t)$ and $t\mapsto \big(x_\alpha(t),y_\alpha(t)\big)$ have the property that their $x$-coordinate is weakly decreasing and their $y$-coordinate is weakly increasing,
and since $t\mapsto \big(x_\alpha(t),y_\alpha(t)\big)$ is continuous, knowing that the bound
$$\left\lVert \frac{\Phi_{n,\alpha}(t)}{\sqrt{n}} - \big(x_\alpha(t),y_\alpha(t)\big) \right\rVert > \epsilon$$
holds for all values of $t$ in a finite set that is sufficiently dense in $[\alpha,1]$ ensures that the same inequality (with $\epsilon$ replaced by, say, $2\epsilon$) will hold
for all $t\in [\alpha,1]$. The details are easy and are left to the reader.
\end{proof}
\begin{proof}[Proof of \eqref{eq:limit2}]
It is now easy to derive \eqref{eq:limit2} from \eqref{eq:stronger}. The idea is that the relation between $t$ and $m$ in \eqref{eq:bumping-reparam} can be inverted, expressing the $m$th point $\big(b_{T_{n-1},\alpha}(m),m\big)$ of the bumping route as $\Phi_{n,\alpha}(t(m))$ where $t(m)$ is the minimal value $t\ge \alpha$ for which the $y$-coordinate of $\Phi_{n,\alpha}(t)$ is equal to $m$. (Note that $t(\cdot)$ also depends on $n$ and $\alpha$, but for convenience we leave this dependence implicit in our notation.) Expressing \eqref{eq:stronger} in terms of $t(m)$ gives the convergence in probability
$$
\max_{1\le m\le k_{T_{n-1},\alpha}} \left\lVert
\frac{\big(b_{T_{n-1},\alpha}(m),m\big)}{\sqrt{n}}-\big(x_\alpha(t(m)),y_\alpha(t(m))\big)\right\rVert \xrightarrow[n\to\infty]{\mathbb{P}} 0,
$$
which can be broken down into two separate convergence relations,
\begin{equation}
\max_{1\le m\le k_{T_{n-1},\alpha}} \left| y_\alpha(t(m)) - \frac{m}{\sqrt{n}} \right| \xrightarrow[n\to\infty]{\mathbb{P}} 0,
\label{eq:first-conv}
\end{equation}
\begin{equation}
\max_{1\le m\le k_{T_{n-1},\alpha}} \left| \frac{b_{T_{n-1},\alpha}(m)}{\sqrt{n}}-x_\alpha(t(m))\right| \xrightarrow[n\to\infty]{\mathbb{P}} 0.
\label{eq:second-conv}
\end{equation}
Now observe that $z\mapsto y_\alpha^{-1}(z\wedge \kappa(\alpha))$ is a continuous function on $[0,2\kappa(\alpha)]$. Combining this with \eqref{eq:first-conv} and the fact that (by \eqref{eq:limit1}) \linebreak $k_{T_{n-1},\alpha}/\sqrt{n} \le 2\kappa(\alpha)$ with asymptotically high probability, we see that
$$
\max_{1\le m\le k_{T_{n-1},\alpha}} \left| t(m) - y_\alpha^{-1}\left(\frac{m}{\sqrt{n}} \wedge \kappa(\alpha) \right) \right| \xrightarrow[n\to\infty]{\mathbb{P}} 0.
$$
Finally, this relation, together with \eqref{eq:second-conv} and the fact that $x_\alpha(\cdot)$ is continuous, implies that
$$
\max_{1\le m\le k_{T_{n-1},\alpha}} \left| \frac{b_{T_{n-1},\alpha}(m)}{\sqrt{n}}-x_\alpha\left(y_\alpha^{-1}\left(\frac{m}{\sqrt{n}} \wedge \kappa(\alpha) \right)\right)\right| \xrightarrow[n\to\infty]{\mathbb{P}} 0,
$$
which is exactly \eqref{eq:limit2}.
\end{proof}
\section*{Acknowledgments}
P.\'S.'s research has been supported by a grant number SN 101/1-1 from \emph{Deutsche Forschungsgemeinschaft}.
Dan Romik's work was supported by the National Science Foundation under grant DMS-0955584 and by grant \#228524 from the Simons Foundation.
\bibliographystyle{alpha}
|
1,116,691,499,280 | arxiv | \section{Introduction}
Throughout the gig economy, numerous digital platforms algorithmically profile, control, and discipline workers that offer on-demand services to consumers. Data collection and predictive modeling are critical for a typical platform’s business as machine learning algorithms power ranking, scoring, and classification tasks of various kinds \citep{woodcock2019gig, gray2019ghost, schor2021after}.
Troves of academic scholarship document the emergence and preponderance of precarity in the gig economy. \cite{wood2019good} argue that platform-based algorithmic control can lead to ``low pay, social isolation, working unsocial and irregular hours, overwork, sleep deprivation and exhaustion.'' This is further exacerbated by ``high levels of inter-worker competition with few labor protections and a global oversupply of labor relative to demand.'' In response, there have been numerous attempts by gig workers to organize in an effort to reconfigure working conditions. A growing repertoire of strategies, as vast as it is eclectic, uses both physical and digital means towards this goal. Indeed, workers have shown significant ingenuity in creating platform-specific infrastructure, such as their own mobile apps, to organize the labor side of the platform \citep{chen2018thrown, rahman2021invisible}. Yet, ``the upsurge of worker mobilization should not blind us to the difficulties of organizing such a diverse and spatially dispersed labor force.'' \citep{vallas2020platforms}
Beyond the gig economy, evidence of consumers seeking to influence the algorithms that power a platform's business is abundant. Examples include social media users attempting to suppress the algorithmic upvoting of harmful content by sharing screenshots rather than original posts \citep{twitter}, or individuals creating bots to influence crowd-sourced navigation systems~\citep{sinai14bots}.
The ubiquity of such strategic attempts calls for a principled study of how coordinated groups can wield control over the digital platforms to which they contribute data.
In this work, we study how a collective of individuals can algorithmically strategize against a learning platform. We envision a collective that pools the data of participating individuals and executes an algorithmic strategy by instructing participants how to modify their own data. The firm in turn solves a machine learning problem over the resulting data. The goal of the collective is to redirect the firm's optimization towards a solution that serves the collective. Notably, coordination is a crucial lever. When data are plentiful, a single individual lacks the leverage to unilaterally change the output of a learning algorithm; in contrast, we show that even small collectives can exert substantial influence.
\subsection{Our contribution}
We initiate a principled study of algorithmic collective action in digital platforms that deploy machine learning algorithms. We propose a simple theoretical model of a collective interacting with a firm’s learning algorithm. The size of the collective is specified by a value~$\alpha>0$ that corresponds to the fraction of participating individuals in a population drawn from a base distribution~$\mathcal P_0.$
The firm observes the mixture distribution $\mathcal P=\alpha \mathcal P^* + (1-\alpha) \mathcal P_0,$ where $\mathcal P^*$ depends on the strategy of the collective, and runs a learning algorithm $\mathcal A$ on $\mathcal P$.
We investigate the consequences of this model in three fundamental learning-theoretic settings, supported by a systematic empirical evaluation on a real resume classification task from a gig platform. In each setting, we come up with algorithmic collective strategies and characterize different success criteria as a function of the collective’s size~$\alpha$. We exhibit \emph{critical thresholds} for the value of~$\alpha$ at which the collective succeeds. Our theory and experiments support the conclusion that collectives of vanishing fractional size can exert significant control over the firm’s learning algorithm.
\paragraph{Classification.}
In line with economic theory, we start with the case of an optimal firm that has full knowledge of the distribution $\mathcal P$. The firm chooses the Bayes optimal predictor~$f=\mathcal A(\mathcal P)$ on the distribution~$\mathcal P$. In the context of classification,
a natural class of objectives for the collective is to correlate a signal $g(\cdot)$ with a target label $y^*$: an individual described by data point $(x,y)$ succeeds if $f(g(x))=y^*$ at test time.
For this goal, a natural strategy is to perturb each participating data point $(x,y)$ by applying the signal $g(\cdot)$ to $x$ and switching the label from $y$ to $y^*$ at training time. That is, $\mathcal P^*$ is the distribution of $(g(x), y^*)$ for a random draw of a labeled data point~$(x, y)$ from~$\mathcal P_0$. We prove that a collective of vanishing fractional size succeeds with high probability by implementing this strategy, provided that the signal $g(x)$ is unlikely to be encountered in the base distribution~$\mathcal P_0$. The success probability is maximized for an optimal classifier and deteriorates gracefully with the suboptimality of the classifier.
In practice, the signal~$g(x)$ may correspond to adding a hidden watermark in image and video content, or subtle syntactic changes in text. It is reasonable to assume that individuals are indifferent to such inconsequential changes to their features. In fact, conventional wisdom in machine learning has it that such hidden signals are easy to come by in practice. The ability to change the label of an instance, however, is a more strenuous requirement. We therefore propose a variant of the strategy where a participating individual does not need to modify the label of their data point. We show that this strategy still succeeds, while quantifying precisely how it diminishes the collective’s success rate as a function of its size and a key parameter of $\mathcal P_0$.
We provide analogous results when the collective's goal is for the firm's predictions to ignore some subset of the feature information. Given a map $g(x)$, the collective succeeds if $f(x)=f(g(x))$. Here, $g(x)$ is a summary of $x$ that, for example, removes private or sensitive information in~$x$.
\paragraph{Experimental evaluation.}
We conduct extensive experiments on a dataset of almost thirty thousand resumes from a gig platform for freelancers. The machine learning task is to tag resumes with a set of ten skills related to work in the IT sector. Through more than two thousand model training runs involving a BERT-like language model, we investigate the predictions made by our theory. What emerges is a striking correspondence between our empirical findings and the theory. The ability to modify both resume and label leads to a near-perfect success rate at collective sizes of only a fraction of a percent of the population, corresponding to fewer than one hundred modified resumes. The label-free strategy still succeeds reliably, albeit at a higher threshold, corresponding to a few hundred resumes. The more well-trained the model is, the lower the threshold for the collective's success. The placement pattern of the signal in the resume is largely irrelevant, so long as the token we plant is unique within the corpus of resumes.
Our theory predicts that collectives, in a certain precise sense, must compete with the strongest alternative signal in the data. The weaker the alternative signals, the lower the threshold for success. To confirm this hypothesis experimentally, we randomize the labels of a random fraction of the data. Confirming our theory, we observe that increasing the fraction of randomized labels, and hence diminishing the strength of alternative signals, indeed lowers the threshold for success of the collective. This observation suggests a blessing of dimensionality: if the data contains many weak signals, as high-dimensional data tends to, algorithmic collectives are especially powerful.
\paragraph{Risk minimization and gradient descent.}
Generalizing beyond classification, we consider the power of collectives in convex risk minimization and gradient-based learning with possibly nonconvex objectives. In the first case, we show that collectives can force the firm to deploy any model with small suboptimality on $\mathcal P_0$ of the collective's choosing.
In the second case, we show that given a more powerful interaction model the collective can influence the firm to deploy any desired model even under nonconvexity, as long as the path from the initial model, computed on $\mathcal P_0$, to the desired model does not traverse large gradients when evaluated on $\mathcal P_0$. Moreover, despite the nonconvexity, convergence to the target is achieved at a convex rate. In both problem settings, the analyzed collective strategies rely on exerting influence on the gradients computed by the firm.
\subsection{Related work}
Our approach to algorithmic collective action is decidedly not adversarial. Instead, the strategic manipulations arise through a misalignment of the firm's and the individuals' objectives. Individuals legitimately optimize their utility through data sharing and coordination. Yet, at a technical level our results relate to topics studied under the umbrella of adversarial machine learning. Most closely related is the line of work on \emph{data poisoning attacks} that seeks to understand how data points can be adversarially ``poisoned'' to degrade the performance of a predictive model at test time.
We refer to recent surveys for an overview of data poisoning attacks~\citep{zhiyi22survey}, and backdoor attacks more specifically~\citep{guo22backdoor}.
While the literature on poisoning attacks focuses predominantly on diminishing the performance of the learning algorithm, documented empirical successes \citep{cherepanova2021lowkey,geiping2021witches} hint at the impact that algorithmic collective action can have on deep learning models.
Despite the increasing number of studies on backdoor attacks and defenses, theoretical work explaining how underlying factors affect the success of backdoor attacks has been limited~\citep{grosse22demystify}.
We point out two recent works that study the relationship between the number of manipulated data points and the success probability of the attack.
\cite{manoj2021excess} show that a fixed number of points is sufficient for backdoor attacks to succeed in binary classification problems, provided that the memorization capacity of the model is sufficiently large.
\cite{cina21learningcurves} empirically investigate backdoor learning curves across many image recognition tasks and they observe curves with diminishing returns, similar in shape to those in our experiments.
Our analysis of Bayes optimal classification provides a new, complementary theoretical perspective and sheds light on the effectiveness of practical data manipulation strategies in a way that is surprisingly predictive of our empirical observations and previous empirical results in adversarial learning.
Our analysis of risk minimization is reminiscent of model-targeted attacks~\citep{suya21model,farhadkhani22grad}, which aim to bias the learner towards a target model. Our gradient-control strategy resembles the counter-gradient attack in~\citep{farhadkhani22grad}.
While the insights from these prior works are valuable to inform the feasibility of collective action in convex risk minimization, our work differs from these existing studies in its focus on the role of collective size and the analysis of nonconvex losses.
Only a handful of works in the adversarial machine learning literature have questioned the institution-centric perspective of the community and discussed the political dimension of adversarial machine learning~\citep{Albert2020PoliticsOA,vincent21leverage,kendra21sop}. In this context, a line of work considers the socially beneficial use of adversarial learning techniques, e.g.,~\citep{delobelle21ethical, shan20fawkes, abebe2022adversarial, kulynych20pot,li2022untargeted}.
Our work can also be viewed as a conceptual reversal of \emph{strategic classification}~\citep{hardt16strategic}. In strategic classification, a firm anticipates the optimal response of a strategic individual to a decision rule. Instead we consider individuals that strategically anticipate the optimizing behavior of the firm, something recently considered by~\cite{zrinc21follow}. Furthermore, our work is conceptually different from strategic classification in its focus on the role and objectives of workers and consumers on online platforms rather than the firm. Another crucial departure from the set of problems considered in strategic classification is our emphasis on collective rather than individual strategies.
The idea of collective action on digital platforms has also been previously studied. \cite{elliot21recourse} show how algorithmic recourse can be improved through coordination. \citet{vincent2019data} examine the effectiveness of \emph{data strikes}. Extending this work to the notion of \emph{data leverage}, \citet{vincent21leverage} describe various ways of ``reducing, stopping, redirecting, or otherwise manipulating data contributions'' for different purposes. See also \citet{vincent21CDC}. Our work provides a theoretical framework for understanding the effectiveness of such strategies, as well as studying more complex algorithmic strategies that collectives may deploy.
Appendix~\ref{sec:extended-related} continues our discussion of related work.
\section{Problem formulation}
We study the strategic interaction of a firm operating a predictive system with a population of individuals. We assume
that the the firm deploys a learning algorithm~$\mathcal A$ that operates on data points in a universe~$\mathcal Z=\mathcal X\times\mathcal Y$. Each individual corresponds to a single data point $z\in\mathcal Z$, typically a feature--label pair. We model the population of individual participants as a distribution $\mathcal P_0$ over $\mathcal Z$.
We say that a fraction $\alpha>0$ of the individuals form a \emph{collective} in order to strategically respond to the firm's learning behavior.
The collective agrees on a potentially randomized strategy $h:\mathcal Z\rightarrow\mathcal Z$ from a space of available strategies $\mathcal H$. The possible strategies $\mathcal H$ capture feasible changes to the data.
For example, content creators on a video streaming platform may be indifferent between videos that differ only in a hidden watermark not visible to human viewers. Freelancers may be indifferent between two resumes that differ only in inconsequential syntactic details.
The firm therefore observes a mixture distribution
\begin{equation*}
\mathcal P = \alpha \mathcal P^* + (1-\alpha) \mathcal P_0,\,
\end{equation*}
where we use $\mathcal P^*$ to denote the distribution of $h(z), z\sim \mathcal P_0$.
The collective strives to choose a strategy $h$ so as to maximize a measure of success over the solution~$f=\mathcal A(\mathcal P)$ chosen by the firm. Here, $f$ is a mapping from features to labels, $f:\mathcal X\rightarrow\mathcal Y$. Given a strategy, we use $S(\alpha)$ to denote the level of success achieved by a collective of size $\alpha$. The central question we study is how the success $S(\alpha)$ grows as a function of collective size~$\alpha,$ and how large $\alpha$ needs to be in order to achieve a target success level.
\begin{definition}[Critical mass] The critical mass for a target success level $S^*$ is defined as the smallest $\alpha$ for which there exists a strategy such that $S(\alpha)\geq S^*$.
\end{definition}
Note that, although motivated from the perspective of labor, our formal model can also serve as a basis for studying collective action on the consumer side of digital platforms.
Before presenting our results we briefly discuss why we focus on collective strategies in this paper.
\paragraph{Why collective action?} By engaging in collective action, individuals can exert influence on the learning algorithm that they could not achieve by acting selfishly. In large-population settings such as online platforms, an individual contributes an infinitesimal fraction of the data used by the learning algorithm. Thus, under reasonable manipulation constraints, individual behavior is largely powerless in systematically changing the deployed model. Instead, individuals are limited to simple adversarial attacks or individual strategies that do not have lasting effects on the learning outcome. By coordinating individuals, however, collectives can wield enough power to steer learning algorithms towards desired goals. In subsequent sections we show that collectives can often do so while only representing a small fraction of the training data.
\section{Collective action in classification}
We start with classification under the assumption that the firm chooses an approximately optimal classifier on the data distribution~$\mathcal P$.
\begin{definition}[$\epsilon$-optimal classifier]
\label{def:clf-opt}
A classifier~$f\colon\mathcal X\to\mathcal Y$ is $\epsilon$-optimal under the distribution~$\mathcal P$ if there exists a $\mathcal P'$ with $\mathrm{TV}(\mathcal P,\mathcal P')\leq \epsilon$ such that
\begin{equation*}
\label{eq:bayes-opt}
f(x) = \mathop{\rm argmax}_{y\in\mathcal Y} \mathcal P'(y|x)\,.
\end{equation*}
\end{definition}
\noindent Note that a $0$-optimal classifier is the Bayes optimal classifier with respect to the zero--one loss.
Under the above assumption, we focus on two general goals for the collective: \emph{planting a signal} and \emph{erasing a signal}.
\subsection{Planting a signal}
\label{sec:trigger}
Assume the collective wants the classifier to learn an association between an altered version of the features $g(x)$ and a chosen target class $y^*$. Formally, given a transformation $g:\mathcal X\rightarrow\mathcal X$, the collective wants to maximize the following measure of success:
\[
S(\alpha)=\Pr_{x\sim \mathcal P_0} \left\{ f(g(x)) = y^*\right\}.
\]
We call this objective ``planting a signal'' and $\mathcal X^*=\{g(x)\colon x\in\mathcal X\}$ the signal set. For example, $g(x)$ could be instance $x$ with an inconsequential trigger (such as a video with an imperceptible watermark or a resume with a unique formatting) and $y^*$ could be a label indicating that the instance is of high quality (e.g., a high-quality video or a highly qualified individual).
As another example, the collective may have an altruistic goal to help individuals in a vulnerable subpopulation $\mathcal X_0\subseteq\mathcal X$ achieve a desired outcome $y^*$. In this case,
$g(x)$ could be a mapping from $x$ to a randomly chosen instance in~$\mathcal X_0$.
We provide natural strategies for planting a signal and characterize their success as a function of $\alpha$. The key parameter that we identify as driving success is the \emph{uniqueness} of the signal.
\begin{definition}[$\xi$-unique signal] We say that a signal is $\xi$-unique if it satisfies $\mathcal P_0(\mathcal X^*)\leq \xi$.
\end{definition}
In addition, success naturally depends on how suboptimal $y^*$ is on the signal set under the base distribution. To formalize this dependence, we define the suboptimality gap of $y^*$ as
$$\Delta = \max_{x\in\mathcal X^*}\left(\max_{y\in\mathcal Y}\,\mathcal P_0(y|x) - \mathcal P_0(y^*|x)\right)\,.$$
We consider two possibilities for the space of available strategies $\mathcal H$. First, we assume that the individuals can modify both features and labels. We call the resulting strategies \emph{feature--label} strategies. Modifying features by, say, planting a trigger often comes at virtually no cost. Changing the label, however, may be hard, costly, or even infeasible. This is why we also consider \emph{feature-only} strategies; such strategies only allow changes to features.
\paragraph{Feature--label signal strategy.} We define the feature--label signal strategy as
\begin{equation}
\label{eq:trigger-label}
h(x,y)=(g(x), y^*)\,.
\end{equation}
The result below quantifies the success of this strategy in terms of the collective size and the uniqueness of the signal.
\begin{theorem}
\label{thm:trigger-label}
Consider the feature--label signal strategy and suppose that the signal is $\xi$-unique. Then, the success against an $\epsilon$-optimal classifier is lower bounded by
\[S(\alpha) \ge 1 - \frac{1-\alpha}{\alpha} \cdot \Delta \cdot \xi - \frac{\epsilon}{1-\epsilon}\,.\]
\end{theorem}
Rearranging the terms, we obtain an upper bound on the critical mass given a desired success probability (e.g., $90\%$).
\begin{corollary}
Suppose the signal is $\xi$-unique. Then, the critical mass for achieving success $S^*\in(0,1)$ with feature--label strategies against an $\epsilon$-optimal classifier is bounded~by
\begin{equation}
\label{eq:feature-label-alpha}
\alpha^* \leq \frac{\Delta\cdot \xi}{1 -S^* - \frac{\epsilon}{1-\epsilon} + \Delta \cdot \xi}.
\end{equation}
\end{corollary}
Therefore, in order to achieve success it suffices to have a collective size proportional to the uniqueness of the signal and the suboptimality of $y^*$ on the signal set, as long as these parameters are sufficiently small relative to the target error rate $1-S^*$. This suggests that planting signals that are exceedingly ``rare'' under the base distribution can be done successfully by small collectives--- a property of feature--label strategies that we empirically validate in Section~\ref{sec:exps}.
\begin{figure}
\begin{center}
\includegraphics[width=0.44\columnwidth]{plots/success-shape-measure.pdf}
\includegraphics[width=0.44\columnwidth]{plots/success-shape-suboptimality.pdf}
\end{center}
\caption{Illustration of the success rate predicted by Theorem~\ref{thm:trigger-label}. In the first we fix $\epsilon=0$ and vary $\xi$, and in the second we fix $\xi$ and vary the classifier's suboptimality, $\epsilon$. We upper bound $\Delta$ by one.}
\end{figure}
In the next result we consider feature-only strategies. An impediment to the success of such strategies is the situation where two likelihoods $\mathcal P_0(x|y)$ and $\mathcal P_0(x|y')$ for distinct labels $y\ne y'$ have no overlapping support. In this case, there is no reason to expect that planting a signal in one class has any effect on the other class. This is the reason why we make one additional assumption that there exists a number $p>0$ such that $\mathcal P_0(y^*|x)\ge p$ for all $x\in\mathcal X$.
\paragraph{Feature-only signal strategy.} We define the feature-only signal strategy as
\begin{equation}
\label{eq:trigger-only}
h(x,y)=
\begin{cases}
(g(x), y^*), & \text{ if } y=y^*,\\
(x, y), & \text{otherwise}.
\end{cases}
\end{equation}
This strategy achieves a similar success rate as the feature--label strategy, but the success diminishes with the constant~$p$.
\begin{theorem}
\label{thm:trigger-only}
Consider the feature-only signal strategy and suppose that the signal is $\xi$-unique. Further, suppose there exists $p>0$ such that $\mathcal P_0(y^*|x)\ge p,\forall x\in\mathcal X$. Then, the success against an $\epsilon$-optimal classifier is lower bounded by
\[S(\alpha) \ge 1 - \frac{1-p}{p\alpha} \cdot \xi - \frac{\epsilon}{1-\epsilon}\,.\]
\end{theorem}
The critical mass for achieving a target success probability is thus bounded as follows.
\begin{corollary}
Suppose the signal is $\xi$-unique. Then, the critical mass for achieving success $S^*\in(0,1)$ with feature-only strategies against an $\epsilon$-optimal classifier is bounded~by
\begin{equation}
\label{eq:feature-only-alpha}
\alpha^* \leq \frac{1-p}{p} \,\frac{\xi}{1 -S^* - \frac{\epsilon}{1-\epsilon}}.
\end{equation}
\end{corollary}
Whenever the positivity constant $p$ is smaller than $0.5$, the critical mass \eqref{eq:feature-only-alpha} that guarantees success of feature-only strategies is at least as large as the critical mass \eqref{eq:feature-label-alpha} for feature--label strategies, as expected.
The positivity constant $p>0$ may be excessively small over the entire data universe. A standard fix to this problem is to restrict $\mathcal P_0$ to a subset where the constant is larger, and pay a penalty for the amount of truncation in the bound. For example, if there exists $R\subseteq\mathcal X$ such that $\mathcal P_0(R)\geq 99\%$, but the positivity constant over $R$ is much larger than $p$, then one can obtain a more powerful version of Theorem~\ref{thm:trigger-only}.
\subsection{Erasing a signal}
Next, we assume the collective wants the classifier to be invariant under a transformation $g:\mathcal X\rightarrow\mathcal X$ of the features. In particular, the success is measured with respect to:
\[S(\alpha)= \Pr_{x\sim\mathcal P_0} \{f(x)=f(g(x))\}.\]
In other words, the collective wants the classifier to output the same predictions for all $x$ and $x'$ that have $g(x)=g(x')$. The map $g$ can be thought of as a summary of $x$ that removes some feature information. We call this objective ``erasing a signal.'' For example, if the collective wants the deployed model to be insensitive to the value of a particular feature $j^*$, then it can use $g(x) = x'$ where $x'_j = x_j$ for $j\neq j^*$ and $x'_{j^*} = 0$. The feature $j^*$ could be the length of a video that content creators do not want to affect the ranking of the content, or it could be a sensitive demographic feature that a collective wants to be independent of the predicted label.
\paragraph{Erasure strategy.} We define the erasure strategy as
\[h(x,y)=\left(x,\mathop{\rm argmax}_{y\in\mathcal Y} \mathcal P_0(y|g(x)) \right).\]
As before, the success of the strategy depends on problem-dependent quantities. In this case, the quantity of interest is the sensitivity of the labels to the erased signal. We capture this sensitivity in the parameter $\tau$, defined as
\[\tau = \E_{x\sim\mathcal P_0} \max_{y\in\mathcal Y}\left|\mathcal P_0(y|x)-\mathcal P_0(y|g(x))\right|.\]
Intuitively, $\tau$ is small if observing the whole feature vector $x$, instead of just the summary $g(x)$, reveals little additional information about the label.
\begin{theorem}\label{thm:dissocation}
Consider the erasure strategy. Then, the success against an $\epsilon$-optimal classifier is lower bounded by
\[S(\alpha)\geq 1 - \frac{2(1-\alpha)}{\alpha} \cdot \tau - \frac{\epsilon}{1-\epsilon}.\]
\end{theorem}
We rearrange the terms and derive a bound on the critical mass that guarantees a signal can be erased with a desired probability.
\begin{corollary}
The critical mass for achieving success $S^*\in~(0,1)$ is bounded by
\[\alpha^*\leq \frac{\tau}{\frac{1}{2}(1-S^*) - \frac{\epsilon}{2(1-\epsilon)}+\tau}.\]
\end{corollary}
The less sensitive the labels to the erased information, the smaller the collective needed to successfully enforce a decision rule independent of the protected information.
In contrast to the strategies in Section~\ref{sec:trigger}, the erasure strategy requires knowledge of statistics about $\mathcal P_0$. This highlights an important benefit of collective action: information sharing. Information about the base distribution is typically difficult to obtain for individual platform users. However, a collective can pool their feature--label information to estimate properties of the distribution from samples; the larger the collective, the better the estimate and consequently the more effective the strategy.
\section{Collective action in risk minimization}
\label{sec:risk-min}
We next study the effect of collective size when the learner is solving parametric risk minimization. Here the firm is choosing a model from a parameterized set $\{f_\theta\}_{\theta\in\Theta}$. We will use $\mathcal A(\mathcal P)$ to denote an element in $\Theta$ that determines the model chosen by the firm.
We begin by studying convex risk minimizers. Then, motivated by nonconvex settings, we look at gradient-descent learners without imposing any convexity assumptions on the objective.
Our main working assumption will be that of a risk-minimizing firm.
\begin{definition}[Risk minimizer]
Fix a loss function $\ell$. The firm is a risk minimizer if under distribution $\mathcal P$ it determines the parameter of the model $f_\theta$ according to
\[\theta = \mathop{\rm argmin}_{\theta'\in\Theta}\; \E_{z\sim \mathcal P} \ell(\theta';z).\]
\end{definition}
\noindent We implicitly assume that $\theta$ is a unique minimizer.
\subsection{Convex risk minimization}
To begin, we assume that $\ell(\theta;z)$ is a convex function of $\theta$, and that the collective's goal is to move the model from $\theta_0$---the optimal model under the base distribution $\mathcal P_0$---to a target model $\theta^*$. To that end, for a given target model $\theta^*\in\Theta$, we measure success in terms of
\[S(\alpha)= - \|\theta-\theta^*\|.\]
Here, $\|\cdot\|$ can be any norm (as long as it is kept fixed in the rest of the section).
In line with first-order optimality conditions for convex optimization, the influence of the collective on the learning outcome depends on the collective's ability to influence the average gradient of $\ell$.
To simplify notation, let $g_{\mathcal P}(\theta') = \E_{z\sim \mathcal P}\nabla \ell(\theta';z)$
denote the expected gradient of the loss over distribution $\mathcal P$ measured at a point $\theta'\in\Theta$.
\paragraph{Gradient-neutralizing strategy.} Define the gradient-neutralizing strategy as follows. Find a \emph{gradient-neutralizing} distribution $\mathcal P'$ for $\theta^*$, meaning $\angle(g_{\mathcal P'}(\theta^*), - g_{\mathcal P_0}(\theta^*))=0$. Then, draw $z'\sim\mathcal P'$ and let
\[
h(z) =\begin{cases}
z', &\text{with probability }\min\left(1,\frac{1}{\alpha} \frac{\|g_{\mathcal P_0}(\theta^*)\|}{\|g_{\mathcal P'}(\theta^*)\| + \|g_{\mathcal P_0}(\theta^*)\|} \right),\\
z, & \text{else}.
\end{cases}
\]
For example, in generalized linear models (GLMs) gradients are given by $\nabla \ell(\theta;(x,y)) = x(\mu_\theta(x) - y)$, where $\mu_\theta(\cdot)$ is a mean predictor (see, e.g., Chapter 3 in \citep{efron2022exponential}). Therefore, one can obtain a gradient-neutralizing distribution by simply letting $h(x,y) = (x',y')$, where $x' = - g_{\mathcal P_0}(\theta^*)$ and $y'$ is any value less than $\mu_\theta(x')$. Alternatively, if the collective is restricted to feature-only strategies, they can set $x' = - g_{\mathcal P_0}(\theta^*)$ only if $y< \mu_\theta(x')$, and $x'=0$ otherwise. As long as the label distribution has sufficiently large support under $\mathcal P_0$, in particular $y< \mu_\theta(- g_{\mathcal P_0}(\theta^*))$ with nonzero probability, this strategy likewise results in a gradient-neutralizing distribution.
\begin{theorem}
\label{thm:convex}
Suppose there exists a gradient-neutralizing distribution $\mathcal P'$ for $\theta^*$. Then, if the loss is $\mu$-strongly convex, the success of the gradient-neutralizing strategy is bounded~by
\[S(\alpha)\geq \frac{1}\mu\min\left(\alpha \|g_{\mathcal P'}(\theta^*)\| - (1-\alpha)\|g_{\mathcal P_0}(\theta^*)\|,0\right).\]
\end{theorem}
The natural target success for the collective is for $\theta^*$ to be reached exactly; this is achieved when $S(\alpha)=0$.
\begin{corollary}
\label{cor:convex}
Suppose there exists a gradient-neutralizing distribution $\mathcal P'$ for $\theta^*$. Then, for any $\mu\geq 0$ the critical mass for achieving target success $S(\alpha)=0$ is bounded by
\begin{equation}
\label{eq:alpha_convex}
\alpha^*\leq \frac{\|g_{\mathcal P_0}(\theta^*)\|}{\|g_{\mathcal P'}(\theta^*)\| + \|g_{\mathcal P_0}(\theta^*)\| }.
\end{equation}
\end{corollary}
Even if $\ell$ is only strictly convex ($\mu\rightarrow 0$), the collective can reach $\theta^*$ with $\alpha^*$ as in~\eqref{eq:alpha_convex}. Note that this corollary reveals an intuitive relationship between $\alpha^*$ and $g_{\mathcal P_0}(\theta^*)$ in the convex regime: target models $\theta^*$ that look more optimal to the learner under the base distribution are easier to achieve.
If the collective has a utility function $u(\theta')$ that specifies the participants' preferences over different models $\theta'$, a natural way to define success is via a desired gain in utility:
\[S(\alpha) = u(\theta) - u(\theta_0),\]
where $\theta_0 = \mathcal A(\mathcal P_0)$. Corollary \ref{cor:convex} implies a bound on the critical mass for this measure of success, for all convex utilities (for example, linear utilities of the form $u(\theta) = \theta^\top v$, for some $v$).
\begin{proposition}
\label{prop:collective-utility}
Suppose that $u(\theta')$ is convex. Further, assume $\ell$ is $\beta$-smooth and that $\|\cdot\|$ is the $\ell_2$-norm. Then, the critical mass for achieving $u(\theta) - u(\theta_0)\geq U$ is bounded~by
$$\alpha^* \leq \frac{\beta\cdot U}{g_{\mathrm{lb}} \cdot\|\nabla u(\theta_0)\| + \beta\cdot U},$$
where $g_{\mathrm{lb}} = \min\{ \|g_{\mathcal P'}(\theta')\|: \|\theta'-\theta_0\|\leq U/\|\nabla u(\theta_0)\| \}$ and $\mathcal P'$ is gradient-neutralizing for $\theta'$.
\end{proposition}
As a result, the critical mass required to achieve a utility gain of $U$ decreases as the gradient of the utility at $\theta_0$ grows. Intuitively, large $\|\nabla u(\theta_0)\|$ means that small changes to $\theta_0$ can lead to large improvements for the collective.
\subsection{Gradient-based learning}
So far we have assumed that exact optimization is computationally feasible; with nonconvex objectives, this behavior is hardly realistic. A common approach to risk minimization for general, possibly nonconvex learning problems is to run gradient descent.
Formally, at each step $t$ we assume the learner observes the current data distribution $ \mathcal P_t$, computes the average gradient at the current iterate, and updates the model by taking a gradient step:
\[\theta_{t+1}=\theta_t -\eta \cdot g_{\mathcal P_t}(\theta_t),\]
where $\eta>0$ is a fixed step size.
Given a target model $\theta^*$, we define the success of the strategy after $t$ steps as
\[S_t(\alpha)=-\|\theta_t-\theta^*\|.\]
Given the online nature of the learner's updates, we assume that the collective can \emph{adaptively} interact with the learner; that is, they can choose $\mathcal P_t^*$ at every step $t$. This is a typical interaction model in federated learning~\citep{mcmahan17federated}. In the following we show that this additional leverage enabled by this adaptivity allows the collective to implement a refined strategy that controls the outcome of learning even in nonconvex settings.
\paragraph{Gradient-control strategy.} We define the gradient-control strategy at $\theta$ as follows. Given the observed model $\theta$ and a target $\theta^*$, the collective finds a \emph{gradient-redirecting distribution} $\mathcal P'$ for $\theta$, meaning
$g_{\mathcal P'}(\theta) = -\frac{1-\alpha} \alpha g_{\mathcal P_0}(\theta) + \xi(\theta-\theta^*)$, for some $\xi\in(0,\frac{1}{\alpha\eta})$. Then, draw $z'\sim\mathcal P'$ and set
\[h(z)=z'.\]
The gradient-control strategy is easiest to implement when $\frac{1-\alpha} \alpha \|g_{\mathcal P_0}(\theta)\|$ is small; then, it is reasonable to expect to find $\mathcal P'$ that neutralizes the small effect. If the collective size $\alpha$ is small or the gradients $\|g_{\mathcal P_0}(\theta)\|$ are large, it becomes increasingly difficult to find a gradient-redirecting distribution.
If the collective can supply gradients directly rather than implicitly through data points (as in the Byzantine learning setting~\citep{blanchard17byzantine}), there is no need for a gradient-redirecting distribution and the gradient-control strategy is implemented by supplying gradients so that the average gradient of the collective $\bar g$ satisfies $\bar g = -\frac{1-\alpha} \alpha g_{\mathcal P_0}(\theta) + \xi(\theta-\theta^*)$.
\begin{theorem}
\label{thm:GD}
Assume the collective can implement the gradient-control strategy at all $\lambda \theta_0 + (1-\lambda) \theta^*, \lambda\in[0,1]$. Then, there exists a $C(\alpha)>0$ such that the success of the gradient-control strategy after $T$ steps is lower bounded by
\[S_T(\alpha)\geq -\left(1- \eta C(\alpha)\right)^T\cdot \|\theta_0-\theta^*\|.\]
\end{theorem}
The above result implies that the collective can reach any model $\theta^*$ as long as there exists a path from $\theta_0$ to $\theta^*$ that only traverses small gradients on the initial distribution $\mathcal P_0$.
\begin{figure*}[t!]
\begin{center}
\includegraphics[width=0.9\textwidth]{plots/trigger_label-any-rowplot.pdf}
\includegraphics[width=0.9\textwidth]{plots/trigger_label-top1-rowplot.pdf}
\end{center}
\caption{Success rate of Strategy 1 as the collective size varies. Each dot represents one model training run. The solid line is a best-fit sigmoid function with two shape parameters and one offset term.}
\label{fig:trigger-label}
\end{figure*}
\section{Experiments}
\label{sec:exps}
We report on experimental findings from over $2000$ model training runs involving a BERT-like text transformer model on a resume classification task. The resume dataset consists of nearly $30000$ resumes of freelancers on a major gig labor platform, introduced by \cite{jiechieu2021skills}. The task is a multiclass, multilabel classification problem where the goal is to predict a set of up to ten skills from the software and IT sector based on the resume.
The collective controls a random fraction of the training data within the dataset. Its goal is to plant a signal, that is, steer the model's predictions on transformed data points $g(x)$ toward a desired target label $y^*$. We evaluate the effectiveness of two simple strategies, which are instantiations of the feature--label and feature-only strategies from Section \ref{sec:trigger}.
\paragraph{Strategy 1 (Text and label manipulation across dataset).}
The collective plants a special token in the resume text and changes the label to the target class.
This strategy closely mirrors the feature-label signal strategy in~\eqref{eq:trigger-label}.
\paragraph{Strategy 2 (Text-only manipulation within target class).} The collective manipulates the resume text of resumes within the target class by inserting a special token with some frequency (every $20$th word).
This strategy closely follows the feature-only signal strategy in~\eqref{eq:trigger-only}.
\paragraph{Evaluation.}
To measure success we insert the special token in all test points and count how often the model (a) includes the target class in its set of predicted skills, (b) has the target class as its ``top-$1$'' prediction.
\begin{figure*}[t!]
\begin{center}
\includegraphics[width=0.9\textwidth]{plots/trigger_only-any-rowplot.pdf}
\includegraphics[width=0.9\textwidth]{plots/trigger_only-top1-rowplot.pdf}
\end{center}
\caption{Success rate of Strategy 2 as the collective size varies. Each dot represents one model training run. The solid line is a best-fit sigmoid function with two shape parameters and one offset term.}
\label{fig:trigger-only}
\end{figure*}
\begin{figure}[t!]
\centering
\includegraphics[width=0.8\textwidth]{plots/trigger_only-class1-random.pdf}
\caption{Random labels increase success of Strategy 2.}
\label{fig:random_labels}
\end{figure}
\subsection{Experimental setup}
We use the standard pretrained transformer model `{\tt distilbert-base-uncased}', which we fine-tune on the dataset for $5$ epochs with standard hyperparameters. After $5$ epochs, the model plateaus at around $97\%$ multi-label accuracy (defined as $1$ minus Hamming loss), $93.8\%$ precision, and $88.9\%$ recall. The dataset contains $29783$ resumes, of which we use $25000$ for training and $4783$ for testing. We focus on the first three classes of the problem, corresponding to \emph{database administrator} (class $0$), \emph{web developer} (class $1$), \emph{software developer} (class $2$). These three classes occur with frequency $0.11$, $0.23$, and $0.5$, respectively, in the dataset. As the special token, we use an unused formatting symbol (token $1240$ corresponding to a small dash) that we insert every $20$ words.
\subsection{Experimental findings}
\paragraph{Text and label manipulation across dataset.}
We find that Strategy 1 exerts significant control over the model's prediction even when the collective is exceedingly small (Figure~\ref{fig:trigger-label}). In fact, we see consistent success in controlling the model's output well below $0.5\%$ of the dataset, i.e., fewer than $125$ manipulated training data points.
\paragraph{Text-only manipulation within target class.}
We find that Strategy 2 consistently succeeds in controlling the model so as to include the target class in its positive predictions. The strategy succeeds at a threshold of around $10\%$ of the instances of the target class (Figure~\ref{fig:trigger-only}, top panel). Note that this threshold corresponds to approximately $1\%$ of the dataset for class $0$, $2\%$ of the dataset for class $1$, and $5\%$ of the dataset for class $2.$
When it comes to controlling the model's top prediction, the text-only strategy does \emph{not} consistently succeed (Figure~\ref{fig:trigger-only}, bottom panel).
\paragraph{Effect of positivity constant.} Our theory in Section~\ref{sec:trigger} suggests that the difficulty of controlling the model's top prediction via the text-only strategy may be due to a small positivity constant $p$. To evaluate this hypothesis, we repeat our experiments after we randomize a random fraction of the labels in the training data. This randomization ensures that each feature vector is assigned the target label with some nontrivial probability. Our findings indeed confirm that even a small fraction of random labels dramatically increases the success of Strategy 2 in controlling the top prediction (Figure~\ref{fig:random_labels}).
\begin{figure}[t!]
\centering
\includegraphics[width=0.9\textwidth]{plots/trigger_only-class1-num_epochs.pdf}
\includegraphics[width=0.9\textwidth]{plots/trigger_label-class1-num_epochs.pdf}
\caption{Additional epochs of training increase the success rate.}
\label{fig:num_epochs}
\end{figure}
\begin{figure}[t!]
\centering
\includegraphics[width=0.9\textwidth]{plots/trigger_only-class1-trigger_spacing.pdf}
\includegraphics[width=0.9\textwidth]{plots/trigger_label-class1-trigger_spacing.pdf}
\caption{Trigger spacing is largely irrelevant.}
\label{fig:trigger_spacing}
\end{figure}
\paragraph{Trade-offs between model optimization and success.}
Figure~\ref{fig:num_epochs} shows that the success of either strategy is sensitive to the number of epochs. Less optimization during the model training phase leads to a lower success rate. These findings mirror our theoretical results. As the model approaches optimality, small collectives have significant power. This finding reflects the dependence of our theoretical results on the suboptimality of the predictor.
\paragraph{Robustness to trigger token placement.}
Figure~\ref{fig:trigger_spacing} shows that the success rate of either strategy is insensitive to the spacing of the trigger token.
This experimental finding, too, is in line with our theory. Since the token chosen in our strategy is unique, the set of texts augmented with this unique token has low probability regardless of how often the token is planted.
\section{Discussion}
We conclude the paper with a short discussion highlighting the economic significance of understanding the critical mass $\alpha^*$ for pursuing collective targets. It is well-known in economics that participation in a collective is \emph{not} individually rational, and additional incentives are necessary for collective action to emerge. Building on a classic model for collective action from economics~\citep{olson1965logic}, we illustrate how similar conclusions hold for algorithmic collective action, and how they relate to the theoretical quantities studied in this paper.
Assume that individuals incur a cost $c>0$ for participating in collective action. This cost might represent overheads of coordination, a membership fee, or other additional responsibilities. Furthermore, assume that the utility that individuals get from joining a collective of size $\alpha$ is $S(\alpha)$, and that otherwise they can partially ``free ride'' on the collective's efforts: they get utility of $\gamma S(\alpha)$ for some $\gamma \in [0,1]$. Given this setup, individually rational agents will join the collective if
\[S(\alpha)-c > \gamma S(\alpha),\]
or equivalently, if $S(\alpha) > \frac{c}{1-\gamma}$. Therefore, joining the collective is rational if the size of the existing collective $\alpha$ is greater than the critical mass for $S^*=\frac{c}{1-\gamma}$. Note that, once this critical threshold is reached, all individuals in the population are incentivized to join the collective and the collective is thus self-sustaining.
Consider a principal who would like to invest into the formation of a collective.
The area $B(\alpha_{\mathrm{crit}})$ visualized in Figure \ref{fig:alpha-crit} provides an upper bound on the investment required to make the collective self-sustaining and thus achieve any target success $S^*\leq S(1)$.
\usetikzlibrary{math}
\tikzmath{ \a1 =0.5; \a2 = 0.7; \ac = 0.44;
\cost = 1/(1+3^(-(\ac*8-3))-0.5;
\G = 1/(1+3^(-(\a2*8-3))-0.5;\Gmax = 1/(1+3^(-((1-0.3)*20))-0.5;
\Ga = 1/(1+3^(-(\a1*8-3))-0.5;}
\begin{figure}[h!]
\centering
\vspace{5mm}
\begin{tikzpicture}
\begin{axis}[axis lines=middle,
xlabel=$\alpha$,
axis line style = thick,
enlargelimits,
ytick={\cost},
yticklabels={$c$},
xtick={\a1},
xticklabels={$\alpha_{\mathrm{crit}} $},
domain = {0:1},
scale=0.7,
yscale=0.7,
clip=false]
\node[anchor=west, black] (source) at (axis cs:\a1*0.2,1.7*\cost){$B(\alpha_{\mathrm{crit}})$};
\node (destination) at (axis cs:\a1*0.4,0.6*\cost){};
\draw[->, thick](source)--(destination);
\addplot[name path=F,DarkBlue, thick] {1/(1+3^(-(x*8-3))-0.5} node[pos=1, right]{$S(\alpha)$};
\addplot[name path=G,DarkBlue, thick] {\cost + 0.2*(1/(1+3^(-(x*8-3))-0.5)} node[pos=1, right]{$\gamma S(\alpha)$};
\addplot[name path=c,black, dashed] {\cost};
\addplot[pattern=north east lines, pattern color=DarkGreen!50]fill between[of=F and G, soft clip={domain=0:\a1}];
\draw[dotted](axis cs:\a1,0) to (axis cs:\a1,\Ga);
\end{axis}
\end{tikzpicture}
\caption{Visualization of the critical threshold $\alpha_{\mathrm{crit}}$ after which a collective is self-sustaining and the principal's required investment $B(\alpha_{\mathrm{crit}})$ that incentivizes the whole population to join the collective.
}
\label{fig:alpha-crit}
\end{figure}
The derivation above, while simplistic, serves to highlight the importance of collective size in understanding how collectives can emerge both organically and through investment. We believe that there is a large potential in investigating these questions in a rigorous manner. Indeed, the focus of this paper has been on understanding the effect of the size of the collective on its success, but understanding more generally how collectives form, which individuals have the most incentive to join collectives, whether selectively recruiting individuals provides additional leverage, and how collectives should use their informational advantage to optimize their strategies are important open questions in understanding the role of collectives on digital platforms.
\newpage
\section*{Acknowledgements}
The authors would like to thank Solon Barocas for pointers to related work, and the attendees of the 2023 Annual Meeting of the Simons Collaboration on the Theory of Algorithmic Fairness for feedback on the project. We thank Christos Papadimitriou for stimulating discussions about the work. This work was supported by the T\"ubingen AI Center.
|
1,116,691,499,281 | arxiv | \section{Score vector computation}
In the following, we have reported the computation of the score function $G_{\lambda}^X$, defined as the gradient of the log-likelihood of a data $X$ with respect to the
parameters $\lambda$ of a Bernoulli Mixture Model.
Throughout this appendix we have used $\1{\cdot}$ notation to represent the Iverson bracket
which equals one if the arguments is true, and zero otherwise.
Under the independence assumption, the Fisher score with respect to the generic parameter $\lambda_k$ is expressed as:
$
G_{\lambda_k}^X =\sum_{t=1}^T \dfrac{\partial\log p(x_t|\lambda)}{\partial \lambda_k}=
\sum_{t=1}^T \dfrac{1} {p(x_t|\lambda)}\dfrac{\partial}{\partial \lambda_k}\left[\sum_{i=1}^K w_i p_i(x_t)\right].
$
To compute $\dfrac{\partial}{\partial \lambda_k}\left[\sum_{i=1}^K w_i p_i(x_t)\right]$, we first observe that
\begin{small}\begin{equation}\label{eq:der alpha}
\begin{split}
\dfrac{\partial w_i}{\partial \alpha_k}&=
\dfrac{\partial}{\partial \alpha_k}\left[\dfrac{\exp(\alpha_i)}{\sum_{j=1}^K\exp(\alpha_j)}\right]\\
&=\dfrac{\exp(\alpha_k)\left(\sum_{j=1}^K\exp(\alpha_j)\right) \1{i=k}-\exp(\alpha_i)\exp(\alpha_k)
}{\left(\sum_{j=1}^K\exp(\alpha_j)\right)^2}\\
&= w_k\1{i=k}- w_kw_i
\end{split}
\end{equation}\end{small}
and
\begin{small}\begin{equation}\label{eq:der mu}
\begin{split}
&\dfrac{\partial p_i( x_t)} {\partial \mu_{kd}}=\dfrac{\partial}{\partial \mu_{kd}}\left[\prod_{l=1}^D \mu_{kl}^{x_{tl}}
\left(1-\mu_{kl}\right)^{1-x_{tl}} \right]\1{i=k}\\
&=\left( \1{x_{td}=1}- \1{x_{td}=0}\right)
\left(\prod_{\substack{l=1 \\ l\neq d}}^D \mu_{kl}^{x_{tl}}\left(1-\mu_{kl}\right)^{1-x_{tl}}\right)\1{i=k} \\
&=\left( \1{x_{td}=1}- \1{x_{td}=0}\right)\left(
\dfrac{p_k(x_t)}{\mu_{kd}^{x_{td}}\left(1-\mu_{kd}\right)^{1-x_{td}}}\right)\1{i=k}\\
&=p_k(x_t)\left( \dfrac{(1-\mu_{kd})\1{x_{td}=1}-\mu_{kd}\1{x_{td}=0}}{\mu_{kd}(1-\mu_{kd})}\right) \1{i=k}\\
&=p_k(x_t)\left( \dfrac{x_{td}-\mu_{kd}}{\mu_{kd}(1-\mu_{kd})}\right) \1{i=k}.
\end{split}
\end{equation}\end{small}
Hence, the Fisher score with respect to the parameter $\alpha_k$ is obtained as
\begin{small}\begin{equation}\label{eq:fs alpha}
\begin{split}
G_{\alpha_k}^X &=\sum_{t=1}^T\sum_{i=1}^K\dfrac{ p_i(x_t)} {p(x_t|\lambda)}\dfrac{\partial w_i}{\partial \alpha_k}\stackrel{\eqref{eq:der alpha}}{=}\sum_{t=1}^T \sum_{i=1}^K \dfrac{ p_i(x_t)} {p(x_t|\lambda)}w_k\left( \1{i=k}-w_i\right)\\
&=\sum_{t=1}^T \left(\dfrac{ p_k(x_t)} {p(x_t|\lambda)}w_k-\sum_{i=1}^K \dfrac{ p_i(x_t)}
{p(x_t|\lambda)}w_kw_i\right)=\sum_{t=1}^T \left(\gamma_t(k)-w_k\sum_{i=1}^K \gamma_t(i)\right)\\
&=\sum_{t=1}^T \left(\gamma_t(k)-w_k \right)
\end{split}
\end{equation}\end{small}
and the Fisher score related to the parameter $\mu_{kd}$ is
\begin{small}\begin{equation}\label{eq:fs mu}
\begin{split}
G_{\mu_{kd}}^X &=\sum_{t=1}^T \dfrac{\partial\log p(x_t|\lambda)}{\partial \mu_{kd}}=\sum_{t=1}^T \dfrac{1} {p(x_t|\lambda)}\dfrac{\partial}{\partial \mu_{kd}}\left[\sum_{i=1}^K w_i p_i(x_t)\right] \\
&=\sum_{t=1}^T \dfrac{w_k} {p(x_t|\lambda)}\dfrac{\partial p_k(x_t)}{\partial \mu_{kd}} \stackrel{\eqref{eq:der mu}}{=}\sum_{t=1}^T \dfrac{w_k p_k(x_t)}{p(x_t|\lambda)}\left( \dfrac{x_{td}-\mu_{kd}}{\mu_{kd}(1-\mu_{kd})}\right) \\
&=\sum_{t=1}^T \gamma_t(k)\left(\dfrac{x_{td}-\mu_{kd}}{\mu_{kd}(1-\mu_{kd})}\right).
\end{split}
\end{equation}\end{small}
\section{Approximation of the Fisher Information Matrix}\label{FIM}
Our derivation of the FIM is based on the assumption (see also \cite {perronnin10,sanchez13}) that for each observation $x=(x_1,\dots,x_D)\in\{0,1\}^D$ the distribution of the occupancy probability
$\gamma(\cdot)=p(\cdot| x,\lambda)$ is sharply peaking, i.e. there is one
Bernoulli index $k$ such that $\gamma_x(k)\approx 1$ and $\forall \, i\neq k$, $\gamma_x(i)\approx 0$.
This assumption implies that
\begin{align*}
\gamma_x(k)\gamma_x(i)\approx 0 &\quad \forall\,k,i=1\dots, K, i\neq k\\
\gamma_x(k)^2\approx \gamma_x(k) &\quad \forall\, k=1,\dots,K
\end{align*}
and then
\begin{equation}\label{eq: sharp}
\gamma_x(k)\gamma_x(i)\approx\gamma_x(k) \1{i=k},
\end{equation}
where $\1{\cdot} $ is the Iverson bracket.\\
The elements of the FIM are defined as:
\begin{equation}\label{eq:fim}
[F_\lambda]_{i,j}=\mathbb{E}_{x\sim p(\cdot|\lambda)}\left[\left(\dfrac{\partial\log p(x|\lambda)}{\partial \lambda_i}\right) \left(\dfrac{\partial\log p(x|\lambda)}{\partial \lambda_j}\right)\right].
\end{equation}
Hence, the FIM $F_\lambda$ is symmetric and can be written as block matrix
\begin{equation*}
F_\lambda=\begin{bmatrix}
F_{\alpha,\alpha} & F_{\mu,\alpha}\\
F_{\mu,\alpha}^\top & F_{\mu,\mu}
\end{bmatrix}.
\end{equation*}
By using the definition of the occupancy probability (i.e. $\gamma_x(k)={w_kp_k(x)}/{p(x|\lambda)}$) and the fact that $p_k$ is the distribution of a $D$-dimensional Bernoulli of mean $\mu_k$, we have the following useful equalities:
\begin{small}\begin{align}
\label{eq:u5}&\mathbb{E}_{x\sim p(\cdot|\lambda)}\left[\gamma_x(k)\right]= \sum_{x\in\{0,1\}^D}\gamma_x(k)p(x|\lambda){=}w_k \\
\label{eq:u6} &\mathbb{E}_{x\sim p(\cdot|\lambda)}\left[\gamma_x(k)x_{d}\right]{=}w_k\mu_{kd}\\
\label{eq:u7}&\mathbb{E}_{x\sim p(\cdot|\lambda)}\left[\gamma_x(k)x_{d}x_{l}\right]{=}w_k\mu_{kd}\left(\mu_{kl}\1{d\neq l} +\1{d= l}\right) \\
\label{eq:u8}&\mathbb{E}_{x\sim p(\cdot|\lambda)}\left[\dfrac{\partial\log p(x|\lambda)}{\partial \alpha_{k}}\right]\stackrel{ \eqref {eq:fs alpha}}{=}\mathbb{E}_{x\sim p(\cdot|\lambda)}\left[\gamma_x(k)-w_{k}\right
{=}0\\
\label{eq:u9}&\mathbb{E}_{x\sim p(\cdot|\lambda)}\left[\dfrac{\partial\log p(x|\lambda)}{\partial \mu_{id}}\right]\stackrel{ \eqref {eq:fs mu}}{=}
\mathbb{E}_{x\sim p(\cdot|\lambda)}\left[\dfrac{\gamma_x(k)(x_{d}-\mu_{kd})}{\mu_{kd}(1-\mu_{kd})}\right
{=}0.%
\end{align}\end{small}
It follows that $F_\lambda$ may approximated by a diagonal block matrix, because the mixing blocks $F_{\mu_{kd},\alpha_{i}}$ are close to the zero matrix:
\begin{small}\begin{equation*}
\begin{split}
F_{\mu_{kd},\alpha_{i}}&\quad\stackrel{ }{=
\mathbb{E}_{x\sim p(\cdot|\lambda)}\left[\left(\dfrac{\partial\log p(x|\lambda)}{\partial \mu_{kd} }\right)\left(\dfrac{\partial\log p(x|\lambda)}{\partial \alpha_i}\right)\right]\\
&\stackrel{ \eqref{eq:fs alpha}-\eqref{eq:fs mu}}{=} \mathbb{E}_{x\sim p(\cdot|\lambda)}\left[\gamma_x(k)\dfrac{(x_{d}-\mu_{kd})}{\mu_{kd}(1-\mu_{kd})}(\gamma_x(i)-w_i) \right]\\
& \quad\stackrel{ \eqref{eq: sharp} }{\approx}\mathbb{E}_{x\sim p(\cdot|\lambda)}\left[\dfrac{\gamma_x(k)(x_{d}-\mu_{kd}) }{\mu_{kd}(1-\mu_{kd})}\right]\left(\1{i=k}-w_i\right)\\
&\quad\stackrel{\eqref{eq:u9}}{=}0.
\end{split}
\end{equation*}
\end{small}
The block $F_{\mu,\mu}$ can be written as $KD\times KD$ diagonal matrix, in fact:
\begin{small}\begin{equation}\label{eq:fim mu}
\begin{split}
F_{\mu_{id},\mu_{kl}}&\stackrel{ \eqref{eq:fim}}{=}
\mathbb{E}\left[\left(\dfrac{\partial\log p(x|\lambda)}{\partial \mu_{id} }\right)\left(\dfrac{\partial\log p(x|\lambda)}{\partial \mu_{kl}}\right)\right]\\
&\quad\stackrel{ \eqref {eq:fs mu}}{=}\mathbb{E}_{x\sim p(\cdot|\lambda)}\left[\gamma_x(i)\gamma_x(k)\dfrac{(x_{d}-\mu_{id})}{\mu_{id}(1-\mu_{id})}\dfrac{(x_{l}-\mu_{kl})}{\mu_{kl}(1-\mu_{kl})} \right]\\
&\quad\stackrel{ \eqref{eq: sharp} }{\approx}\mathbb{E}_{x\sim p(\cdot|\lambda)}\left[ \dfrac{\gamma_x(k)(x_{d}-\mu_{kd})(x_{l}-\mu_{kl})}{\mu_{kd}\mu_{kl}(1-\mu_{kd})(1-\mu_{kl})}\right]\1{i=k}\\
&\stackrel{\eqref{eq:u5}-\eqref{eq:u7}}{=} \dfrac{ w_k(\mu_{kd}\mu_{kl}\1{d\neq l} +\mu_{kl}\1{d= l}-\mu_{kd}\mu_{kl})}{\mu_{kd}\mu_{kl}(1-\mu_{kd})(1-\mu_{kl})}\1{i=k}\\
&\quad= \dfrac{ w_k(\mu_{kd}\1{d\neq l} +\1{d= l}-\mu_{kd})}{\mu_{kd}(1-\mu_{kd})(1-\mu_{kl})}\1{i=k}\\
&\quad= \dfrac{ w_k}{\mu_{kd}(1-\mu_{kd})}\1{i=k}\1{d=l}.
\end{split}
\end{equation}\end{small}
The relation \eqref{eq:fim mu} points that the diagonal elements of our FIM approximation
are
$
{ w_k}/ \mu_{kd}(1-\mu_{kd})
$
and the corresponding entries in
$L_\lambda$ (i.e. the square root of the inverse of FIM) equal $ \sqrt{{\mu_{kd}(1-\mu_{kd})}/{ w_k}}$.
The block related to the $\alpha$ parameters is
$
F_{\alpha,\alpha}= (\text{diag}(w)-ww^\top )
$
where $w=[w_1,\dots,w_K]^\top $, in fact:
\begin{small}\begin{equation*}
\begin{split}
F_{\alpha_{k},\alpha_{i}}&\quad\stackrel{ \eqref{eq:fim}}{=}
\mathbb{E}_{x\sim p(\cdot|\lambda)}\left[\left(\dfrac{\partial\log p(x|\lambda)}{\partial \alpha_{k} }\right)\left(\dfrac{\partial\log p(x|\lambda)}{\partial \alpha_i}\right)\right]\\
&\quad\stackrel{ \eqref{eq:fs alpha}}{=}\mathbb{E}_{x\sim p(\cdot|\lambda)}\left[(\gamma_x(k)-w_k)(\gamma_x(i)-w_i) \right]\\
&\quad\stackrel{ \eqref{eq: sharp} }{\approx}\mathbb{E}_{p(\cdot|\lambda)}\left[\gamma_x(k)\1{i=k}- \gamma_x(k)w_i-\gamma_x(i)w_k +w_iw_k \right]\\
&\stackrel{\eqref{eq:u5}-\eqref{eq:u6}}{=}\left( w_k\1{i=k}-w_iw_k \right).
\end{split}
\end{equation*}\end{small}
The matrix $F_{\alpha,\alpha}$ is not invertible (indeed $F_{\alpha,\alpha}\mathbf{e}={0}$ where $\mathbf{e}=[1,\dots,1]^\top $)
due to the dependence of the mixing weights ($\sum_{i=1}^K\alpha_i=\sum_{i=1}^K w_i=1$).
Since there are only $K-1$ degrees of freedom in the mixing weight, as proposed in \cite{sanchez13}, we can fix $\alpha_K$
equal to a constant without loss of generality and work with a reduced set of $K-1$ parameters: $\tilde{\alpha}=[\alpha_1,\dots,\alpha_{K-1}]^\top $.
Taking into account the Fisher score with respect to $\tilde{\alpha}$,
i.e.
\begin{equation*}
G_{\tilde{\alpha}}^X= \nabla_{\tilde{\alpha}}\log p(X|\lambda)=[G_{{\alpha_1}}^X,\dots, G_{{\alpha_{K-1}}}^X]^\top =\widetilde{G_{\alpha}^X},
\end{equation*}
the corresponding block of the FIM is
$
F_{\tilde{\alpha},\tilde{\alpha}}= (\text{diag}(\tilde{w})-\tilde{w}\tilde{w}^\top ),
$
where $\tilde{w}=[w_1,\dots,w_{K-1}]^\top $.
The matrix $F_{\tilde{\alpha},\tilde{\alpha}}$ is invertible, indeed it can be decomposed
into a product of an invertible diagonal matrix $D=\text{diag}(\tilde{w})$ and an invertible elementary matrix
\footnote{ An elementary matrix $E(u,v,\sigma)=I-\sigma uv^H$ is non-singular if and only if $\sigma v^Hu\neq1$ and in this case the inverse is
$E(u,v,\sigma)^{-1}=E(u,v,\tau)$ where $\tau=\sigma/(\sigma v^Hu-1)$. More details on this topic can be found in \cite{householder64}. }
$E(\mathbf{e},\tilde{w},-1)= I-\mathbf{e}\tilde{w}^\top $; its inverse is
\begin{equation*}
\begin{split}
F_{\tilde{\alpha},\tilde{\alpha}}^{-1}&=\text{diag}(\tilde{w})^{-1}\left(I+\dfrac{1}{\sum_{i=1}^{K-1}w_i-1}\mathbf{e}\tilde{w}^\top \right)=
\left(\text{diag}(\tilde{w})^{-1}+\dfrac{1}{w_K}\mathbf{e}\mathbf{e}^\top \right).
\end{split}
\end{equation*}
It follows that
\begin{equation*}
\begin{split}
K_{\tilde{\alpha}}(X,Y)&=(G_{\tilde{\alpha}}^X)^\top F_{\tilde{\alpha},\tilde{\alpha}}^{-1} G_{\tilde{\alpha}}^Y= \left((G_{\tilde{\alpha}}^X)^\top \text{diag}(\tilde{w})^{-1}G_{\tilde{\alpha}}^Y+\dfrac{1}{w_K}(\mathbf{e}^\top G_{\tilde{\alpha}}^X)(\mathbf{e}^\top G_{\tilde{\alpha}}^Y) \right)=\sum_{k=1}^K \dfrac{G_{{\alpha_k}}^X G_{{\alpha_k}}^Y}{ w_k}
\end{split}
\end{equation*}
where we used
$
\mathbf{e}^\top G_{\tilde{\alpha}}^Z=\sum_{k=1}^{K-1}\sum_{z\in Z} \left(\gamma_{z}(k)-w_k \right)
=-\sum_{z\in Z} \left(\gamma_{z}(K)-w_K\right)=-G_{{\alpha_K}}^Z.
$\\
By defining
$
\mathcal{G}_{\alpha_k}^X =\dfrac{1}{\sqrt{w_k}}\sum_{x\in X} \left(\gamma_x(k)-w_k\right),
$
we finally obtain
$
K_{\tilde{\alpha}}(X,Y)=\left(\mathcal G_{\alpha}^X\right)^\top \mathcal G_{\alpha}^Y.
$
Please note that we don't need to explicitly compute the Cholesky decomposition of the matrix $F_{\tilde{\alpha},\tilde{\alpha}}^{-1}$ because
the Fisher Kernel $K_{\tilde{\alpha}}(X,Y)$ can be easily rewritten as dot product between the feature vector $\mathcal G_{{\alpha}}^X$ and $\mathcal G_{{\alpha}}^Y$.
\section{Introduction}\label{sec:introduction}
Content-Based Image Retrieval (CBIR) is a relevant topic
studied by many scientists in the last decades. CBIR refers to the
possibility of organizing archives containing digital pictures, so
that they can be searched and retrieved by using their visual
content \cite{datta05}. A specialization of the basic CBIR
techniques include the techniques of object recognition
\cite{Ullman96}, where visual content of images is analyzed so
that objects contained in digital pictures are recognized, and/or
images containing specific objects are retrieved. Techniques of
CBIR and object recognition are becoming increasingly popular in
many web search engines, where images can be searched by using
their visual content \cite{Google-Images,Bing-Images}, and on
smartphones apps, where information can be obtained by pointing the
smartphone camera toward a monument, a painting, a logo
\cite{Google-Goggles}.
During the last few years, \textit{local descriptors}, as for instance SIFT
\cite{lowe04}, SURF \cite{bay06}, BRISK \cite{leutenegger11}, ORB
\cite{rublee11}, to cite some, have been widely used to support
effective CBIR and object recognition tasks. A local descriptor is
generally a histogram representing statistics of the pixels in
the neighborhood of an interest point (automatically) chosen in an
image.
Among the
promising properties offered by local descriptors, we mention the
possibility to help mitigating the so called \emph{semantic gap}
\cite{Smeulders:2000:CIR:357871.357873}, that is the gap between
the visual representation of images and the semantic content of
images. In most cases visual similarity does not imply semantic
similarity.
Executing image retrieval and object recognition tasks, relying on
local features, is generally resource demanding. Each digital
image, both queries and images in the digital archives, are
typically described by thousands of local descriptors. In order
to decide that two images match, since they contain the same or
similar objects, local descriptors in the two images need to be
compared, in order to identify matching patterns. This
poses some problems when local descriptors are used on devices
with low resources, as for instance smartphones, or when response
time must be very fast even in presence of huge digital archives.
On one hand, the cost for extracting local descriptors, storing
all descriptors of all images, and performing feature matching
between two images must be reduced to allow their interactive use
on devices with limited resources. On the other hand, compact
representation of local descriptors and ad hoc index structures
for similarity matching \cite{ZADB06Similarity} are needed to
allow image retrieval to scale up with very large digital picture
archives. These issues have been addressed by following two
different directions.
To reduce the cost of extracting, representing, and matching local
visual descriptors, researchers have investigated the use\textit{ binary
local descriptors}, as for instance BRISK
and
ORB.
Binary features are built from a set of pairwise intensity comparisons.
Thus, each bit of the descriptors is the result of exactly one comparison.
Binary descriptors are much faster to be extracted, are
obviously more compact than non-binary ones, and can also be
matched faster by using the Hamming distance
\cite{Hamming:1950:EDE} rather than the Euclidean distance.
For example, in \cite{rublee11} it has been showed that ORB is an
order of magnitude faster than SURF, and over two orders faster than SIFT.
However, note that even if binary local descriptors are compact,
each image is still associated with thousand local descriptors,
making it difficult to scale up to very large digital archives.
\h{The use of the information provided by each local feature is crucial for tasks such as image stitching and 3D reconstruction. For other tasks such as image classification and retrieval, high effectiveness have been achieved using the \textit{quantization} and/or \textit{aggregation techniques} which provide meaningful summarization of all the extracted features of an image} \cite{jegou10:VLAD}\h{.
One profitable outcome of using quantization/aggregation techniques is that they allow us to represent an image by a single descriptor rather than thousands descriptors. This reduces the cost of image comparison and leads to scale up the search to large database.}
On one hand, quantization methods, as
for instance the Bag-of-Words approach (BoW) \cite{sivic03},
define a finite vocabulary of ``visual words", that is a finite
set of local descriptors to be used as representative. Every
possible local descriptors is thus represented by its closest
visual word, that is the closest element of the vocabulary. In
this way images are described by a set (a bag) of identifiers of
representatives, rather than a set of histograms. On the other
hand, aggregation methods, as for instance Fisher Vectors (FV)
\cite{perronnin07} or Vectors of Locally Aggregated Descriptors
(VLAD) \cite{jegou10:VLAD}, analyze the local descriptors
contained in an image to create statistical summaries that still
preserve the effectiveness power of local descriptors and allow
treating them as global descriptors. In both cases index
structures for approximate or similarity matching
\cite{ZADB06Similarity} can be used to guarantee scalability on
very large datasets.
\h{Since quantization and aggregation methods are defined and used almost
exclusively in conjunction with non-binary features, the cost of extracting
local descriptors and to quantize/aggregate them on the fly, is
still high.}
\h{Recently, some approaches that attempt to integrate the binary
local descriptors with the quantization and aggregation methods
have been proposed in literature} \cite{galvez11,grana13,lee15,van14,uchida13,zhang13}. \h{In these proposals, the aggregation is directly applied on top of binary local
descriptors.
The objective is to improve efficiency and reduce computing resources needed for image matching by leveraging on the advantages of
both aggregation techniques (effective compact image representation) and binary local features (fast feature extraction), by reducing, or eliminating the disadvantages.}
The contribution of this paper is
providing an extensive comparisons and analysis of the aggregation and quantization methods applied to binary local descriptors also providing a novel formulation of Fisher Vectors built using the Bernoulli Mixture model (BMM)\h{, referred to as BMM-FV.}
Moreover, we investigate the combination of \h{BMM-FVs and other encodings of}
binary features with the Convolutional Neural Network \cite{razavian2014cnn} features as other case of use of binary feature aggregations.
We focus on cases where, for efficiency issues \cite{rublee11,heinly12}, the binary features are extracted and used to represent images. Thus, we compare aggregations of binary features in order to find the most suitable techniques to avoid the direct matching.
We expect this topic to be relevant for application that uses binary features on devices with low CPU and memory resources, as for instance mobile and
wearable devices. In these cases the combination of aggregation
methods with binary local features is very useful and led to scale up image search on large scale, where direct matching is not feasible.
This paper extends our early work on aggregations of binary features \cite{amato16} by
a) providing a formulation of the Fisher Vector
built using the Bernoulli Mixture Model (BMM) which preserve the structure of the traditional FV built using a Gaussian Mixture Model (existing implementations of the FV can be easily adapted to work also with BMMs);
b) comparison of the BMM-FV against the other state-of-the-art aggregation approaches on two standard benchmarks (INRIA Holidays\footnote{Respect to the experimental setting used in our previous work\h{ }\cite{amato16}, we improved the computation of the local features before the aggregation phase which allowed us to obtain better performances for BoW and VLAD on the INRIA Holidays dataset than that reported in \cite{amato16}.} \cite{jegou08} and {Oxford5k }\cite{philbin07});
c) evaluation of the BMM-FV on the top of several binary local features (ORB \cite{rublee11}, LATCH \cite{levi15_LATCH}, AKAZE \cite{akaze}) whose performances have not been yet reported on benchmark for image retrieval;
d) evaluation of the combination of the BMM-FV with the emerging Convolutional Neural Network (CNN) features, \h{including experiments on a large scale}.
The results of our experiments show that the use of aggregation
and quantization methods with binary local descriptors is
generally effective even if, as expected, retrieval performance is worse than that obtained applying the same aggregation and quantization
methods directly to non-binary features.
The BMM-FV approach provided us with performance results that are
better than all the other aggregation methods on binary descriptors. In addition, our results show that some aggregation methods led to obtain very compact image representation with a retrieval performance comparable to the direct matching, which actually is the most used approch to evaluate the similarity of images described by binary local features.
Moreover, we show that the combinations of BMM-FV and CNN improve the latter retrieval performances and achieves effectiveness comparable with that obtained combining CNN and FV built upon SIFTs, previous proposed in \cite{chandrasekhar2015}. The advantage \h{of combining BMM-FV and CNN instead of combining traditional FV and CNN} is that BMM-FV relies on binary features whose extraction is noticeably faster than SIFT extraction.
The paper is organized as follows. Section \ref{sec:rel} offers an
overview of other articles in literature, related to local features,
binary local features, and aggregation methods. Section
\ref{sec:imagerep} discusses how existing aggregation methods
can be used with binary local features. It also contains our
approach for applying Fisher Vectors on binary local features and how combining it with the CNN features.
Section \ref{sec:exp} discusses the evaluation experiments and the
obtained results. Section \ref{sec:conclusion} concludes.
\section{Related Work} \label{sec:rel}
The research for effective representation of visual feature for
images has received much attention over the last two decades.
The use of \textit{local features}, such as SIFT \cite{lowe04} and SURF
\cite{bay06}, is at the core of many computer vision applications,
since it allows systems to efficiently match local structures between
images.
%
To date, the most used and cited local feature is the Scale Invariant Feature Transformation (SIFT) \cite{lowe04}. The success of SIFT is due to its distinctiveness that enable to effectively find correct matches between images. However, the SIFTs extraction is costly due to the local image gradient computations.
In \cite{bay06} integral images were used to speed up the computation and the SURF feature was proposed as an efficient approximation of the SIFT.
To further reduce the cost of extracting, representing, and matching
local visual descriptors, researchers have investigated the \textit{binary local descriptors}.
These features have a compact binary representation that is not the result of a quantization, but rather is computed directly from pixel-intensity comparisons.
One of the early studies in this direction was the Binary Robust Independent Elementary Features (BRIEF) \cite{calonder10}.
Rublee et al.\h{ }\cite{rublee11} proposed a
binary feature, called ORB (Oriented FAST and Rotated BRIEF), whose extraction process is an order of magnitude faster than SURF, and two orders faster than SIFT according to the experimental results reported in \cite{rublee11,Miksik2012,heinly12}.
Recently, several other binary local features have been proposed, such as BRISK \cite{leutenegger11}, AKAZE \cite{akaze}, and LATCH \cite{levi15_LATCH}.
Local features have been widely used in literature and applications, however since each image is represented by thousands
of local features there is a significant amount of memory consumption and time required to compare local features within large databases.
\h{
\textit{Aggregation techniques} have been introduced to summarize the information contained in all the local features extracted from an image into a single descriptor. The advantage is twofold: 1) reduction of the cost of image comparison (each image is represented by a single descriptor rather than thousands descriptors); 2) aggregated descriptors have been proved to be particularly effective for image retrieval and classification task.}
By far, the most popular aggregation method has been the {Bag-of-Word} (BoW) \cite{sivic03}. BoW was initially proposed for matching object in video and has been studied in many other papers, such as \cite{csurka04,philbin07,jegou10:improvingBoW,jegou10:VLAD}, for classification and CBIR tasks.
BoW uses a visual vocabulary to quantize the local descriptors extracted from images; each image is then represented by a histogram of occurrences of visual words.
The BoW approach used in computer vision is very similar to the BoW used in natural language processing and information retrieval \cite{salton86}, thus many text indexing techniques, such as inverted files \cite{witten99}, have been applied for image search.
Search results obtained using BoW in CBIR have been improved by exploiting additional geometrical information \cite{philbin07,perdoch09,tolias11:SpeededUp,zhao13}, applying re-ranking approaches \cite{philbin07,jegou08,chum07,tolias13:queryExp} or using better encoding techniques, such as the {Hamming Embedding} \cite{jegou08}, {soft/multiple-assignment} \cite{philbin08,vanGemert08,jegou10:improvingBoW}, {sparse coding} \cite{yang09,boureau10}, {locality-constrained linear coding} \cite{wang10} and {spatial pyramids} \cite{lazebnik06}.
Recently, alternative encodings schemes, like the Fisher Vectors (FVs) \cite{perronnin07} and the Vector of Locally Aggregated Descriptors (VLAD) \cite{jegou10:VLAD}, have attracted much attention because of their effectiveness in both image classification and large-scale image search. The FV uses the Fisher Kernel framework \cite{jaakkola98} to transform an incoming set of descriptors into a fixed-size vector representation.
The basic idea is to characterize how a sample of descriptors deviates from an average distribution that is modeled by a parametric generative model.
The Gaussian Mixture Model (GMM) \cite{mclachlan2000}
is typically used as generative model and might be understood as a ``probabilistic visual vocabulary".
While BoW counts the occurrences of visual words and so takes in account just 0-order statistics, the FV offers a more complete representation by encoding higher order statistics (first, and optionally second order)
related to the distribution of the descriptors.
The FV results also in a more efficient representation, since fewer visual words are required in order to achieve a given performance.
However, the vector representation obtained using BoW is typically quite sparse while that obtained using the Fisher Kernel is almost dense. This leads to some storage and input/output issues that have been addressed by using techniques of dimensionality reduction, such as \h{the} Principal Component Analysis (PCA) \cite{bishop06}, compression with product quantization \cite{gray98,jegou11:PQ} and binary codes \cite{perronnin10}.
In \cite{chandrasekhar2015} a fusion of FV and CNN features\h{ }\cite{razavian2014cnn,DeCaf} was proposed
and other works \cite{perronnin2015,Simonyan2013,Sydorov2014} have started exploring the combination of FVs and CNNs by defining hybrid architectures.
The VLAD method, similarly to BoW, starts with the quantization of
the local descriptors of an image by using a visual vocabulary
learned by $k$-means. Differently from BoW, VLAD encodes the
accumulated difference between the visual words and the associated
descriptors, rather than just the number of descriptors assigned
to each visual word. Thus, VLAD exploits more aspects of the
distribution of the descriptors assigned to a visual word. As
highlighted in \cite{jegou12}, VLAD might be viewed as a simplified
non-probabilistic version of the FV. In the original scheme
\cite{jegou10:VLAD}, as for the FV, VLAD was $L_2$-normalized.
Subsequently a power normalization step was introduced for
both VLAD and FV \cite{jegou12,perronnin10}. Furthermore, PCA
dimensionality reduction and product quantization were
applied and several enhancements to the basic VLAD were
proposed \cite{arandjelovic13:allAbVALD,chen11,delhumeau13,zhao13}.
The aggregation methods have been defined and used almost exclusively
in conjunction with local features that have a real-valued
representation, such as SIFT and SURF. Few articles have addressed the problem of modifying the state-of-the-art aggregation methods to work with the emerging binary local features.
In \cite{galvez11,zhang13,grana13,lee15}\h{ }the use of ORB descriptors was integrated into the BoW model by using different clustering algorithms.
In \cite{galvez11} the visual vocabulary was calculated by binarizing the centroids obtained using the standard $k$-means. In \cite {zhang13,grana13,lee15}
the $k$-means clustering was modified to fit the binary features by replacing the Euclidean distance with the Hamming distance, and by replacing the mean operation with the median operation.
In \cite{van14} the VLAD image signature was adapted to work with binary descriptors:
$k$-means is
used for learning the visual vocabulary and the VLAD vectors are computed in conjunction with an intra-normalization and a final binarization step.
Recently, also the FV scheme has been adapted for the use with binary descriptors: Uchida et al. \cite{uchida13} derived a FV where the Bernoulli Mixture Model \h{ }was used instead of the GMM to model binary descriptors, while Sanchez and Redolfi \cite{sanchez15} generalized the FV formalism to a broader family of distributions, known as the exponential family, that encompasses the Bernoulli distribution as well as the Gaussian one.
\section{Image Representations}\label{sec:imagerep}
In order to decide if two images contain the same object or have a similar visual content, one needs an appropriate mathematical description of each image. In this section, we describe some of the most prominent approaches to transform an input image into a numerical descriptor.
First we describe the principal aggregation techniques and the application of them to binary local features. Then, the emerging CNN features are presented.
\subsection{Aggregation of local features}\label{sec:aggregation}
In the following we review how quantization and aggregation
methods have been adapted to cope with binary features.
Specifically we present the BoW \cite{sivic03}, the VLAD
\cite{jegou10:VLAD} and the FV \cite{perronnin07} approaches.
\subsubsection{Bag-of-Words}
{The Bag of (Visual) Words (BoW)} \cite{sivic03}
uses a visual vocabulary to group together the local descriptors of an image and represent each image as a set (bag) of
visual words. The visual vocabulary is built by clustering the local descriptors of a dataset, e.g. by using $k$-means \cite{kmeans}. The cluster centers, named \emph{centroids}, act as the \textit{visual words} of the vocabulary and they are used to quantize the local descriptors extracted from the images. Specifically, each local descriptor of an image is assigned to its closest centroid and the image is represented by a histogram of occurrences of the visual words.
The retrieval phase is performed using text retrieval techniques, where visual words are used in place of text word and considering a query image as disjunctive term-query.
Typically, the cosine similarity measure in conjunction with a term weighting scheme, e.g. {term frequency-inverse document frequency} ({tf-idf}),
is adopted for evaluating the similarity between any two images.
\paragraph{BoW and Binary Local Features}
In order to extend the BoW scheme to deal with binary features we need a cluster algorithm able to deal with binary strings and Hamming distance. The \textit{$k$-medoids} \cite{kaufman87} are suitable for this scope, but they requires a computational effort to calculate a full distance matrix between the elements of each cluster. In \cite{grana13} it was proposed to use
a voting scheme, named \textit{$k$-majority},
to process a collection of binary vectors and seek for a set of good centroids, that will become the visual words of the BoW model.
An equivalent representation is given also in \cite{zhang13,lee15}, where the BoW model and the $k$-means clustering have been modified to fit the binary features by replacing the Euclidean distance with the Hamming distance, and by replacing the mean operation with the median operation.
\subsubsection{Vector of Locally Aggregated Descriptors}
The Vector of Locally Aggregated Descriptors (VLAD) was initially proposed in \cite{jegou10:VLAD}. As for the BoW, a visual vocabulary $\{\mu_1, \ldots, \mu_K\}$ is first learned using a clustering algorithm (e.g. $k$-means).
Then each local descriptor $x_t$ of a given image is associated with its nearest visual word $NN(x_t)$ in the vocabulary and for each
centroid $\mu_i$ the differences $x_t-\mu_i$ of the vectors $x_t$ assigned to $\mu_i$ are accumulated:
$v_i=\sum_{x_t:NN(x_t)=\mu_i} x_t - \mu_i.$
The VLAD is the concatenation of the residual vectors $v_i$, i.e. $V=[v_1^\top \ldots v_K^\top]$.
All the residuals have the same size $D$ which is equal to the size of the used local features.
Thus the dimensionality of the whole vector $V$
is fixed too and it is equal to $DK$.
\paragraph{VLAD and Binary Local Features}
A naive way to apply the VLAD scheme to binary local descriptors is treating binary vectors as a particular case of real-valued vectors. In this way, the \textit{$k$-means} algorithm can be used to build the visual vocabulary and the difference between the centroids and the descriptors can be accumulated as usual. This approach has also been used in \cite{van14}, where a variation to the VLAD image signature, called BVLAD, has been defined to work with binary features.
Specifically, the BVLAD is the binarization (by thresholding) of a VLAD obtained using power-law, intra-normalization, $L_2$ normalization and multiple PCA.
Thereafter we have not evaluated the performance of the BVLAD because the binarization of the final image signature is out of the scope of this paper.
Similarly to BoW, various binary-cluster algorithms (e.g. \textit{$k$-medoids} and \textit{$k$-majority}) and the Hamming distance can be used to build the visual vocabulary and associate each binary descriptor to its nearest visual word. However, as we will see, the use of binary centroid\h{s} may provide less discriminant information during the computation of the residual vectors
\subsubsection{Fisher Vector}
The Fisher Kernel \cite{jaakkola98} is a powerful framework adopted in the context of image classification in \cite{perronnin07} as efficient tool to
{encode} image local descriptors into a fixed-size vector representation.
The main idea
is to derive a kernel function to measure the similarity between two sets of data, such as the sets of local descriptors extracted from two images.
The similarity of two sample sets $X$ and $Y$ is measured by analyzing the difference between the statistical properties of $X$ and $Y$, rather than comparing directly $X$ and $Y$.
To this scope a probability distribution $p(\cdot|\lambda)$ with some parameters $\lambda\in\R{m}$ is first estimated on a training set and it is used as generative model over the
the space of all the possible data observations. Then each set $X$ of observations is represented by a vector, named \textit{Fisher Vector}, that indicates the direction in which the parameter $\lambda$ of the probability distribution $p(\cdot|\lambda)$ should be modified to best fit the data in $X$.
In this way, two samples are considered similar if the directions given by their respective Fisher Vectors are similar.
Specifically, as proposed in \cite{jaakkola98}, the similarity between two sample sets $X$ and $Y$ is measured using the \textit{Fisher Kernel}, defined as
$ K(X,Y)= (G_{\lm}^X)^\top F_\lambda^{-1} G_{\lambda}^Y,$
where $F_\lambda$ is the \textit{Fisher Information Matrix} (FIM) and
${G}_{\lambda}^X=\nabla_\lambda\log p(X|\lambda)$ is referred to as the \textit{score function}.
The computation of the Fisher Kernel is costly due the multiplication by the inverse of the FIM. However, by using the Cholesky decomposition $F_\lambda^{-1}=L_\lambda^\top L_\lambda$, it is possible to re-written the Fisher Kernel as an Euclidean dot-product, i.e.
$ K(X,Y)=(\mathcal{G}_{\lambda}^X)^\top \mathcal{G}_{\lambda}^Y,$
where
$\mathcal{G}_{\lambda}^X=L_\lambda{G}_{\lambda}^X$
is the \emph{Fisher Vector} (FV) of $X$ \cite{perronnin10}.
Note that the FV is a fixed size vector whose dimensionality only depends on the dimensionality $m$ of the parameter $\lambda$.
The FV is further divided by $|X|$ in order to avoid the dependence on the sample size \cite{sanchez13} and $L_2$-normalized because, as proved in \cite{perronin10:improvingFK,sanchez13}, this is a way to cancel-out the fact that different images contain different amounts of image-specific information (e.g. the same object at different scales).
The distribution $p(\cdot|\lambda)$, which models the generative process in the space of the data observation, can be chosen in various way.
The Gaussian Mixture Model (GMM) is typically used to model the distribution of non-binary features
considering that, as pointed in \cite{mclachlan2000}, any continuous
distribution can be approximated arbitrarily well by an
appropriate finite Gaussian mixture.
%
Since the Bernoulli distribution models an experiment that has
only two possible outcomes (0 and 1), a reasonable alternative to characterize the distribution of a set of binary features is to use a
Bernoulli Mixture Model (BMM).
\paragraph{FV and Binary Local Features}
In this work we derive and test an extension of the FV built using BMM, called \textit{BMM-FV}, to encode binary features.
Specifically, we chose $p(\cdot|\lambda)$ to be multivariate Bernoulli mixture with
$K$ components and parameters $\lambda=\{w_k, \mu_{kd},\, k=1,\dots,
K,\, d=1,\dots, D\}$:
\begin{equation}
p(x_t|\lambda)=\sum_{k=1}^K w_k p_k(x_t)
\end{equation}
where
\begin{equation}
p_k(x_t)=\prod_{d=1}^D \mu_{kd}^{x_{td}}(1-\mu_{kd})^{1-x_{td}}
\end{equation}
and
\begin{equation}\label{eq:constrain}
\sum_{k=1}^K w_k=1, \quad w_k>0 \qquad \forall\, k=1, \dots, K.
\end{equation}
To avoid enforcing explicitly the constraints in \eqref{eq:constrain}, we used the soft-max formalism \cite{krapac11,sanchez13} for the weight parameters:
$
w_k={\exp(\alpha_k)}/{\sum_{i=1}^K \exp(\alpha_i)}.
$
Given a set $X=\{ x_t,\, t=1,\dots, T\} $ of $D$-dimensional binary vectors $x_t\in \{0,1\}^D$
and assuming that the samples are independent
we have that the score vector $G_{\lm}^X$ with respect to the parameter $\lambda=\{\alpha_k, \mu_{kd},\, k=1,\dots, K, \, d=1,\dots, D\}$
is calculated (see
Appendix A) as the concatenation of
\begin{align*}
G_{\alpha_k}^X &=\sum_{t=1}^T \dfrac{\partial\log p(x_t|\lambda)}{\partial \alpha_k}=\sum_{t=1}^T\left( \gamma_t(k)-w_k \right)\\
G_{\mu_{kd}}^X &=\sum_{t=1}^T \dfrac{\partial\log p(x_t|\lambda)}{\partial \mu_{kd}} =\sum_{t=1}^T
\gamma_t(k)\left(\dfrac{x_{td}-\mu_{kd}}{\mu_{kd}(1-\mu_{kd})}\right)
\end{align*}
where $\gamma_t(k)=p(k| x_t,\lambda)$ is the \emph{occupancy probability} (or posterior probability). The occupancy probability $\gamma_t(k)$ represents the probability for the observation $x_t$ to be generated by the $k$-th Bernoulli and it is calculated as
$
\gamma_t(k)={w_k p_k(x_t)}/{\sum_{j=1}^K w_j p_j(x_t)}.
$
The FV of $X$ is then obtained by normalizing the score $G_{\lm}^X$ by the matrix $L_\lambda$, which is the square root of the inverse of the FIM, and by the sample size $T$. In the
Appendix B we provide an approximation of FIM under the assumption that the occupancy probability $\gamma_t(k)$ is sharply peaked on a single value of $k$ for each descriptor $x_t$, obtained following an approach very similar to that used in \cite{sanchez13} for the GMM case.
By using our FIM approximation,
we got the following normalized gradient:
\begin{align*}
\mathcal G_{\alpha_k}^X &=\dfrac{1}{T\sqrt{w_k}}\sum_{t=1}^T \left(\gamma_t(k)-w_k\right) \\
\mathcal G_{\mu_{kd}}^X
&=\dfrac{1}{T\sqrt{w_k}}\sum_{t=1}^T \gamma_t(k)\left(\dfrac{x_{td}-\mu_{kd}}{\sqrt{\mu_{kd}(1-\mu_{kd})}}\right)
\end{align*}
The final BMM-FV is the concatenation of $\mathcal G_{\alpha_k}^X$ and $\mathcal G_{\mu_{kd}}^X$ for $k=1,\dots, K$, $d=1,\dots, D$ and is
therefore of dimension $K(D+1)$.
\begin{table}[bt]
\renewcommand{\arraystretch}{1.3}
\small
\caption{Comparison of the structure of the FVs derived using BMM with that derived using GMM. Parameters for BMM are $\lambda^B=\{w_k^B, \mu_{kd}^B,\, k=1,\dots, K,\, d=1,\dots, D\}$ and for GMM are $\lambda^G=\{w_k^G, \mu_{kd}^G,\Sigma_k^G=\text{diag}(\sigma_{k1}^G,\dots,\sigma_{kD}^G,), \, k=1,\dots, K,\, d=1,\dots, D\}$, where $w_{k}^B$, $\mu_{k}^B$ are the mixture weight and the mean vector of the $k$-th Bernoulli and $w_{k}^G$ $\mu_{k}^G$, $\Sigma_{k}^G$ are respectively the mixture weight, mean vector and covariance matrix of Gaussian $k$.}
\label{tab:BMM.GMM}
\centering
\begin{tabular}{l}
\noalign{\smallskip}\hline\noalign{\smallskip}
\multicolumn{1}{c}{\textbf{GMM-FV \cite{sanchez13}}}\\
\noalign{\smallskip}\hline\noalign{\smallskip}
$ \displaystyle
\mathcal G_{\alpha_k^G}^X =\frac{1}{T\sqrt{w_k^G}}\sum_{t=1}^T \left(\gamma_t^G(k)-w_k^G\right)
$\\
$\displaystyle
\mathcal G_{\mu_{kd}^G}^X =\frac{1}{T\sqrt{w_k^G}}\sum_{t=1}^T \gamma_t^G(k)\frac{x_{td}-\mu_{kd}^G}{{\sigma_{kd}^G}}
$ \\
$\displaystyle
\mathcal G_{\sigma_{kd}^G}^X =\frac{1}{T\sqrt{w_k^G}}\sum_{t=1}^T \gamma_t^G(k)\frac{1}{\sqrt{2}}\left[
\frac{(x_{td}-\mu_{kd}^G)^2}{(\sigma_{kd}^G)^2} -1
\right]
$\\
\noalign{\smallskip} \hline\noalign{\smallskip}
\multicolumn{1}{c}{\textbf{BMM-FV} (our formalization)} \\
\noalign{\smallskip}\hline\noalign{\smallskip}
$\displaystyle
\mathcal G_{\alpha_k^B}^X =\frac{1}{T\sqrt{w_k^B}}\sum_{t=1}^T \left(\gamma_t^B(k)-w_k^B\right)
$
\\
$\displaystyle
\mathcal G_{\mu_{kd}^B}^X =\frac{1}{T\sqrt{w_k^B}}\sum_{t=1}^T \gamma_t^B(k)\frac{x_{td}-\mu_{kd}^B}{\sqrt{\mu_{kd}^B(1-\mu_{kd}^B)}}\qquad
$ \\
\noalign{\smallskip} \hline\noalign{\smallskip}
\multicolumn{1}{c}{\textbf{BMM-FV} (Uchida et. al \cite{uchida13})} \\
\noalign{\smallskip}\hline\noalign{\smallskip}
($\displaystyle
\mathcal G_{\alpha_k^B}^X $ not explicitly derived in \cite{uchida13})
\\
$\displaystyle
\mathcal G_{\mu_{kd}^B}^X =\dfrac{\sum_{t=1}^T \gamma_t(k)\frac{(-1)^{1-x_{td}}}{({\mu_{kd}^B})^{x_{td}}\left(1-\mu_{kd}^B\right)^{1-x_{td}}}}{T\sqrt{T w_k^B\left(\frac{\sum_{i=1}^K w_i^B\mu_{id}^B}{(\mu_{kd}^B)^2} +\frac{\sum_{i=1}^K w_i^B(1-\mu_{id}^B)}{(1-\mu_{kd}^B)^2}\right)}} \qquad
$ \\
\noalign{\smallskip}\hline
\end{tabular}
\end{table}
An extension of the FV by using the BMM has been also carried in \cite{uchida13,sanchez15}.
Our approach differs from the one proposed in \cite{uchida13} in the approximation of the square root of the inverse of the FIM (i.e., $L_\lambda$) . It is worth noting that our formalization preserves the structure of the traditional FV derived by using the GMM, where Gaussian means and variances are replaced by Bernoulli means $\mu_{kd}$ and variances $\mu_{kd}(1-\mu_{kd})$ (see Table \ref{tab:BMM.GMM}).
In \cite{sanchez15}, the FV formalism was generalized to a broaden family of distributions knows as\textit{ exponential family} that encompasses the Bernoulli distribution as well as the Gaussian one. However, \cite{sanchez15} lacks in an explicit definition of the FV and of the FIM approximation in the case of BMM which was out of the scope of their work.
Our formulation differs from that of \cite{sanchez15} in the choice of the parameters used in the gradient computation of the score function \footnote{A Bernoulli distribution $p(x)=\mu^x(1-\mu)^{1-x}$ of parameter $\mu$ can be written as exponential distribution $p(x)=exp(\eta x- log(1+e^\eta))$ where $\eta= \log(\frac{\mu}{1-\mu})$ is the \textit{natural parameter}. In \cite{sanchez15} the score function is computed considering the gradient w.r.t. the {natural parameters} $\eta$ while in this paper we used the gradient w.r.t. the standard parameter $\mu$ of the Bernoulli (as also done in \cite{uchida13} ). }. A similar difference holds also for the FV computed on the GMM, given that in \cite{sanchez15} the score function is computed w.r.t. the {natural parameters} of the Gaussian distribution rather than the mean and the variance parameters which are typically used in literature for the FV representation
\h{ }\cite{perronnin10,perronnin07,sanchez13}.
Unfortunately, the authors of \cite{sanchez15} didn't experimentally compare the FVs obtained using or not the natural parameters.
S\`{a}nchez \cite{sanchez13} highlights that the FV derived from GMM can be computed in terms of the following $0$-order and $1$-order statistics: $ S_k^0=\sum_{t=1}^T \gamma_t(k) \, \in\R{}$, $S_k^1=\sum_{t=1}^T \gamma_t(k)x_t\, \in\R{D}$.
Our BMM-FV can be also written in terms of these statistics as
\begin{align*}
\mathcal G_{\alpha_k}^X &=\dfrac{1}{T\sqrt{w_k}}(S_k^0-Tw_k) \\%T
\mathcal G_{\mu_{kd}}^X &=\dfrac{S_{kd}^1 -\mu_{kd}S_k^0}{T\sqrt{w_k \mu_{kd}(1-\mu_{kd})}}
\end{align*}
We finally used power-law and $L_2$ normalization to improve the effectiveness of the BMM-FV approach.
\subsection{Combination of \h{Convolutional Neural Network Features and Aggregations of Binary Local Feature}}\label{sec:cnn}
Convolutional \h{N}eural \h{N}etworks (CNNs) \cite{LeCun2015} have brought breakthroughs in the computer vision area by improving the state-of-the-art in several domains, such as image retrieval, image classification, object recognition, and action recognition.
Depp CNN allows a machine to automatically learn representations of data with multiple levels of abstraction which can be used for detection or classification tasks. CNNs are neural networks specialized for data that has a grid-like topology as image data. The applied discrete convolution operation results in a multiplication by a matrix which has several entries constrained to be equal to other entries. Three important ideas are behind the success CNNs: sparse connectivity, parameter sharing, and equivariant representations \cite{DeepLearning2016}.
In image retrieval, the activations produced by an image within the top layers of the CNN have been successfully used as a high-level descriptors of the visual content of the image \cite{DeCaf}.
The results reported in \cite{razavian2014cnn} shows that
these CNN features, compared by using the Euclidean distance, achieve state-of-the-art quality in terms of mAP.
Most of the papers reporting results obtained using the CNN features
maintain the Rectified Linear Unit (ReLU) transform \cite{DeCaf,razavian2014cnn,chandrasekhar2015}, i.e., negative activations values are discarded replacing them with 0.
Values are typically $L_2$ normalized \cite{babenko2014neural,razavian2014cnn,chandrasekhar2015} and we did the same in this work.
In Section \ref{sec:expsetting} we describe the CNN model used in our experiments.
Recently, in \cite{chandrasekhar2015} it has been shown that the information provided by the FV built upon SIFT helps to further improve the retrieval performance of the CNN features and a combination of FV and CNN features has been used as well \cite{chandrasekhar2015,amato16:JOCCH}. However, the benefits of such combinations are clouded by the cost of extracting SIFTs that can be considered to high with respect to the cost of computing the CNN features (see Table \ref{tab:efficiencyComparison}).
Since the extraction of binary local features is up two times faster than SIFT, in this work we also investigate the combination of CNN features with
\h{aggregations of binary local feature, including BMM-FV.}
We combined BMM-FV and CNN using the following approach. Each image was represented by a couple $(c, f)$, where $c$ and $f$ were respectively the CNN descriptor and the BMM-FV of the image. Then, we evaluated the distance $d$ between two couples $(c_1, f_1)$ and $(c_2, f_2)$ as the convex combination between the $L_2$ distances of the CNN descriptors (i.e. $\|c_1-c_2\|_2$) and the BMM-FV descriptors (i.e. $\|f_1-f_2\|_2$). In other words, we defined the distance between two couples $(c_1, f_1)$ and $(c_2, f_2)$ as
\begin{equation}\label{eq:combination}
d\big((c_1, f_1), (c_2, f_2)\big)= \alpha\,\|c_1-c_2\|_2 +(1-\alpha)\,\|f_1-f_2\|_2
\end{equation}
with $0\leq \alpha\leq1$. Choosing $\alpha=0$ corresponds to use only FV approach, while $\alpha=1$ correspond to use only CNN features.
Please note that in our case both the FV and the CNN features are $L_2$ normalized \h{so the distance function between the CNN descriptors has the same range value of the distance function between the BMM-FV descriptors.}
\h{
Similarly, combinations between CNN features and other image descriptors, such as GMM-FV, VLAD, and BoW can be considered by using the convex combination of the respective distances. Please note that whenever the range of the two used distances is not the same, the distances should be rescaled before the convex combination (e.g. divide each distance function by its maximum value).}
\begin{table*}[bt]\footnotesize
\renewcommand{\arraystretch}{1}
\caption{Average time costs for computing various image representations
using a CPU implementation. The cost of computing the CNN feature of an image was estimated using \h{a} pre-learned AlexNet model and the Caffe framework \cite{caffe2014} \h{on an Intel i7 3.5 GHz}. The values related to the FV refers only to the cost of aggregating the local descriptors of an image into a single vector and do not encompass the cost of extracting the local features, neither the learning of the Gaussian or the Bernoulli Mixture Model which is calculated off-line. The cost of computing FV varies \h{proportionally with $TKD$, where $T$ is the number of local features extracted from an image, $K$ is the number of mixtures of Gaussian/Bernoulli, and $D$ is the dimensionality of each local feature}; we reported the approximate cost for \h{$T=2,000$ and $KD=64*64$ and $KD=64*256$ on an Intel i7 3.5 GHz}. The cost of SIFT/ORB local feature extraction was estimated according to \cite{heinly12} by considering about $2,000$ features per image.}\label{tab:efficiencyComparison}
\centering
\arrayrulecolor{gray}
\setlength\tabcolsep{2pt}
\begin{tabular}{@{}
>{\arraybackslash} p{0.25\textwidth}|
>{\centering\arraybackslash} p{0.15\textwidth}|
>{\centering\arraybackslash} p{0.25\textwidth} |
>{\centering\arraybackslash} p{0.15\textwidth} |
>{\centering\arraybackslash} p{0.15\textwidth}
@{}}
& CNN & {FV} Encoding & SIFT & ORB\\
\hline
\multirow{2}{0.2\textwidth}{\centering Computing time \\ per image }&
\multirow{2}{0.15\textwidth}{\centering ${\sim}300$ ms}&
${\sim}40$ ms [\h{$KD=64*64$}]&
\multirow{2}{0.15\textwidth}{\centering ${\sim}1200$ ms}& \multirow{2}{0.1\textwidth}{\centering ${\sim}26$ ms}\\
& & ${\sim}160$ ms [\h{$KD=256*64$}]& &\\
\end{tabular}
\end{table*}
\section{Experiments} \label{sec:exp}
In this section we evaluate and compare the performance of the
techniques described in this paper to aggregate binary local
descriptors. Specifically, in the Subsection \ref{sec:expCompEncoding} we compare the BoW, the VLAD, the FV based on the GMM,
and the BMM-FV approach to aggregate ORB binary features.
Since the BMM-FV achieved the best results over the other tested approaches, in the Subsection \ref{sec:expCompFVandCNN} we further evaluate the performance of the BMM-FVs using different binary features (ORB, LATCH, AKAZE) and combining them with the CNN features. \h{Finally, in the Subsection} \ref{sec:largescale}\h{, we report experimental results on large scale.}
In the following, we first introduce the datasets used in the evaluations (Subsection \ref{sec:datasets}) and we describe our experimental setup (Subsection \ref{sec:expsetting}). We then report results and their analysis.
\subsection{Datasets} \label{sec:datasets}
The experiments were conducted using two benchmark datasets, namely \textit{INRIA Holidays} \cite{jegou08} and \textit{Oxford5k }\cite{philbin07}, that are publicly available and often used in the context of image retrieval \cite{jegou10:VLAD,zhao13,jegou08,arandjelovic12:rootsift,perronnin10,jegou12,tolias14}.
INRIA Holidays \cite{jegou08}
is a collection of $1,491$ images which mainly contains personal holidays photos. The images are of high resolution and represent a large variety of scene type (natural, man-made, water, fire effects,
etc). The dataset contains 500 queries, each of which represents a distinct scene or object. For each query a list of positive results is provided. As done by the authors of the dataset, we resized the images to a maximum of $786,432$ pixels ($768$ pixels for the smaller dimension) before \h{extracting} the local descriptors.
Oxford5k \cite{philbin07}
consists of $5,062$ images collected from Flickr. The dataset comprise $11$ distinct Oxford buildings together with distractors.
There are $55$ query images: $5$ queries for each building.
The collection is provided with a comprehensive ground truth. For each query there are four image sets: \emph{Good} (clear pictures of the object represented in the query), \emph{OK} (images where more that $25\%$ of the object is clearly visible), \emph{Bad} (images where the object is not present) and \emph{Junk} (images where less than $25\%$ of the object is visible or images with high level of distortion).
As in many other articles, e.g. \cite{jegou10:VLAD,jegou08,philbin08,jegou12}, all the learning stages (clustering, etc.) were performed off-line using independent image collections.
\emph{Flickr60k} dataset \cite{jegou08} was used as training set for INRIA Holidays. It is composed of $67,714$ images \h{randomly extracted} from Flickr.
The experiments on Oxford5k were conducted performing the learning stages on \emph{Paris6k} dataset \cite{philbin08}, that contains $6,300$ high resolution images obtained from Flickr by searching for famous Paris landmarks.
\h{For large-scale experiments we combined the Holidays dataset with the 1 million MIRFlickr dataset} \cite{huiskes08},\h{ used as distractor set as also done in} \cite{jegou08,Amato2016}. \h{
Compared to Holidays, the Flickr datasets is slightly biased, because it includes low-resolution images and more photos of humans.}
\subsection{Experimental settings} \label{sec:expsetting}
In the following we report some details on how the features for the various approaches were extracted.
\begin{description}
\item[\textbf{Local features.}]
In the experiments we used ORB \cite{rublee11}, LATCH \cite{levi15_LATCH}, and AKAZE \cite{akaze} binary local features that \h{were} extracted by using OpenCV (Open Source Computer Vision Library)\footnote{\url{http://opencv.org/}}.
We detected up to $2,000$ local features per image.
\item[\textbf{Visual Vocabularies and Bernoulli/Gaussian Mixture Models.}] The visual vocabularies used for building the BoW and VLAD representations were computed using several clustering algorithms, i.e. $k$-medoids, $k$-majority and $k$-means.
The $k$-means algorithm was applied to the binary features by treating the binary vectors as real-valued vectors.
The parameters
$\lambda^B=\big\{w_k^B, \mu_{kd}^B\big\}_{k=1,\dots, K,\,
d=1,\dots, D}$ of the BMM and $\lambda^G=\big\{w_k^G,\mu_{kd}^G,\sigma_{kd}^G\big\}_{k=1,\dots, K,\, d=1,\dots, D}$ of the
GMM (where $K$ is the number of mixture components and $D$ is the dimension of each local descriptor) were learned independently by optimizing a maximum-likelihood
criterion with the Expectation Maximization (EM) algorithm
\cite{bishop06}.
EM is an iterative method that is deemed to have
converged when the change in the likelihood function, or
alternatively in the parameters $\lambda$, falls below some
threshold $\epsilon$.
As stopping criterion we used the convergence in $L_2$-norm of the mean
parameters, choosing $\epsilon=0.05$.
As suggested in \cite{bishop06}, the BMM/GMM parameters used in EM algorithm were initialized with: (a) $1/K$ for the mixing coefficients $w_{k}^B$ and $w_{k}^G$; (b) random values chosen uniformly in the range $(0.25, 0.75)$, for the BMM means $\mu_{kd}^B$; (c) centroids precomputed using $k$-means for the GMM means $\mu_{kd}^G$; (d) mean variance of the clusters found using $k$-means for the diagonal elements $\sigma_{kd}^G$ of the GMM covariance matrices.
All the learning stages, i.e. $k$-means, $k$-medoids, $k$-majority and the estimation of GMM/BMM, were performed using in order of 1M descriptors randomly selected from the local features extracted on the training sets (namely Flickr60k for INRIA Holidays and Paris6k for Oxford5k).
\item[\textbf{BoW, VLAD, FV.}]
The various encodings of the local feature (as well as the visual vocabularies and the BMM/GMM) were computed by using our Visual Information Retrieval library that is publicly available on GitHub\footnote{https://github.com/ffalchi/it.cnr.isti.vir}.
These representations are all parametrized by a single integer $K$. It corresponds to the number of centroids (visual words) used in BoW and VLAD, and to the number of mixture components of GMM/BMM used in FV representations.
For the FVs, we used only the components
$\mathcal G_{\mu}$ associated with the mean vectors because,
as happened in the non-binary case, we observed that the components
related to the mixture weights do not improve the results.
As a common post-processing step \cite{perronin10:improvingFK,jegou12}, both the FVs and the VLADs were power-law normalized and subsequently $L_2$-normalized. The power-law normalization is parametrized by a constant $\beta$ and it is defined as $x\to |x|^\beta \text{sign}(x)$. In our experiments we used $\beta=0.5$.\\
We also applied PCA to reduce VLAD and FV dimensionality. The projection matrices were estimated on the training datasets.
\item[\textbf{CNN features.}]
We used the pre-trained HybridNet \cite{zhou2014} model, downloaded from the Caffe Model Zoo\footnote{\url{https://github.com/BVLC/caffe/wiki/Model-Zoo}}. The architecture of HybridNet is the same as the BVLC Reference CaffeNet\footnote{\url{https://github.com/BVLC/caffe/tree/master/models/bvlc_reference_caffenet}} which mimics the original AlexNet \cite{krizhevsky2012}, with minor variations as described in \cite{caffe2014}. It has $8$ weight layers ($5$ convolutional + $3$ fully-connected). The model has been trained on $1,183$ categories ($205$ scene categories from Places Database \cite{zhou2014} and $978$ object categories from ImageNet \cite{deng2009}) with about $3.6$ million images.
In the test phase we used Caffe and we extracted the output of the first fully-connected layer (\textit{fc6}) after applying the Rectified Linear Unit (\textit{ReLU}) transform. The resulting $4,096$-dimensional descriptors were $L_2$ normalized.
As preprocessing step we warped the input images to the canonical resolution of $227\times 227$ RGB (as also done in \cite{DeCaf}).
\item[\textbf{Feature comparison and performance measure.}] The cosine similarity in conjunction with a term weighting scheme (e.g., tf-idf) is adopted for evaluating the similarity between BoW representations, while the Euclidean distance is used to compare VLAD, FV and CNN-based image signatures.\h{ Please note that the Euclidean distance is equivalent to the cosine similarity whenever the vectors are $L_2$-normalized, as in our case}\footnote{\h{
To search a database for the objects similar to a query we can use either a \textit{similarity} function or a \textit{distance} function. In the first case, we search for the objects with greatest similarity to the query. In the latter case, we search for the objects with lowest distance from the query. A similarity function is said to be equivalent to a distance function if the ranked list of the results to query is the same.
For example, the Euclidean distance between two vectors ($\ell_2(x_1,x_2)={\|x_1-x_2\|}_2$) is equivalent to the cosine similarity ($s_{\text{cos}}(x_1,x_2)=(x_1\cdot x_2)/({\|x_1\|}_2{\|x_2\|}_2)$) whenever the vectors are $L_2$- normalized (i.e. $\|x_1\|_2=\|x_2\|_2=1$). In fact, in such case, $s_{\text{cos}}(x_1,x_2)=1-\frac{1}{2}{\ell_2(x_1,x_2)}^2$, which implies that the ranked list of the results to a query is the same (i.e., $\ell_2(x_1,x_2)\le \ell_2(x_1,x_3)$ iff $s_{\text{cos}}(x_1,x_2)\geq s_{\text{cos}}(x_1,x_3)\, \forall\, x_1,x_2,x_3 $).}}.
The image comparison based on the \textit{direct matching} of the local features \h{(}i.e. without aggregation) was performed adopting the
distance ratio criterion proposed in \cite{lowe04,heinly12}.
Specifically, candidate matches to local features of the image
query are identified by finding their nearest neighbors in the
database of images. Matches are discarded if the ratio of the
distances between the two closest neighbors is above the 0.8
threshold. Similarity between two images is computed as the
percentage of matching pairs with respect to the total local
features in the query image.
The retrieval performance of each method was measured by the \textit{mean average precision} (mAP). In the experiments on INRIA Holidays, we computed the average precision after removing the query image from the ranking list.
In the experiments on Oxford5k, we removed the \textit{junk} images from the ranking before computing the average precision, as recommended in \cite{philbin07} and in the evaluation package provided with the dataset.
\end{description}
\subsection{Comparison of Various Encodings of Binary Local Features}\label{sec:expCompEncoding}
In Table \ref{tab:tab1} we summarize the retrieval performance of
various aggregation methods applied to ORB features, i.e. the BoW, the VLAD, the FV based on the GMM,
and the BMM-FV. In addition, in the last line of the table we reports the results obtained without
any aggregation, that we refer to as the \emph{direct matching} of local features, which was performed adopting the
distance ratio criterion as previously described in the Subsection \ref{sec:expsetting}.
In our experiments the FV derived as in \cite{uchida13} obtained very similar performance to that of our BMM-FV, thus we have reported just the results obtained by using our formulation.
Furthermore, we have not experimentally evaluated the FVs computed using the gradient with respect to the \textit{natural parameters} of a BMM or a GMM as described in \cite{sanchez15}, because the evaluation of the retrieval performance obtained using or not the natural parameters in the derivation of the score function is a more general topic which reserve to be further investigated outside the specific context of the encodings binary local features.
\begin{table}[tbp]
\renewcommand{\arraystretch}{1}
\caption{Performance evaluation of various aggregation methods applied on ORB binary features. \textit{$K$} indicates the number of centroids (visual words) used in BoW and VLAD and the number of mixture components of GMM/BMM used in FV; \textit{dim} is the number of components of each vector representation.\h{Bold numbers denote maxima in the respective column.}
}\label{tab:tab1}
\centering
\arrayrulecolor{gray}
\scriptsize
\begin{tabular}{
>{\centering\arraybackslash} p{0.1\columnwidth}
>{\centering\arraybackslash}p{0.17\columnwidth}
>{\centering\arraybackslash}p{0.13\columnwidth}
>{\centering\arraybackslash}p{0.06\columnwidth}
>{\raggedleft\arraybackslash}p{0.12\columnwidth}
>{\centering\arraybackslash}p{0.087\columnwidth}
>{\centering\arraybackslash}p{0.087\columnwidth}@{}}\toprule[1.1pt]
\multirow{2}{0.1\columnwidth}{\centering\textbf{Method}} & \multirow{2}{0.17\columnwidth}{\centering\textbf{Local Feature}} & \multirow{2}{0.13\columnwidth}{\centering\textbf{Learning method}}
& \multirow{2}{0.06\columnwidth}{\centering{$\mathbf{K}$}} &
\multirow{2}{0.12\columnwidth}{\centering{\textbf{dim}}} & \multicolumn{2}{c}{\centering\textbf{mAP}} \\
& & & & & \multirow{1}{0.087\columnwidth}{\scriptsize Holidays} & \multirow{1}{0.087\columnwidth}{\scriptsize Oxford5k} \\
\midrule[1.1pt]
\scriptsize BoW & \scriptsize ORB & $k$-means & 20,000 & 20,000 & 44.9 & 22.2 \\
\scriptsize BoW & \scriptsize ORB & $k$-majority & 20,000 & 20,000 & 44.2 & 22.8 \\
\scriptsize BoW & \scriptsize ORB & $k$-medoids & 20,000 & 20,000 & 37.9 & 18.8 \\
\midrule
\scriptsize VLAD & \scriptsize ORB & {\centering $k$-means} & 64 & 16,384 & 47.8 & 23.6 \\
& & & &{\tiny PCA$\rightarrow$}\,1,024 & 46.0 & 23.2 \\
& & & &{\tiny PCA$\rightarrow$}\,\,\,\,\,128 & 30.9 & 19.3\\
\scriptsize VLAD & \scriptsize ORB & $k$-majority & 64 & 16,384 & 32.4 & 16.6\\
\scriptsize VLAD & \scriptsize ORB & $k$-medoids & 64 & 16,384 & 30.6 & 15.6\\
\midrule
\scriptsize FV & \scriptsize ORB & \scriptsize GMM & 64 & 16,384 & 42.0 & 20.4\\
\scriptsize & & & &{\tiny PCA$\rightarrow$}\,1,024 & 42.6 & 20.3 \\
\scriptsize & & & &{\tiny PCA$\rightarrow$}\,\,\,\,\,128 & 35.5 & 19.6 \\
\midrule
\scriptsize FV & & \scriptsize BMM & 64 & 16,384 & 49.6 & \textbf{24.3} \\
\scriptsize & & & &{\tiny PCA$\rightarrow$}\,1,024 & \textbf{51.3} & 23.4 \\
\scriptsize & & & &{\tiny PCA$\rightarrow$}\,\,\,\,\,128 & 44.6 & 19.1 \\
\midrule
\scriptsize No-aggr. & \scriptsize ORB & \-- & \-- & & 38.1 & 31.7 \\
\bottomrule[1.1pt]
\end{tabular}
\end{table}
\begin{table}[tbp]
\renewcommand{\arraystretch}{1}
\caption{Aggregation methods on non-binary local features. Results are reported from \cite{jegou10:VLAD,jegou12}.
}\label{tab:res-literature}
\centering
\arrayrulecolor{gray}
\scriptsize
\begin{tabular}{
>{\centering\arraybackslash}p{0.1\columnwidth}
p{0.17\columnwidth}
>{\centering\arraybackslash}p{0.12\columnwidth}
>{\centering\arraybackslash}p{0.06\columnwidth}
>{\raggedleft\arraybackslash}p{0.12\columnwidth}
>{\centering\arraybackslash}p{0.087\columnwidth}
>{\centering\arraybackslash}p{0.087\columnwidth}@{}}\toprule[1.1pt]
\multirow{2}{0.1\columnwidth}{\centering\textbf{Method}} & \multirow{2}{0.17\columnwidth}{\textbf{Local Feature}} & \multirow{2}{0.12\columnwidth}{\centering\textbf{Learning method}}
& \multirow{2}{0.06\columnwidth}{\centering{$\mathbf{K}$}} &
\multirow{2}{0.12\columnwidth}{\centering{\textbf{dim}}} & \multicolumn{2}{c}{\centering\textbf{mAP}} \\
& & & & & \multirow{1}{0.087\columnwidth}{\scriptsize Holidays} & \multirow{1}{0.087\columnwidth}{\scriptsize Oxford5k} \\
\midrule[1.1pt]
\scriptsize BoW & \scriptsize SIFT & $k$-means & 20,000 & 20,000 & 40.4 & - \\
\scriptsize BoW & {\scriptsize{SIFT}\,PCA\,64} & $k$-means & 20,000 & 20,000 & 43.7 & 35.4 \\
\midrule
\scriptsize VLAD & \scriptsize SIFT & $k$-means & 64 & 8,192 & 52.6 & - \\
& & & & {\tiny PCA$\rightarrow$}\,128 & 51.0 & - \\
\scriptsize VLAD & {\scriptsize{SIFT}\,PCA\,64} & $k$-means & 64 & 4,096 & 55.6 & 37.8 \\
& & & & {\tiny PCA$\rightarrow$}\,128 & 55.7 & 28.7 \\
\midrule
\scriptsize FV & \scriptsize SIFT & \scriptsize GMM & 64 & 8,192 & 49.5 & - \\
& & & & {\tiny PCA$\rightarrow$}\,128 & 49.2 & - \\
\scriptsize FV & {\scriptsize{SIFT}\,PCA\,64} & \scriptsize GMM & 64 & 4,096 & 59.5 & 41.8 \\
& & & & {\tiny PCA$\rightarrow$}\,128 & 56.5 & 30.1 \\
\bottomrule[1.1pt]
\end{tabular}}
\end{table}
Among the various baseline aggregation methods (i.e. without using PCA), the
{BMM-FV} approach achieves the best retrieval performance, that is
a mAP of \emph{49.6\%} on Holidays and \textbf{24.3\% }on
Oxford. PCA dimensionality reduction from $16,384$ to $1,024$
components, applied on BMM-FV, marginally reduces the mAP on
Oxford5k, while on Holiday allows us to get \textbf{51.3\%} that is,
for this dataset, the best result achieved between all the other
aggregation techniques tested on ORB binary features.
Good results are also achieved using VLAD in conjunction with
$k$-means, which obtains a mAP of \emph{47.8\%} on Holidays and
\emph{23.6\%} on Oxford5k.
The BOW representation allows to get a mAP of
\emph{44.9\%}/\emph{44.2\%}/\emph{37.9\%} on Holidays and
\emph{22.2\%}/\emph{22.8\%}/\emph{18.8\%} on Oxford5k using
respectively $k$-means/$k$-majority/$k$-medoids for the learning
of a visual vocabulary of $20,000$ visual words.
The GMM-FV method gives results slight worse than BoW: \emph{42.0\%}
of mAP on Holidays and \emph{20.4\%} of mAP on Oxford5k. The use
of PCA to reduce dimensions from $16,384$ to $1,024$ lefts
the results of GMM-FV on Oxford5k substantially unchanged while slightly
improved the mAP on Holidays (\emph{42.6\%}).
Finally, the worst performance are that of VLAD in combination with
vocabularies learned by $k$-majority (\emph{32.4\%} on Holidays
and \emph{16.6\%} on Oxford) and $k$-medoids (\emph{30.6\%} on
Holidays and \emph{15.6\%} on Oxford).
It is generally interesting to note that on INRIA Holidays, the
VLAD with $k$-means, the BoW with $k$-means/$k$-majority, and the FVs are better than direct match. In
fact, mAP of direct matching of ORB descriptors is
\emph{38.1\%} while on Oxford5k the direct matching
reached a mAP of \emph{31.7\%}.
\begin{table}[tbp]
\renewcommand{\arraystretch}{1}
\caption{Retrieval performance of our BMM-FV on INRIA Holidays and Oxford5k. \textit{$K$} is the number of BMM mixtures. \textit{dim} is the number of components of the final vector representation. \h{Bold numbers denote maxima in the respective column.} }\label{tab3}
\centering
\arrayrulecolor{gray}
\scriptsize
\subtable[Performance evaluation for increasing number $K$ of Bernullian mixture components\label{tab:fvBmm}]{
\begin{tabular}{ >{\centering\arraybackslash}p{0.06\columnwidth} >{\raggedleft\arraybackslash}p{0.175\columnwidth}| >{\centering\arraybackslash}p{0.15\columnwidth} >{\centering\arraybackslash}p{0.15\columnwidth}@{}}\toprule[1.1pt]
\multirow{2}{*}{\centering{$\mathbf{K}$}} &
\multirow{2}{0.17\columnwidth}{\centering{\textbf{dim}}} & \multicolumn{2}{c}{\centering\textbf{mAP}} \\
& & \multirow{1}{*}{Holidays} & \multirow{1}{*}{Oxford5k} \\
\hline
4 & 1,024 & 32.0 & 14.3 \\
8 & 2,048 & 38.2 & 17.4 \\
16 & 4,096 & 41.9 & 19.4 \\
32 & 8,192 & 45.9 & 21.3 \\
64 & 16,384 & 49.6 & 24.3 \\
128 & 32,768 & 52.3 & 26.4 \\
256 & 65,536 & 53.0 & {27.3 } \\
512 & 131,072 & \textbf{54.7} & \textbf{27.4} \\
\bottomrule[1.1pt]
\end{tabular}}\\
\subtable[Results after dimensionality reduction when $K=64$ Bernoulli are used\label{tab:PCA}]{
\begin{tabular}{ >{\centering\arraybackslash}p{0.06\columnwidth} >{\raggedleft\arraybackslash}p{0.175\columnwidth}| >{\centering\arraybackslash}p{0.15\columnwidth} >{\centering\arraybackslash}p{0.15\columnwidth}@{}}\toprule[1.1pt]
\multirow{2}{*}{\centering{$\mathbf{K}$}} &
\multirow{2}{0.17\columnwidth}{\centering{\textbf{dim}}} & \multicolumn{2}{c}{\centering\textbf{mAP}} \\
& & \multirow{1}{*}{Holidays} & \multirow{1}{*}{Oxford5k} \\
\hline
64 & 16,384 & 49.6 & 24.3\\
64 & {\scriptsize PCA$\rightarrow$}\,4,096 &\textbf{52.6} & \textbf{25.1} \\
64 & {\scriptsize PCA$\rightarrow$}\,2,048 &{51.8} & 24.3 \\
64 & {\scriptsize PCA$\rightarrow$}\,1,024 & 51.3 & 23.4 \\
64 & {\scriptsize PCA$\rightarrow$}\,\,\,\,\,512 & 48.2 & 21.7 \\
64 &{\scriptsize PCA$\rightarrow$}\,\,\,\,\,256 & 45.9 & 20.3 \\
64 & {\scriptsize PCA$\rightarrow$}\,\,\,\,\,128 & 44.6 & 19.1 \\
64 & {\scriptsize PCA$\rightarrow$}\,\,\,\,\,\,\,\,64 & 42.9 & 17.2 \\
\bottomrule[1.1pt]
\end{tabular}}
\end{table}
In Table \ref{tab3} we also report the performance of our derivation of the BMM-FV
varying the number $K$ of Bernoulli mixture components
and investigating the impact of the PCA
dimensionality reduction in the case of $K=64$.
In Table (\ref{tab:fvBmm}) we can see that with the Holidays
dataset, the mAP grows from \emph{32.0\%} when using only 4
mixtures to \textbf{54.7\%} when using $K=512$. On Oxford5k, mAP
varies from \emph{14.3\%} to \textbf{27.4\%}, respectively, for
$K=4$ and $K=512$.
Table (\ref{tab:PCA}) shows that the best results are achieved
when reducing the full size BMM-FV to $4,096$ with a mAP of \textbf{52.6\%} for
Holidays and \textbf{25.1\%} for Oxfrod5k.
\paragraph{Analysis of the results}
Summing up, the results show that in the context of binary local features the BMM-FV outperforms the compared aggregation methods, namely \h{the} BoW, \h{the} VLAD and \h{the} GMM-FV.
The performance of the BMM-FV is an increasing function of the number $K$ of Benoulli mixtures. However, for large $K$, the improvement tends to be smaller and the dimensionality of the FV becomes very large (e.g. $65,536$ dimensions using $K=256$). Hence, for high values of $K$, the benefit of the improved accuracy is not worth the computational overhead (both for the BMM estimation and for the cost of storage/comparison of FVs).
The PCA reduction of BMM-FV is effective since it can provide a very compact image signature
with just a slight loss in accuracy, as shown in the case of $K=64$ (Table \ref{tab:PCA}). Dimension reduction does not necessarily reduce the accuracy. Conversely, limited reduction tend to improve the retrieval performance of the FV representations. %
For the computation of VLAD, the $k$-means results are more effective than $k$-majority/$k$-medoids clustering, since the use of non-binary centroids gives more discriminant information during the computation of the residual vectors used in VLAD.
For the BoW approach, $k$-means and $k$-majority performs equally better than $k$-medoids.
However, the $k$-majority is preferable in this case because the cost of the quantization process is significantly reduced by using the Hamming distance, rather than Euclidean one, for the comparison between centroids and binary local features.
Both BMM-FV and VLAD, with only $K=64$, outperform BoW. However, as happens for non-binary features (see Table \ref{tab:res-literature}), the loss in accuracy of BoW representation is comparatively lower when the variability of the images is limited, as for the Oxford5k dataset.
As expected, BMM-FV outperforms GMM-FV, since the probability distribution of binary local features is better described using mixtures of Bernoulli rather than mixtures of Gaussian.
The results of our experiments also show that the use of BMM-FV is still effective even if compared with the direct matching strategy.
In fact, the retrieval performance of BMM-FV on Oxford5k is just slightly worse than traditional direct matching of local feature, while on INRIA Holidays the BMM-FV even outperforms the direct matching result.
For completeness, in Table \ref{tab:res-literature}, we also
report the results of the same base-line encodings approaches applied to
non-binary features (both full-size SIFT and PCA-reduced to 64
components) taken from literature \cite{jegou10:VLAD,jegou12}. As
expected, aggregation methods in general exhibit better performance in
combination with SIFT/SIFTPCA then with ORB, expecially for the Oxford5k dataset.
However, it is worth noting that on the INRIA Holidays the BMM-FV outperforms the BoW on SIFT/SIFTPCA and reach similar performance of the FV built upon SIFTs.
The FV and VLAD get considerable benefit from performing PCA of SIFT local descriptors before the aggregation phase as the PCA rotation decorrelate the descriptors components. This suggest that techniques, such as VLAD with k-means and GMM-FV, which treat binary vectors as real-valued vectors, may also benefit from the use of PCA before the aggregation phase.
In conclusion, it is
important to point-out that there are several applications where
binary features need to be used to improve efficiency, at the cost
of some effectiveness reduction \cite{heinly12}. We showed that in this case the use of the encodings techniques represent a valid alternative to the direct matching.
\subsection{\h{Combination of CNNs and Aggregations of Binary Local Feature}}\label{sec:expCompFVandCNN}
\begin{figure}[tbp]
\centering
{\includegraphics[trim=20mm 95mm 15mm 80mm, width=0.8\linewidth]{combinationCNNs-all}}
\caption{\h{Retrieval performance on the INRIA Holidays dataset of the combination of HybridNet \textit{fc6} and various aggregations of ORB binary feature (BMM-FV, VLAD, BoW, and GMM-FV). Only the full-sized descriptors are considered (i.e., no PCA) and for each aggregation technique we selected the corresponding best setting (e.g learning method) according with results reported in Table} \ref{tab:comb varying local feature}.
\h{$\alpha$ is the parameter used in the combination: $\alpha = 0$ corresponds to use only the aggregated descriptor, while $\alpha=1$ correspond to use only the HybridNet feature. }
}
\label{fig:combinationCNNs-all}
\end{figure}
In this section we evaluate the retrieval performance of the combination of
\h{CNN features with the aggregations of binary local feature, following the approach described in Section} \ref{sec:cnn}.
We considered the INRIA Holidays dataset and we used the
the output of the first fully-connected layer (\textit{fc6}) of the HybridNet \cite{zhou2014} model as CNN feature. In fact, in \cite{chandrasekhar2015} several experiments on the INRIA Holidays have shown that \textit{HybridNet fc6} achieve better mAP result than other outputs (e.g. \textit{pool5, fc6, fc7, fc8}) of several pre-trained CNN models: \h{the} OxfordNet \cite{simonyan2014}, \h{the} AlexNet \cite{krizhevsky2012}, \h{the} PlacesNet \cite{zhou2014} and \h{the} HybridNet itself.
\h{Figure }\ref{fig:combinationCNNs-all}\h{ shows the mAP obtained by combining HybridNet \textit{fc6} with different aggregations of ORB binary features, namely the BMM-FV, the GMM-FV, the VLAD, and the BoW.
Interestingly, with the exception of the GMM-FV, the retrieval performance obtained after the combination is very similar for the various aggregation techniques. This, on the one hand confirms that the GMM-FV is not the best choice for encoding \textit{binary} features, and on the other hand, since each aggregation technique computes statistical summaries of the \textit{same set} of the local descriptors, suggests that the additional information provided by the various aggregated descriptors
helps almost equally to improve the retrieval performance of the CNN feature.
Thus, in the following we further investigate combinations of CNNs and the BMM-FV that, even for a shot, reaches the best performance for all the tested parameter $\alpha$.
}
\begin{table*}[tbp]
\renewcommand{\arraystretch}{1}
\caption{Retrieval performance of various combinations of BMM-FV and HybridNet CNN feature. The BMM-FV representations were computed for three different binary local features (ORB, LATCH, and AKAZE) using $K=64$ mixtures of Bernoulli. The CNN feature was computed as the output the HybriNet fc6 layer after applying the ReLU transform. \textit{Dim} is the number of components of each vector representation.
$\alpha$ is the parameter used in the combination of FV and CNN: $\alpha = 0$ corresponds to use only FV, while $\alpha=1$ correspond to use only the HybridNet feature.
\h{Bold numbers denote maxima in the respective column.}
}\label{tab:comb varying local feature}
\centering
\scriptsize
\arrayrulecolor{gray}
\begin{tabular}{@{}
>{\centering\arraybackslash} p{0.23\textwidth}
>{\centering\arraybackslash} p{0.075\textwidth}
>{\centering\arraybackslash} p{0.075\textwidth}
>{\centering\arraybackslash} p{0.075\textwidth}
>{\centering\arraybackslash}p{0.075\textwidth}
>{\centering\arraybackslash} p{0.075\textwidth}
>{\centering\arraybackslash}p{0.075\textwidth}
>{\centering\arraybackslash}p{0.075\textwidth}
@{}}\toprule[1.1pt]
\multirow{2}{0.23\textwidth}{\centering\textbf{Method}} &\multicolumn{3}{c}{\textbf{Dim}} & \multirow{2}{0.07\textwidth}{\centering{$\boldsymbol{\alpha}$}} & \multicolumn{3}{c}{\textbf{mAP}}\\
\cmidrule{2-4} \cmidrule{6-8}
&\scriptsize ORB & \scriptsize LATCH & \scriptsize AKAZE & & \scriptsize ORB &\scriptsize LATCH &\scriptsize AKAZE \\ \midrule[1.1pt]
BMM-FV (K=64) & 16,384 & 16,384 & 32,768 & 0 & 49.6 & 46.3 & 43.7 \\
\midrule
\multirow{9}{0.25\textwidth}{\centering{Combination of \\ \textit{BMM-FV (K=64)}\\ and\\ \textit{HybridNet fc6}}} & \multirow{9}{0.08\textwidth}{\centering 20,480}& \multirow{9}{0.08\textwidth}{\centering 20,480}&\multirow{9}{0.08\textwidth}{\centering 36,864} & 0.1 & 66.4 & 64.7 & 59.2 \\
& & & & 0.2 & 74.8 & 73.8 & 68.7 \\
& & & & 0.3 & 77.4 & 76.8 & 74.3 \\
& & & & 0.4 & 79.1 & 77.5 & 77.3 \\
& & & & 0.5 & \textbf{79.2 } & 78.3 & 78.0 \\
& & & & 0.6 & 79.0 & \textbf{78.5 } & \textbf{79.2 } \\
& & & & 0.7 & 78.7 & 77.7 & 78.7 \\
& & & & 0.8 & 77.8 & 76.7 & 77.5 \\
& & & & 0.9 & 76.4 & 76.3 & 76.2 \\
\midrule
HybridNet fc6 & \multicolumn{3}{c}{4,096} & 1 & \multicolumn{3}{c}{75.5 } \\ %
\bottomrule[1.1pt]
\end{tabular}
\end{table*}
In Table \ref{tab:comb varying local feature}
we report the mAP obtained combining the HybridNet fc6 feature with the BMM-FV computed for three different kind of binary local features, namely ORB, LATCH and AKAZE, using $K=64$ mixtures of Bernoulli.
It is worth noting that all the three BMM-FVs
give a similar improvement when combined with the HybridNet fc6, although they have rather different mAP results (see first row of Table \ref{tab:comb varying local feature}) which are substantially lower than that of CNN (last row of Table \ref{tab:comb varying local feature}). The intuition is that the additional information provided by using a specific BMM-FV rather than using the CNN feature alone, do not depend very much on the used binary feature.
For each tested BMM-FV seems that exist an optimal $\alpha$ to be used in the convex combination (equation \eqref{eq:combination}). When ORB binary features were used, the optimal $\alpha$ was obtained around $0.5$, which correspond to give the same importance to both FV and CNN feature. For the less effective BMM-FVs built upon LATCH and AKAZE, the optimal $\alpha$ was $0.6$, which means that the CNN feature is used with slightly more importance than BMM-FV during the convex combination.
The use of ORB or AKAZE led to obtain the best performance that was $\textbf{79.2\%}$ of mAP.
This results in a relative improvement of $\textbf{4.9\%}$ respect to the single use of the CNN feature, that in our case was $\textbf{75.5\%}$. So we obtain the same relative improvement of \cite{chandrasekhar2015} but using a less expensive FV representation.
Indeed, in \cite{chandrasekhar2015} the fusion of HybridNet \textit{fc6} and a FV computed on 64-dimensional PCA-reduced SIFTs, using $K=256$ mixtures of Gaussian, have led to obtain a relative improvement of ${4.9\%}$ respect to the use of the CNN feature alone (see also Table \ref{tab:comparison}).
\h{However, the cost for integrating traditional FV built upon SIFTs with CNN features may be considered too high, especially for systems that need to process image in real time. For example, according to}
\cite{heinly12}
\h{and as showed in the table} \ref{tab:efficiencyComparison},
\h{the SIFTs extraction (about $2,000$ features per image), the PCA-reduction to $D=64$ dimensions, and the FV aggregation with $K=256$ requires more than $1.3$ seconds per image, while the CNN feature extraction is 4 times faster (i.e., about $300$ ms per image).
On the other hand, extracting ORB binary features (about $2,000$ features per image, each of dimension $D=256$) and aggregating them using a BMM-FV with $K=64$ requires less than $190$ ms that is in line with the cost of CNN extraction ($300$ ms).
In our tests, the cost for integrating the already extracted BMM-FV and the CNN features was negligible in the search phase, using a sequential scan to search a dataset, also thanks to the fact that both BMM-FV and CNN features are computed using the not too costly Euclidean distance.
}
\begin{figure}[tbp]
\centering
{\includegraphics[trim=20mm 90mm 15mm 80mm, width=0.8\linewidth]{combinationFVCNNs-ORB}}
\caption{Retrieval performance of the combinations of BMM-FV and HybridNet \textit{fc6} for various number $K$ of Bernoulli mixtures. The BMM-FVs were computed using ORB binary features.
$\alpha$ is the parameter used in the combination: $\alpha = 0$ corresponds to use only FV, while $\alpha=1$ correspond to use only the HybridNet feature.
}
\label{fig:bmm-fvORB}
\end{figure}
Since as observed in \cite{levi15_LATCH,akaze} the ORB extraction is faster than LATCH and AKAZE, in the following we focus just on ORB binary feature.
In figure \ref{fig:bmm-fvORB} we show the results obtained by combining HybridNet \textit{fc6} with the BMM-FVs obtained using $K=32,64,128$. We observed that the performance of the CNN feature is improved also when it is combined with the less effective BMM-FV built using $K=32$ Bernoulli. The BMM-FV with $K=128$ achieve the best effectiveness (mAP of $\textbf{79.5\%}$) for $\alpha=0.4$. However, since the cost for computing and storing FV increase with the number $K$ of Bernoulli, the improvement obtained using $K=128$ respect to that of $K=64$ doesn't worth the extra cost of using a bigger value of $K$.
The BMM-FV with $K=64$ is still high dimensional, so to reduce the cost of storing and comparing FV, we also evaluated the combination after the PCA dimensionality reduction.
As already observed, limited dimensionality reduction tends to improve the accuracy of the single FV representation. In fact, the BMM-FV with $K=64$ achieved a mAP of $52.6\%$ when reduced from $16,384$ to $4,096$ dimensions. However, as shown in Table \ref{tab:pcanopca} and Table \ref{tab:comparison}, when the PCA-reduced version of the BMM-FV was combined with HybriNet \textit{fc6}, the overall relative improvement in mAP was $3.9\%$, which is less than that obtained using the full-sized BMM-FV. These result is not surprising given that after the dimensionality reduction we may have a loss of the additional information provided by the FV representation during the combination with the CNN feature.
Finally, in Table \ref{tab:comparison} we summarizes the relative improvement achieved by combining BMM-FV and HybriNet \textit{fc6}, and we compare the obtained results with the relative improvement achieved in \cite{chandrasekhar2015}, where the more expensive FV built upon SIFTs was used. We observed that BMM-FV led to achieve similar or even better relative improvements with an evident advantage from the computational point of view, because it uses binary local features
\begin{table*}[tbp]
\renewcommand{\arraystretch}{1}
\caption{Comparison of the results obtained combining HybridNet fc6 feature with the full-sized and the PCA-reduced version of the BMM-FV. The BMM-FV was computed on ORB binary feature using $K=64$ mixtures of Bernoulli. \textit{Dim} is the number of components of each vector representation. $\alpha$ is the parameter used in the combination of FV and CNN.
\h{Bold numbers denote maxima in the respective column.}
}\label{tab:pcanopca}
\centering
\scriptsize
\arrayrulecolor{gray}
\begin{tabular}{@{}
>{\centering\arraybackslash} p{0.23\textwidth}
>{\centering\arraybackslash} p{0.1\textwidth}
>{\centering\arraybackslash} p{0.1\textwidth}
>{\centering\arraybackslash}p{0.1\textwidth}
>{\centering\arraybackslash} p{0.1\textwidth}
>{\centering\arraybackslash}p{0.1\textwidth}
@{}}\toprule[1.1pt]
\multirow{2}{0.23\textwidth}{\centering\textbf{Method}} & \multicolumn{2}{c}{\textbf{Dim}} & \multirow{2}{0.1\textwidth}{\centering{$\boldsymbol{\alpha}$}} & \multicolumn{2}{c}{\textbf{mAP}}\\ \cmidrule{2-3} \cmidrule{5-6}
& \scriptsize{FV full dim} & \scriptsize{FV PCA-reduced} & & \scriptsize FV full dim &\scriptsize FV PCA-reduced \\ \midrule[1.1pt]
BMM-FV (K=64) & 16,384 & 4,096 & 0 & 49.6 & 52.6 \\
\midrule
\multirow{9}{0.25\textwidth}{\centering{Combination of \\ \textit{BMM-FV (K=64)}\\ and\\ \textit{HybridNet fc6}}} & \multirow{9}{0.085\textwidth}{\centering 20,480} & \multirow{9}{0.085\textwidth}{\centering 8,192} & 0.1 & 66.4 & 66.3 \\
& & & 0.2 & 74.8 & 73.9 \\
& & & 0.3 & 77.4 & 77.3 \\
& & & 0.4 & 79.1 & \textbf{78.5} \\
& & & 0.5 & \textbf{79.2 } & 78.4 \\
& & & 0.6 & 79.0 & {78.5 } \\
& & & 0.7 & 78.7 & 78.1 \\
& & & 0.8 & 77.8 & 77.7 \\
& & & 0.9 & 76.4 & 76.4 \\
\midrule
HybridNet fc6 & \multicolumn{2}{c}{4,096} & 1 & \multicolumn{2}{c}{75.5 } \\
\bottomrule[1.1pt]
\end{tabular}
\end{table*}
\begin{table}[tbp]
\centering
\scriptsize
\caption{Relative mAP improvement obtained after combining FV with HybridNet \textit{fc6}. Each relative improvements was computed respect to the use of the CNN feature alone, that is: $\left(\text{mAP}_\text{after combination}-\text{mAP}_\text{HybridNet fc6}\right)/\text{mAP}_\text{HybridNet fc6}$. The relative improvements obtained using the FV computed on 64-dimensional PCA-reduced SIFTs (SIFTPCA64) was computed according to the results reported in \cite{chandrasekhar2015}. }\label{tab:comparison}
{%
\begin{tabular}{
>{\raggedright\arraybackslash}p{0.2\columnwidth}
>{\centering\arraybackslash}p{0.17\columnwidth}
>{\centering\arraybackslash}p{0.1\columnwidth}
>{\centering\arraybackslash}p{0.15\columnwidth}
>{\centering\arraybackslash}p{0.17\columnwidth}
@{}}\toprule[1.1pt]
\multirow{2}{0.15\textwidth}{\centering\textbf{FV\\ method}} &
\multirow{2}{0.2\textwidth}{\centering\textbf{Local \\Feature}} &
\multirow{2}{0.1\textwidth}{\centering\textbf{K}} &
\multirow{2}{0.15\textwidth}{\centering\textbf{dim}} &
\multirow{2}{0.17\textwidth}{\centering\textbf{Relative improvement}} \\
\\\midrule[1.1pt]
BMM-F
& ORB & 128 & 32,768 & 5.2 \\
BMM-F
& ORB & 64 & 16,384 & 4.9 \\
BMM-F
& AKAZE & 64 & 32,768 & 4.9 \\
BMM-F
& LATCH & 64 & 16,384 & 4.0 \\
BMM-F
+ PCA & ORB & 64 & 4,096 & 3.9 \\
BMM-F
& ORB & 32 & 8,192 & 3.5 \\ \midrule
F
\cite{chandrasekhar2015} & SIFTPCA64 & 256 & 32,768 & 4.9 \\
\bottomrule[1.1pt]
\end{tabular}}
\end{table}
\subsection{\h{Large-Scale Experiments}}\label{sec:largescale}
\begin{table*}[tbp
\renewcommand{\arraystretch}{1}
\caption{
\h{Comparison of the results obtained combining HybridNet fc6 feature and BMM-FV on the INRIA Holidays dataset with the distractor dataset MIRFlickr-1M.
The results related to the INRIA Holidays alone are reported from Table} \ref{tab:pcanopca} \h{for reference.
The BMM-FV was computed on ORB binary feature using $K=64$ mixtures of Bernoulli; both full-sized and the PCA-reduced features are considered.
\textit{Dim} is the number of components of each vector representation. $\alpha$ is the parameter used in the combination of FV and CNN.
Bold numbers denote maxima in the respective column. The last row reports the maximum relative mAP improvement obtained after combining FV with HybridNet \textit{fc6}; relative improvements are computed respect to the use of the CNN feature alone, that is $\left(\text{mAP}_\text{after combination}-\text{mAP}_\text{HybridNet fc6}\right)/\text{mAP}_\text{HybridNet fc6}$. }
}\label{tab:largescaleALL}
\centering
\scriptsize
\begin{tabular}
>{\centering\arraybackslash} p{0.18\textwidth}|
>{\centering\arraybackslash} p{0.08\textwidth}
>{\centering\arraybackslash} p{0.08\textwidth}
>{\centering\arraybackslash}p{0.06\textwidth}
>{\centering\arraybackslash} p{0.065\textwidth}
>{\centering\arraybackslash}p{0.065\textwidth}
>{\centering\arraybackslash}p{0.0051\textwidth}
>{\centering\arraybackslash}p{0.065\textwidth}
>{\centering\arraybackslash}p{0.065\textwidth
}\toprule[1.1pt]
\multirow{3}{0.18\textwidth}{\centering\textbf{Method}} & \multicolumn{2}{c}{\textbf{Dim}}& \multirow{3}{0.06\textwidth}{\centering{$\boldsymbol{\alpha}$}} & \multicolumn{5}{c}{\textbf{mAP}}\\ \cmidrule{2-3} \cmidrule{5-9}
& {\multirow{2}{0.08\textwidth}{\centering\scriptsize{FV full dim}}} & {\multirow{2}{0.08\textwidth}{\scriptsize{FV PCA-reduced}}} & & \multicolumn{2}{c}{\scriptsize{FV full dim}} & & \multicolumn{2}{c}{\scriptsize{FV PCA-reduced}} \\ \cmidrule{5-6} \cmidrule{8-9}
& & & & \rot{{Holidays}} & \rot{\multirow{1}{0.06\textwidth}{Holidays+ MIRFlickr}} && \rot{{Holidays}} & \rot{\multirow{1}{0.06\textwidth}{Holidays+ MIRFlickr}} \\
\midrule
BMM-FV (K=64) & 16,384 & 16,384 & 0 & 49.6 & 31.0 && 52.6 & 34.9 \\ \midrule
\multirow{9}{0.18\textwidth}{\centering{Combination of \\ \textit{BMM-FV (K=64)}\\ and\\ \textit{HybridNet fc6}}} & 20,480 & 8,192 & 0.1 & 66.4 & 47.0 && 66.3 & 50.7 \\
& & & 0.2 & 74.8 & 59.3 && 73.9 & 61.9 \\
& & & 0.3 & 77.4 & 64.0 && 77.3 & 65.6 \\
& & & 0.4 & 79.1 & \textbf{67.1} && \textbf{78.5} & \textbf{67.2} \\
& & & 0.5 &\textbf{79.2}& 66.5 && 78.4 & 66.9 \\
& & & 0.6 & 79.0 & 65.7 && 78.5 & 65.7 \\
& & & 0.7 & 78.7 & 64.4 && 78.1 & 64.4 \\
& & & 0.8 & 77.8 & 62.5 && 77.7 & 62.8 \\
& & & 0.9 & 76.4 & 60.7 && 76.4 & 60.8 \\ \midrule
HybridNet fc6 & \multicolumn{2}{c}{4,096} & 1 & 75.5 & 59.1 && 75.5 & 59.1 \\ \bottomrule[1.1pt] \\
\multicolumn{4}{r}{ \textbf{Maximum relative mAP improvement} $\rightarrow$}& 4.9\%&13.4\%&&4.0\%&13.7\%
\end{tabular}
\end{table*}
\h{ In order to evaluate the behavior of feature combinations on a large scale, we have used a set of up to one million images. More precisely, as in }\cite{jegou08}\h{, we merged the INRIA Holidays dataset with a public large-scale dataset (MIRFlickr-1M} \cite{huiskes08}\h{) used as distraction set; the mAP was measured using the Holidays ground-truth.}
\h{Table} \ref{tab:largescaleALL}\h{ reports results obtained using both the BMM-FV alone and the combinations with the \textit{HybridNet fc6} CNN feature. Given the results reported in the previous section we focus on the BMM-FV encoding of ORB binary features. All the feature combinations show an improvement with respect to the single use of the CNN feature (mAP of $59.1\%$) or BMM-FV (mAP of $31.0\%/34.9\%$ respectively using the full length/PCA-reduced descriptor).
This reflects the very good behavior of feature combinations also in the large-scale case.}
\h{The mAP reaches a maximum using $\alpha$ between 0.4 and 0.5, that is giving (quite) the same weight to BMM-FV and CNN feature during the combination. The results obtained using the full length BMM-FV and the PCA-reduced version are similar. The latter performs slightly better and achieved a maximum of $\textbf{67.2\%}$ of mAP that correspond to $\textbf{13.7\%}$ of relative mAP improvement respect to use the CNN feature alone.
It is worth noting that the relative mAP improvement obtained in the large-scale setting is much greater than that obtained without the distraction set. This suggests that the information provided by the BMM-FV during the combination helps in discerning the visual content of images particularly in presence of distractor images.
Since the computational time of extracting binary features is much faster than others, the computational gain of combining CNN features with BMM-FV encodings of ORB over traditional FV encodings of SIFT is
especially notable in the large-scale scenario. For example, the process for extracting SIFTs from the INRIA Holidays+ MIRFlickr dataset ($1,001,491$ images) would have required more than 13 days (about 1,200 ms per image) while ORB extraction took less than 8 hours (about 26 ms per image).
}
\section{Conclusion}
\label{sec:conclusion}
Motivated by recent results obtained on one hand with the use of
aggregation methods applied to local descriptors\h{,} and on the
other with the definition of binary local features, this paper
has performed an extensive comparisons of techniques that mix the two approaches by using aggregation methods on binary
local features.
The use of aggregation methods on binary local features is
\h{motivated}
by the need for increasing efficiency and reducing computing resources for image matching on a large scale, at the
expense of some degradation in the accuracy of retrieval
algorithms. Combining the two approaches lead
to execute image retrieval on a very large scale and reduce the
cost for feature extraction and representation.
Thus we expect that the results of our empirical evaluation are useful for people working with binary local descriptors.
Moreover, we investigated how
\h{aggregations of binary local features work}
in conjunction with the CNN pipeline in order to improve the latter retrieval performance. We showed that the BMM-FV built upon ORB binary features can be profitable use to this scope, even if a relative small number of Bernoulli is used. In fact, the relative improvement in the retrieval performance obtained \h{combining CNN features with}
the BMM-FV is similar to that previously obtained in \cite{chandrasekhar2015} \h{where a combination of the CNN features with}
the more expensive FV built on SIFT \h{was proposed}.
\h{Experimental evaluation on large scale confirms the effectiveness and scalability of our proposal.}
It is also worth mentioning that the BMM-FV approach is
very general and could be applied to any binary feature.
Recent works based on CNNs suggest that binary features aggregation technique could be further
applied to deep features. In fact, on one hand, local
features based on CNNs, aggregated with VLAD and FV approaches,
have been proposed to obtain robustness to geometric deformations \cite{Ng_2015_CVPR_Workshops,Uricchio_2015_ICCV_Workshops}. On
the other hand, binarization of global CNN features have been also
proposed in \cite{Lin_2015_CVPR_Workshops,Lai_2015_CVPR}. Thus, as
a future work, we plan to test the BMM-FV approach over binary
deep local descriptors leveraging on the local and binary
approaches mentioned above.
\begin{acknowledgements}
This work was partially founded by: EAGLE, Europeana network of Ancient Greek and Latin Epigraphy, co-founded by the European Commission, CIP-ICT-PSP.2012.2.1 - Europeana and creativity, Grant Agreement n. 325122; and Smart News, Social sensing for breakingnews, co-founded by the Tuscany region under the FAR-FAS 2014 program, CUP CIPE D58C15000270008.
\end{acknowledgements}
\bibliographystyle{spmpsci}
|
1,116,691,499,282 | arxiv | \section{Introduction}
Lorentz violating theories of gravity have attracted much attention recently. This is mainly due to the fact that some quantum gravity theories, such as string theory and loop quantum gravity, predict that the spacetime structure at very high energies--typically at the Planck scale--may not be smooth and continuous, as assumed by relativity. This means that the rules of relativity do not apply and Lorentz symmetry must break down at or below the Planck distance. Indeed, in recent years, astrophysical observations on high-energy cosmic rays seem to have provided evidence supporting this idea (see, e.g., \cite{matt}).
The simplest way to study Lorentz violation in the context of gravity is to assume that there is a vector field with fixed norm coupling to gravity at each point of spacetime. In other words, the spacetime is locally endowed with a metric tensor and a dynamical vector field with constant norm. The vector field defined in this way is referred to as the \textquotedblleft aether" because it establishes a preferred direction at each point in spacetime and thereby explicitly breaks local Lorentz symmetry. The existence of such a vector field would affect the propagation of particles--such as electrons and photons--through spacetime, which manifests itself at very high energies and can be observed by studying the spectrum of high energy cosmic rays. For example, the interactions of these particles with the field would restrict the electron's maximum speed or cause polarized photons to rotate as they travel through space over long distances. Any observational evidence in these directions would be a direct indication of Lorentz violation, and therefore new physics, at or beyond the Planck scale.
So vector-tensor theories of gravity are of physical importance today because they may shed some light on the internal structure of quantum gravity theories. One such theory is Einstein-Aether theory \cite{jm,jac} in which the aether field is assumed to be timelike and therefore breaks the boost sector of the Lorentz symmetry. This theory has been investigated over the years from various respects \cite{ej1,ej2,gej,tm,bjs,bbm,gs,bs,dww,bsv,gur,gsen,cl,zfs1,bdfsz,zfs2,zzbfs,bl,ab}. There also appeared some related works \cite{rizzo,acw,ct,cpw} which discuss the possibility of a spacelike aether field breaking the rotational invariance of space. The internal structure and dynamics of such theories are still under examination; for example, the stability problem of the aether field has been considered in \cite{cdgt,dj}.\footnote{Breaking of Lorentz symmetry is discussed also in \cite{cg}.} Of course, to gain more understanding in these respects, one also needs explicit analytic solutions to the fairly complicated equations of motion that these theories possess.
In this paper, we propose yet another possibility, namely, the possibility of a \textit{null} aether field which dynamically couples to the metric tensor of spacetime. Such a vector field picks up a preferred null direction in spacetime and violates the local Lorentz invariance explicitly. From now on, we shall refer to the theory constructed in this way as Null Aether Theory (NAT). This construction enables us to naturally introduce an extra degree of freedom, i.e. the spin-0 part of the aether field, which is a scalar field that has a mass in general. By using this freedom, we show that it is possible to construct exact black hole solutions and nonlinear wave solutions in the theory.\footnote{In the context of Einstein-Aether theory, black hole solutions were considered in \cite{ej1,ej2,gej,tm,bjs,bbm,gs,bs,dww,bsv} and linearized plane wave solutions were studied in \cite{jm1}.} Indeed, assuming the null aether vector field ($v_\m$) is proportional to the one of the principal null direction(s) of the Weyl tensor ($l_\m$), i.e. $v_\m=\p(x)l_\m$, where $\p(x)$ is the spin-0 aether field, we first discuss the Newtonian limit of NAT and show that the theory exactly recovers general relativity with the meaning that the solar system observations put no constraints on the parameters of the theory in this approximation. We then proceed to construct exact spherically symmetric black hole solutions to the full nonlinear theory in four dimensions. Among these, there are Vaidya-type nonstationary solutions which do not need the existence of any extra matter field: the null aether field present in the very foundation of the theory behaves, in a sense, as a null matter to produce such solutions. For special values of the parameters of the theory, there are also stationary Schwarzschild-(A)dS type solutions that exist even when there is no explicit cosmological constant in the theory, Reissner-Nordstr\"{o}m-(A)dS type solutions with some \ql charge" sourced by the aether, and solutions of conformal gravity that contain a term growing linearly with radial distance and so associated with the flatness of the galaxy rotation curves. Our exact solutions perfectly match the solutions in the Newtonian limit when the aether field is on the order of the Newtonian potential.
On the other hand, the same construction, $v_\m=\p(x)l_\m$, also permits us to obtain exact solutions describing gravitational waves in NAT. In searching for such solutions, the Kerr-Schild-Kundt class of metrics \cite{ggst,gst1,ghst,gst2,gst3,gst4} was shown to be a valuable tool to start with: Indeed, recently, it has been proved that these metrics are universal in the sense that they constitute solutions to the field equations of \textit{any theory} constructed by the contractions of the curvature tensor and its covariant derivatives at any order \cite{gst4}. In starting this work, one of our motivations was to examine whether such universal metrics are solutions to vector-tensor theories of gravity as well. Later on, we perceived that this is only possible when the vector field in the theory is null and aligned with the propagation direction of the waves. Taking the metric in the Kerr-Schild-Kundt class with maximally symmetric backgrounds and assuming further $l^\m\pa_\m\p=0$, we show that the AdS-plane waves and $pp$-waves form a special class of exact solutions to NAT. The whole set of field equations of the theory are reduced to two coupled differential equations, in general, one for a scalar function related to the profile function of the wave and one for the \ql massive" spin-0 aether field $\p(x)$. When the background spacetime is AdS, it is possible to solve these coupled differential equations exactly in three dimensions and explicitly construct plane waves propagating in the AdS spacetime. Such constructions are possible also in dimensions higher than three but with the simplifying assumption that the profile function describing the AdS-plane wave does not depend on the transverse $D-3$ coordinates. The main conclusion of these computations is that the mass corresponding to the spin-0 aether field acquires an upper bound (the Breitenlohner-Freedman bound \cite{bf}) determined by the value of the cosmological constant of the background spacetime. In the case of $pp$-waves, where the background is flat, the scalar field equations decouple and form one Laplace equation for a scalar function related to the profile function of the wave and one massive Klein-Gordon equation for the spin-0 aether field in $(D-2)$-dimensional Euclidean flat space. Because of this decoupling, plane wave solutions, which are the subset of $pp$-waves, can always be constructed in NAT. Being exact, all these solutions might provide important insights into the internal dynamics of Lorentz violating vector-tensor theories of gravity.
The paper is structured as follows. In Sec. 2, we introduce NAT and present the field equations. In Sec. 3, we study the Newtonian limit of the theory to see the effect of the null vector field on the solar system observations. In Sec. 4, we construct exact spherically symmetric black hole solutions in their full generality in four dimensions. In Sec. 5, we study the nonlinear wave solutions of NAT propagating in nonflat backgrounds, which are assumed to be maximally symmetric, by taking the metric in the Kerr-Schild-Kundt class. In Sec. 6, we specifically consider AdS-plane waves describing plane waves moving in the AdS spacetime in $D\geq3$ dimensions. In Sec. 7, we briefly describe the $pp$-wave spacetimes and show that they provide exact solutions to NAT. We also discuss the availability of the subclass plane waves under certain conditions. Finally, in Sec. 8, we summarize our results.
We shall use the metric signature $(-,+,+,+,\ldots)$ throughout the paper.
\section{Null Aether Theory}
The theory we shall consider is defined in $D$ dimensions and described by, in the absence of matter fields, the action
\begin{equation}
I={1 \over 16 \pi G}\, \int d^{D}\,x\sqrt{-g}\,[R-2\Lambda-K^{\m\n}\,_{\a\b}\nabla_{\m}v^{\a}\nabla_{\n}v^{\b}
+\lambda(v_{\mu}v^{\mu}+\varepsilon)],\label{action}
\end{equation}
where
\begin{equation}
K^{\mu \nu}\,_{\alpha \beta}=c_{1}g^{\mu \nu}g_{\alpha
\beta}+c_{2}\delta^{\mu}_{\alpha}
\delta^{\nu}_{\beta}+c_{3}\delta^{\mu}_{\beta}
\delta^{\nu}_{\alpha}-c_{4}v^{\mu}v^{\nu}g_{\alpha
\beta}.\label{Ktensor}
\end{equation}
Here $\Lambda$ is the cosmological constant and $v^\mu$ is the so-called aether field which dynamically couples to the metric tensor $g_{\m\n}$ and has the fixed-norm constraint
\begin{equation}\label{con}
v_{\mu}v^{\mu}=-\varepsilon,~~(\varepsilon=0,\pm1)
\end{equation}
which is introduced into the theory by the Lagrange multiplier $\lambda$ in (\ref{action}). Accordingly, the aether field is a timelike (spacelike) vector field when $\varepsilon=+1$ ($\varepsilon=-1$), and it is a null vector field when $\varepsilon=0$.\footnote{The case with $\varepsilon=+1$ is associated with Einstein-Aether theory \cite{jm,jac}.} The constant coefficients $c_{1}, c_{2}, c_{3}$ and $c_{4}$ appearing in (\ref{Ktensor}) are the dimensionless parameters of the theory.\footnote{In Einstein-Aether theory, these parameters are constrained by some theoretical and observational arguments \cite{jm,jac,cl,ej3,ems,fj,jac2,zfz,ybby}.}
The equations of motion can be obtained by varying the action (\ref{action}) with respect to the independent variables: Variation with respect to $\lambda$ produces the constraint equation (\ref{con}) and variation with respect to $g^{\mu\nu}$ and $v^\mu$ produces the respective, dynamical field equations
\begin{eqnarray}
&&G_{\mu \nu}+\Lambda g_{\mu\nu}=\nabla_{\alpha}\l[ J^{\alpha}\,_{(\mu}
\,v_{\nu)}-J_{(\mu}\,^{\alpha}\, v_{\nu)}+J_{(\mu \nu )}\,
v^{\alpha}\r] \nonumber\\
&&~~~~~~~~~~~~~~~~~~+c_{1}\l(\nabla_{\mu}v_{\alpha}\nabla_{\nu}
v^{\alpha}-\nabla_{\alpha}v_{\mu}\nabla^{\alpha}
v_{\nu}\r) \nonumber \\
&&~~~~~~~~~~~~~~~~~~+c_{4}\dot{v}_{\mu}\dot{v}_{\nu}+\lambda v_{\mu}v_{\nu}-{1 \over 2} L g_{\mu \nu}, \label{eqn01}\\
&&\nonumber\\
&&c_{4} \dot{v}^{\alpha} \nabla_{\mu}
v_{\alpha}+\nabla_{\alpha}J^{\alpha}\,_{\mu}+\lambda
v_{\mu}=0,\label{eqn02}
\end{eqnarray}
where $\dot{v}^{\mu}\equiv v^{\alpha}\nabla_{\alpha}v^{\mu}$ and
\begin{eqnarray}
&&J^{\mu}\,_{\a}\equiv K^{\mu \n}\,_{\a \b}\nabla_{\n}v^{\beta},\label{J}\\
&&L\equiv J^{\mu}\,_{\a}\nabla_{\mu}v^{\a}.\label{L}
\end{eqnarray}
In writing (\ref{eqn01}), we made use of the constraint (\ref{con}). From now on, we will assume that the aether field $v^\m$ is null (i.e., $\varepsilon=0$) and refer to the above theory as Null Aether Theory, which we have dubbed NAT. This fact enables us to obtain $\la$ from the aether equation (\ref{eqn02}) by contracting it by the vector $u^\m=\d^\m_0$; that is,
\begin{equation}\label{}
\la=-\frac{1}{u^\n v_\n}\l[c_4u^\m\dot{v}^{\alpha} \nabla_{\mu}v_{\alpha}+u^\m\nabla_{\alpha}J^{\alpha}\,_{\mu}\r],
\end{equation}
since it is always the case that $u^\n v_\n\neq0$ for a null vector. It is obvious that flat Minkowski metric ($\eta_\mn$) and a constant null vector ($v_\m=const.$), together with $\la=0$, constitute a solution to NAT.
Null Aether Theory, to our knowledge, is introduced for the first time in this paper. There are some number of open problems to be attacked such as Newtonian limit, black holes, exact solutions, stability, etc. In this work, we investigate the Newtonian limit, the spherically symmetric black hole solutions (in $D=4$) and the AdS wave and $pp$-wave solutions of NAT. In all these cases, we assume that $v_\m=\p(x)l_\m$, where $l_\m$ is a null vector defining the principal null direction(s) of the Weyl tensor and $\p(x)$ is a scalar field defined as the spin-0 aether field that has a mass in general. The covariant derivative of the null vector $l_\m$ can always be decomposed in terms of the optical scalars: expansion, twist, and shear \cite{step}.
\section{Newtonian Limit of Null Aether Theory}
Now we shall examine the Newtonian limit of NAT to see whether there are any contributions to the Poisson equation coming from the null aether field. For this purpose, as usual, we shall assume that the gravitational field is weak and static and produced by a nonrelativistic matter field. Also, we know that the cosmological constant--playing a significant role in cosmology--is totally negligible in this context.
Let us take the metric in the Newtonian limit as
\begin{equation}\label{}
ds^2=-[1+2\Phi(\vec{x})]dt^2+[1-2\Psi(\vec{x})][(dx^1)^2+(dx^2)^2+\ldots+(dx^{D-1})^2],
\end{equation}
where $x^\m=(t,\vec{x})$ with $\vec{x}=(x^1,x^2,\ldots,x^{D-1})$. In this spacetime, a null vector at linear order can be defined, up to a multiplicative function of $\vec{x}$, as
\begin{equation}\label{}
l_\m=\d_\m^0+(1-\Phi-\Psi)\frac{x^i}{r}\d_\m^i,
\end{equation}
where $r^2=\vec{x}\cdot\vec{x}=\d_{ij}x^ix^j=x^ix^i$ with $i=1,\ldots,D-1$. Now we write the null aether field as $v_\m=\p(\vec{x})l_\m$ (since we are studying with a null vector, we always have this freedom) and assume that $\p(\vec{x})$ is some arbitrary function at the same order as $\Phi$ and/or $\Psi$. Then the zeroth component of the aether equation (\ref{eqn02}) gives, at the linear order,
\begin{equation}\label{NewA0}
c_1\nabla^2\p+\la\p=0,
\end{equation}
where $\nabla^2\equiv\pa_i\pa_i$, and the $i$th component gives, at the linear order,
\begin{eqnarray}
&&(c_2+c_3)r^2x^j\pa_j\pa_i\p-(2c_1+c_2+c_3)x^ix^j\pa_j\p\nn\\
&&~~~~~~~~+[2c_1+(D-1)(c_2+c_3)]r^2\pa_i\p-(D-2)(c_1+c_2+c_3)x^i\p=0,\label{Newcon}
\end{eqnarray}
after eliminating $\la$ using (\ref{NewA0}). It can also be shown that at the linear order the aether contribution to the equation (\ref{eqn01}) is zero and so the only contribution comes from the nonrelativistic matter for which
\begin{equation}\label{}
T_{\mn}^{matter}=\rho_m(\vec{x})\d_\m^0\d_\n^0.
\end{equation}
Here we are assuming that the matter fields do not couple to the aether field at the linear order. Therefore, the only nonzero components of (\ref{eqn01}) are the 00 and the $ij$ component (the $0i$ component is satisfied identically). Taking the trace of the $ij$ component produces
\begin{equation}\label{}
\nabla^2(\Phi-\Psi)=0,
\end{equation}
which enforces
\begin{equation}\label{}
\Phi=\Psi,
\end{equation}
for the spacetime to be asymptotically flat. Using this fact, we can write, from the 00 component of (\ref{eqn01}),
\begin{equation}\label{Pois}
\nabla^2\Phi=4\pi G\rho_m.
\end{equation}
Thus we see that the Poisson equation is unaffected by the null aether field at the linear order in $G$.
Outside of a spherically symmetric mass distribution, the Poisson equation (\ref{Pois}) reduces to the Laplace equation which gives
\begin{equation}\label{}
\Phi(r)=-\frac{GM}{r^{D-3}},
\end{equation}
for dimensions $D>3$. Here we have absorbed all the constants into $G$. In $D=3$, since the force goes with inverse $r$, the Newtonian potential should go with the logarithm of $r$. On the other hand, for spherical symmetry, the condition (\ref{Newcon}) can be solved and yields
\begin{equation}\label{sigma}
\p(r)=a_1r^{\a_1}+a_2r^{\a_2},
\end{equation}
where $a_1$ and $a_2$ are arbitrary constants and
\begin{equation}\label{}
\a_{1,2}=-\frac{1}{2}\l[(D-3)\pm\sqrt{(D-1)^2+4(D-2)\frac{c_1}{c_2+c_3}}\,\r].
\end{equation}
This solutions immediately puts the following condition on the parameters of the theory
\begin{equation}\label{}
\frac{c_1}{c_2+c_3}\geq-\frac{(D-1)^2}{4(D-2)},
\end{equation}
since we always have $D>2$. Specifically, when $c_1=-(D-1)^2(c_2+c_3)/4(D-2)$, we have
\begin{equation}\label{}
\p(r)=\frac{a_1+a_2}{r^{(D-3)/2}};
\end{equation}
when $c_1=0$, we have
\begin{equation}\label{}
\p(r)=\frac{a_1}{r^{D-2}}+a_2r;
\end{equation}
or when $c_1=-(c_2+c_3)$, we have
\begin{equation}\label{}
\p(r)=\frac{a_1}{r^{D-3}}+a_2.
\end{equation}
To sum up, in $D=4$, we have exactly the same Newtonian approximation as that of general relativity. Hence our NAT is in agreement with the solar system observations and this fact puts no constraints on the parameters \{$c_1,c_2,c_3,c_4$\} of the theory.
\section{Black Hole Solutions in Null Aether Theory}
In this section, we shall construct spherically symmetric black hole solutions to NAT in $D=4$. Let us start with the generic spherically symmetric metric in the following form with $x^{\mu}=(u,r,\theta, \vartheta)$:
\begin{equation}\label{BHKS}
ds^2=-\l(1-\frac{\La}{3}r^2\r)du^2+2dudr+r^2d\theta^2+r^2\sin^2\theta d\vartheta^2+2 f(u,r)du^2,
\end{equation}
where $\La$ is the cosmological constant. For $f(u,r)=0$, this becomes the metric of the usual (A)dS spacetime. Since the aether field is null, we take it to be $v_\m=\p(u,r)l_\m$ with $l_\m=\delta^{u}_\m$ being the null vector of the geometry.
With the metric ansatz (\ref{BHKS}), from the $u$ component of the aether equation (\ref{eqn02}), we obtain
\begin{eqnarray}
&&\la=-\frac{1}{3r^2\p}\bl\{3(c_1+c_3)\l[\La r^2+(r^2f')'\r]\p+c_1\l[(3-\La r^2-6f)(r^2\p')'+6r(r\dot{\p})'\r]\nn\\
&&~~~~~~+3(c_2+c_3)(r^2\dot{\p})'-3c_4\l[2r^2\p'^2+\p(r^2\p')'\r]\p\br\},\label{AEu}
\end{eqnarray}
and from the $r$ component, we have
\begin{equation}\label{AEr}
(c_2+c_3)(r^2\p''+2r\p')-2(c_1+c_2+c_3)\p=0,
\end{equation}
where the prime denotes differentiation with respect to $r$ and the dot denotes differentiation with respect to $u$. The equation (\ref{AEr}) can easily be solved and the generic solution is
\begin{equation}\label{}
\p(u,r)=a_1(u)r^{\a_1}+a_2(u)r^{\a_2},
\end{equation}
for some arbitrary functions $a_1(u)$ and $a_2(u)$, where
\begin{equation}\label{alpha}
\a_{1,2}=-\frac{1}{2}\l[1\pm\sqrt{9+8\frac{c_1}{c_2+c_3}}\,\r].
\end{equation}
Note that when $c_1=-9(c_2+c_3)/8$, the square root vanishes and the roots coincide to give $\a_1=\a_2=-1/2$. Inserting this solution into the Einstein equations (\ref{eqn01}) yields, for the $ur$ component,
\begin{equation}\label{ur}
(1+2\a_1)a_1(u)^2b_1r^{2\a_1}+(1+2\a_2)a_2(u)^2b_2r^{2\a_2}-(rf)'=0,
\end{equation}
with the identifications
\begin{equation}
b_1\equiv-\frac{1}{4}[2c_2+(c_2+c_3)\a_1],~~b_2\equiv-\frac{1}{4}[2c_2+(c_2+c_3)\a_2].
\end{equation}
Thus we obtain
\begin{equation}\label{}
f(u,r)=\left\{\begin{array}{ll}
\displaystyle a_1(u)^2b_1r^{2\a_1}+a_2(u)^2b_2r^{2\a_2}+\frac{\tilde{\m}(u)}{r},
&\mbox{for $\displaystyle \a_1\neq-\frac{1}{2}~\&~\displaystyle \a_2\neq-\frac{1}{2}$,}\\
&\\
\displaystyle \frac{\m(u)}{r},
&\mbox{for $\displaystyle \a_1=\a_2=-\frac{1}{2}$,}\label{fm}
\end{array} \right.
\end{equation}
where $\tilde{\m}(u)$ and $\m(u)$ are arbitrary functions. Notice that the last case occurs only when $c_1=-9(c_2+c_3)/8$. If we plug (\ref{fm}) into the other components, we identically satisfy all the equations except for the $uu$ component which, together with $\la$ from (\ref{AEu}), produces
\begin{equation}\label{uu}
[2c_2+(c_2+c_3)\a_1]\dot{a}_1a_2+[2c_2+(c_2+c_3)\a_2]a_1\dot{a}_2+2\dot{\tilde{\m}}=0,
\end{equation}
for $\a_1\neq-\frac{1}{2}$ and $\a_2\neq-\frac{1}{2}$, and
\begin{equation}\label{uu1o2}
(3c_2-c_3)\dot{\overline{(a_1+a_2)^2}}+8\dot{\m}=0,
\end{equation}
for $\a_1=\a_2=-\frac{1}{2}$. The last case immediately leads to
\begin{equation}\label{}
\m(u)=\frac{1}{8}(c_3-3c_2)(a_1+a_2)^2+m,
\end{equation}
where $m$ is the integration constant. Thus we see that Vaidya-type solutions can be obtained in NAT without introducing any extra matter fields, which is unlike the case in general relativity. Observe also that when $f(u,r)=0$, we should obtain the (A)dS metric as a solution to NAT [see (\ref{BHKS})]. Then it is obvious from (\ref{ur}) that this is the case only if $\a_1=\a_2=-\frac{1}{2}$ corresponding to
\begin{equation}\label{}
\p(u,r)=\left\{\begin{array}{ll}
\displaystyle \frac{d}{\sqrt{r}},&\mbox{for $\displaystyle c_1=-\frac{9}{8}(c_2+c_3)$,}\\
&\\
\displaystyle \frac{a(u)}{\sqrt{r}},
&\mbox{for $\displaystyle c_1=-\frac{9}{8}(c_2+c_3)$ \& $c_3=3c_2$,}\label{AdS}
\end{array} \right.
\end{equation}
where $d$ is an arbitrary constant and $a(u)$ is an arbitrary function.
Defining a new time coordinate $t$ by the transformation
\begin{equation}\label{utTrans}
du=g(t,r)dt+\frac{dr}{1-\frac{\La}{3}r^2-2f(t,r)},
\end{equation}
one can bring the metric (\ref{BHKS}) into the Schwarzschild coordinates
\begin{equation}\label{BHSch}
ds^2=-\l(1-\frac{\La}{3}r^2-2f\r)g^2dt^2+\frac{dr}{\l(1-\frac{\La}{3}r^2-2f\r)}+r^2d\theta^2+r^2\sin^2\theta d\vartheta^2,
\end{equation}
where the function $g(t,r)$ should satisfy
\begin{equation}\label{gf}
\frac{\pa g}{\pa r}=2\l(1-\frac{\La}{3}r^2-2f\r)^{-2}\frac{\pa f}{\pa t}.
\end{equation}
When $a_1(u)$ and $a_2(u)$ are constants, since $f=f(r)$ then, the condition (\ref{gf}) says that $g=g(t)$ and so it can be absorbed into the time coordinate $t$, meaning that $g(t,r)$ can be set equal to unity in (\ref{utTrans}) and (\ref{BHSch}). In this case, the solution (\ref{BHSch}) will describe a spherically symmetric stationary black hole spacetime. The horizons of this solution should then be determined by solving the equation
\begin{equation}\label{}
0=h(r)\equiv1-\frac{\La}{3}r^2-2f
=\left\{\begin{array}{ll}
\displaystyle 1-\frac{\La}{3}r^2-\frac{2}{r}\l(a_1^2b_1r^{-q}+a_2^2b_2r^{q}\r)-\frac{2\tilde{m}}{r}
&\mbox{(for $q\neq0$),}\\
&\\
\displaystyle 1-\frac{\La}{3}r^2-\frac{2m}{r}
&\mbox{(for $q=0$),}\label{hm}
\end{array} \right.
\end{equation}
where $\tilde{m}=const.$, $m=const.$, and
\begin{equation}\label{}
q\equiv\sqrt{9+8\frac{c_1}{c_2+c_3}},~~b_1=\frac{1}{8}[c_3-3c_2+(c_2+c_3)q],~~b_2=\frac{1}{8}[c_3-3c_2-(c_2+c_3)q].
\end{equation}
The last case ($q=0$) is the usual Schwarzschild-(A)dS spacetime. At this point, it is important to note that when $a_1$ and $a_2$ are in the order of the Newton's constant $G$, i.e. $a_1\sim G$ and $a_2\sim G$, since $h(r)$ depends on the squares of $a_1$ and $a_2$, we recover the Newtonian limit discussed in Sec. 3 for $\La=0$, $\tilde{m}=GM$ and $D=4$. For special values of the parameters of the theory, the first case ($q\neq0$) of (\ref{hm}) becomes a polynomial of $r$; for example,
\begin{itemize}
\item When $c_1=0$ ($q=3$), $h(r)\equiv1-A/r^4-Br^2-2\tilde{m}/r$: This is a Schwarzschild-(A)dS type solution if $A=0$. Solutions involving terms like $A/r^4$ can be found in, e.g., \cite{bbm,gser}.
\item When $c_1=-(c_2+c_3)$ ($q=1$), $h(r)\equiv1-A-\La r^2/3-B/r^2-2\tilde{m}/r$: This is a Reissner-Nordstr\"{o}m-(A)dS type solution if $A=0$.
\item When $c_1=-5(c_2+c_3)/8$ ($q=2$), $h(r)\equiv1-\La r^2/3-A/r^3-Br-2\tilde{m}/r$: This solution with $A=0$ has been obtained by Mannheim and Kazanas \cite{mk} in conformal gravity who also argue that the linear term $Br$ can explain the flatness of the galaxy rotation curves.
\end{itemize}
Here $A$ and $B$ are the appropriate combinations of the constants appearing in (\ref{hm}). For such cases, the equation $h(r)=0$ may have at least one real root corresponding to the event horizon of the black hole. For generic values of the parameters, however, the existence of the real roots of $h(r)=0$ depends on the signs and values of the constants $\La$, $b_1$, $b_2$, and $\tilde{m}$ in (\ref{hm}). When $q$ is an integer, the roots can be found by solving the polynomial equation $h(r)=0$, as in the examples given above. When $q$ is not an integer, finding the roots of $h(r)$ is not so easy, but when the signs of $\lim_{r\rightarrow0^+}h(r)$ and $\lim_{r\rightarrow\infty}h(r)$ are opposite, we can say that there must be at least one real root of this function. Since the signs of these limits depends on the signs of the constants $\La$, $b_1$, $b_2$, and $\tilde{m}$, we have the following cases in which $h(r)$ has at least one real root:
\begin{itemize}
\item If $0<q<3$, $\La<0$, $b_1>0$~~ $\Rightarrow$~~$\displaystyle\lim_{r\rightarrow0^+}h(r)<0~~~\&~~\displaystyle\lim_{r\rightarrow\infty}h(r)>0$;
\item If $0<q<3$, $\La>0$, $b_1<0$~~ $\Rightarrow$~~$\displaystyle\lim_{r\rightarrow0^+}h(r)>0~~~\&~~\displaystyle\lim_{r\rightarrow\infty}h(r)<0$;
\item If $q>3$, $b_1>0$, $b_2<0$~~ $\Rightarrow$~~$\displaystyle\lim_{r\rightarrow0^+}h(r)<0~~~\&~~\displaystyle\lim_{r\rightarrow\infty}h(r)>0$;
\item If $q>3$, $b_1<0$, $b_2>0$~~ $\Rightarrow$~~$\displaystyle\lim_{r\rightarrow0^+}h(r)>0~~~\&~~\displaystyle\lim_{r\rightarrow\infty}h(r)<0$.
\end{itemize}
Of course, these are not the only possibilities, but we give these examples to show the existence of black hole solutions of NAT in the general case.
\section{Plane Wave Solutions in Null Aether Theory: Kerr-Schild-Kundt Class of Metrics}
Now we shall construct exact plane wave solutions to NAT by studying in generic $D\geq3$ dimensions. For this purpose, we first write the aether field in the form $v^\m=\p(x)l^\m$, where $\p(x)$ is a scalar function representing the spin-0 aether field and $l^\m$ is a null vector, and assume the relations
\begin{eqnarray}
&&l_\m l^\m=0,~~\nabla_\m l_\n=\frac{1}{2}(l_\m \xi_\n+l_\n \xi_\m),~~l_\m \xi^\m=0,\label{lxi}\\
&&l^\m\pa_\m\p=0,\label{lphi}
\end{eqnarray}
where $\xi^\m$ is an arbitrary vector field for the time being. It should be noted that $l^\m$ is not a Killing vector. From these relations it follows that
\begin{equation}\label{}
l^\mu\nabla_\m l_\n=0,~~l^\mu\nabla_\n l_\m=0,~~\nabla_\m l^\m=0,~~\dot{v}_{\m}=0,
\end{equation}
and (\ref{J}) and (\ref{L}) are worked out to be
\begin{equation}\label{JKSK}
J^{\m}~_{\n}=c_1l_\n\nabla^\m\p+c_3l^\m\nabla_\n\p+(c_1+c_3)\p\nabla^\m l_\n,~~L=0.
\end{equation}
Then one can compute the field equations (\ref{eqn01}) and (\ref{eqn02}) as
\begin{eqnarray}
&&G_{\mu \nu}+\Lambda g_{\mu\nu}=\bl[-c_3\nabla_\a\p\nabla^\a\p+(c_1-c_3)\p\Box\p-2c_3\p\xi^\a\pa_\a\p\nn\\
&&~~~~~~~~~~~~~~+\l(\la-\frac{c_1+c_3}{4}\,\xi_\a\xi^\a\r)\p^2\br]l_\m l_\n-(c_1+c_3)\p^2R_{\m\a\n\b}l^\a l^\b,\label{Einfull}\\
&&\nonumber\\
&&[c_1(\Box\p+\xi^\a\pa_\a\p)+\la\p]l_\m+(c_1+c_3)\p R_{\m\n}l^\n=0,\label{Aetherfull}
\end{eqnarray}
where $\Box\equiv\nabla_\m\nabla^\m$ and use has been made of the identity $[\nabla_\m,\nabla_\n]l_\a=R_{\m\n\a\b}l^\b$.
Now suppose that the spacetime metric has the generalized Kerr-Schild form \cite{ks}
\begin{equation}\label{KS}
g_{\m\n}=\bar{g}_{\m\n}+2Vl_\m l_\n,
\end{equation}
where $\bar{g}_{\m\n}$ is the background metric assumed to be maximally symmetric; i.e. its curvature tensor has the form
\begin{equation}\label{}
\bar{R}_{\m\a\n\b}=K(\bar{g}_{\m\n}\bar{g}_{\a\b}-\bar{g}_{\m\b}\bar{g}_{\n\a})
\end{equation}
with
\begin{equation}\label{}
K=\frac{\bar{R}}{D(D-1)}=const.
\end{equation}
It is therefore either Minkowski, de Sitter (dS), or anti-de Sitter (AdS) spacetime, depending on whether $K=0$, $K>0$, or $K<0$.
The vector $l^\m$ in (\ref{KS}) satisfies (\ref{lxi}) and (\ref{lphi}) and additionally
\begin{equation}\label{lV}
l^\m\pa_\m V=0.
\end{equation}
Kerr-Schild metrics with $l_\m$ satisfying the relations (\ref{lxi}) and (\ref{lV}) are called Kerr-Schild-Kundt metrics \cite{ggst,gst1,ghst,gst2,gst3,gst4}. All the properties (\ref{lxi}), (\ref{lphi}), and (\ref{lV}), together with the inverse metric
\begin{equation}\label{KSinv}
g^{\m\n}=\bar{g}^{\m\n}-2Vl^\m l^\n,
\end{equation}
imply that (see, e.g., \cite{gst1})
\begin{eqnarray}
&&\G^\m_{\m\n}=\bar{\G}^\m_{\m\n},~~l_\m\G^\m_{\a\b}=l_\m\bar{\G}^\m_{\a\b},~~l^\a\G^\m_{\a\b}=l^\a\bar{\G}^\m_{\a\b},\label{Gamma}\\
&&\bar{g}^{\a\b}\G^\m_{\a\b}=\bar{g}^{\a\b}\bar{\G}^\m_{\a\b},\\
&&R_{\m\a\n\b}l^\a l^\b=\bar{R}_{\m\a\n\b}l^\a l^\b=-Kl_\m l_\n,\\
&&R_{\m\n}l^\n=\bar{R}_{\m\n}l^\n=(D-1)Kl_\m,\\
&&R=\bar{R}=D(D-1)K,
\end{eqnarray}
and the Einstein tensor is calculated as
\begin{equation}\label{}
G_{\m\n}=-\frac{(D-1)(D-2)}{2}K\bar{g}_{\m\n}-\rho l_\m l_\n,\\
\end{equation}
with
\begin{equation}\label{}
\rho\equiv\bar{\Box}V+2\xi^\a\pa_\a V+\l[\frac{1}{2}\xi_\a\xi^\a+(D+1)(D-2)K\r]V,
\end{equation}
where $\bar{\Box}\equiv\bar{\nabla}_\m\bar{\nabla}^\m$ and $\bar{\nabla}_\m$ is the covariant derivative with respect to the background metric $\bar{g}_{\m\n}$.
Thus, for the metric ansatz (\ref{KS}), the field equations (\ref{Einfull}) and (\ref{Aetherfull}) become
\begin{eqnarray}
&&\l[-\frac{(D-1)(D-2)}{2}K+\La\r]\bar{g}_{\m\n}-(\rho-2\La V)l_\m l_\n\nn\\
&&~~~~~~~~~~~~~~=\bl\{-c_3\bar{\nabla}_\a\p\bar{\nabla}^\a\p+(c_1-c_3)\p\bar{\Box}\p-2c_3\p\xi^\a\pa_\a\p\nn\\
&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~+\l[\la+(c_1+c_3)\l(K-\frac{1}{4}\,\xi_\a\xi^\a\r)\r]\p^2\br\}l_\m l_\n,\label{EinKS}\\
&&\nonumber\\
&&\{c_1(\bar{\Box}\p+\xi^\a\pa_\a\p)+\l[\la+(c_1+c_3)(D-1)K\r]\p\}l_\m=0.\label{AetherKS}
\end{eqnarray}
From these, we deduce that
\begin{eqnarray}
&&\La=\frac{(D-1)(D-2)}{2}K,\\
&&\nn\\
&&\bar{\Box}V+2\xi^\a\pa_\a V+\l[\frac{1}{2}\xi_\a\xi^\a+2(D-2)K\r]V\nn\\
&&~~~~~~~~=c_3\l[\bar{\nabla}_\a\p\bar{\nabla}^\a\p-\frac{\la}{c_1}\p^2\r]+(c_1+c_3)\p\xi^\a\pa_\a\p\nn\\
&&~~~~~~~~~~~~~~~+\frac{c_1+c_3}{c_1}\l\{\l[c_1(D-2)-c_3(D-1)\r]K+\frac{c_1}{4}\,\xi_\a\xi^\a\r\}\p^2,\label{EinKS1}\\
&&\nonumber\\
&&c_1(\bar{\Box}\p+\xi^\a\pa_\a\p)+\l[\la+(c_1+c_3)(D-1)K\r]\p=0,\label{AetherKS1}
\end{eqnarray}
where we eliminated the $\p\bar{\Box}\p$ term that appears in (\ref{EinKS}) by using the aether equation (\ref{AetherKS1}) and assuming $c_1\neq0$.
Now let us make the ansatz
\begin{equation}\label{Vcurved}
V(x)=V_0(x)+\a\p(x)^2,
\end{equation}
for some arbitrary constant $\a$. With this, we can write (\ref{EinKS1}) as
\begin{eqnarray}
&&\bar{\Box}V_0+2\xi^\a\pa_\a V_0+\l[\frac{1}{2}\xi_\a\xi^\a+2(D-2)K\r]V_0\nn\\
&&~~~~~=(c_3-2\a)\l\{\bar{\nabla}_\a\p\bar{\nabla}^\a\p-\frac{1}{c_1}\l[\la+(c_1+c_3)(D-1)K\r]\p^2\r\}\nn\\
&&~~~~~+(c_1+c_3-2\a)\l\{\p\xi^\a\pa_\a\p+\l[(D-2)K+\frac{1}{4}\,\xi_\a\xi^\a\r]\p^2\r\}.\label{EinKS2}
\end{eqnarray}
Here there are two possible choices for $\a$. The first one is $\a=c_3/2$, as in the previous section, for which (\ref{EinKS2}) becomes
\begin{eqnarray}
&&\bar{\Box}V_0+2\xi^\a\pa_\a V_0+\l[\frac{1}{2}\xi_\a\xi^\a+2(D-2)K\r]V_0\nn\\
&&~~~~~~~~~~~~~=c_1\l\{\p\xi^\a\pa_\a\p+\l[(D-2)K+\frac{1}{4}\,\xi_\a\xi^\a\r]\p^2\r\},\label{EinKS3}
\end{eqnarray}
and reduces to
\begin{equation}\label{}
\bar{\Box}V_0=0
\end{equation}
when $K=0$ and $\xi^\m=0$, which is the $pp$-wave case to be discussed in Sec. \ref{pp:flat}. The other choice, $\a=(c_1+c_3)/2$, drops the second term in (\ref{EinKS2}) and produces
\begin{eqnarray}
&&\bar{\Box}V_0+2\xi^\a\pa_\a V_0+\l[\frac{1}{2}\xi_\a\xi^\a+2(D-2)K\r]V_0\nn\\
&&~~~~~~~~~~~~~=-c_1\bar{\nabla}_\a\p\bar{\nabla}^\a\p+\l[\la+(c_1+c_3)(D-1)K\r]\p^2.\label{EinKS4}
\end{eqnarray}
Here it should be stressed that this last case is present only when the background metric is nonflat (i.e. $K\neq0$) and/or $\xi^\m\neq0$.
On the other hand, the aether equation (\ref{AetherKS1}) can be written as
\begin{equation}\label{KGcurved}
(\bar{\Box}+\xi^\a\pa_\a)\p-m^2\p=0,
\end{equation}
where, assuming $\la$ is constant, we defined
\begin{equation}\label{mnonflat}
m^2\equiv-\frac{1}{c_1}\l[\la+(c_1+c_3)(D-1)K\r]
\end{equation}
since $c_1\neq0$. The equation (\ref{KGcurved}) can be considered as the equation of the spin-0 aether field $\p$ with $m$ being the \ql mass" of the field. The definition (\ref{mnonflat}) requires that
\begin{equation}\label{}
\frac{1}{c_1}\l[\la+(c_1+c_3)(D-1)K\r]\leq0,
\end{equation}
the same constraint as in (\ref{c3}) when $K=0$. Obviously, the field $\p$ becomes \ql massless" if
\begin{equation}\label{}
\la=-(c_1+c_3)(D-1)K.
\end{equation}
Thus we have shown that, for any solution $\p$ of the equation (\ref{KGcurved}), there corresponds a solution $V_0$ of the equation (\ref{EinKS3}) for $\a=c_3/2$ or of the equation (\ref{EinKS4}) for $\a=(c_1+c_3)/2$, and we can construct an exact wave solution with nonflat background given by (\ref{KS}) with the profile function (\ref{Vcurved}) in NAT.
\section{AdS-Plane Waves in Null Aether Theory}
In this section, we shall specifically consider AdS-plane waves for which the background metric $\bar{g}_{\m\n}$ is the usual $D$-dimensional AdS spacetime with the curvature constant
\begin{equation}\label{curv}
K\equiv-\frac{1}{\ell^2}=-\frac{2|\La|}{(D-1)(D-2)},
\end{equation}
where $\ell$ is the radius of curvature of the spacetime. We shall represent the spacetime by the conformally flat coordinates for simplicity; i.e. $x^\m=(u,v,x^i,z)$ with $i=1,\ldots,D-3$ and
\begin{equation}\label{back}
d\bar{s}^2=\bar{g}_{\m\n}dx^\m dx^\n=\frac{\ell^2}{z^2}(2dudv+dx_idx^i+dz^2),
\end{equation}
where $u$ and $v$ are the double null coordinates. In these coordinates, the boundary of the AdS spacetime lies at $z=0$.
Now if we take the null vector in the full spacetime of the Kerr-Schild form (\ref{KS}) as $l_\m=\d^u_\m$, then using (\ref{KSinv}) along with $l_\m l^\m=0$,
\begin{equation}\label{}
l^\m=g^{\m\n}l_\n=\bar{g}^{\m\n}l_\n=\frac{z^2}{\ell^2}\d^\m_v~~\Rightarrow~~l^\a\pa_\a V=\frac{z^2}{\ell^2}\frac{\pa V}{\pa v}=0~~\&~~
l^\a\pa_\a \p=\frac{z^2}{\ell^2}\frac{\pa \p}{\pa v}=0,
\end{equation}
so the functions $V$ and $\p$ are independent of the coordinate $v$; that is, $V=V(u,x^i,z)$ and $\p=\p(u,x^i,z)$. Therefore the full spacetime metric defined by (\ref{KS}) will be
\begin{equation}\label{AdSwave}
ds^2=[\bar{g}_{\m\n}+2V(u,x^i,z)l_\m l_\n]dx^\m dx^\n=d\bar{s}^2+2V(u,x^i,z)du^2,
\end{equation}
with the background metric (\ref{back}). It is now straightforward to show that (see also \cite{gst1})
\begin{equation}\label{lxiAdS}
\nabla_\m l_\n=\bar{\nabla}_\m l_\n=\frac{1}{z}(l_\m\d^z_\n+l_\n\d^z_\m),
\end{equation}
where we used the second property in (\ref{Gamma}) to convert the full covariant derivative $\nabla_\m$ to the background one $\bar{\nabla}_\m$, and $l_\m=\d^u_\m$ with $\pa_\m l_\n=0$. Comparing (\ref{lxiAdS}) with the defining relation in (\ref{lxi}), we see that
\begin{equation}
\left.\begin{array}{l}
\displaystyle\xi_\m=\frac{2}{z}\d^z_\m,\\%[2ex]
\displaystyle\xi^\m=g^{\m\n}\xi_\n=\bar{g}^{\m\n}\xi_\n=\frac{2z}{\ell^2}\d^\m_z,
\end{array} \right\}~~\Rightarrow~~\xi_\m\xi^\m=\frac{4}{\ell^2},
\end{equation}
where we again used (\ref{KSinv}) together with $l_\m \xi^\m=0$.
Thus, for the AdS-plane wave ansatz (\ref{AdSwave}) with the profile function
\begin{equation}\label{}
V(u,x^i,z)=V_0(u,x^i,z)+\a\,\p(u,x^i,z)^2
\end{equation}
to be an exact solution of NAT, the equations that must be solved are the aether equation (\ref{KGcurved}), which takes the form
\begin{equation}\label{KGAdS}
z^2\hat{\pa}^2\p+(4-D)z\,\pa_z\p-m^2\ell^2\p=0,
\end{equation}
where $\hat{\pa}^2\equiv\pa_i\pa^i+\pa_z^2$ and
\begin{equation}\label{mAdS}
m^2\equiv-\frac{1}{c_1}\l[\la-(c_1+c_3)\frac{D-1}{\ell^2}\r],
\end{equation}
and the equation (\ref{EinKS3}) for $\a=c_3/2$, which becomes
\begin{equation}
z^2\hat{\pa}^2V_0+(6-D)z\,\pa_zV_0+2(3-D)V_0=c_1[2z\p\pa_z\p+(3-D)\p^2],\label{EinAdS1}
\end{equation}
or the equation (\ref{EinKS4}) for $\a=(c_1+c_3)/2$, which becomes
\begin{equation}
z^2\hat{\pa}^2V_0+(6-D)z\,\pa_zV_0+2(3-D)V_0
=-c_1[z^2(\hat{\pa}\p)^2+m^2\ell^2\p^2],\label{EinAdS2}
\end{equation}
where $(\hat{\pa}\p)^2\equiv\pa_i\p\pa^i\p+(\pa_z\p)^2$.
\subsection{AdS-Plane Waves in Three Dimensions}
It is remarkable that the equations (\ref{KGAdS}), (\ref{EinAdS1}), and (\ref{EinAdS2}) can be solved exactly in $D=3$. In that case $x^\m=(u,v,z)$, and so, $V_0=V_0(u,z)$ and $\p=\p(u,z)$. Then (\ref{KGAdS}) becomes
\begin{equation}\label{KGAdS3D}
z^2\pa_z^2\p+z\pa_z\p-m^2\ell^2\p=0,
\end{equation}
with
\begin{equation}\label{m3D}
m^2\equiv-\frac{1}{c_1}\l[\la-\frac{2(c_1+c_3)}{\ell^2}\r],
\end{equation}
and has the general solution, when $m\neq0$,
\begin{equation}\label{p3D}
\p(u,z)=a_1(u)z^{m\ell}+a_2(u)z^{-m\ell},
\end{equation}
where $a_1(u)$ and $a_2(u)$ are arbitrary functions. With this solution, (\ref{EinAdS1}) and (\ref{EinAdS2}) can be written compactly as
\begin{equation}
z^2\pa_z^2V_0+3z\pa_zV_0=E_1(u)z^{2m\ell}+E_2(u)z^{-2m\ell},\label{EinAdS3D}
\end{equation}
where
\begin{eqnarray}
&&\left.\begin{array}{l}
\displaystyle E_1(u)\equiv2c_1m\ell\,a_1(u)^2,\\
\displaystyle E_2(u)\equiv-2c_1m\ell\,a_2(u)^2,
\end{array} \right\}~\mbox{for $\displaystyle\a=\frac{c_3}{2}$,}\\
&&\nn\\
&&\left.\begin{array}{l}
\displaystyle E_1(u)\equiv-2c_1m^2\ell^2\,a_1(u)^2,\\
\displaystyle E_2(u)\equiv-2c_1m^2\ell^2\,a_2(u)^2,
\end{array} \right\}~\mbox{for $\displaystyle\a=\frac{c_1+c_3}{2}$.}
\end{eqnarray}
The general solution of (\ref{EinAdS3D}) is
\begin{equation}
V_0(u,z)=b_1(u)+b_2(u)z^{-2}+\frac{1}{4m\ell}\bl[\frac{E_1(u)}{m\ell+1}\,z^{2m\ell}+\frac{E_2(u)}{m\ell-1}\,z^{-2m\ell}\br],\label{sol3D1}
\end{equation}
with the arbitrary functions $b_1(u)$ and $b_2(u)$. Note that the second term $b_2(u)z^{-2}$ can always be absorbed into the AdS part of the metric (\ref{AdSwave}) by a redefinition of the null coordinate $v$, which means that one can always set $b_2(u)=0$ here and in the following solutions without loosing any generality. In obtaining (\ref{sol3D1}), we assumed that $m\ell\pm1\neq0$. If, on the other hand, $m\ell+1=0$, then the above solution becomes
\begin{equation}
V_0(u,z)=b_1(u)+b_2(u)z^{-2}-\frac{E_1(u)}{2}\,z^{-2}\ln z+\frac{E_2(u)}{8}\,z^{2},\label{sol3D2}
\end{equation}
and if $m\ell-1=0$, it becomes
\begin{eqnarray}
V_0(u,z)=b_1(u)+b_2(u)z^{-2}+\frac{E_1(u)}{8}\,z^{2}-\frac{E_2(u)}{2}\,z^{-2}\ln z.\label{sol3D3}
\end{eqnarray}
At this point, a physical discussion must be made about the forms of the solutions (\ref{p3D}) and (\ref{sol3D1}): As we pointed out earlier, the point $z=0$ represents the boundary of the background AdS spacetime; so, in order to have an asymptotically AdS behavior as we approach $z=0$, we should have (the Breitenlohner-Freedman bound \cite{bf})
\begin{equation}\label{}
-1< m\ell<1.
\end{equation}
Since $\ell^2=1/|\La|$ in three dimensions, this restricts the mass to the range
\begin{equation}\label{min3D}
0< m<\sqrt{|\La|},
\end{equation}
which, in terms of $\la$ through (\ref{m3D}), becomes
\begin{eqnarray}
(c_1+2c_3)|\La|<\la<2(c_1+c_3)|\La|&\mbox{if $c_1>0$},\\
2(c_1+c_3)|\La|<\la<(c_1+2c_3)|\La|&\mbox{if $c_1<0$}.
\end{eqnarray}
Thus we have shown that the metric
\begin{equation}\label{}
ds^2=g_{\m\n}dx^\m dx^\n=\frac{\ell^2}{z^2}(2dudv+dz^2)+2V(u,z)du^2,
\end{equation}
with the profile function
\begin{equation}\label{sol3D}
V(u,z)=V_0(u,z)+\a\p(u,z)^2,
\end{equation}
describes an exact plane wave solution, propagating in the three-dimensional AdS background, in NAT.
Up to now, we considered the case $m\neq0$. The case $m=0$, which corresponds to the choice $\la=2(c_1+c_3)/\ell^2$ in (\ref{m3D}), needs special handling. The solution of (\ref{KGAdS3D}) when $m=0$ is
\begin{equation}\label{p3Dm0}
\p(u,z)=a_1(u)+a_2(u)\ln z,
\end{equation}
with the arbitrary functions $a_1(u)$ and $a_2(u)$. Inserting this into (\ref{EinAdS1}) and (\ref{EinAdS2}) for $D=3$ produces
\begin{equation}
z^2\pa_z^2V_0+3z\pa_zV_0=E_1(u)+E_2(u)\ln z,\label{EinAdS3Dm0}
\end{equation}
where
\begin{eqnarray}
&&\left.\begin{array}{l}
\displaystyle E_1(u)\equiv2c_1a_1(u)a_2(u),\\
\displaystyle E_2(u)\equiv2c_1a_2(u)^2,
\end{array} \right\}~\mbox{for $\displaystyle\a=\frac{c_3}{2}$,}\\
&&\nn\\
&&\left.\begin{array}{l}
\displaystyle E_1(u)\equiv-c_1a_2(u)^2,\\
\displaystyle E_2(u)\equiv0,
\end{array} \right\}~\mbox{for $\displaystyle\a=\frac{c_1+c_3}{2}$.}
\end{eqnarray}
The general solution of (\ref{EinAdS3Dm0}) can be obtained as
\begin{equation}
V_0(u,z)=b_1(u)+b_2(u)z^{-2}+\frac{E_1(u)}{2}\ln z+\frac{E_2(u)}{4}\ln z(\ln z-1).\label{sol3D1m0}
\end{equation}
\subsection{AdS-Plane Waves in $D$ Dimensions: A Special Solution}
Let us now study the problem in $D$ dimensions. Of course, in this case, it is not possible to find the most general solutions of the coupled differential equations (\ref{KGAdS}), (\ref{EinAdS1}), and (\ref{EinAdS2}). However, it is possible to give a special solution, which may be thought of as the higher-dimensional generalization of the previous three-dimensional solution (\ref{sol3D}).
The $D$-dimensional spacetime has the coordinates $x^\m=(u,v,x^i,z)$ with $i=1,\ldots,D-3$. Now assume that the functions $V_0$ and $\p$ are homogeneous along the transverse coordinates $x^i$; i.e., take
\begin{equation}\label{}
V_0=V_0(u,z)~~\&~~\p=\p(u,z)~~\Rightarrow~~V(u,z)=V_0(u,z)+\a\p(u,z)^2.
\end{equation}
In that case, the differential equation (\ref{KGAdS}) becomes
\begin{equation}\label{KGAdSD}
z^2\pa_z^2\p+(4-D)z\pa_z\p-m^2\ell^2\p=0,
\end{equation}
where $m$ is given by (\ref{mAdS}), whose general solution is, for $D\neq3$,
\begin{equation}\label{phiD}
\p(u,z)=a_1(u)z^{r_+}+a_2(u)z^{r_-},
\end{equation}
where $a_1(u)$ and $a_2(u)$ are two arbitrary functions and
\begin{equation}\label{r12}
r_\pm=\frac{1}{2}\l[D-3\pm\sqrt{(D-3)^2+4m^2\ell^2}\r].
\end{equation}
Inserting (\ref{phiD}) into (\ref{EinAdS1}) and (\ref{EinAdS2}) yields
\begin{equation}
z^2\pa_z^2V_0+(6-D)z\pa_zV_0+2(3-D)V_0=E_1(u)z^{2r_+}+E_2(u)z^{2r_-},\label{EinAdSD}
\end{equation}
where
\begin{eqnarray}
&&\left.\begin{array}{l}
\displaystyle E_1(u)\equiv c_1(2r_++3-D)\,a_1(u)^2,\\
\displaystyle E_2(u)\equiv c_1(2r_-+3-D)\,a_2(u)^2,
\end{array} \right\}~\mbox{for $\displaystyle\a=\frac{c_3}{2}$,}\\
&&\nn\\
&&\left.\begin{array}{l}
\displaystyle E_1(u)\equiv -c_1(r_+^2+m^2\ell^2)\,a_1(u)^2,\\
\displaystyle E_2(u)\equiv -c_1(r_-^2+m^2\ell^2)\,a_2(u)^2,
\end{array} \right\}~\mbox{for $\displaystyle\a=\frac{c_1+c_3}{2}$.}
\end{eqnarray}
The general solution of (\ref{EinAdSD}) can be obtained as
\begin{equation}
V_0(u,z)=b_1(u)z^{D-3}+b_2(u)z^{-2}+\frac{E_1(u)}{d_+}\,z^{2r_+}+\frac{E_2(u)}{d_-}\,z^{2r_-},\label{solD1}
\end{equation}
where $b_1(u)$ and $b_2(u)$ are arbitrary functions. This solution is valid only if
\begin{eqnarray}
&&d_+\equiv4r_+^2+2(5-D)r_++2(3-D)\neq0,\\
&&d_-\equiv4r_-^2+2(5-D)r_-+2(3-D)\neq0.
\end{eqnarray}
When $d_+=0$, we have
\begin{equation}
V_0(u,z)=b_1(u)z^{D-3}+b_2(u)z^{-2}+\frac{E_1(u)}{4r_++5-D}\,z^{2r_+}\ln z+\frac{E_2(u)}{d_-}\,z^{2r_-},\label{solD2}
\end{equation}
and, when $d_-=0$, we have
\begin{equation}
V_0(u,z)=b_1(u)z^{D-3}+b_2(u)z^{-2}+\frac{E_1(u)}{d_+}\,z^{2r_+}+\frac{E_2(u)}{4r_-+5-D}\,z^{2r_-}\ln z.\label{solD3}
\end{equation}
For $m\neq0$, all these expressions reduce to the corresponding ones in the previous section when $D=3$.
As we discussed in the previous subsection, these solutions should behave like asymptotically AdS as we approach $z=0$. This means that
\begin{equation}\label{}
r_->-1.
\end{equation}
With (\ref{r12}) and (\ref{curv}), this condition gives
\begin{equation}\label{}
m<\sqrt{\frac{2|\La|}{D-1}},
\end{equation}
where $D>3$. For $D=4$ and taking the present value of the cosmological constant, $|\La|<10^{-52}$ m$^{-2}\approx 10^{-84}$ (GeV)$^2$, we obtain the upper bound $m<10^{-42}$ GeV for the mass of the spin-0 aether field $\p$.
Therefore the metric
\begin{equation}\label{}
ds^2=g_{\m\n}dx^\m dx^\n=\frac{\ell^2}{z^2}(2dudv+dx_idx^i+dz^2)+2V(u,z)du^2,
\end{equation}
with the profile function
\begin{equation}\label{}
V(u,z)=V_0(u,z)+\a\p(u,z)^2,
\end{equation}
describes an exact plane wave, propagating in the $D$-dimensional AdS background, in NAT.
\section{$pp$-Waves in Null Aether Theory} \label{pp:flat}
As a last example of Kerr-Schild-Kundt class of metrics, we shall consider $pp$-waves, \textit{plane-fronted waves with parallel rays}. These are defined to be spacetimes that admit a covariantly constant null vector field $l^\mu$; i.e.,
\begin{equation}\label{ccnull1}
\nabla_\m l_\n=0,~~l_\m l^\m=0.
\end{equation}
These spacetimes are of great importance in general relativity in that they constitute exact solutions to the full nonlinear field equations of the theory, which may represent gravitational, electromagnetic, or some other forms of matter waves \cite{step}.
In the coordinate system $x^\m=(u,v,x^i)$ with $i=1,\ldots,D-2$ adapted to the null Killing vector $l_\m=\d^u_\m$, the $pp$-wave metrics take the Kerr-Schild form \cite{ks,gg}
\begin{equation}\label{pp1}
ds^2=2dudv+2V(u,x^i)du^2+dx_idx^i,
\end{equation}
where $u$ and $v$ are the double null coordinates and $V(u,x^i)$ is the profile function of the wave. For such metrics, the Ricci tensor and the Ricci scalar become
\begin{equation}\label{}
R_\mn=-(\nabla^2_\bot V)l_\m l_\n~~\Rightarrow~~R=0,
\end{equation}
where $\nabla^2_\bot\equiv\pa_i\pa^i$. A particular subclass of $pp$-waves are plane waves for which the profile function $V(u,x^i)$ is quadratic in the transverse coordinates $x^i$, that is,
\begin{equation}\label{plane}
V(u,x^i)=h_{ij}(u)x^ix^j,
\end{equation}
where the symmetric tensor $h_{ij}(u)$ contains the information about the polarization and amplitude of the wave. In this case the Ricci tensor takes the form
\begin{equation}\label{}
R_\mn=-2\mbox{Tr}(h)l_\m l_\n,
\end{equation}
where $\mbox{Tr}(h)$ denotes the trace of the matrix $h_{ij}(u)$.
Now we will show that $pp$-wave spacetimes described above constitute exact solutions to NAT. As before, we define the null aether field as $v^\m=\p(x)l^\m$, but this time we let the scalar function $\p(x)$ and the vector field $l^\m$ satisfy the following conditions
\begin{equation}\label{ccnull}
l_\m l^\m=0,~~\nabla_\m l_\n=0,~~l^\m \pa_\m V,~~l^\m \pa_\m\p=0.
\end{equation}
Note that this is a special case of the previous analysis achieved by taking the background is flat (i.e. $K=0$) and $\xi^\m=0$ there. Then it immediately follows from (\ref{JKSK}), (\ref{Einfull}), and (\ref{Aetherfull}) that
\begin{equation}\label{}
J^{\m}~_{\n}=c_1l_\n\nabla^\m\p+c_3l^\m\nabla_\n\p,~~L=0,
\end{equation}
and the field equations are
\begin{eqnarray}
&&G_{\mu \nu}+\Lambda g_{\mu\nu}=-c_3\l[\nabla_\a\p\nabla^\a\p-\frac{\la}{c_1}\p^2\r]l_\m l_\n,\label{Ein-dust}\\
&&\nonumber\\
&&(c_1\Box\p+\la\p)l_\m=0,\label{KG}
\end{eqnarray}
where we have eliminated the $\p\Box\p$ term that should appear in (\ref{Ein-dust}) by using the aether equation (\ref{KG}) assuming $c_1\neq0$. The right-hand side of the equation (\ref{Ein-dust}) is in the form of the energy-momentum tensor of a null dust, i.e. $T_{\m\n}=\mathcal{E} l_\m l_\n$ with
\begin{equation}\label{}
\mathcal{E}\equiv-c_3\l[\nabla_\a\p\nabla^\a\p-\frac{\la}{c_1}\p^2\r].
\end{equation}
The condition $\mathcal{E}\geq0$ requires that\footnote{At this point, it is worth mentioning that, although the Null Aether Theory being discussed here is inherently different from the Einstein-Aether theory \cite{jm,jac} with a unit timelike vector field, the constraint $c_3\leq0$ in (\ref{c3}) is not in conflict with the range given in the latter theory. Indeed, imposing that the PPN parameters of Einstein-Aether theory are identical to those of general relativity, the stability against linear perturbations in Minkowski background, vacuum-\v{C}erenkov, and nucleosynthesis constraints require that (see, e.g., \cite{fj}) $$0<c_+<1,~~~~~~0<c_-<\frac{c_+}{3(1-c_+)},$$ where $c_+\equiv c_1+c_3$ and $c_-\equiv c_1-c_3$. Thus, for any fixed value $c_+$ in the range $2/3<c_+<1$, $c_3$ is restricted to the range $$-\frac{c_+(3c_+-2)}{6(1-c_+)}<c_3<\frac{c_+}{2}.$$ So there is always a region where $c_3$ is negative; for example, when $c_+=4/5$, we have $-4/15<c_3<2/5$.}
\begin{equation}\label{c3}
c_3\leq0,~~\frac{\la}{c_1}\leq0.
\end{equation}
On the other hand, the equation (\ref{KG}) gives Klein-Gordon equation for the field $\p(x)$:
\begin{equation}\label{}
\Box\p-m^2\p=0,
\end{equation}
where we defined the \ql mass" by
\begin{equation}\label{}
m^2\equiv-\frac{\la}{c_1},
\end{equation}
which is consistent with the constraint (\ref{c3}).
With the $pp$-wave ansatz (\ref{pp1}), the field equations (\ref{Ein-dust}) and (\ref{KG}) become
\begin{eqnarray}
&&-\,(\nabla^2_\bot V-2\Lambda V)\,l_\m l_\n+\Lambda \eta_{\mu\nu}=-c_3\l[\pa_i\p\pa^i\p+m^2\p^2\r]l_\m l_\n,\label{}\\
&&\nonumber\\
&&\nabla^2_\bot\p-m^2\p=0.\label{KG1}
\end{eqnarray}
Therefore, the profile function of $pp$-waves should satisfy
\begin{equation}\label{ddV}
\nabla^2_\bot V=c_3\l[\pa_i\p\pa^i\p+m^2\p^2\r],
\end{equation}
since it must be that $\La=0$. At this point, we can make the following ansatz
\begin{equation}\label{V}
V(u,x^i)=V_0(u,x^i)+\a\p(u,x^i)^2,
\end{equation}
where $\a$ is an arbitrary constant. Now plugging this into (\ref{ddV}), we obtain
\begin{equation}\label{ddV0}
\nabla^2_\bot V_0=(c_3-2\a)\l[\pa_i\p\pa^i\p+m^2\p^2\r],
\end{equation}
and since we are free to choose any value for $\a$, we get
\begin{equation}\label{Lap}
\nabla^2_\bot V_0=0~~\mbox{for}~~\a=\frac{c_3}{2}.
\end{equation}
Thus, any solution $\p(u,x^i)$ of the equation (\ref{KG1}) together with the solution $V_0(u,x^i)$ of the Laplace equation (\ref{Lap}) constitutes a $pp$-wave metric (\ref{pp1}) with the profile function $V(u,x^i)$ given by (\ref{V}).
Let us now consider the plane wave solutions described by the profile function (\ref{plane}). In that case, we can investigate the following two special cases.\\
\ni\textbf{The $c_3=0$ case:}
\vspace{0.3cm}
\ni When $c_3=0$ [or, $\a=0$ through (\ref{Lap})], it is obvious from (\ref{V}) that the function $\p$, satisfying (\ref{KG1}), detaches from the function $V$ and we should have $V=V_0$. This means that the profile function satisfies the Laplace equation, i.e.,
\begin{equation}\label{LapV}
\nabla^2_\bot V=0,
\end{equation}
which is solved by $V(u,x^i)=h_{ij}(u)x^ix^j$ only if $\mbox{Tr}(h)=0$. Thus we have shown that plane waves are solutions in NAT provided the equation (\ref{KG1}) is satisfied independently. For example, in four dimensions with the coordinates $x^\m=(u,v,x,y)$, the metric
\begin{equation}\label{pp2}
ds^2=2dudv+2[h_{11}(u)(x^2-y^2)+2h_{12}(u)xy]du^2+dx^2+dy^2
\end{equation}
describes a plane wave propagating along the null coordinate $v$ [related to the aether field through $v^\m=\p\d^\m_v$ with $\p(u,x^i)$ satisfying (\ref{KG1})] in flat spacetime. Here the function $h_{12}(u)$ is related to the polarization of the wave and, for a wave with constant linear polarization, it can always be set equal to zero by performing a rotation in the transverse plane coordinates $x$ and $y$.\\
\ni\textbf{The $c_3\neq0$ \& $V_0(u,x^i)=t_{ij}(u)x^ix^j$ case:}
\vspace{0.3cm}
\ni In this case, the Laplace equation (\ref{Lap}) says that $\mbox{Tr}(t)=0$, and from (\ref{V}) we have
\begin{equation}\label{}
\p=\sqrt{\frac{2}{c_3}[h_{ij}(u)-t_{ij}(u)]x^ix^j}.
\end{equation}
Inserting this into (\ref{KG1}), we obtain
\begin{equation}\label{c3not0}
\l[h^k_{~k}(h_{ij}-t_{ij})-(h_{ki}-t_{ki})(h^k_{~j}-t^k_{~j})\r]x^ix^j-m^2\l[(h_{ij}-t_{ij})x^ix^j\r]^2=0.
\end{equation}
This condition is trivially satisfied if $h_{ij}=t_{ij}$, but this is just the previous $c_3=0$ case in which $V=V_0$. Nontrivially, however, the condition (\ref{c3not0}) can be satisfied by setting the coefficient of the first term and the mass $m$ (or, equivalently, the Lagrange multiplier $\la$) equal to zero. Then again plane waves occur in NAT.
\section{Conclusion}
In this work, we introduced the Null Aether Theory (NAT) which is a vector-tensor theory of gravity in which the vector field defining the aether is assumed to be null at each point of spacetime. This construction allows us to take the aether field ($v_\m$) to be proportional to a principal null direction of the Weyl tensor ($l_\m$), i.e. $v_\m=\p(x)l_\m$ with $\p(x)$ being the spin-0 part of the aether field. We first investigated the Newtonian limit of this theory and showed that it is exactly the same as that of general relativity; that is to say, NAT reproduces the solar system phenomenology in this approximation and this fact puts no constraints on the free parameters \{$c_1,c_2,c_3,c_4$\} of the theory. We then constructed exact spherically symmetric black hole solutions in $D=4$ and nonlinear wave solutions in $D\geq3$ in the theory. Among the black hole solutions, we have Vaidya-type nonstationary solutions which do not need any extra matter fields for their very existence: the aether behaves in a sense as a null matter field to produce such solutions. Besides these, there are also (i) Schwarzschild-(A)dS type solutions with $h(r)\equiv1-Br^2-2m/r$ for $c_1=0$ that exist even when there is no explicit cosmological constant in the theory, (ii) Reissner-Nordstr\"{o}m-(A)dS type solutions with $h(r)\equiv1-\La r^2/3-B/r^2-2m/r$ for $c_1=-(c_2+c_3)$, (iii) solutions with $h(r)\equiv1-\La r^2/3-Br-2m/r$ for $c_1=-5(c_2+c_3)/8$, which were also obtained and used to explain the flatness of the galaxy rotation curves in conformal gravity, and so on. All these solutions have at least one event horizon and describe stationary black holes in NAT. We also discussed the existence of black hole solutions for arbitrary values of the parameters \{$c_1,c_2,c_3,c_4$\}.
As for the wave solutions, we specifically studied the Kerr-Schild-Kundt class of metrics in this context and showed that the full field equations of NAT reduce to just two, in general coupled, partial differential equations when the background spacetime takes the maximally symmetric form. One of these equations describes the massive spin-0 aether field $\p(x)$. When the background is AdS, we solved these equations explicitly and thereby constructed exact AdS-plane wave solutions of NAT in three dimensions and in higher dimensions than three if the profile function describing the wave is independent of the transverse $D-3$ coordinates. When the background is flat, on the other hand, the $pp$-wave spacetimes constitute exact solutions, for generic vaules of the coupling constants, to the theory by reducing the whole set of field equations to two decoupled differential equations: one Laplace equation for a scalar function related to the profile function of the wave and one massive Klein-Gordon equation for the spin-0 aether field in $(D-2)$-dimensional Euclidean flat space. We also showed that the plane waves, subset of $pp$-waves, are solutions to the field equations of NAT provided that the parameter $c_3$ vanishes. When $c_3$ is nonvanishing, however, the solution of the Laplace equation should satisfy certain conditions and the spin-0 aether field must be massless, i.e., $\la=0$. The main conclusion of these computations is that the spin-0 part of the aether field has a mass in general determined by the cosmological constant and the Lagrange multiplier given in the theory and in the case of AdS background this mass acquires an upper bound (the Breitenlohner-Freedman bound) determined by the value of the background cosmological constant.
\section*{Acknowledgements}
This work is partially supported by the Scientific and Technological Research Council of Turkey (TUBITAK).
|
1,116,691,499,283 | arxiv | \section{Introduction}
We consider degenerate convection-diffusion equations perturbed by a fractional diffusion (L\'{e}vy) operator; more precisely, problems of the form
\begin{align}\label{1}
\begin{cases}
u_{t}+f(u)_{x}=(a(u)u_x)_x+b\Levy[u]&\quad(x,t)\in Q_T=\mathbb{R}\times(0,T),\\
u(x,0)=u_{0}(x)&\quad x\in\mathbb{R},
\end{cases}
\end{align}
where $f,a:\mathbb{R}\rightarrow\mathbb{R}$ ($a\geq0$ and bounded) are Lipschitz continuous functions, $b\geq0$ is a constant, and $\Levy$ is a nonlocal
operator whose singular integral representation reads (cf.~\cite{Landkof,Droniou/Imbert})
\begin{align*}
\Levy[u(x,t)]=c_{\lambda}\int_{|z|>0}\frac{u(x+z,t)-u(x,t)}{|z|^{1+\lambda}}\dz,\ \lambda\in(0,1)\text{ and }c_{\lambda}>0.
\end{align*}
For sake of simplicity, we assume $f(0)=0$. The initial datum $u_0:\R\rightarrow\R$ is chosen in different spaces (cf. Theorems \ref{DDGlin}, \ref{LDGlin} and
\ref{th:existence}) depending on whether the equations are linear or nonlinear.
The operator $\Levy$ is known as the fractional Laplacian (a nonlocal generalization of the Laplace operator) and can also be defined in terms of its Fourier
transform:
\begin{align}\label{fourier}
\widehat{\Levy[u(\cdot,t)]}(\xi)=-|\xi|^{\lambda}\hat{u}(\xi,t).
\end{align}
As pointed out in \cite{Alibaud,Droniou/Imbert,Landkof}, $u(\cdot,t)$ has to be rather smooth with suitable growth at infinity for the quantity $\Levy[u]$ to
be pointwise well defined. However, smooth solutions of \eqref{1} do not exist in general (shocks may develop), and weak entropy solutions have to be
considered, cf.~Definition \ref{def:entropy} and Lemma \ref{lemmanonloc} below.
Nonlocal equations like \eqref{1} appear in different areas of research. For instance, in mathematical finance, option pricing models based on jump processes
(cf.~\cite{Cont/Tankov}) give rise to linear partial differential equations with nonlocal terms. Nonlinear equations appear in dislocation dynamics,
hydrodynamics and molecular biology \cite{Espedal/Karlsen}; applications to semiconductors devices and explosives can also be found \cite{Matalon}. For more
information about the possible applications of such equations we refer the reader to the detailed discussions in \cite{Alibaud},
\cite{Alibaud/Droniou/Vovelle}, and \cite{Droniou}.
Equation \eqref{1} consists of three different terms: nonlinear convection $f(u)_x$, nonlinear diffusion $(a(u)u_x)_x$, and fractional diffusion $\Levy[u]$. It
is expected that the effect of a diffusion operator is that solutions become smoother than the prescribed initial data. In our case, however, $a$ can be
strongly degenerate (i.e., vanish on intervals of positive length), and hence solutions can exhibit shocks. We refer to \cite{Evje/Karlsen,Espedal/Karlsen} for
the case when $b=0$, and to \cite{Alibaud/Droniou/Vovelle,Cifani/ERJ/KHK} for the case when $\lambda\in(0,1)$ and $a\equiv0$. The issue at stake here is that
the fractional diffusion operator may not be strong enough to prevent solutions of \eqref{1} from developing discontinuities. However, and as expected, in the
linear case ($f(u)=cu$, $a(u)=au$ with $c\in\R$, $a>0$), some regularity can be proved (cf.~Lemma \ref{smoothsol}).
An ample literature is available on numerical methods for computing entropy solutions of degenerate convection-diffusion equations,
cf.~\cite{Cockburn/Shu,Espedal/Karlsen,Evje/Karlsen,Evje/Karlsen2,Holden/Karlsen/Lie1,Holden/Karlsen/Lie2,Karlsen/Risebro1,Karlsen/Risebro2,Hailiang/Jue}. To
the best of our knowledge, there are no works on nonlocal versions of these equations. However, for the special case of fractional conservation laws ($a\equiv
0$) there are a few recent works \cite{Dedner/Rohde,Droniou,Cifani/ERJ/KHK}. Dedner and Rohde \cite{Dedner/Rohde} introduced a general class of difference
methods for equations appearing in radiative hydrodynamics. Droniou \cite{Droniou} devised a classs difference method for \eqref{1} ($a=0$) and proved
convergence. Cifani \emph{et al.}~\cite{Cifani/ERJ/KHK} applied the discontinuous Galerkin method to \eqref{1} ($a=0$) and proved error estimates. Finally, let
us mention that the discontinuous Galerkin method has also been used to numerically solve nonlinear convection problems appended with possibly nonlocal
dissipative terms in \cite{Jenny,Jenny/Rodhe}.
The discontinuous Galerkin (DG hereafter) method is a well established method for approximating solutions of convection \cite{Cockburn} and
convection-diffusion equations \cite{Cockburn/Shu,Hailiang/Jue}. To obtain a DG approximation of a nonlinear equation, one has to pass to the weak formulation,
do integration by parts, and replace the nonlinearities with suitable numerical fluxes (fluxes which enforce numerical stability and convergence). Available DG
methods for convection-diffusion equations are the local DG (LDG hereafter) \cite{Cockburn/Shu} and the direct DG (DDG hereafter) \cite{Hailiang/Jue}. In the
LDG method, the convection-diffusion equation is rewritten as a first order system and then approximated by the DG method for conservation laws. In the DDG
method, the DG method is applied directly to the convection-diffusion equation after a suitable numerical flux has been derived for the diffusion term.
This paper is a continuation of our previous work on DG methods for fractional conservation laws \cite{Cifani/ERJ/KHK}. We devise and study DDG and LDG
approximations of \eqref{1}, we prove that both approximations are $L^2$-stable and, whenever linear equations are considered, high-order accurate. In the
nonlinear case, we work with an entropy formulation for \eqref{1} which generalizes the one in \cite{Wu/Yin,Evje/Karlsen}, and we show that the DDG method
converges toward an entropy solution when piecewise constant elements are used. To do so, we extend the results in \cite{Evje/Karlsen} to our nonlocal setting.
Finally, we present numerical experiments shedding some light on the qualitative behavior of solutions of fractional, strongly degenerate convection-diffusion
equations.
\section{A semi-discrete method}
Let us choose a spatial grid $x_{i}=i\Delta x$ ($\Delta x>0$, $i\in\mathbb{Z}$), and label $I_{i}=(x_{i},x_{i+1})$. We denote by $P^{k}(I_{i})$ the space of
all polynomials of degree at most $k$ with support on $I_i$, and let
\begin{align*}
V^k=\{v:v|_{I_i}\in P^k(I_i),\ i\in\Z\}.
\end{align*}
Let us introduce the Legendre polynomials $\{\varphi_{0,i},\varphi_{1,i},\ldots,\varphi_{k,i}\}$, where $\varphi_{j,i}\in P^{j}(I_{i})$. Each function in
$P^{k}(I_{i})$ can be written as a linear combination of these polynomials.
We recall the following well known properties of the Legendre polynomials: for all $i\in\mathbb{Z}$,
\begin{align*}
\int_{I_{i}}\varphi_{p,i}\varphi_{q,i}\dx&=\left\{
\begin{array}{cl}
\frac{\Delta x}{2q+1}&\text{for }p=q\\
0&\text{otherwise}
\end{array}
\right.,\ \varphi_{p,i}(x_{i+1}^{-})=1\text{ and }\varphi_{p,i}(x_{i}^{+})=(-1)^{p},
\end{align*}
where $\varphi(x_i^{\pm})=\lim_{s\rightarrow x_i^{\pm}}\varphi(s)$.
The following fractional Sobolev space is also needed in what follows (see, e.g., \cite{Abels/Kassmann} or \cite[Section 6]{Folland}):
\begin{align*}
\|u\|_{H^{\lambda/2}(\mathbb{R})}^{2}=\|u\|_{L^{2}(\mathbb{R})}^{2} +|u|_{H^{\lambda/2}(\mathbb{R})}^2,
\end{align*}
with semi-norm $|u|_{H^{\lambda/2}(\mathbb{R})}^2= \int_\R\int_\R\frac{(u(z)-u(x))^2}{|z-x|^{1+\lambda}}\dz\dx$. Finally, let us introduce the operators
\begin{align*}
[p(x_i)]=p(x_i^+)-p(x_i^-),\quad \overline{p(x_i)}=\frac{1}{2}(p(x_i^+)+p(x_i^-)).
\end{align*}
From now on we split our exposition into two parts, one dedicated to
the DDG method and another one dedicated to the LDG method.
\subsection{DDG method}
Let us multiply \eqref{1} by an arbitrary $v\in P^k(I_i)$, integrate over $I_i$, and use integration by parts, to arrive at
\begin{align}\label{Ds1}
\begin{split}
\int_{I_{i}}u_{t}v\ -&\int_{I_{i}}f(u)v_{x}+f(u_{i+1})v_{i+1}^{-}-f(u_{i})v_{i}^{+}\\
&+\int_{I_i}a(u)u_xv_x-h(u_{i+1},u_{x,i+1})v_{i+1}^-+h(u_{i},u_{x,i})v_{i}^+ =b\int_{I_{i}}\Levy[u]v,
\end{split}
\end{align}
where $f(u_i)=f(u(x_i))$, $h(u,u_x)=a(u)u_x$ and $(u_i,u_{x,i})=(u(x_i),u_x(x_i))$. Let us introduce the Lipschitz continuous E-flux (a consistent and monotone
flux),
\begin{align}\label{eflux}
\hat{f}(u_i)=\hat{f}(u(x_i^-),u(x_i^+)).
\end{align}
Note that since $\hat{f}$ is consistent ($\hat{f}(u,u)=f(u)$) and monotone (increasing w.r.t.~ its first variable and decreasing w.r.t~ its second variable),
\begin{align}\label{eeeflux}
\int_{u_i^-}^{u_i^+}\big[f(x)-\hat{f}(u_i^-,u_i^+)\big]\dx\geq0.
\end{align}
Following Jue and Liu \cite{Hailiang/Jue}, let us also introduce the flux
\begin{align*}
\hat{h}(u_i)&=\hat{h}(u(x_i^-),\ldots,\partial_x^ku(x_i^-),u(x_i^+),\ldots,\partial_x^ku(x_i^+))\\
&=\beta_0\frac{[A(u_i)]}{\Delta x}+\overline{A(u_i)_x} +\sum_{m=1}^{\lfloor k/2\rfloor}\beta_m\Delta x^{2m-1}[\partial_x^{2m}A(u_i)],
\end{align*}
where $A(u)=\int^{u}a$ and the weights $\{\beta_0,\ldots,\beta_{\lfloor k/2\rfloor}\}$ fulfill the following admissibility condition: there exist
$\gamma\in(0,1)$ and $\alpha\geq0$ such that
\begin{align}\label{admiss}
\sum_{i\in\Z}\hat{h}(u_i)[u_i]\geq\alpha \sum_{i\in\Z}\frac{[A(u_i)]}{\Delta x}[u_i] -\gamma\sum_{i\in\Z}\int_{I_i}a(u)(u_x)^2.
\end{align}
Note that the numerical flux $\hat h$ is an approximation of $A(u_i)_x=a(u(x_i))u_x(x_i)$ involving the average $\overline{A(u_i)_x}$ and the jumps of even
order derivatives of $A(u_i)$ up to $m=k/2$. For example, if $k=0$ and $\beta_0=1$, then
$$
\hat{h}(u_i)=\frac{1}{\Delta x} [A(u_i)]=\frac{A(u(x_i^+))-A(u(x_i^-))}{\Delta x},$$ and this function satisfies condition \eqref{admiss}. In this case
($k=0$),
$$\overline{A(u_i)_x}=\overline{a(u(x_i))\partial_xu(x_i)}
=\frac{1}{2}\Big(a(u(x_i^+))\partial_xu(x_i^+)+a(u(x_i^-))\partial_xu(x_i^-)\Big)=0.$$ When $k\geq2$, some extra differentiability on $a$ is required. For
example, with $k=2$,
\begin{align*}
\sum_{m=1}^{\lfloor k/2\rfloor}\beta_m\Delta x^{2m-1}[\partial_x^{2m}A(u_i)]&=\beta_1\Delta x[\partial_x^{2}A(u_i)]=\beta_1\Delta
x[a'(u_i)(\partial_xu_i)^2+a(u_i)\partial_x^{2}u_i].
\end{align*}
We see that the flux $\hat h$ is locally Lipschitz if $a$ is sufficently regular, and that $\hat h(0)=0$ for all $k$. Let us rewrite \eqref{Ds1} as
\begin{align}\label{Ds2}
\begin{split}
\int_{I_{i}}u_{t}v\ -&\int_{I_{i}}f(u)v_{x}+\hat{f}(u_{i+1})v_{i+1}^{-}-\hat{f}(u_{i})v_{i}^{+}\\
&+\int_{I_i}a(u)u_xv_x-\hat{h}(u_{i+1})v_{i+1}^-+\hat{h}(u_{i})v_{i}^+=b\int_{I_{i}}\Levy[u]v,
\end{split}
\end{align}
and use the initial condition
\begin{align}\label{initial_condition}
\begin{split}
\int_{I_{i}}u(x,0)v(x)\dx=\int_{I_{i}}u_0(x)v(x)\dx.
\end{split}
\end{align}
The DDG method consists of finding functions $\hat{u}:Q_T\rightarrow\mathbb{R}$, $\hat{u}(\cdot,t)\in V^k$, and
\begin{align}\label{udef}
\hat{u}(x,t)=\sum_{i\in\mathbb{Z}}\sum_{p=0}^{k}U_{p,i}(t)\varphi_{p,i}(x),
\end{align}
which satisfy \eqref{Ds2}-\eqref{initial_condition} for all $v\in P^{k}(I_i)$, $i\in\mathbb{Z}$.
\subsection{LDG method}
Let us write $a(u)u_x=\sqrt{a(u)}g(u)_x$, where $g(u)=\int^{u}\sqrt{a}$, and turn equation \eqref{1} into the following system of equations
\begin{align}\label{prob}
\left\{
\begin{array}{ll}
u_t+(f(u)-\sqrt{a(u)}q)_x=b\Levy[u],\\
q-g(u)_x=0.
\end{array}
\right.
\end{align}
Let us introduce the notation $\mathbf{w}=(u,q)'$ (here $'$ denotes the transpose), and write
\begin{align*}
\mathbf{h}(\mathbf{w})=\mathbf{h}(u,q)=\left(
\begin{array}{c}
h_u(\mathbf{w})\\
h_q(u)
\end{array}
\right)=\left(
\begin{array}{c}
f(u)-\sqrt{a(u)}q\\
-g(u)
\end{array}
\right).
\end{align*}
Let us multiply each equation in \eqref{prob} by arbitrary $v_u,v_q\in P^{k}(I_{i})$, integrate over the interval $I_{i}$, and use integration by parts, to
arrive at
\begin{align*}
&\int_{I_{i}}\partial_tuv_{u}-\int_{I_{i}}h_u(\mathbf{w})\partial_{x}v_u +h_u(\mathbf{w}_{i+1})v_{u,i+1}^{-}-h_u(\mathbf{w}_{i})v_{u,i}^{+}
=b\int_{I_{i}}\Levy[u]v_u,\\
&\int_{I_{i}}qv_q-\int_{I_{i}}h_q(u)\partial_xv_{q}+h_q(u_{i+1}) v_{q,i+1}^{-}-h_q(u_{i})v_{q,i}^{+}=0,
\end{align*}
where $h_u(\mathbf{w}_{i})=h_u(u_{i},q_i)$, $u_i=u(x_i)$, $q_i=q(x_i)$, $v_{u,i}^-=v_u(x_i^-)$ and $v_{u,i}^+=v_u(x_i^+)$. Following Cockburn and Shu
~\cite{Cockburn/Shu}, we introduce the numerical flux
\begin{align}\label{Lflux}
\hat{\mathbf{h}}(\mathbf{w}_i^-,\mathbf{w}_i^+)=\left(
\begin{array}{c}
\hat{h}_u(\mathbf{w}_i^-,\mathbf{w}_i^+)\\
\hat{h}_q(u_i^-,u_i^+)
\end{array}
\right)=\left(
\begin{array}{c}
\frac{[F(u_i)]}{[u_i]}-\frac{[g(u_i)]}{[u_i]}\overline{q_i}\\
-\overline{g(u_i)}
\end{array}
\right)-\C[\mathbf{w}_i],
\end{align}
where $F(u)=\int^{u}f$, $\C=\left(
\begin{array}{cc}
c_{11} & c_{12}\\
-c_{12} & 0
\end{array}
\right)$,
\begin{align*}
\begin{split}
c_{11}=\frac{1}{[u_i]}\left(\frac{[F(u_i)]}{[u_i]}-\hat{f}(u_i^-,u_i^+)\right),
\end{split}
\end{align*}
$c_{12}=c_{12}(\mathbf{w}_i^-,\mathbf{w}_i^+)$ is Lipschitz continuous in all its variables, and $c_{12}=0$ whenever $a=0$ or
$\mathbf{w}_i^-,\mathbf{w}_i^+=0$. Note that $c_{11}\geq0$ since $\hat{f}$ is an E-flux and, thus, the matrix $\C$ is semipositive definite.
The LDG method consists of finding $\tilde{\mathbf{w}}=(\tilde{u},\tilde{q})'$, where
\begin{align*}
\tilde{u}(x,t)=\sum_{i\in\mathbb{Z}}\sum_{p=0}^{k}U_{p,i}(t)\varphi_{p,i}(x)\quad\text{and}\quad
\tilde{q}(x,t)=\sum_{i\in\mathbb{Z}}\sum_{p=0}^{k}Q_{p,i}(t)\varphi_{p,i}(x)
\end{align*}
are functions satisfying
\begin{align}\label{systdisc}
\begin{split}
&\int_{I_{i}}\partial_tuv_{u}-\int_{I_{i}}h_u(\mathbf{w})\partial_{x}v_u+\hat{h}_u(\mathbf{w}_{i+1})v_{u,i+1}^{-}
-\hat{h}_u(\mathbf{w}_{i})v_{u,i}^{+}=b\int_{I_{i}}\Levy[u]v_u,\\
&\int_{I_{i}}qv_q-\int_{I_{i}}h_q(u)\partial_xv_{q}+\hat{h}_q(u_{i+1})v_{q,i+1}^{-}-\hat{h}_q(u_{i})v_{q,i}^{+}=0,
\end{split}
\end{align}
for all $v_u,v_q\in P^{k}(I_i)$, $i\in\mathbb{Z}$, and initial conditions for $u$ and $q$ given by \eqref{initial_condition}.
\section{$L^2$-stability for nonlinear equations}
We will show that in the semidiscrete case (no time discretization) both the DDG and LDG methods are $L^2$-stable, for linear and nonlinear equations.
In this section and the subsequent one, we assume the existence of solutions $\hat{u}$ and $\tilde{\mathbf{w}}=(\tilde{u},\tilde{q})'$ of the DDG and LDG
methods \eqref{Ds2} and \eqref{systdisc}, respectively, satisfying $\hat{u},\tilde{u},\tilde{q}\in C^1([0,T];V^k\cap L^{2}(\mathbb{R}))$, in which case the
integrals containing the nonlocal operator $\Levy[\cdot]$ are all well defined. Indeed, by Lemma \ref{Hs}, $V^k\cap L^{2}(\mathbb{R})\subseteq
H^{\lambda/2}(\mathbb{R})$, and hence all integrals of the form
$$
\int_{\mathbb{R}}\varphi_1\,\Levy[\varphi_2]\quad\text{for$ \quad\varphi_1,\varphi_2\in V^k\cap L^{2}(\mathbb{R})$,}
$$
can be interpreted as the pairing between $\varphi_1\in H^{\lambda/2}(\mathbb{R})$ and $\Levy[\varphi_2]\in H^{-\lambda/2}(\mathbb{R})$. Here
$H^{-\lambda/2}(\mathbb{R})$ is the dual space of $H^{\lambda/2}(\mathbb{R})$, and $\Levy[\varphi]\in H^{-\lambda/2}(\mathbb{R})$ whenever $\varphi\in
H^{\lambda/2}(\mathbb{R})$ (cf.~Corollary A.3 and proof in \cite{Cifani/ERJ/KHK}).
\begin{remark}
The existence and uniqueness of solutions in $C^1([0,T];V^k\cap L^{2}(\mathbb{R}))$ can be proved using the Picard-Cauchy-Lipschitz theorem. The argument
outlined in \cite[Section 3]{Cifani/ERJ/KHK}, can be adapted to the current setting since all numerical fluxes are (locally) Lipschitz (cf. \cite{Cockburn/Shu}
for the LDG case). For the DDG method with $k>2$, additional differentiability on $a$ is needed for this proof to work.
\end{remark}
\subsection{DDG method}
Let us sum over all $i\in\Z$ in \eqref{Ds2}, integrate over $t\in(0,T)$, and introduce the functional
\begin{align}\label{functionalM}
\begin{split}
M_{DDG}[u,v]=\int_0^T&\int_{\mathbb{R}}u_{t}v
-\int_0^T\sum_{i\in\mathbb{Z}}\Big[\hat{f}(u_{i})[v_{i}]+\int_{I_{i}}f(u)v_{x}\Big]\\
&+\int_0^T\sum_{i\in\mathbb{Z}}\Big[\hat{h}(u_i)[v_i] +\int_{I_i}a(u)u_xv_x\Big]-b\int_0^T\int_{\mathbb{R}}\Levy[u]v.
\end{split}
\end{align}
Let us define
\begin{align*}
\Gamma_T[u]=(1-\gamma)\int_0^T\sum_{i\in\Z}\int_{I_i}a(u)(u_x)^2 +\alpha\int_0^T \sum_{i\in\Z}\frac{[A(u_i)]}{\Delta x}[u_i],
\end{align*}
where $\gamma\in(0,1)$ and $\alpha>0$. Note that $\Gamma_T\geq0$ since $a\geq0$ and, using the Taylor's formula, $[A(u_i)][u_i]=a(\xi_i)[u_i]^2\geq0$ where and
$\xi_i\in[u(x_i^-),u(x_i^+)]$, $i\in\Z$.
\begin{theorem}\label{stab1}
\emph{(Stability)} Let $\hat{u}$ be a solution of \eqref{Ds2} such that both $\hat u,A(\hat u)$ and their first $k$ derivatives are sufficiently integrable.
Then
\begin{align*}
\|\hat{u}(\cdot,T)\|_{L^{2}(\mathbb{R})}^2+2\Gamma_{T}[\hat{u}] +bc_\lambda\int_0^T|\hat{u}(\cdot,t)|^2_{H^{\lambda/2}(\mathbb{R})}\dt
\leq\|u_{0}\|_{L^{2}(\mathbb{R})}^2.
\end{align*}
\end{theorem}
\begin{remark}
Since $\tilde u\in C^1([0,T];V^k\cap L^{2}(\mathbb{R}))$ and $f(0)=0$, all terms in \eqref{main} below are well defined -- except for
\begin{align*}
\int_0^T\Big[\sum_{i\in\Z}\hat{h}(\hat u_i)[\hat{u}_i] +\int_{I_i}a(\hat{u})(\hat{u}_x)^2\Big].
\end{align*}
When $k\geq2$, additional integrability of $\hat u,A(\hat u)$, and their first $k$ derivatives, is required in order to give meaning to the $\hat h$-term.
\end{remark}
\begin{proof}
By construction, $M_{DDG}[\hat{u},v]=0$ for all $v\in V^k\cap L^{2}(\R)$. If we set $v=\hat{u}$, we obtain
\begin{align}\label{main}
\begin{split}
\int_0^{T}\int_{\mathbb{R}}\hat{u}_{t}\hat{u}&-\int_0^{T}\sum_{i\in\mathbb{Z}}\Big[\hat{f}(\hat{u}_{i})
[\hat{u}_{i}]+\int_{I_{i}}f(\hat{u})\hat{u}_{x}\Big]\\
&+\int_0^T\sum_{i\in\Z}\Big[\hat{h}(\hat u_i)[\hat{u}_i] +\int_{I_i}a(\hat{u})(\hat{u}_x)^2\Big]-b\int_0^{T}\int_{\mathbb{R}}\Levy[\hat{u}]\hat{u}=0.
\end{split}
\end{align}
Next, as a direct consequence of \eqref{eeeflux} and a change of variables, we see that
\begin{align}\label{e1}
\int_0^{T}\sum_{i\in\mathbb{Z}}\Big[\hat{f}(\hat{u}_{i})[\hat{u}_{i}] +\int_{I_{i}}f(\hat{u})\hat{u}_{x}\Big]\leq0.
\end{align}
Since $\hat{h}$ satisfies the expression \eqref{admiss},
\begin{align}\label{e2}
\int_0^{T}\sum_{i\in\Z}\hat{h}_i(\hat u_i)[\hat{u}_i]\geq \alpha\int_0^T\sum_{i\in\Z}\frac{[A(\hat{u}_i)]}{\Delta x}[\hat{u}_i]
-\gamma\int_0^T\sum_{i\in\Z}\int_{I_i}a(\hat{u})(\hat{u}_x)^2.
\end{align}
Finally, using Lemma \ref{lemmanonloc},
\begin{align}\label{e3}
\int_{\mathbb{R}}\Levy[\hat{u}]\hat{u}= -\frac{c_\lambda}{2}|\hat{u}|_{H^{\lambda/2}(\mathbb{R})}^2.
\end{align}
We conclude by inserting \eqref{e1}, \eqref{e2}, and \eqref{e3} into \eqref{main}.
\end{proof}
\subsection{LDG method}
By summing over all $i\in\Z$, we can rewrite \eqref{systdisc} as
\begin{align*}
&\int_{\R}\partial_{t}uv_u-\sum_{i\in\Z}\bigg(\hat{h}_u(\mathbf{w}_{i})[v_{u,i}]
+\int_{I_{i}}h_u(\mathbf{w})\partial_{x}v_u\bigg)=b\int_{\R}\Levy[u]v_u,\\
&\int_{\R}q v_q-\sum_{i\in\Z}\bigg(\hat{h}_q(u_{i})[v_{q,i}] +\int_{I_{i}}h_{q}(u)\partial_xv_q\bigg)=0.
\end{align*}
We add the two equations and integrate over $t\in(0,T)$ to find $M_{LDG}[\mathbf{w},\mathbf{v}]=0$ for
\begin{align}\label{functionalB}
\begin{split}
M_{LDG}[\mathbf{w},\mathbf{v}]=&\int_{0}^{T}\int_{\mathbb{R}}u_tv_u+\int_0^T\int_{\R}qv_q\\
&-\int_0^T\sum_{i\in\Z}\bigg(\hat{\mathbf{h}}(\mathbf{w}_i)'[\mathbf{v}_i]
+\int_{I_i}\mathbf{h}(\mathbf{w})'\partial_x\mathbf{v}\bigg)-b\int_0^T\int_{\R}\Levy[u]v_u,
\end{split}
\end{align}
where $\hat{\mathbf{h}}(\mathbf{w}_i)= (\hat{h}_u(\mathbf{w}_{i}),\hat{h}_q(u_{i}))'$, $\mathbf{v}=(v_u,v_q)'$ and $\mathbf{v}_i=(v_{u,i},v_{q,i})'$. Moreover,
let (remember that, as noted earlier, the matrix $\C$ is semipositive definite)
\begin{align*}
\Theta_{T}[\mathbf{w}]=\int_0^T\sum_{i\in\Z}[\mathbf{w}_i]'\C[\mathbf{w}_i]\ (\geq0).
\end{align*}
\begin{theorem}
\emph{(Stability)} If $\tilde{\mathbf{w}}=(\tilde{u},\tilde{q})'$ is a $C^1([0,T];(V^k\cap L^2)^2 )$ solution of \eqref{systdisc}, then
\begin{align*}
\|\tilde{u}(\cdot,T)\|_{L^{2}(\mathbb{R})}^2+2\|\tilde{q}\|_{L^2(Q_T)}^2 +2\Theta_{T}(\tilde{\mathbf{w}})
+bc_\lambda\int_0^T|\tilde{u}(\cdot,t)|^2_{H^{\lambda/2}(\mathbb{R})}\dt \leq\|u_{0}\|_{L^{2}(\mathbb{R})}^2.
\end{align*}
\end{theorem}
Here, as opposed to Theorem \ref{stab1}, no further integrability of the first $k$ derivatives of the numerical solution
$\tilde{\mathbf{w}}=(\tilde{u},\tilde{q})'$ is needed. The reason is that the numerical flux $\hat{\mathbf{h}}$ has been built without the use of derivatives
of $\tilde{\mathbf{w}}=(\tilde{u},\tilde{q})'$. Each term in expression \eqref{main2} below is well defined thanks to \eqref{e4}, the fact that $f(0)=0$ (which
implies that $c_{11}(0)=0$), $c_{12}(0)=0$, and $\tilde{u},\tilde{q}\in C^1([0,T];V^k\cap L^{2}(\mathbb{R}))$.
\begin{proof}
By construction, $M_{LDG}(\hat{\mathbf{w}},\mathbf{v})=0$ for all $\mathbf{v}=(v_{u},v_{q})'$, $v_u,v_q\in V^{k}\cap L^{2}(\R)$. We set
$\mathbf{v}=\hat{\mathbf{w}}$ and find that
\begin{align}\label{main2}
\begin{split}
\int_0^T\int_{\R}\tilde{u}_t\tilde{u}+\int_0^T\int_{\R}\tilde{q}^2 -\int_0^T\sum_{i\in\Z}\bigg( \hat{\mathbf{h}}(\tilde{\mathbf{w}}_i)'[\tilde{\mathbf{w}}_i]
+\int_{I_{i}}\mathbf{h}(\tilde{\mathbf{w}})'
\partial_x\tilde{\mathbf{w}}\bigg)&\\-b\int_0^T
\int_{\R}\Levy[\tilde{u}]\tilde{u}&=0.
\end{split}
\end{align}
Here we also used the fact that
\begin{align}\label{e4}
\begin{split}
-\int_0^T\sum_{i\in\Z}\bigg(\hat{\mathbf{h}}(\tilde{\mathbf{w}}_i)'[\tilde{\mathbf{w}}_i]
+\int_{I_{i}}\mathbf{h}(\tilde{\mathbf{w}})'\partial_x\tilde{\mathbf{w}}\bigg) =\int_0^T\sum_{i\in\Z}[\tilde{\mathbf{w}}_i] \C[\tilde{\mathbf{w}}_i],
\end{split}
\end{align}
see \cite{Cockburn/Shu} for a proof. To conclude, insert \eqref{e4} and \eqref{e3} into \eqref{main2}.
\end{proof}
\section{High-order convergence for linear equations}
In this section we consider the linear problem
\begin{align}\label{linear}
\begin{cases}
u_{t}+cu_{x}=u_{xx}+b\Levy[u]&(x,t)\in Q_T,\\
u(x,0)=u_0(x)&x\in\R,
\end{cases}
\end{align}
with the aim of proving that the DDG and LDG methods converge to a regular solution of \eqref{linear} with high-order accuracy.
\begin{lemma}\label{smoothsol}
Let $u_{0}\in H^{k+1}(\mathbb{R})$, with $k\geq0$. There exists a
unique function $u\in H^{k+1}_{\mathrm{par}}(Q_T)$ solving \eqref{linear}, where
$$ H^{k+1}_{\mathrm{par}}(Q_T):=\Big\{\phi\in L^2(Q_T):
\|\del_t^m\del_x^{r}u\|_{L^2(Q_T)}<\infty \text{ for all }
0\leq r+2m\leq k+1\Big\}.$$
Moreover, $\|u(\cdot,t)\|_{H^{k+1}(\mathbb{R})}
\leq\|u_{0}\|_{H^{k+1}(\mathbb{R})}.$
\end{lemma}
\begin{proof}
Since the equation is linear, we can pass to the Fourier space. In view of \eqref{fourier}, the Fourier transform of \eqref{linear} is $\hat{u}_{t}+i\xi
c\hat{u}= -\xi^2\hat{u}-b|\xi|^{\lambda}\hat{u}$. It follows that
\begin{align*}
\hat{u}(\xi,t)=\hat{u}_{0}(\xi)e^{-(i\xi c+\xi^2+b|\xi|^{\lambda})t}.
\end{align*}
By the properties of the Fourier transform, the above expression implies the existence of a unique $L^{2}$-stable weak solution of \eqref{linear}. The
$L^{2}$-stability for higher derivatives can be obtained by iteration as follows: take the derivative of \eqref{linear}, use the Fourier transform to get
stability, and iterate up to the $k$th derivative. Regularity in time
follows from the regularity in space since equation \eqref{linear}
implies that $\del_t^ku=(-c\del_x+\del_x^2+b\Levy)^ku$.
\end{proof}
In the following two theorems we obtain $L^2$-type error estimates for
the DDG and LDG methods in the case that equation \eqref{linear} has
$H^{k+1}_{\mathrm{par}}$-regular solutions. (Note that the time regularity does
not play any role here). To do so, we combine estimates for the local terms derived in \cite{Cockburn/Shu,Hailiang/Jue} with estimates for the nonlocal term derived by the
authors in \cite{Cifani/ERJ/KHK}. In \cite{Cockburn} it was observed that most relevant numerical $\hat f$ fluxes reduce to
\begin{align*}
\hat{f}(u_i^-,u_i^+)&=c\overline{u_i}-|c|\frac{[u_i]}{2}
\end{align*}
in the linear case. In this section we only consider this $\hat f$ flux.
\subsection{DDG method}
\begin{theorem}\label{DDGlin}
\emph{(Convergence)} Let $u\in H^{k+1}_{\mathrm{par}}(Q_T)$, $k\geq0$, be a solution of \eqref{linear} and $\hat{u}\in C^1([0,T];V^k\cap L^2(\R))$ be a solution of
\eqref{Ds2}. With $e=u-\hat{u}$,
\begin{align*}
\begin{split}
\int_\R e^2(x,T) & +\frac{|c|}{2}\int_0^T \sum_{i\in\Z}[e_i]^2+(1-\gamma) \int_0^T\int_{\R}(e_x)^2 +\alpha\int_0^{T}
\sum_{i\in\Z}\frac{[e_i]^2}{\Delta x}\\
& +bc_\lambda\int_0^T|e|^2_{H^{\lambda/2}(\R)}
=\mathcal{O}(1)\Delta x^{2k}.
\end{split}
\end{align*}
\end{theorem}
\begin{remark}
The error $\mathcal{O}(1)\Delta x^{2k}$ is due to the diffusion term $u_{xx}$. The errors from the convection term $cu_x$ and the fractional diffusion term
$b\Levy[u]$ are of the form $\mathcal{O}(1)\Delta x^{2k+1}$ and $\mathcal{O}(1)\Delta x^{2k+2-\lambda}$ respectively.
\end{remark}
\begin{proof}
Let us set
\begin{align*}
M_{a}[u,v]&=\int_0^T\int_{\mathbb{R}}u_{t}v+\int_0^T\int_{\R}u_xv_x
+\int_0^{T}\sum_{i\in\Z}\hat h(u_i)[v_i],\\
M_{f}[u,v]&=-\int_0^T\sum_{i\in\mathbb{Z}}\Big[\hat{f}(u_{i})[v_{i}]
+\int_{I_{i}}cuv_{x}\Big],\\
M_\Levy[u,v]&=-b\int_0^T\int_{\mathbb{R}}\Levy[u]v.
\end{align*}
With this notation in hand, we can write \eqref{functionalM} as
\begin{align*}
\begin{split}
M_{DDG}[u,v]=M_{a}[u,v]+M_{f}[u,v]+M_\Levy[u,v].
\end{split}
\end{align*}
Let $\p e$ be the $L^{2}$-projection of $e$ into $V^k$, i.e., $\p e$ is the $V^k\cap L^2(\mathbb{R})$ function satisfying
\begin{align*}
\int_{I_{i}}\big(\p e(x)-e(x)\big)\varphi_{ji}(x)\dx=0 \text{ for all $i\in\Z$ and $j=\{0,\ldots,k$\}.}
\end{align*}
Note that $\p e\in H^{\lambda/2}(\mathbb{R})$ since $V^k\cap L^2(\mathbb{R})\subset H^{\lambda/2}(\mathbb{R})$ by Lemma \ref{Hs}. For all $v\in V^k\cap
L^2(\mathbb{R})$, we have $M_{DDG}[\hat{u},v]=0$ since $\hat{u}$ is a DDG solution of \eqref{linear}, while $M_{DDG}[u,v]=0$ since $u$ is a continuous (by
Sobolev imbedding) solution of \eqref{1} and hence a solution of \eqref{linear}. Thus $M_{DDG}[e,v]=0$, and by bilinearity ($\hat h$ is linear
since $a\equiv1$),
\begin{align}\label{iii}
M_{DDG}[\p e,\p e]=M_{DDG}[\p e-e,\p e].
\end{align}
One can proceed as in \cite{Hailiang/Jue} (in that paper, combine
the last inequality of the proof of Lemma 3.3 with Lemma 3.2 and
(3.5)) to obtain
\begin{align}\label{i1}
\begin{split}
M_{a}[\p e-e,\p e]=\frac{1}{2}\int_0^T\int_{\R}(\p e_x)^2 +\frac{1}{2}\int_0^{T}\sum_{i\in\Z}\hat h(\p e_i)[\p e_i]+\mathcal{O}(1)\Delta x^{2k}.
\end{split}
\end{align}
Moreover, proceeding as in \cite[Lemma 2.17]{Cockburn},
\begin{align}\label{i2}
\begin{split}
M_{f}[\p e-e,\p e]=\frac{|c|}{4}\int_0^T \sum_{i\in\Z}[\p e_i]^2+\mathcal{O}(1)\Delta x^{2k+1}.
\end{split}
\end{align}
As shown by the authors in \cite{Cifani/ERJ/KHK},
\begin{align}\label{i3}
\begin{split}
&M_\Levy[\p e-e,\p e]-M_\Levy[\p e,\p e]=b\int_{0}^{T}\int_{\mathbb{R}}\Levy[e]\p e\\
&=\frac{b}{2}\int_{0}^{T}\int_{\mathbb{R}}\Levy[\p e]\p e +\frac{b}{2}\int_{0}^{T}\int_{\mathbb{R}}\Levy[e]e
-\frac{b}{2}\int_{0}^{T}\int_{\mathbb{R}}\Levy[e-\p e](e-\p e)\\
&\leq-\frac{bc_\lambda}{4}\int_{0}^{T}|\p e|_{H^{\lambda/2}(\mathbb{R})}^{2} -\frac{bc_\lambda}{4}\int_{0}^{T}|e|_{H^{\lambda/2}(\mathbb{R})}^{2}
+\frac{bc_\lambda}{4}\int_{0}^{T}\|e-\p e\|_{H^{\lambda/2}(\mathbb{R})}^{2},
\end{split}
\end{align}
where $M_\Levy[\p e,\p e]=\frac{bc_\lambda}{2}\int_{0}^{T}|\p e|_{H^{\lambda/2}(\mathbb{R})}^{2}$ (Lemma \ref{lemmanonloc}) and
\begin{align}\label{aaa}
\|e-\p e\|_{H^{\lambda/2}(\mathbb{R})}^{2}\leq\mathcal{O}(1)\Delta x^{2k+2-\lambda}.
\end{align}
By \eqref{functionalM}, Lemma \ref{lemmanonloc}, and the definition of $\hat f$,
\begin{align*}
\begin{split}
M_{DDG}[\p e,\p e]=&\int_\R (\p e^2)_t+\frac{|c|}{2}\int_0^T\sum_{i\in\Z}[\p e_i]^2 +\int_0^T\int_{\R}(\p e_x)^2\\&+\int_0^{T}\sum_{i\in\Z}\hat h(\p e_i)[\p
e_i]+\frac{bc_\lambda}{2}\int_0^T|\p e|^2_{H^{\lambda/2}(\R)}.
\end{split}
\end{align*}
Inserting this equation along with \eqref{i1}, \eqref{i2}, and \eqref{i3} into \eqref{iii} then shows that
\begin{align*}
\begin{split}
\int_0^T\int_\R (\p e^2)_t+\frac{|c|}{4}\int_0^T\sum_{i\in\Z}[\p e_i]^2 +\frac{1}{2}\int_0^T\int_{\R}(\p e_x)^2
+\frac{1}{2}\int_0^{T}\sum_{i\in\Z}\hat h(\p e_i)[\p e_i]&\\
+\frac{bc_\lambda}{4}\int_0^T|\p e|^2_{H^{\lambda/2}(\R)} +\frac{bc_\lambda}{4}\int_0^T| e|^2_{H^{\lambda/2}(\R)}=\mathcal{O}(1)\Delta x^{2k},&
\end{split}
\end{align*}
and, using the admissibility condition \eqref{admiss},
\begin{align*}
\begin{split}
\int_0^T\int_\R (\p e^2)_t+\frac{|c|}{4}\int_0^T\sum_{i\in\Z}[\p e_i]^2 +\frac{1-\gamma}{2}\int_0^T\int_{\R}(\p e_x)^2
+\frac{\alpha}{2}\int_0^{T}\sum_{i\in\Z}\frac{[\p e_i]^2}{\Delta x}&\\
+\frac{bc_\lambda}{4}\int_0^T|\p e|^2_{H^{\lambda/2}(\R)} +\frac{bc_\lambda}{4}\int_0^T| e|^2_{H^{\lambda/2}(\R)} =\mathcal{O}(1)\Delta x^{2k}.&
\end{split}
\end{align*}
To conclude, we need to pass form $\p e$ to $e$ in the above expression. This has already been done for the diffusion term in Section 3 in \cite{Hailiang/Jue}
and for the convection term in the proof of Lemma 2.4 in \cite{Cockburn/Shu}. For the nonlocal term, we see that by \eqref{aaa}
\begin{align*}
\begin{split}
|\p e|^2_{H^{\lambda/2}(\R)}=|e|^2_{H^{\lambda/2}(\R)} -\mathcal{O}(1)\Delta x^{2k+2-\lambda},
\end{split}
\end{align*}
and the conclusion follows.
\end{proof}
\subsection{LDG method}
\begin{theorem}\label{LDGlin}
\emph{(Convergence)} Let $u\in H^{k+1}_{\mathrm{par}}(Q_T)$, $k\geq0$, be a solution of \eqref{linear} and $\tilde{\mathbf{w}}=(\tilde{u},\tilde{q})'\in C^1([0,T];V^k\cap L^2
)$ be a solution of \eqref{Ds2}. With $e_u=u-\tilde{u}$ and $e_q=q-\tilde{q}$,
\begin{align*}
\begin{split}
\int_\R e_u^2(x,T)+\int_0^T\int_{\R}e_q^2 +\Theta_T[\mathbf{e}] +bc_\lambda\int_0^T|e_u|^2_{H^{\lambda/2}(\R)} =\mathcal{O}(1)\Delta x^{2k}.
\end{split}
\end{align*}
\end{theorem}
\begin{proof}
Let us choose a test function $\mathbf{v}=(v_{u},v_{q})'$, $v_u,v_q\in V^{k}\cap L^{2}(\R)$, and define
\begin{align*}
\begin{split}
M_l[\mathbf{w},\mathbf{v}]&=\int_{0}^{T}\int_{\mathbb{R}}u_tv_u+\int_0^T\int_{\R}qv_q -\int_0^T\sum_{i\in\Z}\bigg(\hat{\mathbf{h}}(\mathbf{w}_i)'[\mathbf{v}_i]
+\int_{I_i}\mathbf{h}(\mathbf{w})'\partial_x\mathbf{v}\bigg).
\end{split}
\end{align*}
With this notation at hand, we can write \eqref{functionalB} as
\begin{align*}
\begin{split}
M_{LDG}[\mathbf{w},\mathbf{v}]=M_l[\mathbf{w},\mathbf{v}] +M_\Levy[\mathbf{w},\mathbf{v}],
\end{split}
\end{align*}
where $M_\Levy$ is defined in the previous proof. Proceeding as in the proof of Theorem \ref{DDGlin}, we find that
\begin{align}\label{mmm}
\begin{split}
M_{LDG}[\p\mathbf{e},\p\mathbf{e}]=M_{LDG}[\p\mathbf{e}-\mathbf{e},\p\mathbf{e}].
\end{split}
\end{align}
In \cite{Cockburn/Shu} (Lemma 2.4) it is proved that
\begin{align}\label{k1}
M_l(\p \mathbf{e}-\mathbf{e},\p \mathbf{e})=\frac{1}{2}\Theta_T[\p \mathbf{e}] +\frac{1}{2}\int_0^T\int_\R\p e_q^2+\mathcal{O}(1)\Delta x^{2k}.
\end{align}
By \eqref{functionalB}, \eqref{e4}, and Lemma \ref{lemmanonloc},
\begin{align*}
\begin{split}
M_{LDG}[\p \mathbf{e},\p \mathbf{e}]=\int_{0}^{T}\int_{\mathbb{R}}(\p e^2_u)_t +\int_0^T\int_{\R}\p e_q^2+\Theta_T[\p
\mathbf{e}]+\frac{bc_\lambda}{2}\int_0^T|\p e_u|^2_{H^{\lambda/2}(\R)}.
\end{split}
\end{align*}
By inserting this inequality along with \eqref{k1} and \eqref{i3} into \eqref{mmm}, we find that
\begin{align*}
\begin{split}
\int_{0}^{T}\int_{\mathbb{R}}(\p e^2_u)_t&
+\frac{1}{2}\int_0^T\int_{\R}\p e_q^2+\frac{1}{2}\Theta_T[\p\mathbf{e}]\\
&+\frac{bc_\lambda}{4}\int_0^T|\p e_u|^2_{H^{\lambda/2}(\R)} +\frac{bc_\lambda}{4}\int_0^T|e_u|^2_{H^{\lambda/2}(\R)}=\mathcal{O}(1)\Delta x^{2k}.
\end{split}
\end{align*}
The conclusion now follows as in the proof of Theorem \ref{DDGlin}.
\end{proof}
\section{Convergence for nonlinear equations}
In the nonlinear case we will show that the DDG method converges towards an appropriately defined entropy solution of \eqref{1} whenever piecewise constant
elements are used. In what follows we need the functions
\begin{align*}
\eta_k(s)&=|s-k|,\\
\eta'_k(s)&=\text{sgn}(s-k),\\
q_k(s)&=\eta'_k(s)(f(s)-f(k)),\\
r_k(s)&=\eta'_k(s)(A(s)-A(k)).
\end{align*}
Remember that $A(u)=\int^ua$, and let $C^{1,\frac{1}{2}}(Q_T)$ denote the H\"{o}lder space of bounded functions $\phi:Q_T\rightarrow\R$ for which there is a
constant $c_\phi>0$ such that
\begin{align*}
\begin{split}
|\phi(x,t)-\phi(y,\tau)|\leq c_\phi\left[|x-y|+\sqrt{|t-\tau|}\right]\quad \text{for all}\quad (x,t),(y,\tau)\in Q_T.
\end{split}
\end{align*}
We now introduce the entropy formulation for \eqref{1}.
\begin{definition}\label{def:entropy}
A function $u\in L^{\infty}(Q_T)$ is a {\em $BV$ entropy solution} of the initial value problem \eqref{1} provided that the following conditions hold:
\begin{Definitions}
\item $u\in L^1(Q_T)\cap BV(Q_T)$;
\label{def:bv}
\item $A(u)\in C^{1,\frac{1}{2}}(Q_T)$;
\label{def:spacetimereg}
\item for all non-negative test functions $\varphi\in C_c^{\infty}(\R\times[0,T))$ and all $k\in\R$,
\begin{align*}
\begin{split}
\int_{Q_T}\eta_k(u)\varphi_t+q_k(u)\varphi_{x}
+r_{k}(u)\varphi_{xx}+\eta'_k(u)\Levy[u]\varphi\ \dx\dt&\\
+\int_{\R}\eta_k(u_0(x))\varphi(0,x)\dx&\geq 0.
\end{split}
\end{align*}
\label{def:entr}
\end{Definitions}
\end{definition}
This definition is a straightforward combination of the one of Wu and Yin \cite{Wu/Yin} (cf.~also \cite{Evje/Karlsen}) for degenerate convection-diffusion
equations ($b=0$) and the one of Cifani \emph{et al.}~\cite{Cifani/ERJ/KHK} for fractional conservation laws ($a\equiv0$). By the regularity of $\varphi$ and
$u$ and Lemma \ref{lemmanonloc}, each term in the entropy inequality \ref{def:entr} is well defined.
\begin{remark}
The $L^1$-contraction property (uniqueness) for $BV$ entropy solutions follows along the lines of \cite{KHK/Ulusoy}, since the $BV$-regularity of $u$ and the
$L^\infty$-bound on $A(u)_x$ makes it possible to recover from \ref{def:entr} the more precise entropy inequality utilized in \cite{KHK/Ulusoy} for $L^1\cap
L^\infty$ entropy solutions.
\end{remark}
We will now prove, under some additional assumptions, that the explicit DDG method with piecewise constant elements (i.e., $k=0$) converges to the $BV$ entropy
solution of \eqref{1}. In addition to convergence for the numerical method, this also gives the first existence result for entropy solutions of \eqref{1}.
\subsection{The explicit DDG method with piecewise constant elements}
When piecewise constant elements are used ($k=0$ in \eqref{udef}), equation \eqref{Ds2} takes the form
\begin{align*}
\int_{I_{i}}\hat{u}_{t}+\hat{f}(\hat{u}_{i+1})-\hat{f}(\hat{u}_{i})-\hat{h}(\hat{u}_{i+1}) +\hat{h}(\hat{u}_{i})=b\int_{I_{i}}\Levy[\hat{u}].
\end{align*}
Since $\hat{u}(x,t)=\sum_{i\in\mathbb{Z}} U_{i}(t)\mathbf{1}_{i}(x)$ (i.e., $\varphi_{0,i}=\mathbf{1}_i$, the indicator function of the interval $I_i$), we can
and will use the admissible flux $\hat{h}(u_i)=\frac{1}{\Delta x}[A(u_i)]$ (which satisfies \eqref{admiss} with $k=0$ and $\beta_0=1$) to rewrite the above
equation as
\begin{align*}
\Delta x\frac{d}{\dt}U_i+\hat{f}(U_i,U_{i+1})-\hat{f}(U_{i-1},U_{i}) -\frac{[A(U_{i+1})]}{\Delta x}+\frac{[A(U_{i})]}{\Delta
x}=b\sum_{j\in\Z}U_j\int_{I_{i}}\Levy[\mathbf{1}_{I_j}].
\end{align*}
For $\Delta t>0$ we set $t_n=n\Delta t$ for $n=\{0,\ldots,N\}$, $T=t_N$, and $\phi^n_i=\phi(x_i,t_n)$ for any function $\phi$. By a forward difference
approximation in time, we obtain the explicit numerical method
\begin{align}\label{Ds3}
\begin{split}
\frac{U_i^{n+1}-U_i^{n}}{\Delta t}&+\frac{\hat{f}(U_i^n,U_{i+1}^n) -\hat{f}(U_{i-1}^n,U_{i}^n)}{\Delta x}\\&-\frac{A(U^n_{i+1})-A(U^n_{i})}{\Delta
x^2}+\frac{A(U^n_{i})-A(U^n_{i-1})}{\Delta x^2}=\frac{b}{\Delta x}\sum_{j\in\Z}G^i_jU_j^n,
\end{split}
\end{align}
where the weights $G^i_j=\int_{I_{i}}\Levy[\mathbf{1}_{I_j}]\text{ for
all $(i,j)\in\Z\times\Z$.}$ All relevant properties of these
weights are collected in Lemma \ref{matG}. Next we define
\begin{gather*}
D_\pm U_i=\pm\frac{1}{\Delta x}\left(U_{i\pm 1}-U_{i}\right)\quad\text{and}\quad \Levy\langle U^{n}\rangle_{i}=\frac{1}{\Delta
x}\int_{I_{i}}\Levy[\bar{U}^{n}]\dx =\frac{1}{\Delta x}\sum_{j\in\mathbb{Z}}G^i_{j}U_{j}^n,
\end{gather*}
where $\bar{U}^n$ is the piecewise constant interpolant of $U^n$:
$$
\bar{U}^n(x)=U^n_i, \qquad x\in[x_i,x_{i+1}).
$$
The explicit numerical method we study can then be written as
\begin{align}\label{scheme}
\begin{cases}
\frac{U_i^{n+1}-U_i^{n}}{\Delta t}+D_{-} \Big[\hat{f}(U_i^n,U_{i+1}^n)-D_{+}A(U^n_i)\Big]
=b\Levy\langle U^{n}\rangle_{i},\\[0.2cm]
U^0_i=\frac{1}{\Delta x}\int_{I_i}u_0(x)\dx.
\end{cases}
\end{align}
As we will see in what follows, the low-order difference method \eqref{scheme} allows for a complete convergence analysis for general nonlinear equations of
the form \eqref{1}.
Let us now prove that the difference scheme \eqref{scheme} is conservative (\textbf{P}.1), monotone (\textbf{P}.2), and translation invariant (\textbf{P}.3).
\begin{itemize}
\item[(\textbf{P}.1)] Assume $\bar U^n\in L^1(\R)\cap BV(\R)$. By
Lemma \ref{lemmanonloc}
\begin{align}\label{crucial}
\sum_{i\in\mathbb{Z}}\sum_{j\in\mathbb{Z}}|G^i_jU_{j}^n|\leq \int_\R|\Levy[\bar{U}^n(x)]|\dx\leq c_\lambda
C\|\bar{U}^n\|_{L^1(\R)}^{1-\lambda}|\bar{U}^n|_{BV(\R)}^{\lambda},
\end{align}
and hence we can revert the order of summation to obtain
\begin{align*}
\sum_{i\in\mathbb{Z}}\sum_{j\in\mathbb{Z}}G^i_{j}U_{j}^n= \sum_{j\in\mathbb{Z}}U_{j}^n\sum_{i\in\mathbb{Z}}G^i_{j}=0
\end{align*}
since $\sum_{i\in\mathbb{Z}}G^i_{j}=0$ by Lemma \eqref{matG}. By summing over all $i\in\mathbb{Z}$ on each side of \eqref{scheme}, we then find that
\begin{align*}
\begin{split}
\sum_{i\in\Z}U_i^{n+1}=\sum_{i\in\Z}\Big(U_i^{n} +\frac{\Delta t}{\Delta x}\sum_{j\in\Z}G^i_jU_j^n\Big)=\sum_{i\in\Z}U_i^{n}.
\end{split}
\end{align*}
\item[(\textbf{P}.2)]We show that $U^{n+1}_i$ is an increasing function
of all $\{U^{n}_i\}_{i\in\Z}$. First note that
\begin{align*}
\begin{split}
\frac{\partial U^{n+1}_i}{\partial U^{n}_j}\geq0 \quad\text{for} \quad i\neq j,
\end{split}
\end{align*}
since $\hat{f}$ is monotone and $G^i_j\geq0$ for $i\neq j$ by Lemma \ref{matG}. By Lemma \ref{matG} we also see that $G_i^i=-d_\lambda\Delta
x^{1-\lambda}\leq0,$ and hence
\begin{align*}
\begin{split}
\frac{\partial U^{n+1}_i}{\partial U^{n}_i}=&\ 1-\frac{\Delta t}{\Delta
x}\Big[\partial_{u_1}\hat{f}(U_i^n,U_{i+1}^n)-\partial_{u_2}\hat{f}(U_{i-1}^n,U_{i}^n)\Big]\\
&-2\frac{\Delta t}{\Delta x^2}a(U^n_{i})-\frac{\Delta t}{\Delta x^{\lambda}}d_\lambda.
\end{split}
\end{align*}
Here $\partial_{u_i}\hat{f}$ denotes the derivative of $\hat{f}(u_1,u_2)$ w.r.t.~$u_i$ for $i=1,2$. Therefore the following CFL condition makes the explicit
method \eqref{scheme} monotone:
\begin{align}\label{cfl}
\begin{split}
\frac{\Delta t}{\Delta x}\bigg(\|\partial_{u_1}\hat{f}\|_{L^\infty(\R)} +\|\partial_{u_2}\hat{f}\|_{L^\infty(\R)}\bigg) +\frac{2\Delta t}{\Delta
x^2}\|a\|_{L^\infty(\R)} +d_{\lambda}\frac{\Delta t}{\Delta x^\lambda}\leq1.
\end{split}
\end{align}
\item[(\textbf{P}.3)] Translation invariance ($V^0_i=U^0_{i+1}$ implies
$V_i^n=U^n_{i+1}$) is straightforward since \eqref{scheme} does not depend explicitly on a grid point $x_i$.
\end{itemize}
\begin{remark}
For several well known numerical fluxes $\hat f$ (i.e.~Godunov, Engquist-Osher, Lax-Friedrichs, \emph{etc.}), we may replace
$$\|\partial_{u_1}\hat{f}\|_{L^\infty(\R)}
+\|\partial_{u_2}\hat{f}\|_{L^\infty(\R)}$$ in the above CFL condition by the Lipschitz constant of the original flux $f$.
\end{remark}
In the following, we always assume that the CFL condition \eqref{cfl} holds.
\subsection{Further properties of the explicit DDG method \eqref{scheme}}
Define
\begin{align*}
\begin{split}
\|U\|_{L^{1}(\mathbb{Z})}=\sum_{i\in\Z}|U_i|,\quad
\|U\|_{L^{\infty}(\mathbb{Z})}=\sup_{i\in\Z}|U_i|,\quad\text{and}\quad|U|_{BV(\mathbb{Z})}=\sum_{i\in\Z}|U_{i+1}-U_{i}|.
\end{split}
\end{align*}
\begin{lemma}\label{stab}\*
\begin{itemize}
\item[\emph{i)}]$\|U^n\|_{L^{1}(\Z)}\leq\|u_0\|_{L^{1}(\R)}$,
\item[\emph{ii)}]$\|U^n\|_{L^{\infty}(\Z)}\leq\|u_0\|_{L^{\infty}(\R)}$,
\item[\emph{iii)}]$|U^n|_{BV(\Z)}\leq|u_0|_{BV(\R)}$.
\end{itemize}
\end{lemma}
\begin{proof}
Since the numerical method \eqref{scheme} is conservative monotone and translation invariant, the results due to Crandall-Tartar
\cite{Crandall/Tartar,Evje/Karlsen} and Lucier \cite{Lucier,Evje/Karlsen} apply.
\end{proof}
For all $(x,t)\in R^n_i=[x_i,x_{i+1})\times[t_n,t_{n+1})$, let $\hat{u}_{\Delta x}(x,t)$ be the time-space bilinear interpolation of $U^n_i$, i.e.
\begin{align}\label{interp}
\begin{split}
\hat{u}_{\Delta x}(x,t)=U_i^n&+(U_{i+1}^n-U_i^n)\left(\frac{x-i\Delta x}{\Delta x}\right)+(U_i^{n+1}-U_i^n)\left(\frac{t- n\Delta t}{\Delta t}\right)\\
&+(U_{i+1}^{n+1}-U_i^{n+1}-U_{i+1}^n+U_i^n)\left(\frac{x-i\Delta x}{\Delta x}\right)\left(\frac{t-n\Delta t}{\Delta t}\right).
\end{split}
\end{align}
Note that $\hat{u}_{\Delta x}$ is continuous and a.e.~differentiable on $Q_T$. We need the above bilinear interpolation -- rather than a piecewise constant one
-- to prove the H\"{o}lder regularity in \ref{def:spacetimereg}. We will show that the functions $A(\hat{u}_{\Delta x})$ enjoy H\"{o}lder regularity as in
\ref{def:spacetimereg}, and then via an Ascoli-Arzel\`{a} type of argument, so does the limit $A(u)$.
The following lemmas which are needed in the proof of Theorem \ref{th:existence}, are nonlocal generalizations of the ones proved in \cite{Evje/Karlsen}. In
what follows we assume $f\in C^1(\R)$, and note that the general case follows by approximation as in \cite{Evje/Karlsen}.
\begin{lemma}\label{lemmaAP1}
\begin{align}\label{first}
\begin{split}
\bigg\|\hat{f}(U_i^n,U_{i+1}^n)~-~&D_{+}A(U^n_i)-\sum_{k=-\infty}^i\sum_{j\in\Z}G^k_jU_j^n\bigg\|_{L^{\infty}(\Z)}\\
&\leq\bigg\|\hat{f}(U_i^0,U_{i+1}^0)-D_{+}A(U^0_i)-\sum_{k=-\infty}^i\sum_{j\in\Z}G^k_jU_j^0\bigg\|_{L^{\infty}(\Z)},
\end{split}
\end{align}
\begin{align}\label{second}
\begin{split}
\bigg|\hat{f}(U_i^n,U_{i+1}^n)~-~&D_{+}A(U^n_i)-\sum_{k=-\infty}^i\sum_{j\in\Z}G^k_jU_j^n\bigg|_{BV(\Z)}\\
&\leq\bigg|\hat{f}(U_i^0,U_{i+1}^0)-D_{+}A(U^0_i)-\sum_{k=-\infty}^i\sum_{j\in\Z}G^k_jU_j^0\bigg|_{BV(\Z)}.
\end{split}
\end{align}
\end{lemma}
\begin{proof}
\emph{Inequality \eqref{first}.} Let us start by defining $V_i^n=\frac{\Delta x}{\Delta t}\sum_{k=-\infty}^i(U_k^n-U_k^{n-1})$. This sum is finite since
$U^n\in L^1(\Z)$ for all $n\geq0$. If we use \eqref{scheme}, we can write
\begin{align}\label{a10}
\begin{split}
V_i^{n+1}=-\Big[\hat{f}(U_i^n,U_{i+1}^n)-D_{+}A(U^n_i)\Big] +\sum_{k=-\infty}^i\sum_{j\in\Z}G^k_jU_j^n.
\end{split}
\end{align}
Here we have used that $U^n\in L^1(\Z)\cap BV(\Z)$, $f$ and $A$ are Lipschitz continuous, and $f(0)=0$ to conclude that the sum
$\sum_{k=-\infty}^iD_{-}[\hat{f}(U_j^n,U_{j+1}^n) -D_{+}A(U^n_j)]$ is finite and has value $[\hat{f}(U_i^n,U_{i+1}^n)-D_{+}A(U^n_i)]$. Next we rewrite the
right-hand side of \eqref{a10} in terms of $\{V_i^n\}_{i\in\Z}$. By \eqref{a10},
\begin{align}\label{a1}
\begin{split}
V_i^{n+1}&=V^n_i-\Big[\hat{f}(U^n_i,U^n_{i+1})-\hat{f}(U^{n-1}_i,U^{n-1}_{i+1})
-D_+(A(U^n_i)-A(U^{n-1}_i))\Big]\\
&\qquad+\sum_{k=-\infty}^{i}\sum_{j\in\Z}G^k_j(U_j^{n}-U_j^{n-1}).
\end{split}
\end{align}
We prove that
\begin{align}\label{rights}
\begin{split}
\sum_{k=-\infty}^{i}\sum_{j\in\Z}G^k_j(U_j^{n}-U_j^{n-1})= \frac{\Delta t}{\Delta x}\sum_{j\in\Z}G^i_jV_j^{n}.
\end{split}
\end{align}
Indeed, note that $D_-V^n_j=\frac{1}{\Delta t}\left(U_j^{n}-U_j^{n-1}\right)$ and
\begin{align*}
\begin{split}
\sum_{j\in\Z}G^k_jV_{j-1}^{n}=\sum_{j\in\Z}G^{k}_{j+1}V_{j}^{n}=\sum_{j\in\Z}G^{k-1}_jV_{j}^{n}
\end{split}
\end{align*}
since $G^{k}_{j+1}=G^{k-1}_j$. Thus,
\begin{align*}
\begin{split}
\sum_{k=-\infty}^{i}\sum_{j\in\Z}G^k_j(U_j^{n}-U_j^{n-1})&=
\Delta t\sum_{k=-\infty}^{i}\sum_{j\in\Z}G^k_jD_-V^n_j\\
&=\frac{\Delta t}{\Delta x}\sum_{k=-\infty}^{i}\sum_{j\in\Z}G^k_j(V_j^{n}-V_{j-1}^{n})\\
&=\frac{\Delta t}{\Delta x}\sum_{k=-\infty}^{i}
\sum_{j\in\Z}G^k_jV_j^{n}-\frac{\Delta t}{\Delta x}\sum_{k=-\infty}^{i}\sum_{j\in\Z}G^k_jV_{j-1}^{n}\\
&=\frac{\Delta t}{\Delta x}\sum_{k=-\infty}^{i}
\sum_{j\in\Z}G^k_jV_j^{n}-\frac{\Delta t}{\Delta x}\sum_{k=-\infty}^{i}\sum_{j\in\Z}G^{k-1}_jV_{j}^{n}\\
&=\frac{\Delta t}{\Delta x}\sum_{k=-\infty}^{i}
\sum_{j\in\Z}G^k_jV_j^{n}-\frac{\Delta t}{\Delta x}\sum_{k=-\infty}^{i-1}\sum_{j\in\Z}G^{k}_jV_{j}^{n}\\
&=\frac{\Delta t}{\Delta x}\sum_{j\in\Z}G^i_jV_j^{n}.
\end{split}
\end{align*}
Using Taylor expansions, we can replace the nonlinearities $\hat{f}, A$ with linear approximations as follows. We write
\begin{align}\label{r1}
\begin{split}
\hat{f}(U^n_i,U^n_{i+1})-\hat{f}(U^{n-1}_i,U^{n-1}_{i+1}) =\Delta t\hat{f}_{1,i}^nD_-V^n_i+\Delta t\hat{f}_{2,i}^nD_-V^n_{i+1},
\end{split}
\end{align}
where $\hat{f}_{1,i}^n=\partial_{1}\hat{f}(\alpha_i^n,U^n_{i+1})$, $\hat{f}_{2,i}^n =\partial_{2}\hat{f}(U^{n-1}_{i},\tilde{\alpha}_{i+1}^n)$ and
$\alpha_i^n,\tilde{\alpha}_i^n\in(U^{n-1}_i,U^n_i)$. Similarly, we write
\begin{align}\label{r2}
\begin{split}
A(U^n_i)-A(U^{n-1}_i)=a(\beta_i^n)(U^n_i-U^{n-1}_i)=\Delta ta_i^nD_-V^n_i,
\end{split}
\end{align}
where $a_i^n=a(\beta_i^n)$ and $\beta_i^n\in(U^{n-1}_i,U^n_i)$. Inserting \eqref{rights} and \eqref{r1}-\eqref{r2} into expression \eqref{a1} returns
\begin{align}\label{888}
\begin{split}
V_i^{n+1}&=V^n_i-\Delta t(\hat{f}_{1,i}^nD_-V^n_i +\hat{f}_{2,i}^nD_-V^n_{i+1})+\Delta tD_+(a_i^nD_-V_i^n) +\frac{\Delta t}{\Delta x}\sum_{j\in\Z}G^i_jV_j^{n}
\end{split}
\end{align}
or
\begin{align}\label{a2}
\begin{split}
V^{n+1}_i=A^n_iV^n_{i-1}+B^n_iV^n_i+C^n_iV^n_{i+1} +\frac{\Delta t}{\Delta x}\sum_{j\in\Z}G^i_jV_j^{n},
\end{split}
\end{align}
where
\begin{align*}
\begin{split}
A^n_i&=\left[\frac{\Delta t}{\Delta x}\hat{f}_{1,i}^n+\frac{\Delta t}{\Delta x^2}a_i^n\right],\\
B^n_i&=\left[1-\frac{\Delta t}{\Delta x}(\hat{f}_{1,i}^n-\hat{f}_{2,i}^n)
-\frac{\Delta t}{\Delta x^2}(a_i^n+a^n_{i+1})\right],\\
C^n_i&=\left[\frac{\Delta t}{\Delta x^2}a_{i+1}^n-\frac{\Delta t}{\Delta x}\hat{f}_{2,i}^n\right].
\end{split}
\end{align*}
Since $\hat{f}$ is monotone and $a\geq0$, $A^n_i,C^n_i\geq0$. Moreover, $B^n_i+\frac{\Delta t}{\Delta x}G^i_i\geq0$ since the CFL condition \eqref{cfl} holds
true. Thus, since \eqref{a2} is conservative, monotone, and translation invariant (cf.~the proof of Lemma \ref{stab}),
$\|V^{n}\|_{L^\infty(\Z)}\leq\ldots\leq\|V^{1}\|_{L^\infty(\Z)}$, and the conclusion follows from \eqref{a10}.
\emph{Inequality \eqref{second}.} Let us introduce $Z^n_i=V^n_i-V^n_{i-1}$. Note that, since $G^{i-1}_{j-1}=G_j^i$ for all $(i,j)\in\Z\times\Z$,
\begin{align*}
\begin{split}
\sum_{j\in\Z}\left(G^i_jV_j^{n}-G^{i-1}_jV_j^{n}\right)= \sum_{j\in\Z}\left(G^i_jV_j^{n}-G^{i}_jV_{j-1}^{n}\right)=\sum_{j\in\Z}G^i_jZ_j^{n}.
\end{split}
\end{align*}
Thus, \eqref{888} can be rewritten as
\begin{align*}
\begin{split}
Z_i^{n+1}&=Z^n_i-\Delta tD_-(\hat{f}_{1,i}^nZ^n_i +\hat{f}_{2,i}^nZ^n_{i+1})+\Delta t D_-D_+(a_i^nZ_i^n) +\frac{\Delta t}{\Delta x}\sum_{j\in\Z}G^i_jZ_j^{n}
\end{split}
\end{align*}
or
\begin{align}\label{a11}
\begin{split}
Z^{n+1}_i=\bar{A}^n_iZ^n_{i-1}+\bar{B}^n_iZ^n_i+\bar{C}^n_iZ^n_{i+1} +\frac{\Delta t}{\Delta x}\sum_{j\in\Z}G^{i}_jZ_j^{n},
\end{split}
\end{align}
where $\bar{A}^n_i,\bar{B}^n_i,\bar{C}^n_i$ have similar properties as $A^n_i,B^n_i,C^n_i$. Proceeding as in the first part of the proof, \eqref{a11} can be
shown to be conservative, monotone, and translation invariant. Thus $\|Z^{n}\|_{L^1(\Z)}\leq\ldots\leq\|Z^{1}\|_{L^1(\Z)}$, and the conclusion follows from
\eqref{a10}. We refer to \cite{Evje/Karlsen} for the precise details concerning $\bar{A}^n_i,\bar{B}^n_i,\bar{C}^n_i$.
\end{proof}
The next lemma ensures that the numerical solutions are uniformly $L^1$-Lipschitz in time (and hence $BV$ in both space and time by Lemma \ref{stab}).
\begin{lemma}\label{Liptime}
\begin{align*}
\begin{split}
\sum_{i\in\Z}|U_i^m-U_i^n|\leq\Big|\hat{f}(U_i^0,U_{i+1}^0)-D_{+}A(U^0_i) -\sum_{k=-\infty}^{i}\sum_{j\in\Z}G^k_jU^0_j\Big|_{BV(\R)}\frac{\Delta t}{\Delta
x}|m-n|.
\end{split}
\end{align*}
\end{lemma}
\begin{proof}
Let us assume that $m>n$, the case $m<n$ is analogous. Note that
\begin{align*}
\begin{split}
&\sum_{i\in\Z}|U_i^m-U_i^n|\leq\sum_{l=n}^{m-1}\sum_{i\in\Z}|U_i^{l+1}-U_i^{l}|\\
&\leq\Delta t\sum_{l=n}^{m-1}\sum_{i\in\Z}\Bigg|D_{-}\Big[\hat{f}(U_i^l,U_{i+1}^l) -D_{+}A(U^l_i)\Big]-\frac{1}{\Delta x}\sum_{j\in\Z}G^i_jU_j^l\Bigg|.
\end{split}
\end{align*}
Since $D_-\left(\sum_{k=-\infty}^{i}\sum_{j\in\Z}G^k_jU_j\right) =\frac{1}{\Delta x}\sum_{j\in\Z}G^i_jU_j$,
\begin{align*}
\begin{split}
&\sum_{i\in\Z}|U_i^m-U_i^n|\\
&\leq\Delta t\sum_{l=n}^{m-1}\sum_{i\in\Z} \Bigg|D_{-}\Big[\hat{f}(U_i^l,U_{i+1}^l)-D_{+}A(U^l_i) -\sum_{k=-\infty}^{i}\sum_{j\in\Z}G^k_jU^l_j\Big]\Bigg|.
\end{split}
\end{align*}
To conclude, use \eqref{second}.
\end{proof}
We now show that the numerical solutions satisfy a discrete version of \ref{def:spacetimereg}.
\begin{lemma}\label{lemmalip}
If $|\hat{f}(U_i^0,U_{i+1}^0)-D_{+}A(U^0_i) -\sum_{k=-\infty}^i\sum_{j\in\Z}G^k_jU_j^0|_{BV(\Z)}<\infty$, then
\begin{align*}
\begin{split}
|A(U^m_i)-A(U^n_j)|=\mathcal{O}(1) \left[|i-j|\Delta x+\sqrt{|m-n|\Delta t}\right].
\end{split}
\end{align*}
\end{lemma}
\begin{proof}
Let us write
\begin{align*}
\begin{split}
\left|A(U^m_i)-A(U^n_j)\right|\leq\left|A(U^m_i)-A(U^m_j)\right| +\left|A(U^m_j)-A(U^n_j)\right|=I_1+I_2.
\end{split}
\end{align*}
We first estimate the term $I_1$, then the term $I_2$.
\emph{Estimate of $I_1$.} Using \eqref{first}, \eqref{crucial}, Lemma \ref{stab} ii), and the fact that $f$ is Lipschitz continuous,
\begin{align*}
\begin{split}
\left\|D_{+}A(U^m_i)\right\|_{L^{\infty}(\Z)}\leq\ & \bigg\|\hat{f}(U_i^0,U_{i+1}^0)-D_{+}A(U^0_i)
-\sum_{k=-\infty}^i\sum_{j\in\Z}G^k_jU_j^0\bigg\|_{L^{\infty}(\Z)}\\
&+\bigg\|\hat{f}(U_i^m,U_{i+1}^m)\bigg\|_{L^{\infty}(\Z)} +\bigg\|\sum_{k=-\infty}^i\sum_{j\in\Z} G^k_jU_j^m\bigg\|_{L^{\infty}(\Z)}=\mathcal{O}(1).
\end{split}
\end{align*}
Hence $I_1=\mathcal{O}(1)|i-j|\Delta x$.
\emph{Estimate of $I_2$.} Take a test function $\phi\in C^1_c(\R)$, and let $\phi_i=\phi(i\Delta x)$. Let us assume $m>n$ (the case $m<n$ is analogous). Using
\eqref{888} we find that
\begin{align*}
\begin{split}
&\left|\Delta x\sum_{i\in\Z}\phi_i\left(V_i^m-V_i^n\right)\right|
\\ & =\Delta x\left|\sum_{l=n}^{m-1}\sum_{i\in\Z}\phi_i\left(V^{l+1}_i-V^l_i\right)\right|\\
&=\Delta x\sum_{l=n}^{m-1}\sum_{i\in\Z}\phi_i
\Big|(\hat{f}_{1,i}^nD_-V^l_i+\hat{f}_{2,i}^nD_-V^l_{i+1})+D_+(a_i^nD_-V_i^l)\Big|\\
&\qquad\qquad +\Delta t\left|\sum_{l=n}^{m-1} \sum_{i\in\Z}\phi_i\sum_{j\in\Z}G^i_jV_j^{l}\right|=C_1+C_2.
\end{split}
\end{align*}
We use summation by parts to move $D_+$ onto $\phi_i$ and the fact that $\hat{f}_{1,i}^n$ $\hat{f}_{2,i}^n$ $a_i^n$ and $|V^l|_{BV(\Z)}$ are uniformly bounded
to arrive that
\begin{align*}
\begin{split}
C_1=\mathcal{O}(1)\Delta t(m-n)\left(\|\phi\|_{L^\infty(\R)} +\|\phi'\|_{L^\infty(\R)}\right).
\end{split}
\end{align*}
For more details, see \cite{Evje/Karlsen}. Then by \eqref{crucial}, $\sum_{i\in\Z}\sum_{j\in\Z}|G^i_jV_j^{l}|=\mathcal{O}(1)$, and hence
\begin{align}\label{kkk}
\begin{split}
C_2\leq\Delta t\|\phi\|_{L^\infty(\R)}\sum_{l=n}^{m-1} \sum_{i\in\Z}\sum_{j\in\Z}|G^i_jV_j^{l}| =\mathcal{O}(1)\Delta t(m-n)\|\phi\|_{L^\infty(\R)}.
\end{split}
\end{align}
Therefore,
\begin{align}\label{n1}
\begin{split}
\left|\Delta x\sum_{i\in\Z}\phi_i\left(V_i^m-V_i^n\right)\right| =\mathcal{O}(1)\Delta t(m-n)\Big[\|\phi\|_{L^\infty(\R)}+\|\phi'\|_{L^\infty(\R)}\Big].
\end{split}
\end{align}
The above inequality is exactly expression (40) in \cite{Evje/Karlsen}. From now on the proof continues as in \cite{Evje/Karlsen}. Loosely speaking we take an
appropriate sequence of test functions $\phi_\eps\in C^1_c(\R)$ to deduce from \eqref{n1} that
\begin{align*}
\begin{split}
\Delta x\sum_{i\in\Z}|V_i^{m}-V_i^{n}|=\mathcal{O}(1)\sqrt{(m-n)\Delta t}.
\end{split}
\end{align*}
By \eqref{a10}, Lemma \ref{Liptime}, and inequality \eqref{kkk} we also find that
$$\Delta
x\sum_{i\in\Z}|V_i^{m}-V_i^{n}|=\mathcal{O}(1)(m-n)\Delta t+\Delta x\sum_{i\in\Z}|D_+A(U_i^{m})-D_+A(U_i^{n})|,$$ and hence $\Delta
x\sum_{i\in\Z}|D_+A(U_j^{m})-D_+A(U_j^{n})|=\mathcal{O}(1)\sqrt{(m-n)\Delta
t}$. We conclude by noting that
\begin{align*}
\begin{split}
I_2=|A(U^m_j)-A(U^n_j)|&=\Delta x\left|\sum_{i=-\infty}^jD_+A(U_i^m)
-\sum_{i=-\infty}^jD_+A(U_i^n)\right|\\
&\leq\Delta x\sum_{i\in\Z}|D_+A(U_i^{m})-D_+A(U_i^{n})| =\mathcal{O}(1)\sqrt{(m-n)\Delta t}.
\end{split}
\end{align*}
\end{proof}
Next we show that the numerical method \eqref{scheme} satisfies a cell entropy inequality, which is a discrete version of \ref{def:entr}.
\begin{lemma}
Let $k\in\R$ and $\eta_i^n=|U^n_i-k|$. Then
\begin{align}\label{entropcell}
\begin{split}
\eta_i^{n+1}-\eta_i^{n}+\Delta tD_{-}Q_i^n-\Delta tD_{-}D_{+}|A(U^n_i)-A(k)| &\leq\Delta t\eta'_k(U^{n+1}_i)\Levy\langle U^n\rangle_i,
\end{split}
\end{align}
where $Q_i^n=\hat{f}(U_i^n\vee k,U_{i+1}^n\vee k) -\hat{f}(U_i^n\wedge k,U_{i+1}^n\wedge k)$.
\end{lemma}
\begin{proof}
Let us introduce the notation $a\wedge b=\min\{a,b\}$ and $a\vee b=\max\{a,b\}$. Note that $\eta_i^n=(U_i^{n}\vee k)-(U_i^{n}\wedge k)$. Since the numerical
method \eqref{scheme} is monotone,
\begin{align*}
\begin{split}
&\frac{(U_i^{n+1}\vee k)-(U_i^{n}\vee k)}{\Delta t} +\frac{\hat{f}(U_i^n\vee k,U_{i+1}^n\vee k)-\hat{f}(U_{i-1}^n\vee k,U_{i}^n\vee k)}{\Delta
x}\\&-\frac{A(U^n_{i+1}\vee k)-A(U^n_{i}\vee k)}{\Delta x^2} +\frac{A(U^n_{i}\vee k)-A(U^n_{i-1}\vee k)}{\Delta x^2}\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\leq\Delta t\mathbf{1}_{(k,+\infty)}(U_{i}^{n+1})\Levy\langle U^{n}\rangle_{i}
\end{split}
\end{align*}
and
\begin{align*}
\begin{split} &\frac{(U_i^{n+1}\wedge k)-(U_i^{n}\wedge k)}{\Delta t} +\frac{\hat{f}(U_i^n\wedge k,U_{i+1}^n\wedge k) -\hat{f}(U_{i-1}^n\wedge
k,U_{i}^n\wedge k)}{\Delta
x}\\&-\frac{A(U^n_{i+1}\wedge k)-A(U^n_{i}\wedge k)}{\Delta x^2} +\frac{A(U^n_{i}\wedge k)-A(U^n_{i-1}\wedge k)}{\Delta x^2}\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\geq\Delta t\mathbf{1}_{(-\infty,k)}(U_{i}^{n+1})\Levy\langle U^{n}\rangle_{i}.
\end{split}
\end{align*}
To conclude, subtract the above inequalities.
\end{proof}
\subsection{Convergence of the DDG method}
We are now in position to prove convergence of the fully explicit numerical method \eqref{scheme} to a $BV$ entropy solution of \eqref{1}. Let us introduce
$\mathcal{B}$ (cf.~\cite{Evje/Karlsen}), the space of all functions $z:\R\rightarrow\R$ such that
\begin{align*}
\begin{split}
\left|f(z)-\partial_xA(z)-\int^x\Levy[z]\right|_{BV(\R)}<\infty.
\end{split}
\end{align*}
In the following theorem we choose the initial datum to be in $L^1(\R)\cap BV(\R)\cap \mathcal{B}$, which is done to make sense to the right-hand side of
\eqref{second}. Note that whenever $z\in L^1(\R)\cap BV(\R)$, $\Levy[z]\in L^1(\R)$ by Lemma \ref{lemmanonloc}, and hence
$$
\left|\int^x\Levy[z]\right|_{BV(\R)} =\left\|\frac{d}{\dx}\int^x\Levy[z]\right\|_{L^1(\R)} =\|\Levy[z]\|_{L^1(\R)}<\infty.$$
\begin{theorem}[Convergence for DDG]\label{th:existence}
Suppose $u_0\in L^1(\R)\cap BV(\R)\cap \mathcal{B}$, and let $\hat{u}_{\Delta x}$ be the interpolant \eqref{interp} of the solution of the explicit DGG scheme
\eqref{scheme}. Then there is a subsequence of $\{\hat{u}_{\Delta x}\}$ and a function $u\in L^1(Q_T)\cap BV(Q_T)$ such that (a) $\hat{u}_{\Delta x}\to u$ in
$L_{loc}^1(Q_T)$ as $\Delta x\ra0$; (b) $u$ is a $BV$ entropy solution of \eqref{1}.
\end{theorem}
\begin{corollary}[Existence]
If $u_0\in L^1(\R)\cap BV(\R)\cap \mathcal{B}$, then there exists a BV entropy solution of \eqref{1}.
\end{corollary}
\begin{proof}[Proof of Theorem \ref{th:existence}]
We will prove strong $L_{loc}^1$ compactness, and hence we need the following estimates uniformly in $\Delta x>0$:
\begin{itemize}
\item[\emph{i)}]$\|\hat{u}_{\Delta x}\|_{L^\infty(Q_T)}\leq C,$
\item[\emph{ii)}]$\|\hat{u}_{\Delta x}\|_{BV(Q_T)}\leq C.$
\end{itemize}
Estimate $\emph{i)}$ is a consequence of Lemma \ref{stab} and \eqref{interp}, while estimate $\emph{ii)}$ comes from the following computations
(cf.~\cite{Evje/Karlsen} for more details). Using the interpolation \eqref{interp}, we find that
\begin{align*}
\begin{split}
\int_{Q_T}|\hat{u}_x| &\leq\frac{\Delta t}{2}\sum_{n=0}^{N-1}\sum_{i\in\Z}\left|U^n_{i+1} -U^n_{i}\right|+\frac{\Delta t}{2}\sum_{n=0}^{N-1}
\sum_{i\in\Z}\left|U^{n+1}_{i+1}-U^{n+1}_{i}\right|\\
&\leq T|U^0|_{BV(\Z)}.
\end{split}
\end{align*}
Note that Lemma \ref{stab} has been used in the second inequality. Similarly,
\begin{align*}
\begin{split}
\int_{Q_T}|\hat{u}_t| &\leq\frac{\Delta x}{2}\sum_{n=0}^{N-1} \sum_{i\in\Z}\left|U^{n+1}_{i}-U^n_{i}\right|+\frac{\Delta x}{2}
\sum_{n=0}^{N-1}\sum_{i\in\Z}\left|U^{n+1}_{i+1}-U^{n}_{i+1}\right|\\
&\leq T\Big|\hat{f}(U_i^0,U_{i+1}^0)-D_{+}A(U^0_i) -\sum_{h=-\infty}^{i}\sum_{j\in\Z}G^h_jU^0_j\Big|_{BV(\Z)},
\end{split}
\end{align*}
where Lemma \ref{Liptime} has been used in the second inequality. Hence, there exists a sequence $\{\hat{u}_{\Delta x_i}\}_{i\in\N}$ which converges in
$L^1_{loc}(Q_T)$ to a limit
$$
u\in L^1(Q_T)\cap BV(Q_T).
$$
Next we check that the limit $u$ satisfies \ref{def:spacetimereg}. We define $w_{\Delta x}=A(\hat{u}_{\Delta x})$. Note that $A(\hat{u}_{\Delta x})\rightarrow
A(u)$ a.e. since $\hat{u}_{\Delta x}\rightarrow u$ a.e. (up to a subsequence) and $A$ is continuous.
Now choose $(x,t),(y,\tau),(j,n),(i,m)$ such that
$(x,t)\in R_j^n$ and $(y,\tau)\in R_i^m$ for $R^n_i=[x_i,x_{i+1})\times[t_n,t_{n+1})$. Then,
\begin{align*}
\begin{split}
|w_{\Delta x}(y,\tau)-w_{\Delta x}(x,t)|&\leq|w_{\Delta x}(y,\tau)-w_{\Delta x}(i\Delta x,m\Delta t)|\\
&\ +|w_{\Delta x}(i\Delta x,m\Delta t)-w_{\Delta x}(j\Delta x,n\Delta t)|\\
&\ +|w_{\Delta x}(j\Delta x,n\Delta t)-w_{\Delta x}(x,t)|\\
&=I_1+I_2+I_3.
\end{split}
\end{align*}
Note that by Lemma \ref{lemmalip}, $I_2=\mathcal{O}(1)(|i-j|\Delta x+\sqrt{|m-n|\Delta t})$, while by Lemma \ref{lemmalip} again, \eqref{interp}, and $A'=a\in
L^\infty$, $I_1+I_3=\mathcal{O}(1)(\Delta x+\sqrt{\Delta t})$. Thus
\begin{align*}
|w_{\Delta x}(y,\tau)-w_{\Delta x}(x,t)|=\mathcal{O}(1)\left[|y-x|+\sqrt{|\tau-t|}+\Delta x+\sqrt{\Delta t}\right].
\end{align*}
We also have that $w_{\Delta x}=A(\hat{u}_{\Delta x})$ is uniformly bounded since $A$ is Lipschitz and $\hat u_{\Delta x}$ is uniformly bounded. By
essentially repeating the proof of the Ascoli-Arzel\`{a} compactness theorem, we can now deduce the existence of a subsequence $\{w_{\Delta x}\}$ converging
locally uniformly towards the limit $A(u)$. By the estimates on $w_{\Delta x}$, it then follows that
\begin{align}\label{kkjj}
A(u)\in C^{1,\frac{1}{2}}(Q_T).
\end{align}
Finally, let us check that the limit $u$ satisfies \ref{def:entr} in Definition \ref{def:entropy}. Here we need to introduce a piecewise constant inteporlation
of our data points $U^n_i$. We call
\begin{align*}
\bar{u}_{\Delta x}(x,t)=U_{i}^{n}\text{ for all $(x,t)\in[x_i,x_{i+1})\times[t_n,t_{n+1})$}.
\end{align*}
We do this since the discontinuous sign function $\eta_k'$ makes it difficult to work with the bilinear interpolant $\hat u_{\Delta x}$ in what follows. The
need for the piecewise linear interpolation was dictated by the condition \ref{def:spacetimereg}: continuity of the functions $A(\hat{u}_{\Delta x})$ were
needed to prove H\"{o}lder space-time regularity for the limit $A(u)$ (cf.~the proof of \eqref{kkjj}). To verify that the limit $u$ also satisfies
\ref{def:entr} the piecewise constant interpolation $\bar{u}_{\Delta x}$ suffices since, as we already have strong convergence for the piecewise linear
interpolation, strong convergence toward the same limit $u$ for the piecewise constant interpolation is ensured thanks to the fact that
$$\|\bar{u}_{\Delta x}(\cdot,t)- \hat{u}_{\Delta x}(\cdot,t)\|_{L^1(Q_T)}\leq c|U^n|_{BV(\Z)}\Delta x.$$ We now take a positive test function $\varphi\in
C_c^{\infty}(\R\times[0,T))$, and let $\varphi_i^n=\varphi(x_i,t_n)$. We multiply both sides of \eqref{entropcell} by $\varphi_i^n$, and sum over all $(i,n)$.
Using summation by parts, we obtain
\begin{align}\label{ex}
\begin{split}
\Delta x\Delta t\sum_{n=0}^{N-1}&\sum_{i\in\mathbb{Z}}\eta_{i}^{n}\frac{\varphi_{i}^{n+1}-\varphi_{i}^{n}}{\Delta t}\\
&+\Delta x\Delta t\sum_{n=0}^{N-1}\sum_{i\in\mathbb{Z}}Q_{i}^{n}D_{+}\varphi_i^n\\
&+\Delta x\Delta t\sum_{n=0}^{N-1}\sum_{i\in\mathbb{Z}}|A(U^n_i)-A(k)|D_{-}(D_{+}\varphi_i^n)\\
&+\Delta x\Delta t\sum_{n=0}^{N-1}\sum_{i\in\mathbb{Z}}\eta'_k(U^{n+1}_i)\Levy\langle U^{n}\rangle_i\varphi_{i}^{n}\\
&+\Delta x\sum_{i\in\mathbb{Z}}\varphi_{i}^{0}\eta_{i}^{0}\geq0.
\end{split}
\end{align}
A standard argument shows that all the local terms in the above expression converge to the ones appearing in the entropy inequality \ref{def:entr}, see
e.g.~\cite{Holden/Risebro,Evje/Karlsen}. Let us look the term containing the nonlocal operator $\Levy\langle\cdot\rangle$. We can rewrite it as
\begin{align*}
\begin{split}
\int_{\Delta t}^{T+\Delta t}\int_{\mathbb R}\eta'_{k}(\bar{u}_{\Delta x}) \Levy[\bar{u}_{\Delta x}]\bar{\varphi}\dx\dt+R,
\end{split}
\end{align*}
where $R\stackrel{\Delta x\rightarrow0}{\longrightarrow}0$ and $\bar\varphi$ is the piecewise constant interpolant of $\varphi_i^n$. Indeed, let us write
$\eta_k'(U^{n+1}_i)\Levy\langle U^{n}\rangle_i=\eta_k'(U^{n+1}_i)\Levy\langle U^n-U^{n+1}\rangle_i +\eta_k'(U^{n+1}_i)\Levy\langle U^{n+1}\rangle_i$. Note that
\begin{align*}
\begin{split}
\Delta x\Delta t\sum_{n=0}^{N-1}\sum_{i\in\mathbb{Z}} |\Levy\langle U^n-U^{n+1}\rangle_i|\varphi_{i}^{n}\leq\Delta
t\|\bar{\varphi}\|_{L^\infty(Q_T)}\sum_{n=0}^{N-1}\int_{\R}|\Levy[\bar{U}^n(x)-\bar{U}^{n+1}(x)]|\dx,
\end{split}
\end{align*}
where the last quantity vanishes as $\Delta x\rightarrow0$ by $L^1$-Lipschitz continuity in time (cf. Lemma \ref{Liptime}, and also Lemmas \ref{stab} and
\ref{lemmanonloc}). Next,
\begin{align*}
\begin{split}
&\sum_{n=0}^{N}\sum_{i\in\mathbb{Z}}\eta'_{k}(U^{n+1}_i)
\Levy\langle{U}^{n+1}\rangle_i\varphi_i^{n}\\
&=\sum_{n=0}^{N}\sum_{i\in\mathbb{Z}}\eta'_{k}(U^{n+1}_i) \Levy\langle{U}^{n+1}\rangle_i(\varphi_i^{n}-\varphi_i^{n+1})
+\sum_{n=0}^{N}\sum_{i\in\mathbb{Z}}\eta'_{k}(U^{n+1}_i) \Levy\langle{U}^{n+1}\rangle_i\varphi_i^{n+1},
\end{split}
\end{align*}
where the first term on the right-hand side vanishes as $\Delta x\rightarrow0$ since there exists a constant $c_{\varphi}>0$ such that
$|\varphi_i^{n}-\varphi_i^{n+1}|\leq c_{\varphi}\Delta x$ for all $(i,n)$. To conclude, we prove that up to a subsequence and for a.e.~$k\in\mathbb{R}$,
\begin{align}\label{hhh}
\begin{split}
\int_{\Delta t}^{T+\Delta t}\int_{\mathbb R}\eta'_{k}(\bar{u}_{\Delta x}) \Levy[\bar{u}_{\Delta x}]\bar{\varphi}\dx\dt\stackrel{\Delta x\rightarrow
0}{\longrightarrow}\int_{Q_T}\eta'_k(u)\Levy[u]\varphi \dx\dt.
\end{split}
\end{align}
This is a consequence of the dominated convergence theorem since the left hand side integrand converges pointwise a.e.~to the right hand side integrand.
Indeed, first note that $\bar{\varphi}\rightarrow\varphi$ pointwise on $Q_T$, while a.e.~up to a subsequence, $\bar{u}_{\Delta x}\rightarrow u$ on $Q_T$. We
also have $\eta_k'(\bar{u}_{\Delta x})\rightarrow\eta'_k(u)$ a.e.~in $Q_T$ since for a.e.~$k\in\mathbb{R}$ the measure of $\{(x,t)\in Q_T:u(x,t)=k\}$ is zero
and $\eta'_k$ is continuous on $\mathbb{R}\backslash \{k\}$. Finally if the (compact) support of $\varphi$ is containd in $[-R,R]\times[0,T]$, $R>0$, then a
trivial extension of Lemma \ref{lemmanonloc} implies that
\begin{align*}
\begin{split}
\int_{[-R,R]\times[0,T]}|\Levy[\bar{u}_{\Delta x}-u]|\dx\dt\leq c_\lambda C\int_{0}^{T}\|\bar{u}_{\Delta x}-u\|_{L^1(-R,R)}^{1-\lambda} |\bar{u}_{\Delta
x}-u|_{BV(-R,R)}^{\lambda},
\end{split}
\end{align*}
where the last quantity vanishes as $\Delta x\rightarrow0$ since $\bar{u}_{\Delta x}\rightarrow u$ in $L^1_{loc}(Q_T)$. Then $\Levy[\bar{u}_{\Delta
x}]\rightarrow \Levy[u]$ a.e.~in $[-R,R]\times(0,T)$ up to a
subsequence. Convergence for all $k\in\mathbb{R}$ can be proved along the lines of \cite[Lemmas 4.3 and 4.4]{Karlsen:2003lz}.
\end{proof}
\subsection{Remarks on the LDG method}\label{sec:remarksldg}
The derivation of the LDG method in the piecewise constant case is not as straightforward as the one for the DDG method. Indeed, the numerical fluxes
introduced in \eqref{Lflux} depend on the choice of the function $c_{12}$, and computations cannot be performed until this function has been defined. Our aim
now is to show that the LDG method reduces to a numerical method similar to \eqref{Ds3} for a suitable choice of the function $c_{12}$.
Let us for the time being ignore the nonlinear convection and
fractional diffusion terms and focus on the problem
\begin{align*}
\left\{
\begin{array}{ll}
u_t-\partial_x\sqrt{a(u)}q=0,\\
q-\partial_xg(u)=0,\\
u(x,0)=u_{0}(x).
\end{array}
\right.
\end{align*}
The LDG method \eqref{systdisc} then takes the form
\begin{align}\label{systpwc}
\begin{cases}
\int_{I_{i}}\tilde{u}_t+\hat{h}_u(\tilde{\mathbf{w}}_{i+1})
-\hat{h}_u(\tilde{\mathbf{w}}_{i})=0,\\[0.2cm]
\int_{I_{i}}\tilde{q}+\hat{h}_q(\tilde{u}_{i+1})-\hat{h}_q(\tilde{u}_{i})=0,
\end{cases}
\end{align}
where $\tilde{u}(x,t)=\sum_{i\in\mathbb{Z}}U_{i}(t)\mathbf{1}_{I_i}(x)$, $\tilde{q}(x,t)=\sum_{i\in\mathbb{Z}}Q_{i}(t)\mathbf{1}_{I_i}(x)$, and the fluxes
$(\hat h_u,\hat h_q)$ are defined in \eqref{Lflux}. Let us insert $\tilde{u}$ and $\tilde{q}$ into the system \eqref{systpwc}, and use the flux \eqref{Lflux}
to get
\begin{align}\label{ggg}
\begin{cases}
\frac{d}{dt}U_{i}\Delta x-\frac{g(U_{i+1})-g(U_{i})}{U_{i+1}
-U_{i}}\frac{Q_{i+1}+Q_{i}}{2}-c_{12}(Q_{i+1}-Q_{i})\\[0.2cm]
\qquad\qquad+\frac{g(U_{i})-g(U_{i-1})}{U_{i}-U_{i-1}}\frac{Q_{i}
+Q_{i-1}}{2}+c_{12}(Q_{i}-Q_{i-1})=0,\\[0.2cm]
Q_{i}\Delta x-\frac{g(U_{i+1})+g(U_{i})}{2}+c_{12}(U_{i+1}-U_{i})\\[0.2cm]
\qquad\qquad+\frac{g(U_{i})+g(U_{i-1})}{2}-c_{12}(U_{i}-U_{i-1})=0.
\end{cases}
\end{align}
Let us choose the function $c_{12}$ to be
\begin{align}\label{c12}
c_{12}(U_{i},U_{i-1})=\frac{1}{2}\frac{g(U_{i})-g(U_{i-1})}{U_{i}-U_{i-1}}.
\end{align}
Inserting \eqref{c12} into \eqref{ggg} then leads to
\begin{align*}
\begin{cases}
\frac{d}{dt}U_{i}\Delta x-\frac{g(U_{i+1})-g(U_{i})}{U_{i+1}-U_{i}}Q_{i+1}
+\frac{g(U_{i})-g(U_{i-1})}{U_{i}-U_{i-1}}Q_{i}=0,\\[0.2cm]
Q_{i}=\frac{g(U_{i})-g(U_{i-1})}{\Delta x},
\end{cases}
\end{align*}
or
\begin{align*}
\frac{d}{dt}U_{i}\Delta x-\frac{1}{\Delta x}\frac{(g(U_{i+1})-g(U_{i}))^2}{U_{i+1}-U_{i}} +\frac{1}{\Delta x}\frac{(g(U_{i})-g(U_{i-1}))^2}{U_{i}-U_{i-1}}=0.
\end{align*}
For the full equation \eqref{1}, this choice of $c_{12}$ along with a forward difference approximation in time, lead to the following piecewise constant LDG
approximation:
\begin{align}\label{Ls1}
\begin{split}
&\frac{U_i^{n+1}-U_i^{n}}{\Delta t}+\frac{\hat{f}(U_i^n,U_{i+1}^n) -\hat{f}(U_{i-1}^n,U_{i}^n)}{\Delta x}\\&-\frac{1}{\Delta
x^2}\frac{(g(U^n_{i+1})-g(U^n_{i}))^2}{U_{i+1}^n-U_{i}^n} +\frac{1}{\Delta x^2}\frac{(g(U^n_{i})-g(U^n_{i-1}))^2}{U_{i}^n-U_{i-1}^n} =\frac{1}{\Delta
x}\sum_{j\in\Z}G^i_jU_j^n.
\end{split}
\end{align}
\begin{remark}
We do not prove convergence for the numerical method \eqref{Ls1}. However, we note that
\begin{align*}
\begin{split}
\left(\frac{dg}{du}\right)^2=\frac{dA}{du},
\end{split}
\end{align*}
since $g=\int^{u}\sqrt{a}$ and $A=\int^{u}a$. Roughly speaking this means that
\begin{align*}
\begin{split}
\frac{(g(U^n_{i+1})-g(U^n_{i}))^2}{U_{i+1}-U_{i}}\approx A(U^n_{i+1})-A(U^n_{i}),
\end{split}
\end{align*}
and hence that \eqref{Ds3} and \eqref{Ls1} are closely related. Experiments indicates that the two methods produce similar solutions (cf.~Figure
\ref{figure4}).
\end{remark}
\section{Numerical experiments}
We conclude this paper by presenting some experimental results obtained using the fully explicit (piecewise constant) numerical methods \eqref{scheme} and
\eqref{Ls1}, and the DDG method \eqref{Ds2} with fully explicit third order Runge-Kutta time discretization and piecewise constant, linear, and quadratic
elements. In the computations we have imposed a zero Dirichlet boundary condition on the whole exterior domain $\{|x|>1\}$. In all the plots, the dotted line
represents the initial datum while the solid one (or the dashed-dotted one in Figure \ref{figure4}) the numerical solution at $t=T$.
\begin{remark}
The operator $\Levy[\hat u]$ requires the evaluation of the discrete solution $\hat u$ on the whole real axis, thus making necessary the use of some
localization procedure. In our numerical experiments we have confined the nonlocal operator $\Levy[\cdot]$ to the domain $\Omega=\{|x|\leq 1\}$. That is to
say, for each grid point $(x_i,t_n)\in\Omega\times (0,T)$ we have computed the value of $\hat u$ at time $t_{n+1}$ by using only the values $\hat u(x_i,t_n)$
with $x_i\in\Omega$.
\end{remark}
We consider two different sets of data taken from \cite{Evje/Karlsen}. In Example 1 we take
\begin{align}\label{ex1}\tag{\textbf{Ex}.1}
\begin{split}
f_1(u)&=u^{2},\\
a_1(u)&=
\begin{cases}
0&\text{for} \ u\leq0.5\\
2.5u-1.25&\text{for} \ 0.5<u\leq0.6\\
0.25&\text{for} \ u>0.6,
\end{cases}\\
u_{0,1}(x)&=
\begin{cases}
0&\text{for} \ x\leq-0.5\\
5x+2.5&\text{for} \ -0.5<x\leq-0.3\\
1&\text{for} \ -0.3<x\leq0.3\\
2.5-5x&\text{for} \ 0.3<x\leq0.5\\
0&\text{for} \ x>0.5.
\end{cases}
\end{split}
\end{align}
In Example 2 we choose
\begin{align}\label{ex2}\tag{\textbf{Ex}.2}
\begin{split}
f_2&=\frac{1}{4}\,f_1,\\
a_2&=4\,a_1,\\
u_{0,2}(x)&=
\begin{cases}
1&\text{for} \ x\leq-0.4\\
-2.5x&\text{for} \ -0.4<x\leq0\\
0&\text{for} \ x>0.
\end{cases}
\end{split}
\end{align}
Furthermore, in Example 3 we use
\begin{align}\label{ex3}\tag{\textbf{Ex}.3}
\begin{split}
f_3(u)&=u,\\
a_3(u)&=0.1,\\
u_{0,3}(x)&=\exp{\left(-\left(\frac{x}{0.1}\right)^2\right)}.
\end{split}
\end{align}
The numerical results are presented in Figure \ref{fig1}, \ref{fig2}, \ref{figure4}, and \ref{fig_add}. The results confirm what we expected: the solutions of
the initial value problem \eqref{1} can develop shocks in finite time (this feature has been proved in \cite{Alibaud/Droniou/Vovelle} for the case $a=0$). In
Figure \ref{fig1} and \ref{fig2} you can see how the presence of the fractional diffusion $\Levy$ influences the shock's size and speed. In Figure
\ref{fig_add} you can see how the accuracy of DDG method \eqref{Ds2} improves when high-order polynomials are used ($k=0,1,2$).
\begin{figure}[h]
\subfigure[$u_t+f(u)_x=(a(u)u_x)_x$]{
\includegraphics[scale=0.4]{ffig2}}
\subfigure[Equation \eqref{1} with $\lambda=0.5$]{
\includegraphics[scale=0.4]{ffig3}}
\caption{(\ref{ex1}): $T=0.15$ and $\Delta x=1/640$.}\label{fig1}
\end{figure}
\begin{figure}[h]
\subfigure[$u_t+f(u)_x=(a(u)u_x)_x$]{
\includegraphics[scale=0.4]{qqqfig2}}
\subfigure[Equation \eqref{1} with $\lambda=0.5$]{
\includegraphics[scale=0.4]{qqqfig3}}
\caption{(\ref{ex2}): $T=0.25$ and $\Delta x=1/640$.}\label{fig2}
\end{figure}
\begin{figure}[h]
\subfigure[$T=0.0625$]{
\includegraphics[scale=0.4]{fig1}}
\subfigure[$T=1$]{
\includegraphics[scale=0.4]{fig3}}
\caption{(\ref{ex2}): solutions of $u_t+f(u)_x=(a(u)u_x)_x$ at different times using methods \eqref{scheme} and \eqref{Ls1} ($\Delta x=1/160$).}\label{figure4}
\end{figure}
\begin{figure}[ht]
\subfigure[Piecewise constant ($k=0$) with $\Delta x=1/20$]{
\includegraphics[scale=0.4]{constant}}
\subfigure[Piecewise linear ($k=1$) with $\Delta x=1/20$]{
\includegraphics[scale=0.4]{linear}}\\
\subfigure[Piecewise quadratic ($k=2$) with $\Delta x=1/20$]{
\includegraphics[scale=0.4]{quadrat}}
\subfigure[Solution computed using $\Delta x=1/640$]{
\includegraphics[scale=0.4]{exact}}
\caption{(\ref{ex3}): solutions at $T=0.1$ using $k=0,1,2$.}\label{fig_add}
\end{figure}
In Figure \ref{figure4}, the dashed-dotted curve represents method \eqref{scheme}, while the solid one represents method \eqref{Ls1}. The two numerical
solutions stay close, and numerical convergence has been observed for finer grids. Note that here we have set $b=0$ (no fractional diffusion) in order to
stress the differences between the two methods.
The numerical rate of convergence for the solutions in Figure \ref{fig1} (b), \ref{fig2} (b), and \ref{fig_add} (b) are presented in Table \ref{table1}. We
have measured the $L^p$-error
$$E_{\Delta x,p}=\|\hat{u}_{\Delta
x}(\cdot,T)-\hat{u}_{e}(\cdot,T)\|^p_{L^{p}(\mathbb{R})},$$
where $\hat{u}_{e}$ is the numerical solution which has been computed using a very fine grid ($\Delta x=1/640$), the relative error
$$R_{\Delta
x,p}=\left(\frac{1}{\|\hat{u}_{e}(\cdot,T)\|^p_{L^{p}(\mathbb{R})}}\right)\,E_{\Delta x,p},$$ and the approximate rate of convergence $$\alpha_{\Delta
x,p}=\left(\frac{1}{\log 2}\right)\Big(\log E_{\Delta x,p}-\log E_{\Delta x/2,p}\Big).$$
\begin{table}[h]
\caption{Error, relative error, and numerical rate of convergence for
the solutions in Figure \ref{fig1} (b), \ref{fig2} (b), and \ref{fig_add} (b).}\label{table1}
\begin{tabular}{c| c c c | c c c|c c c }
&\multicolumn{3}{c|}{Figure \ref{fig1} (b)}&\multicolumn{3}{c}{Figure
\ref{fig2} (b)} & \multicolumn{3}{c}{Figure \ref{fig_add} (b)}\\
$\Delta x$ & $E_{\Delta x,1}$ & $R_{\Delta x,1}$ & $\alpha_{\Delta x,1}$ & $E_{\Delta x,1}$ & $R_{\Delta x,1}$ & $\alpha_{\Delta x,1}$& $E_{\Delta x,2}$ &
$R_{\Delta x,2}$ & $\alpha_{\Delta x,2}$
\\
\hline
$\mathbf{1/10}$ & 0.0706 & 0.0942 & 0.97 & 0.0474 & 0.0550 & 0.86& 0.009000 & 0.093595 & 2.00 \\
$\mathbf{1/20}$ & 0.0361 & 0.0482 & 0.92 & 0.0261 & 0.0302 & 0.49& 0.002300 & 0.023493 & 1.85 \\
$\mathbf{1/40}$ & 0.0191 & 0.0255 & 0.57 & 0.0186 & 0.0216 & 0.52 & 0.000626 & 0.006518 & 1.54 \\
$\mathbf{1/80}$ & 0.0128 & 0.0171 & 0.60 & 0.0130 & 0.0150 & 0.42 & 0.000216 & 0.002248 & 1.10 \\
$\mathbf{1/160}$ & 0.0084 & 0.0113 & 0.76 & 0.0097 & 0.0112 & 0.77 & 0.000101 & 0.001052 & 1.04 \\
$\mathbf{1/320}$ & 0.0050 & 0.0066 & - & 0.0057 & 0.0066 & - & 0.000049 & 0.000510 & - \\
\hline
\end{tabular}
\end{table}
Our simulations seem to indicate numerical convergence of order less than one for the solutions depicted in Figure \ref{fig1} (b) and \ref{fig2} (b) (nonlinear
equations and piecewise constant elements), and numerical convergence of order higher than one for the solution depicted in Figure \ref{fig_add} (b) (linear
equation and piecewise linear elements). In the last case we do not seem to reach the expected value 2 (cf.~the statement of Theorem \ref{DDGlin}). This
deterioration of the numerical order of convergence for high-order polynomials has already been observed by the authors in \cite{Cifani/ERJ/KHK}. The reasons
behind this deterioration are still not clear.
Finally, let us remind the reader that no general results concerning the rate of convergence of numerical methods for nonlinear equations like \eqref{1} have
been produced so far. For more details, cf.~\cite{Chen/Karlsen}.
|
1,116,691,499,284 | arxiv | \section{Introduction}
Our understanding of percolation properties of networks has expanded significantly in recent years. Percolation theory, a classical model in statistical physics, has been applied in a number of different network science topics, such as network structure~\cite{PhysRevE.80.036105,PhysRevLett.96.040601}, network robustness~\cite{PhysRevE.64.026118,PhysRevLett.85.5468,PhysRevLett.85.4626}, node ranking \cite{Ji2017Effective}, community detection \cite{palla2005uncovering}, as well as in studies of network dynamics, such as information spreading \cite{PhysRevE.66.016128}, and highway traffic flows \cite{Li20012015}.
Although we often assume that the underlying structure of a network is complex, the rules for the percolation process are comparatively simple. Typically, each link or node is occupied with a given probability $p$, independent of the states of other links and nodes. By contrast, real-world network processes are often ``history-dependent'': for example, the spread of a particular disease can depend on the spread of other diseases \cite{Newman2013} and it can be also influenced by the availability of immunization information~\cite{Sneppen2010}. Network topology itself can be affected by cascading failures \cite{Buldyrev2010,Gao2012} or by recovery processes \cite{majdandzic2014spontaneous} in other networks. The universality class of a percolation transition depends on the quenched disorder topology induced by the previous percolation transition \cite{PhysRevE.92.010103}.
A second complication is due to the presence of multiple interaction channels that are often involved in history-dependent processes \cite{Boccaletti20141,Gao2014}. Such systems can be naturally described in terms of multiplex networks where nodes are connected through different types of links\cite{bianconi2018multilayer}, such as the social networks with different types of interactions which can be either online or offline \cite{Szell2010PNAS,PhysRevLett.111.128701}, the multilayer transportation network with various means of vehicles \cite{Morris2012Transport}, and the brain network with both functional correlation and morphological similarity \cite{Bullmore2009Complex}. One of the typical examples of iterative interactions on multiplex networks is the interplay between the spreading of an epidemic and the information awareness that prevents its further spreading \cite{PhysRevLett.111.128701}. Another example is that of cascading failures on coupled networks of power distribution and communications \cite{Buldyrev2010,Bianconi2014}. Although these works consider a similar mechanism (\emph{i.e.}, a history-dependent process), the former features a continuous percolation transition, while the latter features a discontinuous phase transition\cite{PhysRevLett.109.248701}. We address here the question, whether there is a general model of history-dependent percolation on multiplex networks where both continuous and discontinuous percolation transitions can emerge.
To answer this question, we introduce an iterative percolation model on multiplex networks. The percolation of each generation is based on the resulting state of the previous generation, which is referred to as history-dependent percolation here. Different with the previous percolation models with some iterative processes, our work focuses also on the intermediate states of the iterative process; these intermediate states are referred to as generations here. The benefits of doing so are twofold. First, while individual intermediate generations can have their direct real counterparts, they are overlooked by focusing on the infinite (steady-state) generation. Second, by examining generations in succession, we gain understanding of the origin of the discontinuous transition in the steady state and its relation with the continuous transition.
Theoretical analysis indicates that the intermediate states of the recursive process are not cluttered, hence the percolation transition can be observed in any generations. Monte Carlo simulations on Erd\H{o}s--R\'{e}nyi (ER) networks further suggest that all these continuous transitions belong to the same universality class. Although the size of the giant cluster becomes smaller and smaller as the generations progress, endless iterations cannot completely destroy the network when it is initially dense enough. Instead, a non-vanished cluster suddenly appears above the threshold indicating a discontinuous percolation transition. Specifically, scale-free (SF) networks with exponent $2<\gamma<3$ have a vanished critical point for any finite generations, and the non-trivial critical point can suddenly emerge when the number of percolation generations diverges. With the example of human brain networks, our model shows that to find a meaningful structure (such as the abnormal structures of human brain networks), it's not always necessary to evolve the recursive process into the steady states. Our model thus provides a novel approach to analyze the network structure.
\section{Results}
\subsection{History-dependent Model}
An undirected multiplex network is formed by a set of $N$ nodes and multiple layers with links. Each layer is described by its adjacency matrix whose unit elements correspond to links between the corresponding nodes. For simplicity and without loss of generality, we consider here only the case of a multiplex network with two layers which we refer to as layers $A$ and $B$, respectively (see Fig.~\ref{fig1}(a) as an example). An extension to the general case with more layers is straightforward, and some discussion on this can be found in Supplementary Information (SI).
\begin{figure}
\centering
\scalebox{1}[1]{\includegraphics{fig1}}
\caption{Sketch of the history-dependent percolation on a small multiplex network. (a) A multiplex network with two layers $A$ and $B$ separated by the horizontal gray line. (b) Generation $n=1$. The configuration $C^1$ induced by layer $A$ is connected. (c) Generation $n=2$. The configuration induced by layer $B$ on $C^1$ has two clusters indicated by blue and red. (d) Generation $n=3$. The configuration $C^3$ induced by layer $A$ on $C^2$ has three clusters indicated by blue, orange and red, respectively. (e) Generation $n=4$. Applying layer $B$ on configuration $C^3$, no new clusters can be found; the percolation process has reached a steady state. If the system is large enough, this process can be done to any number of generations.}
\label{fig1}
\end{figure}
To model the history-dependent iterative process, we use the two network layers alternately to investigate the percolation process in every generation. For generation $n=1$, we use layer $A$ to check the percolation process among all the nodes. Then, nodes form a configuration $C^1$, see Fig.~\ref{fig1}(b). The percolation of generation $n=2$ is induced by layer $B$ based on the configuration $C^1$ formed in generation $n=1$. Specifically, if two nodes $i$ and $j$ who are in the same cluster in $C^1$ are connected in layer $B$, then they will be connected in this generation. Traverse all node pairs in the same clusters, we obtain a new configuration $C^2$, see Fig.~\ref{fig1}(c). Similarly, for generation $n=3$, we use layer $A$ to perform a percolation on configuration $C^2$. If two nodes $i$ and $j$ who are in the same cluster in $C^2$ are connected in layer $A$, they will be connected in this generation. Then, a new configuration $C^3$ is obtained, see Fig.~\ref{fig1}(d), and so on, to any number of generations. Note that the two layers $A$ and $B$ are used cyclically in this progressive process.
Note that once the initial configuration of the two layers is given, the constructions of clusters in all generations are deterministic. To study the percolation transition in each generation, a natural choice of the control parameter is the average degree $z$ of the initial network layers. For a network ensemble with a fixed average degree (or for a given real multiplex network), one can study the percolation transition by introducing the link occupation probability $p$ as the control parameter: the fraction $1-p$ of links in each layer are chosen at random and removed, and the remaining links are then used in the iterated percolation processes.
\subsection{Network with ER layers}
We first consider the case where the two network layers are both ER networks with the same average degree $z$, for which the model can be solved exactly (see Section Method). The results indicate that the first generation and also all finite generations of iterative percolation demonstrate a continuous percolation transition. Figure~\ref{fig2}(a) shows that the computed order parameter $\psi^n$ agrees well with the numerical simulations of the process. The theoretical solution also shows that the critical point $z_c^n$ does not diverge with the increasing generation $n$, but rather trends to a fixed value $z_c^{\infty}\approx2.455$, at which $\psi_c^\infty\approx 0.512$. This means that the percolation transition becomes discontinuous when $n\to\infty$.
\begin{figure}
\centering
\scalebox{1}{\includegraphics{fig2}}
\caption{The simulation results for ER networks. (a) The size of the giant cluster $\psi^n$ as a function of the average degree $z$ for different generations. The solid lines are the theoretical results obtained by our method. The system size in simulations is $N=2^{16}$. (b) The order parameter at the critical point $\psi_c^n$ for finite generations as a function of the network size $N$. The simulation results are obtained by averaging over all the model realizations. (c) The distribution of the order parameters obtained in each individual realization $f(\psi_c^n)$ for generation $n=10$. (d) The order parameter at the critical point $\psi_c^\infty$ for the infinite generation as a function of the network size $N$. The simulation results are the average over the roughly $60\%$ of model realizations that percolate. The fitted curve has the form $\psi_c^{\infty}= \psi_{c0}^\infty+O(N^{-\varepsilon})$ with $\psi_{c0}^\infty=0.514\pm0.001$ and $\varepsilon=0.233\pm0.005$.}
\label{fig2}
\end{figure}
To clarify the types of the percolation transition from simulations, we study the finite-size scaling of the order parameter $\psi^n_c$ at the critical point (see Section Method). Figure~\ref{fig2}(b) shows that the simulation results of the first several generations have the same scaling for large $N$, which indicates the order parameter $\psi^n_c$ will vanish when $N\to\infty$. This implies these transitions are all continuous.
\begin{figure}
\centering
\scalebox{1}{\includegraphics{fig3}}
\caption{The cluster size distribution $p_s$ at the critical point. (a) $p_s$ for different generations $n$; the network size is $N=2^{20}$. (b) $p_s$ of generation $n=10$ for different network sizes $N$.}
\label{fig3}
\end{figure}
However, for a large $n$, the simulation results appear to deviate from the finite-size scaling of a continuous phase transition, see Fig.~\ref{fig2} (a). For a better understanding of this, we further show the distribution of $\psi^n_c$ obtained in each individual realization. Figure ~\ref{fig2} (c) takes generation $n=10$ as an example, see Section C of SI for other generations. The results show a heavy-tailed distribution instead of being confined to a small region as that of the classical percolation transition\cite{aharony2003introduction}. Especially when the system size is small (see the case $N=2^{16}$), the bimodal distribution as that of the discontinuous percolation transition can be found, namely, $\psi^n_c$ around zero corresponds to the non-percolating realizations and the larger values for the percolating realizations. That is why we cannot observe a finite-size scaling in Fig.~\ref{fig2} (b) for large $n$. In addition, Fig.~\ref{fig2} (c) also shows that the heavy tail will disappear with the increasing of the system size. Thus, the same scaling as that of the classical percolation is expected for very large systems.
For $n=\infty$, the distribution of $\psi^n_c$ becomes a standard bimodal distribution (see Section C of SI), which allows us to identify the non-percolating realizations and remove them from the subsequent analysis. Then, the fitted result of the percolating realizations $\psi_{c0}^{\infty}\approx 0.514$ shown in Fig.~\ref{fig2}(d) is agreement well with the theoretical analysis, indicating a discontinuous percolation transition.
Moreover, for a finite system the infinite generation just corresponds to a generation $n_c$, for which the late generations do not further alter the results. When $n\to n_c$, the system will demonstrate a much sharped percolation transition as that for the thermodynamic limit. However, $n_c$ is varied for different network realizations, and the corresponding largest clusters are thus much different. As a result, the simulation results of the largest clusters at the theoretical threshold are distributed broadly. Consequently, the heavy-tailed distribution is found (see Fig.~\ref{fig2}(c)), and the order parameters obtained by averaging over these become larger than the expectation of the finite-size scaling (see Fig.\ref{fig2} (b)). Note that $n_c$ generally increases with the system size, thus the heavy-tailed distribution shown in Fig.\ref{fig2} (c) is more obvious for smaller systems.
Since $p_c^{n-1}<p_c^{n}$, the percolation transition of generation $n$ just corresponds to a classical percolation process (with a rescaled control parameter) in the supercritical phase of generation $n-1$. This indicates that there is no essential difference between the percolation transition of two consecutive generations. As an immediate consequence, all the finite generations must belong to the same universality as the classical percolation, and the scaling behavior changes abruptly from the class shown in Fig.~\ref{fig2}(b) to that of Fig.~\ref{fig2}(d) when $n\to\infty$.
To further confirm the universality class of the history-dependent percolation model, we study the cluster size distribution $p_s$. As pointed above, the largest clusters around the critical point are distributed broadly for a large $n$. A direct result of this is increasing the probability that finding large clusters in the system. For the cluster size distribution, this results in a cocked tail before a normal exponential cutoff, and becomes more and more apparent as the increasing of generation $n$ (see Fig.\ref{fig3}(a)). However, from Fig.~\ref{fig3}(b), we can find this phenomenon weakens with the increasing of the system size $N$, so the distribution as that of the classical percolation can be expected for larger systems, \emph{i.e.}, a pow-law with an exponential cutoff (without a cocked tail). This also suggests that the universality class is the same for all finite generations of the iterated percolation process.
\subsection{Network with SF layers}
We now study the case where both layers are SF networks. In particular, we assume the power-law degree distribution~\cite{Cohen2010Complex}
\begin{equation}
\label{SF-p_k}
p_k =ck^{-\gamma},\qquad k=m, m+1, \dots, K,
\end{equation}
where $c$ is a normalization factor, and $m$ and $K$ are the lower and upper bounds of degree, respectively. If $K$ is large enough and $\gamma>1$, the normalization factor is approximately $c\approx(\gamma-1)m^{\gamma-1}$. In the simulation, networks are constructed by generating node degree values with Eq.~(\ref{SF-p_k}) and then connecting the nodes with the configuration model. Since the average degree is fixed by Eq.~(\ref{SF-p_k}), we activate a fraction $p$ of links of both network layers to control the effective mean degree and trigger the percolation process. In this section, we therefore seek the critical point in terms of critical probability $p_c$, not critical average degree $z_c$ as in the previous section.
It is known that when $\gamma>3$, such scale-free network also has a non-trivial critical point as that of ER networks~\cite{PhysRevE.64.026118}. Consequently, the results are similar to that found for ER networks. Here we focus on the case $\gamma\in(2,3)$ which is realized in many real-world networks~\cite{barabasi2016network}. Previous studies have demonstrated that the standard percolation in this case has zero critical point~\cite{PhysRevLett.85.4626,PhysRevLett.85.5468}.
The simulation results shown in Fig.~\ref{fig4}(a) show that the percolation transition becomes sharper and sharper as the generation increases. However, Fig.~\ref{fig4}(b) demonstrates that the pseudo-critical point indicated by the maximum of the second largest cluster decreases with the system size, which suggests a vanished critical point can also be found for infinite system. From the theoretical analysis (see Section Method and SI), we found that all the finite generations on such networks have zero critical point as the classical percolation, which is further confirmed by the finite-size scaling of pseudo-critical point shown in Fig.~\ref{fig4}(c).
\begin{figure}
\centering
\scalebox{1}{\includegraphics{fig4}}
\caption{The simulation results for SF networks with the degree distribution given by Eq.~(\ref{SF-p_k}) where $m=2$, $K=\sqrt{N}$, and $\gamma = 2.5$. (a) The size of the giant cluster $\psi^n$ as a function of probability $p$ for different generations. The network size is $N=2^{16}$. (b) The size of the giant cluster $\psi^n$ of generation $n=3$ as a function of probability $p$ for different network sizes $N$. The solid lines are the number of nodes in the second largest cluster $N_2$. (c) The finite-size scaling of the pseudo-critical points for generation $n=3$. The fitting curve takes the form $p_c\varpropto N^{-\alpha}$ with $\alpha=0.075\pm0.003$. (d) The size of the giant cluster $\psi^n$ as a function of probability $p$ for infinite generations with different minimum degrees $m$. The network size is $N=2^{16}$.}
\label{fig4}
\end{figure}
We also find that to observe a non-trivial critical point on such networks, it requires endless iterations, large $m$ and broad degree distribution, otherwise the infinite generation will destroy the whole network (see Section A-C in SI for details). Generally speaking, SF networks with more connections and border degree distributions are more likely to survive in the infinite iterated processes. Figure~\ref{fig4}(d) demonstrates that when $m>1$, a discontinuous percolation transition with non-trivial critical point can also be found for SF networks.
\subsection{Real multiplex networks}
Our brain is a complex system and network neuroscience holds great promise for expanding our understanding of a healthy brain functioning, brain diseases, brain development, and brain aging\cite{sporns2005human,bassett2017network}. Simply, the brain can be modeled by a network, where the brain regions and their connections constitute the set of nodes and the set of links, respectively. Here, we consider the human brain networks which are constructed using the high-resolution brain atlas with $1024$ Regions-of-Interest (ROIs) \cite{Zalesky2010Whole}, \emph{i.e.}, the network thus has $1024$ nodes. This multiplex network has two layers, which capture functional correlations and morphological similarity of the human brain based on ROIs, respectively. For the functional brain layer, the mean time series is extracted for each ROI by averaging the time series of all voxels (small measured volumes in three-dimensional space) within it. Then, we calculate the Pearson correlation for each pair of ROIs and generated a $1024\times1024$ correlation matrix. For the morphological brain layer, we estimate the interregional similarity in the distribution of regional gray matter volume in terms of the Kullback-Leibler divergence measure~\cite{Hao2016Single}. By this construction, both layers are represented by weighted complete networks. We tune the resulting networks by choosing the average degree in each layer, $z$, and thus obtain the corresponding unweighted networks where only links with the highest weights are kept. See the last section of SI for a detailed description of the data and the MRI data preprocessing strategy.
\begin{figure}
\centering
\scalebox{1}{\includegraphics{fig5}}
\caption{History-dependent percolation on human functional-morphological brain networks for different average degree values ($z$); $p$ is the link occupation probability; $\psi^n$ is the size of the giant cluster given the percolation generation $n$. (a) The HC participant's bilayer brain network. (b) The MDD participant's bilayer brain network.}
\label{fig5}
\end{figure}
In Fig.~\ref{fig5}, we compare the results of our percolation model on two different human functional-morphological brain networks, one from a major depressive disorder (MDD) participant and a healthy control (HC) participant. For the MDD data, both the critical point and the giant cluster for a given $p$ are smaller than that of HC, indicating that the brain network of the MDD participant is more vulnerable. For the infinite generation of the model, the pattern of the remaining nodes and links (the giant cluster) of the MDD data is sparser and more dispersed. These nodes mainly locate in frontal, parietal, and occipital lobes, and don’t form obvious community structure, whereas, for the HC participant, these nodes are close together and mainly locate in the highly myelinated brain regions (i.e., the motor-somatosensory strip in the central sulcus, the visual cortex in the occipital lobe). This suggests that for the HC participant, the giant cluster identified by a finite-generation percolation process may reflect the biological meanings. See Fig.7 in SI for details.
\begin{figure}
\centering
\scalebox{1}{\includegraphics{fig6}}
\caption{The visualization of human brain bilayer networks at degree $z=20$ and $1024$ parcellation templates when $p$ is slightly larger than the corresponding $p_c$. (a) The HC's bilayer brain network. (b) The MDD's bilayer brain network. We show the lateral and medial brain of each hemisphere. The visualization is done with BrainNet viewer~\cite{Xia_2013}.}
\label{fig6}
\end{figure}
When we increase the average degree to $z = 20$, both brain networks become robust, and the percolation transition of infinite generation becomes sharper (see Fig.~\ref{fig5}). As shown in Fig.~\ref{fig6}, for the HC participant, the giant cluster of the infinite generation diffuses across frontal, parietal, occipital, temporal, and subcortical lobes. Compared with HC data, the MDD network shows an apparent deficiency of nodes in frontal, occipital, and temporal lobes, some of which belong to the default mode network (DMN) \cite{Raichle_2001}. These DMN regions play an important role in our advanced cognitive abilities, such as executive control, visual and auditory sense and some pieces of evidence support the decreased functional connectivity \cite{Veer,Zhang} and frontal cortical thinning \cite{Bos} in these regions for MDD participants, see Fig.~\ref{fig6}. Therefore, our model opens a new avenue toward detecting the abnormal nodes or components between the healthy and the diseases participants from a comprehensive functional-morphological perspective, which can help us detect the potential connectome-based MRI biomarker and gain new insights on the mechanism for some brain disorders. Importantly, these findings are beyond the single modal imaging/layer and traditional percolation, thus provide the novel understanding and convergent results for MDD. Furthermore, our model is easy to expand to the networks with three or more layers and could be used to investigate the similarities, differences and comprehensive understanding of various brain disorders. Note that the current study is a single case validation. Besides, our brain is highly personalized and a large sample size is needed to infer robust conclusions \cite{Termenon,Turner}. Further study can utilize large sample size or different brain disorders to verify our model.
We also applied our model to a bilayer social network composed of users who are active on both Twitter and FriendFeed~\cite{Magnani2011The}. Among the $150,684$ common users of the two networks, there are $8,308,326$ and $5,270,665$ links in the Twitter and FriendFeed layer, respectively. The results can be found in Fig.6 in SI. We find that the percolation transition occurs at much lower activation probability $p$ than that of the brain network data. This is a direct consequence of the average degree in the social network ($z\approx110$ and $70$ in the two network layers, respectively) being substantially higher than in the brain network ($z=6,8,10,20$). The Twitter-FriendFeed network reaches the steady state after $4$ generations, and the discontinuous percolation transition is also absent. This can be due to many links (about $37\%$ of the total number) that occur in both network layers which renders subsequent process generations equivalent to the first two or three generations of the case discussed in Fig.~\ref{fig4}.
These results suggest that to observe non-trivial behaviors in iterative percolation, one needs to study multiplex networks with limited layer overlap. To fully understand the relation between the layer overlap and the dynamics of the iterative percolation remains a future challenge. The human brain network reaches the steady state after $10$ generations. At this point, the percolation transition is more abrupt than that for smaller iteration values. With respect to the small size of the studied system ($1024$ nodes), the possibility that a larger brain network (achievable with a higher imaging resolution) would display yet more abrupt transition, and thus suggest a discontinuous percolation transition in the thermodynamic limit, remains open. However, whether the percolation transition is discontinuous or not, the process provides a series of method to analyze network structures.
\section{Discussion}
In summary, we introduce a history-dependent percolation model to study the critical behavior of the percolation transition on multiplex networks. The percolation process is ran iteratively on different layers. Instead of focusing solely on the steady state, we pay more attention to finite generations, each of which can be considered as an independent model. For example, $n=1$ corresponds to the common model of information or disease spreading on monopartite networks; $n=2$ can be used to describe the interplay between the spreading of disease and immunity information \cite{PhysRevLett.111.128701}; $n=3$ can be used to model the information spreading on multiplex social networks where users communicate through multiple channels. The general example of multilayer propagation is the one where online communication influences user offline behavior and then couples back to online communication. A typical example of $n=\infty$ is the study of cascading failures in coupled networks \cite{Buldyrev2010}. In this sense, our model provides a unified framework to study the percolation process on multiplex networks.
We investigate both ER networks and SF networks with power-law exponent $2<\gamma<3$. The results reveal that the intermediate state of the recursive process can be also used to uncover meaningful structures, and therefore the percolation transition should be defined and studied in each generation. For any finite generations, the percolation transition on random networks show a continuous transition and belong to the same universality class, while scale-free networks have the critical point trending to zero as the network size grows. When $n=\infty$, a discontinuous transition exists for both networks \cite{Buldyrev2010}. In essence, this is due to that the percolation transition of generation $n$ emerges from the supercritical phase of the percolation of generation $n-1$. As a result, it inherently cannot generate a new universality class\cite{PhysRevE.92.010103}. However, when $n$ diverges, $n-1$ is also divergent, this relation of the critical state is broken. Then, the new phenomenon, \emph{i.e.}, discontinuous transition, emerges.
Furthermore, on one hand, our result indicates that the continuous transition found in real systems could be the result of a combination of many sequential processes, see an example in Ref.~\cite{PhysRevLett.111.128701}. On the other hand, to observe the abrupt percolation transitions in real multiplex networks, for which the generation cannot go to infinite actually, the model may need to be extended by, for example, considering the space embedding of network layers \cite{bashan2013extreme}, modeling relative importance of inter- and intra-layer connections \cite{Radicchi2015Percolation}, or introducing cores of ``high quality" edges~\cite{radicchi2013abrupt}. Our model can be easily extended to a general case with more layers. This would make it possible to use the generalized model to analyze the percolation transition on a temporal network comprising several layers corresponding to different times points \cite{Holme2012Temporal}.
Beyond the theoretical analysis, we showed that the outcomes of the iterative percolation process can be used to characterize real networks whose percolation properties differ markedly between systems (such as the used brain scan and social network data) as well as between various samples of networks from the same class (such as the brain scan data of a healthy participant and a participant with a mental disorder). The proposed model can thus become a useful tool for evaluating and, more importantly, comparing structural properties of multiplex networks. The model represents an important step towards understanding the history-dependent dynamic processes on multiplex networks, and may prove useful in important practical applications like link prediction \cite{L2011Link}, vital node identification \cite{L2016Vital} and community detection \cite{PhysRevX.7.031056} in multiplex networks.
\section{Method}
\label{sec:method}
\subsection{Mean-field theory analysis}
To obtain the exact analytical solution of the critical point, we consider the giant cluster in the infinite system~\cite{Buldyrev2010}. We introduce first the function $\mathfrak{F}(x)$ which returns the size of the giant cluster of a network ensemble with given degree distribution, when a fraction $x$ of nodes are chosen at random and used to construct the giant cluster. Note that the fraction obtained by function $\mathfrak{F}(x)$ is with respect to the actually used nodes. The size of the giant cluster with respect to the original network is thus $x\mathfrak{F}(x)$.
Leaving the specific form of $\mathfrak{F}(x)$ aside, which may be a group of equations or a dataset obtained by Monte Carlo simulations, we now lay out the general analytical framework for the history-dependent percolation process. Since the network's layers are in general different, we assume that layers $A$ and $B$ have functions $\mathfrak{F}_A(x)$ and $\mathfrak{F}_B(x)$, respectively. In addition, we label the size of the giant cluster in generation $n$ as $\psi^n$, and the fraction of the nodes that can be used to construct the giant cluster in generation $n$ as $S^{n-1}$. The function $\mathfrak{F}(x)$ allows us to write
\begin{equation}
\psi^n=S^{n-1}\mathfrak{F}(S^{n-1}), \label{gnpsi}
\end{equation}
where $\mathfrak{F}(x)$ is $\mathfrak{F}_A(x)$ and $\mathfrak{F}_B(x)$ for odd and even generations, respectively. All we need to do now is to find $S^{n-1}$ for each generation $n$.
For an infinite system, the giant cluster of generation $n$ can only emerge from the giant cluster of generation $n-1$. So, leveraging this recursive relationship, the fraction of nodes that can be used to construct the giant cluster $S^{n}$ for an odd $n$ satisfies
\begin{equation}
S^{n} = S^0\mathfrak{F}_A(S^{n-1}), \label{gns1}
\end{equation}
and for an even $n$,
\begin{equation}
S^{n} = S^0\mathfrak{F}_B(S^{n-1}). \label{gns2}
\end{equation}
In addition, if one removes a fraction $1-p$ of nodes in the initial configuration to trigger the iterated percolation, then $S^0=p$. If the removal is for links, such as the ones used in Figs.~\ref{fig4} and \ref{fig5}, the function $\mathfrak{F}(x)$ for the diluted network should be replaced with $\mathfrak{F}(px)$ since the degree distribution has changed~\cite{PhysRevE.66.016128}.
If functions $\mathfrak{F}_{A,B}(x)$ are known, Eqs. (\ref{gnpsi})-(\ref{gns2}) can be used to calculate the size of the giant cluster for any generation and also the critical point. For ER networks $\mathfrak{F}(x)$ can be obtained by solving a self-consistent equation, however, there is no closed form for SF networks and the theoretical analysis can be only done around the critical point, see Section A of SI for details.
If function $\mathfrak{F}(x)$ has a non-trivial critical point $x_c$ below which $\mathfrak{F}(x)=0$ and $\mathfrak{F}(x_c)=0$, such as ER networks. Equations (\ref{gns1}) and (\ref{gns2}) shows that the critical point of generation $n$ corresponds to a non-zero $S^{n-1}$, that is the supercritical state of generation $n-1$. This indicates that there is no essential difference between the percolation transition of the two generations, and consequently the first generation and also all finite generations of iterative percolation demonstrate a continuous percolation transition. Figure~\ref{fig2}(a) takes ER networks as example to shows that the resulting $\psi^n$ agrees well with numerical simulations of the process.
The recursive relations Eqs.(\ref{gns1}) and (\ref{gns2}) have their fixed point $S=\mathfrak{F}(S)$, which corresponds to the infinite generation, and allows us to find the fixed point for ER layers $z_c^{\infty}\approx2.455$ and $S_c=\left(1+\sqrt{1-z_c^\infty/2}\right)/2\approx 0.715$, see Section A-C of SI for details. The critical order parameter that corresponds to the found $S_c$ is $\psi_c^\infty=(S_c)^2\approx 0.512$, which also agrees well with numerical simulations shown in Fig.\ref{fig2}(d). The giant cluster size thus undergoes a discontinuous phase transition at $z_c^{\infty}$.
For SF networks, the recursive relations Eqs.~(\ref{gns1}) and (\ref{gns2}) can also be used to obtain the critical point of each generation. Since $\mathfrak{F}(px)$ has a zero critical point for $2<\gamma<3$, we can find that all the finite generations give a vanished critical point $p_c=0$ by using Eqs.~(\ref{gns1}) and (\ref{gns2}), recursively (see Section A-B of SI for details). By examining the fixed point of Eqs.~(\ref{gns1}) and (\ref{gns2}), we can also find the infinite generation demonstrates a discontinuous percolation transition, see Section A-C of SI for details.
\subsection{The order parameter at the critical point}
As a topological phase transition, there is no free energy can be used to determine the type of the percolation transition. Therefore, the one with a step-like changing of the order parameter at the critical point is often treated as a discontinuous percolation transition \cite{Buldyrev2010,Boccaletti20141,PhysRevLett.96.040601,PhysRevLett.109.248701,Radicchi2015Percolation,radicchi2013abrupt}, and the continuous percolation transition is for that with continuous changing at the critical point. That is to say, the discontinuous percolation transition has a non-zero order parameter at the critical point, and a zero order parameter can be found for continuous percolation transition.
However, due to finite sizes of the systems used in the simulation, both the two types of percolation transitions could give a non-zero order parameter at the critical point. In this way, we use the finite-size scaling to check the types of percolation transition in simulations. For a continuous transition, the order parameter $\psi_c$ must decrease with increasing system size and becomes zero for an infinite system, which corresponds to $\psi_c\propto N^{-\epsilon}$ where $\epsilon=1/3$ for random networks~\cite{aharony2003introduction}. For a discontinuous transition, the order parameter $\psi_c$ takes a bimodal distribution, the values around zero correspond to the non-percolating realizations and the larger values for the percolating realizations. Excluding the non-percolating realizations, the finite-size scaling thus takes the form $\psi_c\sim \psi_{c0}+O(N^{-\varepsilon})$, where $\psi_{c0}$ is the order parameter at the critical point for an infinite system. Therefore, Figs.~\ref{fig2} (b)-(d) just indicate that all the finite generations take the continuous percolation transition, and the infinite generation takes the discontinuous percolation transition. However, due to finite-size effects, larger systems are needed to observe a better finite-size scaling.
\begin{acknowledgments}
The research was partially supported by the National Natural Science Foundation of China under grant Nos.11622538, 61773148, 61673150, 11625522 and 11672289. L.L. also acknowledges Zhejiang Provincial Natural Science Foundation of China under grant No.LR16A05000, and the Science Strength Promotion Programme of UESTC. Y.D. also acknowledges the Ministry of Science and Technology of China under grant No.2016YFA0301604.
\end{acknowledgments}
|
1,116,691,499,285 | arxiv | \section{Introduction}
\emph{Probabilistic programming languages} (PPLs) are programming languages extended with statements to
(1) draw values at random from a given probability distribution, and
(2) perform conditioning due to observation.
Probabilistic programs yield, instead of a deterministic outcome, a probability distribution over possible outcomes.
PPLs greatly simplify representation of, and reasoning with rich probabilistic models.
Interest in PPLs has increased in recent years,
mainly in the context of Bayesian machine learning.
Examples of modern PPLs include \emph{Church}, \emph{Venture} and \emph{Figaro}~\cite{DBLP:conf/uai/GoodmanMRBT08,DBLP:journals/corr/MansinghkaSP14,pfeffer2009figaro},
while early work goes back to Kozen~\cite{DBLP:journals/jcss/Kozen81}.\looseness=-1
\emph{Ranking theory} is a qualitative abstraction of probability theory in which events receive discrete degrees of surprise called \emph{ranks}~\cite{DBLP:books/daglib/0035277}.
That is, events are ranked 0 (not surprising), 1 (surprising), 2 (very surprising), and so on, or $\infty$ if impossible.
Apart from being computationally simpler,
ranking theory permits meaningful inference without requiring precise probabilities.
Still, it provides analogues to powerful notions known from probability theory, like conditioning and independence.
Ranking theory has been applied in logic-based AI (e.g. belief revision and non-monotonic reasoning~\cite{DBLP:dblp_journals/ai/DarwicheP97,goldszmidt1996qualitative})
as well as formal epistemology~\cite{DBLP:books/daglib/0035277}.\looseness=-1
In this paper we develop a language called \emph{RankPL}.
Semantically, the language draws a parallel with probabilistic programming in terms of ranking theory.
We start with a minimal imperative programming language (\textbf{if-then-else}, \textbf{while}, etc.)
and extend it with statements to
(1) draw choices at random from a given ranking function and
(2) perform ranking-theoretic conditioning due to observation.
Analogous to probabilistic programs, a RankPL programs yields, instead of a deterministic outcome, a ranking function over possible outcomes.
Broadly speaking, RankPL can be used to represent and reason about processes
whose input or behavior exhibits uncertainty expressible
by distinguishing normal (rank $0$) from surprising (rank $> 0$) events.
Conditioning in RankPL amounts to the (iterated) revision of rankings over alternative program states.
This is a form of revision consistent with the well-known AGM and DP postulates for (iterated) revision~\cite{DBLP:dblp_journals/ai/DarwicheP97,Gardenfors:1995:BR:216136.216138}.
Various types of reasoning can be modeled, including abduction and causal inference.
Like with PPLs, these reasoning tasks can be modeled without having to write inference-specific code.
The overview of this paper is as follows.
Section~\ref{sec:rankingtheory} deals with the basics of ranking theory.
In section~\ref{sec:rpl} we introduce RankPL and present its syntax and formal semantics.
In section~\ref{sec:noisy} we discuss two generalized conditioning schemes (L-conditioning and J-conditioning) and show how they can be implemented in RankPL.
All the above will be demonstrated by practical examples.
In section~\ref{sec:implementation} we discuss our RankPL implementation.
We conclude in section~\ref{sec:conclusion}.
\section{Ranking Theory}\label{sec:rankingtheory}
Here we present the necessary basics of ranking theory, all of which is due to Spohn~\cite{DBLP:books/daglib/0035277}.
The definition of a \emph{ranking function} presupposes a finite set $\Omega$ of possibilities
and a boolean algebra $\mathcal{A}$ over subsets of $\Omega$, which we call \emph{events}.
\begin{definition}
A ranking function is a function $\kappa: \Omega \rightarrow \mathbb{N} \cup \{\infty\}$ that associates every possibiltiy with a \emph{rank}.
$\kappa$ is extended to a function over events by defining $\kappa(\emptyset) = \infty$ and $\kappa(A) = min( \{ \kappa(w) \mid w \in A \} )$ for each $A \in \mathcal{A} \setminus \emptyset$.
A ranking function must satisfy $\kappa(\Omega) = 0$.
\end{definition}
As mentioned in the introduction, ranks can be understood as degrees of surprise or, alternatively, as inverse degrees of plausibility.
The requirement that $\kappa(\Omega) = 0$ is equivalent to the condition that at least one $w \in \Omega$ receives a rank of 0.
We sometimes work with functions $\lambda: \Omega \rightarrow \mathbb{N} \cup \{\infty\}$ that violate this condition.
The \emph{normalization} of $\lambda$ is a ranking function denoted by $|| \lambda ||$ and defined by $|| \lambda ||(w) = \lambda(w) - \lambda(\Omega)$.
Conditional ranks are defined as follows.
\begin{definition}\label{def:conditional}
Given a ranking function $\kappa$, the rank of $A$ conditional on $B$ (denoted $\kappa(A \mid B)$ is defined by
\[
\kappa(A \mid B) = \left\{ \begin{array}{ll}
\kappa(A \cap B) - \kappa(B)& \mbox{if $\kappa(B) \not = \infty$,} \\
\infty & \mbox{otherwise.} \\
\end{array}
\right. \\
\]
We denote by $\kappa_{B}$ the ranking function defined by $\kappa_{B}(A) = \kappa(A \mid B)$.
\end{definition}
In words, the effect of conditioning on $B$ is that the rank of $B$ is shifted down to zero (keeping the relative ranks of the possibilities in $B$ constant)
while the rank of its complement is shifted up to $\infty$.
How do ranks compare to probabilities?
An important difference is that ranks of events do not add up as probabilities do.
That is, if $A$ and $B$ are disjoint, then $\kappa(A \cup B) = min(\kappa(A), \kappa(B))$, while $P(A \cup B) = P(A) + P(B)$.
This is, however, consistent with the interpretation of ranks as degrees of surprise (i.e., $A \cup B$ is no less surprising than $A$ or $B$).
Furthermore, ranks provide deductively closed beliefs, whereas probabilities do not.
More precisely, if we say that $A$ is believed with firmness $x$ (for some $x > 0$) with respect to $\kappa$ iff $\kappa(\overline A) > x$,
then if $A$ and $B$ are believed with firmness $x$ then so is $A \cap B$.
A similar notion of belief does not exist for probabilities, as is demonstrated by the Lottery paradox~\cite{kyburg1961probability}.
Finally, note that $\infty$ and $0$ in ranking theory can be thought of as playing the role of $0$ and $1$ in probability,
while $min$, $-$ and $+$ play the role, respectively, of $+$, $\div$ and $\times$.
Recall, for example, the definition of conditional probability, and compare it with definition~\ref{def:conditional}.
This correspondence also underlies notions such as (conditional) independence and ranking nets (the ranking-based counterpart of Bayesian networks)
that have been defined in terms of rankings~\cite{DBLP:books/daglib/0035277}.
\section{RankPL}\label{sec:rpl}
We start with a brief overview of the features of RankPL.
The basis is a minimal imperative language
consisting of integer-typed variables, an \textbf{if}-\textbf{then}-\textbf{else} statement and a \textbf{while-do} construct.
We extend it with the two special statements mentioned in the introduction.
We call the first one \emph{ranked choice}.
It has the form
$\{s_1\} \Rank{e} \{s_2\}.$
Intuitively, it states that either $s_1$ or $s_2$ is executed,
where the former is a normal (rank 0) event and the latter a typically surprising event whose rank is the value
of the expression~$e$.
Put differently, it represents a draw of a statement to be executed, at random, from a ranking function over two choices.
Note that we can set $e$ to zero to represent a draw from two equally likely choices,
and that larger sets of choices can be represented through nesting.
The second special statement is called the \emph{observe} statement
$\textbf{observe }b.$
It states that the condition $b$ is observed to hold.
Its semantics corresponds to ranking-theoretic conditioning.
To illustrate, consider the program
$$\texttt{x} := 10;\mbox{ } \{ \texttt{y} := 1 \} \Rank{1} \{ \{\texttt{y} := 2\} \Rank{1} \{\texttt{y} := 3\} \};\mbox{ } \texttt{x} := \texttt{x} \times \texttt{y};$$
This program has three possible outcomes: $\texttt{x} = 10$, $\texttt{x} = 20$ and $\texttt{x} = 30$, ranked 0, 1 and 2, respectively.
Now suppose we extend the program as follows:
$$\texttt{x} := 10;\mbox{ } \{ \texttt{y} := 1 \} \Rank{1} \{ \{\texttt{y} := 2\} \Rank{1} \{\texttt{y} := 3\} \};\mbox{ } \textbf{observe } \texttt{y} > 1;\mbox{ } \texttt{x} := \texttt{x} \times \texttt{y};$$
Here, the observation rules out the event $\texttt{y} = 1$,
and the ranks of the remaining possibilities are shifted down,
resulting in two outcomes $\texttt{x} = 20$ and $\texttt{x} = 30$, ranked 0 and 1, respectively.
A third special construct is the \emph{rank expression} $\textbf{rank }b.$, which evaluates to the rank of the boolean expression $b$.
Its use will be demonstrated later.
\subsection{Syntax}
We fix a set $\textit{Vars}$ of variables (ranged over by $x$) and denote by $\textit{Val}$ the set of integers including $\infty$ (ranged over by $n$).
We use $e$, $b$ and $s$ to range over the numerical expressions, boolean expressions, and statements.
They are defined by the following BNF rules:
\noindent \begin{center}
\begin{tabular}{rcl}
$\textit{e}$ & $:=$ & $n \mid x \mid \textbf{rank }b \mid (e_1 \hspace{2pt} \lozenge \hspace{2pt} e_2)$ (for $\lozenge \in \{ -, +, \times, \div \}$)\\
$\textit{b}$ & $:=$ & $\neg b \mid (b_1 \vee b_2) \mid (e_1 \hspace{2pt} \blacklozenge \hspace{2pt} e_2)$ (for $\blacklozenge \in \{ =, < \}$)\\
$\textit{s}$ & $:=$ & $\{s_0; s_1 \} \mid x := e \mid \textbf{if }b\textbf{ then }\{ s_1 \}\textbf{ else } \{ s_2 \} \mid $\\
& & $\textbf{while } b \textbf{ do } \{ s \} \mid \{ s_1 \} \Rank{e} \{ s_2 \} \mid \textbf{observe }b \mid \textbf{skip}$\\
\end{tabular}
\end{center}
We omit parentheses and curly brackets when possible and define $\wedge$ in terms of $\vee$ and $\neg$.
We write $\textbf{if }b\textbf{ then } \{ s \}$ instead of $\textbf{if }b\textbf{ then } \{ s \}$ $\textbf{else }\{\textbf{skip}\}$,
and abbreviate statements of the form $\{x := e_1\} \Rank{e} \{x := e_2\}$ to $x := e_1 \Rank{e} e_2$.
Note that the \textbf{skip} statement does nothing and is added for technical convenience.
\subsection{Semantics}
\label{sec:formalsemantics}
The denotational semantics of RankPL defines the meaning of a statement $s$ as a function $\dn{s}$ that maps prior rankings into posterior rankings.
The subjects of these rankings are program states represented by \emph{valuations}, i.e., functions that assign values to all variables.
The \emph{initial valuation}, denoted by $\sigma_0$, sets all variables to 0.
The \emph{initial ranking}, denoted by $\kappa_0$, assigns 0 to $\sigma_0$ and $\infty$ to others.
We denote by $\sigma[x \rightarrow n]$ the valuation equivalent to $\sigma$ except for assigning $n$ to $x$.\looseness=-1
From now on we associate $\Omega$ with the set of valuations and denote the set of rankings over $\Omega$ by $K$.
Intuitively, if $\kappa(\sigma)$ is the degree of surprise
that $\sigma$ is the actual valuation \emph{before} executing $s$,
then $\dn{s}(\kappa)(\sigma)$ is the degree of surprise
that $\sigma$ is the actual valuation \emph{after} executing $s$.
If we refer to the result of running the \emph{program} $s$, we refer to the ranking $\dn{s}(\kappa_0)$.
Because $s$ might not execute successfully, $\dn{s}$ is not a total function over $K$.
There are two issues to deal with.
First of all, non-termination of a loop leads to an undefined outcome.
Therefore $\dn{s}$ is a partial function whose value $\dn{s}(\kappa)$ is defined only if $s$ terminates given $\kappa$.
Secondly, observe statements may rule out all possibilities.
A program whose outcome is empty because of this is said to \emph{fail}.
We denote failure with a special ranking $\kappa_{\infty}$ that assigns $\infty$ to all valuations.
Since $\kappa_{\infty} \not \in K$, we define the range of $\dn{s}$ by $K^{*} = K \cup \{\kappa_{\infty}\}$.
Thus, the semantics of a statement $s$ is defined by a partial function $\dn{s}$ from $K^{*}$ to $K^{*}$.\looseness=-1
But first, we define the semantics of expressions.
A numerical expression is evaluated w.r.t. both a ranking function (to determine values of \textbf{rank} expressions) and a valuation (to determine values of variables).
Boolean expressions may also contain \textbf{rank} expressions and therefore also depend on a ranking function.
Given a valuation $\sigma$ and ranking $\kappa$, we denote by $\sigma_{\kappa}(e)$ the value of the numerical expression $e$ w.r.t. $\sigma$ and $\kappa$,
and by $\mods{\kappa}{b}$ the set of valuations satisfying the boolean expression $b$ w.r.t. $\kappa$.
These functions are defined as follows.\looseness=-1\footnote{We omit explicit treatment of undefined operations (i.e. division by zero and some operations involving $\infty$). They lead to program termination.}
\noindent\begin{minipage}{.40\columnwidth}
\begin{eqnarray*}
\sigma_\kappa(n) & = & n \\
\sigma_\kappa(x) & = & \sigma(x) \\
\sigma_\kappa(\textbf{rank }b) & = & \kappa(\mods{\kappa}{b}) \\
\sigma_\kappa(a_1 \hspace{2pt} \lozenge \hspace{2pt} a_2) & = & \sigma_\kappa(a_1) \hspace{2pt} \lozenge \hspace{2pt} \sigma_\kappa(a_2) \\
\end{eqnarray*}
\end{minipage}
\begin{minipage}{.60\columnwidth}
\begin{eqnarray*}
\mods{\kappa}{\neg b} & = & \Omega \setminus \mods{\kappa}{b} \\
\mods{\kappa}{b_1 \vee b_2} & = & \mods{\kappa}{b_1} \cup \mods{\kappa}{b_2} \\
\mods{\kappa}{a_1 \hspace{2pt} \blacklozenge \hspace{2pt} a_2} & = & \{ \sigma \in \Omega \mid \sigma_\kappa(a_1) \hspace{2pt} \blacklozenge \hspace{2pt} \sigma_\kappa(a_2) \} \\
\end{eqnarray*}
\end{minipage}
Given a boolean expression $b$ we will write $\kappa(b)$ as shorthand for $\kappa(\mods{\kappa}{b})$ and $\kappa_{b}$ as shorthand for $\kappa_{\mods{\kappa}{b}}$.
We are now ready to define the semantics of statements.
It is captured by seven rules, numbered \textbf{D1} to \textbf{D7}.
The first deals with the \textbf{skip} statement, which does nothing and therefore maps to the identity function.
\begin{align}
\dn{\textbf{skip}}(\kappa) & = \kappa \tag{\textbf{D1}}.
\end{align}
The meaning of $s_1; s_2$ is the composition of $\dn{s_1}$ and $\dn{s_2}$.
\begin{align}
\dn{s_1 ; s_2}(\kappa) & = \dn{s_2}(\dn{s_1}(\kappa))\tag{\textbf{D2}}
\end{align}
The rank of a valuation $\sigma$ after executing an assignment $x := e$
is the minimum of all ranks of valuations that equal $\sigma$ after assigning the value of $e$ to $x$.
\begin{align}
\dn{x := e}(\kappa)(\sigma) & = \kappa( \{ \sigma' \in \Omega \mid \sigma = \sigma'[x \rightarrow \sigma'_\kappa(e)] \})\tag{\textbf{D3}}
\end{align}
To execute $\textbf{if }b\textbf{ then }\{ s_1 \}\textbf{ else }\{ s_2 \}$ we first
execute $s_1$ and $s_2$ conditional on $b$ and $\neg b$,
yielding the rankings $\dn{s_1}(\kappa_{b})$ and $\dn{s_2}(\kappa_{\neg b})$.
These are adjusted by adding the prior ranks of $b$ and $\neg b$ and
combined by taking the minimum of the two.
The result is normalized to account for the case where one branch fails.\looseness=-1
\begin{align*}
\dn{\textbf{if }e\textbf{ then }\{ s_1 \}\textbf{ else }\{ s_2 \}}(\kappa) & = || \lambda ||, \\
& \hspace{-30pt} \mbox{ where }\lambda(\sigma) = min\begin{pmatrix} \dn{s_1}(\kappa_{b})(\sigma) + \kappa(b), \tag{\textbf{D4}} \\ \dn{s_2}(\kappa_{\neg b})(\sigma) + \kappa(\neg b) \end{pmatrix}
\end{align*}
Given a prior $\kappa$, the rank of a valuation after executing $s_1 \Rank{e} s_2$
is the minimum of the ranks assigned by $\dn{s_1}(\kappa)$ and $\dn{s_2}(\kappa)$,
where the latter is increased by $e$.
The result is normalized to account for the case where one branch fails.\looseness=-1
\begin{align}
\dn{\{ s_1 \} \Rank{e} \{ s_2 \}}(\kappa) & = || \lambda ||, \mbox{ where }\lambda(\sigma) = min \begin{pmatrix} \dn{s_1}(\kappa)(\sigma), \\ \dn{s_2}(\kappa)(\sigma) + \sigma_{\kappa}(e) \end{pmatrix} \tag{\textbf{D5}}
\end{align}
The semantics of $\textbf{observe }b$ corresponds to conditioning on the set of valuations satisfying $b$,
unless the rank of this set is $\infty$ or the prior ranking equals $\kappa_{\infty}$.
\begin{align}
\dn{\textbf{observe }b}(\kappa) & =
\left\{
\begin{array}{ll}
\kappa_{\infty} & \mbox{if }\kappa = \kappa_\infty\mbox{ or }\kappa(b) = \infty\mbox{, or}\\
\kappa_{b} & \mbox{otherwise.}\tag{\textbf{D6}}
\end{array}
\right.
\end{align}
We define the semantics of $\textbf{while }b\textbf{ do }\{s\}$
as the iterative execution of $\textbf{if }b$ $\textbf{then}$ $\{s\}$ $\textbf{else } \{\textbf{skip}\}$
until the rank of $b$ is $\infty$ (the loop terminates normally) or the result is undefined ($s$ does not terminate).
If neither of these conditions is ever met (i.e., if the $\textbf{while}$ statement loops endlessly) then the result is undefined.
\begin{align}
\hspace{-5pt}\dn{\textbf{while }b\textbf{ do }\hspace{-2pt}\{ s \}}(\kappa) &=
\left\{
\begin{array}{ll}
F_{b,s}^{n}(\kappa) \hspace{4pt} & \mbox{for the first }n\mbox{ s.t. }F_{b,s}^{n}(\kappa)(b) = \infty \mbox{, or }\\
\mbox{undef.} & \mbox{if there is no such }n,
\end{array}\hspace{-5pt}\tag{\textbf{D7}}
\right.
\end{align}
where $F_{b,s}: K_{\bot} \rightarrow K_{\bot}$ is defined by $F_{b,s}(\kappa) = \dn{\textbf{if }b\textbf{ then }\{ s \}\textbf{ else } \{ \textbf{skip} \}}(\kappa)$.
Some remarks.
Firstly, the semantics of RankPL can be thought of as a ranking-based variation of the
Kozen's semantics of probabistic programs~\cite{DBLP:journals/jcss/Kozen81} (i.e., replacing $\times$ with $+$ and $+$ with $min$).
Secondly, a RankPL implementation does not need to compute complete rankings.
Our implementation discussed in section~\ref{sec:implementation} follows a \emph{most-plausible-first} execution strategy:
different alternatives are explored in ascending order w.r.t. rank,
and higher-ranked alternatives need not be explored if knowing the lowest-ranked outcomes is enough,
as is often the case.\looseness=-1
\subsubsection{Example}
Consider the \emph{two-bit full adder} circuit shown in figure~\ref{fig:adder}.
It contains two \emph{XOR} gates $X_1, X_2$, two \emph{AND} gates $A_1, A_2$ and an \emph{OR} gate $O_1$.
The function of this circuit is to generate a binary representation $(\texttt{b}_1,\texttt{b}_2)$
of the number of inputs among $\texttt{a}_1$, $\texttt{a}_2$, $\texttt{a}_3$ that are high.
The \emph{circuit diagnosis problem} is about explaining observed incorrect behavior by finding minimal sets of gates that, if faulty, cause this behavior.\footnote{See Halpern~\cite[Chapter~9]{DBLP:books/daglib/0014219} for a similar treatment of this example.}
\begin{figure}
\centering
\begin{tikzpicture}[circuit logic US, minimum height=4mm]
\matrix[column sep=10mm,row sep=-1.5mm]
{ \node[yshift=3pt] (i0) {$\texttt{a}_1$};& \node [xor gate] (xor1) {\small$X_1$}; & \node[yshift=-3pt,xor gate] (xor2) {\small$X_2$}; & & \node[yshift=-3pt] (o1) {$\texttt{b}_2$}; \\
\node[yshift=2pt] (i1) {$\texttt{a}_2$};& & \node (cin) {}; & & \\
\node[yshift=-3pt] (i2) {$\texttt{a}_3$};& & \node[and gate] (and1) {\small$A_2$}; & & \\
& \node[and gate,yshift=-3pt] (and2) {\small$A_1$}; & & \node[or gate] (or1) {\small$O_1$}; & \node (o2) {$\texttt{b}_1$};\\ };
\draw[>-] (i0.east) -- ++(right:3mm) |- (xor1.input 1);
\draw[*-] (xor1.input 1) ++(-5mm,0.8mm) |- (and2.input 2);
\draw[*-] (xor1.input 2) ++(-2.5mm,0.8mm) |- (and2.input 1) ;
\draw[>-] (i1.east) -- ++(right:3mm) |- (xor1.input 2);
\draw (xor1.output) -- ++(right:3mm) |- (xor2.input 1) node[midway,above,xshift=4pt,yshift=0pt] {$\texttt{l}_1$};
\draw[*-] (and1.input 1) ++(-5mm,-2.8mm) |- (xor2.input 2);
\draw[*-] (xor1.output) ++(3mm,0.9mm) |- (and1.input 1);
\draw[>-] (i2.east) -- ++(right:15mm) |- (and1.input 2);
\draw[->] (xor2.output) -- ++(right:3mm) |- (o1.west);
\draw (and1.output) -- ++(right:3mm) |- (or1.input 1) node[midway, above right ] {$\texttt{l}_3$};
\draw (and2.output) -- ++(right:3mm) |- (or1.input 2) node[near end,below right,xshift=+16pt] {$\texttt{l}_2$};
\draw[->] (or1.output) -- (o2.west);
\end{tikzpicture}
\caption{A two-bit full adder}
\label{fig:adder}
\end{figure}
The listing below shows a RankPL solution.
On line 1 we set the constants \texttt{L} and \texttt{H} (representing a \emph{low} and \emph{high} signal); and $\texttt{OK}$ and $\texttt{FAIL}$ (to represent the state of a gate).
Line 2 encodes the space of possible inputs (\texttt{L} or \texttt{H}, equally likely).
The failure variables $\texttt{fa}_1$, $\texttt{fa}_2$, $\texttt{fo}_1$, $\texttt{fx}_2$ and $\texttt{fx}_2$ represent the events of individual gates failing and are set on line 3.
Here, we assume that failure is surprising to degree 1.
The circuit's logic is encoded on lines 4-8, where the output of a failing gate is arbitrarily set to \texttt{L} or \texttt{H}.
Note that $\oplus$ stands for XOR.
At the end we observe $\phi$.
\noindent \begin{center}
\small
\begin{tabular}{rl}
\hline
1 & $\texttt{L} := 0; \texttt{H} := 1; \texttt{OK} := 0; \texttt{FAIL} := 1;$ \\
2 & $\texttt{a}_1 := (\texttt{L} \Rank{0} \texttt{H}); \texttt{a}_2 := (\texttt{L} \Rank{0} \texttt{H}); \texttt{a}_3 := (\texttt{L} \Rank{0} \texttt{H});$ \\
3 & $\texttt{fx}_1 := (\texttt{OK} \Rank{1} \texttt{FAIL}); \texttt{fx}_2 := (\texttt{OK} \Rank{1} \texttt{FAIL}); \texttt{fa}_1 := (\texttt{OK} \Rank{1} \texttt{FAIL}); $ \\
& \hspace{20pt}$\texttt{fa}_2 := (\texttt{OK} \Rank{1} \texttt{FAIL}); \texttt{fo}_1 := (\texttt{OK} \Rank{1} \texttt{FAIL}); $ \\
4 & $\textbf{if }\texttt{fx}_1 = \texttt{OK}\textbf{ then }\texttt{l}_1 := \texttt{a}_1 \oplus \texttt{a}_2\textbf{ else } \texttt{l}_1 := (\texttt{L} \Rank{0} \texttt{H});$ \\
5 & $\textbf{if }\texttt{fa}_1 = \texttt{OK}\textbf{ then }\texttt{l}_2 := \texttt{a}_1 \wedge \texttt{a}_2\textbf{ else } \texttt{l}_2 := (\texttt{L} \Rank{0} \texttt{H});$ \\
6 & $\textbf{if }\texttt{fa}_2 = \texttt{OK}\textbf{ then }\texttt{l}_3 := \texttt{l}_1 \wedge \texttt{a}_3\textbf{ else } \texttt{l}_3 := (\texttt{L} \Rank{0} \texttt{H});$ \\
7 & $\textbf{if }\texttt{fx}_2 = \texttt{OK}\textbf{ then }\texttt{b}_2 := \texttt{l}_1 \oplus \texttt{a}_3\textbf{ else } \texttt{b}_2 := (\texttt{L} \Rank{0} \texttt{H});$ \\
8 & $\textbf{if }\texttt{fo}_1 = \texttt{OK}\textbf{ then }\texttt{b}_1 := \texttt{l}_3 \vee \texttt{l}_2\textbf{ else } \texttt{b}_1 := (\texttt{L} \Rank{0} \texttt{H});$ \\
9 & $\textbf{observe}\textbf{ }\phi;$\\
\hline
\end{tabular} \end{center}
The different valuations of the failure variables produced by this program
represent explanations for the observation $\phi$, ranked according to plausibility.
Suppose we observe that the input $(\texttt{a}_1,\texttt{a}_2,\texttt{a}_3)$ is valued (\texttt{L},\texttt{L},\texttt{H}) while the output $(\texttt{b}_1,\texttt{b}_2)$ is incorrectly valued (\texttt{H},\texttt{L}) instead of (\texttt{L},\texttt{H}).
Thus, we set $\phi$ to $\texttt{a}_1 = \texttt{L} \wedge \texttt{a}_2 = \texttt{L} \wedge \texttt{a}_3 = \texttt{H} \wedge \texttt{b}_1 = \texttt{H} \wedge \texttt{b}_2 = \texttt{L}$.
The program then produces one outcome ranked 0, namely
$(\texttt{fa}_1,\texttt{fa}_2,\texttt{fo}_1,\texttt{fx}_2,\texttt{fx}_2) = (\texttt{OK},\texttt{OK},\texttt{OK},\texttt{FAIL},\texttt{OK})$.
That is, $\phi$ is most plausibly explained by failure of $X_1$.
Other outcomes are ranked higher than 0 and represent explanations involving more than one faulty gate.\looseness=-1
\section{Noisy Observation and Iterated Revision}\label{sec:noisy}
Conditioning by $A$ means that $A$ becomes believed with infinite firmness.
This is undesirable if we have to deal with iterated belief change or noisy or untrustworthy observations,
since we cannot, after conditioning on $A$, condition on events inconsistent with $A$.
\emph{J-conditioning}~\cite{goldszmidt1996qualitative} is a more general form of conditioning that addresses this problem.
It is parametrized by a rank $x$ that indicates the firmness with which the evidence must be believed.
\begin{definition}\label{defn:resultoriented}
Let $A \in \Omega$, $\kappa$ a ranking function over $\Omega$ such that $\kappa(A), \kappa(\overline A) < \infty$, and $x$ a rank.
The \emph{J-conditioning} of $\kappa$ by $A$ with strength $x$, denoted by $\kappa_{A \rightarrow x}$, is defined by
$\kappa_{A \rightarrow x}(B) = min ( \kappa(B | A), \kappa(B | \overline A) + x ).$
\end{definition}
In words, the effect of J-conditioning by $A$ with strength $x$ is that $A$ becomes believed with firmness $x$.
This permits iterated belief change, because the rank of $\overline A$ is shifted up only by a finite number of ranks
and hence can be shifted down again afterwards.
Instead of introducing a special statement representing J-conditioning, we show that we can already express it in RankPL,
using ranked choice and observation as basic building blocks.
Below we write $\kappa_{b \rightarrow x}$ as shorthand for $\kappa_{\mods{\kappa}{b} \rightarrow x}$.
Proofs are omitted due to space constraints.
\begin{theorem}
Let $b$ be a boolean expression and $\kappa$ a ranking function s.t. $\kappa(b) < \infty$ and $\kappa(\neg b) < \infty$.
We then have
$\kappa_{b \rightarrow x} = \dn{\{\emph{\textbf{observe }}b \}\Rank{x} \{\emph{\textbf{observe }}\neg b}(\kappa)\}.$
\end{theorem}
\emph{L-conditioning}~\cite{goldszmidt1996qualitative}
is another kind of generalized conditioning.
Here, the parameter $x$ characterizes the `impact' of the evidence.
\begin{definition}\label{defn:evidenceoriented}
Let $A \in \Omega$, $\kappa$ a ranking function over $\Omega$ such that $\kappa(A), \kappa(\overline A) < \infty$, and $x$ a rank.
The \emph{L-conditioning} of $\kappa$ is denoted by $\kappa_{A \uparrow x}$ and is defined by
$\kappa_{A \uparrow x}(B) = min ( \kappa(A \cap B) - y, \kappa(\neg A \cap B) + x - y ),$
where $y = min(\kappa(A), x).$
\end{definition}
Thus, L-conditioning by $A$ with strength $x$ means that $A$ improves by $x$ ranks w.r.t. the rank of $\neg A$.
Unlike J-conditioning, L-conditioning satisfies two properties
that are desirable for modeling noisy observation:
\textit{reversibility} ($(\kappa_{A \uparrow x})_{\overline{A} \uparrow x} = \kappa$) and
\textit{commutativity} ($(\kappa_{A \uparrow x})_{B \uparrow x} = (\kappa_{B \uparrow x})_{A \uparrow x}$)~\cite{DBLP:books/daglib/0035277}.
We can expression L-conditioning in RankPL using ranked choice, observation and the rank expression as basic building blocks.
Like before, we write $\kappa_{b \uparrow x}$ to denote $\kappa_{\mods{\kappa}{b} \uparrow x}$.
\begin{theorem}\label{thm:evidenceoriented}
Let $b$ be a boolean expression, $\kappa$ a ranking function over $\Omega$ such that $\kappa(b), \kappa(\neg b) < \infty$, and $x$ a rank.
We then have:
$$\kappa_{b \uparrow x} = D
\left\llbracket\begin{tabular}{c}
\mbox{$\emph{\textbf{if }}(\emph{\textbf{rank}}(b) \leq x)\emph{\textbf{ then}}$} \\
\mbox{$\{\emph{\textbf{observe }}b \} \Rank{x - \emph{\textbf{rank}}(b) + \emph{\textbf{rank}}(\neg b)} \{ \emph{\textbf{observe }}\neg b\}$} \\
\mbox{$\emph{\textbf{else}}$} \\
\mbox{$\{\emph{\textbf{observe }}\neg b\} \Rank{\emph{\textbf{rank}}(b) - x} \{\emph{\textbf{observe }}b\}$} \\
\end{tabular}\right\rrbracket(\kappa)$$
\end{theorem}
In what follows we use the statement $\textbf{observe}_{\textbf{L}}(x, b)$ as shorthand for the statement that represents L-conditioning as defined in theorem~\ref{thm:evidenceoriented}.
\textbf{Example} This example involves both iterated revision and noisy observation.
A robot navigates a grid world and has to determine its location using a map and two distance sensors.
Figure~\ref{fig:localization} depicts the map that we use.
Gray cells represent walls and other cells are empty (ignoring, for now, the red cells and dots).
The sensors (oriented north and south) measure the distance to the nearest wall or obstacle.
To complicate matters, the sensor readings might not match the map.
For example, the $X$ in figure~\ref{fig:localization} marks an obstacle that affects sensor readings, but as far as the robot knows, this cell is empty.\looseness=-1
The listing below shows a RankPL solution.
The program takes as input:
(1) A map, held by an array $\texttt{map}$ of size $\texttt{m}\times \texttt{n}$,
storing the content of each cell (0 = empty, 1 = wall);
(2) An array $\texttt{mv}$ (length $\texttt{k}$) of movements (\texttt{N}/\texttt{E}/\texttt{S}/\texttt{W} for north/east/south/west) at given time points; and
(3) Two arrays $\texttt{ns}$ and $\texttt{ss}$ (length $\texttt{k}$) with distances read by the north and south sensor at given time points.
Note that, semantically, arrays are just indexed variables.\looseness=-1
\noindent \begin{center}
\small
\begin{tabular}{rl}
\hline
Input: & $\texttt{k}$: number of steps to simulate\\
Input: & $\texttt{mv}$: array (size $\geq \texttt{k}$) of movements (\texttt{N}/\texttt{E}/\texttt{S}/\texttt{W}) \\
Input: & \texttt{ns} and \texttt{ss}: arrays (size $\geq \texttt{k}$) of north and south sensor readings \\
Input: & $\texttt{map}$: 2D array (size \texttt{m} $\times$ \texttt{n}) encoding the map \\
1 & $\texttt{t} := 0; \texttt{ x} := 0 \Rank{0} \{1 \Rank{0} \{ 2 \Rank{0} \ldots \texttt{m}\} \}; \texttt{ y} := 0 \Rank{0} \{1 \Rank{0} \{ 2 \Rank{0} \ldots \texttt{n}\} \};$ \\
2 & $\textbf{while }(\texttt{t} < \texttt{k})\textbf{ do } \{$ \\
3 & \hspace{20pt} $\textbf{if }(\texttt{mv}[\texttt{t}] = \texttt{N})\textbf{ then } \texttt{y} := \texttt{y} + 1$ \\
4 & \hspace{20pt} $\textbf{else if }(\texttt{mv}[\texttt{t}] = \texttt{S})\textbf{ then } \texttt{y} := \texttt{y} - 1$ \\
5 & \hspace{20pt} $\textbf{else if }(\texttt{mv}[\texttt{t}] = \texttt{W})\textbf{ then } \texttt{x} := \texttt{x} - 1$ \\
6 & \hspace{20pt} $\textbf{else if }(\texttt{mv}[\texttt{t}] = \texttt{E})\textbf{ then } \texttt{x} := \texttt{x} + 1 \textbf{ else skip;}$ \\
7 & \hspace{20pt} $\texttt{nd} := 0; \textbf{ while }\texttt{map}[\texttt{x}][\texttt{y} + \texttt{nd} + 1] = 0\textbf{ do }\texttt{nd} := \texttt{nd} + 1;$ \\
8 & \hspace{20pt} $\textbf{observe}_{\textbf{L}}(1, \texttt{nd} = \texttt{ns}[\texttt{t}]);$ \\
9 & \hspace{20pt} $\texttt{sd} := 0; \textbf{ while }\texttt{map}[\texttt{x}][\texttt{y} - \texttt{sd} - 1] = 0\textbf{ do }\texttt{sd} := \texttt{sd} + 1;$ \\
10 & \hspace{20pt} $\textbf{observe}_{\textbf{L}}(1, \texttt{sd} = \texttt{ss}[\texttt{t}]);$ \\
11 & \hspace{20pt} $\texttt{t} := \texttt{t} + 1;$\\
12 & $\}$ \\
\hline
\end{tabular}
\end{center}
The program works as follows.
On line 1 the time point $\texttt{t}$ is set to 0 and the robot's location $(\texttt{x}$, $\texttt{y})$ is set to a randomly chosen coordinate (all equally likely) using nested ranked choice statements.
Inside the \textbf{while} loop, which iterates over $\texttt{t}$,
we first process the movement $\texttt{mv}[\texttt{t}]$ (lines 3-6).
We then process (lines 7-8) the north sensor reading,
by counting empty cells between the robot and nearest wall,
the result of which is observed to equal $\texttt{ns}[\texttt{t}]$---and likewise for the south sensor (lines 9-10).
We use L-conditioning with strength 1 to account for possible incorrect observations.
On line 11 we update $\texttt{t}$.\looseness=-1
\noindent \begin{figure}[t]
\centering
\begin{subfigure}[t]{0.24\columnwidth}
\label{fig:localizationA}
\centering
\begin{tikzpicture}[box/.style={rectangle,draw=black,thin, minimum size=1cm,scale=0.22},scale=0.22]
\foreach \x in {0,1,...,10}{
\foreach \y in {0,1,...,7}
\node[box] at (\x,\y){};
}
\foreach \i in {0,...,10} \node[below=3pt] at (\i,\yMin) {\scriptsize$\i$};
\foreach \i in {0,...,7} \node[left=3pt] at (\xMin,\i) {\scriptsize$\i$};
\node[box,fill=gray] at (0,0){};\node[box,fill=gray] at (1,0){};\node[box,fill=gray] at (2,0){};\node[box,fill=gray] at (3,0){};\node[box,fill=gray] at (4,0){};\node[box,fill=gray] at (5,0){};\node[box,fill=gray] at (6,0){};\node[box,fill=gray] at (7,0){};\node[box,fill=gray] at (8,0){};\node[box,fill=gray] at (9,0){};\node[box,fill=gray] at (10,0){};\node[box,fill=gray] at (0,1){};\node[box,fill=gray] at (1,1){};\node[box,fill=gray] at (2,1){};\node[box,fill=gray] at (3,1){};\node[box,fill=gray] at (4,1){};\node[box,fill=gray] at (5,1){};\node[box,fill=gray] at (6,1){};\node[box,fill=gray] at (7,1){};\node[box,fill=gray] at (8,1){};\node[box,fill=gray] at (9,1){};\node[box,fill=gray] at (10,1){};\node[box,fill=gray] at (0,2){};\node[box,fill=gray] at (1,2){};\node[box,fill=gray] at (2,2){};\node[box,fill=gray] at (3,2){};\node[box,fill=gray] at (6,2){};\node[box,fill=gray] at (7,2){};\node[box,fill=gray] at (8,2){};\node[box,fill=gray] at (9,2){};\node[box,fill=gray] at (10,2){};\node[box,fill=gray] at (5,6){};\node[box,fill=gray] at (6,6){};\node[box,fill=gray] at (7,6){};\node[box,fill=gray] at (8,6){};\node[box,fill=gray] at (9,6){};\node[box,fill=gray] at (10,6){};\node[box,fill=gray] at (0,7){};\node[box,fill=gray] at (1,7){};\node[box,fill=gray] at (2,7){};\node[box,fill=gray] at (3,7){};\node[box,fill=gray] at (4,7){};\node[box,fill=gray] at (5,7){};\node[box,fill=gray] at (6,7){};\node[box,fill=gray] at (7,7){};\node[box,fill=gray] at (8,7){};\node[box,fill=gray] at (9,7){};\node[box,fill=gray] at (10,7){};Iteration took 1 ms
\node[font=\scriptsize] at (2,3) {X};
\node[box,fill=red] at (1,5){};\node[box,fill=red] at (5,4){};\node[box,fill=red] at (2,5){};\node[box,fill=red] at (3,5){};
\fill (1,5) circle[radius=5pt] node {};
\end{tikzpicture}
\caption{$k = 1$}
\end{subfigure}
\begin{subfigure}[t]{0.24\columnwidth}
\label{fig:localizationB}
\centering
\begin{tikzpicture}[box/.style={rectangle,draw=black,thin, minimum size=1cm,scale=0.22},scale=0.22]
\foreach \x in {0,1,...,10}{
\foreach \y in {0,1,...,7}
\node[box] at (\x,\y){};
}
\foreach \i in {0,...,10} \node[below=3pt] at (\i,\yMin) {\scriptsize$\i$};
\foreach \i in {0,...,7} \node[left=3pt] at (\xMin,\i) {\scriptsize$\i$};
\node[box,fill=gray] at (0,0){};\node[box,fill=gray] at (1,0){};\node[box,fill=gray] at (2,0){};\node[box,fill=gray] at (3,0){};\node[box,fill=gray] at (4,0){};\node[box,fill=gray] at (5,0){};\node[box,fill=gray] at (6,0){};\node[box,fill=gray] at (7,0){};\node[box,fill=gray] at (8,0){};\node[box,fill=gray] at (9,0){};\node[box,fill=gray] at (10,0){};\node[box,fill=gray] at (0,1){};\node[box,fill=gray] at (1,1){};\node[box,fill=gray] at (2,1){};\node[box,fill=gray] at (3,1){};\node[box,fill=gray] at (4,1){};\node[box,fill=gray] at (5,1){};\node[box,fill=gray] at (6,1){};\node[box,fill=gray] at (7,1){};\node[box,fill=gray] at (8,1){};\node[box,fill=gray] at (9,1){};\node[box,fill=gray] at (10,1){};\node[box,fill=gray] at (0,2){};\node[box,fill=gray] at (1,2){};\node[box,fill=gray] at (2,2){};\node[box,fill=gray] at (3,2){};\node[box,fill=gray] at (6,2){};\node[box,fill=gray] at (7,2){};\node[box,fill=gray] at (8,2){};\node[box,fill=gray] at (9,2){};\node[box,fill=gray] at (10,2){};\node[box,fill=gray] at (5,6){};\node[box,fill=gray] at (6,6){};\node[box,fill=gray] at (7,6){};\node[box,fill=gray] at (8,6){};\node[box,fill=gray] at (9,6){};\node[box,fill=gray] at (10,6){};\node[box,fill=gray] at (0,7){};\node[box,fill=gray] at (1,7){};\node[box,fill=gray] at (2,7){};\node[box,fill=gray] at (3,7){};\node[box,fill=gray] at (4,7){};\node[box,fill=gray] at (5,7){};\node[box,fill=gray] at (6,7){};\node[box,fill=gray] at (7,7){};\node[box,fill=gray] at (8,7){};\node[box,fill=gray] at (9,7){};\node[box,fill=gray] at (10,7){};Iteration took 1 ms
\node[font=\scriptsize] at (2,3) {X};
\node[box,fill=red] at (6,4){};
\fill (2,5) circle[radius=5pt] node {};
\end{tikzpicture}
\caption{$k = 2$}
\end{subfigure}
\begin{subfigure}[t]{0.24\columnwidth}
\label{fig:localizationC}
\centering
\begin{tikzpicture}[box/.style={rectangle,draw=black,thin, minimum size=1cm,scale=0.22},scale=0.22]
\foreach \x in {0,1,...,10}{
\foreach \y in {0,1,...,7}
\node[box] at (\x,\y){};
}
\foreach \i in {0,...,10} \node[below=3pt] at (\i,\yMin) {\scriptsize$\i$};
\foreach \i in {0,...,7} \node[left=3pt] at (\xMin,\i) {\scriptsize$\i$};
\node[box,fill=gray] at (0,0){};\node[box,fill=gray] at (1,0){};\node[box,fill=gray] at (2,0){};\node[box,fill=gray] at (3,0){};\node[box,fill=gray] at (4,0){};\node[box,fill=gray] at (5,0){};\node[box,fill=gray] at (6,0){};\node[box,fill=gray] at (7,0){};\node[box,fill=gray] at (8,0){};\node[box,fill=gray] at (9,0){};\node[box,fill=gray] at (10,0){};\node[box,fill=gray] at (0,1){};\node[box,fill=gray] at (1,1){};\node[box,fill=gray] at (2,1){};\node[box,fill=gray] at (3,1){};\node[box,fill=gray] at (4,1){};\node[box,fill=gray] at (5,1){};\node[box,fill=gray] at (6,1){};\node[box,fill=gray] at (7,1){};\node[box,fill=gray] at (8,1){};\node[box,fill=gray] at (9,1){};\node[box,fill=gray] at (10,1){};\node[box,fill=gray] at (0,2){};\node[box,fill=gray] at (1,2){};\node[box,fill=gray] at (2,2){};\node[box,fill=gray] at (3,2){};\node[box,fill=gray] at (6,2){};\node[box,fill=gray] at (7,2){};\node[box,fill=gray] at (8,2){};\node[box,fill=gray] at (9,2){};\node[box,fill=gray] at (10,2){};\node[box,fill=gray] at (5,6){};\node[box,fill=gray] at (6,6){};\node[box,fill=gray] at (7,6){};\node[box,fill=gray] at (8,6){};\node[box,fill=gray] at (9,6){};\node[box,fill=gray] at (10,6){};\node[box,fill=gray] at (0,7){};\node[box,fill=gray] at (1,7){};\node[box,fill=gray] at (2,7){};\node[box,fill=gray] at (3,7){};\node[box,fill=gray] at (4,7){};\node[box,fill=gray] at (5,7){};\node[box,fill=gray] at (6,7){};\node[box,fill=gray] at (7,7){};\node[box,fill=gray] at (8,7){};\node[box,fill=gray] at (9,7){};\node[box,fill=gray] at (10,7){};Iteration took 1 ms
\node[font=\scriptsize] at (2,3) {X};
\node[box,fill=red] at (3,5){};\node[box,fill=red] at (7,4){};
\fill (3,5) circle[radius=5pt] node {};
\end{tikzpicture}
\caption{$k = 3$}
\end{subfigure}
\begin{subfigure}[t]{0.24\columnwidth}
\label{fig:localizationD}
\centering
\begin{tikzpicture}[box/.style={rectangle,draw=black,thin, minimum size=1cm,scale=0.22},scale=0.22]
\foreach \x in {0,1,...,10}{
\foreach \y in {0,1,...,7}
\node[box] at (\x,\y){};
}
\foreach \i in {0,...,10} \node[below=3pt] at (\i,\yMin) {\scriptsize$\i$};
\foreach \i in {0,...,7} \node[left=3pt] at (\xMin,\i) {\scriptsize$\i$};
\node[box,fill=gray] at (0,0){};\node[box,fill=gray] at (1,0){};\node[box,fill=gray] at (2,0){};\node[box,fill=gray] at (3,0){};\node[box,fill=gray] at (4,0){};\node[box,fill=gray] at (5,0){};\node[box,fill=gray] at (6,0){};\node[box,fill=gray] at (7,0){};\node[box,fill=gray] at (8,0){};\node[box,fill=gray] at (9,0){};\node[box,fill=gray] at (10,0){};\node[box,fill=gray] at (0,1){};\node[box,fill=gray] at (1,1){};\node[box,fill=gray] at (2,1){};\node[box,fill=gray] at (3,1){};\node[box,fill=gray] at (4,1){};\node[box,fill=gray] at (5,1){};\node[box,fill=gray] at (6,1){};\node[box,fill=gray] at (7,1){};\node[box,fill=gray] at (8,1){};\node[box,fill=gray] at (9,1){};\node[box,fill=gray] at (10,1){};\node[box,fill=gray] at (0,2){};\node[box,fill=gray] at (1,2){};\node[box,fill=gray] at (2,2){};\node[box,fill=gray] at (3,2){};\node[box,fill=gray] at (6,2){};\node[box,fill=gray] at (7,2){};\node[box,fill=gray] at (8,2){};\node[box,fill=gray] at (9,2){};\node[box,fill=gray] at (10,2){};\node[box,fill=gray] at (5,6){};\node[box,fill=gray] at (6,6){};\node[box,fill=gray] at (7,6){};\node[box,fill=gray] at (8,6){};\node[box,fill=gray] at (9,6){};\node[box,fill=gray] at (10,6){};\node[box,fill=gray] at (0,7){};\node[box,fill=gray] at (1,7){};\node[box,fill=gray] at (2,7){};\node[box,fill=gray] at (3,7){};\node[box,fill=gray] at (4,7){};\node[box,fill=gray] at (5,7){};\node[box,fill=gray] at (6,7){};\node[box,fill=gray] at (7,7){};\node[box,fill=gray] at (8,7){};\node[box,fill=gray] at (9,7){};\node[box,fill=gray] at (10,7){};Iteration took 1 ms
\node[font=\scriptsize] at (2,3) {X};
\node[box,fill=red] at (4,5){};
\fill (4,5) circle[radius=5pt] node {};
\end{tikzpicture}
\caption{$k = 4$}
\end{subfigure}
\caption{Most plausible inferred locations during four iterations}
\label{fig:localization}
\end{figure}
Suppose we want to simulate a movement from $(0,5)$ to $(4, 5)$.
Thus, we use the inputs $\texttt{mv} = \{ E, E, E, E \}$, $\texttt{ns} = \{ 1, 1, 1, 1 \}$, $\texttt{ss} = \{ 2, 1, 2, 3 \}$,
while $\texttt{map}$ is set as shown in figure~\ref{fig:localization} (i.e., 1 for every cell containing a wall, 0 for every other cell).
Note that the observed distances stored in $\texttt{ns}$ and $\texttt{ss}$ are consistent with the distances observed along this path,
where, at $t = 1$, the south sensor reads a distance of 1 instead of 2, due to the obstacle $X$.
The different values of $\texttt{x},\texttt{y}$ generated by this program encode possible locations of the robot, ranked according to plausibility.
The dots in figure~\ref{fig:localization} show the actual locations, while the red cells represent the inferred most plausible (i.e., rank zero) locations generated by the program.
Terminating after $t = 0$ (i.e., setting $k$ to $1$) yields four locations, all consistent with the observed distances 1 (north) and 2 (south).
If we terminate after $t = 1$, the robot wrongly believes to be at (6, 4), due to having observed the obstacle.
However, if we terminate after $t = 3$, the program produces the actual location.\looseness=-1
Note that using L-conditioning here is essential.
Regular conditioning would cause failure after the third iteration.
We could also have used J-conditioning, which gives different rankings of intermediate results.
\section{Implementation}\label{sec:implementation}
A RankPL interpreter written in Java can be found at \url{http://github.com/tjitze/RankPL}.
It runs programs written using the syntax described in this paper,
or constructed using Java classes that map to this syntax.
The latter makes it possible to embed RankPL programs inside Java code
and to make it interact with and use classes and methods written Java.
The interpreter is faithful to the semantics described in section~\ref{sec:rpl} and
implements the \emph{most-plausible-first} execution strategy discussed in section~\ref{sec:formalsemantics}.
All examples discussed in this paper are included, as well as a number of additional examples.
\section{Conclusion and Future Work}\label{sec:conclusion}
We have introduced RankPL, a language semantically similar to probabilistic programming,
but based on Spohn's ranking theory,
and demonstrated its utility using examples involving abduction and iterated revision.
We believe that the approach has great potential for applications
where PPLs are too demanding due to their computational complexity and dependence on precise probability values.
Moreover, we hope that our approach will generate a broader and more practical scope for the topics of
ranking theory and belief revision which, in the past, have been studied mostly from purely theoretical perspectives.
A number of aspects were not touched upon and will be addressed in future work.
This includes a more fine grained characterization of termination
and a discussion of the relationship with nondeterministic programming,
which is a special case of RankPL.
Furthermore, we have omitted examples to show that RankPL
subsumes ranking networks
and can be used to reason about causal rules and actions~\cite{goldszmidt1996qualitative}.
We also did not contrast our approach
with default reasoning formalisms that use ranking theory as a semantic foundation (see, e.g.,~\cite{DBLP:conf/tark/Pearl90}).
Even though we demonstrated that RankPL is expressive enough to solve
fairly complex tasks in a compact manner, it is a very basic language that is best regarded as proof of concept.
In principle, the approach can be applied to any programming language,
whether object-oriented, functional, or LISP-like.
Doing so would make it possible to reason about ranking-based models expressed using, for example, recursion and complex data structures.
These features are also supported by PPLs such as Church~\cite{DBLP:conf/uai/GoodmanMRBT08},
Venture~\cite{DBLP:journals/corr/MansinghkaSP14} and Figaro~\cite{pfeffer2009figaro}.
|
1,116,691,499,286 | arxiv |
\section{}
\subsection{}
\subsubsection{}
|
1,116,691,499,287 | arxiv | \section{Introduction}
It is well known that Poisson-Nijenhuis (PN) manifolds \cite{MagriMorosiRagnisco85,KM} are an important notion in the theory of integrable systems. Roughly speaking, they are Poisson manifold $({\mathcal M},\pi)$ endowed with a tensor field of type $(1,1)$, say $N:T{\mathcal M}\to T{\mathcal M}$, which is torsionless and compatible (see Section \ref{subsec:traces}) with the Poisson tensor $\pi$. They turn out to be bi-Hamiltonian manifolds, with the traces of the powers of $N$ satisfying the Lenard-Magri relations and thus being in involution with respect to the Poisson brackets induced by the Poisson tensors. An example of integrable system that can be studied in the context of PN manifolds is the open (or non periodic) $n$-particle Toda lattice. (For both the periodic and the non periodic Toda system, see \cite{Perelomov-book} and references therein; see also \cite{Damianou04,MorosiPizzocchero96,Okubo90}.)
The PN structure of the open Toda lattice was presented in \cite{DO}. Its Poisson tensor is non degenerate, so that the PN manifold is a symplectic manifold (sometimes it is called an $\omega$N-manifold). This kind of geometrical structure was shown to play an important role in the bi-Hamiltonian interpretation of the separation of variable method (see, e.g., \cite{FMP01,FP03}).
Poisson quasi-Nijenhuis (PqN) manifolds are an interesting generalization of PN manifolds. They were introduced in \cite{SX}, where the requirement about the vanishing of the (Nijenhuis) torsion of $N$ is weakened in a suitable sense, and the relations with quasi-Lie bialgebroid and symplectic Nijenhuis groupoids are investigated. In their Remark 3.13, the authors write: ``Poisson Nijenhuis structures arise naturally in the study of integrable systems. It would be interesting to find applications of Poisson quasi-Nijenhuis structures in integrable systems as well."
As far as we know, no progress in this direction was made until now.
The aim of this paper is to interpret the well known integrability of the closed Toda lattice in the framework of PqN manifolds. More precisely, we introduce a tensor field $N$ of type $(1,1)$ which is compatible with the canonical Poisson tensor and endows ${{\mathbb{R}}}^{2n}$ with the structure of a PqN manifold, and we show that the traces $I_k$ of the powers of $N$ are integrals of motion in involution. However, we discuss a class
of PqN manifolds clarifying that the involutivity of the $I_k$ does not hold in every PqN manifold.
The organization of this paper is the following. In Section \ref{subsec:traces} we recall the definitions of PN and PqN manifold, and we show how the classical Lenard-Magri recursion relations among the $I_k$ are modified in the PqN case. Section \ref{laclassediesempidiGiovanni} is devoted to a class of PqN structures on ${{\mathbb{R}}}^6$ depending on a potential $V$ and showing that the $I_k$ are in involution only for special choices of $V$. In Section \ref{toda-4} we consider the 4-particle closed Toda system with its PqN structure, performing some computations on the $I_k$ to prove that they are in involution. These results are generalized in Section \ref{FS}, while in
Section \ref{igor-connection} we present general results clarifying the relation between the PN structure of the open Toda lattice and the PqN structure of the closed one.
\par\smallskip\noindent
{\bf Acknowledgments.}
We wish to thank Yvette Kosmann-Schwarzbach, Franco Magri and, especially, Orlando Ragnisco for useful discussions.
MP thanks the {Dipartimento di Matematica e Applicazioni} of
Universit\`a Milano-Bicocca for its hospitality.
This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sk{\l}odowska-Curie grant no 778010 {\em IPaDEGAN}. All authors gratefully acknowledge the auspices of the GNFM Section of INdAM under which part of this work was carried out.
\section{Nijenhuis torsion and Poisson quasi-Nijenhuis manifolds}
\label{subsec:traces}
It is well known that the {\it Nijenhuis torsion\/} of a $(1,1)$ tensor field $N:T{\mathcal M}\to T{\mathcal M}$ on a manifold ${\mathcal M}$ is defined as
\begin{equation}
\label{tndef1}
T_N(X,Y)=[NX,NY]-N\left([NX,Y]+[X,NY]-N[X,Y]\right)\, .
\end{equation}
It can be written as
\begin{equation}
\label{tndef2}
T_N(X,Y)=(L_{NX}N-NL_XN)
Y\,,
\end{equation}
where, hereafter, $L_X$ denotes the Lie derivative with respect to the vector field $X$. Hence one arrives at the formula
\begin{equation}
\label{ff1}
N\, L_X N=L_{NX}N-i_XT_N\, ,
\end{equation}
where $i_XT_N$ is the $(1,1)$ tensor field obviously defined as $(i_XT_N)(Y)=T_N(X,Y)$. We recall that, given a $p$-form $\alpha$, with $p\ge 1$, one can construct another $p$-form $i_N\alpha$ as
\begin{equation}
\label{iNalpha}
i_N\alpha(X_1,\dots,X_p)=\sum_{i=1}^p \alpha(X_1,\dots,NX_i,\dots,X_p)\,,
\end{equation}
and that $i_N$ is a derivation
of degree zero (if $i_N f=0$ for all function $f$).
We also remind \cite{MagriMorosiRagnisco85} that $N:T{\mathcal M}\to T{\mathcal M}$ and a Poisson bivector $\pi:T^*{\mathcal M}\to T{\mathcal M}$ are said to be {\it compatible\/} if
\begin{equation}
\label{N-P-compatible}
\begin{split}
&N\pi=\pi N^*\,,\qquad
\mbox{where $N^*:T^*{\mathcal M}\to T^*{\mathcal M}$ is the transpose of $N$;}\\
&L_{\pi\alpha}(N) X-\pi L_{X}(N^*\alpha)+\pi L_{NX}\alpha=0\,,\qquad\mbox{for all 1-forms $\alpha$ and vector fields $X$.}
\end{split}
\end{equation}
Some nice interpretations of these compatibility conditions
were given in \cite{YKS96}. We will use one of them in Section \ref{igor-connection}.
In \cite{SX} a {\it Poisson quasi-Nijenhuis (PqN) manifold\/} was defined as a quadruple $({\mathcal M},\pi,N, \phi)$ such that
\begin{itemize}
\item the Poisson bivector $\pi$ and the $(1,1)$ tensor field $N$
are compatible;
\item the 3-forms $\phi$ and $i_N\phi$ are closed;
\item $T_N(X,Y)=\pi\left(i_{X\wedge Y}\phi\right)$ for all vector fields $X$ and $Y$, where $i_{X\wedge Y}\phi$ is the 1-form defined as
$\langle i_{X\wedge Y}\phi,Z\rangle=\phi(X,Y,Z)$.
\end{itemize}
The bivector field $\pi'=N\pi$ turns out to satisfy the conditions
\begin{equation}
\label{schouten-conditions}
[\pi,\pi']=0\,,\qquad [\pi',\pi']=2\pi(\phi)\, ,
\end{equation}
where $[\cdot,\cdot]$ is the Schouten bracket (see, e.g., \cite{Izu}) between bivectors and
$\pi(\phi)(\alpha,\beta,\gamma)=\phi(\pi\alpha,\pi\beta,\pi\gamma)$
for any triple of 1-forms $(\alpha,\beta,\gamma)$.
The following result, also proved in \cite{SX}, will be used in this paper.
\begin{prop}
\label{prop:piprime}
Let ${\mathcal M}$ be a manifold endowed with a non degenerate Poisson tensor $\pi$, a tensor field $N$ of type $(1,1)$, and a closed 3-form $\phi$.
If $N\pi=\pi N^*$ and conditions (\ref{schouten-conditions}) are satisfied (with $\pi'=N\pi$), then $({\mathcal M},\pi,N, \phi)$ is a PqN manifold.
\end{prop}
If $\phi=0$, then the torsion of $N$ vanishes and ${\mathcal M}$ becomes a {\it Poisson-Nijenhuis manifold} (see \cite{KM} and references therein). The bivector field $\pi'=N\pi$ is in this case a Poisson tensor compatible with $\pi$.
Moreover, the functions
\begin{equation}
\label{tracce}
I_k=\frac1{k}\operatorname{Tr}(N^k)\,,\qquad k=1,2,\dots\, ,
\end{equation}
satisfy $dI_{k+1}=N^* dI_{k}$, entailing the so-called {\it Lenard-Magri relations\/}
\begin{equation}
\label{LM-rel}
\pi dI_{k+1}=\pi' dI_{k}
\end{equation}
and therefore the involutivity of the $I_k$ (with respect to both Poisson brackets induced by $\pi$ and $\pi'$).
For a general PqN manifold ${\mathcal M}$, we will see in the next section
that such involutivity (with respect to the unique Poisson bracket defined on ${\mathcal M}$, i.e., the one associated with $\pi$) does not hold. Anyway, we have that, for $k\ge 2$ and for a generic vector field $X$ on ${\mathcal M}$,
\begin{equation}
\label{req1}
\begin{split}
\brke{dI_{k+1}}{X}&=L_X\left(\frac{1}{k+1} \operatorname{Tr}(N^{k+1})\right)=\operatorname{Tr}\left((NL_X N)N^{k-1}\right)\\
&\buildrel(\ref{ff1})\over=\operatorname{Tr}\left(L_{NX}(N)N^{k-1}\right)-\operatorname{Tr}\left((i_XT_N)\, N^{k-1}\right)\\
&=L_{NX}\left(\frac1{k} \operatorname{Tr}(N^{k})\right)-\operatorname{Tr}\left((i_XT_N)\, N^{k-1}\right)\\ &=\brke{dI_{k}}{NX}-\operatorname{Tr}\left((i_XT_N)\, N^{k-1}\right)\\
&=\brke{N^*dI_{k}}{X}-\operatorname{Tr}\left((i_XT_N)\, N^{k-1}\right)\, .
\end{split}
\end{equation}
So we arrive at the generalized Lenard-Magri relations
\begin{equation}
\label{ff2}
dI_{k+1}=N^* dI_{k}-{\phi_{k-1}}
\, ,
\end{equation}
where we used the definition
\begin{equation}
\label{varphi}
\brke{\phi_l}{ X}= \operatorname{Tr}\left((i_XT_N)\, N^{l}\right)= \operatorname{Tr}\left(N^{l}\, (i_XT_N)\right)\, ,\qquad l\ge 0\, .
\end{equation}
Notice that this definition, along with (\ref{ff2}), was used in \cite{Bogo96-180,Bogo96-182} for different purposes.
Let us compute now the Poisson bracket $\parpo{I_k}{I_j}$ for $k>j\ge 1$:
\begin{equation}
\label{rectr}
\begin{split}
\parpo{I_k}{I_j}&=\brke{dI_k}{\pi dI_j}\buildrel{(\ref{ff2})}\over=\brke{N^*d I_{k-1}}{ \pi dI_j}-\brke{\phi_{k-2}}{ \pi dI_j}=\brke{dI_{k-1}}{N\pi dI_j}-\brke{\phi_{k-2}}{ \pi dI_j}\\ &=\brke{dI_{k-1}}{\pi\, N^* dI_{j}}-\brke{\phi_{k-2}}{ \pi dI_j}\buildrel{(\ref{ff2})}\over=
\brke{dI_{k-1}}{\pi dI_{j+1}}+\brke{dI_{k-1}}{\pi \phi_{j-1}}-\brke{\phi_{k-2}}{ \pi dI_j}
\\ \noalign{\medskip}
&=\parpo{I_{k-1}}{I_{j+1}}-\left(\brke{ \phi_{j-1}}{\pi dI_{k-1}}+\brke{\phi_{k-2}}{ \pi dI_j}\right)\, .
\end{split}
\end{equation}
Thus, the usual formula
\begin{equation}
\label{LMrec}
\parpo{I_k}{I_j}=\parpo{I_{k-1}}{I_{j+1}}\,,
\end{equation}
entailed by the Lenard-Magri relations (\ref{LM-rel}), in the non vanishing torsion case is modified as follows:
\begin{equation}
\label{recadd}
\parpo{I_k}{I_j}-\parpo{I_{k-1}}{I_{j+1}}=-\left(\brke{ \phi_{j-1}}{\pi dI_{k-1}}+\brke{\phi_{k-2}}{ \pi dI_j}\right)\, .
\end{equation}
Actually, one can see that the 1-forms $\phi_l$ compute the Poisson brackets between the $I_j$.
Indeed, if we consider $k=j+1$, we obtain from
(\ref{recadd})
\begin{equation}
\label{exk=j+1}
\parpo{I_{j+1}}{I_j}=-\brke{ \phi_{j-1}}{\pi dI_{j}}\, .
\end{equation}
A necessary condition for the traces of the powers of $N$ to be in involution is thus $\brke{ \phi_{j-1}}{\pi dI_{j}}=0$ for all $j\ge 1$,
which explicitly reads
\begin{equation}\label{jj+1}
\operatorname{Tr}\left((i_{\pi dI_j} T_N)\, N^{j-1}\right)=0\, .
\end{equation}
However, imposing the condition that
\begin{equation}
\label{nocond}
\brke{ \phi_{k}}{\pi dI_{j}}=
\operatorname{Tr}\left((i_{\pi dI_j} T_N)\, N^{k}\right)=0
\end{equation}
for all $k,j$ (although being clearly
sufficient), is too restrictive: indeed, it fails in the simplest non trivial
case, namely, the closed Toda system with $4$ particles (see Section \ref{Toda4}).
Some further conditions can be written, which explain the above sentence in general. For example, if we take $k=j+2$ we obtain, still from
(\ref{recadd}),
\begin{equation}
\label{exk=j+2}
\parpo{I_{j+2}}{I_j}={\parpo{I_{j+1}}{I_{j+1}}}-\left(\brke{ \phi_{j-1}}{\pi dI_{j+1}}+\brke{\phi_{j}}{ \pi dI_j}\right)\, .
\end{equation}
To ensure that $\parpo{I_{j+2}}{I_j}$ be zero, no need that the last two terms in the right-hand side of the above equation be simultaneously vanishing. Indeed, the Toda closed chain with $4$ particles is already an example in which these two terms {\em cancel\/} each other without vanishing on their own.
\section{A class of non involutive PqN manifolds}
\label{laclassediesempidiGiovanni}
In this section we present a wide class of examples of PqN manifolds such that the traces (\ref{tracce}) are not in involution. Let us consider, on ${\mathcal M}={{\mathbb{R}}}^6$ with (canonical) variables $(q_{{1}},q_{{2}},q_{{3}},p_{{1}},p_{{2}},p_{{3}})$, the canonical Poisson tensor $\pi$ and the $(1,1)$ tensor field given by
\begin{equation}
\label{qN3-V}
N=\left[ \begin {array}{cccccc}
p_{{1}}&0&0&0&1&1\\ \noalign{\medskip}
0&p_{{2}}&0&-1&0&1\\ \noalign{\medskip}
0&0&p_{{3}}&-1&-1&0\\ \noalign{\medskip}
0&-V(q_1-q_2)&-V(q_3-q_1)&p_{{1}}&0&0\\ \noalign{\medskip}
V(q_1-q_2)&0&-V(q_2-q_3)&0&p_{{2}}&0\\ \noalign{\medskip}
V(q_3-q_1)&V(q_2-q_3)&0&0&0&p_{{3}}
\end {array} \right]\,,
\end{equation}
where $V$ is an arbitrary (differentiable) function of one variable.
First of all, we use Proposition \ref{prop:piprime} to show that $\pi$ and $N$ define, together with a suitable 3-form $\phi$, a PqN structure on
${{\mathbb{R}}}^6$. Indeed, if
\begin{equation}
\label{P1-V}
\pi'=N\pi=\left[ \begin {array}{cccccc}
0&-1&-1&p_{{1}}&0&0\\ \noalign{\medskip}
1&0&-1&0&p_{{2}}&0\\ \noalign{\medskip}
1&1&0&0&0&p_{{3}}\\ \noalign{\medskip}
-p_{{1}}&0&0&0&-V(q_1-q_2)&-V(q_3-q_1)\\ \noalign{\medskip}
0&-p_{{2}}&0&V(q_1-q_2)&0&-V(q_2-q_3)\\ \noalign{\medskip}
0&0&-p_{{3}}&V(q_3-q_1)&V(q_2-q_3)&0
\end {array}\right]\,,
\end{equation}
then one can easily show that $[\pi,\pi']=0$, so that the first of (\ref{schouten-conditions}) holds. Moreover, we have that
\begin{equation}
\label{Scbr-V}
\begin{split}
[\pi',\pi']&=2\left(V(q_1-q_2)-V'(q_1-q_2)\right)\partial_{p_2}\wedge \partial_{p_1}\wedge \left(\partial_{q_1}+\partial_{q_2}\right)\\
&+2\left(V(q_2-q_3)-V'(q_2-q_3)\right)\partial_{p_3}\wedge \partial_{p_2}\wedge \left(\partial_{q_2}+\partial_{q_3}\right)\\
&+2\left(V(q_3-q_1)+V'(q_3-q_1)\right)\partial_{p_3}\wedge \partial_{p_1}\wedge \left(\partial_{q_1}+\partial_{q_3}\right)\\
&+4 V'(q_3-q_1)\partial_{p_3}\wedge \partial_{p_1}\wedge \partial_{q_2}\,.
\end{split}
\end{equation}
Therefore, the 3-form $\phi$ such that $[\pi',\pi']=2\pi(\phi)$ turns out to be
\begin{equation}
\label{Phi-V}
\begin{split}
\phi &=\left(V'(q_1-q_2)-V(q_1-q_2)\right)d(p_1+p_2)\wedge d{q_2}\wedge d{q_1}\\
&+\left(V'(q_2-q_3)-V(q_2-q_3)\right)d(p_2+p_3)\wedge d{q_3}\wedge d{q_2}\\
&-\left(V'(q_3-q_1)+V(q_3-q_1)\right)d(p_1+p_3)\wedge d{q_3}\wedge d{q_1}\\
&-2 V'(q_3-q_1)dp_2\wedge d{q_3}\wedge d{q_1}\,,
\end{split}
\end{equation}
which is clearly closed. Hence we can conclude by Proposition \ref{prop:piprime} that $({{\mathbb{R}}}^6, \pi, N, \phi)$ is a PqN manifold for every choice of the function $V$.
One can check that $T_N(X,Y)=\pi\left(i_{X\wedge Y}\phi\right)$ for all vector fields $X,Y$, as stated in \cite{SX}.
Consider now the functions $H_k=\frac12 I_k=\frac1{2k}\operatorname{Tr}(N^k)$. We have that $H_1=p_1+p_2+p_3$,
\begin{equation}
\label{ham-V}
H_2=\frac12({p_1}^2+{p_2}^2+{p_3}^2)+V(q_1-q_2)+V(q_2-q_3)+V(q_3-q_1)\,,
\end{equation}
which can be obviously thought of as the Hamiltonian of three interacting particles of equal mass, and
\begin{equation}
\label{ham3-V}
\begin{split}
H_3&=\frac13({p_1}^3+{p_2}^3+{p_3}^3)+p_1\left(V(q_1-q_2)+V(q_3-q_1)\right)
+p_2\left(V(q_2-q_3)+V(q_1-q_2)\right)\\
&+p_3\left(V(q_3-q_1)+V(q_2-q_3)\right)\,.
\end{split}
\end{equation}
It is clear that $\parpo{H_1}{H_2}=\parpo{H_1}{H_3}=0$, while the Poisson bracket
\begin{equation}
\label{invH2H3-V}
\begin{split}
\parpo{H_2}{H_3}&=V(q_1-q_2) \left(V'(q_2-q_3)-V'(q_3-q_1)\right)
+V(q_2-q_3) \left(V'(q_3-q_1)-V'(q_1-q_2)\right)\\
&+V(q_3-q_1) \left(V'(q_1-q_2)-V'(q_2-q_3)\right)\,
\end{split}
\end{equation}
does not vanish for any function $V$ (for example, one can easily check that it is different from zero if $V(x)=1/x$). However, involutivity holds in the cases
$V(x)=\mathrm{e}^x$ (to be discussed in the next sections) and $V(x)=1/x^2$ (corresponding to the Calogero model).
In conclusion, given a PqN manifold, further conditions on $(\pi,N,\phi)$ are needed to guarantee that the functions $I_k$ are in involution.
\section{The 4-particle closed Toda case}\label{toda-4}
In this section we consider the closed (or periodic) Toda system with $n=4$ particles. In the canonical variables
$(q_{{1}},q_{{2}},q_{{3}},q_{{4}},p_{{1}},p_{{2}},p_{{3}},p_{{4}})$,
the Hamiltonian is given by
\begin{equation}
\label{Htoda}
H_{\mbox{\scriptsize Toda}}=\sum_{i=1}^4 \left(\frac12 p_i^2+\mathrm{e}^{q_i-q_{i+1}}\right),\qquad\mbox{where $q_{5}
=q_1$.}
\end{equation}
Let us introduce the $(1,1)$
tensor field on ${\mathcal M}={{\mathbb{R}}}^8$ given by
\begin{equation}
\label{qN4}
N=\left[ \begin {array}{cccccccc} p_{{1}}&0&0&0&0&1&1&1
\\ \noalign{\medskip}0&p_{{2}}&0&0&-1&0&1&1\\ \noalign{\medskip}0&0&p_
{{3}}&0&-1&-1&0&1\\ \noalign{\medskip}0&0&0&p_{{4}}&-1&-1&-1&0
\\ \noalign{\medskip}0&-{{\rm e}^{q_{{1}}-q_{{2}}}}&0&-{{\rm e}^{q_{{4}
}-q_{{1}}}}&p_{{1}}&0&0&0\\ \noalign{\medskip}{{\rm e}^{q_{{1}}-q_{{2}
}}}&0&-{{\rm e}^{q_{{2}}-q_{{3}}}}&0&0&p_{{2}}&0&0
\\ \noalign{\medskip}0&{{\rm e}^{q_{{2}}-q_{{3}}}}&0&-{{\rm e}^{q_{{3}
}-q_{{4}}}}&0&0&p_{{3}}&0\\ \noalign{\medskip}{{\rm e}^{q_{{4}}-q_{{1
}}}}&0&{{\rm e}^{q_{{3}}-q_{{4}}}}&0&0&0&0&p_{{4}}\end {array}
\right]\,
\end{equation}
and the traces of its powers, $I_k=\frac1{k}\operatorname{Tr}(N^k)$.
As we will see, these functions are in involution with respect to the (canonical) Poisson bracket $\parpo{\cdot}{\cdot}$ induced by the
canonical Poisson tensor $\pi$. If we put $H_k=\frac12 I_k$, then it is easy to check that $H_1=\sum_{i=1}^4 p_i$ and $H_2=H_{\mbox{\scriptsize Toda}}$, while $H_3$ is the third constant of the motion of the $4$-particle Toda chain, and $H_4$ coincides with the fourth one up to a constant. Here, by ``constants of the motion of the $4$-particle Toda chain'' we mean those obtained by taking traces of the powers of the well known Lax matrix (see, e.g., \cite{Perelomov-book})
\begin{equation}
L=\left[
\begin{array}{cccc}\medskip
p_1&{{\rm e}^{\frac12(q_{{1}}-q_{{2}) }}}&0&{{\rm e}^{\frac12(q_{{4}}-q_{{1}) }}}\\ \medskip
{{\rm e}^{\frac12(q_{{1}}-q_{{2}) }}}&p_2&{{\rm e}^{\frac12(q_{{2}}-q_{{3}) }}}&0\\ \medskip
0&{{\rm e}^{\frac12(q_{{2}}-q_{{3}) }}}&p_3&{{\rm e}^{\frac12(q_{{3}}-q_{{4}) }}}\\ \medskip
{{\rm e}^{\frac12(q_{{4}}-q_{{1}) }}}&0&{{\rm e}^{\frac12(q_{{3}}-q_{{4}) }}}&p_4
\end{array}\right]\>.
\end{equation}
We can use Proposition \ref{prop:piprime} to show that $\pi$ and $N$ define a PqN structure on ${{\mathbb{R}}}^8$.
Indeed, $N$ differs from the {\em torsion free\/} $(1,1)$ tensor field of the {\em open\/} Toda chain (see \cite{DO}),
\begin{equation}
\label{Nopen}
N_{(O)}=\left[ \begin {array}{cccccccc} p_{{1}}&0&0&0&0&1&1&1
\\ \noalign{\medskip}0&p_{{2}}&0&0&-1&0&1&1\\ \noalign{\medskip}0&0&p_
{{3}}&0&-1&-1&0&1\\ \noalign{\medskip}0&0&0&p_{{4}}&-1&-1&-1&0
\\ \noalign{\medskip}0&-{{\rm e}^{q_{{1}}-q_{{2}}}}&0&0&p_{{1}}&0&0&0
\\ \noalign{\medskip}{{\rm e}^{q_{{1}}-q_{{2}}}}&0&-{{\rm e}^{q_{{2}}-
q_{{3}}}}&0&0&p_{{2}}&0&0\\ \noalign{\medskip}0&{{\rm e}^{q_{{2}}-q_{{
3}}}}&0&-{{\rm e}^{q_{{3}}-q_{{4}}}}&0&0&p_{{3}}&0
\\ \noalign{\medskip}0&0&{{\rm e}^{q_{{3}}-q_{{4}}}}&0&0&0&0&p_{{4}}
\end {array} \right]\,,
\end{equation}
by the rank $2$ tensor
\begin{equation}
\label{delta}
\Delta N=\mathrm{e}^{q_4-q_1}\left(\partial_{p_4}\otimes dq_1-\partial_{p_1}\otimes dq_4\right)\, .
\end{equation}
It can be checked that the torsion of $\Delta N$ vanishes, while that
of $N$ turns out to be
\begin{equation}
\label{TN4}
T_N=\mathrm{e}^{q_4-q_1}\left(\partial_{p_1}\otimes dq_4\wedge
dI_1
-\partial_{p_4}\otimes dq_1\wedge
dI_1
-X_1\otimes dq_1\wedge dq_4\right)\, ,
\end{equation}
where $X_1=\pi dI_1=2\sum_{i=1}^4\partial_{q_i}$ is (twice) the translation vector field. In other words,
\begin{equation}
\label{Igor-TN}
T_N(X,Y)=\langle dI_1,Y\rangle\Delta N(X)-\langle dI_1,X\rangle\Delta N(Y)+\Omega(X,Y)X_1,
\end{equation}
where $\Omega=\mathrm{e}^{q_4-q_1}dq_4\wedge dq_1$. It is easily seen that (\ref{Igor-TN}) holds for the general $n$-particle case, with
$\Omega=\mathrm{e}^{q_n-q_1}dq_n\wedge dq_1$.
We also have that
\begin{equation}
\label{P1}
\pi'=N\pi=\left[ \begin {array}{cccccccc} 0&-1&-1&-1&p_{{1}}&0&0&0
\\ \noalign{\medskip}1&0&-1&-1&0&p_{{2}}&0&0\\ \noalign{\medskip}1&1&0
&-1&0&0&p_{{3}}&0\\ \noalign{\medskip}1&1&1&0&0&0&0&p_{{4}}
\\ \noalign{\medskip}-p_{{1}}&0&0&0&0&-{{\rm e}^{q_{{1}}-q_{{2}}}}&0&
-{{\rm e}^{q_{{4}}-q_{{1}}}}\\ \noalign{\medskip}0&-p_{{2}}&0&0&{{\rm e}
^{q_{{1}}-q_{{2}}}}&0&-{{\rm e}^{q_{{2}}-q_{{3}}}}&0
\\ \noalign{\medskip}0&0&-p_{{3}}&0&0&{{\rm e}^{q_{{2}}-q_{{3}}}}&0&-{
{\rm e}^{q_{{3}}-q_{{4}}}}\\ \noalign{\medskip}0&0&0&-p_{{4}}&{
{\rm e}^{q_{{4}}-q_{{1}}}}&0&{{\rm e}^{q_{{3}}-q_{{4}}}}&0\end {array}
\right]\,,
\end{equation}
while the corresponding {\em Poisson\/} tensor for the open Toda lattice is
\begin{equation}
\label{P1O}
\pi'_{(O)}=N_{(O)}\pi= \left[ \begin {array}{cccccccc} 0&-1&-1&-1&p_{{1}}&0&0&0
\\ \noalign{\medskip}1&0&-1&-1&0&p_{{2}}&0&0\\ \noalign{\medskip}1&1&0
&-1&0&0&p_{{3}}&0\\ \noalign{\medskip}1&1&1&0&0&0&0&p_{{4}}
\\ \noalign{\medskip}-p_{{1}}&0&0&0&0&-{{\rm e}^{q_{{1}}-q_{{2}}}}&0&0
\\ \noalign{\medskip}0&-p_{{2}}&0&0&{{\rm e}^{q_{{1}}-q_{{2}}}}&0&-{
{\rm e}^{q_{{2}}-q_{{3}}}}&0\\ \noalign{\medskip}0&0&-p_{{3}}&0&0&{
{\rm e}^{q_{{2}}-q_{{3}}}}&0&-{{\rm e}^{q_{{3}}-q_{{4}}}}
\\ \noalign{\medskip}0&0&0&-p_{{4}}&0&0&{{\rm e}^{q_{{3}}-q_{{4}}}}&0
\end {array} \right]\,.
\end{equation}
It holds
\begin{equation}
\label{delta P1}
\pi'=\pi'_{(O)}+\mathrm{e}^{q_4-q_1}
\partial_{p_4}\wedge\partial_{p_1}\,,
\end{equation}
and the Schouten bracket of $\pi'$ with itself is
\begin{equation}
\label{Scbr}
[\pi',\pi']
=2\mathrm{e}^{q_4-q_1}\left(X_1\wedge \partial_{p_4}\wedge \partial_{p_1}\right)\,.
\end{equation}
Then we find that the second of (\ref{schouten-conditions}) is satisfied with
\begin{equation}
\label{Phi}
\phi=\mathrm{e}^{q_4-q_1}\left(dI_1\wedge dq_1\wedge dq_4\right)
= dI_1\wedge d \mathrm{e}^{q_4}\wedge d\mathrm{e}^{-q_1}=d \left( I_1\, d \mathrm{e}^{q_4}\wedge d\mathrm{e}^{-q_1}\right)\, ,
\end{equation}
which is obviously closed. Hence one is left with showing that the first of (\ref{schouten-conditions}) holds, which is a quite easy task. Using Proposition \ref{prop:piprime}, we can then conclude that $({{\mathbb{R}}}^8, \pi, N, \phi)$ is a PqN manifold.
\begin{rem}
As we have already seen in Section \ref{subsec:traces}, many features of the usual picture of Poisson-Nijenhuis manifolds are lost in the case. Not only $\pi'$ is not Poisson, but the Hamiltonians $I_k$ do not fulfill the Lenard-Magri relations. For example, $N^*dI_1\not=dI_2$, so that $N^*dH_1\not=dH_2$ and
$N X_1\not=X_2$, where $X_2=\pi dI_2$ is twice the ``physical" Toda vector field $X_{\mbox{\scriptsize Toda}}=\pi dH_2$. However, we will show
that the $I_k$ are in involution. This is not true for an arbitrary PqN manifold, as we have seen in Section \ref{laclassediesempidiGiovanni}.
\end{rem}
The generalization of these patterns to the $n$-particle case is clear. In particular, to obtain the corresponding formulas of
(\ref{delta}, \ref{TN4}, \ref{delta P1}, \ref{Scbr},
\ref{Phi}), one simply has to make the replacement $4 \mapsto n$.
\subsection{Some computations on the traces}\label{Toda4}
This subsection is devoted to some computations, still for the case $n=4$. First of all, we rewrite formula (\ref{Igor-TN}) as
\begin{equation}
i_XT_N=\Delta N (X)\otimes dI_1-\langle dI_1,X\rangle\Delta N+X_1\otimes i_X\Omega\,,\label{eq:5}
\end{equation}
where $\Delta N$ is given by (\ref{delta}) and $\Omega=\mathrm{e}^{q_4-q_1}dq_4\wedge dq_1$. We notice that
$\Delta N=-\pi\,\Omega^\flat$, where $ \Omega^\flat:T{\mathcal M}\to T^*{\mathcal M}$ is defined as usual by $\Omega^\flat(X)=i_X\Omega$.
If we call
$X_j=\pi\, dI_j$ the vector fields of the hierarchy, then we have that $\langle dI_1,X_j\rangle=-\brke{dI_j}{X_1}=0$, since $N$ and hence its traces depend only on the differences $q_{i}-q_{i+1}$ of the coordinates. Therefore
\begin{equation}
\label{eq:5b}
i_{X_j}T_N=\Delta N (X_j)\otimes dI_1+X_1\otimes i_{X_j}\Omega\,,
\end{equation}
so that
\begin{equation}
\label{eq-pre-tr}
\begin{split}
\brke{\phi_k}{X_j} &=\operatorname{Tr}\left(N^k(i_{X_j}T_N)\right)=\operatorname{Tr}\left(N^k(\Delta N (X_j)\otimes dI_1+X_1\otimes i_{X_j}\Omega)\right)\\
&=\operatorname{Tr}\left((N^k\Delta N )(X_j)\otimes dI_1\right)+\operatorname{Tr}\left((N^k X_1)\otimes i_{X_j}\Omega)\right)
\,.
\end{split}
\end{equation}
Both summands coincide with $\Omega(X_j,N^k X_1)$. This is easily seen for the second summand,
since $\operatorname{Tr}(X\otimes\alpha)=\brke{\alpha}{X}$ for all vector fields $X$
and 1-forms $\alpha$. As far as the first one is concerned,
\begin{equation}
\label{eqtr-b}
\begin{split}
\operatorname{Tr}\left((N^k\Delta N )(X_j)\otimes dI_1\right) &=\brke{dI_1}{(N^k\Delta N )(X_j)}
=-\brke{dI_1}{(N^k\pi\, \Omega^\flat)(X_j)}
=-\brke{dI_1}{(\pi {N^*}^k \Omega^\flat)(X_j)}\\
&=\brke{({N^*}^k \Omega^\flat)(X_j)}{X_1}
=\brke{\Omega^\flat(X_j)}{N^k X_1}
=\Omega(X_j,N^k X_1)\,.
\end{split}
\end{equation}
Therefore we have obtained the final formula
\begin{equation}
\label{eqtr}
\brke{\phi_k}{X_j}=2\Omega(X_j,N^k X_1)
=2\mathrm{e}^{q_4-q_1} \left(\brke{dq_4}{X_j} \brke{dq_1}{N^kX_1}-\brke{dq_1}{X_j} \brke{dq_4}{N^kX_1}\right)
\,.
\end{equation}
We are now ready for the explicit computations of the Poisson brackets between the $I_j$. We have just seen that $\parpo{I_1}{I_j}=\brke{dI_1}{X_j}=0$ for all $j$,
therefore we have to check that
\begin{equation}\label{3eq}
\parpo{I_3}{I_2}=0\,,\qquad \parpo{I_4}{I_3}=0\,,\qquad \parpo{I_4}{I_2}=0\, .
\end{equation}
Taking
(\ref{exk=j+1}), (\ref{exk=j+2}) and (\ref{eqtr}) into account, these three relations translate respectively into:
\begin{eqnarray}
\label{neq3-1} &&\brke{\phi_1}{X_2}=2\Omega(X_2,N X_1)=
0\\
\label{neq3-2} &&\brke{\phi_2}{X_3}=2\Omega(X_3,N^2 X_1)=
0\\
\label{neq3-3} &&\brke{\phi_1}{X_3}+\brke{\phi_2}{X_2}=2\Omega(X_3,N X_1)+2\Omega(X_2,N^2 X_1)=
0\,.
\end{eqnarray}
If we call $Y_k=N^{k-1}X_1-X_k$, it turns out (see next section) that
\begin{equation}\label{Y14eq}
i_{Y_k}\Omega=0\,,\qquad\mbox{that is,}\qquad \brke{dq_1}{Y_k}=\brke{dq_4}{Y_k}=0\, .
\end{equation}
Hence we can show that (\ref{neq3-1}) holds by replacing $NX_1$ with $X_2$. Similarly, (\ref{neq3-2}) reduces to $\Omega(X_3,X_3)=0$,
so we have shown that $\parpo{I_3}{I_2}=\parpo{I_4}{I_3}=0$.
We are left with $\parpo{I_4}{I_2}$, that is, with (\ref{neq3-3}).
In the light of (\ref{Y14eq}), this can be written as
\begin{equation}
\label{I2I4n-bis}
\Omega(X_3,X_2)+\Omega(X_2,X_3)=0\,,
\end{equation}
which clearly holds. Notice however that, e.g.,
$$
\operatorname{Tr}\left(N\, (i_{X_3}T_N)\right)=\brke{\phi_1}{X_3}=2\Omega(X_3,NX_1)=2\Omega(X_3,X_2)
$$
is {\em not\/} vanishing by itself, as anticipated in Section \ref{subsec:traces}.
\section{The $n$-particle closed Toda case}\label{FS}
In this section we show that the results obtained for the 4-particle case hold in the general ($n$-particle) case.
\begin{theorem}
\label{teo:toda-n}
Let us consider the PqN structure $({{\mathbb{R}}}^{2n},\pi,N,\phi)$, where $\pi$ is the canonical Poisson tensor
and $N$, $\phi$ are given by the obvious generalizations
of (\ref{qN4},\ref{Phi}). Then
\begin{enumerate}
\item For all $k\ge 1$, we have that $i_{Y_k}\Omega=0$, where $\Omega=\mathrm{e}^{q_n-q_1}dq_n\wedge dq_1$
and $Y_k=N^{k-1}X_1-X_k$.
\item The functions $I_k=\frac1k\operatorname{Tr}(N^k)$ are in involution.
\end{enumerate}
\end{theorem}
{\bf Proof.}
1. {\color{black} Applying $\pi$ to both members of (\ref{ff2}), one easily finds that $NX_l-NX_{l+1}=\pi\,\phi_{l-1}$. Then we have
\begin{equation}
Y_k=\sum_{l=1}^{k-1}\left(N^{k-l}X_l-N^{k-l-1}X_{l+1}\right)=\sum_{l=1}^{k-1}N^{k-l-1}\left(N X_l-X_{l+1}\right)
=\sum_{l=1}^{k-1}N^{k-l-1}\pi\,\phi_{l-1}\,,
\label{pre-eq:1}
\end{equation}
so that
\begin{equation}
Y_k=\pi\left(\sum_{l=1}^{k-1}(N^*)^{k-l-1}\phi_{l-1}\right)=\pi\left(\sum_{l=0}^{k-2}{(N^*)}^{k-l-2}\phi_l\right)\,.
\label{eq:1}
\end{equation}}
Therefore, the condition $i_{Y_k}\Omega=0$, that is, $\langle dq_n,Y_k\rangle=\langle dq_1,Y_k\rangle=0$, becomes
\begin{equation}
\sum_{l=0}^{k-2}\brke{\phi_l}{N^{k-l-2}\partial_{p_n}}=\sum_{l=0}^{k-2}\brke{\phi_l}{N^{k-l-2}\partial_{p_1}}=0\,.\label{eq:2}
\end{equation}
Recall now the definition
\begin{equation}
\label{eq:3}
\brke{\phi_l}{X}=\operatorname{Tr}\left(N^l (i_XT_N)\right)
\end{equation}
of the 1-forms $\phi_l$ and formula (\ref{eq:5}), that is,
\begin{equation}
i_XT_N=\Delta N (X)\otimes dI_1-\langle dI_1,X\rangle\Delta N+X_1\otimes i_X\Omega\,,\label{eq:5c}
\end{equation}
where $\Delta N$ is given by the obvious generalization of (\ref{delta}),
\begin{equation}
\label{delta-n}
\Delta N=\mathrm{e}^{q_n-q_1}\left(\partial_{p_n}\otimes dq_1-\partial_{p_1}\otimes dq_n\right)\, .
\end{equation}
Then, for all $k\geq 2$ and $l=0,\dots,k-2$, we have that
\begin{equation}
\label{eq:4}
\begin{split}
&\brke{\phi_l}{N^{k-l-2}\partial_{p_n}}=\operatorname{Tr}\left(N^l(i_{N^{k-l-2}\partial_{p_n}}T_N)\right)\\
&\quad =\operatorname{Tr} \left[N^l\left(\Delta N(N^{k-l-2}\partial_{p_n})\otimes dI_1-\langle dI_1,N^{k-l-2}\partial_{p_n}\rangle\Delta N+
X_1\otimes i_{N^{k-l-2}\partial_{p_n}}\Omega\right)\right]\\
&\quad=\langle dI_1,N^l\Delta N(N^{k-l-2}\partial_{p_n})\rangle-\langle dI_1,N^{k-l-2}\partial_{p_n}\rangle\operatorname{Tr}(N^l\Delta N)+\Omega(N^{k-l-2}\partial_{p_n},N^lX_1)\\
&\quad=2\Omega(N^{k-l-2}\partial_{p_n},N^lX_1)-\langle dI_1,N^{k-l-2}\partial_{p_n}\rangle\operatorname{Tr}(N^l\Delta N),
\end{split}
\end{equation}
where the last equality follows from the identity $\Delta N=-\pi\, \Omega^\flat$.
Let us compute the three terms appearing in (\ref{eq:4}):
\begin{enumerate}
\item[(i)]
$\Omega(N^{k-l-2}\partial_{p_n},N^lX_1)=
e^{q_n-q_1}\left[\langle dq_n,N^{k-l-2}\partial_{p_n}\rangle\langle dq_1, N^lX_1\rangle-\langle dq_1,N^{k-l-2}\partial_{p_n}\rangle\langle dq_n,N^lX_1\rangle\right].
$
\item[(ii)]
$\langle dI_1,N^{k-l-2}\partial_{p_n}\rangle=-\langle dI_1,N^{k-l-2}(\pi dq_n)\rangle=\langle dq_n,N^{k-l-2}X_1\rangle.$\
\item[(iii)]
$\operatorname{Tr} (N^l\Delta N)
=\langle dq_1,N^l\Delta N(\partial_{q_1})\rangle+\langle dq_n,N^l\Delta N(\partial_{q_n})\rangle
=e^{q_n-q_1}\big(\langle dq_n,N^l\partial_{p_1}\rangle-\langle dq_1,N^l\partial_{p_n}\rangle\big)
=-2e^{q_n-q_1}\langle dq_1,N^l\partial_{p_n}\rangle.$
\end{enumerate}
Then we proved that
\begin{eqnarray*}
\brke{\phi_l}{N^{k-l-2}\partial_{p_n}}&=&2e^{q_n-q_1}\big[\langle dq_n,N^{k-l-2}\partial_{p_n}\rangle\langle dq_1,N^lX_1\rangle\\
&-&\langle dq_1,N^{k-l-2}\partial_{p_n}\rangle\langle dq_n,N^lX_1\rangle\\
&+&\langle dq_1,N^{l}\partial_{p_n}\rangle\langle dq_n, N^{k-l-2}X_1\rangle].
\end{eqnarray*}
It follows that, for all $k\geq 2$,
\begin{equation}
\begin{split}
\langle dq_n,Y_k\rangle &=\brke{dq_n}{\pi
\sum_{l=0}^{k-2}(N^*)^{k-l-2}\phi_l
}= \sum_{l=0}^{k-2}\brke{\phi_l}{N^{k-l-2}\partial_{p_n}}\\
&=2e^{q_n-q_1}\sum_{l=0}^{k-2}
\langle dq_n,N^{k-l-2}\partial_{p_n}\rangle \langle dq_1,N^l X_1\rangle \,,
\end{split}
\end{equation}
proving that if $\langle dq_n,N^j\partial_{p_n}\rangle=0$ for all $j\geq 1$, then $\langle d q_n,Y_k\rangle=0$ for all $k\ge 1$.
A similar computation shows that $\langle dq_1, Y_k\rangle=0$ is implied by $\langle dq_1,N^{j}\partial_{p_1}\rangle=0$. Hence we are left with proving that the entries
$(1,n+1)$ and $(n,2n)$ of $N^k$ vanish for all $k\geq 1$. But this follows from the fact that the $n\times n$ block in the upper right corner of $N^k$ is skewsymmetric,
since $N^k\pi=\pi\, {N^*}^k$.
2. It suffices to show that the additional term, appearing in (\ref{recadd}), to the usual Lenard-Magri recursion relations for the Poisson
brackets between the traces of the powers of $N$
vanishes. Actually, this additional term is
\begin{equation}
\Delta_{j,k-1}=-\brke{ \phi_{j-1}}{\pi\, dI_{k-1}}-\brke{\phi_{k-2}}{ \pi\, dI_j}
\end{equation}
and it reads, thanks to (the generalization to arbitrary $n$ of) equation (\ref{eqtr}),
\begin{equation}\label{qf2}
\Delta_{j,k-1}=-2\Omega(X_{k-1},N^{j-1}X_1)-2\Omega(X_{j},N^{k-2}X_1)\,.
\end{equation}
Now, thanks to the first part of this theorem, we can substitute $N^{i-1}X_1$ with $X_{i}$ in the previous formula for $\Delta_{j,k-1}$, showing that it vanishes.
Hence we obtain that the Lenard-Magri recursion relations (\ref{LMrec}) hold also in this case, leading to the involutivity of the $I_k$.
\hfill$\square$ \medskip
We notice that in many points of the previous proof (see, e.g., item (iii)) very peculiar properties of the tensor field $\Delta N$ have been exploited.
\section{A relation between PN and PqN manifolds}
\label{igor-connection}
In this section we present a general result concerning the connection between PN and PqN structures. When applied to the PqN manifold we have studied in the previous section, this result allows us to establish a relation between the geometrical structures of the closed and open Toda lattices.
First of all, we recall that, given a tensor field $N:T{\mathcal M}\to T{\mathcal M}$, the usual Cartan differential can be modified as follows,
\begin{equation}
\label{eq:dN}
\begin{split}
(d_N\alpha)(X_0,\dots,X_q)&=
\sum_{j=0}^q(-1)^j L_{NX_j}\left(\alpha(X_0,\dots,\hat{X}_j,\dots,X_q)\right)\\&
+\sum_{i<j}(-1)^{i+j}\alpha([X_i,X_j]_N,X_0,\dots,\hat{X}_i,\dots,\hat{X}_j,\dots,X_q)\,,
\end{split}
\end{equation}
where $\alpha$ is a $q$-form, the $X_i$ are vector fields,
and
$[X,Y]_N=[NX,Y]+[X,NY]-N[X,Y]$. Note that
$d_Nf=N^* df$ for all $f\in C^\infty({\mathcal M})$. Moreover,
\begin{equation}
\label{eq:dNd}
d_N=i_N\circ d-d\circ i_N\,,
\end{equation}
where $i_N$ is given by (\ref{iNalpha}), and consequently $d\circ d_N+d_N\circ d=0$.
Finally, $d_N^2 =0$ if and only if the torsion of $N$ vanishes.
We also remind that one can define a Lie bracket between the 1-forms on a Poisson manifold $({\mathcal M},\pi)$ as
\begin{equation}
\label{eq:liealgpi}
[\alpha,\beta]_\pi=L_{\pi\alpha}\beta-L_{\pi\beta}\alpha-d\langle\beta,\pi\alpha\rangle\,,
\end{equation}
and that this Lie bracket
can be uniquely extended to all forms on ${\mathcal M}$ in such a way that
\begin{itemize}
\item[(K1)] $[\eta,\eta']_\pi=-(-1)^{(q-1)(q'-1)}[\eta',\eta]_\pi$ if $\eta$ is a $q$-form and $\eta'$ is a $q'$-form;
\item[(K2)] $[\alpha,f]_\pi=i_{\pi df}\,\alpha=\langle\alpha,\pi df\rangle
$ for all $f\in C^\infty(M)$ and for all 1-forms $\alpha$;
\item[(K3)] if $\eta$ is a $q$-form, then $[\eta,\cdot]_\pi$
is a derivation of degree $q-1$ of the wedge product, that is,
\begin{equation}
\label{deriv-koszul}
[\eta,\eta'\wedge\eta'']_\pi=[\eta,\eta']_\pi\wedge\eta''+(-1)^{(q-1)q'}\eta'\wedge[\eta,\eta'']_\pi
\end{equation}
if $\eta'$ is a $q'$-form and $\eta''$ is any differential form.
\end{itemize}
This extension is a {\it graded\/} Lie bracket, in the sense that (besides (K1)) the graded Jacobi identity holds:
\begin{equation}
\label{graded-jacobi}
(-1)^{(q_1-1)(q_3-1)}[\eta_1,[\eta_2,\eta_3]_\pi]_\pi+(-1)^{(q_2-1)(q_1-1)}[\eta_2,[\eta_3,\eta_1]_\pi]_\pi+(-1)^{(q_3-1)(q_2-1)}[\eta_3,[\eta_1,\eta_2]_\pi]_\pi=0
\end{equation}
if $q_i$ is the degree of $\eta_i$.
It is sometimes called the Koszul bracket --- see, e.g., \cite{FiorenzaManetti2012} and references therein.
It was proved in \cite{YKS96} that the compatibility conditions (\ref{N-P-compatible}) between a Poisson tensor $\pi$ and a tensor field $N:T{\mathcal M}\to T{\mathcal M}$ hold if and only if
$d_N$ is a derivation of $[\cdot,\cdot]_\pi$,
that is,
\begin{equation}
\label{deriv-wedge}
d_N[\eta,\eta']_\pi=[d_N\eta,\eta']_\pi+(-1)^{(q-1)}[\eta,d_N\eta']_\pi
\end{equation}
if $\eta$ is a $q$-form and $\eta'$ is any differential form. In particular, taking $N=Id$, one has that the Cartan differential $d$ is always a derivation of
$[\cdot,\cdot]_\pi$. Moreover,
if $\phi$ is any 3-form,
\begin{equation}
\label{iff:SX}
d_{N}^2=[\phi,\cdot]_\pi\quad\mbox{if and only if}\quad
\left\{\begin{array}{l}
T_{N}(X,Y)=\pi\left(i_{X\wedge Y}\phi\right)\quad\mbox{ for all vector fields $X,Y$}\\
\noalign{\medskip}
i_{(\pi\alpha)\wedge(\pi\beta)\wedge(\pi\gamma)}(d\phi)=0\quad\mbox{ for all 1-forms $\alpha,\beta,\gamma$,}
\end{array}\right.
\end{equation}
see \cite{SX}. We are now ready to state
\begin{theorem}
\label{thm:NEW}
Suppose that $({\mathcal M},\pi,\phi,N)$ is a PqN manifold and that there exists a closed 2-form $\omega$ such that
\begin{equation}
\label{Mau-Car}
d_N\omega+\frac{1}{2}[\omega,\omega]_\pi=-\phi\,.
\end{equation}
If $N'=N-\pi\,\omega^{\flat}$, then
$({\mathcal M},\pi,N')$ is a PN manifold.
\end{theorem}
{\bf Proof.}
First of all we show that $d_{\pi\,\omega^{\flat}}=-[\omega,\cdot]_\pi$. This follows from the fact that both are derivations (with respect to the wedge product)
anti-commuting with $d$, and they coincide on functions. Indeed, for all $f\in C^\infty({\mathcal M})$,
\[
d_{\pi\,
\omega^{\flat}}f=(\pi\,
\omega^{\flat})^*df=(\omega^{\flat}
\pi)df=i_{\pi df}\,\omega=-[\omega,f]_\pi,
\]
where the last equality holds for every 2-form $\omega$ and can be easily checked to be a consequence of (K2) and (K3).
Hence $d_{N'}=d_N-d_{\pi\,
\omega^{\flat}}=d_N+[\omega,\cdot]_\pi$
is a derivation of $[\cdot,\cdot]_\pi$ (since $\pi$ and $N$ are compatible and $[\cdot,\cdot]_\pi$ satisfies (\ref{graded-jacobi})), so that
$\pi$ and $N'$ are compatible too.
Finally, equivalence (\ref{iff:SX}) and formula (\ref{Mau-Car}) imply that $d_{N'}^2=0$, meaning that the torsion of $N'$ vanishes. We conclude that $({\mathcal M},\pi,N')$ is a PN manifold.
\hfill$\square$ \medskip
In the terminology of \cite{ILX}, Theorem \ref{thm:NEW} describes how to deform a quasi-Lie bialgebroid into a Lie bialgebroid by means of the so called {\it twist}.
\begin{rem}
\label{ex:toda}
Twisting the PqN structure of the Toda closed system with
\begin{equation}
\label{eq:todatwist}
{\color{black}\omega=-\Omega=-e^{q_n-q_1}dq_n\wedge dq_1=-d\left(e^{q_n-q_1} dq_1\right)\,,}
\end{equation}
one obtains the the PN structure of Toda open system. In fact,
$[\omega,\omega]_\pi={\color{black}[\Omega,\Omega]_\pi=}0$ and $d_N\omega={\color{black}-d_N\Omega=}-\phi$,
so that (\ref{Mau-Car}) is satisfied. Moreover, using the notations of Section \ref{toda-4} and and \ref{FS}, we have
that $N'=N_{(O)}=N-\Delta N$, where $\Delta N=\pi\,\omega^{\flat}{\color{black}=-\pi\,\Omega^{\flat}}$.
\end{rem}
A kind of converse of Theorem \ref{thm:NEW} is given by
\begin{theorem}
Let $(M,\pi,N)$ be a PN manifold. Then:
\begin{enumerate}
\item
For every closed 2-form $\omega$ such that
\begin{equation}
d_N\omega+\frac{1}{2}[\omega,\omega]_\pi=0\,,\label{eq:MComega}
\end{equation}
defining $N'=N-\pi\,
\omega^\flat$, we have that $(M,\pi,N')$ is a PN manifold.
\item
Let $\omega$
be a closed 2-form such that
\begin{equation}
\label{eq:cond}
[d_N\omega,\omega]_\pi=0\,.
\end{equation}
If
\begin{equation}
\label{eq:condiphi}
\phi=d_N\omega+\frac{1}{2}[\omega,\omega]_\pi
\end{equation}
and $N'=N-\pi\,\omega^\flat$, then $(M,\pi,N',\phi)$ is a PqN manifold.
\end{enumerate}
\end{theorem}
{\bf Proof.}
Part 1 is Theorem \ref{thm:NEW} with $\phi=0$.
As far as part 2
is concerned, note that condition \eqref{eq:cond} guarantees that the 3-form $\phi$ defined by \eqref{eq:condiphi} satisfies $d_N\phi=0$ and $d\phi=0$. Thanks to (\ref{eq:dNd}), it follows that $i_N\phi$ is closed. Since $d_{N'}=d_N-d_{\pi\,
\omega^{\flat}}=d_N+[\omega,\cdot]_\pi$, the compatibility between $\pi$ and $N'$ can be shown as in the proof of Theorem \ref{thm:NEW}.
Finally, using \eqref{eq:condiphi} and $d_N^2=0$, we can prove that $d_{N'}^2=[\phi,\cdot]_\pi$. To conclude, it suffices to use equivalence (\ref{iff:SX}).
\hfill$\square$\medskip
\begin{rem} We conclude this section with a couple of remarks.
\begin{enumerate}
\item In the notations of Sections \ref{toda-4} and \ref{FS}, starting from the open Toda system, we can consider the closed 2-form
$\omega=\Omega=e^{q_n-q_1}dq_n\wedge dq_1$.
{\color{black} As we have seen in Remark \ref{ex:toda}, it satisfies $[\Omega, \Omega]_\pi = 0$. One can also show that}
$d_{N_{(O)}}\Omega=\phi$, so that {\color{black} (\ref{eq:condiphi}) is fulfilled and} condition \eqref{eq:cond} becomes $[\phi,\Omega]_\pi=0$, {\color{black} which is a direct consequence of
$[\Omega, \Omega]_\pi = 0$.}
Note that this computation involves the (torsionless) tensor $N_{(O)}$ of the (PN structure of the) Toda open system, while the analogue computation in Remark \ref{ex:toda} involves the tensor $N$ of the (PqN structure of the) Toda closed system, whose torsion does not vanish.
\item \textcolor{black}{To the best of our knowledge equation \eqref{eq:MComega} was first introduced and studied by Liu, Weinstein and Xu in their work on the theory of Manin triples for Lie algebroids, see Section 6 of \cite{LWX}. These authors, starting from a Poisson manifold $(\mathcal M,\pi)$ and the corresponding \emph{standard} Courant algebroid structure on $T^\ast\mathcal M\oplus T\mathcal M$, showed that for $N=Id$ every solution of \eqref{eq:MComega} defines a Dirac subbundle
$\Gamma_\omega\subset T^\ast\mathcal M\oplus T\mathcal M$ transversal to $T^\ast\mathcal M$. Moreover they proved that every solution of
\begin{equation}
d\omega=0\quad\text{and}\quad[\omega,\omega]_\pi=0\label{complementary}
\end{equation}
defines a new Poisson structure $\pi'$ on $\mathcal M$ compatible with $\pi$ and induced by a torsionless operator, defining in this way a Poisson-Nijenhuis structure on $\mathcal M$. It is worth to mention that the second equation in \eqref{complementary} was studied in depth by Vaisman in \cite{Complementary}, where its solutions were named complementary $2$-forms of the (underlying) Poisson structure.}
\end{enumerate}
\end{rem}
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\end{document}
|
1,116,691,499,288 | arxiv | \section{Introduction}
The box-ball system (BBS) is an interacting particle system introduced in the 1990s by physicists Takahashi and Satsuma as a model to understand solitons, that is, travelling waves \cite{takahashi1990}. In particular, it is connected with the Korteweg-de Vries (KdV) equation, which describes shallow water waves; see \cite{TTMS} for background. The BBS is briefly described as follows. Initially, each site of the integer lattice $\mathbb{Z}$ contains a ball (or particle -- we will use the two terms interchangeably) or is vacant. For simplicity at this point, suppose there are only a finite number of particles. The system then evolves by means of a `carrier', which moves along the integers from left to right (negative to positive). When the carrier sees a ball it picks it up, and when it sees a vacant site it puts a ball down (unless it is not carrying any already, in which case it does nothing). See Figure \ref{bbsfig} for an example realisation.
To date, much of the interest in the BBS has come from applied mathematicians/theoretical physicists, who have established many beautiful combinatorial properties of the BBS, see \cite{IKT, T, TT} for introduction to such work. What has only recently started to be explored, however, are the probabilistic properties of the BBS resulting from a random initial starting configuration, see \cite{CKSS, Ferrari, FG, Lev} for essentially the only current literature on this topic. One particularly natural question in this direction is that of invariance, namely, which random configurations have a distribution that is invariant under the action of the box-ball system? In this article, we describe the invariant measures based on two-sided stationary Markov chains that were identified in \cite{CKSS}, and also introduce a family of new ones for periodic configurations that are expressed in terms of Gibbs measures.
\begin{figure}[b]
\begin{center}
{\circle{15}$\:$\circle*{15}$\:$\circle{15}$\:$\circle*{15}$\:$\circle*{15}$\:$\circle*{15}$\:$\circle{15}$\:$\circle{15}$\:$\circle*{15}$\:$\circle{15}$\:$\circle{15}$\:$\circle{15}$\:$\circle{15}$\:$\circle{15}$\:$\circle{15}\\
\vspace{5pt}
\circle{15}$\:$\circle{15}$\:$\circle*{15}$\:$\circle{15}$\:$\circle{15}$\:$\circle{15}$\:$\circle*{15}$\:$\circle*{15}$\:$\circle{15}$\:$\circle*{15}$\:$\circle*{15}$\:$\circle{15}$\:$\circle{15}$\:$\circle{15}$\:$\circle{15}\\
\vspace{5pt}
\circle{15}$\:$\circle{15}$\:$\circle{15}$\:$\circle*{15}$\:$\circle{15}$\:$\circle{15}$\:$\circle{15}$\:$\circle{15}$\:$\circle*{15}$\:$\circle{15}$\:$\circle{15}$\:$\circle*{15}$\:$\circle*{15}$\:$\circle*{15}$\:$\circle{15}}
\end{center}
\caption{Two evolutions of the BBS. Black circles represent particles, white circles represent vacant sites.}\label{bbsfig}
\end{figure}
Given the transience of the system, i.e.\ each particle moves at least one position to the right on each time step of the dynamics, the question of invariance in distribution immediately necessitates the consideration of configurations $\eta=(\eta_n)_{n\in\mathbb{Z}}\in\{0,1\}^{\mathbb{Z}}$, where we write $\eta_n=1$ if there is a particle at location $n$ and $\eta_n=0$ otherwise, that incorporate an infinite number of particles on both the negative and positive axes. Of course, for such configurations, the basic description of the BBS presented above is no longer applicable, as one has to consider what it means for the carrier to traverse the integers from $-\infty$. This issue was addressed systematically in \cite{CKSS}, and at the heart of this study was a link between the BBS dynamics and the transformation of reflection in the past maximum that Pitman famously used to connect a one-dimensional Brownian motion with a three-dimensional Bessel process in \cite{Pitman}. We now describe this connection. Given a configuration $\eta\in\{0,1\}^{\mathbb{Z}}$, introduce a path encoding $S:\mathbb{Z}\rightarrow\mathbb{Z}$ by setting $S_0:=0$, and
\[S_n-S_{n-1}:=1-2\eta_n,\qquad \forall n\in\mathbb{Z},\]
and then define $TS:\mathbb{Z}\rightarrow\mathbb{Z}$ via the relation
\[(TS)_n:=2M_n-S_n-2M_0,\qquad \forall n\in\mathbb{Z},\]
where $M_n:=\sup_{m\leq n}S_m$ is the past maximum of $S$. Clearly, for the above formula to be well-defined, we require $M_0<\infty$. If this is the case, then we let $T\eta\in\{0,1\}^\mathbb{Z}$ be the configuration given by
\begin{equation}\label{teta}
(T\eta)_n:=\mathbf{1}_{\{(TS)_n-(TS)_{n-1}=-1\}},\qquad \forall n\in\mathbb{Z},
\end{equation}
(so that $TS$ is the path encoding of $T\eta$). It is possible to check that the map $\eta\mapsto T\eta$ coincides with the original definition of the BBS dynamics in the finite particle case \cite[Lemma 2.3]{CKSS}, and moreover is consistent with an extension to the case of a bi-infinite particle configuration satisfying $M_0<\infty$ from a natural limiting procedure \cite[Lemma 2.4]{CKSS}. We thus restrict to configurations for which $M_0<\infty$, and take \eqref{teta} as the definition of the BBS dynamics in this article. We moreover note that the process $W=(W_n)_{n\in\mathbb{Z}}$ given by
\[W_n:=M_n-S_n\]
can be viewed as the carrier process, with $W_n$ representing the number of balls transported by the carrier from $\{\dots,n-1,n\}$ to $\{n+1,n+2,\dots\}$; see \cite[Section 2.5]{CKSS} for discussion concerning the (non-)uniqueness of the carrier process.
Beyond understanding the initial step of the BBS dynamics, in the study of invariant random configurations it is natural to look for measures supported on the set of configurations for which the dynamics are well-defined for all times. Again, such an issue was treated carefully in \cite{CKSS}, with a full characterisation being given of the sets of configurations for which the one-step (forwards and backwards) dynamics are reversible (i.e.\ invertible), and for which the dynamics can be iterated for all time. Precisely, in \cite[Theorem 1.1]{CKSS} explicit descriptions were given for the sets:
\[\mathcal{S}^{rev}:=\left\{S\in \mathcal{S}^0\::\:\mbox{$TS$, $T^{-1}S$, $T^{-1}TS$, $TT^{-1}S$ well-defined, $T^{-1}TS=S$, $TT^{-1}S=S$}\right\},\]
where we have written $\mathcal{S}^0:=\{S:\mathbb{Z}\rightarrow \mathbb{Z}:\: S_0=0,\:|S_n-S_{n-1}|=1,\:\forall n\in\mathbb{Z}\}$ for the set of two-sided nearest-neighbour paths started from 0 (i.e.\ path encodings for configurations in $\{0,1\}^\mathbb{Z}$), and $T^{-1}$ for the inverse operation to $T$ that is given by `reflection in future minimum', see \cite[Section 2.6]{CKSS} for details; and also the invariant set
\[\mathcal{S}^{inv}:=\left\{S\in \mathcal{S}^{0}\::\:\mbox{$T^kS\in \mathcal{S}^{rev}$ for all $k\in\mathbb{Z}$}\right\}.\]
Whilst in this article we do not need to make full use of the treatment of these sets from \cite{CKSS}, we note the following important subset of path encodings
\begin{equation}\label{slin}
\mathcal{S}^{lin}:=\left\{S\in \mathcal{S}^{0}\::\:\lim_{|n|\rightarrow\infty}\frac{S_n}{n}=c\mbox{ for some }c>0\right\},
\end{equation}
consisting of asymptotically linear functions with a strictly positive drift. It is straightforward to check from the description given in \cite[Theorem 1.1]{CKSS} that $\mathcal{S}^{lin}\subseteq\mathcal{S}^{inv}\subseteq\mathcal{S}^{rev}$.
With the preceding preparations in place, we are ready to discuss directly the topic of invariance in distribution for random configurations, or equivalently particle encodings. In \cite{CKSS}, two approaches were pursued. The first was to relate the invariance of the BBS dynamics to the stationarity of the particle current across the origin, see \cite[Theorem 1.6]{CKSS}. Whilst the latter viewpoint does also provide an insight into the ergodicity of the transformation $\eta\mapsto T\eta$, in checking invariance in examples a more useful result was \cite[Theorem 1.7]{CKSS}, which relates the distributional invariance of $\eta$ under $T$ to two natural symmetry conditions -- one concerning the configuration itself, and one concerning the carrier process. In particular, to state the result in question, we introduce the reversed configuration $\overleftarrow{\eta}$, as defined by setting
\[\overleftarrow{\eta}_n=\eta_{-(n-1)},\]
and the reversed carrier process $\bar{W}$, given by
\[\bar{W}_n=W_{-n}.\]
\begin{thm}[See {\cite[Theorem 1.7]{CKSS}}]\label{mrd} Suppose $\eta$ is a random particle configuration, and that the distribution of the corresponding path encoding $S$ is supported on $\mathcal{S}^{rev}$. It is then the case that any two of the three following conditions imply the third:
\begin{equation}\label{threeconds}
\overleftarrow{\eta}\buildrel{d}\over{=}\eta,\qquad \bar{W}\buildrel{d}\over{=}W,\qquad T\eta\buildrel{d}\over{=}\eta.
\end{equation}
Moreover, in the case that two of the above conditions are satisfied, then the distribution of $S$ is actually supported on $\mathcal{S}^{inv}$.
\end{thm}
As an application of the previous result, the following fundamental examples of invariant random configurations were presented in \cite[Theorem 1.8]{CKSS}:
\begin{itemize}
\item The particle configuration $(\eta_n)_{n\in\mathbb{Z}}$ given by a sequence of independent identically distributed (i.i.d.) Bernoulli random variables with parameter $p\in[0,\frac{1}{2})$.
\item The particle configuration $(\eta_n)_{n\in\mathbb{Z}}$ given by a two-sided stationary Markov chain on $\{0,1\}$ with transition matrix
\[\left(
\begin{array}{cc}
1-p_0 & p_0 \\
1-p_1 & p_1 \\
\end{array}
\right)\]
where $p_0\in (0,1)$, $p_1\in[0,1)$ satisfy $p_0+p_1<1$.
\item For any $K\in\mathbb{Z}_+$, the particle configuration $(\eta_n)_{n\in\mathbb{Z}}$ given by conditioning a sequence of i.i.d.\ Bernoulli random variables with parameter $p\in(0,1)$ on the event $\sup_{n\in\mathbb{Z}}W_n\leq K$.
\end{itemize}
Further details of these are recalled in Subsections \ref{iidsec}-\ref{boundedsec}, respectively. Another easy example, discussed in \cite[Remark 1.13]{CKSS}, arises from a consideration of the periodic BBS introduced in \cite{YT} -- that is, the BBS that evolves on the torus $\mathbb{Z}/N\mathbb{Z}$. As commented in \cite{CKSS}, if we repeat a configuration of length $N$ with strictly fewer than $N/2$ balls in a cyclic fashion, then we obtain a configuration with path encoding contained in $\mathcal{S}^{lin}$, and, by placing equal probability on each of the distinct configurations that we see as the BBS evolves, we obtain an invariant measure for the system.
Now, it should be noted that \cite{CKSS} was not the first study to identify the first two configurations above (i.e.\ the i.i.d.\ and Markov configurations) as invariant under $T$. Such results had previously been established in queueing theory -- the invariance of the i.i.d.\ configuration can be seen as a discrete time analogue of the classical theorem of Burke \cite{Burke}, and the invariance of the Markov configuration was essentially proved in \cite{HMOC}. However, in the study of invariants for Pitman's transformation, the BBS does add an important new perspective -- the central role of solitons. Indeed, in the original study of \cite{takahashi1990}, it was observed that configurations can be decomposed into a collection of ‘basic strings’ of the form (1,0), (1,1,0,0), (1,1,1,0,0,0), etc., which act like solitons in that they are preserved by the action of the carrier, and travel at a constant speed (depending on their length) when in isolation, but experience interactions when they meet. Moreover, in the enlightening recent work of \cite{Ferrari} (where the invariance of the i.i.d.\ configuration was again observed), it was conjectured that any invariant measure on configurations can be decomposed into independent measures on solitons of different sizes. (The latter study investigated the speeds of solitons in invariant random configurations under continued evolution of the BBS system.) See also \cite{FG} for a related follow-up work.
Motivated in part by \cite{Ferrari}, in this article we introduce a class of invariant periodic configurations whose laws are described in terms of Gibbs measures involving a soliton decomposition. (These were already described formally in \cite[Remark 1.12]{CKSS}, and are closely paralleled by the measures studied in \cite{FG}.) Specifically, we first fix a cycle length $N\in\mathbb{N}$, and then define a random variable $(\eta^N_n)_{n=1}^{N}$ taking values in $\{0,1\}^N$ by setting
\begin{equation}\label{gibbs}
\mathbf{P}\left((\eta^N_n)_{n=1}^{N}=(x_n)_{n=1}^{N}\right)=\frac{1}{Z}\exp\left(-\sum_{k=0}^\infty\beta_kf_k\left((x_n)_{n=1}^{N}\right)\right)\mathbf{1}_{\{f_0\left((x_n)_{n=1}^{N}\right)<N/2\}},
\end{equation}
for $(x_n)_{n=1}^{N}\in\{0,1\}^N$, where $\beta_k\in\mathbb{R}\cup\{\infty\}$ for each $k\geq 0$,
\[f_0\left((x_n)_{n=1}^{N}\right):=\#\left\{\mbox{particles in $(x_n)_{n=1}^{N}$}\right\},\]
\[f_k\left((x_n)_{n=1}^{N}\right):=\#\left\{\mbox{solitons of size $\geq k$ in $(x_n)_{n=1}^{N}$}\right\},\qquad \forall k\in\mathbb{N},\]
and $Z$ is a normalising constant. We then extend to $\eta^N=(\eta^N_n)_{n\in\mathbb{Z}}$ by cyclic repetition; the law of $\eta^N$ is our Gibbs measure. (Further details are provided in Subsection \ref{gibbssec}.) The invariance under $T$ of such a random configuration is checked as Corollary \ref{gibbsinv} below. Moreover, in Proposition \ref{infvollim}, it is shown that each of the three configurations of \cite[Theorem 1.8]{CKSS} can be obtained as an infinite volume ($N\rightarrow\infty$) limit of these periodic configurations.
\begin{rem} In this article, we are using the term `Gibbs measure' in a loose sense. Given that the expression at \eqref{gibbs} incorporates the infinite number of conserved quantities for the integrable system that is the BBS, following \cite{RDYO, RMO} (see also the review \cite{VR}), it might rather be seen as a `generalised Gibbs measure'. Since we plan to present a more comprehensive study of Gibbs-type measures for the BBS in a following article, we leave further discussion of this point until the future.
\end{rem}
The description of the path encoding of a configuration and its evolution under the BBS dynamics provides a convenient framework for deriving scaling limits. In \cite{CKSS}, the most natural example from the point of view of probability theory, in which the path encodings of a sequence of i.i.d.\ configurations of increasing density were rescaled to a two-sided Brownian motion with drift, was presented. Not only did the latter result provide a means to establishing the invariance of two-sided Brownian motion with drift under Pitman's transformation (a result which was already known from the queuing literature, see \cite[Theorem 3]{OCY}, and \cite{HW} for an even earlier proof), but it provided motivation to introduce a model of BBS on $\mathbb{R}$. (A particular version of the latter model is checked to be integrable in \cite{CST}.) Specifically, this was given by applying Pitman's transformation to elements $S$ of $C(\mathbb{R},\mathbb{R})$ satisfying $S_0=0$ and $\sup_{t\leq 0}S_t<\infty$. In this article, we recall the aforementioned scaling limit (see Subsection \ref{BMsec}), and also give its periodic variant (see Subsection \ref{perbmsec}), as well as discuss a continuous version of the bounded soliton example (see Subsection \ref{contbounded}). As another important example, we describe a parameter regime in which the Markov configuration can be rescaled to the zigzag process, which consists of straight line segments of random length and alternating gradient $+1$ or $-1$ (see Subsection \ref{zzsec}). The description of the latter process as a scaling limit readily yields its invariance under Pitman's transformation (this result also appears in the queueing literature, see \cite{HMOC}). We also give a periodic version of zigzag process, show it is a scaling limit of cyclic Markov configurations, and establish its invariance under Pitman's transformation -- a result that we believe is new (see Subsection \ref{perzzsec}). From the point of view of integrable systems, the transformation of the zigzag process (and its periodic counterpart) under Pitman's transformation can be seen as describing the dynamics of the ultra-discrete Toda lattice (and its periodic counterpart, respectively) started from certain random initial conditions. By considering certain Palm measures associated with the zigzag process, the results of this article give natural invariant probability measures for the latter system as well (see Section \ref{palmsec}).
The remainder of this article is organised as follows. In Section \ref{discsec}, we present our examples of discrete invariant measures for the transformation $\eta\mapsto T\eta$. In Section \ref{contsec}, we detail the scaling limit framework, and explain how this can be applied to deduce invariance under Pitman's transformation of various random continuous stochastic processes. In Section \ref{palmsec}, we introduce Palm measures for the zigzag process, and use these to derive invariant measures for the ultra-discrete Toda lattice. Finally, in Section \ref{condsec}, we give a brief presentation concerning the connection between invariance under $T$ for a two-sided process and the laws of a conditioned versions of the corresponding one-sided process. NB. Regarding notational conventions, we write $\mathbb{N}=\{1,2,3,\dots\}$ and $\mathbb{Z}_+=\{0,1,2,\dots\}$.
\section{Discrete invariant measures}\label{discsec}
In the first part of this section (Subsections \ref{iidsec}-\ref{boundedsec}), we recall the invariant measures for the box-ball system (or equivalently the discrete-space version of Pitman's transformation) that were studied in \cite{CKSS}. As established in \cite{CKSS}, these represent all the invariant measures whose path encodings are supported on $\mathcal{S}^{rev}$ for which either the configuration $\eta$ or the carrier process $W$ is a two-sided stationary Markov chain (see \cite[Remark 1.10]{CKSS} in particular). Following this, in Subsection \ref{gibbssec}, we introduce a family of new invariant measures on periodic configurations based on certain Gibbs measures, and show that all the earlier examples can be obtained as infinite volume limits of these.
\subsection{Independent and identically distributed initial configuration}\label{iidsec} Perhaps the most fundamental invariant measure for the box-ball system is the case when $\eta$ is given by a sequence of independent and identically distributed Bernoulli random variables with parameter $p$. To ensure the law of the associated path encoding has distribution supported on $\mathcal{S}^{lin}$ (as defined at \eqref{slin}), we require $p<\frac12$. It is also clear that $\overleftarrow{\eta}\buildrel{d}\over{=}\eta$, and so the first of the conditions at \eqref{threeconds} is fulfilled. Moreover, the second of the conditions at \eqref{threeconds}, i.e.\ that $\bar{W}\buildrel{d}\over{=}W$, readily follows from the following description of the carrier process $W$ as a Markov chain. Indeed, the equations \eqref{Wprobs} and \eqref{pidef} below imply that detailed balance is satisfied by $W$, and thus it is reversible. As a result, Theorem \ref{mrd} can immediately be applied to deduce the invariance of the i.i.d.\ configuration, which we state precisely as Corollary \ref{revcor}.
{\lem[See {\cite[Lemma 3.13]{CKSS}}] If $\eta$ is given by a sequence of i.i.d.\ Bernoulli($p$) random variables with $p\in[0,\frac12)$, then $W$ is a two-sided stationary Markov chain with transition probabilities given by
\begin{equation}\label{Wprobs}
\mathbf{P}\left(W_{n}=W_{n-1}+j\:\vline\: W_{n-1}\right)=\left\{\begin{array}{ll}
p, & \mbox{if }j=1,\\
1-p, & \mbox{if }W_{n-1}>0\mbox{ and }j=-1,\\
1-p, & \mbox{if }W_{n-1}=0\mbox{ and }j=0.\\
\end{array}\right.
\end{equation}
The stationary distribution of this chain is given by $\pi=(\pi_x)_{x\in\mathbb{Z}_+}$, where
\begin{equation}\label{pidef}
\pi_x=\left(\frac{1-2p}{1-p}\right)\left(\frac{p}{1-p}\right)^x,\qquad \forall x\in\mathbb{Z}_+.
\end{equation}
}
{\cor[See {\cite[Corollary 3.14]{CKSS}}]\label{revcor} If $\eta$ is a sequence of i.i.d.\ Bernoulli($p$) random variables with $p\in[0,\frac12)$, then the three conditions of \eqref{threeconds} are satisfied. In particular, $\eta$ is invariant in distribution under $T$.}
\subsection{Markov initial configuration}\label{markovsec} As a generalisation of the i.i.d.\ configuration of the previous section, we next consider the case when $\eta$ is a two-sided stationary Markov chain on $\{0,1\}$ with transition matrix
\begin{equation}\label{pdef}
P=\left(
\begin{array}{cc}
1-p_0 & p_0 \\
1-p_1 & p_1 \\
\end{array}
\right),
\end{equation}
by which we mean
\[\mathbf{P}\left(\eta_{n+1}=1\:\vline\:\eta_n=j\right)=p_j,\qquad j\in\{0,1\},\]
for some parameters $p_0\in (0,1)$, $p_1\in[0,1)$. Note that we recover the i.i.d.\ case when $p_0=p_1=p$. The stationary distribution of this chain is given by
\begin{equation}\label{rhodef}
\rho=\mathbf{P}\left(\eta_0=1\right)=\frac{p_0}{1-p_1+p_0},
\end{equation}
and so to ensure the associated path encoding has distribution supported on $\mathcal{S}^{lin}$, we thus need to assume $p_0+p_1<1$. Since detailed balance is satisfied by $\eta$, we have that $\overleftarrow{\eta}\buildrel{d}\over{=}\eta$. Moreover, although $W$ is not a Markov chain, it is a stationary process whose marginal distributions are given by the following lemma, and \cite[Theorem 2]{HMOC} gives that $\bar{W}\buildrel{d}\over{=}W$. Thus we obtain from another application of Theorem \ref{mrd} the generalisation of Corollary \ref{revcor} to the Markov case, see Corollary \ref{markovcor} below.
{\lem[See {\cite[Lemma 3.15]{CKSS}}]\label{w0pm} If $\eta$ is the two-sided stationary Markov chain described above with
$p_0\in (0,1)$, $p_1\in[0,1)$ satisfying $p_0+p_1<1$, then
\[\mathbf{P}\left(W_{0}=m\right)=\left\{\begin{array}{ll}
\frac{1-p_0-p_1}{(1-p_0)(1+p_0-p_1)}, & \mbox{if }m=0,\\
\frac{p_0(1-p_0+p_1)(1-p_0-p_1)}{(1-p_0)^2(1+p_0-p_1)}\left(\frac{p_1}{1-p_0}\right)^{m-1}, & \mbox{if }m\geq 1.\\
\end{array}\right.\]}
{\cor[See {\cite[Corollary 3.16]{CKSS}}]\label{markovcor} If $\eta$ is the two-sided stationary Markov chain described above with
$p_0\in (0,1)$, $p_1\in[0,1)$ satisfying $p_0+p_1<1$, then the three conditions of (\ref{threeconds}) are satisfied. In particular, $\eta$ is invariant in distribution under $T$.}
\subsection{Conditioning the i.i.d.\ configuration to have bounded solitons}\label{boundedsec} In the two previous examples, it is possible to check that $\sup_{n\in\mathbb{Z}}W_n=\infty$, $\mathbf{P}$-a.s., which can be interpreted as meaning that the configurations admit solitons of an unbounded size. The motivation for the introduction of the example we present in this section came from the desire to exhibit a random initial configuration that contained solitons of a bounded size. To do this, the approach of \cite{CKSS} was to condition the i.i.d.\ configuration of Section \ref{iidsec} to not contain any solitons of size greater than $K$, or equivalently that $\sup_{n\in\mathbb{Z}}W_n\leq K$, for some fixed $K\in\mathbb{Z}_+$. Since the latter is an event of 0 probability whenever $\eta$ is Bernoulli($p$), for any $p\in (0,1)$, a limiting argument was used to make the this description rigourous. In particular, applying the classical theory of quasi-stationary distributions for Markov chains, we were able to show that the resulting configuration $\tilde{\eta}^{(K)}$ is stationary, ergodic, has path encoding with distribution supported on $\mathcal{S}^{lin}$, and moreover the three conditions at (\ref{threeconds}) hold.
To describe the construction of $\tilde{\eta}^{(K)}$ precisely, we start by defining the associated carrier process. Let $P=(P(x,y))_{x,y\in\mathbb{Z}_+}$ be the transition matrix of $W$, as defined in (\ref{Wprobs}) (where we now allow any $p\in (0,1)$). For $K\in \mathbb{Z}_+$ fixed, let $P^{(K)}=(P^{(K)}(x,y))_{x,y\in\{0,\dots,K\}}$ be the restriction of $P$ to $\{0,\dots,K\}$. Since $P^{(K)}$ is a finite, irreducible, substochastic matrix, it admits (by the Perron-Frobenius theorem) a unique eigenvalue of largest magnitude, $\lambda_K$ say. Moreover, $\lambda_K\in(0,1)$ and has a unique (up to scaling) strictly positive eigenvector $h_K=(h_K(x))_{x\in\{0,\dots,K\}}$. Let $\tilde{P}^{(K)}=(\tilde{P}^{(K)}(x,y))_{x,y\in\{0,\dots,K\}}$ be the stochastic matrix defined by
\[\tilde{P}^{(K)}(x,y)=\frac{{P}^{(K)}(x,y)h_K(y)}{\lambda_Kh_K(x)},\qquad \forall x,y\in \{0,\dots,K\}.\]
The associated Markov chain is reversible, and has stationary probability measure given by $\tilde{\pi}^{(K)}=(\tilde{\pi}^{(K)}_x)_{x\in \{0,\dots,K\}}$, where $\tilde{\pi}^{(K)}_x = c_1 h_K(x)^2{\pi}_x$ for some constant $c_1\in(0,\infty)$ (which may depend on $K$), and $\pi$ is defined as at (\ref{pidef}). Thus the Markov chain in question admits a two-sided stationary version, and we denote this by $\tilde{W}^{(K)}=(\tilde{W}^{(K)}_n)_{n\in\mathbb{Z}}$. We view $\tilde{W}^{(K)}$ as a random carrier process, and write the associated particle configuration $\tilde{\eta}^{(K)}=(\tilde{\eta}^{(K)}_n)_{n\in\mathbb{Z}}$.
To justify the claim that $\tilde{\eta}^{(K)}$ is the i.i.d.\ configuration of Section \ref{iidsec} conditioned to have solitons of size no greater than $K$, we have the following result. (An alternative description of the limit that is valid for $p\in(0,\frac12)$ is given in \cite[Remark 3.18]{CKSS}.)
{\lem[See {\cite[Lemma 3.17]{CKSS}}] Fix $K\in \mathbb{Z}_+$. Let ${\eta}=({\eta}_n)_{n\in\mathbb{Z}}$ be an i.i.d.\ Bernoulli($p$) particle configuration for some $p\in (0,1)$. Write $\eta^{[-N,N]}=(\eta^{[-N,N]}_n)_{n\in\mathbb{Z}}$ for the truncated configuration given by $\eta^{[-N,N]}_n=\eta_n\mathbf{1}_{\{-N<n\leq N\}}$. If $W^{[-N,N]}$ is the associated carrier process, then we have the following convergence of conditioned processes:
\[W^{[-N,N]}\:\vline\: \left\{\sup_{n\in\mathbb{Z}}W^{[-N,N]}_n\leq K\right\}\rightarrow\tilde{W}^{(K)}\]
in distribution as $N\rightarrow \infty$. In particular, this implies
\[\eta^{[-N,N]}\:\vline\: \left\{\sup_{n\in\mathbb{Z}}W^{[-N,N]}_n\leq K\right\}\rightarrow\tilde{\eta}^{(K)}\]
in distribution as $N\rightarrow \infty$.}
\bigskip
As a consequence of the construction of $\tilde{\eta}^{(K)}$, it is possible to check the following result.
{\cor[See {\cite[Corollary 3.19]{CKSS}}]\label{boundedcor} If $\tilde{\eta}^{(K)}$ and $\tilde{W}^{(K)}$ are as described above, then, for any $p\in (0,1)$, $K\in\mathbb{Z}_+$, $\tilde{\eta}^{(K)}$ is a stationary, ergodic process satisfying
\[\mathbf{P}\left(\tilde{\eta}^{(K)}_0=1\right)<\frac12,\]
and also the three conditions of (\ref{threeconds}). In particular, $\tilde{\eta}^{(K)}$ is invariant in distribution under $T$.}
\subsection{Initial configurations given by periodic Gibbs measures}\label{gibbssec} To define the Gibbs measures of interest, we start by introducing functions to count the number of solitons of certain sizes within the cycle of a periodic configuration. In particular, we first fix $N\in\mathbb{N}$ to represent our cycle length, and define
\[\begin{array}{rcl}
f_0 : \{0,1\}^N& \to& \mathbb{Z}_+\\
(x_n)_{n=1}^{N}&\mapsto& \sum_{n=1}^{N}x_n,
\end{array}\]
which will count the number of particles within a cycle of a periodic configuration. Next, we introduce
\[\begin{array}{rcl}
f_1 : \{0,1\}^N& \to& \mathbb{Z}_+\\
(x_n)_{n=1}^{N}&\mapsto& \sum_{n=1}^{N}\mathbf{1}_{\{x_{n-1}=1,x_{n}=0\}},
\end{array}\]
where we suppose $x_0:=x_N$ for the purposes of the above formula; this function will count the number of solitons within a cycle of a periodic configuration. To define $f_k$ for higher values of $k$, we introduce a contraction operation on particle configurations. Specifically, given a finite length configuration $(x_n)_{n=1}^{m}$ of $0$s and $1$s, define a new configuration $H((x_n)_{n=1}^{m})$ by removing all $(1,0)$ strings from $(x_n)_{n=1}^{m}$, including the pair $(x_{m},x_1)$ if relevant. For $k\geq 2$, we then set
\[\begin{array}{rcl}
f_k : \{0,1\}^N& \to& \mathbb{Z}_+\\
(x_n)_{n=1}^{N}&\mapsto& f_1\left(H^{k-1}\left((x_n)_{n=1}^{N}\right)\right),
\end{array}\]
where the definition of $f_1$ is extended to finite strings of arbitrary length in the obvious way; this function will count the number of solitons of length at least $k$ within a cycle of a periodic configuration. That $f_k$ describe conserved quantities for the box-ball system and indeed have the desired soliton interpretation, see \cite{YYT} (cf.\ the corresponding description in the non-periodic case of \cite{Torii}, and the description of the number of solitons of certain lengths via the `hill-flattening' operator of \cite{Lev}). We subsequently define a random variable $(\eta^N_n)_{n=1}^{N}$ taking values in $\{0,1\}^N$ by setting, as initially presented at \eqref{gibbs},
\[\mathbf{P}\left((\eta^N_n)_{n=1}^{N}=(x_n)_{n=1}^{N}\right)=\frac{1}{Z}\exp\left(-\sum_{k=0}^\infty\beta_kf_k\left((x_n)_{n=1}^{N}\right)\right)\mathbf{1}_{\{f_0\left((x_n)_{n=1}^{N}\right)<N/2\}}\]
for $(x_n)_{n=1}^{N}\in\{0,1\}^N$, where $\beta_k\in\mathbb{R}\cup\{\infty\}$ for each $k\geq 0$ and $Z$ is a normalising constant. NB.\ To ensure the measure is well-defined, we adopt the convention that if $\beta_k=\infty$ and $f_k((x_n)_{n=1}^{N})=0$, then their product is zero. We then extend to $\eta^N=(\eta^N_n)_{n\in\mathbb{Z}}$ by cyclic repetition; the law of $\eta^N$ is our Gibbs measure. Clearly, the inclusion of the term $\mathbf{1}_{\{f_0((x_n)_{n=1}^{N})<N/2\}}$ yields that the distribution of the path encoding of the configuration $\eta^N$ is supported on $\mathcal{S}^{lin}$.
We next check the spatial stationarity and distributional symmetry of $\eta^N$, and the distributional symmetry of the associated carrier process $W^N$.
\begin{lem}\label{etalem} The law of the periodic configuration $\eta^N$, as described by the Gibbs measure at \eqref{gibbs}, is stationary under spatial shifts. Moreover, $\overleftarrow{\eta}^N\buildrel{d}\over{=}\eta^N$.
\end{lem}
\begin{proof} For $x=(x_n)_{n=1}^{N}\in\{0,1\}^N$, it is straightforward to check from the definitions of the relevant functions that
\begin{equation}\label{per}
f_k(x)=f_k(\theta_{Per}x),\qquad \forall k\geq 0,
\end{equation}
where $\theta_{Per}$ is the periodic shift operator given by $\theta_{Per}x:=(x_2,\dots,x_N,x_1)$. Hence we obtain from \eqref{gibbs} that
\[\mathbf{P}\left(\theta_{Per}\left((\eta^N_n)_{n=1}^{N}\right)=x\right)=\mathbf{P}\left((\eta^N_n)_{n=1}^{N}=x\right),\qquad \forall x\in\{0,1\}^N.\]
It readily follows that $\theta\eta^N\buildrel{d}\over{=}\eta^N$, where $\theta$ is the left-shift on doubly infinite sequences, i.e.\ $\theta((x_n)_{n\in\mathbb{Z}})=(x_{n+1})_{n\in\mathbb{Z}}$. This establishes the first claim of the lemma.
We now check the second claim. For $x=(x_n)_{n=1}^{N}\in\{0,1\}^N$, write $\overleftarrow{x}$ for the reversed sequence $(x_{N+1-n})_{n=1}^N$. We clearly have that
\[f_0\left(x\right)=f_0\left(\overleftarrow{x}\right).\]
Moreover, recall that $f_1$ counts the number of $(1,0)$ strings in $x$, including the $(x_N,x_1)$ pair. The latter periodicity readily implies that this is equal to the number of $(0,1)$ strings in $x$ (cf.\ \cite[Lemma 2.1]{YYT}). Hence
\begin{equation}\label{f1prop}
f_1\left(x\right)=f_1\left(\overleftarrow{x}\right).
\end{equation}
Next, further recall that the configuration $H(x)$ is obtained by removing all $(1,0)$ strings from $x$, including the pair $(x_{N},x_1)$ if relevant. Since this operation simply reduces the lengths of all the strings of consecutive $0$ strings of consecutive $1$s by one, it is the same (up to a periodic shift) as the $(0,1)$-removal operation; this observation was made in \cite{YYT} (below Lemma 2.1 of that article), and also in the proof of \cite[Lemma 2.1]{Lev} in the non-periodic case. In particular, we have that
\[H(x)=\theta_{Per}^{l_x}\overleftarrow{H\left(\overleftarrow{x}\right)}\]
for some integer $l_x$ (where the definition of the periodic shift operator is extended to finite sequences of arbitrary length in the obvious way). Hence, applying this observation in conjunction with \eqref{per} and \eqref{f1prop}, we find that
\[f_k\left(\overleftarrow{x}\right)=f_1\left(H^{k-1}\left(\overleftarrow{x}\right)\right)=
f_1\left(\overleftarrow{H^{k-1}\left({x}\right)}\right)=f_1\left({H^{k-1}\left({x}\right)}\right)=f_k(x).\]
As a consequence of these observations, we thus obtain
\[\mathbf{P}\left((\eta^N_n)_{n=1}^{N}=x\right)=\mathbf{P}\left((\eta^N_n)_{n=1}^{N}=\overleftarrow{x}\right),\qquad \forall x\in\{0,1\}^N,\]
which implies $\overleftarrow{\eta}^N\buildrel{d}\over{=}\eta^N$, as desired.
\end{proof}
\begin{lem}\label{wlem} If $\eta^N$ is the periodic configuration with law given by the Gibbs measure at \eqref{gibbs}, then $\bar{W}^N\buildrel{d}\over{=}W^N$.
\end{lem}
\begin{proof} For a sequence $w:\{1,\dots,N\}\rightarrow \mathbb{Z}_+$, define the associated periodic increment process $\Delta(w)=(\Delta(w)_n)_{n=1}^N$ by setting
\[\Delta(w)_n=w_n-w_{n-1},\qquad \forall n\in\{1,\dots,N\},\]
where we define $w_0:=w_N$. Moreover, let $\mathcal{W}$ be the set of $w:\{1,\dots,N\}\rightarrow \mathbb{Z}_+$ such that $\Delta(w)\in\{-1,0,1\}^N$, $\Delta(w)_n=0$ if and only if $w_n=w_{n-1}=0$, and $\Delta(w)_n=0$ for at least one $n\in\{1,\dots,N\}$. Note that, on this set, $w$ is uniquely determined by $\Delta(w)$.
Now, since the configuration is $N$-periodic and $S_N>0$, $W^N$ is also $N$-periodic and moreover $(W_n)_{n=1}^N$ takes values in $\mathcal{W}$, $\mathbf{P}$-a.s. Since $\Delta(W^N)_n=1$ if and only if $\eta^N_n=1$, it follows that, for all $w\in\mathcal{W}$,
\[\mathbf{P}\left((W^N_n)_{n=1}^N=w\right)=\mathbf{P}\left((\Delta(W^N)_n)_{n=1}^N=\Delta(w)\right)=\mathbf{P}\left((\eta^N_n)_{n=1}^N=x\right),\]
where $x=(x_n)_{n=1}^N$ is defined by setting $x_n:=\mathbf{1}_{\{\Delta(w)_n=1\}}$. Moreover, using the notation $\bar{w}=(w_{N-1},w_{N-2},\dots,w_1,w_N)$ (which is also an element of $\mathcal{W}$), we have that
\[\mathbf{P}\left((\bar{W}^N_n)_{n=1}^N=w\right)=\mathbf{P}\left(({W}^N_n)_{n=1}^N=\bar{w}\right)=
\mathbf{P}\left((\Delta(W^N)_n)_{n=1}^N=\Delta(\bar{w})\right).\]
A simple calculation yields that $\Delta(\bar{w})=-\overleftarrow{\Delta(w)}$, and so we find that
\[\mathbf{P}\left((\bar{W}^N_n)_{n=1}^N=w\right)=\mathbf{P}\left((\eta^N_n)_{n=1}^N=\bar{x}\right),\]
where $\bar{x}=(\bar{x}_n)_{n=1}^N$ is defined by setting $\bar{x}_n:=\mathbf{1}_{\{\overleftarrow{\Delta(w)}_n=-1\}}$. In particular, the result will follow from the above observations and \eqref{gibbs} if we can show that $f_k(x)=f_k(\bar{x})$ for each $k\geq 0$.
Clearly, periodicity implies that the number of up-jumps of $w$ equals the number of down-jumps, and so
\[f_0(\bar{x})=\sum_{n=1}^N\mathbf{1}_{\left\{\overleftarrow{\Delta(w)}_n=-1\right\}}=
\sum_{n=1}^N\mathbf{1}_{\left\{{\Delta(w)}_n=-1\right\}}=\sum_{n=1}^N\mathbf{1}_{\left\{{\Delta(w)}_n=1\right\}}=f_0(x).\]
Furthermore, since $\Delta(w)$ can not contain the substrings $(0,-1)$ or $(1,0)$,
\begin{eqnarray*}
f_1(\bar{x})&=&\sum_{n=1}^N\mathbf{1}_{\left\{\overleftarrow{\Delta(w)}_{n-1}=-1,\:\overleftarrow{\Delta(w)}_n\in\{0,1\}\right\}}\\
&=&\sum_{n=1}^N\mathbf{1}_{\left\{{\Delta(w)}_{n-1}\in\{0,1\},\:{\Delta(w)}_n=-1\right\}}\\
&=&\sum_{n=1}^N\mathbf{1}_{\left\{{\Delta(w)}_{n-1}=1,\:{\Delta(w)}_n\in\{-1,0\}\right\}}\\
&=&f_1(x)
\end{eqnarray*}
Finally, observe that the $(1,-1)$ substrings of $\Delta(w)$ (including the one at $(w_N,w_1)$ if relevant) precisely correspond to the $(1,0)$ substrings of $x$. Moreover, if we suppose $H_W$ is the operation which removes these substrings, then it is an easy exercise to check that $H_W(\Delta(w))$ is the element of $\mathcal{W}$ representing the periodic increment process of the carrier associated with the configuration given by $H(x)$. We can iterate this argument to further obtain that $H_W^{k-1}(\Delta(w))$ is the element of $\mathcal{W}$ representing the periodic increment process of the carrier associated with the configuration given by $H^{k-1}(x)$ for any $k\geq 2$. Hence we can write
\begin{equation}\label{fk1}
f_k({x})=f_1\left(H^{k-1}({x})\right)=f_1\left(\left(\mathbf{1}_{\{H^{k-1}_W(\Delta({w}))_n=1\}}\right)_{n=1}^l\right),
\end{equation}
where $l$ is the length of the sequence $H_W^{k-1}(\Delta(w))$. Applying the same logic to $\bar{w}$, we similarly have that $H_W^{k-1}(\Delta(\bar{w}))$ is the element of $\mathcal{W}$ representing the periodic increment process of the carrier associated with the configuration given by $H^{k-1}(\bar{x})$ for any $k\geq 2$, and moreover the definition of $H_W$ readily implies that
\[H_W^{k-1}(\Delta(\bar{w}))=H_W^{k-1}\left(-\overleftarrow{\Delta({w})}\right)=-\overleftarrow{H_W^{k-1}(\Delta({w}))}.\]
Hence
\begin{equation}\label{fk2}
f_k(\bar{x})=f_1\left(\left(\mathbf{1}_{\{-\overleftarrow{H^{k-1}_W(\Delta({w}))}_n=1\}}\right)_{n=1}^l\right),
\end{equation}
and the argument for $f_1$ above shows the right-hand side of \eqref{fk1} and \eqref{fk2} are equal, which completes the proof.
\end{proof}
As a consequence of the previous two lemmas and Theorem \ref{mrd}, we readily obtain the main result of this section.
\begin{cor}\label{gibbsinv} If $\eta^N$ is the periodic configuration with law given by the Gibbs measure at \eqref{gibbs}, then the three conditions of (\ref{threeconds}) are satisfied. In particular, $\eta^N$ is invariant in distribution under $T$.
\end{cor}
\begin{rem} We now discuss an alternative, direct proof of Corollary \ref{gibbsinv}. Let $x\in\{0,1\}^N$ be such that $f_0(x)<N/2$, and $Tx=((Tx)_n)_{n=1}^N$ be the image of $x$ under the action of the periodic BBS. The definitions readily yield that if $w$ is the carrier path associated with $x$, then
\[Tx=\left(\mathbf{1}_{\Delta(w)_n=-1}\right)_{n=1}^N=\overleftarrow{\bar{x}},\]
where we are using the notation of the proofs of Lemmas \ref{etalem} and \ref{wlem}. Moreover, the arguments applied in these proofs imply that
\begin{equation}\label{fkpres}
f_k(Tx)=f_k(x),\qquad \forall k\geq 0.
\end{equation}
It clearly follows that the Gibbs measure at (\ref{gibbs}) is invariant under $T$, and we arrive at Corollary \ref{gibbsinv}. We note that the identity at (\ref{fkpres}) was previously proved as \cite[Proposition 2.1]{YYT}, see also \cite{Torii} for a proof in the non-periodic case.
\end{rem}
To conclude this section, we relate the Gibbs measures of this section with the i.i.d., Markov and bounded soliton configurations of Subsections \ref{iidsec}, \ref{markovsec} and \ref{boundedsec}, respectively. In particular, in the following examples we introduce three specific parameter choices for the Gibbs measures, and then show in Proposition \ref{infvollim} below that the aforementioned configurations can be obtained as infinite volume limits of these. Moreover, in Subsections \ref{perbmsec}, \ref{perzzsec} and \ref{contbounded}, we present scaling limits for certain sequences of periodic configurations based on these examples.
\begin{exmp}[Periodic i.i.d. initial configuration]\label{periidex} Similarly to \cite[Remark 1.12]{CKSS}, let $p\in (0,1)$, and consider the parameter choice
\[\beta_0=\log\left(\frac{1-p}{p}\right),\qquad \beta_k=0,\:\forall k \ge 1.\]
(Figure \ref{iidfig} shows a typical realisation of a configuration chosen according the associated Gibbs measure, and its subsequent evolution.) It is then an elementary exercise to check that
\begin{equation}\label{iidexp}
\mathbf{P}\left((\eta^N_n)_{n=1}^{N}=(x_n)_{n=1}^{N}\right)=\mathbf{P}\left((\eta_n)_{n=1}^{N}=(x_n)_{n=1}^{N}\:\vline\:S_N>0\right),
\end{equation}
where $\eta$ is an i.i.d.\ sequence of Bernoulli($p$) random variables. Note that the restriction $p<\frac12$ of Subsection \ref{iidsec} is equivalent to taking $\beta_0>0$, and in this regime we will check that $\eta^N$ converges in distribution to $\eta$ as $N\rightarrow\infty$ (see Proposition \ref{infvollim}(a)). We also describe the infinite volume limit in the case $\beta_0\leq 0$ (see Proposition \ref{infvollimhighdens}).
\end{exmp}
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.9\textwidth,height=0.3\textwidth]{iid.eps}
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\rput(-12.33,2.4833){$20$}
\rput(-12.33,3.3){$30$}
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\end{center}
\caption{Initial path encoding and first 25 steps of the dynamics for particle configuration sampled from the periodic i.i.d.\ configuration of Example \ref{periidex} with $p=0.35$, i.e.\ $\beta_0=0.62$.}\label{iidfig}
\end{figure}
\begin{exmp}[Periodic Markov initial configuration]\label{permarex} Again similarly to \cite[Remark 1.12]{CKSS}, let $p_0,p_1\in (0,1)$, and consider the parameter choice
\[\beta_0=\log \left(\frac{1-p_0}{p_1}\right),\qquad \beta_1=\log \left(\frac{p_1(1-p_0)}{p_0(1-p_1)}\right),\qquad\beta_k=0,\:\forall k \ge 2.\]
(Figure \ref{markovfig} shows a typical realisation of a configuration chosen according the associated Gibbs measure, and its subsequent evolution.) For these parameters, one can check that
\[\mathbf{P}\left((\eta^N_n)_{n=1}^{N}=(x_n)_{n=1}^{N}\right)\propto\prod_{n=1}^{N}P(x_{n-1},x_{n})\mathbf{1}_{\left\{\sum_{i=1}^{N}{x_i}<N/2\right\}},\]
where, as above, we have supposed that $x_0:=x_N$ in the preceding formula, and the matrix $P=(P(x,y))_{x,y\in\{0,1\}}$ is given by \eqref{pdef}. It follows that one has the following alternative characterisation of the law of $\eta^N$ via the formula
\begin{equation}\label{cyclicmarkov}
\mathbf{E}\left(F\left((\eta^N_n)_{n=1}^{N}\right)\right)=\frac{\mathbf{E}\left(\nu(\eta_0)^{-1}F\left((\eta_n)_{n=1}^{N}\right)\:\vline\:\eta_N=\eta_0,\:S_N>0\right)}{\mathbf{E}\left(\nu(\eta_0)^{-1}\:\vline\:\eta_N=\eta_0,\:S_N>0\right)},
\end{equation}
where $\eta$ is the two-sided stationary Markov configuration of Subsection \ref{markovsec} (noting that we now allow an increased range of parameters $p_0,p_1$), $\nu$ is its invariant measure, and the above formula holds for any function $F:\{0,1\}^{N}\rightarrow\mathbb{R}$. In particular, the initial segment of $\eta^N$ is obtained from $\eta$ by conditioning the latter process to return to its starting state at time $N$ and on seeing less than $N/2$ particles by that time, as well as weighting probabilities by $\nu(\eta_0)^{-1}$. Note that the latter step has the effect of removing the distributional influence of the initial state, thus ensuring the law of $\eta^N$ is stationary under spatial shifts (which is checked more generally as part of Lemma \ref{etalem} below). We note that a similar definition, without the $\nu(\eta_0)^{-1}$ term and $S_N>0$ conditioning, of a (non-stationary) cyclic Markov chain was given in \cite{Albenque}. Finally, the restriction $p_0+p_1<1$ of Subsection \ref{markovsec} is equivalent to taking $\beta_0>0$, and, similarly to the previous example, we will check that $\eta^N$ converges in distribution to $\eta$ as $N\rightarrow\infty$ in this regime (see Proposition \ref{infvollim}(b)).
\end{exmp}
\begin{figure}[t]
\begin{center}
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\end{center}
\caption{Initial path encoding and first 25 steps of the dynamics for particle configuration sampled from the periodic Markov configuration of Example \ref{permarex} with $N=100$ and parameters $p_0=0.11$, $p_1=0.80$, i.e.\ $\beta_0=0.11$, $\beta_1=3.48$. (Note these parameters correspond to a density of $\rho=0.35$ for the non-periodic version of the configuration, matching that for the non-periodic version of the i.i.d.\ example shown in Figure \ref{iidfig}.)}\label{markovfig}
\end{figure}
\begin{exmp}[Periodic bounded soliton configuration]\label{perboundedex} Once again similarly to \cite[Remark 1.12]{CKSS}, let $p\in (0,1)$ and $K\in\mathbb{N}$, and consider the parameter choice
\[\beta_0=\log \left(\frac{1-p}{p}\right),\qquad \beta_k=0,\:\forall k\in\{1,\dots,K\},\qquad \beta_k=\infty,\:\forall k> K.\]
For these parameters, one can check that
\begin{equation}\label{bexp0}
\mathbf{E}\left(F\left((\eta^N_n)_{n=1}^{N}\right)\right)=\mathbf{E}\left(F\left((\eta_n)_{n=1}^{N}\right)\:\vline\:\mathcal{A}_{N,K},\:S_N>0\right),
\end{equation}
where $\eta$ is an i.i.d.\ sequence of Bernoulli($p$) random variables, and
\begin{equation}\label{ankdef}
\mathcal{A}_{N,K}=\left\{\max_{0\leq n\leq N}\max\left\{\max_{0\leq m\leq n}(S_m-S_n),\:\max_{n\leq m\leq N}(S_m-S_N-S_n)\right\}\leq K\right\}.
\end{equation}
(Note the expression involving nested maxima simply describes the supremum of the carrier corresponding to the cyclic repetition of $(\eta_n)_{n=1}^{N}$.) We will check that, for any parameters $p\in (0,1)$ and $K\in\mathbb{N}$, $\eta^N$ converges in distribution to $\tilde{\eta}^{(K)}$, the example of Subsection \ref{boundedsec}, as $N\rightarrow\infty$ (see Proposition \ref{infvollim}(c)).
\end{exmp}
We now give the infinite volume limits for the previous three examples.
\begin{prop}\label{infvollim} (a) Let $p\in(0,\frac12)$, and $\eta^{N,iid}$ be the periodic i.i.d.\ configuration of Example \ref{periidex} (i.e.\ with law given by \eqref{iidexp}). Then
\[\eta^{N,iid}\buildrel{d}\over\rightarrow\eta^{iid}\]
as $N\rightarrow\infty$, where $\eta^{iid}$ is the i.i.d.\ configuration of Subsection \ref{iidsec}.\\
(b) Let $p_0,p_1\in(0,1)$ be such that $p_0+p_1<1$, and $\eta^{N,Mar}$ be the periodic Markov configuration of Example \ref{periidex} (i.e.\ with law given by \eqref{cyclicmarkov}). Then
\[\eta^{N,Mar}\buildrel{d}\over\rightarrow\eta^{Mar}\]
as $N\rightarrow\infty$, where $\eta^{Mar}$ is the configuration of Subsection \ref{markovsec}.\\
(c) Let $p\in(0,1)$ and $K\in\mathbb{N}$, and $\eta^{N,b}$ be the periodic bounded soliton configuration of Example \ref{perboundedex} (i.e.\ with law given by \eqref{bexp0}). Then
\[\eta^{N,b}\buildrel{d}\over\rightarrow\tilde{\eta}^{(K)}\]
as $N\rightarrow\infty$, where $\tilde{\eta}^{(K)}$ is the bounded soliton example of Subsection \ref{boundedsec}.
\end{prop}
\begin{proof} The proof of (a) is straightforward. Indeed, starting from \eqref{iidexp}, and applying that $\mathbf{P}(S_N>0)\rightarrow 1$, we obtain: for any $M\in\mathbb{N}$, $x\in\{0,1\}^M$,
\[\mathbf{P}\left((\eta^{N,iid}_n)_{n=1}^{M}=(x_n)_{n=1}^{M}\right)=\frac{\mathbf{P}\left((\eta^{iid}_n)_{n=1}^{M}=(x_n)_{n=1}^{M},\:S_N>0\right)}{\mathbf{P}\left(S_N>0\right)}\rightarrow\mathbf{P}\left((\eta^{iid}_n)_{n=1}^{M}=(x_n)_{n=1}^{M}\right).\]
For (b), we start from \eqref{cyclicmarkov} to deduce: for any $M\in\mathbb{N}$, $x\in\{0,1\}^M$,
\begin{equation}\label{ratio}
\mathbf{P}\left((\eta^{N,Mar}_n)_{n=1}^{M}=(x_n)_{n=1}^{M}\right)=
\frac{\mathbf{E}\left(\nu(\eta^{Mar}_0)^{-1}\mathbf{1}_{\left\{(\eta^{Mar}_n)_{n=1}^{M}=(x_n)_{n=1}^{M},\:\eta^{Mar}_N=\eta^{Mar}_0,\:S_N>0\right\}}\right)}
{\mathbf{E}\left(\nu(\eta^{Mar}_0)^{-1}\mathbf{1}_{\left\{\:\eta^{Mar}_N=\eta^{Mar}_0,\:S_N>0\right\}}\right)}.
\end{equation}
Now, by the definition of the Markov chain, the numerator can be written
\[\sum_{x_0\in\{0,1\}}\prod_{n=1}^{M}P(x_{n-1},x_n)\mathbf{P}\left(\eta^{Mar}_{N-M}=x_0,\:S_{N-M}+\sum_{n=1}^M(1-2x_n)>0\:\vline\:\eta^{Mar}_0=x_M\right).\]
Since $N^{-1}S_N\rightarrow1-2\rho>0$, $\mathbf{P}$-a.s., where $\rho$ was defined at \eqref{rhodef}, it readily follows that this expression converges as $N\rightarrow\infty$ to
\[\sum_{x_0\in\{0,1\}}\nu(x_0)\prod_{n=1}^{M}P(x_{n-1},x_n)=\nu(x_1)\prod_{n=2}^{M}P(x_{n-1},x_n)=\mathbf{P}\left((\eta^{Mar}_n)_{n=1}^{M}=(x_n)_{n=1}^{M}\right).\]
Summing over $x\in\{0,1\}^M$ shows that the denominator of \eqref{ratio} converges to one, and hence we have established the result in this case.
Finally, we prove (c) for $p\in(0,1)$, $K\in \mathbb{N}$. To this end, we first provide an alternative characterisation of \eqref{bexp0}. In particular, let $(\eta_n)_{n\geq 1}$ be i.i.d.\ with parameter $p\in(0,1)$, and $\hat{W}$ be the associated carrier process started from the initial condition that $\hat{W}_0$ is uniform on $\{0,1,\dots,K\}$. (Note the latter process is a Markov chain on $\mathbb{Z}_+$ with transition probabilities as at \eqref{Wprobs}.) We then claim that
\begin{equation}\label{bexp}
\mathbf{E}\left(F\left((\eta^{N,b}_n)_{n=1}^{N}\right)\right)=
\mathbf{E}\left(F\left((\eta_n)_{n=1}^{N}\right)\:\vline\:\hat{W}_N=\hat{W}_0,\:\max_{1\leq n\leq N}\hat{W}_n\leq K,\:S_N>0\right).
\end{equation}
To prove this, observe that for any sequence $x\in\{0,1\}^N$
\begin{eqnarray*}
\lefteqn{\mathbf{P}\left((\eta_n)_{n=1}^{N}=(x_n)_{n=1}^{N}\:\vline\:\hat{W}_N=\hat{W}_0,\:\max_{1\leq n\leq N}\hat{W}_n\leq K,\:S_N>0\right)}\\
&=&c
\sum_{w_0=0}^K\mathbf{P}\left((\eta_n)_{n=1}^{N}=(x_n)_{n=1}^{N},\:\hat{W}_0=w_0\right)\mathbf{1}_{\{w^{x,w_0}_N=w_0,\:
\max_{1\leq n\leq N}w^{x,w_0}_n\leq K,\:\sum_{n=1}^Nx_n<N/2\}},
\end{eqnarray*}
where $c:=\mathbf{P}(\hat{W}_N=\hat{W}_0,\:\max_{1\leq n\leq N}\hat{W}_n\leq K,\:S_N>0)^{-1}$ is the required normalising constant, and $(w^{x,w_0}_n)_{n=1}^N$ is the path of the carrier process corresponding to initial carrier value $w_0$ and particle configuration $x$. Since we are assuming the initial distribution of $\hat{W}$ is uniform, and it also holds that $\hat{W}_0$ is independent of $(\eta_n)_{n=1}^{N}=(x_n)_{n=1}^{N}$, we thus have that the above expression is equal to
\[c(K+1)^{-1}\mathbf{P}\left((\eta_n)_{n=1}^{N}=(x_n)_{n=1}^{N}\right)\mathbf{1}_{\{\sum_{n=1}^Nx_n<N/2\}}
\sum_{w_0=0}^K\mathbf{1}_{\{w^{x,w_0}_N=w_0,\:
\max_{1\leq n\leq N}w^{x,w_0}_n\leq K\}}.\]
Now, under the conditions that $w^{x,w_0}_N=w_0$ and $\sum_{n=1}^Nx_n<N/2$, it is straightforward to check that $\max_{1\leq n\leq N}w^{x,w_0}_n\leq K$ is equivalent to $x\in\mathcal{A}_{N,K}$ (in the sense that the associated path encoding satisfies the condition given in the definition of $\mathcal{A}_{N,K}$ at \eqref{ankdef}). And, it is moreover possible to show that under $\sum_{n=1}^Nx_n<N/2$ and $x\in\mathcal{A}_{N,K}$, the condition $w^{x,w_0}_N=w_0$ holds for exactly one $w_0$ (corresponding to $\max_{0\leq n\leq N}S_n-S_N$ for the relevant path encoding). Hence we conclude that
\begin{eqnarray*}
\lefteqn{\mathbf{P}\left((\eta_n)_{n=1}^{N}=(x_n)_{n=1}^{N}\:\vline\:\hat{W}_N=\hat{W}_0,\:\max_{1\leq n\leq N}\hat{W}_n\leq K,\:S_N>0\right)}\\
&=&c(K+1)^{-1}\mathbf{P}\left((\eta_n)_{n=1}^{N}=(x_n)_{n=1}^{N}\right)\mathbf{1}_{\{x\in\mathcal{A}_{N,K},\:
\sum_{n=1}^Nx_n<N/2\}},
\end{eqnarray*}
and hence \eqref{bexp} follows from the characterisation of the law of $\eta^{N,b}$ at \eqref{bexp0}. To study the limit of \eqref{bexp} as $N\rightarrow\infty$, we start by considering the corresponding formula without the $S_N>0$ conditioning. That is, given a sequence $x\in\{0,1\}^M$ representing a particle configuration, we will deduce the $N\rightarrow\infty$ asymptotics of
\begin{equation}\label{bbb}
\mathbf{P}\left((\eta_n)_{n=1}^{M}=(x_n)_{n=1}^{M}\:\vline\:\hat{W}_N=\hat{W}_0,\:\max_{1\leq n\leq N}\hat{W}_n\leq K\right).
\end{equation}
Decomposing over the value of $\hat{W}_0$, we have that the above probability can be written
\begin{eqnarray*}
\lefteqn{\sum_{w_0=0}^K\mathbf{P}\left(\hat{W}_0=w_0,\:(\hat{W}_n)_{n=1}^M=(w^{x,w_0}_n)_{n=1}^M\:\vline\:\hat{W}_N=\hat{W}_0,\:\max_{1\leq n\leq N}\hat{W}_n\leq K\right)}\\
&=&\frac{\sum_{w_0=0}^K\prod_{n=1}^MP_W(w^{x,w_0}_{n-1},w^{x,w_0}_{n})
\mathbf{P}\left(\hat{W}_{N-M}=w_0,\:\max_{1\leq n\leq {N-M}}\hat{W}_n\leq K\:\vline\:\hat{W}_{0}=w^{x,w_0}_M\right)}{(K+1)\mathbf{P}\left(\hat{W}_N=\hat{W}_0,\:\max_{1\leq n\leq N}\hat{W}_n\leq K\right)},
\end{eqnarray*}
where $P_W$ is the transition matrix of $\hat{W}$, as given by \eqref{Wprobs}. Similarly decomposing the numerator, this equals
\begin{equation}\label{aaa}
\frac{\sum_{w_0=0}^K\prod_{n=1}^MP_W(w^{x,w_0}_{n-1},w^{x,w_0}_{n})
\mathbf{P}\left(\hat{W}_{N-M}=w_0,\:\max_{1\leq n\leq {N-M}}\hat{W}_n\leq K\:\vline\:\hat{W}_{0}=w^{x,w_0}_M\right)}{\sum_{w_0=0}^K\mathbf{P}\left(\hat{W}_N=w_0,\:\max_{1\leq n\leq N}\hat{W}_n\leq K\:\vline\:\hat{W}_{0}=w_0\right)}.
\end{equation}
Now, applying \cite[Proposition 1]{GT}, we have that
\[\mathbf{P}\left(\hat{W}_{N-M}=w_0,\:\max_{1\leq n\leq {N-M}}\hat{W}_n\leq K\:\vline\:\hat{W}_{0}=w^{x,w_0}_M\right)\sim \frac{\lambda_K^{N-M}h_K(w^{x,w_0}_M)\tilde{\pi}^{(K)}_{w_0}}{h_K(w_0)},\]
where we have applied the notation of Subsection \ref{boundedsec}, and similarly
\[\mathbf{P}\left(\hat{W}_{N}=w_0,\:\max_{1\leq n\leq {N}}\hat{W}_n\leq K\:\vline\:\hat{W}_{0}=w_0\right)\sim
\lambda_K^{N}\tilde{\pi}^{(K)}_{w_0}.\]
It follows that \eqref{aaa} converges as $N\rightarrow\infty$ to
\begin{eqnarray*}
\lefteqn{\sum_{w_0=0}^K\tilde{\pi}^{(K)}_{w_0}\lambda_K^{-M}h_K(w^{x,w_0}_M)h_K(w_0)^{-1}\prod_{n=1}^MP_W(w^{x,w_0}_{n-1},w^{x,w_0}_{n})}\\
&=&\sum_{w_0\in\mathbb{Z}_+}\tilde{\pi}^{(K)}_{w_0}\prod_{n=1}^M\tilde{P}^{(K)}(w^{x,w_0}_{n-1},w^{x,w_0}_{n})\\
&=&\mathbf{P}\left(\left(\tilde{W}^{(K)}_n\right)_{n=1}^M=\left(w^{x,\tilde{W}^{(K)}_0}_n\right)_{n=1}^M\right)\\
&=&\mathbf{P}\left(\left(\tilde{\eta}^{(K)}_n\right)_{n=1}^M=\left(x_n\right)_{n=1}^M\right).
\end{eqnarray*}
In order to complete the proof, we need to show the same limit when the $S_N>0$ conditioning is reintroduced. To this end, first suppose $\tilde{\eta}^{N,b}$ is a random configuration chosen such that $\mathbf{P}((\tilde{\eta}^{N,b}_n)_{n=1}^N=(x_n)_{n=1}^N)$ is given by \eqref{bbb} (with $M=N$), so that ${\eta}^{N,b}$ has the law of $\tilde{\eta}^{N,b}$ conditioned on $\sum_{n=1}^N(1-2\tilde{\eta}^{N,b}_n)>0$. Moreover, observe that, for any $M\in\mathbb{N}$,
\begin{eqnarray*}
\limsup_{N\rightarrow\infty}\mathbf{P}\left(\sum_{n=1}^N(1-2\tilde{\eta}^{N,b}_n)\leq 0\right)&\leq & \limsup_{N\rightarrow\infty}\mathbf{P}\left(\sum_{n=1}^M(1-2\tilde{\eta}^{N,b}_n)
\leq K\right)\\
&=&\mathbf{P}\left(\sum_{n=1}^M(1-2\tilde{\eta}^{(K)}_n)\leq K\right),
\end{eqnarray*}
and, by Corollary \ref{boundedcor}, the final expression here can be made arbitrarily small by choosing $M$ large. Hence, in conjunction with the previous part of the proof, we obtain that
\begin{eqnarray*}
\mathbf{P}\left(\left({\eta}^{N,b}_n\right)_{n=1}^M=\left(x_n\right)_{n=1}^M\right)
&=&\mathbf{P}\left(\left(\tilde{\eta}^{N,b}_n\right)_{n=1}^M=\left(x_n\right)_{n=1}^M\:\vline\:\sum_{n=1}^N(1-2\tilde{\eta}^{N,b}_n)>0\right)\\
&\rightarrow&\mathbf{P}\left(\left(\tilde{\eta}^{(K)}_n\right)_{n=1}^M=\left(x_n\right)_{n=1}^M\right),
\end{eqnarray*}
as desired.
\end{proof}
In the final result of this section, we demonstrate that if we take the infinite volume limit in the periodic i.i.d.\ initial configuration (Example \ref{periidex}) for a parameter $\beta_0\leq0$ (corresponding to $p\geq \frac12$), then the limit is independent of the particular parameter chosen, being equal to the configuration consisting of i.i.d.\ Bernoulli parameter $\frac12$ random variables. Note that, whilst the latter configuration can be thought of as lying on the boundary of a collection of random configurations that are invariant for $T$, the two-sided dynamics are not even defined in this case (since obviously $M_0=\infty$). Moreover, we observe that its density is critical, in the sense that any infinite volume limit of a periodic Gibbs measure can be no greater than $\frac12$. Whilst we do not pursue this point further, we expect similar phenomena for other choices of parameter $(\beta_k)_{k\geq 0}$ that, beyond the $S_N>0$ restriction, favour configurations of density greater than or equal to $\frac12$.
\begin{prop}\label{infvollimhighdens} Let $p\geq \frac12$, and $\eta^{N,iid}$ be the periodic i.i.d.\ configuration of Example \ref{periidex} (i.e.\ with law given by \eqref{iidexp}). Then
\[\eta^{N,iid}\buildrel{d}\over\rightarrow\eta^{\frac12}\]
as $N\rightarrow\infty$, where $\eta^{\frac12}$ is the i.i.d.\ configuration of Subsection \ref{iidsec} with $p=\frac12$.
\end{prop}
\begin{proof} We first deal with the case when $p=\frac12$. For this parameter choice, we have that
\begin{eqnarray*}
\mathbf{P}\left((\eta^{N,iid}_n)_{n=1}^{M}=(x_n)_{n=1}^{M}\right)&=&\frac{\mathbf{P}\left((\eta^{\frac12}_n)_{n=1}^{M}=(x_n)_{n=1}^{M}\right)\mathbf{P}\left(S_{N-M}+\sum_{n=1}^M(1-2x_n)>0\right)}{\mathbf{P}\left(S_N>0\right)}\\
&\rightarrow&\mathbf{P}\left((\eta^{\frac12}_n)_{n=1}^{M}=(x_n)_{n=1}^{M}\right),
\end{eqnarray*}
where in the above $S$ is the path encoding of $\eta^\frac12$, and the limit is a ready consequence of the fact that ${N}^{-1/2}S_N$ converges in distribution to a standard normal as $N\rightarrow\infty$.
We now consider the case when $p>\frac12$. Conditioning on the value of $S_N$, we have that
\begin{eqnarray*}
\lefteqn{\mathbf{P}\left((\eta^{N,iid}_n)_{n=1}^{M}=(x_n)_{n=1}^{M}\right)}\\
&=&\sum_{k>0}\mathbf{P}\left((\eta^{\frac12}_n)_{n=1}^{M}=(x_n)_{n=1}^{M}\:\vline\:S_N=k\right)
\mathbf{P}\left(S_N=k\:\vline\:S_N>0\right)\\
&=&\sum_{k>0}\frac{(N-M)!\left(\frac{N-k}{2}\right)!\left(\frac{N+k}{2}\right)!}{
N!\left(\frac{N-k}{2}-\sum_{n=1}^Mx_n\right)!\left(\frac{N+k}{2}-M+\sum_{n=1}^Mx_n\right)!}\mathbf{P}\left(S_N=k\:\vline\:S_N>0\right),
\end{eqnarray*}
where the summands should be interpreted as $0$ wherever the arguments of the terms involving factorials are not all non-negative integers. We next note that Cramer's theorem for an i.i.d.\ sequence (e.g.\ \cite[Theorem 2.2.3]{DZ}) yields that, for any $\varepsilon>0$,
\[\mathbf{P}\left(S_N>\varepsilon N\:\vline\:S_N>0\right)\rightarrow 0.\]
Moreover, straightforward calculations give that, uniformly over the relevant $k\in[0,\varepsilon N]$,
\[2^{-M}(1-\varepsilon)^M
\leq \frac{(N-M)!\left(\frac{N-k}{2}\right)!\left(\frac{N+k}{2}\right)!}{
N!\left(\frac{N-k}{2}-\sum_{n=1}^Mx_n\right)!\left(\frac{N+k}{2}-M+\sum_{n=1}^Mx_n\right)!}\leq
2^{-M}(1+\varepsilon)^M\left(\frac{1}{1-\frac{M-1}{N}}\right)^M.\]
It thus follows that
\[\mathbf{P}\left((\eta^{N,iid}_n)_{n=1}^{M}=(x_n)_{n=1}^{M}\right)\rightarrow 2^{-M}=\mathbf{P}\left((\eta^{\frac12}_n)_{n=1}^{M}=(x_n)_{n=1}^{M}\right),\]
as desired.
\end{proof}
\section{Continuous invariant measures}\label{contsec}
In \cite{CKSS}, a continuous state space version of the BBS was formulated to describe scaling limits of the discrete system. This was based on a two-sided version of Pitman's transformation for continuous functions, which had been studied previously in the probabilistic literature, particularly in the context of queuing (see, for example, \cite{OCY}). The main example given in \cite{CKSS} was the two-sided Brownian with drift (this is recalled in Subsection \ref{BMsec}), which had previously been shown to be invariant for Pitman's transformation in \cite{HW}. Here we further show that the zigzag process, which also appears in the queueing literature \cite{HMOC}, naturally arises as a limit of the Markov initial configuration, see Subsection \ref{zzsec}. Whilst it is possible to check that Brownian motion and the zigzag process are both invariant under Pitman's transformation directly, our approach is to deduce the latter results by establishing that the processes in question are scaling limits of discrete systems, and showing that the invariance under $T$ transfers to the limit. In addition to the examples already mentioned, we follow this line of argument for the periodic models described in Examples \ref{periidex} and \ref{permarex}, see Subsections \ref{perbmsec} and \ref{perzzsec}, respectively. We also discuss continuous versions of the bounded soliton examples of Subsection \ref{boundedsec} and Example \ref{perboundedex} in Subsection \ref{contbounded}.
Prior to introducing the specific models, let us summarise the scaling approach we will use. The following assumption describes the framework in which we are working.
\begin{assu}\label{sassu} It holds that $\eta^\varepsilon=(\eta^\varepsilon_n)_{n\in \mathbb{Z}}$, $\varepsilon>0$, is a collection of random configurations such that
\begin{equation}\label{invarassu}
T\eta^\varepsilon\buildrel{d}\over{=}\eta^\varepsilon
\end{equation}
for each $\varepsilon>0$. The corresponding path encodings $S^\varepsilon$, $\varepsilon>0$, satisfy
\begin{equation}\label{scaling}
\left(a_\varepsilon S^\varepsilon_{t/b_\varepsilon}\right)_{t\in\mathbb{R}}\buildrel{d}\over{\rightarrow} \left(S_t\right)_{t\in\mathbb{R}},
\end{equation}
in $C(\mathbb{R},\mathbb{R})$, where: $(a_\varepsilon)_{\varepsilon>0}$ and $(b_\varepsilon)_{\varepsilon>0}$ are deterministic sequences in $(0,\infty)$; $S^\varepsilon$ is extended to an element of $C(\mathbb{R},\mathbb{R})$ by linear interpolation; and $S$ is a random element of $C(\mathbb{R},\mathbb{R})$. Moreover, for any $t\in\mathbb{R}$, it holds that
\begin{equation}\label{cond1}
\lim_{s \to -\infty} \limsup_{\varepsilon\rightarrow 0} \mathbf{P}\left(M^{\varepsilon}_{s/b_{\varepsilon}} > S^{\varepsilon}_{t/b_\varepsilon}\right)=0,
\end{equation}
and
\begin{equation}\label{cond2}
\lim_{s \to -\infty} \mathbf{P}\left(M_{s} > S_{t}\right)=0,
\end{equation}
where $M^{\varepsilon}$ and $M$ are the past maximum processes associated with $S^{\varepsilon}$ and $S$, respectively.
\end{assu}
We note that the conditions at \eqref{cond1} and \eqref{cond2} ensure the simultaneous convergence of the rescaled past maximum processes with the convergence of path encodings given at \eqref{scaling}, and as a consequence we obtain the following result concerning the invariance under $T$ of the limiting path encoding.
\begin{prop}[cf.\ {\cite[Lemma 5.11]{CKSS}}]\label{invarlimit} If Assumption \ref{sassu} holds, then
\[TS\buildrel{d}\over{=}S.\]
\end{prop}
\subsection{Brownian motion with drift}\label{BMsec} Perhaps the simplest, and most fundamental, (non-trivial) example of a scaling limit for the path encoding of the box-ball system is seen in the high-density regime. Specifically, fix a constant $c>0$, and consider the configuration $\eta^\varepsilon$ generated by an i.i.d.\ sequence of Bernoulli random variables, with parameter
\begin{equation}\label{peps}
p_\varepsilon:=\frac{1-\varepsilon c}{2}.
\end{equation}
(We assume $\varepsilon<c^{-1}$ for the above to make sense.) By Corollary \ref{revcor}, we have that \eqref{invarassu} holds. Moreover, it is an elementary application of the classical invariance principle that \eqref{scaling} holds with $a_\varepsilon=\varepsilon$, $b_\varepsilon=\varepsilon^2$, and $S$ a two-sided Brownian motion with drift $c$, i.e.\
\[S_t=\left\{\begin{array}{ll}
ct+S^{(1)}_t, & t\geq 0,\\
ct+S^{(2)}_{-t}, & t<0,
\end{array}\right.\]
where $S^{(1)}$ and $S^{(2)}$ are independent standard Brownian motions (starting from 0). Also, \eqref{cond1} and \eqref{cond2} were checked as \cite[Lemma 5.12]{CKSS}. Hence Assumption \ref{sassu} is satisfied in this setting, and we conclude from Proposition \ref{invarlimit} the following result.
\begin{prop} If $S$ is a two-sided Brownian motion with drift $c>0$, then $TS\buildrel{d}\over{=}S$.
\end{prop}
\begin{rem}\label{bmcar} In this case, the carrier $W=M-S$ is the stationary version of Brownian motion with drift $-c$, reflected at the origin. In particular, $W_0$ is exponentially distributed with parameter $2c$, so that $\mathbf{E}W_0=(2c)^{-1}$.
\end{rem}
\subsection{Zigzag process}\label{zzsec} It is not difficult to extend the result of the previous section to show that Brownian motion with drift can also be obtained from a more general class of Markov configurations in the high-density limit. In this section, however, we study a different scaling regime for the Markov configurations of Section \ref{markovsec}. Indeed, we will consider the case when the adjacent states are increasingly likely to be the same, and explain how we can see the so-called zigzag process (we take the name from \cite{DMOC}, though there the name was applied to the carrier process $M-S$; our version is also a generalisation of the so-called telegrapher's process \cite{Kac}) as a scaling limit.
Concerning the details, in this section we fix $\lambda_0,\lambda_1>0$, and suppose $\eta^\varepsilon$ is a two-sided stationary Markov chain on $\{0,1\}$ with transition matrix
\begin{equation}\label{Peps}
P_\varepsilon=\left(
\begin{array}{cc}
1- \varepsilon \lambda_0 & \varepsilon \lambda_0 \\
\varepsilon \lambda_1 & 1- \varepsilon \lambda_1 \\
\end{array}
\right).
\end{equation}
(We assume $\varepsilon$ is small enough so that the entries of this matrix are strictly positive.) We note that the invariant measure for $\eta^{\varepsilon}$ is independent of $\varepsilon$, being given by
\[\mathbf{P}\left(\eta^\varepsilon_0=1\right)=\frac{\lambda_0}{\lambda_0+\lambda_1},\]
and so to ensure the associated path encoding has distribution supported on $\mathcal{S}^{lin}$, we thus need to assume $\lambda_0<\lambda_1$, as we will do henceforth. From Corollary \ref{markovcor}, we then have that \eqref{invarassu} holds in this setting.
By definition, the numbers of spatial locations for which $\eta^{\varepsilon}$ takes the value $0$ or $1$ before a change are given by geometric random variables with parameters $\varepsilon \lambda_0$ or $\varepsilon \lambda_1$, respectively. Noting that, when multiplied by $\varepsilon$, the latter random variables converge to exponential, parameter $\lambda_0$ or $\lambda_1$, random variables, it is an elementary exercise to check that
\begin{equation}\label{etaconv}
\left(\eta^\varepsilon_{\lfloor t/\varepsilon \rfloor}\right)_{t\in\mathbb{R}}\buildrel{d}\over{\rightarrow} \left(\eta_t\right)_{t\in\mathbb{R}}
\end{equation}
in $D(\mathbb{R},\{0,1\})$, where the limiting process is the two-sided, stationary continuous-time Markov chain on $\{0,1\}$ that jumps from $0$ to $1$ with rate $\lambda_0$, and from $1$ to $0$ with rate $\lambda_1$. As a consequence, we find that \eqref{scaling} holds with $a_\varepsilon=b_{\varepsilon}=\varepsilon$, and the limiting process being given by $S=(S_t)_{t\in\mathbb{R}}$, where
\begin{equation}\label{sdef}
S_t:=\int_0^t(1-2\eta_s)ds;
\end{equation}
this is the zigzag process. Since $\lambda_0<\lambda_1$, it is an elementary to exercise to check that $t^{-1}S_t\rightarrow \frac{\lambda_1-\lambda_0}{\lambda_1+\lambda_0}>0$ as $|t|\rightarrow \infty$, $\mathbf{P}$-a.s., from which \eqref{cond2} readily follows. The remaining condition we need to apply Proposition \ref{invarlimit} is given by the following lemma.
\begin{lem} If $\eta^{\varepsilon}$ are the random configurations described above with $\lambda_0<\lambda_1$, then \eqref{cond1} holds with $b_\varepsilon=\varepsilon$.
\end{lem}
\begin{proof} Applying the Markov property for $\eta^\varepsilon$ it will suffice to show that
\[\lim_{t \to -\infty} \limsup_{\varepsilon\rightarrow 0} \mathbf{P}\left(M^{\varepsilon}_{t/{\varepsilon}} > 0\:\vline\:\eta^{\varepsilon}_0=i\right)=0,\qquad i=0,1.\]
To this end, observe that, for any $x\geq 0$,
\[\mathbf{P}\left(M^{\varepsilon}_{t/{\varepsilon}} > 0\:\vline\:\eta^{\varepsilon}_0=i\right)\leq
\mathbf{P}\left(\varepsilon S^{\varepsilon}_{t/{\varepsilon}} > -x\:\vline\:\eta^{\varepsilon}_0=i\right)+\sup_{j\in\{0,1\}}\mathbf{P}\left(\varepsilon M^{\varepsilon}_{0} > x\:\vline\:\eta^{\varepsilon}_0=j\right).\]
The first term on the right-hand side here is readily checked to converge to
$\mathbf{P}(S_{t} > -x\:\vline\:\eta_0=i)$ as $\varepsilon\rightarrow 0$, and this limit converges to 0 as $t\rightarrow-\infty$. As for the second term, from \eqref{w0pm} we have that
\begin{eqnarray*}
\sup_{j\in\{0,1\}}\mathbf{P}\left(\varepsilon M^{\varepsilon}_{0} > x\:\vline\:\eta^{\varepsilon}_0=j\right)&\leq & \frac{\lambda_0+\lambda_1}{\lambda_0}\mathbf{P}\left(\varepsilon M^{\varepsilon}_{0} > x\right)\\
&\leq & C\varepsilon\sum_{m>x/\varepsilon}\left(\frac{1-\lambda_1\varepsilon}{1-\lambda_0\varepsilon}\right)^{m}\\
&\leq & C\varepsilon\sum_{m>x/\varepsilon}e^{-(\lambda_1-\lambda_0)\varepsilon m}\\
&\leq & Ce^{-(\lambda_1-\lambda_0)x},
\end{eqnarray*}
where $C$ is a constant not depending on $\varepsilon$ that might vary from line to line. This expression can be taken arbitrarily small by choosing $x$ large, and so the proof is complete.
\end{proof}
\begin{prop} If $S$ is the zigzag process with parameters $\lambda_0<\lambda_1$, then $TS\buildrel{d}\over{=}S$.
\end{prop}
\begin{rem} In this case, the carrier $W=M-S$ is a stationary, non-Markov process. It is possible to compute its marginal distribution by taking the appropriate scaling limit of the distribution given in Lemma \ref{w0pm}, yielding
\[W_0\sim\frac{\lambda_1-\lambda_0}{\lambda_1+\lambda_0}\delta_0+\frac{2\lambda_0}{\lambda_1+\lambda_0}\mathrm{Exp}(\lambda_1-\lambda_0),\]
where $\delta_0$ is the probability measure placing all its mass at $0$, and $\mathrm{Exp}(\lambda_1-\lambda_0)$ is the law of an exponential random variable with parameter $\lambda_1-\lambda_0$. In particular, $\mathbf{E}W_0=2\lambda_0(\lambda_1^2-\lambda_0^2)^{-1}$.
\end{rem}
\subsection{Periodic Brownian motion}\label{perbmsec} In this subsection, we describe the periodic version of the scaling argument of Subsection \ref{BMsec}. Let $\eta^{\varepsilon}$ be again an i.i.d.\ sequence of Bernoulli random variables, with parameter $p_\varepsilon$ as given by \eqref{peps}, for some constant $c\in \mathbb{R}$. (Note that we no longer need to assume $c>0$.) Moreover, for $L>0$, set $N_\varepsilon:=\lfloor L/\varepsilon^2\rfloor$, and let $(\eta^{\varepsilon,L}_n)_{n=1}^{N_\varepsilon}$ be a random sequence with law given by that of $(\eta^{\varepsilon}_n)_{n=1}^{N_\varepsilon}$ conditioned on $S^{\varepsilon}_{N_{\varepsilon}}>0$. Extend $\eta^{\varepsilon,L}$ to $(\eta^{\varepsilon,L}_n)_{n\in\mathbb{Z}}$ by cyclic repetition. From Proposition \ref{gibbsinv}, we then have that $T\eta^{\varepsilon,L}\buildrel{d}\over{=}\eta^{\varepsilon,L}$, and so \eqref{invarassu} holds for these random configurations. Moreover, it is straightforward to check that \eqref{scaling} holds, in the sense that the associated path encodings satisfy
\[\left(\varepsilon S^{\varepsilon,L}_{t/\varepsilon^2}\right)_{t\in\mathbb{R}}\buildrel{d}\over{\rightarrow}
\left( S^{L}_{t}\right)_{t\in\mathbb{R}},\]
where $(S^L_t)_{t\in[0,L]}$ has the distribution of the initial segment of two-sided Brownian motion with drift $c$, $(S_t)_{t\in[0,L]}$, conditioned on $S_L>0$, and this definition is extended by cyclic repetition to give a process on $\mathbb{R}$. With the latter definition, it is obvious that $t^{-1}S^{L}_t\rightarrow L^{-1}S_L>0$ as $|t|\rightarrow \infty$, $\mathbf{P}$-a.s., and so \eqref{cond2} holds. As for \eqref{cond1}, we simply note
\begin{eqnarray*}
\lim_{s \to -\infty} \limsup_{\varepsilon\rightarrow 0} \mathbf{P}\left(M^{\varepsilon,L}_{s/\varepsilon^2} > S^{\varepsilon,L}_{t/\varepsilon^2}\right)&=&
\lim_{s \to -\infty} \limsup_{\varepsilon\rightarrow 0} \mathbf{P}\left(M^{\varepsilon,L}_{s/\varepsilon^2} > 0\right)\\
&\leq &\lim_{s \to -\infty} \limsup_{\varepsilon\rightarrow 0} \mathbf{P}\left(\sup_{u\in[0,L]}\varepsilon S^{\varepsilon,L}_{u/\varepsilon^2}+\left\lfloor\frac{s}{L}\right\rfloor \varepsilon S^{\varepsilon,L}_{L/\varepsilon^2} > 0\right)\\
&=&\lim_{s \to -\infty} \mathbf{P}\left(\sup_{u\in[0,L]}S_u^{L}+\left\lfloor\frac{s}{L}\right\rfloor S^{L}_{L} > 0\right)\\
&=&0,
\end{eqnarray*}
where $M^{\varepsilon,L}$ is the past maximum process associated with $S^{\varepsilon,L}$. Hence Assumption \ref{invarassu} holds, and we obtain the following.
\begin{prop} Fix $L>0$. If $S^L$ is the periodic extension of $(S_t)_{t\in[0,L]}$ conditioned on $S_L>0$, where $S$ is a two-sided Brownian motion with drift $c\in\mathbb{R}$, then $TS^L\buildrel{d}\over{=}S^L$.
\end{prop}
\subsection{Periodic zigzag process}\label{perzzsec} The periodic analogue of Subsection \ref{zzsec} is checked similarly to the previous subsection. For $L>0$, set $N_\varepsilon:=\lfloor L/\varepsilon\rfloor$, and let $(\eta^{\varepsilon,L}_n)_{n=1}^{N_\varepsilon}$ be a random sequence with law given by \eqref{cyclicmarkov}, where $N=N_\varepsilon$, and $(\eta_n)_{n=1}^{N_\varepsilon}$ is given by the two-sided stationary Markov chain with transition matrix $P_\varepsilon$ from \eqref{Peps} for some $\lambda_0,\lambda_1>0$. Extending $\eta^{\varepsilon,L}$ to $(\eta^{\varepsilon,L}_n)_{n\in\mathbb{Z}}$ by cyclic repetition, we then have from Proposition \ref{gibbsinv} that $T\eta^{\varepsilon,L}\buildrel{d}\over{=}\eta^{\varepsilon,L}$, and so \eqref{invarassu} holds for these random configurations. Moreover, it is not difficult to deduce from \eqref{etaconv} that
\[\left(\eta^{\varepsilon,L}_{\lfloor t/\varepsilon \rfloor}\right)_{t\in\mathbb{R}}\buildrel{d}\over{\rightarrow} \left(\eta^L_t\right)_{t\in\mathbb{R}}\]
in $D(\mathbb{R},\{0,1\})$, with the law of the $L$-periodic process $\eta^L$ being characterised by
\begin{equation}\label{etal}
\mathbf{E}\left(F\left((\eta^L_t)_{t\in[0,L]}\right)\right)=\frac{\mathbf{E}\left(\nu(\eta_0)^{-1}F\left((\eta_t)_{t\in[0,L]}\right)\:\vline\:\eta_L=\eta_0,\:S_L>0\right)}{\mathbf{E}\left(\nu(\eta_0)^{-1}\:\vline\:\eta_L=\eta_0,\:S_L>0\right)},
\end{equation}
where $1-\nu(0)=\frac{\lambda_0}{\lambda_0+\lambda_1}=\nu(1)$, and $\eta$ is the two-sided, stationary continuous-time Markov chain that appears as a limit in \eqref{etaconv}. It follows that the associated path encodings satisfy
\[\left(\varepsilon S^{\varepsilon,L}_{t/\varepsilon }\right)_{t\in\mathbb{R}}\buildrel{d}\over{\rightarrow} \left(S^L_t\right)_{t\in\mathbb{R}},\]
yielding \eqref{scaling} in this case; the limit process can be seen as a periodic version of the zigzag process with stationary increments. By applying identical arguments to those of the previous subsection, we are also able to confirm \eqref{cond1} and \eqref{cond2} both hold with the appropriate scaling, and we subsequently obtain the following.
\begin{prop} Fix $L>0$. If $S^L$ is the path encoding of the $\eta^L$, as given by \eqref{etal}, for some $\lambda_0,\lambda_1>0$, then $TS^L\buildrel{d}\over{=}S^L$.
\end{prop}
\subsection{Brownian motion conditioned to stay close to its past maximum}\label{contbounded}
In this section, we consider the transfer of the bounded soliton examples of Subsection \ref{boundedsec} and Example \ref{perboundedex} to the continuous setting, starting with the periodic case. Let $L>0$, and $S^L$ be the $L$-periodic Brownian motion with drift $c>0$ of Subsection \ref{perbmsec}. If $W^L=M^L-S^L$ is the associated carrier process and $K>0$, we define $S^{L,K}$ to have law equal to that of $S^L$ conditioned on $\sup_{t\in\mathbb{R}}W^L_t\leq K$. (Note the latter event has strictly positive probability.) We then have the following.
\begin{prop} Fix $L,K>0$. If $S^{L,K}$ is the $L$-periodic Brownian motion with drift $c>0$ conditioned to stay within $K$ of its past maximum (i.e.\ the process described above), then $TS^{L,K}\buildrel{d}\over{=}S^{L,K}$.
\end{prop}
\begin{proof} Note that $\sup_{t\in\mathbb{R}}W^L_t\leq K$ can alternatively be expressed as
\begin{equation}\label{alk}
\left\{\max_{0\leq t\leq L}\left\{\max_{0\leq s\leq t}(S^L_s-S^L_t),\:\max_{t\leq s\leq L}(S^L_s-S^L_L-S^L_t)\right\}\leq K\right\}.
\end{equation}
Hence, applying the definitions of $S^{L,K}$ and $S^L$, we find that
\begin{equation}\label{123}
\mathbf{E}\left(F\left((S^{L,K}_t)_{t\in[0,L]}\right)\right)=\mathbf{E}\left(F\left((S_t)_{t\in[0,L]}\right)\:\vline\:\mathcal{A}_{L,K},\:S_L>0\right),
\end{equation}
where $\mathcal{A}_{L,K}$ is defined similarly to \eqref{alk}, but with $S^L$ replaced by $S$. This characterisation of the law of $S^{L,K}$ allows us to show that it can be arrived at as the scaling limit of a sequence of discrete models. Indeed, let $S^{\varepsilon,L,K}$ be the periodic bounded soliton configuration of Example \ref{perboundedex} with $(p,N,K)$ being given by $(\frac{1-\varepsilon c}{2},\lfloor L/\varepsilon^2\rfloor,K/\varepsilon)$. From \eqref{bexp0}, we then have that
\begin{equation}\label{456}
\mathbf{E}\left(F\left((S^{\varepsilon,L,K}_n)_{n=0}^{\lfloor L/\varepsilon^2\rfloor}\right)\right)=\mathbf{E}\left(F\left((S^{\varepsilon}_n)_{n=0}^{\lfloor L/\varepsilon^2\rfloor}\right)\:\vline\:\mathcal{A}_{\varepsilon,L,K},\:S^\varepsilon_{\lfloor L/\varepsilon^2\rfloor}>0\right),
\end{equation}
where $S^{\varepsilon}$ is the path encoding of the i.i.d.\ configuration with density $\frac{1-\varepsilon c}{2}$, and
\[\mathcal{A}_{\varepsilon,L,K}=\left\{\max_{0\leq n\leq {\lfloor L/\varepsilon^2\rfloor}}\left\{\max_{0\leq m\leq n}(S^\varepsilon_m-S^\varepsilon_n),\:\max_{n\leq m\leq {\lfloor L/\varepsilon^2\rfloor}}(S^\varepsilon_m-S^\varepsilon_{\lfloor L/\varepsilon^2\rfloor}-S^\varepsilon_n)\right\}\leq K/\varepsilon\right\}.\]
Since $(\varepsilon S^{\varepsilon}_{t/\varepsilon^2})_{t\in\mathbb{R}}\buildrel{d}\over{\rightarrow}S$, it is an elementary exercise to deduce from \eqref{123} and \eqref{456} that
\[\left(\varepsilon S^{\varepsilon,L,K}_{t/\varepsilon^2}\right)_{t\in\mathbb{R}}\buildrel{d}\over{\rightarrow}\left(S^{L,K}_t\right)_{t\in\mathbb{R}},\]
i.e.\ \eqref{scaling} holds with $a_\varepsilon=\varepsilon$, $b_\varepsilon=\varepsilon^2$. We also have \eqref{invarassu} from Corollary \ref{gibbsinv}, and \eqref{cond1} and \eqref{cond2} can be checked as in Subsection \ref{perbmsec}. Hence Assumption \ref{sassu} holds, and Proposition \ref{invarlimit} yields the result.
\end{proof}
The non-periodic version of the previous result is more of a challenge, and we do not prove it here. Rather we describe a potential proof strategy. Firstly, recall from Remark \ref{bmcar} that the carrier process $W=M-S$ associated with Brownian motion with drift $c>0$ is the stationary version of Brownian motion with drift $-c$, reflected at the origin. By applying \cite[Section 4]{GT}, it is possible to define a stationary Markov process ${W}^{K}$ that can be interpreted as $W$ conditioned on $\sup_{t\in\mathbb{R}}W_t\leq K$ (cf.\ the discussion for reflecting Brownian motion without drift in \cite[Section 7]{GT2}). Letting $L^K$ be the local time at 0 of this process, with boundary condition $L^K_0=0$, then, by analogy with the unconditioned case, set $S^{K}=L^K-W^K+W^K_0$. We expect that this process, which one might interpret as Brownian motion with drift $c>0$ conditioned to stay within $K$ of its past maximum, can alternatively be obtained as a scaling limit of the path encodings of the random configurations described in Subsection \ref{boundedsec}, and make the following conjecture.
\begin{conj} Fix $K>0$. If $S^{K}$ is the Brownian motion with drift $c>0$ conditioned to stay within $K$ of its past maximum (in the sense described above), then $TS^{K}\buildrel{d}\over{=}S^{K}$.
\end{conj}
\section{Palm measures for the zigzag process and the ultra-discrete Toda lattice}\label{palmsec}
In this section we relate the dynamics of the zigzag process under Pitman's transformation to the dynamics of the ultra-discrete Toda lattice, and use this connection to derive natural invariant measures for the latter. The state of the ultra-discrete Toda lattice is described by a vector $((Q_j)_{j=1}^J,({E}_j)_{j=1}^{J-1}) \in (0,\infty)^{2J-1}$ for some $J\in\mathbb{N}$, and its one-step time evolution by the equation
\begin{align}
(\mathcal{T}Q)_j &:=\min \left\{ \sum_{l=1}^j Q_l-\sum_{l=1}^{j-1}(\mathcal{T}Q)_l,{E}_j \right\},\label{udtodaeq}\\
(\mathcal{T}{E})_j &:=Q_{j+1}+{E}_j-(\mathcal{T}Q)_j,\nonumber
\end{align}
where for the purposes of these equations we suppose ${E}_{J} = \infty$. Similarly to the path encoding of the BBS, we can associate a path $S\in C(\mathbb{R},\mathbb{R})$ to the state of the ultra-discrete Toda lattice $((Q_j)_{j=1}^J,({E}_j)_{j=1}^{J-1})$ by setting $S_t=t$ for $t<0$, and for $t\geq 0$, concatenating path segments of gradient $-1,1,-1,1,\dots,-1,1,-1,1$, of lengths $Q_1,{E}_1,Q_2,{E}_2,\dots,Q_{J-1},{E}_{J-1},Q_{J},{E}_J=\infty$, i.e.\
\begin{equation}\label{abpath}
S_{t}=\begin{cases}
t&\mbox{for }t<0,\\
-t+2\sum_{l=1}^j{E}_l,&\mbox{for }\sum_{l=1}^j Q_l+\sum_{l=1}^j{E}_l\leq t\leq\sum_{l=1}^{j+1}Q_l+\sum_{l=1}^j{E}_l,\\
t-2\sum_{l=1}^{j+1}Q_l,&\mbox{for }\sum_{l=1}^{j+1}Q_l+\sum_{l=1}^j{E}_l\leq t\leq\sum_{l=1}^{j+1}Q_l+\sum_{l=1}^{j+1}{E}_l,
\end{cases}
\end{equation}
where $j=0,\dots,J-1$ (interpreting sums of the form $\sum_{l=1}^0$ as zero), and we again suppose ${E}_{J} = \infty$. As is confirmed by the next proposition we present, the dynamics of the ultra-discrete Toda lattice given by \eqref{udtodaeq} are described by Pitman's transformation applied to this path encoding. However, in this case, it is convenient to shift the path after applying $T$ so that $0$ is still a local maximum. In particular, for $t\in\mathbb{R}$, we define $\theta^tS$ by setting
\begin{equation}\label{thetatdef}
(\theta^tS)_s=S_{t+s}-S_t,\qquad\forall s\in\mathbb{R},
\end{equation}
let
\[\tau(S):=\inf\{t\geq 0:\:t\in \mathrm{LM}(S)\},\]
where $\mathrm{LM}(S)$ is the set of local maxima of $S$ (for the elements of $C(\mathbb{R},\mathbb{R})$ that are considered in this section, $\tau(S)$ is always well-defined and finite), and define
\[\theta^\tau(S):=\theta^{\tau(S)}(S).\]
We then introduce an operator $\mathcal{T}$ on the path encoding by the composition of $T$ and $\theta^\tau$, that is
\begin{equation}\label{tstarpath}
\mathcal{T}S:=\theta^\tau(TS).
\end{equation}
The motivation for this definition is the following. (See Figure \ref{todafig} for a graphical representation of the result.)
\begin{prop}[See {\cite[Theorem 1.1]{CST}}] Fix $J\in\mathbb{N}$. Let $((Q_j)_{j=1}^J,({E}_j)_{j=1}^{J-1}) \in (0,\infty)^{2J-1}$, and $S$ be its path encoding, defined as at \eqref{abpath}. It is then the case that the transformed configuration $(((\mathcal{T}Q)_j)_{j=1}^J,((\mathcal{T}{E})_j)_{j=1}^{J-1})$, defined as at \eqref{udtodaeq}, has path encoding given by $\mathcal{T}S$, defined as at \eqref{tstarpath}.
\end{prop}
\begin{figure}[t]
\begin{flushleft}
\hspace{60pt}\includegraphics[width=0.5\textwidth]{udtoda.eps}
\rput[tl](0,6.4){$\left((Q_j)_{j=1}^J,({E}_j)_{j=1}^{J-1}\right)$}
\rput[tl](0,4.7){$S$}
\rput[tl](0,2.65){$TS$}
\rput[tl](0,1.2){$\mathcal{T}S$}
\rput[tl](0,0.4){$\left(((\mathcal{T}Q)_j)_{j=1}^J,((\mathcal{T}{E})_j)_{j=1}^{J-1}\right)$}
\end{flushleft}
\caption{Graphical representation of the dynamics of the ultra-discrete Toda lattice in terms of the associated path encodings. NB. The red line in the graphs for $S$ and $TS$ shows the path of $M$.}\label{todafig}
\end{figure}
Just as for the BBS, the ultra-discrete Toda lattice evolves in a solitonic way. Eventually the configuration orders itself so that $Q_J\geq Q_{J-1}\geq\dots\geq Q_1$, and these quantities -- which can be thought of as representing intervals where particles are present -- remain constant, whilst the $({E}_j)_{j=1}^{J-1}$ -- which can be thought of as representing the gaps between blocks of particles -- grow linearly (see \cite[equations (20), (21)]{NTS}, though note the labelling convention is reversed in the latter article). Thus to see a stationary measure one might consider, as we did for the BBS, a two-sided infinite configuration $((Q_j)_{j\in\mathbb{Z}},({E}_j)_{j\in\mathbb{Z}})$. Under suitable conditions regarding the asymptotic behaviour of these sequences, one might then encode these via piecewise linear paths with intervals of gradient $-1$ or $1$ as at \eqref{abpath} -- extending the definition to the negative axis in the obvious way, and then defining the dynamics via \eqref{tstarpath}. This is our approach in the next part of our discussion. Although $\mathcal{T}$ is a more complicated operator than $T$, we are still able to identify an invariant measure for it by considering the Palm measure of the zigzag process under which $0$ is always a local maximum. As we show in Corollary \ref{abcor}, reading off the lengths of the intervals of constant gradient, from the latter conclusion we obtain a natural invariant measure for the ultra-discrete Toda lattice. Specifically, the invariant configuration we present has that both $(Q_j)_{j\in\mathbb{Z}}$ and $({E}_j)_{j\in\mathbb{Z}}$ are i.i.d.\ sequences of exponential random variables (independent of each other).
The result described in the previous paragraph for the Palm measure of the zigzag process, and the corollary for the ultra-discrete Toda lattice, will be proved in Subsection \ref{zzpalm}. Towards this end, in Subsection \ref{marpalm}, we first establish the BBS analogue of the results for the Markov configuration of Subsection \ref{markovsec}. Finally, in Subsection \ref{perpalm}, we establish periodic versions of the results.
\subsection{Invariance of a Palm measure for the Markov configuration}\label{marpalm} In this subsection, we suppose $\eta$ is the Markov configuration of Subsection \ref{markovsec} with $p_0,p_1\in(0,1)$ and $p_0+p_1<1$. The associated Palm measure we will consider is defined to be the law of the random configuration $\eta^*$, as characterised by
\begin{equation}\label{etastar}
\mathbf{E}\left(f(\eta^*)\right)=\mathbf{E}\left(f(\eta)\:\vline\:\eta_0=0,\:\eta_1=1\right)
\end{equation}
for any bounded functions $f:\{0,1\}^\mathbb{Z}\rightarrow\mathbb{R}$. Equivalently, we can express this in terms of the associated path encodings as
\[\mathbf{E}\left(f(S^*)\right)=\mathbf{E}\left(f(S)\:\vline\:0\in \mathrm{LM}(S)\right).\]
The main result of the subsection is the following, which establishes invariance of $S^*$ under $\mathcal{T}$. The proof is an adaptation of \cite[Lemma 4.5]{Ferrari}, cf.\ the classical arguments of \cite{Har,PS}.
\begin{prop}\label{propperpalmdisc} If $S^*$ is the path encoding of the two-sided stationary Markov chain described in Subsection \ref{markovsec} with $p_0,p_1\in (0,1)$ satisfying $p_0+p_1<1$ conditioned to have a local maximum at $0$, then $\mathcal{T}S^*\buildrel{d}\over{=}S^*$.
\end{prop}
\begin{proof} By definition, writing $c=\mathbf{P}(0\in \mathrm{LM}(S))^{-1}$, we have that
\begin{eqnarray}
\mathbf{E}\left(f(\mathcal{T}S^*)\right)&=&c\mathbf{E}\left(f(\theta^\tau(TS))\mathbf{1}_{\{0\in \mathrm{LM}(S)\}}\right)\nonumber\\
&=&c\sum_{n>0}\mathbf{E}\left(f(\theta^n(TS))\mathbf{1}_{\{0\in \mathrm{LM}(S),\:\tau(TS)=n\}}\right),\label{asf}
\end{eqnarray}
where we note that $\tau(TS)>0$ on the event $0\in \mathrm{LM}(S)$, and $\theta^n$ is defined as at \eqref{thetatdef}. Now, it is an elementary exercise to check that, on $0\in \mathrm{LM}(S)$, the event $\tau(TS)=n$ is equivalent to $\bar{\tau}(S)=n$, where $\bar{\tau}(S):=\inf\{n\geq 0:\:n\in \mathrm{LI}(S)\}$, and $\mathrm{LI}(S)$ is the set of local minima of $S$. Hence we obtain from \eqref{asf} that
\begin{eqnarray*}
\mathbf{E}\left(f(\mathcal{T}S^*)\right)&=&c\sum_{n>0}\mathbf{E}\left(f(\theta^n(TS))\mathbf{1}_{\{0\in \mathrm{LM}(S),\:\bar{\tau}(S)=n\}}\right)\\
&=&c\sum_{n>0}\mathbf{E}\left(f(T\theta^nS)\mathbf{1}_{\{\tau_-(\theta^nS)=-n,\:0\in LI(\theta^nS)\}}\right),
\end{eqnarray*}
where we define $\tau_-(S):=\sup\{n\leq 0:\:n\in LM(S)\}$. Applying the spatial stationarity of $\eta$, it follows that
\begin{eqnarray*}
\mathbf{E}\left(f(\mathcal{T}S^*)\right)&=&c\sum_{n>0}\mathbf{E}\left(f(TS)\mathbf{1}_{\{\tau_-(S)=-n,\:0\in LI(S)\}}\right)\\
&=&c\mathbf{E}\left(f(TS)\mathbf{1}_{\{0\in LI(S)\}}\right).
\end{eqnarray*}
Finally, we note that $0\in LI(S)$ if and only if $0\in LM(TS)$, and so
\[\mathbf{E}\left(f(\mathcal{T}S^*)\right)=c\mathbf{E}\left(f(TS)\mathbf{1}_{\{0\in LM(TS)\}}\right)
=c\mathbf{E}\left(f(S)\mathbf{1}_{\{0\in LM(S)\}}\right)
=\mathbf{E}\left(f(S^*)\right),\]
where the second equality follows from the invariance of $S$ under $T$ (i.e.\ Corollary \ref{markovcor}).
\end{proof}
\subsection{Invariance of a Palm measure for the zigzag process}\label{zzpalm} Via a scaling limit, the result of the previous subsection readily transfers to the zigzag process. In particular, given $\lambda_0<\lambda_1$, now let $\eta^*=(\eta^*_t)_{t\in\mathbb{R}}$ be a continuous time stochastic process taking values on $\{0,1\}$ such that: $(\eta^*_t)_{t\geq0}$ is a continuous time Markov chain that jumps from 0 to 1 with rate $\lambda_0$, and from 1 to 0 with rate $\lambda_1$, started from $\eta^*_0=1$; $(\eta^*_{-t})_{t\geq0}$ is a continuous time Markov chain with the jumps from 0 to 1 with rate $\lambda_0$, and from 1 to 0 with rate $\lambda_1$, started from $\eta^*_0=0$; and the two processes are independent. (NB. To make the process $\eta^*$ right-continuous, we ultimately set $\eta^*_0=1$, and also take the right-limits at all the jump times.) Our Palm measure for the zigzag process is then the law of $S^*=(S^*_t)_{t\in\mathbb{R}}$, where
\[S^*_t:=\int_0^t(1-2\eta^*_s)ds,\]
which can be viewed as the zigzag process $S$ of Subsection \ref{zzsec} conditioned on $0\in\mathrm{LM}(S)$, though in this case we note the conditioning is non-trivial since the event $0\in\mathrm{LM}(S)$ has zero probability. For the process $S^*$, we have the following result.
\begin{prop}\label{uuu} If $S^*$ is the zigzag process with rates $0<\lambda_0<\lambda_1$ conditioned to have a local maximum at 0 (in the sense described above), then $\mathcal{T}S^*\buildrel{d}\over{=}S^*$.
\end{prop}
\begin{proof} Let $\eta^{*,\varepsilon}$ be the process defined at \eqref{etastar} for parameters $p_0=\varepsilon\lambda_0$ and $p_1=1-\varepsilon\lambda_1$. Then, similarly to \eqref{etaconv}, it is straightforward to check that
\[\left(\eta^{*,\varepsilon}_{\lfloor t/\varepsilon \rfloor}\right)_{t\in\mathbb{R}}\buildrel{d}\over{\rightarrow} \left(\eta^*_t\right)_{t\in\mathbb{R}},\]
and hence the associated path encodings satisfy
\[\left(\varepsilon S^{*,\varepsilon}_{t/\varepsilon}\right)_{t\in\mathbb{R}}\buildrel{d}\over{\rightarrow} \left(S^*_t\right)_{t\in\mathbb{R}}.\]
Moreover, the conditions \eqref{cond1} and \eqref{cond2} are readily checked in this setting. From these facts, together with the readily-checked observation that $\varepsilon\tau(TS^{*,\varepsilon})\buildrel{d}\over{\rightarrow} \tau(TS^*)$ (simultaneously with the convergence of path encodings), the result follows by a simple adaptation of the argument of Proposition \ref{invarlimit}.
\end{proof}
Since the lengths of the intervals upon which $S^{*}$ is decreasing are i.i.d.\ parameter $\lambda_1$ exponential random variables, and the lengths of the intervals upon which it is increasing are i.i.d.\ parameter $\lambda_0$ exponential random variables (and the two collections are independent), we immediately deduce the following conclusion from the previous result (and the description of the ultra-discrete Toda lattice given at the start of the section).
\begin{cor}\label{abcor} Let $(Q_j)_{j\in\mathbb{Z}}$ be an i.i.d.\ sequence of parameter $\lambda_1$ exponential random variables, and $({E}_j)_{j\in\mathbb{Z}}$ be an i.i.d.\ sequence of parameter $\lambda_0$ exponential random variables. Suppose further $(Q_j)_{j\in\mathbb{Z}}$ and $({E}_j)_{j\in\mathbb{Z}}$ are independent. If $0<\lambda_0<\lambda_1$, then the distribution of $((Q_j)_{j\in\mathbb{Z}},({E}_j)_{j\in\mathbb{Z}})$ is invariant under the dynamics of the ultra-discrete Toda lattice.
\end{cor}
\subsection{Palm measures in the periodic case}\label{perpalm} The arguments of the previous two subsections are readily adapted to the periodic case. Since few changes are needed, we only present a sketch, beginning with the discrete case. For $N\in\mathbb{N}$, let $\eta^{*N}=(\eta^{*N}_n)_{n\in\mathbb{Z}}$ be the random configuration with law characterised by
\begin{equation}\label{ggg}
\mathbf{E}\left(f(\eta^{*N})\right)=\mathbf{E}\left(f(\eta^N)\:\vline\:\eta^N_0=0,\:\eta^N_1=1\right),
\end{equation}
where $\eta^N$ is the periodic Markov configuration of Example \ref{permarex}, cf.\ \eqref{etastar}. Note that an alternative characterisation of the law of $\eta^{*N}$ is given by
\begin{equation}\label{altchar}
\mathbf{E}\left(f((\eta^{*N}_n)_{n=1}^N)\right)=\mathbf{E}\left(f((\eta_n)_{n=1}^N)\:\vline\:\eta_1=1,\:\eta_N=0,\:S_N>0\right),
\end{equation}
where $\eta$ is the Markov configuration of Subsection \ref{markovsec}. We are then able to check the following result.
\begin{prop} Let $p_0,p_1\in(0,1)$. If $S^{*N}$ is the path encoding of $\eta^{*N}$, then $\mathcal{T}S^{*N}\buildrel{d}\over{=}S^{*N}$.
\end{prop}
\begin{proof} This is identical to the proof of Proposition \ref{propperpalmdisc}. In particular, in view of the Palm description of the law of $\eta^{*N}$ at \eqref{ggg}, it suffices to note that $\eta^N$ is spatially stationary (Lemma \ref{etalem}) and invariant under $T$ (Corollary \ref{gibbsinv}), that $\mathbf{P}(0\in LM(S^N))>0$, and that the terms involving $\tau$, $\bar{\tau}$ and $\tau_-$ are almost-surely finite.
\end{proof}
For the continuous version of this result, first let $\eta^{*L}=(\eta^{*L}_t)_{t\in\mathbb{R}}$ be the $L$-periodic process whose law is characterised by
\begin{equation}\label{altchar2}
\mathbf{E}\left(F\left((\eta^{*L}_t)_{t\in[0,L]}\right)\right)=\mathbf{E}\left(F\left((\eta_t)_{t\in[0,L]}\right)\:\vline\:\eta_0=1,\:\eta_L=0,\:S_L>0\right),
\end{equation}
where $\eta$ is the two-sided stationary continuous time Markov chain of Subsection \ref{zzsec}. (NB. Of course, this definition is problematic in terms of defining $\eta_t^{*L}$ for $t\in L\mathbb{Z}$; we resolve the issue by assuming $\eta^{*L}$ is right-continuous.) If $S^{*L}$ is the corresponding path encoding, defined similarly to \eqref{sdef}, then we have the following result.
\begin{prop}\label{perpropsl} Let $\lambda_0,\lambda_1>0$. If $S^{*L}$ is the path encoding of $\eta^{*L}$, then $\mathcal{T}S^{*L}\buildrel{d}\over{=}S^{*L}$.
\end{prop}
\begin{proof} Similarly to the proof of Proposition \ref{uuu}, we use a scaling argument. Specifically, as in Subsection \ref{perzzsec}, we set $N_\varepsilon:=\lfloor L/\varepsilon\rfloor$, and define the discrete time process $\eta^{*\varepsilon,L}$ by \eqref{ggg}, where the underlying Markov parameters are chosen as in \eqref{Peps}. Comparing \eqref{altchar} and \eqref{altchar2}, it is straightforward to argue from \eqref{etaconv} that
\[\left(\eta^{*\varepsilon,L}_{\lfloor t/\varepsilon \rfloor}\right)_{t\in\mathbb{R}}\buildrel{d}\over{\rightarrow} \left(\eta^{*L}_t\right)_{t\in\mathbb{R}}.\]
The convergence of associated path encodings follows, and the remainder of the proof is identical to Proposition \ref{uuu}.
\end{proof}
We conclude the section be describing the application of the previous result to the ultra-discrete periodic Toda lattice, see \cite{IKT, InTa, KiTo} for background. For this model, we describe the current state by a vector of the form $((Q_j)_{j=1}^J,({E}_j)_{j=1}^{J}) \in (0,\infty)^{2J}$ for some $J\in\mathbb{N}$. Although it appears we have an extra variable to the non-periodic case, this is not so, because we assume that $\sum_{j=1}^J Q_j+\sum_{j=1}^J{E}_j=L$ for some fixed $L\in\mathbb{R}$. Moreover, in order to define the dynamics, we further suppose that $\sum_{j=1}^J Q_j<L/2$, which can be seen as the equivalent condition to requiring fewer than $N/2$ particles in the $N$-periodic BBS model. Introducing the additional notation $({D}_j)_{j=1}^J$ for convenience, the dynamics of the system are given by the following adaptation of \eqref{udtodaeq}:
\begin{align}
(\mathcal{T}Q)_j &:=\min \left\{ Q_j-{D}_j,{E}_j \right\},\label{udtodaper}\\
(\mathcal{T}{E})_j &:=Q_{j+1}+{E}_j-(\mathcal{T}Q)_j,\nonumber,\\
{D}_j&:=\min_{0\leq k\leq J-1}\sum_{l=1}^k\left({E}_{j-l}-Q_{j-l}\right).\nonumber
\end{align}
(In these definitions $((Q_j)_{j=1}^J,({E}_j)_{j=1}^{J})$ are extended periodically to $((Q_j)_{j\in\mathbb{Z}},({E}_j)_{j\in\mathbb{Z}})$.) Given a state vector $((Q_j)_{j=1}^J,({E}_j)_{j=1}^{J})$, we define an associated path encoding $S$ by appealing to the definition at \eqref{abpath} for $t\in[0,L]$, and then concatenating copies of $(S_t)_{t\in[0,L]}$ in such a way that the resulting path is an element of $C(\mathbb{R},\mathbb{R})$. Using this path encoding, the dynamics at \eqref{udtodaper} can be expressed in terms of the operator $\mathcal{T}$ defined as at \eqref{tstarpath}.
\begin{prop}[See {\cite[Theorem 2.3]{CST}}] Fix $J\in\mathbb{N}$ and $L\in(0,\infty)$. Let $((Q_j)_{j=1}^J,({E}_j)_{j=1}^{J}) \in (0,\infty)^{2J}$ satisfy $\sum_{j=1}^J Q_j+\sum_{j=1}^J{E}_j=L$ and $\sum_{j=1}^J Q_j<L/2$, and $S$ be the associated path encoding. It then holds that the periodically transformed configuration $(((\mathcal{T}Q)_j)_{j=1}^J,((\mathcal{T}{E})_j)_{j=1}^{J})$, defined as at \eqref{udtodaper}, has path encoding given by $\mathcal{T}S$, defined as at \eqref{tstarpath}.
\end{prop}
This picture of the ultra-discrete periodic Toda lattice dynamics allows us to deduce the following corollary of Proposition \ref{perpropsl}.
\begin{cor} Fix $J\in\mathbb{N}$, and $A,L\in(0,\infty)$ such that $0<A<L/2$. Let $(\Delta^Q_j)_{j=1}^J$ and $(\Delta^{E}_j)_{j=1}^J$ be independent $\mathrm{Dirichlet}(1,1,\dots,1)$ random variables, and set
\[Q_j:=A\Delta^Q_j,\qquad {E}_j:=(L-A)\Delta^{E}_j,\qquad j=1,\dots,J.\]
It is then the case that $((Q_j)_{j=1}^J,({E}_j)_{j=1}^J)$ is invariant under the dynamics of the ultra-discrete periodic Toda lattice.
\end{cor}
\begin{proof} Fix $\lambda_0,\lambda_1>0$. Let $S^{*L,J}$ be a random path with law equal to that of $S^{*L}$ conditioned on
\begin{equation}\label{tLM}
\#\left\{t\in[0,L):\:t\in LM(S^{*L})\right\}=J,
\end{equation}
and write $Q_1,{E}_1,\dots,Q_J,{E}_J$ for the lengths of the sub-intervals of $[0,L]$ upon which $S^{*L,J}$ has gradient $-1,1,\dots,-1,1$, respectively. Since the left-hand side of \eqref{tLM} is preserved by $\mathcal{T}$ (see \cite[Theorem 2.3]{CST}, for example), it readily follows from Proposition \ref{perpropsl} that $\mathcal{T}S^{*L,J}\buildrel{d}\over{=}S^{*L,J}$, and hence the law of $((Q_j)_{j=1}^J,({E}_j)_{j=1}^J)$ is invariant for the dynamics of the ultra-discrete Toda lattice.
We next aim to identify the distribution of $((Q_j)_{j=1}^J,({E}_j)_{j=1}^J)$ as described in the previous paragraph. By considering the behaviour of the underlying two-sided stationary Markov configuration $\eta$ (that jumps from $i$ to $1-i$ with rate $\lambda_i$, $i=0,1$), it is straightforward to deduce that
\begin{eqnarray}
f_{Q,E}\left((q_j)_{j=1}^J,(e_j)_{j=1}^{J-1}\right)&\propto &\left(\prod_{j=1}^{J}\lambda_1e^{-\lambda_1 q_j}\right)\left(\prod_{j=1}^{J-1}\lambda_0e^{-\lambda_0 e_j}\right)e^{-\lambda_0\left(L-\sum_{j=1}^Jq_j-\sum_{j=1}^{J-1}e_j\right)}\nonumber\\
&\propto&e^{-(\lambda_1-\lambda_0)\sum_{j=1}^Jq_j}\label{abdens}
\end{eqnarray}
for vectors $(q_j)_{j=1}^J$ and $(e_j)_{j=1}^{J-1}$ satisfying $\sum_{j=1}^Jq_j+\sum_{j=1}^{J-1}e_j<L$, $\sum_{j=1}^Jq_j<L/2$ (and the density is zero otherwise). The form of this density suggests the merit of introducing transformed random variables:
\begin{align*}
A&:=\sum_{j=1}^J Q_j,\\
\Delta^Q_j&:=\frac{Q_j}{A},\qquad j=1,\dots,J-1,\\
\Delta^{E}_j&:=\frac{{E}_j}{L-A},\qquad j=1,\dots,J-1.
\end{align*}
Writing $f_{Q,E}$ for the density of the random variables $((Q_j)_{j=1}^J,({E}_j)_{j=1}^{J-1})$, and $f_{\Delta^{Q},A,\Delta^{{E}}}$ for the density of the random variables $((\Delta^Q_j)_{j=1}^{J-1},A,(\Delta^{E}_j)_{j=1}^{J-1})$, we have from a standard change of variable formula:
\[f_{\Delta^{Q},A,\Delta^{{E}}}\left((\delta^Q_j)_{j=1}^{J-1},a,(\delta^{E}_j)_{j=1}^{J-1}\right)=f_{{Q},{E}}\left((q_j)_{j=1}^J,(e_j)_{j=1}^{J-1}\right)\mathrm{Jac}\left((\delta^Q_j)_{j=1}^{J-1},a,(\delta^{E}_j)_{j=1}^{J-1}\right),\]
where
\begin{align*}
(q_1,\dots,q_J)&=\left(a\delta^Q_1,\dots,a\delta^Q_{J-1},a\left(1-\sum_{j=1}^{J-1}\delta^Q_j\right)\right),\\
(e_1,\dots,e_{J-1})&=\left((L-a)\delta^{E}_1,\dots,(L-a)\delta^{E}_{J-1}\right),
\end{align*}
and $\mathrm{Jac}((\delta^Q_j)_{j=1}^{J-1},a,(\delta^{E}_j)_{j=1}^{J-1})$, the Jacobian of the relevant transformation, is given by the modulus of the determinant of the following matrix (all other entries are zero):
\[
\left(
\begin{array}{ccccccccc}
a & & & & \delta^Q_1 & & &&\\
& a & & & \delta^Q_2 & & &&\\
& &\ddots & & \vdots & & &&\\
& & & a & \delta^Q_{J-1} & & &&\\
-a &-a & \dots &-a & 1-\sum_{j=1}^{J-1}\delta^Q_j & & &&\\
& & & & \delta^{E}_1 & L-a & &&\\
& & & & \delta^{E}_2 & & L-a &&\\
& & & & \vdots & & & \ddots & \\
& & & & \delta^{E}_{J-1} & & & & L-a
\end{array}
\right).\]
Now, it is elementary to compute this Jacobian to be equal to $(a(L-a))^{(J-1)}$, and thus we obtain from \eqref{abdens} that
\[f_{\Delta^{Q},A,\Delta^{{E}}}\left((\delta^Q_j)_{j=1}^{J-1},a,(\delta^{E}_j)_{j=1}^{J-1}\right)=C(a(L-a))^{(J-1)}e^{-(\lambda_1-\lambda_0)a}\]
for $a<L/2$, $\sum_{j=1}^{J-1}\delta^Q_j<1$, $\sum_{j=1}^{J-1}\delta^{E}_j<1$ (and the density is zero otherwise). Setting $\Delta_j^\kappa=1-\sum_{j=1}^{J-1}\Delta^\kappa_j$ for $\kappa={Q},{E}$, the above formula implies that $A$, $(\Delta^Q_j)_{j=1}^J$ and $(\Delta^{E}_j)_{j=1}^J$ are independent, with $A$ having density
\[f_A(a)=C(a(L-a))^{(J-1)}e^{-(\lambda_1-\lambda_0)a},\qquad a\in(0,L/2),\]
and $(\Delta^Q_j)_{j=1}^J$ and $(\Delta^{E}_j)_{j=1}^J$ being distributed as $\mathrm{Dirichlet}(1,1,\dots,1)$ random variables.
To complete the proof, it remains to condition on the value of $A$. To this end, we first note that, since $A$ is preserved by the dynamics (see \cite[Section 4]{TComm} or \cite[Corollary 2.4]{CST}, for example), we readily obtain that, for any continuous bounded function $F$:
\[\mathbf{E}\left(F\left((((\mathcal{T}Q)_j)_{j=1}^J,((\mathcal{T}{E})_j)_{j=1}^J)\right)\:\vline\:A\right)=\mathbf{E}\left(F\left(((Q_j)_{j=1}^J,({E}_j)_{j=1}^J)\right)\:\vline\:A\right).\]
Moreover, it is straightforward to check that both sides here are continuous in the value of $A$, which means we can interpret the above equation as holding for any fixed, deterministic value of $A\in(0,L/2)$. The desired result follows.
\end{proof}
Finally, we note that a similar conclusion can be drawn for the ultra-discrete periodic Toda lattice whose states are restricted to integer values. Since its proof is almost identical (but slightly easier) to that of the previous corollary, we simply state the result.
\begin{cor} Fix $J\in\mathbb{N}$, and $A,L\in\mathbb{N}$ be such that $J\leq \min\{A,L-A\}$ and also $A<L/2$. Let $(Q_j-1)_{j=1}^J$ and $({E}_j-1)_{j=1}^J$ be independent multinomial random variables with parameters given by $(A-J; J^{-1},J^{-1},\dots,J^{-1})$ and $(L-A-J; J^{-1},J^{-1},\dots,J^{-1})$, respectively. It is then the case that $((Q_j)_{j=1}^J,({E}_j)_{j=1}^J)$ is invariant under the dynamics of the ultra-discrete periodic Toda lattice.
\end{cor}
\section{Conditioned one-sided processes}\label{condsec}
The introduction of Pitman's transformation in \cite{Pitman} was important as it provided a (simple) sample path construction of a three-dimensional Bessel process from a one-dimensional Brownian motion, where the former process can be viewed as Brownian motion conditioned to stay non-negative. Moreover, in the argument of \cite{Pitman}, a discrete analogue of a three-dimensional Bessel process is constructed, and the relation between such a process and random walk conditioned to stay non-negative is explored in detail in \cite{BD}. In this section, we present a general statement that highlights how a statement of invariance under Pitman's transformation for a two-sided process naturally yields an alternative characterisation of the one-sided process conditioned to stay non-negative. The result is particularly transparent in the case of random walks with i.i.d.\ or Markov increments, as well as the zigzag process (details of these examples are presented below). Whilst the applications are not new (cf.\ \cite{HMOC} in particular), we believe it is still worthwhile to present a simple proof of this unified result.
\begin{prop} Let $S=(S_t)_{t\in\mathbb{R}}$ be a random element of $C(\mathbb{R},\mathbb{R})$ that is almost-surely asymptotically linear with strictly positive drift (cf. $\mathcal{S}^{lin}$, as defined at \ref{slin}), and which satisfies $TS\buildrel{d}\over{=}S$. It then holds that
\begin{equation}\label{fffff}
(S_t)_{t\geq 0}\:\vline\:\left\{\inf_{t\geq 0}S_t=0\right\}\buildrel{d}\over{=}(2\bar{M}_t-S_t)_{t\geq 0}\:\vline\:\left\{M_0=0\right\},
\end{equation}
where $\bar{M}=(\bar{M}_t)_{t\geq0}$ is defined by $\bar{M}_t:=\sup_{0\leq s\leq t}S_s$.
\end{prop}
\begin{proof} We have that
\begin{eqnarray*}
(S_t)_{t\geq 0}\:\vline\:\left\{\inf_{t\geq 0}S_t=0\right\}&\buildrel{d}\over{=}&(TS_t)_{t\geq 0}\:\vline\:\left\{\inf_{t\geq 0}TS_t=0\right\}\\
&\buildrel{d}\over{=}&(TS_t)_{t\geq 0}\:\vline\:\left\{M_0=0\right\}\\
&\buildrel{d}\over{=}&(2M_t-S_t)_{t\geq 0}\:\vline\:\left\{M_0=0\right\}\\
&\buildrel{d}\over{=}&(2\bar{M}_t-S_t)_{t\geq 0}\:\vline\:\left\{M_0=0\right\},
\end{eqnarray*}
where the first equality is a consequence of the assumption $TS\buildrel{d}\over{=}S$; the second follows because $\inf_{t\geq 0}TS_t=M_0$ for asymptotically linear $S$ (see \cite[Theorem 2.14]{CKSS}); the third by the definition of $T$ (and the conditioning on $M_0=0$); and the fourth from the observation that $M_t=\max\{\bar{M}_t,M_0\}$ for $t\geq 0$.
\end{proof}
\begin{rem} The condition of asymptotic linearity is sufficient but not necessary for the above proof to work. The relation between the future infimum of $TS$ and the past maximum of $S$ holds whenever $S$ is in the domain of $T$ and $T^{-1}TS=S$. See \cite[Theorem 2.14]{CKSS} for details.
\end{rem}
\begin{rem} The same result holds for paths $S:\mathbb{Z}\rightarrow\mathbb{R}$ whose increments take values either $-1$ or $1$. For more general increments, the argument does not apply (since the future infimum of $TS$ and the past maximum of $S$ do not necessarily agree).
\end{rem}
\begin{exmp} The simplest non-trivial application of the previous result (and the previous remark) is when $S$ is a simple random walk with i.i.d.\ Bernoulli increments and strictly positive drift (i.e.\ the path encoding of Subsection \ref{iidsec}). In this case, the right-hand side of \eqref{fffff} can be replaced by the unconditioned process.
\end{exmp}
\begin{exmp} We next consider the case when $S$ is a path with Markovian increments of the form described in Subsection \ref{markovsec}. In this case, the conditioning on right-hand side of \eqref{fffff} can be replaced by the initial condition $\eta_0=0$ (using the BBS notation of earlier sections).
\end{exmp}
\begin{exmp} For $S$ the zigzag process of Subsection \ref{zzsec}, the result applies, and the conditioning on right-hand side of \eqref{fffff} can also be replaced by the initial condition $\eta_0=0$ (i.e.\ $S$ has a gradient of $1$ at 0).
\end{exmp}
\section*{Acknowledgements}
DC would like to acknowledge the support of his JSPS Grant-in-Aid for Research Activity Start-up, 18H05832, and MS would like to acknowledge the support of her JSPS Grant-in-Aid for Scientific Research (B), 16KT0021.
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|
1,116,691,499,289 | arxiv | \section{Supplementary Materials}
\subsection{MABK-violation without classical post-processing}
In the main text, we have presented the probability of observing a violation by
considering all equivalent Mermin-Ardehali-Belinski\v{\i}-Klyshko (MABK)
inequalities. If the classical post-processing to optimize over the measurement
labelings is \emph{not} carried out (i.e., we calculate the probability to
violate a particular inequality rather than an equivalent set), then we find
$p^{\ket{\mbox{\tiny GHZ}_n}}_{\mbox{\tiny MABK}}$ decreases steadily with $n$; see
Fig.~\ref{Fig:ProbMABK}. However, in this case, among those choices of bases
that give rise to an MABK-violation, we find that their average violation
increases with $n$ when normalized by the classical threshold
$2^{n-\frac{1}{2}}$ but decreases when normalized by the maximal possible
quantum violation
$2^{\frac{3n}{2}-1}$~\cite{MerminIneq,WW:Complete:npartiteBI}. See also
Ref.~\cite{I.Pitowsky:PRA:022103} in this regard.
\begin{figure}[h!]
\scalebox{0.50}{\includegraphics{MABK.eps}}
\caption{\label{Fig:ProbMABK}
(Color online) Probability of finding a randomly generation correlation from
$\ket{\rm GHZ}_n$ to violate a single MABK inequality.}
\end{figure}
\subsection{Analytic derivation of probability of violation for $n=2$}
For the specific case of $n=2$, the probability of violating the (2-party) MABK
inequality, i.e., the Bell-CHSH inequality~\cite{Bell,CHSH}
\begin{equation}\label{Ineq:CHSH}
\S_{\mbox{\tiny CHSH}} =|E(1,1)+E(1,2)+E(2,1)-E(2,2)|\le 2,
\end{equation}
can be computed analytically. To this end, it is expedient to consider the
two-qubit singlet state
\begin{equation}
\ket{\Psi^-}=\tfrac{1}{\sqrt{2}} \left(\ket{0}_1\ket{1}_2-\ket{1}_1\ket{0}_2 \right),
\end{equation}
instead of the 2-party GHZ state:
\begin{equation}
\ket{{\rm GHZ}}_2=\tfrac{1}{\sqrt{2}}\left(\ket{0}_1\ket{0}_2+\ket{1}_1\ket{1}_2\right).
\end{equation}
Clearly, these states are local unitarily equivalent, and hence their
probability of violating the Bell-CHSH inequality via random isotropic
measurements (ROM) is identical. However, $\ket{\Psi^-}$ is also rotationally
invariant (i.e., $U\otimes U \ket{\Psi^-}=\ket{\Psi^-}$ for arbitrary qubit
unitary transformation $U$), which allows us to simplify our analysis
considerably.
Let us begin by writing the measurement directions for Alice and Bob as
$\vec{a}_{s_1}$ and $\vec{b}_{s_2}$. From some simple calculation, one can show
that the correlation function, i.e., the average of the product of measurement
outcomes for Alice measuring $\vec{a}_{s_1}\cdot\vec{\sigma}$ and Bob measuring
$\vec{b}_{s_s}\cdot\vec{\sigma}$ is
\begin{equation}\label{Eq:Qubit:Correlation}
E\left(\vec{a}_{s_1},\vec{b}_{s_2}\right)=-\vec{a}_{s_1}\cdot\vec{b}_{s_2},
\end{equation}
where $\vec{\sigma}=(\sigma_x, \sigma_y, \sigma_z)$ is the vector of Pauli
matrices. Using this in Eq.~\eqref{Ineq:CHSH}, we are thus interested to know
how often the inequality
\begin{equation}\label{bell_random_directions}
\left|\vec{a}_1\cdot\vec{b}_1+\vec{a}_1\cdot\vec{b}_2+\vec{a}_2\cdot\vec{b}_1-\vec{a}_2\cdot\vec{b}_2\right|>2.
\end{equation}
holds when the unit vectors $\vec{a}_1$, $\vec{a}_2$, $\vec{b}_1$ and
$\vec{b}_2$ are chosen independently, randomly and isotropically. We can
simplify this by defining the two orthogonal vectors,
$\vec{r}_{\perp}=\vec{b}_1-\vec{b}_2$ and $\vec{r}=\vec{b}_1+\vec{b}_2$, in
terms of which inequality~\eqref{bell_random_directions} holds if and only if
\begin{equation}\label{bell_ineq_ar}
\left|\vec{a}_1\cdot\vec{r}+\vec{a}_2\cdot\vec{r}_{\perp}\right|>2.
\end{equation}
Note that as well as being orthogonal, the vectors $\vec{r}_{\perp}$ and
$\vec{r}$ have lengths satisfying,
\begin{align}
\left|\vec{r}\right|=\sqrt{2(1+\vec{b}_1\cdot\vec{b}_2)}
=\sqrt{2(1+x)},\nonumber\\
\left|\vec{r}_{\perp}\right|=\sqrt{2(1-\vec{b}_1\cdot\vec{b}_2)}
=\sqrt{2(1-x)},\label{Eq:r_rp:magnitude}
\end{align}
where $x=\vec{b}_1\cdot\vec{b}_2$.
By rewriting $\vec{r}$ in terms of its unit vector $\hat{r}$ and magnitude
$|\vec{r}|$, i.e., $\vec{r}=\hat{r}|\vec{r}|=\hat{r}\sqrt{2(1+x)}$, and
similarly for $\vec{r}_\perp$, inequality~\eqref{bell_ineq_ar} becomes
\begin{equation}\label{Ineq:arx}
\left|\vec{a}_1\cdot\hat{r}\sqrt{1+x}+\vec{a}_2\cdot\hat{r}_{\perp}\sqrt{1-x}
\right|>\sqrt{2}.
\end{equation}
Now the rotational invariance of dot products ensures that in determining the
fraction of measurement directions that violate inequality~\eqref{Ineq:arx},
the actual direction of $\hat{r}$ and $\hat{r}_\perp$ is irrelevant. With some
thought, one can see that to determine this fraction, and hence the probability
of violating the Bell-CHSH inequality~\eqref{Ineq:CHSH} via random isotropic
measurements, it suffices to sample $x$ and the dot products
$\vec{a}_1\cdot\hat{r}$, $\vec{a}_2\cdot\hat{r}_{\perp}$ uniformly from the
interval $[-1,1]$. Writing $\beta=\vec{a}_1\cdot\hat{r}$ and
$\alpha=\vec{a}_2\cdot\hat{r}_{\perp}$, we therefore wish to find the fraction
of values of $\alpha,\beta,x\in[-1,1]$ satisfying,
\begin{equation}\label{random_bell_alphabetax}
\left|\alpha\sqrt{1-x}+\beta\sqrt{1+x}\right|>\sqrt{2}.
\end{equation}
This problem has a useful geometrical interpretation. The set of all points
$\alpha,\beta,x\in[-1,1]$ define a cube, $\mathfrak{C}$, which encloses a
volume $2^3$ and has faces located at $\alpha,\beta,x=\pm{1}$. To find the
fraction of Alice and Bob's measurement directions that would violate the
Bell-CHSH inequality we need to find what fraction of the volume of
$\mathfrak{C}$ contains points $(\alpha,\beta,x)$ satisfying the constraint of
Eq.~(\ref{random_bell_alphabetax}).
\begin{figure*}
\begin{tabular}{ccccc
\includegraphics[scale=0.15]{rand_bell_xm08}&
\includegraphics[scale=0.15]{rand_bell_xm04}&
\includegraphics[scale=0.15]{rand_bell_x0}&
\includegraphics[scale=0.15]{rand_bell_x04}&
\includegraphics[scale=0.15]{rand_bell_x08}
\end{tabular}
\caption{(Color online) Plots showing the triangular regions containing values
of $\alpha$ and $\beta$ which satisfy inequality~\eqref{random_bell_alphabetax}
(i.e., violating the Bell-CHSH inequality). From left to right, the plots are
for values of $x=-0.8,-0.4,0,0.4,0.8$.\label{FIG:cube_cross_sections} }
\end{figure*}
By separately considering the cases where $\alpha\sqrt{1-x}+\beta\sqrt{1+x}>0$
and $\alpha\sqrt{1-x}+\beta\sqrt{1+x}<0$, the regions of $\mathfrak{C}$
containing points satisfying Eq.~(\ref{random_bell_alphabetax}) are given by,
\begin{equation}\label{triangle_1}
\alpha>\frac{\sqrt{2}-\beta\sqrt{1+x}}{\sqrt{1-x}}
\end{equation}
and,
\begin{equation}\label{triangle_2}
\alpha<-\frac{\sqrt{2}+\beta\sqrt{1+x}}{\sqrt{1-x}}.
\end{equation}
If we take a cross-section through $\mathfrak{C}$ corresponding to a particular
fixed value of $x$, then these two expressions define triangular regions at the
corners of the cube, as is shown in Fig.~\ref{FIG:cube_cross_sections}. From
some straightforward computation, it can be shown that the total area of the
two triangles for a given $x$ is
\begin{equation}
\frac{\left(\sqrt{1+x}+\sqrt{1-x}-\sqrt{2}\right)^2}{\sqrt{1-x^2}}.
\end{equation}
Integrating this expression from $x=-1$ to 1, and dividing the result by $2^3$
(the total volume of $\mathfrak{C}$), we therefore find that the fraction of
Alice and Bob's measurement directions that will violate the Bell-CHSH
inequality is:
\begin{equation}
\frac{\pi-3}{2}\simeq{7.0796\%}.
\end{equation}
Notice that the above expression is the probability of violating
inequality~\eqref{Ineq:CHSH} without taking into account of any possible
relabeling of measurement settings and/or outcomes. To take this into account,
note that the four equivalent Bell-CHSH inequalities are as follows:
\begin{align}\label{Ineq:CHSH:Equivalent}
|E(1,1)+E(1,2)-E(2,1)-E(2,2)|\le 2,\\
|E(1,1)+E(1,2)-E(2,1)+E(2,2)|\le 2,\\
|E(1,1)-E(1,2)+E(2,1)+E(2,2)|\le 2,\\
|-E(1,1)+E(1,2)+E(2,1)+E(2,2)|\le 2.
\end{align}
It is not difficult to see that for any given measurement directions, {\em at
most} one of these inequalities can be violated. To see this, suppose otherwise
and one will find that an inequality like the following
\begin{equation}
E(1,1)+E(1,2)>2,
\end{equation}
has to hold true. However, this is logically impossible as the absolute value
of each of the correlation function $|E(s_1,s_2)|$ is upper bounded by 1.
Moreover, by symmetry, the probability of finding measurement directions that
violate any of these equivalent inequalities is identical. Hence, the
probability of finding measurement directions that violate any of the Bell-CHSH
inequality, with the help of relabeling of measurement settings and/or outcomes
is:
\begin{equation}
4\times\frac{\pi-3}{2}\simeq{28.3185\%}.
\end{equation}
\subsection{Identifying nonclassical correlation via linear programming}
Linear programming is a convex optimization problem~\cite{S.Boyd:Book:2004}
which can be efficiently solved on a computer. For this kind of optimization
problem, both the objective function and the constraints are linear in the
variable $\mathbf{x}\in\mathbb{R}^d$. In its {\em standard form}, a linear
program reads as~\cite{S.Boyd:Book:2004}:
\begin{subequations}
\begin{align}
&\,\text{~minimize}\qquad \mathbf{c}^{\mbox{\tiny T }}\mathbf{x},\label{Eq:Obj}\\
&\text{~subject to}\quad A\mathbf{x}=\mathbf{b},\label{Eq:Const:Eq}\\
&\qquad\qquad\qquad\,\,\, \mathbf{x} \succeq\mathbf{0}\label{Eq:Const:Ineq},
\end{align}
\end{subequations}
where $\mathbf{c}\in\mathbb{R}^{d}$, $\mathbf{b}\in\mathbb{R}^{d'}$, $A$ is a
$d'\times d$ matrix, $\succeq$ represents a component-wise inequality and
$\mathbf{0}$ is a $d\times1$ null vector.
For the experimental scenarios described in the main text, the $k$-th party is
allowed to perform measurements in two different measurement directions
$\Omega^{[k]}_{s_k}$, with $s_k=1,2$. Let us denote by $o^{[k]}_{s_k}$ the
measurement outcome corresponding to the measurement direction
$\Omega^{[k]}_{s_k}$. Now, recall that a given correlation admits a locally
causal/ realistic description if and only if all local measurement outcomes
$o^{[k]}_{s_k}$ are (probabilistically) determined in advance solely by the
choice of the local setting $s_k$. Equivalently, a locally causal correlation
is one where the experimental statistics admit a joint probability distribution
\begin{equation*}
p_{1\ldots n}(o^{[1]}_1, o^{[1]}_2, \ldots, o^{[k]}_1,o^{[k]}_2,\ldots, o^{[n]}_1,
o^{[n]}_2)
\end{equation*}
for all the $2n$ variables $o^{[k]}_{s_k}$, and which recovers all the observed
experimental statistics as marginals~\cite{fn:ExplicitBipartite}. For example,
the joint probability $p_{1n}$ of observing $o^{[1]}_1=1$ and $o^{[n]}_2=-1$,
must be reproducible via:
\begin{align*}
&p_{1n}(o^{[1]}_1=1,o^{[n]}_2=-1)\\
=&\sum_{k=2}^{n-1}\sum_{o^{[1]}_2,o^{[n]}_1,o^{[k]}_{s_k}=\pm1}
p_{1\ldots n}(o^{[1]}_1, o^{[1]}_2, \ldots, o^{[k]}_1,o^{[k]}_2,\ldots, o^{[n]}_1,
o^{[n]}_2);
\end{align*}
likewise for all the correlation functions, such as
\begin{align*}
&E(s_1,s_n)=\sum_{k=1,n}\sum_{o^{[k]}_{s_k}=\pm1}
\prod_{j=1,n} o^{[j]}_{s_j} \,p_{1n}(o^{[1]}_{s_1},o^{[n]}_{s_n}),
\end{align*}
which must also be reproducible from appropriate linear combination of the
entries of the joint probability distribution $p_{1\ldots n}$ (see
Ref.~\cite{D.Gosal:PRA:042106} for a more elaborate exposition of these
constraints).
Clearly, these constraints can be put in the form of Eq.~\eqref{Eq:Const:Eq},
with $\mathbf{x}\in\mathbb{R}^{2^{2n}}$ being the entries of $p_{1\ldots n}$
sought for, with all (but one) entries of $\mathbf{b}\in\mathbb{R}^{3^n}$ being
the measured correlation functions $E(s_1,\ldots, s_k)$, and with entries of
$A$ giving the coefficients required to reproduce each $E(s_1,\ldots, s_k)$
from the entries of $p_{1\ldots n}$. On top of the $3^{n}-1$ correlation
functions, $\mathbf{b}$ must also contain an additional entry that is set to 1;
the same goes for the corresponding row of entries of $A$, which must all be
set to 1. This last requirement ensures that the joint probability distribution
$p_{1\ldots n}$ is normalized while the non-negativity of the probabilities is
taken care of by Eq.~\eqref{Eq:Const:Ineq}.
The problem of determining if a given correlation admits a locally causal
description can then be cast as the following linear programming {\em
feasibility problem}:
\begin{align*}
&\,\text{~minimize}\qquad\quad\, 0,\\
&\text{~subject to}\quad A\mathbf{x}=\mathbf{b},\\
&\qquad\qquad\qquad\,\,\, \mathbf{x} \succeq\mathbf{0}.
\end{align*}
For an alternative formulation of the problem as a linear programming {\em
optimization problem}, see Ref.~\cite{M.B.Elliott:0905.2950}.
\begin{figure}[h!]
\scalebox{0.50}{\includegraphics{Fig3.eps}}
\caption{\label{Fig:ProbPureTwoQubit}
(Color online) Probability of finding the measurement statistics on a pure
two-qubit state $\ket{\Psi}=\cos\theta\ket{0}_1\ket{0}_2
+\sin\theta\ket{1}_1\ket{1}_2$ to violate the class of Bell-CHSH inequalities.
Here, the largest Schmidt coefficient $\cos\theta$ quantifies the entanglement.
The values $\theta=0$ and $\pi/4$ give, respectively, the pure product state
and the maximally entangled two-qubit state. Inset: a magnified plot of the
crossover region.}
\end{figure}
\subsection{Probability of violation vs entanglement}
For $n=2$, a pure quantum state can always be written as
$\ket{\Psi}=\cos\theta\ket{0}_1\ket{0}_2 +\sin\theta\ket{1}_1\ket{1}_2$ for
some local bases. We numerically compute the probability of violation as a
function of entanglement, as shown in Fig.~\ref{Fig:ProbPureTwoQubit}. Note
that, while the probability of violation is greater with ROM in most instances,
it also decreases more rapidly to zero with decreasing $\theta$. In fact, at
$\theta\approx0.068\,\pi$, the probability of violation of $\ket{\Psi}$ is
already close to $10^{-5}\%$ with ROM whereas the chance is still about 0.26\%
with RIM (inset, Fig.~\ref{Fig:ProbPureTwoQubit}).
|
1,116,691,499,290 | arxiv | \section*{Abstract}
{\bf
We compare the efficiency of different matrix product state (MPS) based methods for the calculation of two-time correlation functions in open quantum systems. The methods are the purification approach~\cite{WolffKollath2019} and two approaches~\cite{BreuerPetruccione1997,MolmerDalibard1993} based on the Monte-Carlo wave function (MCWF) sampling of stochastic quantum trajectories using MPS techniques. We consider a XXZ spin chain either exposed to dephasing noise or to a dissipative local spin flip. We find that the preference for one of the approaches in terms of numerical efficiency depends strongly on the specific form of dissipation.}
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\section{Introduction}
\label{sec:intro}
The investigation of open quantum many-body systems has been a very active field of research over the past decades. One motivation is the understanding of destructive effects of environments on quantum processes which are used in quantum technologies, e.g in quantum computing or quantum communication. More recently, another point of view has been taken. Environments are particularly tailored in order to stabilize and control quantum many-body states \cite{MuellerZoller2012,BarreiroBlatt2011, SyassenDuerr2008, CaballarWatanabe2014,BrennenWilliams2005}.
However, it has been shown that the dynamic properties of such states can have very distinct behavior from their Hamiltonian counterparts \cite{SciollaKollath2015}. In particular the response of open quantum systems to external perturbations very different to the Hamiltonian evolution.
Quantities which are of particular importance in this respect are two-time correlation functions $\langle B(t_2) A(t_1) \rangle$. Here $A$ and $B$ are operators, $t_1$ and $t_2$ are two different times, and $\langle \dots \rangle = \text{tr}(\rho \dots)$ is the expectation value over the density matrix $\rho$ of a given system. In isolated systems, such two-time correlation functions are powerful tools to give information on the response of the system to a small perturbation. Many experimental techniques are based on such processes and the observation of the subsequent response is described by these two-time functions. Examples include neutron scattering~\cite{BramwellKeimer2014}, ARPES~\cite{LuShen2012}, conductivity and magnetization measurements in solids~\cite{AshcroftMermin} and spectroscopic measurement as radio-frequency~\cite{StewartJin2008}, Raman, Bragg~\cite{Esslinger2010} or modulation spectroscopy~\cite{StoeferleEsslinger2004} in the field of quantum gases.
Two-time correlations have been studied extensively in isolated many-body quantum systems both in and out of equilibrium. However, in many-body quantum systems coupled to environments their determination is very challenging and only few studies are available mostly using approximate approaches or small systems \cite{BuchholdDiehl2015,MarcuzziLesanovsky2015,SciollaKollath2015,EverestLevi2017,HeDiehl2017,WangPoletti2018}. In particular, a change of the behavior of dynamic correlations has been demonstrated in the presence of dissipation (see for example \cite{CarusottoCiuti2013, RitschEsslinger2013,HoeningFleischhauer2012,SiebererDiehl2013,LesanovskyGarrahan2013b,TaeuberDiehl2014}).
In this work we present a comprehensive study on the application of matrix product state (MPS) algorithms to the determination of two-time correlation functions in open systems. We compare an extension of the purification approach \cite{Nishino1995,MoukouriCaron1996,BatistaAligia1998,
HallbergAligia1999,VerstraeteCirac2004b, FeiguinWhite2005, ZwolakVidal2004,HartmannPlenio2009} to two-time correlations and two different stochastic approaches based on the unraveling of the quantum evolution \cite{BreuerPetruccione1997,CarmichaelBook,DalibardMolmer1992,DumRitsch1992,MolmerDalibard1993, BreuerPetruccione2002, Daley2014}. The first approach has been proposed by Breuer et al.~\cite{BreuerPetruccione1997} and the second approach by M\o lmer et al.~\cite{MolmerDalibard1993}.
The comparison is performed using an XXZ spin model with two different couplings to the environment. The first coupling is a dephasing noise applied globally to the system and the second a local loss of magnetization.
In section \ref{sec:model} we describe the models used. In section \ref{sec:openMPS} we give an introduction to matrix product state based methods for open quantum systems. In section \ref{sec:two_time_methods}, we introduce the three different methods in order to calculate two-time correlation functions and in section \ref{sec:two_time_comparison} we present a comprehensive comparison between the methods.
\section{Model}\label{sec:model}
\begin{figure}[tb]
\centering
\includegraphics[width=0.4\textwidth]{Model_Sketches_paper.pdf}
\caption{The sketch of the spin models with $(a)$ bulk dephasing where all spins are coupled to the bath and $(b)$ local demagnetization where the central spin is disspative.}
\label{fig:Spin_Model}
\end{figure}
We consider spin-$\nicefrac{1}{2}$ chains, with coupling between spins on adjacent sites, as a paradigmatic model for interacting one-dimensional systems. In addition, the system is subjected to an environment which introduces dissipative processes to the system dynamics. Assuming that retroactive influences of earlier dissipation effects on the current dynamics can be neglected, the system dynamics can be described by the Lindblad master equation \cite{BreuerPetruccione2002}
\begin{equation}
\frac{\partial}{\partial t} \rho(t) = \mathcal{L} \rho(t) = -\frac{i}{\hbar}\left[ H_{\mathrm{XXZ}}, \rho(t) \right] + \mathcal{D}[\rho(t)]\label{eq:LindbladEq}
\end{equation}
with the superoperator $\mathcal{L}$ and the system density matrix $\rho$. The first term on the right hand side represents the unitary contribution generated by the Hamiltonian. Here we consider the $\mathrm{XXZ}$ spin-$\nicefrac{1}{2}$ Hamiltonian
\begin{equation}
H_{\mathrm{XXZ}} = \sum_{l=1}^{L-1} \left[ J_x \left(S^x_l S^x_{l+1} + S^y_l S^y_{l+1}\right) + J_z S^z_l S^z_{l+1}\right],
\end{equation}
describing a chain of $L$ spins, where $J_x$ and $J_z$ are exchange couplings according to different spin directions and $S^\alpha_l$ is the spin operator in direction $\alpha$ at site $l$. In equilibrium this model is well understood and exhibits in the ground state, three phases for different ratios of the interaction strengths \cite{MikeskaKolezhuk2004,Giamarchibook}: For $-1 \le J_z/J_x \le 1$ a gapless Tomonaga-Luttinger liquid is formed, whereas $J_z/J_x < -1$ and $J_z/J_x > 1$ present gapped phases showing ferromagnetic and antiferromagnetic nature, respectively. The second term on the right hand side of Eq.~\ref{eq:LindbladEq} represents dissipative noise. We use the Lindblad form of the dissipator which is given by
\begin{equation}
\mathcal{D} [\rho] =\sum_l\Gamma_l\left( L_l \rho L_l^\dagger - \frac{1}{2} L_l^\dagger L_l \rho - \frac{1}{2}\rho L_l^\dagger L_l\right).
\label{eq:dissipator}
\end{equation}
Here $\Gamma_l$ is the effective dissipation strength corresponding to the Lindblad jump operator $L_l$.
For the comparison of methods presented here, two different types of dissipation are considered. As will be shown, each of them have different impacts on various aspects of the performance of the methods.
First we study systems exposed to bulk dephasing (see Fig.~\ref{fig:Spin_Model}($a$)), where the jump operators $L_l$ are given by the set of local $S^z_l$ operators controlled by the dissipation strength $\Gamma_l=\gamma$
\begin{equation}
\mathcal{D}_1(\rho) = \gamma \sum_{l=1}^{L}\left( S^z_l \rho S^z_l - \frac{1}{4} \rho \right).
\label{eq:D1}
\end{equation}
Here the jump operators $S^z_l$ are Hermitian, which implies that the infinite temperature state is a steady state of the model. One can show that this is the unique steady state. The dissipative dynamics arising in this and related models has been studied previously and interesting critical dynamics and aging dynamics have been pointed out to occur \cite{PolettiKollath2012,PolettiKollath2013,CaiBarthel2013, EverestLevi2017,WolffKollath2019}.
Furthermore, we investigate the case in which the coupling to the environment results in a local defect that is represented by the single Lindblad operator $S^-_c$ which only acts on the central site as shown in Fig.~\ref{fig:Spin_Model}($b$). Here $c$ is the index of the central site for a chain with an odd number of sites and the center-left site otherwise. The dissipation strength is $\Gamma_{c}=\gamma$ and $\Gamma_{l\neq c}=0$ which results in the dissipator
\begin{equation}
\mathcal{D}_2(\rho) = \gamma\left( S^-_c \rho S^+_c - \frac{1}{2} S^+_{c} S^-_{c} \rho - \frac{1}{2} \rho S^+_{c} S^-_{c}\right).
\label{eq:D2}
\end{equation}
The steady state of this system is the ferromagnetic state with all spins pointing down. Using the mapping to interacting fermions or hard core bosons, the jump operator corresponds to a local particle loss process. Similar 'lossy' defects have been studied in a variety of models previously and interesting transport effects and meta-stable states have been identified \cite{BarmettlerKollath2011,WolffKollath2020,BrazhnyiOtt2009,ShchesnovichKonotop2010,ShchesnovichMogilevtsev2010,WitthautWimberger2011,ZezyulinOtt2012,KieferEmmanouilidisSirker2017,FroemlDiehl2019,FroemlDiehl2019b, DamanetDaley2019}.
\section{Matrix product state approaches for open quantum system dynamics}\label{sec:openMPS}
A variety of tensor network based algorithms has been successfully used to simulate the dynamics of open one-dimensional quantum systems with a focus on equilibrium or equal time properties.
This section is structured such that we start by giving a short overview of the established concepts of MPS in closed quantum systems in Sec.~\ref{sec:closedMPS}, which are essential for the extension to open systems. Subsequently, we describe two prominent approaches for computing the dissipative dynamics of open quantum systems. We describe first the full evolution of the purified density matrix in Sec.~\ref{sec:methods_purification} and second the Monte-Carlo wave function (MCWF) sampling of stochastic quantum trajectories in Sec.~\ref{sec:methods_MCWF}.
\subsection{Matrix product state formalism for closed systems}
\label{sec:closedMPS}
The description of quantum states in MPS form has become a standard method for the simulation of one-dimensional many-body quantum systems. It has been used for a wide range of models, since it is very efficient and well-controlled approximation \cite{Schollwoeck2011,PaeckelHubiged2019}. In this section and the following on open systems, we describe the basics and key concepts of this technique \cite{Schollwoeck2011} such that in the following we can detail the particularities of the approach to the determination of two-time functions in open quantum systems.
\subsubsection{MPS representation of quantum many-body states}
The idea relies on the approximate representation of the quantum many-body wave function for a one-dimensional lattice system of $L$ sites as a set of local tensors/matrices. In order to achieve this, a singular value decomposition (SVD) is applied to the amplitude matrix of a bipartite system whose subparts $A$ and $B$ are connected by a bond between site $l$ and $l+1$
\begin{align}
\ket{\psi} = \sum_{m,n}\psi_{m,n}\ket{m}_A\otimes \ket{n}_B
\overset{SVD}{=} \sum_{m,n, a_l} U_{m a_l} s_{a_l} V^{\dagger}_{a_l n}\ket{m}_A\otimes \ket{n}_B.
\end{align}
Here $\ket{m}_{A/B}$ are basis states in subsystem $A/B$ and $U$, $s$, $V$ are obtained by the singular value decomposition of the amplitude matrix $\psi_{m,n}$. Iterating this procedure on all bonds then yields the expression of the quantum state by single site tensors
\begin{equation}
\ket{\psi} = \sum_{\sigma_1, \ldots, \sigma_L} \sum_{a_1, \ldots, a_{L-1}} M^{\sigma_1}_{1, a_1} M^{\sigma_2}_{a_1, a_2} \ldots M^{\sigma_L}_{a_{L-1}, 1} \ket{\vec{\sigma}},
\label{eq:MPS_representation}
\end{equation}
where we use $\ket{\vec{\sigma}}=\ket{\sigma_1 \sigma_2 \ldots \sigma_L}$ and $\sigma_l$ labels the local basis states at the site $l\in\{1, 2, \ldots, L\}$. The number of local basis states is called the physical dimension $d$, where for the considered spin-$\nicefrac{1}{2}$ model $d=2$ and $\sigma_l \in\{\uparrow, \downarrow\}$. The representation \ref{eq:MPS_representation} is still exact. The singular values $s_{a_l}$ are the coefficients of the Schmidt decomposition and are thus directly linked to the von Neumann entropy
\begin{equation}
S_{\mathrm{vN}}=-\sum_{a_l} s^2_{a_l} \log\left(s^2_{a_l}\right).
\end{equation}
The von Neumann entropy is a measure for the entanglement between the two subsystems connected by the considered bond. If the entanglement is not too strong, this corresponds to a sufficiently fast decay of the descendingly sorted squared singular values $s_{a_l}^2$. In this case, the dimension of the matrices for the representation of the state can be cut at a maximal value $D$, resulting in a compressed state which is a very good approximation of the exact state. A weak von Neumann entanglement is for example found for ground states of short-ranged one-dimensional Hamiltonians \cite{Schollwoeck2011}.
As the value $D$ limits the extend of the indices $\{a_1, a_2, \ldots, a_{L-1}\}$, connecting two adjacent site tensors, this parameter is known as the bond dimension. The approximation is controlled by the sum of the discarded squared singular values, the so-called truncation weight
\begin{equation}
\varepsilon = \sum_{a_l>D} s^2_{a_l},
\end{equation}
and reduces the computational complexity from exponential to polynomial.
In the following we will use the established graphical notation for tensor networks \cite{Schollwoeck2011}
\begin{equation}
T_{ijk} \equiv
\begin{tikzpicture}[baseline={([yshift=-.5ex]current bounding box.center)},vertex/.style={anchor=base,
circle,fill=black!25,minimum size=18pt,inner sep=2pt, scale=.7}]
\draw [thick] (0,0) -- (0,0.75);
\draw [thick] (0,0) -- (-0.5,-0.5);
\draw [thick] (0,0) -- (0.5,-0.5);
\draw (0.2, 0.75) node {$i$};
\draw (-0.2, -0.5) node {$j$};
\draw (0.7, -0.5) node {$k$};
\node[draw, shape=circle, fill=lightgray] (v0) at (0,0) {};
\end{tikzpicture},
\end{equation}
where a tensor is portrayed by a shape (here circle) with sticking out lines representing the indices. Connecting two lines depicts a tensor contraction with regard to this pair of indices. Using this notation, a MPS is depicted by
\begin{equation}
\ket{\psi} = \sum_{\vec \sigma}
\begin{tikzpicture}[baseline={([yshift=-2.5ex]current bounding box.center)},vertex/.style={anchor=base,
circle,fill=black!25,minimum size=18pt,inner sep=2pt, scale=.8}]
\draw [thick] (0,0) -- (0,0.75);
\draw [thick] (1,0) -- (1,0.75);
\draw [thick] (2,0) -- (2,0.75);
\draw [thick] (4,0) -- (4,0.75);
\draw [thick] (0,0) -- (2.5,0.);
\draw [thick] (3.5,0) -- (4.,0.);
\draw [thick, dotted] (0,0) -- (4.,0.);
\draw (0.3, 0.75) node {$\sigma_1$};
\draw (1.3, 0.75) node {$\sigma_2$};
\draw (2.3, 0.75) node {$\sigma_3$};
\draw (3., 0.75) node {$\ldots$};
\draw (4.3, 0.75) node {$\sigma_L$};
\draw (0.5, 0.2) node {$a_1$};
\draw (1.5, 0.2) node {$a_2$};
\draw (2.5, 0.2) node {$a_3$};
\draw (3.3, 0.2) node {$a_{L-1}$};
\draw [fill=lightgray] (0,0.) circle [radius=0.2] node {};
\draw [fill=lightgray] (1,0.) circle [radius=0.2] node {};
\draw [fill=lightgray] (2,0.) circle [radius=0.2] node {};
\draw [fill=lightgray] (4,0.) circle [radius=0.2] node {};
\end{tikzpicture} \ket{\vec \sigma}.\label{eq:physMPS}
\end{equation}
\subsubsection{Time-dependent matrix product state algorithm}
Time-dependent matrix product state ($t$MPS) algorithms are well-established tools for the efficient computation of the dynamics of closed many-body quantum systems at zero \cite{Vidal2004,DaleyVidal2004, WhiteFeiguin2004,Schollwoeck2011,PaeckelHubiged2019} and finite temperature \cite{KarraschMoore2012,KarraschMoore2013,Barthel2013,PirozenTroyer2014}. In particular, it is possible to calculate the time evolution of a MPS for the considered spin-$\nicefrac{1}{2}$ model with $\gamma=0$, using a Suzuki-Trotter decomposition \cite{Suzuki1976, HatanoSuzuki2005} of the time evolution operator for small time steps $\Delta t$. Here we use the second order Suzuki-Trotter decomposition given by
\begin{equation}
\ee^{-i H \Delta t/\hbar} =\ee^{-i H_{\mathrm{odd}} \Delta t / (2\hbar)}\ee^{-i H_{\mathrm{even}}\Delta t/\hbar} \ee^{-i H_{\mathrm{odd}} \Delta t / (2\hbar)}+ \mathcal{O}(\Delta t^3).
\end{equation}
Here $H_{\mathrm{odd}}$ ($H_{\mathrm{even}}$) only contains parts of the Hamiltonian covering the odd (even) numbered bonds of the lattice, so that all contributing terms are commuting, but $[H_{\mathrm{odd}}, H_{\mathrm{even}}]\neq 0$. In the diagram formalism, this corresponds to a successive application of two-site gate operations and compressions
\begin{align}
\ket{\psi(t+\Delta t)} =\sum_{\vec{\sigma}}\:\:
\begin{tikzpicture}[baseline={([yshift=-.5ex]current bounding box.center)},vertex/.style={anchor=base,
circle,fill=black!25,minimum size=18pt,inner sep=2pt}, scale=.5]
\draw [thick] (0,0) -- (9.,0.);
\draw [thick] (0,0) -- (0,3.5);
\draw [thick] (1,0) -- (1,3.5);
\draw [thick] (2,0) -- (2,3.5);
\draw [thick] (3,0) -- (3,3.5);
\draw [thick] (4,0) -- (4,3.5);
\draw [thick] (5,0) -- (5,3.5);
\draw [thick] (6,0) -- (6,3.5);
\draw [thick] (7,0) -- (7,3.5);
\draw [thick] (8,0) -- (8,3.5);
\draw [thick] (9.,0) -- (9,3.5);
\draw [fill=lightgray] (0,0.) circle [radius=0.2] node {};
\draw [fill=lightgray] (1,0.) circle [radius=0.2] node {};
\draw [fill=lightgray] (2,0.) circle [radius=0.2] node {};
\draw [fill=lightgray] (3,0.) circle [radius=0.2] node {};
\draw [fill=lightgray] (4,0.) circle [radius=0.2] node {};
\draw [fill=lightgray] (5,0.) circle [radius=0.2] node {};
\draw [fill=lightgray] (6,0.) circle [radius=0.2] node {};
\draw [fill=lightgray] (7,0.) circle [radius=0.2] node {};
\draw [fill=lightgray] (8,0.) circle [radius=0.2] node {};
\draw [fill=lightgray] (9.,0.) circle [radius=0.2] node {};
\draw [fill=evengate] (0.-.2, 1-.25) rectangle (1. + .2, 1.+ .25) ;
\draw [fill=evengate] (2.-.2, 1-.25) rectangle (3. + .2, 1.+ .25) ;
\draw [fill=evengate] (4.-.2, 1-.25) rectangle (5. + .2, 1.+ .25) ;
\draw [fill=evengate] (6.-.2, 1-.25) rectangle (7. + .2, 1.+ .25) ;
\draw [fill=evengate] (8.-.2, 1-.25) rectangle (9. + .2, 1.+ .25) ;
\draw [fill=oddgate] (1.-.2, 2-.25) rectangle (2. + .2, 2.+ .25) ;
\draw [fill=oddgate] (3.-.2, 2-.25) rectangle (4. + .2, 2.+ .25) ;
\draw [fill=oddgate] (5.-.2, 2-.25) rectangle (6. + .2, 2.+ .25) ;
\draw [fill=oddgate] (7.-.2, 2-.25) rectangle (8. + .2, 2.+ .25) ;
\draw [fill=evengate] (0.-.2, 3-.25) rectangle (1. + .2, 3.+ .25) ;
\draw [fill=evengate] (2.-.2, 3-.25) rectangle (3. + .2, 3.+ .25) ;
\draw [fill=evengate] (4.-.2, 3-.25) rectangle (5. + .2, 3.+ .25) ;
\draw [fill=evengate] (6.-.2, 3-.25) rectangle (7. + .2, 3.+ .25) ;
\draw [fill=evengate] (8.-.2, 3-.25) rectangle (9. + .2, 3.+ .25) ;
\draw [] (0.15, 3.7) node {$\sigma_1$};
\draw [] (1.15, 3.7) node {$\sigma_2$};
\draw [] (2.15, 3.7) node {$\ldots$};
\draw [] (9.15, 3.7) node {$\sigma_{L}$};
\draw[->,color=black,rounded corners=2mm, line width=1.25, >=stealth', dotted] (-0.5, 1) -- (9.5, 1) -| (9.5, 2) -|
(-0.5, 2) -- (-0.5, 3) -- (9.75, 3);
\end{tikzpicture} \: \ket{\vec{\sigma}} \quad + \mathcal{O}(\Delta t^3) ,
\label{eq:unitary_tMPS}
\end{align}
where the application order is indicated by the dotted arrow and the different colors distinguish gates and subsequent compressions of $H_{\mathrm{odd}}$ and $H_{\mathrm{even}}$. By contracting these gates with the MPS tensors, the bond dimension also increases from $D$ to $d^2D$, so that a subsequent compression via SVD is necessary.
A large reduction of the computational effort can be achieved by including conservation laws. In the present case, the total magnetization $M_{\mathrm{tot}}= \sum_j S^z_j$ is preserved by the closed system evolution under the Hamiltonian $H_{\mathrm{XXZ}}$. This leads to a block-diagonal form of matrices, and thus, tensor operations, such as SVDs or tensor contractions, can be performed much more efficiently.
\subsection{Purification approach for the evolution of the density matrix of an open quantum system}
\label{sec:methods_purification}
\subsubsection{Purification of the density matrix}
\label{sec:purifying_pure_state}
One way to transfer the techniques from the previous section to dissipative systems with finite dissipation strength $\gamma$, is to rewrite the density matrix acting on the physical Hilbert space
$\mathcal{H}_{\mathrm{phys}}$ as a state in a doubled space
$\mathcal{H}_{\mathrm{phys}}\otimes\mathcal{H}_{\mathrm{phys}}$ \cite{Nishino1995,MoukouriCaron1996,BatistaAligia1998,
HallbergAligia1999,VerstraeteCirac2004b,FeiguinWhite2005,ZwolakVidal2004, Schollwoeck2011}
\begin{align}
&\rho = \sum_{\vec{\sigma}, \vec{\sigma}'}\rho_{\vec{\sigma}, \vec{\sigma}'} \ket{\sigma_1 \sigma_2 \ldots \sigma_L}\bra{\sigma_1' \sigma_2' \ldots \sigma_L'} \notag\\ &\qquad \xrightarrow{\hspace*{.5cm}}\quad \vert \rho \rrangle =\sum_{\vec{\sigma}, \vec{\sigma}'}\rho_{\vec{\sigma}, \vec{\sigma}'} \vert \sigma_1^{\phantom{\dagger}} \sigma_1'\sigma_2^{\phantom{\dagger}} \sigma_2'\ldots \sigma_L^{\phantom{\dagger}} \sigma_L'\rrangle
\end{align}
where $\vert \rho \rrangle\in \mathcal{H}_{\mathrm{phys}}\otimes \mathcal{H}_{\mathrm{phys}}$. Thus, the density matrix acting on the physical space becomes a pure state in the 'doubled' space, giving the procedure the name purification. The order of the resorting is in principle arbitrary. We use here the given resorting of the state, since this has the advantage that the time-evolution only acts on four 'neighboring sites'. In this representation of the density matrix in the super space $\mathcal{H}_{\mathrm{phys}}\otimes\mathcal{H}_{\mathrm{phys}}$, the Lindblad master equation (Eq.~\ref{eq:LindbladEq}) is written as,
\begin{eqnarray}
\frac{\partial}{\partial t} \vert \rho(t) \rrangle = \mathds{L} \vert \rho(t) \rrangle \equiv \left(-\frac{i}{\hbar}H\otimes I+\frac{i}{\hbar}I \otimes H^T + \mathds{D}\right)\vert \rho(t) \rrangle
\label{eq:LindbladEq2}
\end{eqnarray}
with the new representation of the dissipator $\mathcal{D}$ in the super space,
\begin{eqnarray}
\mathds{D} \equiv \sum_l\Gamma_l \left(L_l \otimes (L_l^\dagger)^T - \frac{1}{2} L_l^\dagger L_l \otimes I - \frac{1}{2} I\otimes (L_l^\dagger L_l)^T\right).
\label{eq:Loperator}
\end{eqnarray}
Here $I$ is the identity matrix in the Hilbert space $\mathcal{H}_{\mathrm{phys}}$ and $T$ denotes the transpose of an operator.
\subsubsection{Representation of an initial state in the purified form}
Often we are confronted with the situation that we start a Lindblad evolution from a pure state represented in the MPS formalism, for example this can be the ground state of a Hamiltonian obtained by the MPS ground state search. For this purpose we present the steps of purifying a state $\ket{\psi}$ that is given in MPS form [Eq.~\ref{eq:physMPS}]. Since the density matrix of this pure state is given by $\ket{\psi}\bra{\psi}$, first a copy of the state is created which represents the bra contribution. Then the tensors stemming from the ket and the bra corresponding to the same site are contracted, which results in a set of tensors with six indices
\begin{equation}
\begin{tikzpicture}[baseline={([yshift=-2.8ex]current bounding box.center)},vertex/.style={anchor=base,
circle,fill=black!25,minimum size=18pt,inner sep=2pt}]
\draw [thick] (0,0) -- (0,0.75);
\draw [thick] (-.75,0) -- (.75,0.);
\draw (0.25, 0.75) node {$\sigma_l$};
\draw (-0.7, 0.25) node {$a_{l-1}$};
\draw (0.5, 0.25) node {$a_l$};
\draw [fill=lightgray] (0,0.) circle [radius=0.2] node {};
\end{tikzpicture}
\: \cdot \:
\begin{tikzpicture}[baseline={([yshift=-2.8ex]current bounding box.center)},vertex/.style={anchor=base,
circle,fill=black!25,minimum size=18pt,inner sep=2pt}]
\draw [thick] (0,0) -- (0,0.75);
\draw [thick] (-.75,0) -- (.75,0.);
\draw (0.25, 0.75) node {$\sigma_l'$};
\draw (-0.7, 0.25) node {$a_{l-1}'$};
\draw (0.5, 0.25) node {$a_l'$};
\draw [fill=lightgray] (0,0.) circle [radius=0.2] node {};
\end{tikzpicture}
=
\begin{tikzpicture}[baseline={([yshift=-1.5ex]current bounding box.center)},vertex/.style={anchor=base,
circle,fill=black!25,minimum size=18pt,inner sep=2pt}]
\draw [thick] (0,0) -- (0,0.75);
\draw [thick] (1,0) -- (1,0.75);
\draw [thick] (-.75,0.1) -- (1.75,0.1);
\draw [thick] (-.75,-.1) -- (1.75,-0.1);
\draw (0.25, 0.75) node {$\sigma_l$};
\draw (1.+0.25, 0.75) node {$\sigma_l'$};
\draw (-0.7, 0.3) node {$a_{l-1}$};
\draw (-0.7, -0.4) node {$a_{l-1}'$};
\draw (1.5, 0.3) node {$a_l$};
\draw (1.5, -0.4) node {$a_l'$};
\draw [fill=lightgray] (0.-.2, 0-.25) rectangle (1. + .2, 0.+ .25) ;
\end{tikzpicture}\:\: .
\end{equation}
In order to obtain single-site tensors for the ket and the bra part, the bond indices are combined, i.e. $(a_{l-1}^{\phantom{\dagger}}, a_{l-1}')\rightarrow \lambda_{l-1}'$ and $(a_{l}^{\phantom{\dagger}}, a_{l}')\rightarrow \lambda_{l}'$, before the tensor are separated into the respective single-site parts using a singular value decomposition with regard to the indices $(\lambda_{l-1}', \sigma_l^{\phantom{\dagger}})\times(\sigma_l', \lambda_l')$. The new bond index, created by the SVD, is denoted by $\lambda_l$, so that the MPS representation of a purified state reads
\begin{align}
\vert \rho \rrangle = \sum_{\vec{\sigma}, \vec{\sigma}'} \sum_{\substack{\lambda_1^{\phantom{\dagger}}, \ldots, \lambda_L^{\phantom{\dagger}} \\ \lambda_1', \ldots, \lambda_{L-1}'}} M^{\sigma_1}_{1, \lambda_1^{\phantom{\dagger}}}M^{\sigma_1'}_{ \lambda_1^{\phantom{\dagger}}, \lambda_1'} \ldots M^{\sigma_L}_{\lambda_{L-1}', \lambda_L}M^{\sigma_L'}_{ \lambda_L^{\phantom{\dagger}}, 1} \vert \sigma_1^{\phantom{\dagger}} \sigma_1'\ldots \sigma_L^{\phantom{\dagger}} \sigma_L'\rrangle.
\end{align}
This translates to the following diagrammatic expression
\begin{align}
\vert \rho \rrangle = \sum_{\substack{\sigma_1 \ldots \sigma_L \\ \sigma_1' \ldots \sigma_L'}}
\begin{tikzpicture}[baseline={([yshift=-.75ex]current bounding box.center)},vertex/.style={anchor=base,
circle,fill=black!25,minimum size=18pt,inner sep=2pt}, scale=.6]
\draw [thick] (0,0) -- (0,0.75);
\draw [thick] (1,0) -- (1,0.75);
\draw [thick] (2,0) -- (2,0.75);
\draw [thick] (3,0) -- (3,0.75);
\draw [thick] (4,0) -- (4,0.75);
\draw [thick] (5,0) -- (5,0.75);
\draw [thick] (8,0) -- (8,0.75);
\draw [thick] (9,0) -- (9,0.75);
\draw [thick] (9,0) -- (6.5,0.);
\draw [thick] (0,0) -- (5.5,0.);
\draw [thick] (7.5,0) -- (9.,0.);
\draw [thick, dotted] (0,0) -- (9.,0.);
\draw (0.25, 1.1) node {$\sigma_1^{\phantom{\dagger}}$};
\draw (1.25, 1.1) node {$\sigma_1'$};
\draw (2.25, 1.1) node {$\sigma_2^{\phantom{\dagger}}$};
\draw (3.25, 1.1) node {$\sigma_2'$};
\draw (4.25, 1.1) node {$\sigma_3^{\phantom{\dagger}}$};
\draw (5.25, 1.1) node {$\sigma_3'$};
\draw (6., 1.1) node {$\ldots$};
\draw (8.25, 1.1) node {$\sigma_L^{\phantom{\dagger}}$};
\draw (9.25, 1.1) node {$\sigma_L'$};
\draw (0.5, -.5) node {$\lambda_1^{\phantom{\dagger}}$};
\draw (1.5, -.5) node {$\lambda_1'$};
\draw (2.5, -.5) node {$\lambda_2^{\phantom{\dagger}}$};
\draw (3.5, -.5) node {$\lambda_2'$};
\draw (4.5, -.5) node {$\lambda_3^{\phantom{\dagger}}$};
\draw (5.5, -.5) node {$\lambda_3'$};
\draw (7.2, -.5) node {$\lambda_{L-1}'$};
\draw (8.5, -.5) node {$\lambda_L^{\phantom{\dagger}}$};
\draw [fill=lightgray] (0,0.) circle [radius=0.2] node {};
\draw [fill=lightgray] (1,0.) circle [radius=0.2] node {};
\draw [fill=lightgray] (2,0.) circle [radius=0.2] node {};
\draw [fill=lightgray] (3,0.) circle [radius=0.2] node {};
\draw [fill=lightgray] (4,0.) circle [radius=0.2] node {};
\draw [fill=lightgray] (5,0.) circle [radius=0.2] node {};
\draw [fill=lightgray] (8,0.) circle [radius=0.2] node {};
\draw [fill=lightgray] (9,0.) circle [radius=0.2] node {};
\end{tikzpicture} \vert \sigma_1 \sigma_1' \ldots \sigma_L\sigma_L' \rrangle \:\: .
\end{align}
It is important to notice that this approach affects the bond dimension drastically. Assuming the bond dimension of the state to purify is given by $D$, the combining of bond indices results in a dimension $D^2$ for the index set $\{\lambda_l'\}$ and the SVD without compression causes an increase to $dD^2$ for the indices $\{\lambda_l\}$. In order to keep the original specified bond dimension $D$, the state to purify needs to be compressed to a bond dimension $\sqrt{D}$ and the spectrum of SVD in the second step needs to be truncated after the $D$ largest values. This poses a very strong constraint, so that particularly strongly entangled states are difficult to purify. Either the need for computational resources increases quadratically or the accuracy, measured by the truncation weight, becomes significantly worse.
\subsubsection{Time-evolution of the purified density matrix}
\begin{figure*}
\begin{align}
\vert \rho(t+ \Delta t) \rrangle &= \ee^{\mathcal{L}\Delta t} \vert \rho(t)\rrangle = \ee^{\mathcal{L}_{\mathrm{odd}}\Delta t/2}\ee^{\mathcal{L}_{\mathrm{even}}\Delta t}\ee^{\mathcal{L}_{\mathrm{odd}}\Delta t/2} \vert \rho(t)\rrangle + \mathrm{O}(L\Delta t^3) \notag\\
& = \sum_{\substack{\sigma_1 \ldots \sigma_L \\ \sigma_1' \ldots \sigma_L'}}
\begin{tikzpicture}[baseline={([yshift=-.5ex]current bounding box.center)},vertex/.style={anchor=base,
circle,fill=black!25,minimum size=18pt,inner sep=2pt}, scale=.9]
\draw [thick] (0,0) -- (11.,0.);
\draw [thick] (0,0) -- (0,3.5);
\draw [thick] (1,0) -- (1,3.5);
\draw [thick] (2,0) -- (2,3.5);
\draw [thick] (3,0) -- (3,3.5);
\draw [thick] (4,0) -- (4,3.5);
\draw [thick] (5,0) -- (5,3.5);
\draw [thick] (6,0) -- (6,3.5);
\draw [thick] (7,0) -- (7,3.5);
\draw [thick] (8,0) -- (8,3.5);
\draw [thick] (9.,0) -- (9,3.5);
\draw [thick] (10.,0) -- (10,3.5);
\draw [thick] (11.,0) -- (11,3.5);
\draw [fill=lightgray] (0,0.) circle [radius=0.2] node {};
\draw [fill=lightgray] (1,0.) circle [radius=0.2] node {};
\draw [fill=lightgray] (2,0.) circle [radius=0.2] node {};
\draw [fill=lightgray] (3,0.) circle [radius=0.2] node {};
\draw [fill=lightgray] (4,0.) circle [radius=0.2] node {};
\draw [fill=lightgray] (5,0.) circle [radius=0.2] node {};
\draw [fill=lightgray] (6,0.) circle [radius=0.2] node {};
\draw [fill=lightgray] (7,0.) circle [radius=0.2] node {};
\draw [fill=lightgray] (8,0.) circle [radius=0.2] node {};
\draw [fill=lightgray] (9.,0.) circle [radius=0.2] node {};
\draw [fill=lightgray] (10.,0.) circle [radius=0.2] node {};
\draw [fill=lightgray] (11.,0.) circle [radius=0.2] node {};
\draw [fill=evengate] (0.-.2, 1-.25) rectangle (3. + .2, 1.+ .25) ;
\draw [fill=evengate] (4.-.2, 1-.25) rectangle (7. + .2, 1.+ .25) ;
\draw [fill=evengate] (8.-.2, 1-.25) rectangle (11. + .2, 1.+ .25) ;
\draw [fill=oddgate] (2.-.2, 2-.25) rectangle (5. + .2, 2.+ .25) ;
\draw [fill=oddgate] (6.-.2, 2-.25) rectangle (9. + .2, 2.+ .25) ;
\draw [fill=evengate] (0.-.2, 3-.25) rectangle (3. + .2, 3.+ .25) ;
\draw [fill=evengate] (4.-.2, 3-.25) rectangle (7. + .2, 3.+ .25) ;
\draw [fill=evengate] (8.-.2, 3-.25) rectangle (11. + .2, 3.+ .25) ;
\draw [] (0.25, .4) node {$\sigma_1$};
\draw [] (1.25, .4) node {$\sigma_1'$};
\draw [] (2.25, .4) node {$\sigma_2$};
\draw [] (3.25, .4) node {$\sigma_2'$};
\draw [] (4.25, .4) node {$\sigma_3$};
\draw [] (5.25, .4) node {$\sigma_3'$};
\draw [] (6.25, .4) node {$\sigma_4$};
\draw [] (7.25, .4) node {$\sigma_4'$};
\draw [] (8.25, .4) node {$\sigma_5$};
\draw [] (9.25, .4) node {$\sigma_5'$};
\draw [] (10.25, .4) node {$\sigma_6$};
\draw [] (11.25, .4) node {$\sigma_6'$};
\draw[->,color=black,rounded corners=2mm, line width=2, >=stealth', dotted] (-0.5, 1) -- (11.5, 1) -| (11.5, 2) -|
(-0.5, 2) -- (-0.5, 3) -- (11.75, 3);
\end{tikzpicture} \:\: \notag\\ &\notag \\
& \qquad \qquad \qquad \times \vert \sigma_1 \sigma_1' \ldots \sigma_L \sigma_L' \rrangle + \mathcal{O}(L\Delta t^3) \notag
\end{align}
\caption{Dissipative evolution of the purified density matrix in MPS form by a single time step $\Delta t$ using a second order Suzuki-Trotter decomposition for the evolution operator represented by a consecutive application of four-site gates. The two colors mark the affiliation to one of the two parts of the Lindbladian, i.e. either $\mathcal{L}_{\mathrm{odd}}$ or $\mathcal{L}_{\mathrm{even}}$, and the arrow indicates the order of application.}
\label{fig:Methods_PurificationGates}
\end{figure*}
In analogy to the unitary closed system evolution (Eq.~\ref{eq:unitary_tMPS}), it is also possible to split the Lindbladian into two contributions
\begin{equation}
\mathds{L} = \mathds{L}_{\mathrm{odd}} + \mathds{L}_{\mathrm{even}},
\end{equation}
where $\mathds{L}_{\mathrm{odd}}$ ($\mathds{L}_{\mathrm{even}}$) covers the ket and bra sites connected by the odd (even) bonds. Again, the terms within $\mathds{L}_{\mathrm{odd/even}}$ commute, but $[\mathds{L}_{\mathrm{odd}}, \mathds{L}_{\mathrm{even}}]\neq 0$. Consequently, the dissipative evolution can also be approximated by the second order Suzuki-Trotter decomposition, which in this case results in a sequence of applications of four-site gates as shown in Fig.~\ref{fig:Methods_PurificationGates}. This fact raises the challenge, that a gate application causes a stronger increase of the bond dimension
\begin{equation}
\begin{tikzpicture}[baseline={([yshift=-.75ex]current bounding box.center)},vertex/.style={anchor=base,
circle,fill=black!25,minimum size=18pt,inner sep=2pt}, scale=.45]
\draw [thick] (0,0) -- (0,0.75);
\draw [thick] (1.5,0) -- (1.5,0.75);
\draw [thick] (3.,0) -- (3,0.75);
\draw [thick] (4.5,0) -- (4.5,0.75);
\draw [thick] (0,2) -- (0,4);
\draw [thick] (1.5,2) -- (1.5,4);
\draw [thick] (3.,2) -- (3,4);
\draw [thick] (4.5,2) -- (4.5,4);
\draw [thick, dotted] (0., 0.) -- (0., 4.);
\draw [thick, dotted] (1.5, 0.) -- (1.5, 4.);
\draw [thick, dotted] (3., 0.) -- (3., 4.);
\draw [thick, dotted] (4.5, 0.) -- (4.5, 4.);
\draw [thick] (-1,0) -- (5.5,0.);
\draw (0-0.5, 1.7) node {$\sigma_l^{\phantom{\dagger}}$};
\draw (1. + 0., 1.7) node {$\sigma_l'$};
\draw (3.5 + 0.25, 1.7) node {$\sigma_{l+1}^{\phantom{\dagger}}$};
\draw (5. + 0.25, 1.7) node {$\sigma_{l+1}'$};
\draw (-0.75,- 0.6)[blue] node {$D$};
\draw (0.75,- 0.6)[blue] node {$D$};
\draw (2.25,- 0.6)[blue] node {$D$};
\draw (3.75,- 0.6)[blue] node {$D$};
\draw (5.25,- 0.6)[blue] node {$D$};
\draw (.35, 0.7) [blue] node {$d$};
\draw (1.85, 0.7) [blue] node {$d$};
\draw (3.35, 0.7) [blue] node {$d$};
\draw (4.85, 0.7) [blue] node {$d$};
\draw (.35, 4) [blue] node {$d$};
\draw (1.85, 4) [blue] node {$d$};
\draw (3.35, 4) [blue] node {$d$};
\draw (4.85, 4) [blue] node {$d$};
\draw [fill=lightgray] (0.-.2, 3-.5) rectangle (4.5 + .2, 3.+ .5);
\draw [fill=lightgray] (0,0.) circle [radius=0.2] node {};
\draw [fill=lightgray] (1.5,0.) circle [radius=0.2] node {};
\draw [fill=lightgray] (3,0.) circle [radius=0.2] node {};
\draw [fill=lightgray] (4.5,0.) circle [radius=0.2] node {};
\end{tikzpicture}
\overset{SVD}{\longrightarrow}
\begin{tikzpicture}[baseline={([yshift=-.75ex]current bounding box.center)},vertex/.style={anchor=base,
circle,fill=black!25,minimum size=18pt,inner sep=2pt}, scale=.55]
\draw [thick] (0,0) -- (0,0.75);
\draw [thick] (1.5,0) -- (1.5,0.75);
\draw [thick] (3.,0) -- (3,0.75);
\draw [thick] (4.5,0) -- (4.5,0.75);
\draw [thick] (-1,0) -- (5.5,0.);
\draw (0-0., 1.2) node {$\sigma_l^{\phantom{\dagger}}$};
\draw (1.5 - 0.,1.2) node {$\sigma_l'$};
\draw (3. - 0., 1.2) node {$\sigma_{l+1}^{\phantom{\dagger}}$};
\draw (4.5 - 0., 1.2) node {$\sigma_{l+1}'$};
\draw (-0.75,- 0.6)[blue] node {$D$};
\draw (0.75,- 0.6)[blue] node {$d^2D$};
\draw (2.25,- 0.6)[blue] node {$d^4D$};
\draw (3.75,- 0.6)[blue] node {$d^2D$};
\draw (5.25,- 0.6)[blue] node {$D$};
\draw (.25, 0.5) [blue] node {$d$};
\draw (1.75, 0.5) [blue] node {$d$};
\draw (3.25, 0.5) [blue] node {$d$};
\draw (4.75, 0.5) [blue] node {$d$};
\draw [fill=lightgray] (0,0.) circle [radius=0.2] node {};
\draw [fill=lightgray] (1.5,0.) circle [radius=0.2] node {};
\draw [fill=lightgray] (3,0.) circle [radius=0.2] node {};
\draw [fill=lightgray] (4.5,0.) circle [radius=0.2] node {};
\end{tikzpicture}.
\end{equation}
The dimension of the individual indices is marked in blue color, revealing that the bond dimension of a purified state grows from $D$ to up to $d^4D$ by applying a time evolution gate. This requires a stronger truncation to compress the bond dimension to the original size.
Another important feature is that the use of quantum number conserving codes for the computation of the dissipative evolution is only possible if the Lindbladian is subject to a strong symmetry \cite{BucaProsen2012,AlbertJiang2014}, i.e. not only the Hamilton operator but also every single jump operator of the model respects the conservation law. In the cases considered here, this only applies to $\mathcal{D}_1$ (Eq.\ref{eq:D1}), as the jump operator $S^-_c$ in $\mathcal{D}_2$ (Eq.\ref{eq:D2}) does not conserve the total magnetization.
\subsubsection{Calculating expectation values within the purification approach}
Provided with the time evolved purified density matrix, we are left with the task to calculate expectation values of observables to extract information about the system. The trace relation for the expectation value of an operator $A$ translates to a scalar product
\begin{equation}
\expval{A} = \tr(\rho A) = \llangle \mathds{1} \vert A\otimes I \vert \rho \rrangle,
\text{ with} \:\:\: \vert \mathds{1}\rrangle = \bigotimes_{l=1}^L \sum_{\sigma} \vert \sigma\, \sigma \rangle_{l}=\bigotimes_{l=1}^{L} \left( \ket{\uparrow\uparrow}_l+ \ket{\downarrow\downarrow}_l \right).
\label{eq:Methods_TraceVector}
\end{equation}
Here the first and second spin in $\vert \sigma\, \sigma \rangle_{l}$ denote the spin at site $l$ for the ket and the bra part in the purified notation respectively.
The state $\vert \mathds{1}\rrangle$ is the purification of the unnormalized infinite temperature state \cite{CuiBanuls2015}. Unfortunately, the possibility to be able to encode this state as a product state as shown in Eq.~\ref{eq:Methods_TraceVector} gets lost, when using good quantum numbers, where the vector of the purified density matrix cannot be separated into contributions of single sites anymore. As a result, this state can exhibit a potentially large bond dimension. Another obstacle becomes apparent when exploiting good quantum numbers in the algorithm is that $\vert \mathds{1}\rrangle$ spans over all symmetry blocks, so that a restriction to a certain quantum number sector makes a manual selection of basis states fulfilling this condition very cumbersome. Here it is helpful to realize, that the purification procedure is subject to a gauge degree of freedom. The measured expectation value is invariant under local unitary transformations of the states \cite{Schollwoeck2011}. This fact can be used to construct a favorable representation of the bra space of the constituents of $\vert \mathds{1}\rrangle$, defined by
\begin{equation}
\vert \sigma \,\sigma\rangle_l \longrightarrow \left(I\otimes U\right)\vert \sigma\,\sigma \rangle_l.
\end{equation}
To this end, the transformation $U$ is chosen such, that the quantum number of the full initial state is equally distributed over local pairs of ket and bra states. For example, if we consider the $M_{\mathrm{tot}}=0$ symmetry sector in the Lindblad model using the dissipator $\mathcal{D}_1$, one possible transformation is the Pauli matrix in $x$-direction $U=\sigma^x$ on the bra sites, so that we can rewrite the state $\vert \mathds{1}\rrangle$ as a product of local spin pairs
\begin{equation}
\vert \mathds{1} \rrangle \longrightarrow \bigotimes_{l=1}^L \sum_{\sigma} \vert \sigma\, \bar{\sigma} \rangle_{l}=\bigotimes_{l=1}^{L} \left( \ket{\uparrow\downarrow}_l+ \ket{\downarrow\uparrow}_l \right)
\label{eq:trace_state_spinhalf}
\end{equation}
where $\bar{\sigma}=-\sigma$. This transformation automatically guarantees the restriction of the trace generating state $\vert \mathds{1}\rrangle$ to the symmetry sector selected by the initial state. With this, the problems of selecting basis states respecting the quantum number conservation as well as the large bond dimension of an MPS representation of $\ket{\mathds{1}}$ are both solved. Also the initial state and the Lindbladian gates need to be adapted to be consistent with this gauge choice. With $\mathds{U} = \bigotimes_{l=1}^L \left(I \otimes U\right) $, the transformation relations are
\begin{align}
\mathds{L} & \:\:\longrightarrow\:\: \mathds{U}^\dagger \mathds{L} \mathds{U} \notag\\
\vert \rho (t=0)\rrangle & \:\:\longrightarrow\:\: \mathds{U}\vert \rho(t=0) \rrangle.
\label{eq:Methods_PurificationLTransform}
\end{align}
Supposing that the observable can be brought to an efficient tensor representation, we can then proceed to calculate the expectation value. In the exemplary and important case of a local observable the measurement is carried out by the tensor contractions outlined in Fig.\ref{fig:Purification_ExpecationValue}.
As explained before only the case with dissipator $\mathcal{D}_{1}$ conserves the total magnetization. Thus we use this transformation in order to consider the good quantum numbers in the simulation. Applying this transformation to the Lindbladian leaves the Hamiltonian contribution unchanged and only changes the sign of the first part of the dissipator $\mathcal{D}_{1}$. The observables are calculated in the transformed basis (see Fig.~\ref{fig:Purification_ExpecationValue}).
\begin{figure*}
\begin{equation*}
\expval{A_l} = \llangle \mathds{1} \vert A_l \vert \rho(t)\rrangle = \begin{tikzpicture}[baseline={([yshift=-1.75ex]current bounding box.center)},vertex/.style={anchor=base,
circle,fill=black!25,minimum size=18pt,inner sep=2pt}, scale=.85]
\draw [thick, dotted] (0,0) -- (11.,0.);
\draw [thick] (0,0) -- (1.5,0.);
\draw [thick] (2.5,0) -- (8.5,0.);
\draw [thick] (9.5,0) -- (11.,0.);
\draw [thick, dotted] (0,0) -- (0,3.5);
\draw [thick, dotted] (1,0) -- (1,3.5);
\draw [thick, dotted] (3,0) -- (3,3.5);
\draw [thick, dotted] (4,0) -- (4,3.5);
\draw [thick, dotted] (5,0) -- (5,3.5);
\draw [thick, dotted] (6,0) -- (6,3.5);
\draw [thick, dotted] (7,0) -- (7,3.5);
\draw [thick, dotted] (8,0) -- (8,3.5);
\draw [thick, dotted] (10,0) -- (10,3.5);
\draw [thick, dotted] (11,0) -- (11,3.5);
\draw [thick] (0,0) -- (0,.5);
\draw [thick] (1,0) -- (1,.5);
\draw [thick] (3,0) -- (3,.5);
\draw [thick] (4,0) -- (4,.5);
\draw [thick] (5,0) -- (5,.5);
\draw [thick] (6,0) -- (6,.5);
\draw [thick] (7,0) -- (7,.5);
\draw [thick] (8,0) -- (8,.5);
\draw [thick] (10,0) -- (10,.5);
\draw [thick] (11,0) -- (11,.5);
\draw [thick] (0,3.25) -- (0,2.5);
\draw [thick] (1,3.25) -- (1,2.5);
\draw [thick] (3,3.25) -- (3,2.5);
\draw [thick] (4,3.25) -- (4,2.5);
\draw [thick] (5,3.25) -- (5,2.5);
\draw [thick] (6,3.25) -- (6,2.5);
\draw [thick] (7,3.25) -- (7,2.5);
\draw [thick] (8,3.25) -- (8,2.5);
\draw [thick] (10,3.25) -- (10,2.5);
\draw [thick] (11,3.25) -- (11,2.5);
\draw [thick] (5,1.5-0.65) -- (5,1.5+0.65);
\draw [fill=lightgray] (0,0.) circle [radius=0.2] node {};
\draw [fill=lightgray] (1,0.) circle [radius=0.2] node {};
\draw [fill=lightgray] (3,0.) circle [radius=0.2] node {};
\draw [fill=lightgray] (4,0.) circle [radius=0.2] node {};
\draw [fill=lightgray] (5,0.) circle [radius=0.2] node {};
\draw [fill=lightgray] (6,0.) circle [radius=0.2] node {};
\draw [fill=lightgray] (7,0.) circle [radius=0.2] node {};
\draw [fill=lightgray] (8,0.) circle [radius=0.2] node {};
\draw [fill=lightgray] (10,0.) circle [radius=0.2] node {};
\draw [fill=lightgray] (11,0.) circle [radius=0.2] node {};
\draw [] (2., 1.5) node {$\ldots$};
\draw [] (9., 1.5) node {$\ldots$};
\draw [] (0.25, .4) node {$\sigma_1$};
\draw [] (1.25, .4) node {$\sigma_1'$};
\draw [] (3.5, .4) node {$\sigma_{l-1}$};
\draw [] (4.5, .4) node {$\sigma_{l-1}'$};
\draw [] (5.25, .4) node {$\sigma_l$};
\draw [] (6.25, .4) node {$\sigma_l'$};
\draw [] (7.5, .4) node {$\sigma_{l+1}$};
\draw [] (8.5, .4) node {$\sigma_{l+1}'$};
\draw [] (10.3, .4) node {$\sigma_L$};
\draw [] (11.3, .4) node {$\sigma_L'$};
\draw [fill=lightgray] (0.5, 3.25) circle [x radius=.95, y radius=.6] node {{\small$\sum\vert \sigma\,\bar \sigma\rangle_1$}};
\draw [fill=lightgray] (3.5, 3.25) circle [x radius=1.03, y radius=.6] node {{\small$\sum\vert \sigma\,\bar \sigma\rangle_{l-1}$}};
\draw [fill=lightgray] (5.5, 3.25) circle [x radius=.87, y radius=.6] node {{\small$\sum\vert \sigma\,\bar \sigma\rangle_{l}$}};
\draw [fill=lightgray] (7.5, 3.25) circle [x radius=1.03, y radius=.6] node {{\small$\sum\vert \sigma\,\bar \sigma\rangle_{l+1}$}};
\draw [fill=lightgray] (10.5, 3.25) circle [x radius=.95, y radius=.6] node {{\small$\sum\vert \sigma\,\bar \sigma\rangle_{L}$}};
\draw [fill=neutral] (5.-.4, 1.5-.4) rectangle (5. + .4, 1.5+ .4) ;
\draw [] (5., 1.5) node {$A_l$};
\end{tikzpicture} \:\:
\end{equation*}
\caption{Visualization of the computation of local expectation values using the contraction of the MPS with a local operator $A_l$ and the trace generating state $\ket{\mathds{1}}$ in the transformed basis (see Eq.~\ref{eq:trace_state_spinhalf}) represented by a product of spin pairs $\sum_{\sigma}\vert \sigma\,\bar \sigma\rangle_{l}$ covering associated sites corresponding to the ket and bra part of the density matrix at site $l$. The density matrix is also calculated in the transformed basis as Eq.~\ref{eq:Methods_PurificationLTransform}. Note that the observable $A_l$ does not change with this transformation. }
\label{fig:Purification_ExpecationValue}
\end{figure*}
\subsection{Monte-Carlo wave function method}
\label{sec:methods_MCWF}
Instead of evolving the full density matrix, an alternative way is to compute the evolution of wave function trajectories in the original Hilbert space \cite{CarmichaelBook,DalibardMolmer1992,MolmerDalibard1993}. This is at the expense of the need for a sampling over many realizations due to the presence of stochastic processes, originating from the action of the environment on the system. This approach, known as the unraveling of the master equation, can be realized by piece-wise deterministic jump processes, where the deterministic time evolution of a state is interrupted by the application of jump operators. The deterministic evolution is performed with regard to an effective Hamiltonian
\begin{equation}\label{eq:Heff}
H_{\mathrm{eff}} = H_{\mathrm{XXZ}} - \frac{i \hbar\gamma}{2}\sum_l L_l^\dagger L_l,
\end{equation}
where $\{L_l\}$ are the the jump operators of the respective model, i.e. $\{L_l\} = \{S^z_j, j=1, 2, \ldots, L\}$ for $\mathcal{D}_1$ and $\{L_l\}=\{S^-_c, c=\text{central site}\}$ for $\mathcal{D}_2$. As this effective Hamiltonian is not Hermitian, the corresponding evolution is non-unitary, resulting in a decay of the norm of the state over time. The creation of a single time-evolved trajectory sample can be summarized as follows:
\begin{enumerate}
\item Define the initial state, which is either a pure state or is selected according to the probability weights in a mixture of states.
\item \label{it:MCWF_drawRN} Draw a random number $\eta \in [0, 1)$.
\item \label{it:MCWF_timeevolutionstep}Evolve the state under $H_{\mathrm{eff}}$ by a sequence of small time steps $\Delta t$
until $\langle\psi(t)\ket{\psi(t)} \le \eta$.
\item \begin{enumerate}
\item Draw a jump operator $L_{l'}$ according to the probability distribution $\frac{p_{l'}}{\sum_{l} p_l}$, with
\begin{equation}
p_l = \Gamma_l \bra{\psi(t)}L^\dagger_l L_l^{\phantom{\dagger}}\ket{\psi(t)},
\end{equation}
\item apply the selected operator $L_{l'}$ to the state and renormalize it
\begin{equation}
\ket{\psi(t)}=\frac{L_{l'} \ket{\psi(t)}}{\sqrt{ \bra{\psi(t)}L_{l'}^\dagger L_{l'}^{\phantom{\dagger}} \ket{\psi(t)}}}.
\end{equation}
\end{enumerate}
\item Iterate from \ref{it:MCWF_drawRN} until the final time is reached.
\end{enumerate}
Performing the average over many trajectories generated by this scheme ultimately yields an estimate of the density matrix that is accurate up to the first order in the time step. The density matrix can be approximated by a Monte-Carlo (MC) average over a finite number $R$ of trajectory samples $\ket{\psi_r(t)}$
\begin{equation}
\rho
\approx \frac{1}{R} \sum_{r=1}^R \ket{\psi_r(t)}\bra{\psi_r(t)}
\end{equation}
giving the technique name of Monte-Carlo wave function (MCWF) method. Similarly, the expectation values of observables can also be evaluated as
\begin{equation}
\llangle \hat O (t) \rrangle \approx \frac{1}{R}\sum_{r=1}^R\bra{\psi_r(t)} \hat O \ket{\psi_r(t)},
\end{equation}
where $\llangle \ldots \rrangle$ denotes the MC average and the stochastic accuracy of the independent sampling approach is given by the time-dependent standard deviation of the mean
\begin{eqnarray}
\sigma_{\mathrm{mean}}(\hat O(t)) = \sqrt{\frac{1}{R(R-1)} \sum_{r=1}^R \left( \bra{\psi_r(t)} \hat O \ket{\psi_r(t)} - \llangle \hat O(t) \rrangle \right)^2}.
\end{eqnarray}
\begin{figure}[tb]
\centering
\includegraphics[width=0.6\textwidth]{TwoJumpProbability_Poisson.pdf}
\caption{Probability distribution for two or more jump events happening in the same time step to estimate the necessary time step for the model with dissipator $\mathcal{D}_1$ and different system sizes and dissipation strengths. The vertical dashed lines mark the time-steps typically chosen in the calculations and the horizontal dotted lines are the corresponding probability for $L=16$. }
\label{fig:twojump_probability}
\end{figure}
\begin{figure}[tb]
\centering
\includegraphics[width=0.6\textwidth]{MCWF_Average.pdf}
\caption{Dissipative evolution of local equal-time correlation functions $C_d(t)$ (top) and the von Neumann entropy $S_{\mathrm{vN}}$ (bottom) of a spin-$\nicefrac{1}{2}$ chain initially prepared in the Neel state under the effect of bulk dephasing $\mathcal{D}_1$. We show both, the Monte-Carlo average with statistical error bars (not visible, since they are below the line width) and the measured values for a selection of stochastically sampled trajectories for a chain of length $L=32$, dissipation strength $\hbar\gamma/J_x=1$, spin anisotropy $J_z/J_x=1$, a time step $\hbar \Delta t/J_x=0.05$, a truncation goal $\varepsilon=10^{-12}$, a maximal bond dimension $D=100$ and $10^{4}$ samples for the MC average. }
\label{fig:mcwf_average}
\end{figure}
The random sampling of an application time in steps \ref{it:MCWF_drawRN} and \ref{it:MCWF_timeevolutionstep} of the algorithm is equivalent to drawing a waiting time $\tau$ until the next jump occurs according to the distribution
\begin{equation}\label{eq:waittime}
P(t, \tau) = 1 - \langle \psi(t)\vert \ee^{i H_{\mathrm{eff}} \tau/\hbar} \ee^{-i H_{\mathrm{eff}}\tau/\hbar} \vert \psi(t)\rangle.
\end{equation}
This quantity is important for the convergence of the method as it can be used to determine a sufficiently small time step when simulating the non-unitary evolution under the effective Hamiltonian. It is important to choose a time step small enough so that the probability of more than one jump event happening in the span of a time step is negligible.
For the system with dissipator $\mathcal{D}_1$ the imaginary part of the effective Hamiltonian (Eq.~\ref{eq:Heff}) is equal to $-\frac{\gamma}{2}\sum_l S^z_l S^z_l=-\frac{\gamma L}{8}$. Thus the waiting time distribution (Eq.~\ref{eq:waittime}) simplifies to
\begin{equation}
P(t, \tau) = 1 - \ee^{-\gamma L \tau/ 4}.
\end{equation}
The waiting time distribution is the probability of having a jump in the interval $[t, t+\tau)$ which is independent of the initial time $t$ here.
The probability to have two or more jumps in this interval is $P_{n>2}(\tau)
1-\ee^{-\gamma L \tau/ 4}-\frac{\gamma L \tau} {4}\ee^{-\gamma L \tau/ 4}$.
In Fig.~\ref{fig:twojump_probability} we show the probability of two or more jumps taking place in one time step for the model with dissipator $\mathcal{D}_1$.
When a good value for $\Delta t$ has been found for one parameter configuration, the dissipation strength
and the system size are crucial for estimating a suitable time step for another set-up.
For the determination of a single trajectory
applying the introduced $t$MPS algorithm Eq.~\ref{eq:unitary_tMPS} offers a promising solution for computing the deterministic part of the evolution (see \cite{Daley2014} and references therein). Using this procedure, we can access the dissipative time evolution of a spin-$\nicefrac{1}{2}$ chain under the action of the Lindbladian. It is noteworthy that here the use of symmetries to reduce the computational complexity only requires the conservation of the associated quantum numbers by the effective Hamiltonian. Consequently, in contrast to the purification approach, quantum number conserving codes can be used for the simulation of both dissipators $\mathcal{D}_1$ and $\mathcal{D}_2$.
In Fig.~\ref{fig:mcwf_average} we present for the example of the dissipator $\mathcal{D}_1$ the evolution of the local equal-time correlation function
\begin{equation}
C_d(t) = \langle S^z_{L/2} S^z_{L/2+d}\rangle (t),
\end{equation}
and the von Neumann entropy $S_{\mathrm{vN}}$ starting from an alternating spin configuration $\ket{\psi_{\mathrm{Neel}}} = \ket{\uparrow \downarrow \uparrow \downarrow\dots}$, known as the classical Neel state. For both quantities we show the Monte-Carlo average as well as a selection of measurements of individual trajectory samples. Within the MPS, the von Neumann entropy is connected to the required maximally allowed bond dimension in order to obtain a good description of the state. Here, we find that during the evolution a spreading around the mean develops so that the maximum entanglement in individual samples can be significantly larger than the mean value suggests. The relationship between entropy and the necessary bond dimension is highlighted further for a single trajectory created from the same random seed for different maximal bond dimensions $D$ in Fig.~\ref{fig:entropy_singlesample}. It is evident that calculations with a fixed maximum value for the bond dimension can only provide a sufficiently accurate representation of the entanglement up to a certain time.
\begin{figure}[tb]
\centering
\includegraphics[width=0.6\textwidth]{Entropy_Single_trajectory.pdf}
\caption{Evolution of the von Neumann entropy $S_{\mathrm{vN}}$ of a single trajectory sample with different maximal values for the bond dimension for a chain of length $L=32$, dissipation strength $\hbar\gamma/J_x=1$, spin anisotropy $J_z/J_x=1$, time step $\hbar \Delta t/J_x=0.05$ and truncation goal $\varepsilon=10^{-12}$.}
\label{fig:entropy_singlesample}
\end{figure}
\section{Methods for the computation of two-time correlation functions in open quantum systems}
\label{sec:two_time_methods}
We will discuss the different concepts for determining two-time correlation functions in open quantum systems with the methods introduced in Sec.~\ref{sec:openMPS}, which will later form the basis for the comparison. More precisely, the goal is to compute correlation functions of the form
\begin{equation}
g(t_2, t_1) = \langle B(t_2) A(t_1)\rangle,
\end{equation}
where the two operators are applied at different times during the course of a dissipative time evolution.
\subsection{Two-time correlations within the purification approach}
The purification approach can be straight-forwardly extended in order to determine two time correlation functions\cite{WolffKollath2019}. After purifying the initial state, the reshaped density matrix is evolved until time $t_1$, where the operator $A$ is applied, followed by an evolution to $t_2$, where an ordinary measurement of $B$ is carried out
\begin{align}
\expval{B(t_2) A(t_1)} = \llangle \mathds{1} \vert \left(B\otimes I\right) \ee^{\mathds{L}(t_2-t_1)}\left(A\otimes I\right)\ee^{\mathds{L}t_1}\vert \rho(t=0)\rrangle.
\end{align}
Only a single time evolution needs to be calculated. Nevertheless, the fact of the evolution taking place in a doubled Hilbert space, can potentially result in the need for a large bond dimension.
\subsection{Stochastic sampling: two approaches to two-time correlations}
In the following we introduce two different approaches to evaluate two-time correlation functions in the framework of Monte-Carlo wave functions. The first step is to evolve an initially prepared state, which is in the case of a mixed state drawn according to the weights in the density matrix, up to the application time $t_1$ of the first operator. This is performed using the unraveling scheme of piece-wise deterministic jump processes. For each trajectory one defines the state $\ket{\phi(t_1)}\equiv A \ket{\psi(t_1)}$. In principle it seems that we are left with the task of calculating the dissipative time-evolution of the two states $\ket{\phi(t_1)}$ and $\ket{\psi(t_1)}$ up to time $t_2$ and then the expectation value with the operator $B$, i.e.~
\begin{align}
g_2(t_2, t_1) &= \bra{\psi(t_2)} B \ket{\phi(t_2)}.
\label{eq:two_time_function_Heisenberg}
\end{align}
However, while this expression is well-defined for states of a closed system, the transfer to a stochastic sampling approached is more involved.
In the next two sections we outline two methods which are well defined for the stochastic sampling approach. Subsequently, we compare the efficiency of both concepts
\begin{figure}[tb!]
\centering
\begin{tikzpicture}[baseline={([yshift=-.5ex]current bounding box.center)},vertex/.style={anchor=base,
circle,fill=black!25,minimum size=18pt,inner sep=2pt}, ->,>=stealth', scale=1.]
\draw [-] (2, 0) node {$\vert \psi(t_1)\rangle$};
\draw [thick, -] (2.7,0) -- (3.5,0);
\draw [-] (3.6, 1) node {split trajectories};
\draw [ thick, ->] (3.4 , 0.) to[out=0, in=180] (4.5, 0.3);
\draw [ thick, ->] (3.4 , 0.) to[out=0, in=180] (4.5, -.3);
\draw [-] (5.4, 0.3) node {$\vert\psi(t_1)\rangle$};
\draw [-] (5.4, -0.3) node {$\vert\phi(t_1)\rangle$};
\draw [thick, ->] (6.1, .3) -- (8.2, .3);
\draw [thick, ->] (6.1, -.3) -- (8.2, -.3);
\draw [-] (8.9, 0.3) node {$\vert\psi(t_2)\rangle$};
\draw [-] (8.9, -0.3) node {$\vert\phi(t_2)\rangle$};
\draw [fill=red] (6.3, -.6) rectangle (6.4, .6);
\draw [fill=red] (6.6, -.6) rectangle (6.7, .6);
\draw [fill=red] (7.5, -.6) rectangle (7.6, .6);
\draw [-, red] (7.2, 1) node {joint jumps $L_j$};
\end{tikzpicture}
\caption[Sketch of the creation of one MCWF sample for two-time correlations following \cite{BreuerPetruccione1997}]{Sketch of the creation of one sample for the computation of two-time correlators following \cite{BreuerPetruccione1997}. After the evolution up to $t_1$ the trajectory is copied, followed by a time span characterized by an independent deterministic non-unitary evolution interrupted by joint jump operator applications.}
\label{fig:OpenMPS_BreuerTwotime}
\end{figure}
\subsubsection{Joint evolution of two states following Breuer et al. \cite{BreuerPetruccione1997}}
The first idea, developed by Breuer et al. \cite{BreuerPetruccione1997}, uses a doubling of the Hilbert space at time $t_1$ by introducing for each trajectory the vector
\begin{equation}
\ket{\Theta(t_1)} = \begin{pmatrix} \ket{\psi(t_1)} \\ \ket{\phi(t_1)} \end{pmatrix}.
\end{equation}
As the matrix $\tilde\rho(t_1) = \ket{\Theta(t_1)}\bra{\Theta(t_1)}$ again fulfills all properties of a physical density matrix, the task is reduced to recover the time dependence in Eq.~\ref{eq:two_time_function_Heisenberg}.
By defining a new Hamiltonian operator and new jump operators acting on the doubled space
\begin{equation}
\tilde H = \begin{pmatrix} H & 0 \\ 0 & H \end{pmatrix} \text{ and } \tilde L_l = \begin{pmatrix} L_l & 0 \\ 0 & L_l \end{pmatrix},
\end{equation}
and using this in a Lindblad-type equation with Hamiltonian $\tilde H$ and jump operators $\tilde L_l$, we arrive at Lindblad equations for all matrix blocks \cite{BreuerPetruccione1997}. As a result, we can apply the same unraveling approach as before on the doubled space to compute two-time correlation functions. Since the operators do not couple the two subspaces, a separate time evolution of $\ket{\psi(t)}$ and $\ket{\phi(t)}$ is possible, where only the application time and the selection of jump operators is determined based on the joint evolution (see Fig.~\ref{fig:OpenMPS_BreuerTwotime}). The algorithm to create a single trajectory can be condensed to the following steps:
\begin{enumerate}
\item Initialize the wave function $\ket{\psi(t_0)}$ in the original Hilbert space and evolve it until $t_1$ using the introduced piece-wise deterministic process.
\item Make a copy of the state at time $t_1$, apply the operator $A$ to it $\ket{\phi(t_1)}\equiv A\ket{\psi(t_1)}$ and define a state in the doubled space $\ket{\Theta(t_1)} \equiv (\ket{\psi(t_1)}, \ket{\phi(t_1)})^T$, where $T$ denotes the transpose. To ensures the transfer of accumulated jump probability at $t_1$ to the doubled space we normalize this state to $\ket{\tilde{\Theta}(t_1)}\equiv \frac{1}{\sqrt{\Omega}}\ket{\Theta(t_1)} \equiv (\ket{\tilde{\psi}(t_1)}, \ket{\tilde{\phi}(t_1)})^T$ with the normalization factor $\Omega=\bra{\Theta(t_1)} \Theta(t_1)\rangle / \bra{\psi(t_1)} \psi(t_1)\rangle$. Here the norm of the new state is the same as of the initial state ($\bra{\tilde{\Theta}(t_1)} \tilde{\Theta}(t_1)\rangle = \bra{\psi(t_1)} \psi(t_1)\rangle$).
\item Evolve both states independently under the effective non-Hermitian Hamiltonian while sampling jumps simultaneously according to the joint loss of norm of $\ket{\tilde{\Theta}(t)}$.
\item Repeat the procedure until the time $t_2$ is reached where one obtains $\ket{\tilde{\Theta}(t_2)} = \begin{pmatrix} \ket{\tilde{\psi}(t_2)} \\ \ket{\tilde{\phi}(t_2)} \end{pmatrix}$. Then, use the components of this state in order to measure the two-time correlation function by calculating the full overlap
$$\langle B(t_2)A(t_1)\rangle =\frac{\bra{\tilde{\psi}(t_2)} B \ket{\tilde{\phi}(t_2)}}{\bra{\tilde{\Theta}(t_2)} \tilde{\Theta}(t_2)\rangle }\Omega.$$
\end{enumerate}
This method can be further extended to multi-time correlation functions by additional operator applications between the first and the final application time \cite{BreuerPetruccione1997}.
\subsubsection{Separate evolution of four trajectories following M\o lmer et al. \cite{MolmerDalibard1993}}
An alternative strategy, established by M\o lmer et al. \cite{MolmerDalibard1993}, uses the quantum regression theorem \cite{Lax1963, BreuerPetruccione2002} to represent the time dependence of two-time correlation functions in terms of the evolution of quantities, which only involve equal-time measurements. The proof relies on the quantum regression theorem which states that if the evolution of equal-time observables for the set of operators $\{B_i\}$ is described by a closed set of differential equations
\begin{equation}
\frac{\dd }{\dd t_1} \expval{B_i(t_1)} = \sum_j G_{ij} \expval{B_j(t_1)},
\end{equation}
the two-time correlations are generated by the same kernel $G$ as
\begin{equation}
\frac{\dd }{\dd (t_2-t_1)} \expval{B_i(t_2) A(t_1)} = \sum_{j} G_{ij} \expval{B_j(t_2) A(t_1)}.
\end{equation}
The idea of this approach relies on constructing four new states at time $t_1$ for each trajectory such that a combination of the expectation values of the operator applied at $t_2$ for each of these state gives the corresponding two-time correlation. To this end, the four states are defined by applying the first operator $A$ \cite{MolmerDalibard1993},
\begin{align}
\ket{\chi_R^\pm(t_1)} = \frac{1}{\sqrt{\mu^\pm_{R}}} (I \pm A) \ket{\psi(t_1)}, \notag \\
\ket{\chi_I^\pm(t_1)} = \frac{1}{\sqrt{\mu^\pm_{I}}} (I \pm i A) \ket{\psi(t_1)},
\label{eq:4states_Molmer}
\end{align}
where for each trajectory $\ket{\psi(t_1)}$ the state at $t_1$ is obtained by the usual unraveling scheme. Further, $\mu^\pm_{R}$ and $\mu^\pm_{I}$ normalize the respective states. These new states evolve independently from time $t_1$ to $t_2$ with the same unraveling scheme (see Fig.~\ref{fig:OpenMPS_DaleyTwotime}). By properly combining the expectation values of the second operator $B$ for each state the two-time correlation functions can be recovered \cite{MolmerDalibard1993}
\begin{align}
\aver{ B(t_2) A(t_1)} &=\frac{1}{4} \left[ \mu_{R}^+ \bra{\chi^+_R(t_2)} B\ket{\chi^+_R(t_2)}- \mu_{R}^- \bra{\chi^-_R(t_2)} B\ket{\chi^-_R(t_2)} \right. \notag \\
& \qquad - i \mu_{I}^+ \bra{\chi^+_I(t_2)} B\ket{\chi^+_I(t_2)} +i \mu_{I}^- \bra{\chi^-_I(t_2)} B\ket{\chi^-_I(t_2)} \left. \right].
\end{align}
Consequently, it is possible to access the two-time correlation functions by evolving the four states from Eq.~\ref{eq:4states_Molmer} separately. In contrast to the MCWF technique of the doubled Hilbert space, also the jump sampling procedure is completely independent so that it is possible to compute the four trajectories after $t_1$ in parallel. In addition to the average over the four states, the average over many trajectories from the initial time to $t_1$ and from time $t_1$ to $t_2$ needs to be taken. A disadvantage of this approach using four different state is, however, that it does not offer a straight-forward extension to multi-time correlation functions.
\begin{figure}[tb!]
\centering
\begin{tikzpicture}[baseline={([yshift=-.5ex]current bounding box.center)},vertex/.style={anchor=base,
circle,fill=black!25,minimum size=18pt,inner sep=2pt}, ->,>=stealth']
\draw [-] (2, 0) node {$\vert \psi(t_1)\rangle$};
\draw [thick, -] (2.7,0) -- (3.5,0);
\draw [-] (3.6, 1.5) node {split trajectories};
\draw [ thick, ->] (3.5 , 0.) to[out=20, in=180] (4.5, .75);
\draw [ thick, ->] (3.45 , 0.) to[out=0, in=180] (4.5, 0.25);
\draw [ thick, ->] (3.45 , 0.) to[out=0, in=180] (4.5, -.25);
\draw [ thick, ->] (3.5 , 0.) to[out=-20, in=180] (4.5, -.75);
\draw [-] (5.4, 0.75) node {$\vert\chi_R^+(t_1)\rangle$};
\draw [-] (5.4, 0.25) node {$\vert\chi_R^-(t_1)\rangle$};
\draw [-] (5.4, -0.25) node {$\vert\chi_I^+(t_1)\rangle$};
\draw [-] (5.4, -0.75) node {$\vert\chi_I^-(t_1)\rangle$};
\draw [thick, ->] (6.1, .75) -- (8.2, .75);
\draw [thick, ->] (6.1, .25) -- (8.2, .25);
\draw [thick, ->] (6.1, -.25) -- (8.2, -.25);
\draw [thick, ->] (6.1, -.75) -- (8.2, -.75);
\draw [fill=blue] (6.3, .5) rectangle (6.4, 1.);
\draw [fill=blue] (7.3, .5) rectangle (7.4, 1.);
\draw [fill=blue] (6.9, 0.) rectangle (7., .5);
\draw [fill=blue] (7.5, 0.) rectangle (7.6, .5);
\draw [fill=blue] (6.2, -0.5) rectangle (6.3, .0);
\draw [fill=blue] (6.6, -0.5) rectangle (6.7, .0);
\draw [fill=blue] (7.7, -0.5) rectangle (7.8, .0);
\draw [fill=blue] (6.8, -1.) rectangle (6.9, -.5);
\draw [fill=blue] (7.5, -1.) rectangle (7.6, -.5);
\draw [-] (9, 0.75) node {$\vert\chi_R^+(t_2)\rangle$};
\draw [-] (9., 0.25) node {$\vert\chi_R^-(t_2)\rangle$};
\draw [-] (9., -0.25) node {$\vert\chi_I^+(t_2)\rangle$};
\draw [-] (9., -0.75) node {$\vert\chi_I^-(t_2)\rangle$};
\draw [-, blue] (7.2, 1.5) node {independent evolution};
\end{tikzpicture}
\caption[Sketch of the creation of one MCWF sample for two-time correlations following M\o lmer et al. \cite{MolmerDalibard1993}]{Sketch of the creation of one sample for the computation of two-time correlators following Ref. \cite{MolmerDalibard1993}. The initial dynamics up to time $t_1$ is followed by the splitting of the wave function into four trajectories from which the two-time expectation value at $t_1$ can be reconstructed. Then each sub-trajectory is evolved separately.}
\label{fig:OpenMPS_DaleyTwotime}
\end{figure}
There are two special cases, defined by the properties of the operators $A$ and $B$ of the correlation function. If the operator at time $t_1$ is Hermitian ($A^\dagger=A$) and commutes with the operator at $t_2$ ($[A,B]=0$), it is possible to consider two instead of four trajectories and to calculate the two-time correlation as
\begin{equation}
\aver{ B(t_2) A(t_1)}=\frac{1}{4} \left[ \mu_{R}^+ \bra{\chi^+_R(t_2)} B\ket{\chi^+_R(t_2)} - \mu_{R}^- \bra{\chi^-_R(t_2)} B\ket{\chi^-_R(t_2)} \right].
\end{equation}
To ensure that the comparison between the two approaches is as fair as possible, we concentrate on such a special case for the two-time correlator in Eq.~\ref{eq:SpinTwotime_correlation_function}, since for this case the number of trajectories needed in both approaches is equal. We expect that in the more general case where even four trajectories are needed in the approach by M\o lmer et al., this approach will become even more costly. The second case is more subtle and appears in the context of conserved quantum numbers. If the Lindbladian evolution allows the partition of the superoperator into symmetry blocks, and the operator $A$ at $t_1$ couples different blocks, the resulting states $\ket{\chi_{R,I}^\pm}$ cannot longer be addressed to a single block. Instead it is possible to evolve the parts of the different sectors separately as they are decoupled for all times $t>t_1$. A closer look reveals that in this case the introduced scheme is equivalent to the approach by Breuer et al. from the last section.
\section{Comparison of the different methods to determine two-time correlations functions in open quantum systems }
\label{sec:two_time_comparison}
In this section we compare the performance of the different methods to calculate two-time correlations using the spin model described in section \ref{sec:model}. We focus on the two-time correlation function relating two applications of local $S^z$-operators at sites separated by a distance $d$ given by
\begin{equation}
C_d(t_2, t_1) = \langle S^z_{c}(t_2) S^z_{c+d}(t_1)\rangle.
\label{eq:SpinTwotime_correlation_function}
\end{equation}
Recall, $c$ is the central site for a chain with an odd number of sites and the center-left site otherwise. This correlation function has been proven to be essential to uncover interesting dynamical regimes displaying physical phenomena such as aging or hierarchical dynamics \cite{WolffKollath2019}. We start by the comparison of the two stochastic approaches for the dissipator $\mathcal{D}_1$ in subsection \ref{sec:comp_stochastic}. We use as an initial state a classical Neel state.
We find that typically the Breuer et al. approach combined with the MPS methods performs better than the M\o lmer et al. approach. We continue in comparing the better performing stochastic approach, the approach by Breuer et al., to the purification approach in subsection \ref{sec:comp_stochastic_purification_Neel}. For the considered situation the purification approach greatly outperforms the stochastic approach. Reasons of the excellent performance of the purification approach in this situation are the 'easy' initial state and the conservation of the magnetization by the Lindblad dynamics and most importantly, the low matrix dimension needed.
In subsection \ref{sec:comp_D2} we turn to the local dissipator $\mathcal{D}_2$. We consider the situation where the initial state is the ground state of the Hamiltonian and the dissipation is switched on at time $t=0$. We compare again the stochastic approach of Breuer et al. to the purification approach. The entangled initial state and the absence of the conservation of magnetization constitutes an additional difficulty for the purification approach. We find that for most parameters considered the stochastic approach is much more suited to treat this situation efficiently.
\subsection{Comparison of stochastic approaches for the dephasing noise $\mathcal{D}_1$}
\label{sec:comp_stochastic}
In this section we compare the efficiency of the two stochastic approaches in order to calculate two-time correlation functions.
\begin{figure}[tb]
\centering
\includegraphics[width=0.45\textwidth]{Entropy_Evolution.pdf}
\includegraphics[width=0.45\textwidth]{Entropy_Evolution_larger_system.pdf}
\caption{Evolution of the von Neumann entropy as a resource measure for different branches of Breuer and M\o lmer MCWF approach to computing two-time correlation functions for two different sets of model parameter. Results are shown for $t_1 J_x/\hbar=5$, a time step of $\Delta t J_x/\hbar=0.02$ and $10^4$ trajectory samples for (left panel) an exact MPS representation with chains of length $L=14$ and (right panel) non-exact MPS representation for $L=20$.}
\label{fig:single_entropy}
\end{figure}
We calculate the correlation function from Eq.~\ref{eq:SpinTwotime_correlation_function} for the spin model with the dissipator $\mathcal{D}_1$. The choice for the correlation functions allows us to only use two states, i.e. $\ket{\chi_R^\pm}$, in the method by M\o lmer et al.
First, we investigate the cost of generating a single two-time trajectory sample with MPS methods. To this end, the entanglement entropy is evaluated for all times $t>t_1$. As it is directly linked to the needed bond dimension, it serves as an architecture-free measure of the required resources in terms of run time and memory consumption. In Fig.~\ref{fig:single_entropy} we show the von Neumann entropy for the two branches in the Breuer approach, where $\ket{\phi(t_1)} = A\ket{\psi(t_1)}$, as well as for the $\ket{\chi_R^\pm}$ branches of the M\o lmer procedure for different parameter sets and system sizes. As the results for the two M\o lmer branches are indistinguishable in the plot, we only present data for $\ket{\chi^+_R}$. Besides the MC average we also show the maximum measured entropy over all sampled trajectories and the 1$\sigma$ interval of the measured standard deviation. In the left panel we use an exact MPS representation for $L=14$ sites in order to avoid biases introduced by the truncation. We see that the entropy increases over time and the statistical deviation increases in all cases. However, for both parameter sets presented it is evident that the two branches of the Breuer approach generate significantly less entropy than the M\o lmer branches. Furthermore, a strong dependence on the model parameters exists. The von Neumann entropy is much smaller for large anisotropy parameters $J_z/J_x$.
In the right panel of Fig.~\ref{fig:single_entropy} larger system size $L=20$ are shown. The total von Neumann entropy generated ($J_z/J_x=2$) is larger for larger system sizes. However, concerning the behavior of the different approaches the findings of the small system size are confirmed that the von Neumann entropy in the M\o lmer approach grows a bit faster than the one of the approach by Breuer et al..
\begin{figure}[tb]
\centering
\includegraphics[width=0.6\textwidth]{Sample_Convergence.pdf}
\caption{Standard deviation of the mean of the two-time correlation function in dependence of the number of Monte-Carlo samples $R$ measured at four different times $t_2> t_1$. The black dashed line is a guide to the eye and data is shown for the interaction strengths $J_z/J_x= 2$ (top panel) and $J_z/J_x= 20$ (bottom panel), $L=14$, $\hbar \gamma/J_x=2$, $t_1J_x/\hbar=5$, an exact MPS representation and a time step of $\Delta t J_x/\hbar=0.02$.}
\label{fig:sample_convergence}
\end{figure}
Another important factor is the convergence of the Monte-Carlo averages with respect to the number of samples. We present the scaling of the standard deviation of the mean with the number of samples at different time points in Fig.~\ref{fig:sample_convergence}. As the sampling is statistically independent we observe an inverse square root scaling in all cases. However, this is more than one order of magnitude smaller for the joint evolution suggested by Breuer et al. than for the evolution of M\o lmer et al.. In addition, the standard deviation is smaller at later times, which indicates the approach of the unique infinite temperature steady state.
In summary of this section, we can conclude, that for the specific model and parameter sets considered here, the approach of Breuer et al. is favorable over the one by M\o lmer et al. in terms of both, memory and run time.
\subsection{Comparison of Monte-Carlo wave function and purification method for $\mathcal{D}_1$}
\label{sec:comp_stochastic_purification_Neel}
After having identified the approach by Breuer et al. as superior compared to M\o lmer et al. for the considered situation, we compare this approach to the purification method. Since the exact increase of the bond dimension is unknown in both cases, the trade-off between the larger Hilbert space and the cost of sampling many trajectories needs to be evaluated carefully.
\begin{figure}[tb]
\centering
\includegraphics[width=0.65\textwidth]{Convergence_Deviation_Purification.pdf}
\caption{Convergence of the purification approach for spin-$\nicefrac{1}{2}$ chains of different length $L$, different dissipation strengths and $J_z/J_x=2$. We chose a vanishing truncation goal $\varepsilon=0$ to enforce the realization of the bond dimension $D$ and use a time step $\Delta t J_x/\hbar=0.05$ and $t_1J_x/\hbar=5$.}
\label{fig:Convergence_Deviation_Purification}
\end{figure}
We compare the accuracy of the two-time correlation function $C_d(t_2, t_1)$ normalized to the value at $t_1$ during the dissipative evolution of a spin-$\nicefrac{1}{2}$ chain initially prepared in the Neel state with bulk dephasing, i.e. using the dissipator $\mathcal{D}_1$. We previously performed such calculations in Ref.~\cite{WolffKollath2019} and analyzed there the time step required to obtain a good accuracy. In the following we use these time steps for our comparison. Fig.~\ref{fig:Convergence_Deviation_Purification} shows the results obtained by the purification method with different values for the bond dimension. This method converges faster in terms of the bond dimension for larger values of the dissipation strength and smaller system sizes. Since it is not clear at which time the greatest inaccuracy exists, we use the maximum deviation of the normalized two-time correlations from the results of the largest achievable bond dimension $D_{\mathrm{max}}$ over the entire time as a measure for the error
\begin{equation}
\Delta_{\mathrm{max}}(D) = \max_{t_2} \left\{\left\vert \frac{C_{1}(t_2, t_1, D)}{C_{1}(t_1, t_1, D)} - \frac{C_{1}(t_2, t_1, D_{\mathrm{max}})}{C_{1}(t_1, t_1, D_{\mathrm{max}})}\right\vert\right\}.
\label{eq:Delta_max}
\end{equation}
The two-time correlation $C_1(t_2, t_1)$ which is calculated using bond dimension $D$ is denoted by $C_1(t_2, t_1, D)$. Let us note that this measure for the error not only measures the 'pure' truncation error, since the trajectories with a different bond dimension also have a different stochastic nature.
\begin{figure}[h]
\centering
\includegraphics[width=0.65\textwidth]{Convergence_Deviation_MCWF.pdf}
\caption{Convergence of two-time correlations obtained from MCWF using the method by Breuer et al. for different system sizes $L$ and dissipation strengths and a spin interaction anisotropy $J_z/J_x=2$ with $t_1J_x/\hbar=5$. Here we use the truncation goal $\varepsilon=0$ and a time step $\Delta t J_x/\hbar=0.02$ for $\hbar\gamma/J_x=1$ and $\Delta t J_x/\hbar=0.002$ for $\hbar\gamma/J_x=10$. The bond dimension $D$ and the number of MC samples $R$ are given for each curve.}
\label{fig:Convergence_Deviation_MCWF}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[width=0.6\textwidth]{MCWF_Error_Estimation.pdf}
\caption{Error estimation of the MCWF computation of the normalized two-time correlator $C_{d=1}(t_2, t_1)$ for $L=16$ sites, $\hbar \gamma/J_x=1$, $J_z/J_x=2$, $t_1J_x/\hbar=5$ and a time step of $\Delta t J_x/\hbar=0.02$. The solid lines show the MPS truncation error $\Delta_{\text{max}}$ and the dashed lines show the statistical error $\sigma_{\text{max}}$ for different number of trajectories $R$. The statistical error becomes dominant when choosing the bond dimension sufficiently large. Here $\Delta_{\text{max}}$ is calculated with respect to the results with $D=500$ and $R=10^4$.}
\label{fig:Error_Estimation}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[width=0.6\textwidth]{MCWF_Timestep_Convergence.pdf}
\caption{Time step convergence of MCWF approach for different dissipation strengths for a chain of $L=16$ spins, $J_z/J_x=2$, $t_1J_x/\hbar=5$, truncation goal $\varepsilon=0$ and a bond dimension $D=500$. The sample size is $10^4$.}
\label{fig:MCWF_Timestep_Convergence}
\end{figure}
The results for the stochastic sampling approach presented in Fig.~\ref{fig:Convergence_Deviation_MCWF} point in a similar direction, i.e. stronger dissipation and smaller systems require a smaller bond dimension to yield reliable results. Surprisingly, the convergence of the stochastic approach with increasing the bond dimension is generally much slower compared to the purification method. In addition, the accuracy is also influenced by the number of stochastic samples taken. To capture the error from the stochastic sampling, we first define the maximum value of the standard deviation of the mean in time as
\begin{equation}
\sigma_{\mathrm{max}}(D)=\max_{t_2}\left\{ \sigma_{\mathrm{mean}}\left(\frac{C_1(t_2, t_1, D)}{C_1(t_1, t_1, D)}\right)\right\}
\label{eq:sigma_max}
\end{equation}
and then the total error for the MCWF approach as the maximum of $\Delta_{\mathrm{max}}(D)$ and $\sigma_{\mathrm{max}}(D)$. As can be seen in Fig.~\ref{fig:Error_Estimation}, increasing the number of samples used, the sampling error decreases. Further, with increasing bond dimensions, the truncation error becomes less important and $\Delta_{\mathrm{max}}(D)$ becomes of the same order as $\sigma_{\mathrm{max}}(D)$.
Another important fact to consider is the dependence of the choice for the time step on the dissipation strength and the system size in the MCWF approach as mentioned in the discussion of Fig.~\ref{fig:twojump_probability}. Fig.~\ref{fig:MCWF_Timestep_Convergence} confirms that $\Delta t J_x/\hbar=0.02$ is a suitable time step for the dissipation strength $\hbar\gamma/J_x=1$. Using the extrapolation indicated in Fig.~\ref{fig:twojump_probability} we estimate $\Delta t J_x/\hbar\approx0.002$ as a potential step size for $\hbar \gamma/J_x=10$. However, the convergence analysis reveals that this is still not sufficient to obtain high quality simulation results. We find that larger dissipation strengths require a very small time step for MCWF simulations in this cases, which causes substantially longer run times and additionally increases the error caused by the MPS truncations, as more gate application with subsequent singular value decompositions are needed to reach a certain simulation time.
\begin{figure}[tb]
\centering
\includegraphics[width=0.6\textwidth]{MaximalDeviation.pdf}
\caption{Comparison of the two-time correlation functions error ($\text{Error}(D)$) calculated in the purification method and the MCWF approach introduced by Breuer et al. for $J_z/J_x=2$ and $t_1J_x/\hbar=5$. For the purification data a time step of $\Delta t J_x/\hbar=0.05$ has been used and for the MCWF method $\Delta t J_x/\hbar=0.02$ for $\hbar\gamma/J_x=1$ and $\Delta t J_x/\hbar=0.002$ for $\hbar\gamma/J_x=10$. As a reference to calculate the $\Delta_{\text{max}}$ used in $\text{Error}(D)$ we choose $D_{\text{max}}=900$ for $\hbar\gamma/J_x=1$ and $D_{\text{max}}=110$ for $\hbar\gamma/J_x=10$ for the purification method. For the MCWF approach we consider $D_{\text{max}}=1000$ with $R=10^4$ for $L=16$ and $\hbar\gamma/J_x=1$, $D_{\text{max}}=1000$ with $R=2500$ for $L=16$ and $\hbar\gamma/J_x=10$ and $D_{\text{max}}=500$ with $R=300$ for $L=24$ and $\hbar\gamma/J_x=1$. }
\label{fig:Delta_max_deviation}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[width=0.6\textwidth]{Error_Runtime.pdf}
\caption{Run time measurement to achieve certain numerical accuracy provided by the measured error of the two-time correlation functions using machines with clock frequency 2.6 GHz. The run time of the creation of single Monte-Carlo sample is compared to the full run time of a purification based computation. Parameters are the same as in Fig.~\ref{fig:Delta_max_deviation}.}
\label{fig:Delta_max_runtime}
\end{figure}
With the introduced benchmark strategy, i.e. using
\begin{equation}
\mathrm{Error}(D) = \begin{cases} \Delta_{\mathrm{max}}(D)&: \text{purification}\\
\mathrm{max}\left\{\Delta_{\mathrm{max}}(D), \sigma_{\mathrm{max}}(D)\right\}&: \text{MCWF}
\end{cases}
\end{equation}
as an error measure for the two approaches, it is now possible to compare them quantitatively. The dependence of the error on the bond dimension is presented in Fig.~\ref{fig:Delta_max_deviation} for two different system sizes and dissipation strengths. While a weak dependence on the system size is noticeable, the dissipation strength turns out to be the decisive parameter. For the purification calculations the convergence regarding the bond dimension is much faster for strong dissipation. The accuracy of the stochastic sampling is relatively quickly dominated by the statistic error, so that the behavior with the bond dimension appears to be almost constant. Consequently, there are crossing points above which the accuracy of the purification method is better for the same bond dimension.
To put these results into the context of realistic numerical resources, we show in Fig.~\ref{fig:Delta_max_runtime} the required run time for a certain error on a computer cluster consisting of machines with 2.6 GHz clock frequency and sufficient memory resources. In case of strong dissipation, the generation of a single trajectory already takes longer than the evolution using the purification of the reshaped density matrix due to the necessity of a much smaller time step. For weaker dissipation strengths the run time of a single trajectory becomes comparable to the purification approach. However, the total run time of the stochastic sampling computation, requiring a sample size of the order $10^4$ trajectories, is orders of magnitude larger than the full evolution, even with a massive parallelization of the sampling process. This means that the purification approach in these cases is strongly favorable over the stochastic approach.
\subsection{Comparison of purification and stochastic approach for local dissipation $\mathcal{D}_2$}
\label{sec:comp_D2}
To demonstrate the strong influence of the model and the initial state on the method choice, we continue by supplementing the findings of the last chapter with investigating results of another set-up. For this purpose, we turn to the dissipator $\mathcal{D}_2$, which only contains one jump operator given by the spin lowering operator $S_c^-$ acting on the central site of a system with an odd number of spins. As this jump operator violates the conservation of the total magnetization during the Lindbladian evolution, it is not possible to exploit symmetry properties in the evolution of the purified state. Nevertheless, the non-unitary evolution in the deterministic part of the stochastic sampling conserves the total magnetization and only the jump applications switch between different symmetry blocks, so that the MCWF evolution can be calculated using conserved quantum numbers.
\begin{figure}[tb]
\centering
\includegraphics[width=0.45\textwidth]{gspurification_gapless_2.pdf}
\includegraphics[width=0.45\textwidth]{gsstate_gapless_2.pdf}
\caption{(left panel) Deviation of ground state energy after purification from the value obtained by DMRG in the original Hilbert space for different system size $L$. (right panel) The difference between ground state energy obtained by DMRG in the original Hilbert space for two bond dimensions $D$ and $D_{\text{max}}$. The maximum bond dimension fulfills truncation error $\epsilon=10^{-12}$ and varies from $D_{\text{max}}=40$ for system size $L=15$ to $D_{\text{max}}=90$ for system size $L=49$. The ground state has been calculated for $J_z/J_x=0.5$.}
\label{fig:gs_purification}
\end{figure}
While the initial state was a product state in the previous chapter, the system is here initially prepared in the ground state of the equilibrium model accessed with the density matrix renormalization group (DMRG) ground state search algorithm \cite{White1992, Schollwoeck2011}. We have chosen the ground state of an anisotropy of $J_z/J_x=0.5$, which is located in the gapless Tomonaga-Luttinger liquid phase of the Hamiltonian and requires a sizable bond dimension for its representation in MPS form. As a result, a strong truncation is necessary when reshaping the corresponding density matrix to a purified state in the doubled Hilbert space as described in Sec.~\ref{sec:purifying_pure_state}. In Fig.~\ref{fig:gs_purification} (left panel) we show the bond dimension dependence of the deviation of the ground state energy calculated after the reshaping process from the original value obtained by DMRG. For small system sizes with less than 20 sites, the ground state energy can be reproduced after the purification step with a medium sized bond dimension with less than $10^3$ states taken into account. However, even for slightly larger systems with up to 50 spins bond dimensions of several thousand states are needed to achieve an accuracy of the ground state energy of only $10^{-3}$. In this case the purification step alone takes more than three days of run time. On the other hand in Fig.~\ref{fig:gs_purification} (right panel) the energy difference between two ground states in the original space with two different bond dimensions $D$ and $D_{\text{max}}$ is plotted for different system sizes. One sees that without purification the bond dimension below $~100$ is large enough to get the convergence of ground state energy of $10^{-8}$.
\begin{figure}[tb]
\centering
\includegraphics[width=0.6\textwidth]{equaltime_benchmark.pdf}
\caption{Comparison of the evolution of the equal-time correlation function under the dissipator $\mathcal{D}_2$. We present the deviation of the simulation results using the purification approach with different values for the maximal bond dimension from MCWF data using $3\times 10^4$ samples and $D=500$ (solid lines). We evolve a chain of size $L=29$ with $\hbar\gamma/J_x=2$ and $J_z/J_x=0.5$. The time step is chosen as $\Delta t J_x/\hbar = 0.05$ in all cases. The dashed lines mark the time average over the presented time interval and the dashed-dotted line is the time-dependency of the standard deviation of the Monte-Carlo average.}
\label{fig:equal_time_benchmark}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[width=0.6\textwidth]{Demagnetization_2time_MCWF.pdf}
\caption{Evolution of the normalized two-time correlation functions computed by the MCWF method by Breuer et al. The simulation has been done for $\hbar\gamma/J_x=1$, $J_x/J_x=0.5$, $D=500$, $t_1J_x/\hbar=1$, $\Delta t J_x/\hbar=0.05$ and $10^4$ trajectory samples.}
\label{fig:Demagnetization_2time_MCWF}
\end{figure}
To estimate the numerical effort and the influence of inaccuracy in the representation of the ground state on time evolution for the purification method we compare the equal-time correlation functions for different values of the bond dimension in Fig.~\ref{fig:equal_time_benchmark}. As the equal-time correlations represent the initial condition for the two-time correlation functions at time $t_1$, the accuracy of their calculation is crucial in order to obtain the two-time correlation functions. The direct comparison in Fig.~\ref{fig:equal_time_benchmark} shows that the purification method reaches a comparable accuracy to the MCWF approach for $D\geq 600$. Even for these bond dimensions sizable deviations occur of the order of $10^{-3}$. Looking at the associated run times, as summarized in Tab.~\ref{tab:runtimes_demagnetization}, shows that the simulation based on purification takes about ten times as long for such a large bond dimension as compared to a parallel implementation of the stochastic sampling.
\begin{table}[t!]
\centering
\begin{tabular}{ccc}
\hline
\textbf{method} \:\: & \:\:\textbf{bond dimension} \:\: &\:\: \textbf{run time [hours]} \\ \hline
MCWF & 500 & 84.47 \\
purification & 100 & 2.93 \\
purification & 200 & 29.04 \\
purification & 300 & 110.28 \\
purification & 400 & 355.24 \\
purification & 500 & 818.38 \\
purification & 600 & 829.88 \\ \hline
\end{tabular}
\caption{Table of run times for evaluating equal-time correlation functions from Fig.~\ref{fig:equal_time_benchmark}. The MCWF simulation was executed in parallel on 10 cores. Parameters are the same as in Fig.~\ref{fig:equal_time_benchmark}.}
\label{tab:runtimes_demagnetization}
\end{table}
Based on the substantially larger run time (of a few weeks to months), the purification becomes very inefficient for the analysis of a physical situation. Even though the actual run time on a single core would be comparable, here, the MCWF scheme is preferential since the 'waiting time' to obtain the results makes a thorough analysis of a physical question more feasible. The waiting time to obtain the results can be easily shortened by the trivial parallelization of the stochastic approach.
To conclude the analysis, it remains to be demonstrated that the two-time correlation functions are accessible by the MCWF scheme. For this purpose we show in Fig.~\ref{fig:Demagnetization_2time_MCWF} the two-time correlation functions for the system sizes $L=25$ and $L=29$ which would be very inaccurate and time consuming to reach by the purification method. We see that the two time function is rising in time, which signals rising fluctuations. A detailed study of the arising physics goes beyond the present work with a strong technical focus.
\section{Conclusion}
\label{sec:conclusion}
To conclude we have presented a comparison of three different MPS based methods for the calculation of two-time functions in open quantum systems. This comprises the purification approach and two different approaches based on the stochastic unraveling of the Lindblad dynamics. First we compared the two stochastic approaches in the situation of an XXZ spin chain subjected to a dephasing noise starting initially in the classical Neel state. In this situation we find a clear preference for the stochastic approach suggested by Breuer et al. over the approach suggested by M\o mler et al.. This is due to the better convergence of the trajectories used in the approach by Breuer et al.. However, the purification approach is even much more efficient for the considered situation. We would like to emphasize that this conclusion that the purification approach was the most efficient also hold for correlations of the type $S^+S^-$.
Additionally, we considered the dynamics of a XXZ spin chain subjected to a local application of the jump operator $S^-_c$ starting from a Tomonaga-Luttinger liquid. This changes drastically the efficiency of the methods and the stochastic approach becomes more efficient than the purification approach. There are several reasons for this. First, already the representation of the Tomonaga-Luttinger liquid in the purified form is resource demanding. Secondly, in the following time-evolution the conservation of the magnetization is not fulfilled anymore. These reasonings also hold for correlations of different type.
We therefore expect that the purification approach is valuable if the initial state is easily represented within the purified space. Further, a strong symmetry of the Linbladian enabling the use of conserved quantities is of advantage.
In comparison the stochastic wave function approach is well suited also to represent difficult initial states and the following Lindblad evolution. However, the presence of many jumps, as for the case of strong dephasing noise, calls for a very low time-step which makes the trajectory approach less efficient.
We would like to point out that even though the comparison was mainly performed for short range correlations $d=1$, since the computational efficiency up to the application of the operator at time $t_2$ is independent of the chosen distances, all our findings will also hold for larger values of the distances.
\section*{Acknowledgements}
We thank J. Dalibard, M. Fleischhauer, A. L\"auchli, and K. M{\o}lmer, for fruitful discussions.
\paragraph{Funding information}
We acknowledge funding from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) in particular under project number 277625399 - TRR 185 (B4) and project number 277146847 - CRC 1238 (C05) and under Germany's Excellence Strategy – Cluster of Excellence Matter and Light for Quantum Computing (ML4Q) EXC 2004/1 – 390534769 and the European Research Council (ERC) under the Horizon 2020 research and innovation program, grant agreement No.~648166 (Phonton).
|
1,116,691,499,291 | arxiv | \section{Introduction}
The Epoch of Reionization (EoR) extends from the time when the first sources form as a result of
the nonlinear growth of primordial density fluctuations in a fully neutral universe ($z \sim 20$), to the
moment when the intergalactic medium is fully reionized under the effect of UV radiations emitted
by the
sources ($z\sim 6$). This simple picture, however, hides a number of uncertainties. At this time, there are
only two observational constraints on the EoR. The first comes from the
Gunn-Peterson effect in the absorption spectrum of high redshift quasars. Indeed the
transmitted flux drops sharply as a small neutral fraction appears toward high redshifts:
it implies an ionization fraction $x_{\mathrm{HI}}< 10^{-4}$ at $z < 5.5$ (Fan {\sl et al.} 2006).
The second constraint is set by the Thomson scattering of CMB photons by the free electrons produced
during the EoR. The corresponding optical depth, $\tau= 0.09 \pm 0.03$ (Spergel {\sl et al.} 2007),
implies that the intergalactic medium is already significantly reionized at $z=11$. In the
next decade, we will learn a lot more by direct observation of the
redshifted 21-cm emission from the neutral IGM during the EoR. Such instruments as LOFAR, PAST or
MWA will be able to probe the statistical properties of the 21-cm signal, while SKA will be able
to make a full tomography of the IGM up to $z \sim 11$. See Carilli {\sl et al.} (2004) and
Carilli (2006) for detailed prospects.
In the last decade, a lot of work has been done, theoretical and numerical, to predict the
properties of the 21-cm emission and better optimize the design of the future instruments. Madau
{\sl et al.} (1997) and Tozzi {\sl et al.} (2000) present the first theoretical models of 21-cm emission.
The signal can be seen either in emission or in absorption against the CMB, depending on
the spin temperature of the neutral hydrogen. Interaction with CMB photons couples the spin
temperature to the CMB temperature in less than $10^5$ years during the EoR, which would make the
signal undetectable. Fortunately two other processes tend to couple the spin temperature to the
gas kinetic temperature instead. The first is the excitation of the hyperfine transition
through collisions
with electrons or other hydrogen atoms (see Kuhlen {\sl et al.} 2006 for numerical simulations),
which, however, is efficient only in overdense regions (baryonic $\delta\rho/\rho > 5/[(1+z)/20]^2$).
The second is
the pumping of the 21-cm transition by Ly-$\alpha$ photons (Wouthuysen-Field effect,
Wouthuysen 1952, Field 1959) which requires a threshold value for the local Ly-$\alpha$ flux to be
effective. Consequently, the value of the kinetic temperature of the gas relative to the CMB
temperature is crucial for determining the 21-cm emission brightness temperature. During the EoR,
the gas is cooling down due to the expansion of the universe faster than the CMB, but is heated by hydrodynamical
shocks from structure formation, X-ray from pop-III star or quasars and, to a much lesser extend, by
Ly-$\alpha$ photons (see e.g., Furlanetto {\sl et al.} 2006a). Simple analytical models are now
available which take into account various source types and formation history (Furlanetto 2006b).
They usually predict that the signal can be seen in absorption early on, then in emission later
on. The prediction for the typical amplitude of the differential brightness temperature is a few
$10$ mK. Analytical models cannot, however, take into account the full complexity of
the 3D inhomogeneous IGM: numerical simulations including the dynamics of structure formation and,
usually as a post-treatment, a full 3D radiative transfer are required.
Dynamical simulations of structure formation have a long history, but cosmological radiative
transfer simulations are a more recent field of investigation. Simulations showed that the
ionized
bubbles around the first sources are not spherical: indeed the ionization fronts propagate fast in
the void and more slowly in the high density filaments (see Abel {\sl et al.} 1999 for the
first simulation). The geometry of reionization is now studied in large simulation boxes
($\sim 100$ Mpc) to get a statistical sample of ionized bubbles of various sizes
(Iliev {\sl et al.} 2006a, McQuinn {\sl et al.} 2007). However, with such large boxes, even with
very high resolution simulations, small scale density structures, also called minihaloes, are
not resolved and are often taken into account through a simple clumping factor. The global effect
of minihaloes is to slow down reionization by consuming photons during their photoevaporation.
But a simple, uniform, clumping factor may be insufficient to model the role of minihaloes, and
more detailed studies have been performed (Ciardi {\sl et al.} 2006, Iliev {\sl et al.} 2005,
and Shapiro {\sl et al.} 2006 for the impact on the 21-cm signal). Using the density field from
the dynamical simulations and the ionization fraction and gas temperature from the radiative
transfer simulations, it is possible to produce 21-cm emission maps.
A number of authors provide predictions for different aspects of the 21-cm signal: the emission
map at a given redshift, the average signal as a function of redshift or the signal power as a
function of the angular scale. These quantities were first predicted from rather small
simulation boxes ($\sim 10$ Mpc) limiting the angular information to $\theta < 10$ arcmin
(Ciardi \& Madau 2003, Gnedin \& Shaver 2004, Furlanetto et al. 2004, Valdes et al. 2006).
Recently, predictions from larger simulation boxes ($\sim 100$ Mpc) became available (Mellema
{\sl et al.} 2006, Iliev {\sl et al.} 2007). It is a fact that the predictions, in particular
the duration and intensity of the absorption phase, depend crucially on the modeling of the
sources. But other factors have the potential to alter these predictions. Indeed, all predictions
from simulations, at this time, assume a uniform efficiency for the Wouthuysen-Field effect.
However, Barkana \& Loeb (2005) or Furlanetto {\sl et al.} (2006a) recognized that
fluctuations in the local Ly-$\alpha$ flux can produce additional fluctuations in the
brightness temperature. An accurate quantitative modelization of these fluctuations is important. Indeed, the
21-cm signal will be difficult to detect due to brighter foregrounds: the unique signature of its brightness
fluctuations will make the extraction possible. Barkana \& Loeb used simplified analytic models
(neglecting radiative transfer
effects on the local Ly-$\alpha$ flux) to compute this contribution. Our goal in this paper is
to investigate how computing the full radiative transfer in the Ly-$\alpha$ line in a
cosmological inhomogeneous medium can modify the picture presented by these authors.
The 21cm signal will be difficult to detect, due to brighter foregrounds, and the unique character
of its brightness fluctuations is a signature that will make its extraction possible. It is
therefore crucial to precise in more details those fluctuations.
Although Monte-Carlo simulations of Ly-$\alpha$ transfer have a long history, starting with
Avery \& House (1968), only recently has the computing power become sufficient to tackle the
case of a 3D inhomogeneous medium with kinematics, without restrictions on the optical thickness
regime (Ahn {\sl et al.} 2001). Several authors have now developed similar codes to simulate
the Ly-$\alpha$ emission from high redshift galaxies (Zheng \& Miralda-Escud\'e 2002, Cantalupo
{\sl et al.} 2005, Dijkstra {\sl et al.} 2006, Verhamme {\sl et al.} 2006, Tasitsiomi 2006).
In this paper, we will present the implementation and validation of Ly-$\alpha$ radiative transfer
in LICORICE, a dynamics-radiative transfer code, with a special emphasis on the treatment large
Hubble flows, which is specific to the EoR applications.
This work is part of the SKADS\footnote{http://www.skads-eu.org} effort (DS2-T1 task)
to produce simulated emission maps that
can be used to optimize the design of the future SKA telescope.
In section 2, we present which physics of the Ly-$\alpha$ transfer is included in LICORICE,
and justify why some aspects of the physics do not need to be included. In section 3 we detail
some aspects of the algorithms and acceleration schemes implemented in the code. Three validation
tests are presented in section 4, comparing the outputs of LICORICE against analytic solutions
or standard numerical results. Finally, in section 5, we apply the code to some typical EoR
situation to investigate possible fluctuations in the Wouthuysen-Field effect.
\section{The code: LICORICE }
LICORICE consists in three parts. A TreeSPH code with multiphase modeling of the gas computes the
dynamics of structure formation (Semelin \& Combes, 2002). A continuum radiative transfer part is
added to compute reionization. This part uses a Monte-Carlo approach and is similar to CRASH (Maselli
{\sl et al.} 2003). It has the advantage over CRASH of using an adaptative grid. LICORICE is
currently participating in the second part of the Cosmological Radiative Transfer Comparison
Project (see Iliev et al. 2006b for the first part of the project). The third part is the
Ly-$\alpha$ radiative transfer which is described in this paper.
\subsection{Physics of the Ly-$\alpha$ radiative transfer}
LICORICE implements a Monte-Carlo approach to radiative transfer: it propagates
photons on an adaptative grid. Consequently we will describe the physics of
Ly-$\alpha$ radiative transfer from the point of view of a photon traveling
through the simulation box.
\subsubsection{ Computing the optical depth}
A photon propagating through neutral hydrogen has a probability
$P(\tau)= e^{-\tau}$ of not being scattered after traveling through an optical
depth $\tau$ from
its emission point. We consider Ly-$\alpha$ scattering only, so the optical
depth can be computed as:
\begin{equation}
\tau = \!\int_0^l \! \int_{-\infty}^{+\infty} \!\!\!\! ds \, du_{\parallel} \,\, n_{\hbox{\scriptsize HI}} \,\,\,p(u_{\parallel}) \,\, \sigma\left(\nu (1-{v_{\parallel}^{\hbox{\scriptsize macro}}+u_{\parallel} \over c}) \right),
\end{equation}
where $\nu$ is the photon frequency in the global rest frame and
$n_{\hbox{\scriptsize HI}}$ is the
local number density of neutral hydrogen. $u_{\parallel}$ is the scattering
atom velocity along the incoming photon's direction in the moving reference frame of the fluid ,
$p(u_{\parallel})$ is the normalized probability distribution for $u_{\parallel}$ ,
$v_{\parallel}^{\hbox{\scriptsize macro}}$ is the gas macroscopic velocity
along the incoming photon's direction in the global rest frame, and $c$ is the speed of light. Finally
$\sigma(\nu^{\prime})$ is the Ly-$\alpha$ scattering cross section of a photon
with frequency $\nu^{\prime}$ {\sl in the atom rest frame}.
The function $p(u_{\parallel})$ usually results from the thermal distribution of the atoms velocity:
\begin{equation}
p(u_{\parallel}) \,=\, {1 \over \sqrt{\pi} \, v_{th}} \, \exp\left(- { u_{\parallel}^2 \over v_{th}^2 } \right) \,\,\, \hbox{with} \,\,\, v_{th} =\sqrt{ {2 k_B T \over m_p }}\,\,.
\end{equation}
Some astrophysical systems have significant velocity gradiants on scales below the best possible resolution
of the simulations. In these cases it may be relevant to model the small scale velocity contribution by adding
a turbulent componant to the thermal velocity
dispersion. We did not include any turbulent contribution in this paper.
The exact expression of the Ly-$\alpha$ scattering cross section is given in Peebles (1993). It is
well approximated by the usual Lorentzian profile:
\begin{equation} \qquad
\sigma(\nu) \,=\, f_{12} \,{\pi e^2 \over m_e c}\, { \Delta \nu_L / 2 \pi \over (\nu - \nu_0)^2 + (\Delta \nu_L / 2)^2} \,\, ,
\end{equation}
where $f_{12}=0.4162$ is the Ly-$\alpha$ oscillator strength, $\nu_0=2.466 \times 10^{15}$ Hz
is the line center frequency and $\Delta \nu_L=9.936 \times 10^7$ Hz is the natural line width.
We introduce the dimensionless parameters $x$, the relative frequency shift, and $b$, the natural to Doppler line width ratio:
\begin{equation} \qquad
x= {\nu-\nu_0 \over \Delta \nu_D} \quad \hbox{with} \quad \Delta \nu_D= { v_{th} \over c} \nu_0\,\, ,
\end{equation}
\begin{equation} \qquad
b = {\Delta \nu_L \over 2 \Delta \nu_D } \,\, .
\end{equation}
Using these notations, we can write the optical depth increment in the gas local rest
frame ($v_{\parallel}^{\hbox{\scriptsize macro}}=0$) as:
\begin{equation} \qquad
d \tau = ds \, n_{\hbox{\scriptsize HI}} \, {f_{12} \sqrt{\pi} e^2 \over m_e c \Delta \nu_D} H(b,x) \,\, ,
\end{equation}
or in cgs units,
\begin{equation} \qquad
d\tau = ds \, n_{\hbox{\scriptsize HI}} \,\, 6.87 \times 10^{-14} \, \left( { T \over 10^4 } \right)^{-{1 \over 2}} \!\! H(b,x) \,\, ,
\end{equation}
\noindent
where $H$ is the Voigt function defined as:
\begin{equation} \qquad
H(b,x) = {b \over \pi} \int_{-\infty}^{+\infty} \!\! {e^{-y^2} \over (x-y)^2 + b ^2} \,dy
\end{equation}
To compute the Voigt function, we either use the analytic fit given by
Tasitsiomi (\cite{Tasitsiomi06}), or, in cosmological situations where the Hubble
flow on a scale of the order of the simulation spatial resolution produces a
frequency shift larger than the Ly-$\alpha$ line width (see section 2.1.3), we use the simple approximation:
\begin{equation} \qquad
H(x,b) \simeq \mathrm{max} \left(e^{-x^2}, {b \over \sqrt{\pi} x^2 } \right) .
\label{simpleH}
\end{equation}
\subsubsection{ Effect of the n=2 state splitting, dust and deuterium.}
A Ly-$\alpha$ photon can excite an hydrogen atom from the $1S_{1/2}$ ground
state to either the $2P_{1/2}$ or the $2P_{3/2}$ state. As discussed by
Tasitsiomi (\cite{Tasitsiomi06}), the splitting between these two $n=2$ states
is small: only $10$ GHz, or $1$ km.s$^{-1}$. If the thermal velocity dispersion
is much larger than $\sim 1$ km.s$^{-1}$ (i.e $T > 100 K$), the level
splitting is washed out from the radiation spectrum after just one scattering.
In the case of the Ly-$\alpha$ background radiation during the EoR, the gas
temperature drops to $\sim 30 K$. However the optical depth as the photon
redshifts from one side to the other of the Ly-$\alpha$ line is very high:
$ \sim 8 \times 10^5 $ for the average gas density of the universe ($\Omega_b=0.045$)
at $z=9.5$ in a standard
cosmological model. Thus the photon will scatter many times off thermal atoms
and still the splitting will be washed out. Consequently, we did not
distinguish between the two $2P$ states in this paper.
Another issue is the possible reshuffling from $2P$ to $2S$ through collisions
with free protons or electrons. An atom cannot be excited directly to the $2S$
state because of the dipole selection rule. But if the $2P \rightarrow 2S $
transition is induced by a collision, then the atom will de-excite through the
emission of 2 continuum photons: the Ly-$\alpha$ photon is lost.
Tasitsiomi (\cite{Tasitsiomi06}, see eq. 26) computes the probability that an atom
in the $2P$ state will reshuffle to $2S$ before it de-excites normally by
emitting a Ly-$\alpha$ photon, $p \sim 8.5 \times 10^{-13} n_p ({T \over 10^4})^{-0.17}$,
where $n_p$ is the proton number density. What can we expect during the EoR ? First, we are interested in the
Ly-$\alpha$ background in the cold, {\sl neutral} region of the universe, thus $n_p \ll 1$.
Moreover, even if we assume a non negligible ionization fraction,
$p \sim 5 \times 10^{-13}$ at $z \sim 10$ for $T=50K$ and for the critical
density of the universe with
$\Omega_b=0.045$. This probability must be compared with the average number of
scatterings a photon undergoes as it redshifts through the Ly-$\alpha$ line,
which is of the order of the optical depth: $\sim 10^6$. We see that only a
fraction of $5 \times 10^{-7}$ of the Ly-$\alpha$ photons will be degraded
into 2 continuum photons before they redshift out of the line. We conclude
that this process is not relevant during the EoR.
Dust is usually a factor in Ly-$\alpha$ transfer simulations since it absorbs
Ly-$\alpha$ photons. Hansen and Oh (\cite{Hansen06}) study the effect of dust
absorption in a multiphase medium and show that photons can escape such a
medium, while they would be absorbed in a more homogeneous single-phase medium,
for the
same total HI column density. In the cosmological context we are working in
($z>6$), there are no observations to help us constrain the dust abundance and
distribution. We can assume that during the EoR, dust may be found only
around the sources and that the IGM is completely dust free. Under this
assumption, the effect of dust on the Ly-$\alpha$ flux can be modeled with a
simple escape fraction coefficient. Furthermore, if this coefficient is assumed to be independent of
the source, dust will not have any effect on the Ly-$\alpha$ flux {\sl fluctuations} which are the
focus of this paper. Therefore, we did not include the effect of dust.
Dijkstra {\sl et al.} (\cite{Dijkstra06}) show that the presence of deuterium
with an abundance [D/H]$= 3 \times 10^{-5}$ leaves a clear imprint on the
spectrum emerging from a uniform sphere of gas with a central source and a
total optical depth $\tau = 7.3 \times 10^5$. Is deuterium relevant to the
Ly-$\alpha$ flux during the EoR? The deuterium line center is $82$ km.s$^{-1}$
blueward of the hydrogen line. This is equivalent to the redshift of a photon
traveling $\sim 0.5$ comoving Mpc at z=9.5. So the first answer is that
deuterium may have an effect on the Ly-$\alpha$ flux fluctuations only at small
scales ( $< 1$ Mpc comoving). However, let us notice that the total optical depth
for deuterium through the Ly-$\alpha$ line is $\tau \sim 20$ for an abundance
[D/H]$=2 \times 10^{-5}$, and the optical depth in the wing of the Hydrogen Ly-$\alpha$ line, as the
photon redshifts from far into the blue to the center of the deuterium line
happens by chance to be also $\tau \sim 20$. This means that, while the photon
will indeed scatter a few times in the deuterium line, it will also have been
scattered in the hydrogen line wing several times before it reaches the
frequency range where deuterium scattering dominates (a
few 10 km.s$^{-1}$ around the line center). We conclude that the presence of
deuterium is unlikely to affect the Ly-$\alpha$ flux fluctuations noticeably.
This does not mean that the effect of deuterium on Ly-$\alpha$ radiation
cannot be observed during the EoR. Indeed, in the case of an ionizing bubble
around a source with a sharp ionization front, the continuum spectrum of the
source will show a Gunn-Peterson trough with a step at the bubble redshift. Deuterium should
create a secondary small step.
\subsubsection{Dealing with large Hubble flows}
In $z \sim 10$ cosmological simulations, the Ly-$\alpha$ thermal line width is equivalent to
the Hubble flow redshift over only a few 10 comoving kpc. This scale
is usually (much) smaller than the size of cells in simulations. So we
must be careful when we compute the optical depth: using a single comoving
frequency for the photons throughout a cell would result in photons flowing
through the line core without feeling it. We must actually compute an integral
along the path of the photon, with a redshifting comoving frequency, to obtain
the correct optical depth. If we consider that the expansion velocity between
any two points of the same cell is non-relativistic
($ {v_H \over c} \sim 0.01$ for
20 comoving Mpc at $z\sim 10$), the computation gets easier. Let $\nu_{in}$ be the
comoving frequency and $x_{in}$ the local rest frame value of $x$ when the photon enters the cell.
\begin{equation} \qquad
x_{in} = {\nu_{in}(1-{\mathrm{\bf v}^{\mathrm{macro}}\cdot \mathrm{\bf k} \over c}) - \nu_0 \over \Delta \nu_D}\,\, ,
\end{equation}
where $\hbox{\bf v}^{\mathrm{macro}}$ is the macroscopic velocity of the gas (uniform inside the cell),
and \hbox{\bf k} is the direction of the photon. Then, at a given
point inside the cell defined by the vector $ r \hbox{\bf k}$ from the entering
point of the photon, the comoving frequency is:
\begin{equation} \qquad
\nu = {a(\nu_{in}) \over a(\nu)} \nu_{in} \sim { \nu_{in} \over 1 + {Hr \over c} } \sim \nu_{in} (1 - {\mathrm{H}r \over c}) \,\, ,
\end{equation}
where $\mathrm{H}$ is the Hubble constant at the simulation redshift, and $a$ is the
expansion factor. We neglect any variation of $\mathrm{H}$ during the photon travel and
we consider non-relativistic expansion velocities.
For non-relativistic macroscopic velocities of the gas, the corresponding value of $x$ writes:
\begin{equation} \qquad
x= x_{in}- {\mathrm{H} r \over c} {\nu_{in} \over \Delta \nu_D} \,\, .
\end{equation}
So, we {\bf linearized} the relation between $r$, the
current path length inside the cell, and the current local rest frame value of
the $x$ variable. In this approximation, noting $x_{out}$ the value of $x$ when the photon exits
the cell, computing the optical depth reduces to computing the integral Voigt function
$H_{\mathrm{int}}(x_{out})$,
with the following definition for the $H_{\mathrm{int}}$ function:
\begin{equation} \qquad
H_{\mathrm{int}}(x) = \int_{x_{in}}^{x} H(x^{\prime},b) dx^{\prime} \,\, .
\end{equation}
However, in the Monte-Carlo method, to find the location of a scattering event,
we need to solve {the equation $H_{\mathrm{int}}(x) = A $, where $A$ is a constant}. This is simplified if we
can provide an analytic expression for $H_{\mathrm{int}}(x)$, {with an explicit inverse function}. It is the case if we use
the simple approximation of $H(x,a)$ given in eq. \ref{simpleH}. It involves the
$\mathrm{erf}(x)$ function for which we are using an approximation which has an explicit
inverse function.
Modeling the effects of expansion only by a redshift computed from a radial dilatation is the usual
approximation for Ly-$\alpha$ radiative transfer codes. Other effects of the variation of the expansion
factor $a$, such as variation of the average density during the photon flight time, are usualy ignored.
However, let us emphasize that adding the expansion velocity to the other types of
velocities (macro or microscopic) to compute Doppler shifts, either during
scattering events or to compute local rest frame values of $x$, works only if
all velocities are non-relativistic. However, we want to study the
Ly-$\alpha$ flux during the EoR, and at $z \sim 10$, a photon emitted just below
the Ly-$\beta$ frequency will travel $\sim 350$ comoving Mpc and be
redshifted to Ly-$\alpha$ by a Hubble flow
velocity such that $ v_H/c \sim {\delta \nu \over \nu} \sim {\delta a \over a} \sim 0.15$. In this case, the
second order
errors in computing the redshift are not completely negligible. But, what is more important, neglecting the
variations of $a$ along the photons path, in computing gas densities for
example, produces first order error in ${\delta a \over a}$. Consequently, we should limit ourselves to
$\sim 100$ comoving Mpc boxes.
In this work we do neglect variations of $a$, except for the cosmological redshift, and in most
cases we use simulation boxes smaller than $\sim 30$ comoving Mpc. The full effect of
expansion will be introduced in the future to handle larger boxes.
\subsection{Scattering off hydrogen atoms}
In an expanding universe the natural variable is the comoving frame frequency
of the photon. However, as long as the Hubble flow velocities are
non-relativistic, we can use the frequency in the rest frame of the
zero-coordinate point of the simulation, hereafter named {\sl global frame},
and treat the expansion as a simple
radial dilatation with a $ \mathrm{\bf v}_H = H \mathrm{\bf r} $ velocity field to be
added to the peculiar velocities.
In the rest frame of the atom, we will consider the scattering to be resonant.
The effect of the recoil, which would change the frequency of the photon by
transferring part of its momentum to the atom, has been shown to be negligible
in astrophysical situations by several authors (Zheng \& Miralda-Escud\'e \cite{Zheng02} or
Tasitsiomi \cite{Tasitsiomi06}). In the global frame,
however, due to the various contributions to the atoms velocity, the frequency
$\nu$ of the photon will change.
There are three main contributions to the atom
velocity: the thermal motion, the macroscopic peculiar motion and the Hubble
flow. When the photon scatters off an atom, we first compute the frequency in
the atom rest frame:
\begin{equation} \qquad
\nu_{\mathrm{atom}}=\nu(1-({\mathrm{\bf v}_H+\mathrm{\bf v}^{\mathrm{macro}}+\mathrm{\bf u}\over c})\cdot \mathrm{\bf k}_i) \,\, ,
\end{equation}
where $\mathrm{\bf u}$ is the thermal velocity of the atom, and
$\mathrm{\bf k}_i$
is the direction of the incoming photon. Let us split $\mathrm{\bf u}$ into
$\mathrm{\bf u}_{||}$, the component parallel to the incoming photon
direction, and $\mathrm{\bf u}_{\perp}$, the perpendicular component.
${ \mathrm{\bf u}_{||} \over v_{th}}$ obeys the distribution:
\begin{equation} \qquad \label{paralvel}
P_1(y)= {b \over \pi H(b,x)} { e^{-{y^2}} \over (x-y)^2+b^2} \,\, ,
\end{equation}
while each of the 2 components of $\hbox{\bf u}_{\perp} $ obeys:
\begin{equation} \qquad
P_2(y)= {1 \over \sqrt{\pi} v_{th}} e^{- { y^2 \over v_{th}^2}}\, \,.
\end{equation}
Then we compute the direction of
the photon after scattering. As described by Tasitsiomi (\cite{Tasitsiomi06})
the scattering phase function depends on the excitation state and on whether the
photon scatters in the wing or in the core of the line. Several authors (see
e.g. Cantalupo {\sl et al.} \cite{Cantalupo05} or Verhamme
{\sl et al.} \cite{ Verhamme06} ) have shown that, for high optical depth media,
the shape of the phase function does not alter the results of the simulation. In this work,
we use isotropic scattering. Using the new direction of the photon, we then
recompute the frequency in the global frame using a non-relativistic Doppler
effect.
\section{Ly-$\alpha$ radiative transfer: numerical methods}
\subsection{Gas density and velocity field}
LICORICE is meant to use results from dynamical simulations done using the
TreeSPH part of the code (Semelin and Combes \cite{Semelin02}). It uses the same Tree structure as the dynamical
code to build an
adaptative grid. The grid is build in such a way that each cell contains between
1 and $N_{\mathrm{max}}$ gas particles. Values from 1 to 30 are commonly used for $N_{\mathrm{max}}$.
The density and velocity fields are then interpolated from the particle
distribution. The cells limit the resolution of the simulation only by having
uniform dynamical properties. Ly-$\alpha$ transfer inside one cell is still computed
exactly under the assumption that the expansion velocity on the scale of the cell is non-relativistic.
\subsection{ Monte-Carlo method}
Using the Monte-Carlo approach, we send individual photon from the source and
follow them from scattering to scattering and from grid cell to grid cell,
until they exit the simulation box. After one event (emission or scattering),
the algorithm is the following:
\begin{itemize}
\item Step 1: Compute the photon global frame frequency, either from the
scattering atom rest frame frequency, or from the source spectrum.
\item Step 2: Draw the new photon direction (isotropically in this work).
\item Step 3: Draw a variable $p$ from a uniform distribution between $[0,1]$. The photon
will travel an optical depth $\tau=-\ln(p)$ to the next scattering
event.
\item Step 4: Increment optical depth with current cell contribution. Determine if scattering occurs
in this cell, if yes go to step 5, if no, pass on to next cell and repeat step 4.
\item Step 5: Draw scattering atom thermal velocity, and compute frequency in the scattering atom rest frame. Go back to step 1.
\end{itemize}
\subsection{Acceleration scheme}
To generate random variable following eq. \ref{paralvel} distribution, we use the method by
Zheng \& Miralda-Escud\'e (\cite{Zheng02}). These authors introduce a parameter
$u_0$ for which an optimal value is needed. We use the following empirical
fit:
\begin{equation} \quad
$$ u_0=1.85-\log(b)/6.73+\ln(\ln(x)) \quad \hbox{for} \quad x > 3 \, ,$$
\end{equation}
and $u_0=0$ otherwise. This has been determined from a systematic numerical
optimization, and works well for $10^{-4} < b < 10^{-1}$. For large values of
$x$, quite common in cosmological simulations, drawing a value from
$P_1$ is still slow, even with Zheng \& Miralda-Escud\'e method. In this case,
however, scattering will most likely occur in the wing of the line.
Consequently, for $x > 10$, we revert to $u_0=0$ but we truncate the
distribution to the limited range $[-3,3]$.
We use the core-skipping acceleration scheme (see Avery \& House 1968 or Ahn,
Lee and Lee 2002 for first applications). In this scheme, we choose a core value
$x_c$ for the variable $x$. In the regime $x < x_c $ the medium must be thick.
If so, the photon will scatter many times over a very short distance. Only when
$x$ gets larger than $x_c$ (scattering in the wing by a fast moving atom) will
the medium become transparent and will the photon travel a long distance. The
idea is to ignore the insignificant core scatterings: every time the photon
enters the core ($x < x_c $), it leaves again immediately by scattering off an
atom with a thermal velocity $u_{at} > x_c v_{th}$.
{ We use the detailed prescriptions given by Tasitsiomi
(\cite{Tasitsiomi06}) on choosing $x_c$ as a function
of $b\tau_0$, where $b$ is defined in eq. 5 and $\tau_0$ is the optical depth at the line center . The core-skipping scheme works well when the expected output of the code is
an emerging spectrum. We will show that it also works for computing the fluctuations of the
local scattering rate. }
\section{Validation tests}
LICORICE being a complex, multipurpose code, we take care of validating each
part separately. We present here validation tests for the Ly-$\alpha$ part,
against analytical solutions or standard numerical setups.
\subsection{ Static homogeneous sphere: emerging spectrum}
A classical validation test for Ly-$\alpha$ codes is the emerging spectrum for
a monochromatic source in the middle plane of a static homogeneous {\sl slab} of
gas. The main reason is that Neufeld (\cite{Neufeld90}) gives an analytic
solution for the emerging spectrum in the case of an extremely thick system.
However Dijkstra {\sl et al.} (\cite{Dijkstra06}) provide a new analytic
expression in the case of a uniform spherical cloud of gas:
\begin{equation} \qquad
J(x) \,=\, {\sqrt{\pi} \over \sqrt{24} a \tau_0} \left[ { x^2 \over 1+ \cosh\left(\sqrt{{2\pi^3 \over 27}} {|x^3| \over a \tau_0} \right) }\right]
\end{equation}
Since this geometry is more relevant in a cosmological context, we present
results for this case (we also performed the Neufeld test and also found a
good agreement). Fig. \ref{test_Dijkstra} shows the comparison between
numerical and analytic spectra, for a static spherical cloud of gas with
temperature $T=10K$ and optical depths at line center from center to edge equal
to $10^5$, $10^6$ and $10^7$. For each run, $10^5$ photons are injected at
the center of the cloud with frequency $\nu_0$. { The emerging spectrum $J(x)$
is obtained by computing a normalized histogram of the frequencies of the photons
as they leave the cloud.}
The agreement is good in all
cases, although it gets better and better as $\tau_0$ increases. Indeed in the
case $\tau_0=10^5$, we have $a \tau_0 = 1500$ which is close to the lower limit
for an extremely thick medium.
\begin{figure}[t]
\resizebox{\hsize}{!}{\includegraphics{test_Dijkstra.eps}}
\caption{ Emerging spectrum for a uniform and spherical cloud of gas at T$= 10$ K and several
values of $\tau_0$, the total optical depth at line center.
Numerical histograms are computed with $10^5$ photons. The analytic solution
is given in Dijkstra {\sl et al.} (\cite{Dijkstra06}).
}
\label{test_Dijkstra}
\end{figure}
\subsection{ Expanding homogeneous sphere: emerging spectrum}
The second test is also becoming a classic. It was first performed by Zheng and Miralda-Escud\'e
(\cite{Zheng02}), followed by several authors. Verhamme {\sl et al.} (\cite{Verhamme06}) present
especially detailed results. The system is a uniform sphere of gas expanding (or contracting)
with an Hubble-like velocity field: the radial velocity is proportional to the radius. We use
the same physical condition as Zheng and Miralda-Escud\'e (\cite{Zheng02}): a temperature of
$20\,000K$ and a radial velocity of $200$ km.s$^{-1}$ at the edge of the cloud. We use three
different column densities from the center to the edge of the cloud: $N_{\mathrm H}$ equals
$2. \times 10^{18}$ cm$^{-2}$, $2. \times 10^{19}$ cm$^{-2}$ or $2. \times 10^{20}$ cm$^{-2}$
(that is $\tau_0 = 8.3 \times 10^4$, $8.3 \times 10^5$, or $8.3 \times 10^7$). We do not run the tests
for a contracting cloud, and we consider only a central point source emitting at
Ly-$\alpha$ frequency, not the case of uniform
emissivity. An expanding cloud with a central point source is the most relevant to the EoR. The
emerging spectra are shown in fig. \ref{test_Z-ME}. We find results very similar to Zheng and
Miralda-Escud\'e (\cite{Zheng02}) or Verhamme {\sl et al.} (\cite{Verhamme06}). The peak
blueward of the Ly-$\alpha$ frequency is completely suppressed by the expansion (not shown
in fig. \ref{test_Z-ME}). For the case of a contracting cloud, the red peak would be suppressed
\begin{figure}[t]
\resizebox{\hsize}{!}{\includegraphics{test_Z-ME.eps}}
\caption{ Emerging spectrum for a spherical, uniform and expanding cloud of
gas at T$= 20000$ K and several values of $\mathrm{N}_{\mathrm{HI}}$, the column density
from the center to the edge of the cloud. The emission is at Ly-$\alpha$ frequency from the central
point of the cloud. The radial velocity of the gas is proportional to the radius and
equals $200$ km.s$^{-1}$ at the edge of the cloud. The peak
blueward of the Ly-$\alpha$ frequency is completely suppressed by the expansion. }
\label{test_Z-ME}
\end{figure}
\subsection{ Expanding homogeneous sphere at T=0K: mean intensity field}
Our last validation test relies on an analytic solution given by Loeb and Rybicki (\cite{Loeb99})
. The setup is again an expanding uniform hydrogen medium with a central monochromatic source at
Ly-$\alpha$ frequency. The velocity field is a Hubble flow again, but this time, the
gas temperature is $T=0$K. Loeb and Rybicki introduce the dimensionless variable
$\tilde{\nu} = { \nu_0 - \nu \over \nu_{\star}}$, where $\nu_{\star}$ is the comoving frequency
shift from Ly-$\alpha$ at which the optical depth to infinity equals 1, and
$\tilde{r} = {r \over r_{\star}}$, where $r_{\star}$ is the proper radius at which the Doppler
shift from the source due to the Hubble expansion equals $\nu_{\star}$. Loeb and Rybicki give
an analytic expression for the corresponding dimensionless mean intensity $\tilde{J}$, valid in
the diffusion regime:
\begin{equation} \qquad
\tilde{J}(\tilde{r},\tilde{\nu})\,=\,{1 \over 4\pi} \left( {9 \over 4 \pi \tilde{\nu}^3} \right)^{3 \over 2} \exp \left[ - {9 \tilde{r}^2 \over 4 \tilde{\nu}^3} \right].
\end{equation}
Loeb and Rybicki (\cite{Loeb99}) show the comparison between the analytic solution and the
numerical solution given by a dedicated Monte-Carlo code. Tasitsiomi (\cite{Tasitsiomi06})
runs this test with his general purpose code. Our results are presented in fig. \ref{test_Loeb_nu}
and \ref{test_Loeb_r}. They are very similar to those of Loeb and Rybicki and Tasitsiomi:
the numerical results are close to the analytic solution where the diffusion regime is valid. However,
where photons enter the free streaming regime ($\tilde r_0=1$ in fig. \ref{test_Loeb_nu} and $\log \tilde \nu_0 = 0.5$ in fig. \ref{test_Loeb_r}), the numerical solution diverge from the analytic solution which becomes
invalid.
Here are some details on how we ran this test. We used
a temperature $T=2$K for computing the optical depth, which is singular for $T=0$K, but we used a zero thermal velocity for
the atoms in all scattering events. We used a simulation box holding a sphere of gas of radius
$10 r_{\star}$. Since the temperature is not zero in all respects, it reintroduces a dimension
in the problem. Very close to the source the thermal speed is larger than the expansion velocity
and the numerical behaviour should diverge from the analytic solution. For reference, we used a
proper expansion velocity of $200$ km.s$^{-1}$ at $r_{\star}$. In our setup, the thermal
and expansion velocity are of the same order for $r \sim 10^{-3} r_{\star}$.
\begin{figure}[t]
\resizebox{\hsize}{!}{\includegraphics{test_loeb_nu.eps}}
\caption{Mean intensity spectra at various radii for a Ly-$\alpha$ monochromatic source in a
uniform expanding medium at $T=0$K. The numerical result is compared to the analytic solution
given, in the diffusion regime, by Loeb and Rybicki (1999).}
\label{test_Loeb_nu}
\end{figure}
The radii range covered by this test is quite large, from $10^{-3} r_{\star}$ to $10 r_{\star}$.
Using $10^{12}$ cells for the radiative transfer grid is not really an option. Actually we used
only $16^3$ cells, but we took advantage of the integral scheme described in equation 12 and 13
for computing exactly the optical depth between two points in the same cell. This approach is
obviously validated by the good agreement shown in fig \ref{test_Loeb_r} at small radii.
\begin{figure}[t]
\resizebox{\hsize}{!}{\includegraphics{test_loeb_r.eps}}
\caption{Mean intensity profile at various frequencies for a Ly-$\alpha$ monochromatic source in a
uniform expanding medium at $T=0$K. The numerical result is compared to the analytic solution
given, in the diffusion regime, by Loeb and Rybicki (1999). }
\label{test_Loeb_r}
\end{figure}
\section{ Ly-$\alpha$ during the EoR}
\subsection{The role of Ly-$\alpha$ photons in the 21 cm emission }
The 21-cm signal can be seen in emission or absorption against the CMB. The differential brightness
temperature observed at redshift z=0 is:
\begin{equation} \qquad
\delta T_b = {T_S - T_{\mathrm{CMB}} \over 1+z} (1- e^{-\tau_{21}}) \, ,
\end{equation}
where $T_S$ is the neutral hydrogen spin temperature, $T_{\mathrm{CMB}}$ is the CMB radiation blackbody
temperature at redshift $z$, and $\tau_{21}$ is the 21-cm line optical depth. The value of
$\tau_{21}$ is given, among others, by Tozzi {\sl et al.} (2000) or Furlanetto {\sl et al.}
(2006a). Injecting the formula for $\tau_{21} \ll 1$, the differential brightness temperature
can be written:
\begin{equation} \quad
\delta T_b \sim 9. \,\,x_{\mathrm{HI}}\,\, (1+\delta) (1+z)^{1 \over 2}\,\, {T_S -T_{\mathrm{CMB}} \over T_S} \,\,\,\,{\mathrm{mK}} \,,
\end{equation}
where $\delta$ is the local overdensity at redshift $z$, and $x_{\mathrm{HI}}$ is the neutral
hydrogen fraction. { This value of $\delta T_b$ is for a flat $\Lambda$CDM model with
$h_0=0.7$, and $\Omega_b=0.044$. It changes by $\pm \, 0.5$ mK when $\Omega_m$ varies from $0.25$ to $0.3$}. The value of the spin temperature $T_S$ is the result of three competing
processes. The absorption/reemission of CMB photons tends to bring $T_S$ to $T_{\mathrm{CMB}}$ over a
time scale under $10^5$ years during the EoR (Tozzi {\sl et al.}, 2000). As we have seen
in the Introduction section, both collisions between hydrogen atoms
and the pumping by Ly-$\alpha$ photons, also known as the Wouthuysen-Field effect
(Woutuysen 1952, Field 1958), tends, instead, to couple $T_S$ to the kinetic temperature of the
gas. As a result, the spin temperature can be written (Furlanetto {\sl et al.} 2006a):
$$
T_S^{-1}={ T_{\mathrm{CMB}}^{-1} +x_c T_K^{-1} + x_{\alpha} T_C^{-1} \over 1+x_c+x_{\alpha}} \quad \mathrm{with} \quad T_C \simeq T_K \,, $$
where $x_c$ and $x_{\alpha}$ are the coupling coefficients respectively for collisions and
Ly-$\alpha$ pumping, {and $T_C$ is the effective color temperature of the UV radiation field (see Furlanetto {\sl et al.} 2006a)}. The coefficient $x_{\alpha}$, which is the focus of this work, can be
explicitly written as:
$$
x_{\alpha}={4 P_{\alpha} T_{\star} \over 27 A_{10} T_{\mathrm{CMB}} } \,\, ,
$$
where $T_{\star}=0.068$K, $A_{10}= 2.85 \times 10^{-15} s^{-1}$ is the spontaneous emission
factor of the 21-cm transition, and $P_{\alpha}$ is the number of scatterings of Ly-$\alpha$
photons per atom per second. Now usually come two approximations that we will {\bf not} make in this
paper. {\bf The first} is that $P_{\alpha}$ is considered proportional to $J(\nu_\alpha)$,
the angle averaged specific intensity at the local Ly-$\alpha$ frequency, neglecting the
contribution of wing absorptions. We will see that this approximation is valid provided $J(\nu)$
itself has been computed taking into account wing absorptions. {\bf The second}, more drastic
approximation, is to evaluate $J(\nu)$ without performing the full radiative transfer
computation. Actually, to our knowledge, all numerical simulations of 21-cm emission consider a
uniform value of $J(\nu_\alpha)$. However, Barkana and Loeb (2005) have shown that several
factors induce fluctuations in $J(\nu_\alpha)$: the $1/r^2$ scaling of the flux which magnifies
the Poisson noise in the source distribution, the clustering of the sources, and the
contribution of higher Ly-$\alpha$ series photons (also studied in detail by Pritchard and
Furlanetto 2006). They predict the power spectrum of the 21-cm brightness temperature due to the
fluctuation in $J(\nu_\alpha)$. Although a vast improvement over using a uniform Ly-$\alpha$ flux,
they are still neglecting radiative transfer effects: they assume that photons are freely
streaming until they redshift to the local Ly-$\alpha$ frequency. We will show that this
assumption breaks down at scales smaller that $\sim 10$ comoving Mpc.
Potentially, pumping the upper excitation level is not the only way in which Ly-$\alpha$
photons can influence the 21-cm emission: they also heat up the gas. This heating mechanism was
first thought to be efficient by Madau {\sl et al.} (1997). However, Chen \& Miralda-Escud\'e (2004),
taking into account the atoms thermal velocity distribution which had been neglected by
Madau {\sl et al.}, found a much smaller, actually negligible heating rate. Furlanetto \&
Pritchard (2006c), taking into account the effect of higher Lyman series photons confirmed this
result and found, for typical EoR conditions, the Ly-$\alpha$ heating rate to be 140 smaller than
the heating rate from X-rays.
Chuzhoy \& Shapiro (2007a) recently challenged this result, adding in particular the
effect of the deuterium $Ly-\beta$ resonance line, but the strength of this effect
is not yet completely probed.
\subsection{$P_\alpha$ profiles for spherically symmetric configurations}
In the next three cases, we consider a central source in a spherically symmetric medium of neutral
hydrogen at $T_K=30$K, and $z \simeq 10$, which is typical of the EoR . The main difference with
the setup of the tests in section 4.3 is that the source emits a continuous flat spectrum. We
deal only with photons between Ly-$\alpha$ and Ly-$\beta$ frequencies.
\subsubsection{Homogeneous medium}
\begin{figure}[t]
\resizebox{\hsize}{!}{\includegraphics{radial_homog.eps}}
\caption{ Profiles of the scattering rate per atom,$P_\alpha$, in the Ly-$\alpha$ line, around a
central source with a continuous flat spectrum in an homogeneous medium of neutral hydrogen at
30K and the average baryon density of the universe at $z\sim10$.}
\label{radial_homog}
\end{figure}
First, we consider the case where the gas is homogeneous with a density equal to the average
density of the universe at $z \sim 10$. We ran the simulation in a 600 comoving Mpc box with the source in the
center of the box. We present the results up to a radius of 100 comoving Mpc: at larger radii
the effect of the box boundaries (no retrodiffusion) and the variation of $a$ during the photon flight
(see section 2.1.3) alter the validity of the simulation. Fig. \ref{radial_homog} presents the
radial profile of the scattering rate per atom, $P_{\alpha}$, in different cases. Using the real
Voigt line profile, we can see a deviation at small scale from the simple $1/r^2$ profile
expected if photons stream freely from the source until they redshift to the local Ly-$\alpha$
frequency. Here is why. Let us consider a photon emitted above the Ly-$\alpha$ frequency. As it
travels away from the source, it is redshifted toward the local Ly-$\alpha$ frequency. Because of
the contribution of the wings of the Voigt profile to the optical depth, it has a probability to
scatter before reaching the local Ly-$\alpha$ frequency. On average, it will scatter for the first time,
when the optical depth along its path reaches 1. This is achieved when the photon is
redshifted to the frequency $\sim \nu_\alpha+\nu_\star$ (see section 4.3), which occurs at a distance
$r_\star$ before the location where it would reach the local Ly-$\alpha$ frequency
(quantities defined in Loeb \& Rybicki 1999). In our setup, typical of the EoR,
$r_\star \sim 10$ comoving Mpc. After the photons scatter for the first time they change
direction. Consequently, the location where a photon actually enters the core of the local
Ly-$\alpha$ line has a probability to be anywhere within $\sim 10$ comoving Mpc of the location
determined by free streaming alone. This is not crucial at large scales where the expected
$1/r^2$ profile is recovered, but it creates a steeper profile at small scales
(exponent $\sim -2.3$ in our setup). To validate our interpretation, we computed the transfer
with a modified line profile: the core of the line is unchanged, but the wings are set to zero.
As can be seen in Fig. \ref{radial_homog}, we then recover the $1/r^2$ profile.
{After this paper was submitted, Chuzhoy and Zheng (2007b) posted a paper with a similar
result. They considered a very similar setup and computed the transfer with a simple
Monte Carlo code which is 1-D (spherical symmetry) and uses a simplified line profile but does
include Ly-$\alpha$ photons locally injected by cascades from upper Lyman series lines. They find
the same steepening of the scattering rate profile at short scales. However, they show that
photons injected from upper Lyman series lines are much less sensitive to radiative transfer
effect: they follow the $1/r^2$ profile more closely. Consequently the discrepancy between
the full radiative transfer computation and the simple $1/r^2$ evaluation is somewhat reduced.}
Finally we checked the effect of using the core-skipping acceleration scheme for evaluating the
fluctuations of $P_\alpha$: since all core absorptions are avoided, it could have
modified the spatial $P_\alpha$ fluctuation map. As can be seen in Fig. \ref{radial_homog}, it
is not the case. Although the wings modify the location where photons enter the core, the scattering
number is still dominated by the close-to-the-core contribution.
\subsubsection{Central clump}
\begin{figure}[t]
\resizebox{\hsize}{!}{\includegraphics{central_clump.eps}}
\caption{Profile of the scattering rate per atom, $P_\alpha$, in the Ly-$\alpha$ line, around a
central source with a continuous flat spectrum, inside an overdense spherical clump of gas (64
times the density of the surrounding medium) of radius 1 comoving Mpc, }
\label{central_clump}
\end{figure}
Until now, the Ly-$\alpha$ scattering rate has been evaluated during the EoR in an homogeneous
medium only (Barkana and Loeb 2006, Pritchard and Furlanetto 2006). With a general purpose 3D
code such as LICORICE, we can investigate the impact of density fluctuations in the gas on the
Ly-$\alpha$ scattering rate {\bf per baryon}. First we choose a very simple setup: we consider a
box of size 64 comoving Mpc, with a central homogeneous spherical clump of gas of radius
1 comoving Mpc. The clump is 64 times denser than the surrounding medium, which has the average baryon
density of the universe at the redshift of the simulation ($z\sim10$). The source is in the center of the clump and has
a continuous flat spectrum. The radial profile of the Ly-$\alpha$
scattering rate per atom, $P_{\alpha}$, is shown in Fig. \ref{central_clump}. The main feature is
a depletion of $P_{\alpha}$ in the low density medium just outside the clump. Indeed, photons
that should redshift to the Ly-$\alpha$ frequency in this region, have first to travel through the
high density region where they have an enhanced probability to be scattered in the wing of an
atom, and redirected to redshift to local Ly-$\alpha$ while still inside the core. In other
words, the enhanced wing scatterings in the clump draw core scatterings from surrounding
regions to the clump itself. This is somewhat similar to the usual shadowing effect in
radiative transfer, although in this case the process occur in the frequency space so the {\sl shadowing}
can be seen even in a spherical configuration. Obviously, in addition to the $1/r^2$
decline of the flux, and the fluctuations in the source distribution, we can expect a new source
of fluctuations for $P_\alpha$: the density fluctuations of the intergalactic medium.
\subsubsection{Isothermal density profile}
\begin{figure}[t]
\resizebox{\hsize}{!}{\includegraphics{isothermal_profile.eps}}
\caption{Profile of the scattering rate per atom $P_\alpha$ in the Ly-$\alpha$ line, around a
central source with a continuous flat spectrum. Inside a sphere of radius 10 comoving Mpc, the
gas density field is an isothermal sphere, outside it is homogeneous at the average baryon
density of the universe ($z\sim 10$). The two regions connect smoothly. A $1/r^3$ profile is plotted for comparison.}
\label{isothermal_profile}
\end{figure}
Instead of the sharp density transition between the clump and the surrounding medium we now
consider an isothermal density profile ($\rho \sim 1/r^2$) around the source. The isothermal
profile connects to the surrounding homogeneous medium at a radius of 10 comoving Mpc. All
other parameters of the simulation are the same as in the previous setup. The radial profile of
the Ly-$\alpha$ scattering rate per atom, $P_{\alpha}$, is shown in Fig. \ref{isothermal_profile}.
Since the average density is higher than in the other setups, photons scatter many times and even
with a acceleration method, we used only $2 \times 10^4$ photons. To avoid a very high noise level
at large radii we used an adaptative resolution: high in the center, lower in the outer
regions. We can see on the figure that, inside the region with an isothermal density profile,
$P_\alpha$ closely matches a $1/r^3$ profile. It reverts to $1/r^2$ in the homogeneous medium
outside the $10$ Mpc radius. This shows that the brightness temperature fluctuations of the
21-cm emission may be stronger than previously estimated at small scales, at least during the
early EoR when $P_\alpha$ fluctuations are meaningful.
Is this setup more relevant to the prediction of $P_{\alpha}$
fluctuations during the EoR than a uniform medium ? Yes in the sense that the medium should
obviously be denser closer to the source. However the spherical symmetry and the specific
profile used here are oversimplified: the actual intergalactic medium in the EoR is not in an
equilibrium configuration, especially during this early period of sources formation, and the
spherical symmetry is broken by the presence of filaments. In the next section we investigate how
filaments modify $P_{\alpha}$.
\subsection{$P_\alpha$ map for an axisymmetric configuration}
\begin{figure}[t]
\resizebox{\hsize}{!}{\includegraphics{contour.eps}}
\caption{Contours of the scattering rate per atom $P_\alpha$ in the Ly-$\alpha$ line, around a
central source with a continuous flat spectrum. The source is located inside a cylindrical
homogeneous filament of gas (shaded zone in the figure) with a density 64 times the density of
the surrounding medium which is at the average baryon density of the universe ($z \sim 10.$).
The contours are equally spaced on a logarithmic scale with a step of $\sqrt{10}$. }
\label{contour}
\end{figure}
We consider now an axisymmetric density field for the gas: inside a cylinder of radius 1
comoving Mpc the density is 64 times the critical baryon density, outside it is equal to the
critical baryon density. The source is located on the symmetry axis. All other parameters are
identical to the previous setup. This setup is a simplified representation of the real case of
source formation during the EoR at the intersection of several filaments,
where the filaments have a density profile. The $P_\alpha$ contour map is shown in Fig.
\ref{contour}. The shaded area represents the filament, and the contours are equally spaced on a
logarithmic scale (two per decade). The map is integrated over a variation of $\pi$ of the
angular variable. About $5. 10^6$ photons where used for this simulation. There is some boundary
effect due to the finite size of the simulation box: photons that would scatter just outside the simulation
box and possibly reenter the box are lost instead. This affects a few Mpc near the boundary.
We can see a sharp
depletion in the number of scatterings per atom inside the filament. We have checked that a
smaller density contrast creates a smaller depletion, as expected. Once again, we see that
the gas density field fluctuations induce fluctuations in $P_\alpha$. However, the weaker
Wouthuysen-Field effect in denser regions may be balanced by the greater coupling due to
collisions (which is proportional to the density). What may be more relevant to the future
observations than what occurs at small scales inside the filaments, is that the presence of the
filaments modifies the shape of the contours in the low density surrounding medium: they are not
spherical but oblate. With our density contrast, the axis ratio is about 2 in the 5-10 Mpc range
and decreases at large distances.
Fig. \ref{filament_profiles} shows the normalized $P_\alpha$ profiles along filament axis and in the
perpendicular plane containing the source. While inside the filament (distance smaller than 1 comoving Mpc),
the profiles match. At larger distances, fig. \ref{filament_profiles} quantify the $P_\alpha$ ratio between
the two regions: it reaches a maximum value of $\sim 100$ at a distance of $\sim 5 Mpc$.
\begin{figure}[t]
\resizebox{\hsize}{!}{\includegraphics{filament_profiles.eps}}
\caption{Scattering rate per atom $P_\alpha$ in the Ly-$\alpha$ line along the axis of a filament and in a
perpendicular plane. A central source with a continuous flat spectrum is located at the intersection of the
filament and the plane. The filament
is cylindrical with radius 1 comoving Mpc and homogeneous with a density 64 times the density of
the surrounding medium which is at the average baryon density of the universe ($z \sim 10.$).}
\label{filament_profiles}
\end{figure}
\section{ Conclusions}
The main goal of this paper was to investigate a source of fluctuations in the brightness
temperature of the 21-cm emission during the EoR usually neglected in numerical simulations. The
Wouthuysen-Field effect, which couples the hydrogen spin temperature to the kinetic temperature
of the gas, is regulated by the crucial parameter $P_\alpha$: the number of scatterings per atom
per second. While all previous simulations of the 21-cm emission used a uniform value for
$P_\alpha$, Loeb and Barkana (2005) have shown, in a simple theoretical framework, that several
sources of fluctuations in $P_\alpha$ can modify the power spectrum of the 21-cm emission. We
studied how a full 3D radiative transfer treatment of the Ly-$\alpha$ line in a cosmological
context modifies the picture given by Loeb and Barkana.
The first step was to implement and validate the Ly-$\alpha$ radiative transfer in LICORICE. We
used a Monte-Carlo approach, and implemented an algorithm and acceleration schemes similar to
those of other existing codes. We discarded physical processes such as recoil or deuterium
contribution, which have negligible effect for the Ly-$\alpha$ transfer during the EoR. On the
other hand we took care to compute the optical depth accurately in an expanding cosmological
medium, without
any resolution effects. We presented three validation tests for the code. The first two are
classical setups: a monochromatic source in a static uniform sphere of gas, and in an
expanding sphere of gas. The agreement for the emerging spectrum with analytic solutions and
results by other authors is good. The third test, the mean intensity map for a monochromatic
source in an expanding sphere of gas at $T=0$K, focuses on a quantity more closely related to
$P_\alpha$. The comparison with the analytic solution provided by Loeb and Rybicki is good where
the analytic solution is valid: in the diffusion regime. This set of tests strongly suggests
that the LICORICE is valid.
Barkana and Loeb (2005) show that several factors contribute to the fluctuations in the local
Ly-$\alpha$ flux: the $1/r^2$ scaling of the flux, and the Poisson noise and clustering in the
sources distribution. They assumed, however that photons redshift freely from the source until
they reach the local Ly-$\alpha$ frequency, and only then scatter off hydrogen atoms. In other
words, they neglected wing scattering. We computed the $P_\alpha$ profile for a source with a flat
continuous spectrum in a uniform expanding medium. We showed that the effect of taking into
account the wing scatterings in a full radiative transfer code is to steepen the profile at small
scales to a $\sim 1/r^{2.3}$ profile (with our choice of parameters). At large scales, or when
wings are suppressed, we recover the $1/r^2$ profile. Thus we may expect, at small scales, more
power in the 21-cm emission than predicted by Barkana and Loeb. But a yet stronger effect was
obtained when we introduced fluctuations in the density of the gas surrounding the source. We
investigated a central clump setup and an isothermal profile. In both cases we observed
alterations in the $P_\alpha$ profile. In the case of an isothermal density profile we found
a $\sim 1/r^3$ profile for $P_\alpha$. This also suggests stronger fluctuations than predicted
by Barkana and Loeb, but, once again, mainly at small scales since the $\sigma_8$ at $z=10$ is
only $\sim 0.1$: large scale structures had not enough time to grow yet.
Finally, we tried to create a more realistic situation by placing a flat spectrum source inside
a filament of overdense gas. We observed a sharp depletion of $P_\alpha$ inside the filament.
The photons scatter out of the filament before they reach the core of the line. This
fluctuation inside the filament is once again a small scale feature, and may be difficult to
catch with LOFAR or SKA. However, the presence of the filament also produced oblate contours
for $P_\alpha$ at larger scales ($> 10$ comoving Mpc), in the surrounding medium. This effect
may be more within reach of the resolution of these instruments.
{The increased fluctuations of $P_{\alpha}$ due to radiative transfer effects and to the
inhomogeneous distribution of the gas translate linearly into fluctuations of the $x_\alpha$
coefficient. On one hand, we expect these fluctuations to be globally significant and produce
brightness temperature fluctuations only as long as
$x_\alpha \sim 1$, i. e. as long as the coupling does not saturate to $T_s = T_K$. This
occurs in the early phase of reionization. It is not possible to be much more specific in terms of
redshift because the Ly-$\alpha$ pumping efficiency depends strongly on the model for the source
type and formation history. On the other hand we showed that density fluctuations in the
gas can create fluctuations of $P_{\alpha}$ of a factor greater than $10$. The depleted regions
will fill up only when the amount of young sources gets more than $10$ times larger. Cosmological
simulations suggest that this corrresponds to a change of redshift between 1 and 2 around redshift 10. Consequently we expect that the fluctuations of $P_{\alpha}$ due to inhomogeneous gas will
lead to a longer survival of depleted regions where $T_S$ remains coupled to $T_{\mathrm{CMB}}$. One may argue that the strongest fluctuations of the gas density are located around the sources
and are ionized very early. However numerical simulations have shown that the ionization front
is not spherical: it is trapped in the high density regions such as filaments pointing to the
source, where reionization is much delayed (see for example, Gnedin 2000). So, these filaments should be able to play their role
in creating $P_\alpha$ fluctuations. }
We have not investigated the effect on $P_\alpha$ of an anisotropic peculiar velocity field
around the source. In principle, it would also induce non spherical contours. However, cosmological
simulations in a $20$ Mpc box suggest peculiar velocities of the order of $~100$ km.s$^{-1}$
during the EoR, when the Hubble constant is H(z=10) $\sim 1000$ km.s$^{-1}$.Mpc$^{-1}$. Moreover,
only velocity
differences will alter $P_\alpha$ contours, which should be smaller than $~100$ km.s$^{-1}$ at large scales.
We estimate that the impact of the velocity field of the gas is smaller than the impact of the
density fluctuations. However, LICORICE fully implements the effects of the gas peculiar velocity,
and it will be taken into account in the future simulations of a cosmological box.
Another process will have to be included in the future: the effect of higher Lyman series lines.
Barkana \& Loeb (2005) and Pritchard and Furlanetto (2006) have shown that these lines, having
horizons closer to the source than the Ly-$\alpha$ line, add to the power of the $P_\alpha$
fluctuations close to the sources. In a forthcoming paper, we will apply LICORICE to a
cosmological field during the EoR using simulation outputs from the HORIZON
project\footnote{http://www.projet-horizon.fr}, and compute the resulting 21-cm brightness temperature map. If,
as we believe, we find significant modification of the predictions, we will include
higher Lyman series lines.
\begin{acknowledgements}
This work was realized in the context of the SKADS and HORIZON projects.
\end{acknowledgements}
|
1,116,691,499,292 | arxiv | \section{Introduction}
Exactly solvable models are key players in few- and many-body quantum mechanics and are indeed also
fascinating creatures \cite{mattis,sutherland}. The analytical intractability
of general $N$-body systems makes it extremely important to have exact results to allow
benchmark tests of complicated numerical methods. Unfortunately, exactly solvable models are
few and far between and are more often found in low-dimensional systems. In the case of
one spatial dimension, the famous Bethe ansatz \cite{bethe1931} was succesfully
applied to models of bosons with zero-range interactions \cite{lieb1963a,lieb1963b,macguire1964},
and later to interactions with long range \cite{sutherland1971,calogero1971}.
Meanwhile, an exciting direction in the field of
cold atomic gases aims at the study of
low-dimensional systems in general and one-dimensional setups in particular \cite{bloch2008}.
A noticable highlight of this pursuit is the experimental realization of the so-called Tonks-Girardeau
gas \cite{paredes2004,kinoshita2004,haller2009}, where strongly-interacting one-dimensional
bosons become impenetrable objects and behave similar to fermions \cite{tonks1936,girardeau1960}.
Very recently, it has even become possible to study this interesting regime in the limit of
small particle number \cite{serwane2011,zurn2012}.
Here we study an exactly solvable model of an $N$-body system in an external parabolic
confinement. The outer trap is always present in cold atomic gas experiments but can
often be neglected or treated in a local density approximation when large systems are
studied. However, for smaller particle numbers the effect of the outer trap becomes
important for the structure and dynamics of the system. To make the $N$-body problem
tractable we use a harmonic Hamiltonian approximation \cite{calogero1971} with carefully
chosen parameters that reproduce essential features of two atoms interacting via short-range
interactions in a parabolic trap \cite{arms11}.
The paper is organized as follows. After a discussion of the harmonic methods we
provide details on how the parameters of the harmonic interactions are obtained
from knowledge of the exact solution of the two-body problem originally obtained
by Busch {\it et al.} \cite{busc98}. We then present results for the ground-state
energies and the radii. Our focus is on bosonic systems, but we do present a few
results for fermions as well. A comparison to the exact results of MacGuire \cite{macguire1964}
is then made in relevant limits. We also compute the one-body density matrix and its
largest eigenvalue to obtain the condensate fraction at zero temperature. Lastly,
we discuss the thermodynamics of our model and then proceed to conclusion and outlook
for future work.
\section{Method}
We consider a system of $N$ quantum particles interacting pairwise via a delta
function interaction and confined by a harmonic external potential in one-spatial
dimension. The Hamiltonian for this system is
\begin{equation}
H=-\frac{\hbar^2}{2m}\sum_{i=1}^N\frac{\partial^2}{\partial x_i^2} +\frac{1}{2}m\omega_0^2\sum_{i=1}^Nx_i^2-\frac{2\hbar^2}{ma}\sum_{i<k}\delta^{1D}(x_i-x_k),
\label{em1}
\end{equation}
where $m$ is the mass of the particles, $\omega_0$ is the frequency of the
external field, and $a$ is the one-dimensional scattering length which parameterizes the
strength of the two-body interaction (we will discuss its relation to the three-dimensional
scattering length below).
The external field, $\omega_0$, defines the length scale of our
system, $l^2=\hbar/(m\omega_0)$. For two particles, (\ref{em1}) has
been solved in, e.g., \cite{busc98,farr10}, and their results of two-body
energies and wave functions for a given scattering length are used to determine
the parameters of our model. We consider only the bound molecular branch
of the system, i.e., $a>0$.
In our general harmonic approximation scheme we replace Hamiltonian (\ref{em1}) with
\begin{eqnarray}
H&=&-\frac{\hbar^2}{2m}\sum_i^N\frac{\partial^2}{\partial x_i^2}
+\frac{1}{2}m\omega_0^2\sum_i^Nx_i^2\nonumber\\
&& +\frac{1}{2}\mu\omega_{in}^2\sum_{i<k}(x_i-x_k)^2+\sum_{i<k}V_S,
\label{em2}
\end{eqnarray}
where $\mu=m/2$ is the reduced mass of the two-body system, $\omega_{in}$ is
the interacting frequency, and $V_S$ is an energy shift. The solution
to this equation for $N$ particles for general systems is described in detail
in \cite{arms11} and more specifically for identical particles
in \cite{calogero1971} and \cite{arms12}.
The
parameters $\omega_{in}$ and $V_S$ are now chosen to fit pertinent
properties of the original Hamiltonian at the two-body level in the
usual spirit of constructive descriptions of $N$-body systems based on
two-body interactions. Here we impose the condition that at the two-body
level the harmonic oscillator reproduces the energy and average square
radius of the exact two-body solution. In order to fulfil these
constraints, we use the size of the system to determine the
interaction frequency through the relation
\begin{equation}
\frac{\langle\psi|x^2|\psi\rangle}{\langle\psi|\psi\rangle}=\frac{\hbar}{2\mu\sqrt{\omega_{in}^2+\omega_0^2}},
\label{em7}
\end{equation}
where $x=x_1-x_2$ is the relative coordinate in the two-body system.
The energy shift is determined by requiring that the model reproduces the energy of the two-body system
\begin{equation}
E_2=\frac{1}{2}\hbar\sqrt{\omega_{in}^2+\omega_0^2}+V_S.
\label{em8}
\end{equation}
The quantities $E_2$ and $\langle x^2\rangle$ can be easily obtained by numerically solving the
transcendental equations fulfilled by the exact solution for two bosons (or two fermions in different
spin states) interacting via a zero-range interaction in a parabolic trap \cite{busc98}.
Note that when we consider identical fermions below, there can in principle only be a
non-zero two-body interaction in states that are odd under exchange of the two fermions.
In this paper we are mostly interested in the scaling behaviour with particle number
of the fermions as the interaction frequency is varied (through the scattering length) and
we therefore ignore this point and use the same input for fermions as for bosons. The
most important difference is of course the quantum statistics which is taken fully into
account.
We note that in cold atoms, effective one-dimensional setups are created by using a tightly
confining potential in two transverse directions, usually through the application of an
optical lattice \cite{bloch2008}. In a deep transverse lattice, the atoms are then effectively only occupying
the lowest transverse motional degree of freedom. This transverse degree of freedom, however,
has an influence on the effective one-dimensional scattering length that described the inter-atomic
interaction in the system. A mapping exists between the true three-dimensional scattering
length and the effective one-dimensional scattering length that takes the transverse degrees
of freedom explicitly into account \cite{olshanii98}. This mapping should thus be applied
before comparison to realistic experiments.
Once the parameters have been determined, (\ref{em2}) can be solved.
The ground state energy for identical bosons is
\begin{equation}
E_{gs}^{(b)}=\frac{1}{2}N(N-1)V_S+\frac{1}{2}(N-1)\hbar\omega_r+\frac{1}{2}\hbar\omega_0,
\label{em3}
\end{equation}
where
\begin{equation}
\omega_r^2=N\omega_{in}^2/2+\omega_0^2
\label{em5}
\end{equation}
is the $(N-1)$-degenerate frequency that comes out of the solution of (\ref{em2}) \cite{arms11}.
The last term in (\ref{em3}) is the energy related to the center of mass motion
and will be ignored as we are interested in the internal dynamics of the system.
Though most of this work concerns bosons, input for identical fermions is also possible.
Since fermions obey the Pauli Principle, $N-1$ particles must be placed in higher
oscillator levels as there are no degeneracies in one dimension.
We can obtain the amount of energy
in the ground state by filling
consecutive oscillator levels, i.e.
\begin{equation}
\hbar\omega_0\sum_{k=0}^{N-1}\left(k+\frac{1}{2}\right)=\frac{N^2}{2}\hbar\omega_0.
\end{equation}
The complete ground state energy is then
\begin{equation}
E_{gs}^{(f)}=\frac{1}{2}N(N-1)V_S+\frac{N^2}{2}\hbar\omega_r+\frac{1}{2}\hbar\omega_0.
\label{em4}
\end{equation}
This is a different scaling than for the bosons in (\ref{em3}). In the fermion case,
the energy is dominated by the $N^2\hbar\omega_r/2$ term which scales as $N^{5/2}$.
\begin{figure}[ht!]
\includegraphics[width=0.5\textwidth]{fig1.eps}
\caption{ Top panel: Energies per particle for bosons confined on one dimension. The $y$-axis is stopped at -20 in order to show the general behaviour of many curves, though the shortest scattering lengths quickly leave the figure. The scattering lengths plotted, moving from bottom to top are 0.4., 0.5, 1.0, 2.0, 5.0, 10.0, and 100.0
Bottom panel: Energies divided by $N^2$ for the bosons to show their scaling behaviour with $N$. The order of the scattering lengths from bottom to is the same as in the top panel.}
\label{fr1}
\end{figure}
\section{Results}
We now present numerical results for systems with $N=3-30$ particles, studying
their energetics, the radial behaviour of the systems, and the one-body
density matrix. Here we focus on
the case of identical bosons. A particular issue is the ratio of the three-body
energy to the two-body energy. An exact formula of MacGuire \cite{macguire1964}
applies to one-dimensional systems of bosons with zero-range interactions and
we make a comparison of the harmonic results to that model in the strongly-bound
limit where the two-body energy is large and negative. Lastly, we consider also
the thermodynamics of the system within the harmonic approximation.
\subsection{Energies and radii}
The ground state energy for bosons is shown in figure \ref{fr1} and that
for fermions in figure \ref{fr2} for different particle numbers and scattering
legnths. Notice that the upper panels show the energy per particle, while in the
lower panels the energy is divided by a different power that we will discuss below.
A striking feature to notice is the positivity of the energy for fermions, while
that of the bosons can have both signs. This is a consequence of the Pauli
principle which implies that the fermions will have to occupy higher
orbitals for lack of degeneracies in one dimension as discussed above.
Even in the case of very small scattering lengths (and thus large and negative two-body
binding energies) the contribution from the higher orbits makes the overall
ground state energy positive.
Once the two-body problem is solved and the parameters are determined, the ground
state energy for bosons is given by (\ref{em3}). As seen in figure \ref{fr1},
the sign of the energy of the Bose system is mostly negative, but for small
systems at large scattering lengths, the energy turns positive.
This is understood from the fact that the energy shift, $V_S$,
is negative and the shift term in the energy of (\ref{em3}) scales with $N^2$
while the positive oscillator frequency term scales with the lower power $N^{3/2}$.
For bosons, the energy is positive for all particle numbers examined for $a/l=100$.
Using (\ref{em3}), one can derive the critical number for bosons where the
energy changes sign. It is given by
\begin{equation}
N_{crit}=\frac{\hbar^2\omega_{in}^2+2\hbar\sqrt{\hbar^2\omega_{in}^4/4+4V_S^2\omega_0^2)}}{4V_S^2}.
\end{equation}
From this relation one can calculate that for bosons the energy
becomes negative at $a/l=100$ when $N\geq 116$.
\begin{figure}[ht!]
\includegraphics[width=0.5\textwidth]{fig2.eps}
\caption{Top panel: Energies per particle for fermions confined on one dimension.
The plotted scattering lengths are in the opposite order from the bosons as moves
from the bottom to the top of the plot.
Bottom panel: Fermion energies divided $N^{5/2}$ to clearly show the scaling
of the energy with $N$.}
\label{fr2}
\end{figure}
The bottom panels of Figures \ref{fr1} and \ref{fr2}
show the energies divided by the relevant scaling factor ($N^2$ for bosons and $N^{5/2}$
for fermions) to emphasize the scaling behaviour at large $N$. In the figures we
can see that the asymptotic scalings are nicely approached when the scattering length
is not too small. For the case of small scattering length, the two-body energy will
in general behave as $E_2\sim -1/a^2$ while $\langle x^2\rangle\sim a^2$,
since in this limit the trap can be ignored
on the bound state branch in the spectrum that we study here. However, the
relations in (\ref{em7}) and (\ref{em8}) that we use to determine our oscillator
parameters now imply that both $V_S$ and $\omega_{in}$ must scale with $a^{-2}$
to be fulfilled (remember that $E_2$ is negative). This means that there will
be a competition between the terms in the ground state energies given by (\ref{em3}) and
(\ref{em4}). The asymptotic behaviour for large $N$ is therefore approached more slowly
for small $a$.
We also calculate the relative size of the bosonic ground state wave
functions, $\langle \left(X-X_{CM}\right)^2\rangle$, as a function of
particle number for several different scattering lengths. Note here that
$X$ denotes the single particle coordinate of one of the bosons and that no
index is needed since the particles are identical.
The results are
shown in figure \ref{fr5}. One can see for the large scattering lengths that
the radius increases before eventually decreasing for larger particle numbers
(more than 20 for $a/l=100$). For scattering lengths of two and smaller, the
size decreases monotonically with increasing particle number. Again this is
connected to the fact that the small $a$ regime has very strong two-body
binding and thus small $\langle x^2\rangle$ which is imprinted on the $N$-body
system which tends to be very compact and presumably leads to strong clusterization
in the real system followed by loss of atoms from the trap.
\begin{figure}[ht!]
\includegraphics[width=0.5\textwidth]{fig3.eps}
\caption{ Radius $(x-X_{CM})^2$ for bosons as a function of particle number
for several different scattering lengths. The scattering lengths are,
from bottom to top, 0.4, 0.5, 1, 2, 5, 10, and 100.}
\label{fr5}
\end{figure}
For fermions (not shown), the Pauli principle
and the subsequent need to occupy higher orbitals means that the system will
in general be larger for the same scattering length and will not show a decreasing
behaviour as $N$ is increased, but rather a slight increase. This can be
seen from the energetics of the ground state in (\ref{em4}) which is
dominated by the positive term containing the oscillator frequency for
large $N$. This implies that the radius will stay large, constrained
solely by the external trap.
We notice that the
dimensional effect on the radius is quite clear when comparing to previous
studies of bosonic systems in two and three dimensions \cite{arms11}.
In the current one dimensional study, we see a non-monotonic behaviour in the
radii as function of $N$ which can also be seen in two dimensions but which is
almost completely absent in the corresponding three-dimensional system. The
scattering lengths for which the maximum in the size occurs are roughly those
where the three-body energy comes out positive. This implies an increased
radial size, although still restricted by the external trapping potential.
For large $N$, however, the bosons will always become negative in energy
as the shift term dominates in (\ref{em3}) and the radius goes down again.
The fact that lower dimensions show a more pronounced non-monotonicity of the
size can then be traced to the fact that the positive energy contribution
from zero-point motion grows with dimension and washes out the behaviour.
For fermions, as discussed above, the energy is positive from the start
and the radius stays large. However, the degeneracies allowed in higher
dimensions adds extra ingredients and shell-structure to both energy and
to the radial size.
\subsection{Three boson energies}
We now consider the case of three bosons within our harmonic
approximation scheme in order to make a comparison to the
results obtained by MacGuire \cite{macguire1964} in the absence
of external confinement. The exact bound state energy, $E_N$, of $N$ bosons interacting
via attractive pairwise zero-range interactions in a
homogeneous one-dimensional space can be expressed in term
of the two-body energy, $E_2$, as
\begin{equation}
E_{N}=\frac{1}{6}N\left(N^2-1\right)E_2.
\end{equation}
For the case of three particles this becomes $E_3=4E_2$. Since we
have an external trap, we do not expect to reproduce this result.
However, in the limit $a\to 0$ where $E_2\to -\infty$ the external
trap should become negligible and a comparison can be made.
We plot the ratio $E_3/E_2$ as a function of
scattering length in figure \ref{fr3} and as a function of $E_2$ in figure \ref{fr4}.
The plot ends at $a/l\sim 0.25$ at which point it becomes numerically
very challenging to compute the wave function based on the exact
solution of Busch {\it et al.} \cite{busc98}. However, beyond that we
can use the exact solution as discussed below.
One can see in figure \ref{fr3} as the scattering
length decreases (more clearly seen as $E_2\to -\infty$ in figure \ref{fr4}),
that the ratio approach a limit. Using (\ref{em3})
and (\ref{em8}) in the limit where $a\to 0$ and $\omega_r\to\omega_{in}$ (see (\ref{em5})), we get that
\begin{equation}
\frac{E_3}{E_2}=3+\left(\sqrt{\frac{3}{2}}-\frac{3}{2}\right)\frac{\omega_{in}}{E_2}.
\end{equation}
Relating $\omega_{in}$ to $\langle x^2\rangle$ through (\ref{em7}), we obtain
\begin{equation}
\frac{E_3}{E_2}=3+\left(\sqrt{\frac{3}{2}}-\frac{3}{2}\right)\frac{\hbar^2}{2\mu E_2\langle x^2\rangle}.
\end{equation}
We can now use the exact wave function for a delta function potential in one
dimension, $\psi(x)=Ae^{-\kappa x}$ where $\kappa=\sqrt{-2\mu E_2/\hbar^2}$,
to obtain $\hbar^2/(2\mu E_2\langle x^2\rangle)=2$. Our limit thus becomes
$E_3/E_2\approx 3.55$. This is also the number we find numerically for the
smallest value of $a$ that we could access. Comparing to the exact
value of MacGuire, $E_3/E_2=4$, this implies that the energetics of
our model is accurate to around 10\% for small scattering
lengths. The wave function in the harmonic model is a
gaussian which in the limit $a\to 0$ will tend to a delta
function, similar to the exact solution, so we also expect the
harmonic model to provide an accurate structural description in
this limit.
On the other hand, when $a$ becomes very large (the unitarity limit)
the trap plays an important role. Looking back at the basic
Hamiltonian in (\ref{em1}), we see that when $a\to\infty$, the
interaction term tends to zero and we should be dominated by
the trap only. However, the exact result does show that there
is a shift of the energy in this limit so that $E_2=\hbar\omega/2$
for $a\to\infty$ (the non-interacting system has twice the
energy since there is zero-point motion from both particles).
Within our model we have $\omega_r\to\omega_0$ and $\langle x^2\rangle\to l^2/2$
when $a\to\infty$
as the external field provides the only length scale left in the problem.
We have
\begin{equation}
\frac{E_3}{E_2}=3-\frac{\hbar\omega_0}{2E_2},
\end{equation}
and in the large $a$ limit, $E_2\to\hbar\omega_0/2$, so we obtain a limiting
value of $E_3\to 2$. At $a/l=100$, we numerically obtain 1.98 for this ratio so
we reproduce this limit. The wave function becomes essentially exact in this
limit, and we thus expect the structure to be well reproduces also.
In the limit of large $N$, we can also estimate the scalings and compare to the
exact results when $a\to 0$. From (\ref{em3}) we have that the bosons scale
as $E_N\sim N(N-1)E_2$, which means that we have an underbinding by a factor
of $N+1$. Fermions in (\ref{em4}) are different since here we have $E_N\sim N^{5/2}E_2$,
i.e. the underbinding is only by a factor of about $\sqrt{N}$. Notice, however,
that in the opposite limit of $a\to \infty$, the dominant term in (\ref{em3})
and (\ref{em4}) change as $V_S\to 0$ and $\omega_r\to\omega_0$. This is an
essentially non-interacting situation and we have the intuitively obvious
result $E_N\sim N\hbar\omega_0/2$, which is just the scaling dictated by
the trap. For values of $a$ away from these two limits, the scaling is
an interplay of both terms and we get a more complicated behaviour. We note
that one could also turn these arguments upside down and use the exact
results for $N$ particles in the $a\to 0$ limit to fit the parameters
of the model, for instance by using $E_3=4E_2$ instead of either the
frequence of radius condition we impose on the two-body problem. Overall,
we expect this to provide only minor quantitative changes in the harmonic
model and its predictions.
\begin{figure}
\includegraphics[width=0.5\textwidth]{fig4.eps}
\caption{The ratio $E_3/E_2$ as a function of scattering length for bosons.
We plot only the cases where both $E_3$ and $E_2$ are negative.}
\label{fr3}
\end{figure}
\begin{figure}
\includegraphics[width=0.5\textwidth]{fig5.eps}
\caption{The ratio $E_3/E_2$ as a function of $E_2$ for bosons.
We plot only the cases where both $E_3$ and $E_2$ are negative.}
\label{fr4}
\end{figure}
\subsection{One-body density}
The one body density matrix
\begin{equation}
\rho(x_1,x_1')=N\int\psi^*(x_1,\dots,x_N)\psi(x_{1}^{'},\dots,x_N)dx_2\dots dx_N,
\end{equation}
is the most basic correlation function and is the starting point of statistical
property calculations. It contains information about the mean-field nature of
the wave function for a bosonic system.
It is of interest in itself, but we will focus on its
largest eigenvalue, $\lambda$, which directly measures how much a given state
has the structure of a coherent state. Figure \ref{fr6} shows $\lambda$ as a
function of the number of bosons for several scattering lengths. We mainly
see an increase with particle number, though there are slight decreases for small
particle numbers and large scattering lengths. In contrast with our previous
results in \cite{arms11} for higher dimensions, only scattering lengths of order
unity or smaller show a $\lambda$ much different than one, showing that the
persistence of the mean-field structure provided by the external potential is
quite strong in one dimension implying that the mean-field approximation is
very good. The difference to higher dimensions arise from the degeneracies
and from the reduction implied by the fact that one can factorize
$\lambda$, so that for instance $\lambda_{3D}=\lambda^3$ (see \cite{arms11}).
\begin{figure}
\includegraphics[width=0.5\textwidth]{fig6.eps}
\caption{ The value of $\lambda$ at several values of the 1D scattering length.
From bottom to top they are 0.4, 0.5, 1, 2, 5, 10, and 100.}
\label{fr6}
\end{figure}
\subsection{Thermodynamics}
The energy spectrum can be obtained by considering center-of-mass excitations
in the system and also the internal excitations characterized by the
frequency in (\ref{em5}).
To address the thermodynamics of the systems, one can proceed as
described in detail in \cite{arms12}. One naturally starts with the partition function
\begin{equation}
Z(T)=\sum_i g_i\exp[-E_i/(k_BT)],
\end{equation}
where $E_i$ is the energy of the $i$th state, $g_i$ is the degeneracy of that
state, and $k_B$ is Boltzmann's constant.
1D systems are simpler since there is no degeneracy in the energy levels in contrast
to higher dimensions. With the sequence of energies and single-particle degeneracies,
thermodynamic quantities can be calculated. Once the spectrum and degeneracies
of our systems have been worked out,
we can calculate basic thermodynamic quantities in the canonical
ensemble. In 1D, the number of states grows much more slowly than in higher
dimensions, though one does have to climb somewhat high in the spectrum due to
the lack of degeneracy. In fact, in 1D there is no difference between bosons
and fermions in the sequence of the degeneracies of the $N$-body excited states. This
comes from the fact that all the Pauli principle implies for fermions is that the
ground state has them occupying higher orbit which increases the ground state
energy in comparison to bosons which can all go into the lowest orbital. Excitations
on top of these ground states will now cost the same amount of energy and they will all
have the same degeneracy since it is simply a matter of distribution of a number of
particles in levels to obtain a specific total energy cost (the energy of the excitation
above the ground state).
This latter fact implies that
some thermodynamic quantities such as the entropy and heat capacity are identical
for bosons and fermions at the same scattering length. Working in the
canonical ensemble, in figure \ref{fr7} we show the heat capacity $C$, which is
the same for bosons and fermions. The heat capacity is related to the partition
function by
\begin{eqnarray}
C=\frac{\partial E}{\partial T},\\
E=-k_BT^2\frac{\partial \log Z}{\partial T}.
\end{eqnarray}
For all particle numbers, we see an initial increase, which then slows
down at higher temperatures. This initial increase is the saturation
of the center of mass mode. The temperature, where the slope changes,
depends on particle number since more particles present in the system implies that
the center of mass mode comprises a smaller share of the overall excitation energy.
At the larger scattering lengths, the heat capacity then gradually increases
towards the equipartion limit value of one. At small scattering lengths,
the large size of the excitation frequency sharply delays the approach to
the equipartition limit. For the same scattering length, more particles
mean a slower approach to the high temperature limit. This is clear from
(\ref{em5}), which shows that the excitation frequency increases with
particle number.
\begin{figure}
\includegraphics[width=0.5\textwidth]{fig7.eps}
\caption{The heat capacity per particle at several boson numbers and
scattering lengths, identified as $N, a$, as a function of temperature.}
\label{fr7}
\end{figure}
\section{Summary and outlook}
We have presented results for an $N$-body system of particles
moving in one dimension and interacting via pairwise
zero-range interactions under the influence of an external harmonic oscillator
trap. The Hamiltonian is solved analytically within the harmonic approximation
where the parameters of the model are adjusted to reproduce properties of the
exact solution of the corresponding two-body problem. This is along the
lines of the standard approach to many-body problems in many fields of
physics.
The ground state energies for both bosonic and fermionic systems were
determined and we discussed the scaling of these quantities with the particle
number. We also obtained the radial size of the bosonic systems and compared
to previous findings in two- and three-dimensional setups. A careful study
of the energy of three bosons in the harmonic model reveals that in the
limit of large two-body binding energy where the external trap can be
neglected, the harmonic model results are about 10\% below the exact
result abtained by MacGuire \cite{macguire1964}, while in the opposite
limit of negligible two-body binding energy (large scattering length)
the model becomes essentially exact. For larger particle numbers, the
harmonic approximation tends to underbind the system compared to the exact
results when the two-body system is strongly bound.
Additionally, we studied the one-body density matrix. The main goal was
the computation of its largest eigenvalue to characterize the degree
of coherence in the system. The results turn out to be similar to those
obtained in two and three dimensions, except that the one-dimensional
coherence turns out to be more robust except for large two-body binding
energies. Lastly, we studied the thermodynamics of the one-dimensional
system. The specific heat was found to behave in a manner that is also
quite similar to higher dimensions. However, it is worth noticing that
within the harmonic model, the spectrum of excitation for bosons and
fermions turns out to be exactly the same. Due to the lack of degeneracy
of the levels in one dimensions, the only difference is therefore found
in the ground state energy.
The one-dimensional setup studied here provides a very valuable benchmark for
the harmonic models that have been used in higher dimensions in different
contexts \cite{calogero1971,zaluska2000,yan2003,gajda2006}. Another good
test ground for the harmonic models is systems that have long-range
interaction. In particular, systems of dipolar molecules in one-dimensional
tubes have been studied recently due to their interesting few- and
many-body structure \cite{arguelles2007,kollath2008,chang2009,huang2009,wunsch2011,dalmonte2011,zinner2011,fellows2011,knap2012,leche2012}.
The dipole-dipole interaction in two different one-dimensional tubes
is typically characterized by having a pocket which becomes deep
when the dipolar interaction is strong \cite{zinner2011}. This implies
that a harmonic approximation is a good starting point. This approach
has been studied in the case of two-dimensional layers with
dipolar particles \cite{wang2006,jeremy2010,zinner2011d,jeremy2011} and
compares very well to exact results \cite{artem2011a,artem2011b,artem2012}
away from the weak-coupling limit. The application of the harmonic
approximation to one-dimensional arrays containing dipolar particles is
a topic for future work.
Another interesting direction in which the harmonic model can be used
is for the study of higher-order terms in the interaction such as
those coming from effective-range corrections to the model of
Busch {\it et al.} \cite{busc98,zinner2012}. Recent experiments have
shown a potential need for such corrections in both two- \cite{baur2012}
and three-dimensional cold atomic gas systems \cite{ohara2012}. A
possible way to account for such corrections would be to include quartic
interaction terms an then carefully fit the additional parameter to few-body
properties. Within the harmonic model, the quartic interactions can be
computed quite straightforwardly through gaussian integrals. This
could then be compared to mean-field results that include effective-range
corrections in the limit of large particle numbers \cite{fu2003,zinner2009a,zinner2009b,thoger2009}.
|
1,116,691,499,293 | arxiv | \section{Introduction}
After Einstein published the general theory of relativity in 1915,
numerous attempts were made to generalize the theory in such a way
as to encompass both gravitation and electromagnetism in a unique
geometrical structure. Three historically important such
generalizations which are relevant to our present work here are the
Einstein-Schrodinger theory, the Weyl Geometry and the Kaluza-Klein
theory, briefly reviewed below.
The Einstein-schrodinger Theory \cite{1} is a generalization of vacuum general
relativity which allows for non-symmetric fields. Such a theory, without a
cosmological constant, was first proposed by Einstein and Straus \cite{5, 6, 7,
8, 9}. Later on Schrodinger showed that it could be derived from a very simple
Lagrangian density if a cosmological constant was included \cite{10, 11, 12}.
Einstein and Schrodinger suspected that the theory might include
electrodynamics, but no Lorentz force was found \cite{15, 16} when using the
Einstein-Infeld-Hoffmann (EIH) method \cite{15,16b}. For detail study of
non-symmetric fields take a look at \cite{9}.
The Weyl Geometry \cite{2} came into being after Einstein put forth
his general theory of relativity, which provided a geometrical
description of gravitation. Later on, Weyl proposed a more general
setup which also included a geometrical description of
electromagnetism. In the case of general relativity one has a
Riemannian geometry with a metric tensor $g_{\mu\nu}$ \cite{2aa}. If
a vector undergoes a parallel displacement in this geometry, its
direction may change, but not its length. In Weyl geometry, there is
a given vector $k^{\mu}$ which, together with $g_{\mu\nu}$,
characterizes the geometry. For any given vector $\xi^{\mu}$
undergoing parallel displacement in this geometry, not only the
direction but also the length $\xi$ may change and this change
depends on $k_{\mu}$ according to the relation
\begin{equation}
d\xi=\xi k_{\mu} dx^{\mu}, \label{eq1}
\end{equation}
or
\begin{equation}
\xi=\xi_{0}e^{\int{k_{\mu} dx^{\mu}}}, \label{eq2}
\end{equation}
where $\xi_{0}$ is the length of the original vector before
displacement. The change in the length of $\xi$ in going from one
point to another depends on the path followed, i.e., length is not
integrable. Mathematically the Weyl geometry is rich but it seems
not to coincide with nature. In general relativity and other main
theories of physics, measuring tools are the same for every event.
Therefore, We have to use the same clocks and rulers in our
experiments. In the Weyl geometry one can arbitrarily regauge the
measuring tools for every event. Einstein is among those who have
objected the theory. A very detail review on the Weyl geometry can
be found in \cite{2a}.
The Kaluza-Klein (KK) theory \cite{3a, 3aa, 3, 4} is perhaps the
most successful of any generalization of general relativity.
Historically, the unification of gravity and electromagnetism was
first addressed in this model by applying Einstein's general theory
of relativity to a five, rather than four-dimensional space-time
manifold. The field equations would logically be expected to be
$\hat{G}_{AB}=k\hat{T}_{AB}$ in $5D$ with some appropriate coupling
constant $k$ and a $5D$ energy-momentum tensor. However, since the
latter is unknown, from the time of Kaluza and Klein onward much
attention has been paid to the vacuum form of the field equations
$\hat{G}_{AB}=0$, where $\hat{G}_{AB}=\hat{R}_{AB}-\frac{1}{2}
\hat{R} \hat{g}_{AB}$ is the Einstein tensor, $\hat{R}_{AB}$,
$\hat{R}$ are the five-dimensional Ricci tensor and scalar
respectively, and $\hat{g}_{AB}$ is the five-dimensional metric
tensor\footnote{Capital Latin indices $A$, $B$, $\cdots$ assume the
values 0 $\cdots$ 4, Greek indices run through $0\cdots 3$ and the
five-dimensional Ricci tensor and Christoffel symbols are defined in
terms of the metric as in four dimensions.}. Equivalently, the
defining equations are
\begin{equation}
\hat{R}_{AB}=0, \hspace{1 cm} A, B = 0\cdots 4. \label{eq5}
\end{equation}
These 15 relations serve to determine the 15 components of the
metric $\hat{g}_{AB}$, at least in principle. In practice, this is
impossible without some starting assumption on $\hat{g}_{AB}$.
Kaluza was interested in electromagnetism and realized that
$\hat{g}_{AB}$ can be expressed in the form of a 4-potential
$A_{\alpha}$ which appears in Maxwell's theory. He adopted the
cylinder condition, namely the independence of the components of the
metric to the fifth coordinate, but also assumed
$\hat{g}_{44}=\mbox{const.}$ Here we look at the more general case
where $\hat{g}_{44}=-\phi^2 (x^{\alpha}).$ The coordinates or gauges
are chosen so as to write the $5D$ metric tensor in the form
\begin{equation}
g_{AB}=\left(%
\begin{array}{cc}
g_{\alpha\beta}-\kappa^2 \phi^2 A_{\alpha}A_{\beta} & -\kappa \phi^2 A_{\alpha} \\
-\kappa \phi^2 A_{\beta} & -\phi^2 \\
\end{array}%
\right), \label{eq6}
\end{equation}
where $\kappa$ is a coupling constant. The field equations then
reduce to
\begin{equation}
G_{\alpha\beta}=\frac{\kappa^2 \phi^2}{2}
T_{\alpha\beta}-\frac{1}{\phi}(\nabla_{\alpha}\nabla_{\beta}\phi-g_{\alpha\beta}\Box\phi),
\label{eq7}
\end{equation}
\begin{equation}
\nabla^{\alpha}F_{\alpha\beta}=-3\frac{\nabla^{\alpha}\phi}{\phi}F_{\alpha\beta},
\label{eq8}
\end{equation}
and
\begin{equation}
\Box\phi=-\frac{\kappa^2 \phi^3}{4}F_{\alpha\beta}F^{\alpha\beta}.
\end{equation}
Here, $G_{\alpha\beta}$ and $F_{\alpha\beta}$ are the usual $4D$
Einstein and Faraday tensors and $T_{\alpha\beta}$ is the
energy-momentum tensor for an electromagnetic field given by
\begin{equation}
T_{\alpha\beta}=\frac{1}{2}\left(\frac{1}{4}g_{\alpha\beta}F_{\gamma\delta}F^{\gamma\delta}-
F_{\alpha}^{\gamma}F_{\beta\gamma}\right). \label{eq9}
\end{equation}
Equation (\ref{eq7}) gives back the 10 Einstein equations of $4D$
general relativity, but with a right-hand side which in some sense
represents the energy and momentum tensor. Kaluza's case
$\hat{g}_{44}=-\phi^{2}=-1$ together with the identification
$\kappa=(\frac{16\pi G}{c^{4}})^{\frac{1}{2}}$ makes (\ref{eq7})
and (\ref{eq8}) to read
\begin{equation}
G_{\alpha\beta}=\frac{8\pi G}{c^{4}} T_{\alpha\beta},
\end{equation}
and
\begin{equation}
\nabla^{\alpha}F_{\alpha\beta}=0.
\end{equation}
Note that this theory does not contain the two homogeneous Maxwell
equations, $\nabla_{(k}F_{mn)}=0$. Of course, it goes without saying
that the homogeneous Maxwell equations are constraints on the
Faraday tensor and are implicit in the KK theory. However, in what
follows, we show that within the context of the theory presented
here, these equations can be independently obtained form the torsion
free condition, in contrast to the conventional KK theory.
All the studies carried out in this regard in the past have found a
common problem, namely that the Homogeneous Maxwell equations don
not appear naturally. Also in most five dimensional theories there
is no obvious reason for the cylinder condition. We try to obtain
not only the Einstein and non-homogeneous Maxwell equations but also
homogeneous Maxwell equations and a solution to the latter by only
one assumption. In this paper we investigate a five dimensional
space-time and examine its consequences. This is done by forcing the
connection to have certain properties, thus enabling one to define
the resulting space-time quantities. We show how our assumption
leads to cylinder condition and also how the Einstein field
equations in five dimensions separate into the 4-dimensional
Einstein and Maxwell equations and that the homogeneous Maxwell
equations are included in the theory as $\hat{R}_{(nmk)4}=0$. The
corresponding energy-momentum tensor is then defined and presented.
The conclusions are drawn in the last section.
\section{Five Dimensional Riemannian Space-Time}
Let us start with the line-element for a five dimensional Riemannian space-time
\begin{equation}
d\hat{s}^2=\hat{g}_{AB}d\hat{x}^Ad\hat{x}^B,
\end{equation}
where a hat represents five dimensional objects while objects without it belong to
the ordinary $4D$ space-time. Here $\hat{x}^{\alpha}=x^{\alpha}$ are ordinary
$4D$ coordinates. Capital latin indices also run over 0, 1, 2, 3, and 4. The
contravariant components of the metric are defined as inverse of the covariant
components
\begin{equation}
\hat{g}_{AB}\hat{g}^{BC}=\delta_A^C.
\end{equation}
In such a space-time, the covariant derivatives are defined as
\begin{equation}
\hat{\nabla}_{A}\hat{f}^{B}=\partial_{A}\hat{f}^{B}+\hat{\Gamma}^{B}_{AC}\hat{f}^{C}.
\end{equation}
Now we can define a geodesic path as the straightest possible curve, i.e. a curve
whose tangent vectors are connected by parallel transport. The tangent vector to
a curve is given by $\hat{\dot{x}}^{A}\hat{e}_{A}$ where $\hat{e}_A$ are a set of
the basis vectors expanding the space and
$\hat{\dot{x}}^{A}=\frac{d\hat{x}^A}{d\tau}$. The equation representing parallel
transportation may then be written as
\begin{eqnarray}
\hat{\dot{x}}^{A}
\hat{\nabla}_{\hat{e}_{A}}\hat{\dot{x}}^{B}\hat{e}_{B}=0,
\end{eqnarray} or
\begin{eqnarray}
{\hat{\ddot{x}}^{B}+\hat{\Gamma}^{B}_{AC}\hat{\dot{x}}^{A}
\hat{\dot{x}}^{C}}=0.\label{ad}
\end{eqnarray}
The $5D$ curvature tensor, $\hat{R}^D_{_{ABC}}$, can be defined as follows
\begin{equation}
\hat{\nabla}_{A}\hat{\nabla}_{B}\hat{f}_{C}-\hat{\nabla}_{B}\hat{\nabla}_{A}\hat{f}_{C}=\hat{R}^D_{_{ABC}}\hat{f}_{D},\label{ab}
\end{equation}
where $\hat{f}_{C}$ is an arbitrary $5D$ vector. Therefore, the
curvature tensor will be as follows
\begin{equation}
\hat{R}^{D}_{_{CBA}}=\partial_{B}\hat{\Gamma}^{D}_{AC}-\partial_{A}\hat{\Gamma}^{D}_{BC}+\hat{\Gamma}^{D}_{B
E}\hat{\Gamma}^{E}_{AC}-\hat{\Gamma}^{D}_{A E}\hat{\Gamma}^{E}_{B
C}\label{a4}.
\end{equation}
The Ricci tensor in a five dimensional Riemannian space-time can
also be defined as follows
\begin{equation}
\hat{R}_{_{CA}}=\hat{R}^{B}_{_{CBA}}\label{a5}.
\end{equation}
Although the 5D Ricci scalar is defined as $\hat{R}=\hat{g}^{CA}
\hat{R}_{_{CA}}$, we do not need it in our 5D field equations any
more. One may now easily write $5D$ field equations as
\begin{equation}
\hat{R}_{_{AB}}-\frac{1}{2} \hat{\Im} \hat{g}_{_{AB}}=\kappa
\hat{T}_{_{AB}},\label{ac}
\end{equation}
where $\hat{R}_{_{AB}}$, $\hat{g}_{_{AB}}$ and $\hat{T}_{_{AB}}$ are
the 5D Ricci, metric, and energy-momentum tensor respectively.
$\hat{\Im}$ is a $5D$ scalar which can be determined using
conservation laws and will be discussed later.
\subsection{The Setup}
At this point, it is appropriate to make our main assumption
\begin{equation}
\hat{\Gamma}^{4}_{AC}=0,\label{a3}
\end{equation}
in the whole $5D$ space-time. Substituting this equation into
(\ref{ad}) leads to the fact that $\hat{\ddot{x}}^4=0$ for particles
that move on geodesics, meaning that for these particles we have
$\hat{\dot{x}}^4=constant$. The implication is that the fifth
component of the position of a particle in our space is the same as
any other. We hope our space-time to include both gravitational and
electromagnetic forces which means that even charged particles in
presence of electromagnetic waves move on geodesics. As a result, in
macroscopic scales one can say that particles never see the fifth
dimension. It is something like the cylinder condition which means
that the derivative with respect to the fifth coordinate on our $4D$
hypersurface is zero. We are now in a position to investigate the
functional form of the Christoffel symbols which we assume are
symmetric in lower indices. Let us begin by considering the well
known fundamental equation
\begin{equation}
\hat{\nabla}_A\hat{g}_{BC}=0,
\end{equation}
which is a result of
\begin{equation}
\partial_A\hat{g}_{BC}=\hat{\Gamma}^{D}_{AB}\hat{g}_{DC}+
\hat{\Gamma}^{D}_{AC}\hat{g}_{BD}.
\end{equation}
We may then write
\begin{equation}
\partial_A\hat{g}_{BC}+\partial_B\hat{g}_{AC}-\partial_C\hat{g}_{AB}=2\hat{\Gamma}^{D}_{AB}\hat{g}_{DC}.
\end{equation}
Therefore, knowing that $\hat{\Gamma}^{4}_{AC}=0$, we find
\begin{equation}
\partial_A\hat{g}_{BC}+\partial_B\hat{g}_{AC}-\partial_C\hat{g}_{AB}=2\hat{\Gamma}^{\lambda}_{AB}\hat{g}_{\lambda C}.\label{aa}
\end{equation}
Before going any further we should make it clear what we mean by the $4D$
metric. If we write the line-element as
\begin{equation}
d\hat{s}^2=\hat{g}_{\alpha
\beta}d\hat{x}^{\alpha}d\hat{x}^{\beta}+2\hat{g}_{\alpha
4}d\hat{x}^{\alpha}d\hat{x}^4+\hat{g}_{4 4}d\hat{x}^4d\hat{x}^4,
\end{equation}
it would be obvious that $\hat{g}_{\alpha \beta}=g_{\alpha \beta}$
are the covariant components of the metric tensor for the four
dimensional space-time, and the contravariant components of the $4D$
metric tensor are defined as
\begin{equation}
g_{\alpha \beta}g^{\beta \gamma}=\delta_{\alpha}^{\gamma}.
\end{equation}
Now going back to our approach to investigate the functional form of the
Christoffel symbols, we replace $C$ in equation (\ref{aa}) with $\sigma$ and find
\begin{equation}
\partial_A\hat{g}_{B \sigma}+\partial_B\hat{g}_{A \sigma}-\partial_{\sigma} \hat{g}_{AB}=
2\hat{\Gamma}^{\lambda}_{AB}\hat{g}_{\lambda
\sigma},
\end{equation}
which leads to what we were looking for
\begin{equation}
\hat{\Gamma}^{\lambda}_{AB}=\frac{1}{2}g^{\lambda
\sigma}(\partial_A\hat{g}_{B \sigma}+\partial_B\hat{g}_{A
\sigma}-\partial_{\sigma} \hat{g}_{AB}). \label{a2}
\end{equation}
One can simply check that when all capital Latin indices are replaced with Greek
indices, then $\hat{\Gamma}^{\lambda}_{\alpha
\beta}=\Gamma^{\lambda}_{\alpha \beta}$, where $\Gamma^{\lambda}_{\alpha
\beta}$ is the ordinary $4D$ connection. This is our final result. Alternatively, if
$C$ in equation (\ref{aa}) is replaced with $4$, we find
\begin{equation}
\partial_A\hat{g}_{B 4}+\partial_B\hat{g}_{A 4}-\partial_4\hat{g}_{AB}=
2\hat{\Gamma}^{\lambda}_{AB}\hat{g}_{\lambda
4},\label{ae}
\end{equation}
or
\begin{equation}
\partial_A\hat{g}_{B 4}+\partial_B\hat{g}_{A 4}=2\hat{\Gamma}^{\lambda}_{AB}\hat{g}_{\lambda
4}\mid_{on~ the~ hypersurface}.\label{ae}
\end{equation}
Now, taking all the possible values of $A$ and $B$ will arrive at
the following useful results
\begin{equation}
\partial_\alpha \hat{g}_{\beta 4}+\partial_\beta\hat{g}_{\alpha 4}=2\Gamma^{\lambda}_{\alpha \beta}\hat{g}_{\lambda
4}\mid_{on~the~hypersurface},
\end{equation}
\begin{equation}
\partial_\alpha \hat{g}_{4 4}=2\hat{\Gamma}^{\lambda}_{\alpha 4}\hat{g}_{\lambda
4}\mid_{on~the~hypersurface},
\end{equation}
and
\begin{equation}
\hat{\Gamma}^{\lambda}_{4 4}\hat{g}_{\lambda 4}=0\mid_{on~the~hypersurface}.
\end{equation}
Using equations (\ref{a3}) and (\ref{a2}), the Riemann tensor,
(\ref{a4}), will separate into
\begin{equation}
\hat{R}^{\kappa}_{_{CBA}}=\partial_{B}\hat{\Gamma}^{\kappa}_{AC}-\partial_{A}\hat{\Gamma}^{\kappa}_{BC}+\hat{\Gamma}^{\kappa}_{B
\gamma}\hat{\Gamma}^{\gamma}_{AC}-\hat{\Gamma}^{\kappa}_{A
\gamma}\hat{\Gamma}^{\gamma}_{B C}\label{a6},
\end{equation}
and
\begin{equation}
\hat{R}^{4}_{_{CBA}}=0\label{a7}.
\end{equation}
Here again it is clear that $\hat{R}^{\kappa}_{_{\alpha \beta
\gamma}}=R^{\kappa}_{\alpha \beta \gamma}$ where $R^{\kappa}_{\alpha \beta
\gamma}$ is an ordinary $4D$ Riemann tensor. It is now interesting to study the
symmetry properties of the curvature tensor. For this purpose, let us adopt the
geodesic coordinate system in which the curvature tensor can be written as
\begin{equation}
\hat{R}^{D}_{_{CBA}}=\partial_{B}\hat{\Gamma}^{D}_{AC}-\partial_{A}\hat{\Gamma}^{D}_{BC},
\end{equation}
from which we immediately see that
\begin{equation}
\hat{R}^{D}_{_{CBA}}=-\hat{R}^{D}_{_{CAB}}.
\end{equation}
Now, to see if other symmetry properties are satisfied, we write our equation as
\begin{equation}
\hat{R}_{_{ABCD}}=\hat{g}_{_{AE}}\hat{R}^{E}_{_{BCD}}=
\partial_{_{C}}(\hat{g}_{_{AE}}\hat{\Gamma}^E_{_{BD}})-
\partial_{_{D}}(\hat{g}_{_{AE}}\hat{\Gamma}^E_{_{BC}}),
\end{equation}
showing that unlike the ordinary $4D$ space-time, because of the
unusual characteristics of the connections, the other common
symmetries are not satisfied. In general we have
\begin{equation}
\hat{R}_{_{DCBA}} \neq -\hat{R}_{_{CDBA}} \neq \hat{R}_{_{BADC}}.
\end{equation}
In addition, the first and second Bianchi identities can be written as
\begin{equation}
\hat{R}^{A}_{_{BCD}}+\hat{R}^{A}_{_{DBC}}+\hat{R}^{A}_{_{CDB}}=0,\label{af}
\end{equation}
and
\begin{equation}
\hat{\nabla}_{_{E}}\hat{R}^D_{_{ABC}}+
\hat{\nabla}_{_{C}}\hat{R}^D_{_{AEB}}+\hat{\nabla}_{_{B}}\hat{R}^D_{_{ACE}}=0,
\end{equation}
where both can be verified easily in geodesic coordinate. Since not
all the symmetry properties of ordinary $4D$ space-time appear in
our theory, naturally not all the known Bianchi identity forms in
$4D$ are correct here. However, there is another form of the First
Bianchi identity which is of much importance to us, namely
\begin{equation}
\hat{R}_{\alpha \beta \gamma 4}+\hat{R}_{\gamma \alpha \beta
4}+\hat{R}_{\beta \gamma \alpha 4}=0\mid_{on~the~hypersurface},\label{ag}
\end{equation}
which can be easily proved in geodesic coordinates using
$\partial_4=0$. The Ricci tensor in this space-time can be obtained
using equations (\ref{a5}), (\ref{a6}), and (\ref{a7}) as
\begin{equation}
\hat{R}_{_{CA}}=\partial_{\kappa}\hat{\Gamma}^{\kappa}_{AC}-\partial_{A}\hat{\Gamma}^{\kappa}_{\kappa
C}+\hat{\Gamma}^{\kappa}_{\kappa
\gamma}\hat{\Gamma}^{\gamma}_{AC}-\hat{\Gamma}^{\kappa}_{A
\gamma}\hat{\Gamma}^{\gamma}_{\kappa C}.
\end{equation}
Again
\begin{equation}
\hat{R}_{\alpha \gamma}=R_{\alpha \gamma},
\end{equation}
where $R_{\alpha \gamma}$ is the ordinary $4D$ Ricci tensor. It is
very simple to show that the Ricci tensor is symmetric.
Now we can investigate the field equations, (\ref{ac}), and
determine the scalar appearing in it. We know that, as mentioned
before, all the matter is confined to our $4D$ hypersurface and
therefore energy-momentum is a conservative quantity in our $4D$
volume. We can interpret $\hat{J}_{A}=\hat{T}_{A
B}\hat{\dot{x}}^{B}$ as current density. Hence the conservation law
can be written as
\begin{equation}
\oint_{\partial\Omega}\hat{J}_{\alpha}df^{\alpha}=
\oint_{\partial\Omega}\hat{T}_{\alpha B}\hat{\dot{x}}^{B}df^{\alpha}=0,
\end{equation}
where $df^{\alpha}$ is the vector representing the area element
perpendicular to the surface $\hat{x}^{\alpha}=const$. Using Gauss's
law, we will have the following equation
\begin{equation}
\int_{\Omega}\nabla^{\alpha}\hat{J}_{\alpha}\sqrt{-g}d^4x=0,
\end{equation}
where $\nabla^{\alpha}$ is the $4D$ covariant derivative. This
integral implies
\begin{equation}
\nabla^{\alpha}\hat{J}_{\alpha}=\nabla^{\alpha}(\hat{T}_{\alpha B}\hat{\dot{x}}^{B})=0,
\end{equation}
or
\begin{equation}
\nabla^{\alpha}(\hat{T}_{\alpha \beta}\hat{\dot{x}}^{\beta}-\hat{T}_{\alpha 4})=0,
\end{equation}
where we use $\hat{\dot{x}}^4=-1$ on the hypersurface. For any
arbitrary but small region we can find vector fields with
$\nabla^{\alpha}\hat{\dot{x}}^{\beta}\simeq0$ \cite{b2}. As a result
\begin{equation}
\nabla^{\alpha}(\hat{T}_{\alpha \beta})\hat{\dot{x}}^{\beta}-\nabla^{\alpha}\hat{T}_{\alpha 4}=0.
\end{equation}
Since $\hat{\dot{x}}^{\beta}$ can be chosen arbitrarily, this equation gives us two
distinct equations
\begin{equation}
\nabla^{\alpha}(\hat{T}_{\alpha \beta})=0,
\end{equation}
and
\begin{equation}
\nabla^{\alpha}\hat{T}_{\alpha 4}=0.
\end{equation}
Now, it is appropriate to take $\hat{T}_{\alpha \beta}$ as the four
dimensional energy-momentum tensor, $T_{\alpha \beta}$, and
$\hat{T}_{\alpha 4}$ as the four dimensional electromagnetic current
vector, $j_{\alpha}$, and write the five dimensional energy-momentum
tensor in the following form
\begin{equation}
\hat{T}_{AB}=\left(%
\begin{array}{ccccc}
T_{\alpha\beta} & & & & \frac{q}{m}\epsilon j_{0} \\
& & & & \frac{q}{m}\epsilon j_{1} \\
& & & & \frac{q}{m}\epsilon j_{2} \\
& & & & \frac{q}{m}\epsilon j_{3} \\
\frac{q}{m}\epsilon j_{0} & \frac{q}{m}\epsilon j_{1} & \frac{q}{m}\epsilon j_{2} & \frac{q}{m}\epsilon j_{3} & \hat{T_{44}} \\
\end{array}%
\right),
\end{equation}
where $\epsilon$ is a coupling constant. If we replace $A$ by
$\alpha$ and $B$ by $\beta$, equation (\ref{ac}) will reduce to
\begin{equation}
\hat{R}_{_{\alpha \beta}}-\frac{1}{2} \hat{\Im} \hat{g}_{_{\alpha \beta}}=\kappa
\hat{T}_{_{\alpha \beta}},
\end{equation}
or equivalently
\begin{equation}
R_{_{\alpha \beta}}-\frac{1}{2} \hat{\Im} g_{_{\alpha \beta}}=\kappa
T_{_{\alpha \beta}}.
\end{equation}
Since $\nabla^{\alpha}T_{\alpha \beta}=0$, we have to have
$\nabla^{\alpha}(R_{_{\alpha \beta}}-\frac{1}{2} \hat{\Im} g_{_{\alpha \beta}})=0$
which leads to the following equation
\begin{equation}
\hat{\Im}=R.
\end{equation}
Here $R$ is $4D$ Ricci scalar. It is easy to show that the $4D$
Ricci scalar is also a scalar in five dimensions. Therefore, we can
write the field equations (\ref{ac}) as follows
\begin{equation}
\hat{R}_{_{AB}}-\frac{1}{2} R \hat{g}_{_{AB}}=\kappa
\hat{T}_{_{AB}}.\label{ac1}
\end{equation}
\section{Electromagnetism}
As the first step let us derive the Lorentz force which is a part
of the geodesic equation. Knowing that on the geodesic equation
$\hat{\dot{x}}^4=-1$, we can write equation (\ref{ad}) as
\begin{equation}
\hat{\ddot{x}}^{\kappa}+\hat{\Gamma}^{\kappa}_{\alpha \beta}\hat{\dot{x}}^{\alpha}\hat{\dot{x}}^{\beta}-2\hat{\Gamma}^{\kappa}_{\alpha 4}\hat{\dot{x}}^{\alpha}+\hat{\Gamma}^{\kappa }_{4 4}=0.
\end{equation}
The geodesic equation is then given by
\begin{equation}
\hat{\ddot{x}}^{\kappa}+\frac{1}{2}g^{\lambda
\kappa}(\hat{g}_{\lambda\beta,\alpha}+\hat{g}_{\lambda\alpha,\beta}-
\hat{g}_{\alpha\beta,\lambda})\hat{\dot{x}}^{\alpha}\hat{\dot{x}}^{\beta}-g^{\lambda
\kappa}(\hat{g}_{4 \lambda,\alpha}-\hat{g}_{4
\alpha,\lambda})\hat{\dot{x}}^{\alpha
}-\frac{1}{2}g^{\lambda \kappa}\hat{g}_{44,\lambda}=0,\label{eq19}
\end{equation}
noting that partial derivatives with respect to $\hat{x}^{4}$ are
zero, and a comma indicates partial derivative. We may now define
the vector potential, $A_{\mu}$, as
\begin{equation}
A_{\mu}=\frac{m}{q}\hat{g}_{4 \mu},
\end{equation}
where $m$ and $q$ are the mass and electric charge of our test
particle respectively. Therefore, we define the Faraday tensor as
\begin{equation}
F_{\mu \nu}=\hat{g}_{4 \mu,\nu}-\hat{g}_{4 \nu,\mu},
\end{equation}
where by means of this equation, one can simply find the Lorentz
force in equation (\ref{eq19}).
As was mentioned above and is well known, the homogeneous Maxwell
equations appear as constraints on the Faraday tensor in the KK
theory and are implicitly assumed to hold. They do do not appear in
an independent manner. However, the situation is different in the
theory presented here. It is therefore appropriate at this point to
show that this is indeed the case. To begin with we note that in
classical electrodynamics, the Jacobi identities lead to the
homogenous Maxwell equations if we define our connections as
$D_{\mu}=\partial_{\mu}+A_{\mu}$, where $A_{\mu}$ is the four vector
potential, so that the Faraday tensor is written as $F_{\mu
\nu}=[D_{\mu},D_{\nu}]$. Now, use of the second Jacobi identity
leads to the Bianchi identity in the form
\begin{equation}
[D_{\mu},F_{\nu \lambda}]+[D_{\nu},F_{\lambda
\mu}]+[D_{\lambda},F_{\mu \nu}]=0,
\end{equation}
which is equivalent to
\begin{equation}
D_{\mu}F_{\nu \lambda}+D_{\nu}F_{\lambda \mu}+D_{\lambda}F_{\mu
\nu}=0.
\end{equation}
This is, of course, the homogenous Maxwell equations \cite{20}. In General
Relativity on the other hand, we replace $D_{\mu}$ by $\nabla_{\mu}$, so that
the first Jacobi identity results in
\begin{equation}
[\nabla_{\mu},R(\partial_{\nu},
\partial_{\lambda})]+[\nabla_{\nu},R(\partial_{\lambda},
\partial_{\mu})]+[\nabla_{\lambda},R(\partial_{\mu},
\partial_{\nu})]=0,
\end{equation}
or
\begin{equation}
R_{\alpha \beta [\mu \nu ;\lambda]}=0.\label{eq100}
\end{equation}
The torsion free condition also provides the first Bianchi identity
\begin{equation}
R_{\lambda [\beta \gamma \delta]}=0.\label{eq101}
\end{equation}
It is now clear that because of the definition of connections in
general relativity, neither the first Bianchi identity, equation
(\ref{eq101}), nor the second Bianchi identity, equation
(\ref{eq100}), lead to homogenous Maxwell equations. Let us now show
that in the model presented here, the first Bianchi identity leads
to the homogenous Maxwell equations. To write the first Bianchi
identity in our model we have to show that
\begin{equation}
\hat{R}^{\alpha}_{\beta \kappa
4}=\frac{1}{2}\nabla_{\kappa}F^{\alpha}_{\beta},\label{eq33}
\end{equation}
where $F_{\alpha \beta}$ is the Faraday tensor in four dimensional
space which is defined by $g_{\mu \nu}$. We write
\begin{equation}
\frac{1}{2}F^{\kappa}_{\sigma}=\frac{1}{2}g^{\kappa
\lambda}(\hat{g}_{4 \lambda,\sigma}-\hat{g}_{4
\sigma,\lambda})=\hat{\Gamma}^{\kappa}_{4\sigma},
\end{equation}
and
\begin{equation}
\frac{1}{2}\nabla_{\kappa}F^{\alpha}_{\beta}=\partial_{\kappa}\hat{\Gamma}^{\alpha}_{4\beta}+
\Gamma^{\alpha}_{\kappa\lambda}\hat{\Gamma}^{\lambda}_{4\beta}-\Gamma^{\lambda}_{\kappa
\beta}\hat{\Gamma}^{\alpha}_{4\lambda}.
\end{equation}
On the other hand we have
\begin{equation}
\hat{R}^{\alpha}_{\beta \kappa
4}=\partial_{\kappa}\hat{\Gamma}^{\alpha}_{4\beta}+\hat{\Gamma}^{\alpha}_{\kappa
\lambda}\hat{\Gamma}^{\lambda}_{4\beta}-\hat{\Gamma}^{\lambda}_{\kappa
\beta}\hat{\Gamma}^{\alpha}_{4\lambda},
\end{equation}
\begin{equation}
\hat{R}^{\alpha}_{\beta \kappa
4}=\partial_{\kappa}\hat{\Gamma}^{\alpha}_{4\beta}+
\Gamma^{\alpha}_{\kappa\lambda}\hat{\Gamma}^{\lambda}_{4\beta}-\Gamma^{\lambda}_{\kappa
\beta}\hat{\Gamma}^{\alpha}_{4\lambda}.
\end{equation}
Hence
\begin{equation}
\hat{R}^{\alpha}_{\beta \kappa
4}=\frac{1}{2}\nabla_{\kappa}F^{\alpha}_{\beta}\label{a8}.
\end{equation}
Now using equation (\ref{a8}), the first Bianchi identity, equation
(\ref{ag}), leads to
\begin{eqnarray}
\frac{1}{2}\nabla_{\kappa}F_{\alpha
\beta}+\frac{1}{2}\nabla_{\beta}F_{\kappa
\alpha}+\frac{1}{2}\nabla_{\alpha}F_{\beta \kappa}=0, \label{eq39}
\end{eqnarray}
showing that, as expected, the first Bianchi identity results in the
homogeneous Maxwell equations. It is worth mentioning that in other
$5D$ theories, e.g. KK, equations (\ref{af}) or (\ref{ag}) do not
lead to equation (\ref{eq39}).
Now, let us see if our field equations, (\ref{ac1}), reduce to
Maxwell equations. If one, in the field equations replaces $A$ by
$4$ and $B$ by $\lambda$, one finds
\begin{equation} \hat{R}_{4\lambda}-\frac{1}{2}\hat{g}_{4\lambda}R=\kappa
\hat{T}_{4\lambda}. \label{eq50}
\end{equation}
We can also see from equation
(\ref{eq33}) that
\begin{equation}
\hat{R}_{4 \lambda}=\hat{R}_{\lambda 4}=\hat{R}^{\alpha}_{\lambda \alpha
4}=\frac{1}{2}\nabla_{\alpha}F^{\alpha}_{\lambda}.
\end{equation}
Substituting this equation into (\ref{eq50}) yields the Maxwell
equations with a correction term
\begin{equation}
\nabla_{\alpha}F^{\alpha}_{\lambda}-\hat{g}_{4 \lambda}R=2\kappa
\hat{T}_{4\lambda},\label{ah}
\end{equation}
Note that in cases where the $4D$ scalar curvature is small, we may
neglect the correction term in (\ref{ah}). We can move $\hat{g}_{4
\lambda}R$ in (\ref{ah}) to the right hand side and look at it as a
new source. In the case of a large scalar curvature, this term,
under certain conditions, results in an electromagnetic field which
is stronger than that expected from the usual form of the Maxwell
equations. Recent observations may provide sufficient evidence
towards such a prediction. Indeed, it is possible to take account of
the unusually large galactic magnetic fields discovered recently
\cite{19,19a,19aa} for which no satisfactory explanation as yet
exists.
\section{Coordinate transformation}
If we change our coordinate system so that we again observe the space-time
from the $4D$ hypersurface, namely our universe, the main assumption
must be satisfied in order to yield the desired equations. It means
that we must have
\begin{equation}
\hat{\Gamma}^{4'}_{A'C'}=0.
\end{equation}
Therefore we need to examine the change of the above equation under
the transformation introduced as follows
\begin{equation}
\begin{array}{c}
~~~~~~~\frac{\partial \hat{x}^{4'}}{\partial\hat{x}^{4}}=const., \\
*\\
\frac{\partial \hat{x}^{4'}}{\partial\hat{x}^{\alpha}}=0\label{a9}.
\end{array}
\end{equation}
On the other hand we know that under transformations, the
connections change as
\begin{equation}
\hat{\Gamma}^{B'}_{A'E'}=\frac{\partial
\hat{x}^{B'}}{\partial\hat{x}^{C}}\left(\frac{\partial}{\partial\hat{x}^{A'}}\frac{\partial
\hat{x}^{C}}{\partial\hat{x}^{E'}}\right)+\frac{\partial
\hat{x}^{B'}}{\partial\hat{x}^{C}}\frac{\partial
\hat{x}^{D}}{\partial\hat{x}^{A'}}\frac{\partial
\hat{x}^{H}}{\partial\hat{x}^{E'}}\hat{\Gamma}^{C}_{DH}.
\end{equation}
If we now replace $B'$ by $4'$ in this equation
\begin{equation}
\hat{\Gamma}^{4'}_{A'E'}=\frac{\partial
\hat{x}^{4'}}{\partial\hat{x}^{C}}\left(\frac{\partial}{\partial\hat{x}^{A'}}\frac{\partial
\hat{x}^{C}}{\partial\hat{x}^{E'}}\right)+\frac{\partial
\hat{x}^{4'}}{\partial\hat{x}^{C}}\frac{\partial
\hat{x}^{D}}{\partial\hat{x}^{A'}}\frac{\partial
\hat{x}^{H}}{\partial\hat{x}^{E'}}\hat{\Gamma}^{C}_{DH},
\end{equation}
and substitute (\ref{a9}) into the above equation we find
\begin{equation}
\hat{\Gamma}^{4'}_{A'E'}=\frac{\partial \hat{x}^{D}}{\partial\hat{x}^{A'}}\frac{\partial \hat{x}^{H}}{\partial\hat{x}^{E'}}\hat{\Gamma}^{4}_{DH}.
\end{equation}
Hence, our assumption is invariant under the introduced
transformation.
\section{Conclusions} In this paper we have presented a unified
theory of gravity and electromagnetism where the resulting
inhomogeneous Maxwell equations are modified by a term involving the
curvature which, in certain cases, leads to electromagnetic fields
which are stronger than those obtained from the usual Maxwell
equations in cases where a large scalar curvature is present,
contrary to other conventional theories, e.g. Kaluza-Klein. This
could have interesting consequences and may be used to describe the
recently observed unusually strong galactic magnetic fields. This is
currently the focus of a work in progress.\vspace{3mm} \\
{\bf Acknowledgement}\vspace{1mm}\noindent\\
The authors would like to thank Professor Anzhong Wang for enlightening
discussions.
|
1,116,691,499,294 | arxiv | \section{Introduction}
In this paper, we are interested in the sampling of probability distribution named Gibbs measure whose density is $\pi(\mathrm{d}x)= \frac{1}{Z}e^{-\frac{U(x)}{2\sigma^2}} \lambda(\mathrm{d}x)$ where $\lambda $ is the Lebesgue measure, $Z= \int_{\mathbb{R}^d} e^{-\frac{2 U(x)}{\sigma^2}}\lambda(\mathrm{d}x)$ and $U:\mathbb{R}^d\rightarrow \mathbb{R}$ is a coercive function. Many applications require the computation of these measures in high dimension state space including for example machine learning, Bayesian estimation or statistical physics. Methods that are studied in this paper are based on the discretization of over-damped Langevin stochastic differential equation (\textbf{SDE})
\begin{equation}\label{art2:eq:SDE}
\mathrm{d} X_t =- \nabla U(X_t) \mathrm{d}t + \sigma \mathrm{d}B_t ,
\end{equation}
where $ \left(B_t \right)_{t \ge 0} $ is a $d$-dimensional Brownian motion and $\sigma \in \mathbb{R}^+_*$. These methods received a lot of attention in the last few years, in particular when $U$ is strongly convex (in the sense where, in the whole space, the smallest eigenvalue of its Hessian is lower bounded by a positive $\alpha$). This assumption may be certainly constraining in view of applications. It is the reason why, in this paper, we suppose that $U$ is not strongly convex but only weakly convex\footnote{We use in the sequel the usual terminology where ``strongly convex'' and ``weakly convex'' respectively means ``uniformly strongly convex'' and ``non-uniformly strongly convex''.}.
\newline More precisely, we will assume that the potential $U$ is a convex twice differentiable function with Lipschitz gradient. Under these assumptions, strong existence and uniqueness of a solution $\left(X_t\right)_{t\ge0}$ classically hold and the solution to \eqref{art2:eq:SDE} is an ergodic Markov process whose invariant distribution is exactly the Gibbs distribution $\pi \propto e^{-U} \mathrm{d}\lambda$ (for background, see \emph{e.g.} \cite{book:meyn2012markov}, \cite{book:KaratzasShreve}, \cite{book:khasminskiiStab},\cite{note:hairerconvergence}).
\bigskip \newline We respectively denote by $(P_t)_{t\ge 0}$ and $\mathcal{L}$ the related semi-group and infinitesimal generator. We recall that for a twice differentiable function $f : \mathbb{R}^d \mapsto \mathbb{R}$ by
\begin{equation*}
\mathcal{L}f= -\left\langle \nabla U, \nabla f\right\rangle + \frac{\sigma^2}{2}\Delta f.
\end{equation*}
It is also well-known that in this log-concave setting, the distribution $\pi$ satisfies the Poincar\'e inequality (see \emph{e.g.} \cite{art:bakry2008simple}) and that convergence holds to equilibrium in distribution and in ``pathwise average'': for any starting point $x\in\mathbb{R}^d$, the occupation measure converges to $\pi$ in the following sense: for all continuous function $f \in L^2(\pi)$,
\begin{equation}\label{art2:eq:ergod}
\frac{1}{t}\int_0^t f \left(X_s^x\right) \mathrm{d}s \underset{t \rightarrow + \infty}{\longrightarrow} \pi(f)\quad a.s.
\end{equation}
In the continuity of \cite{art:egeapanloup2021multilevel}, our multilevel methods will be based on discretized adaptations of \eqref{art2:eq:ergod}. More precisely, we first choose to approximate the stochastic process $(X_t)_{t\ge 0}$ by the classical Euler-Maruyama scheme. When the related step size $\gamma$ is constant, this discretization scheme is defined by $\bar{X}_{0}=x\in \mathbb{R}^d$ and:
\begin{equation*}
\forall n\ge0,\quad \bar{X}_{(n+1)\gamma}=\bar{X}_{n\gamma} - \gamma \nabla U \left(\bar{X}_{n\gamma}\right) + \sigma \sqrt{\gamma}Z_{n+1},
\end{equation*}
where $\left(Z_n \right)_{n\in \mathbb{N}}$ is an $i.i.d$ sequence of $d$-dimensional standard Gaussian random variables. In the long-time setting, these schemes and there convergence properties to equilibrium were first studied in the nineties by \cite{art:talay1990second} and \cite{art:TweedieRoberts}. Then, some decreasing step Euler schemes were investigated by \cite{art:LambertonPages} (see also \cite{these:lemaire}) in order to manage, in the same time, the discretization and long-time errors. Here, we choose to keep the constant step size point of view in order to avoid some additional technicalities but our ideas could be probably adapted to this setting.
\bigskip \newline We also introduced the continuous-time extension of $(\bar{X}_{n\gamma})_{n\ge0}$ given by: for all $n\in\mathbb{N}$ and for all $t \in [ n\gamma, (n+1)\gamma )$,
\begin{equation*}
\bar{X}_t^{\gamma}: = \bar{X}_{n\gamma} - (t-n\gamma)\nabla U \left( \bar{X}_{n\gamma}\right) + \sigma \left( B_t - B_{(n+1)\gamma} \right).
\end{equation*}
If we denote by $\underline{t}_\gamma$ the discretization time related to a positive number $t$, \emph{i.e.},
\begin{equation}\label{art2eq:tbargamma}
\underline{t}_\gamma=\gamma \sup\{n\ge 1, n\gamma\le t\},
\end{equation}
we remark that
\begin{equation*}
\forall t\ge0, \quad \bar{X}_t = x - \int_0^t \nabla U(\bar{X}_{\underline{s}_\gamma}) \mathrm{d}s+ \sigma \int_0^t \mathrm{d}B_s.
\end{equation*}
This ``pseudo-diffusion'' form is usually convenient for proofs but it is worth noting that the procedure is only based on the discrete-time Euler scheme. If no confusion arises, we will sometimes write $\underline{t}$ instead of $\underline{t}_\gamma$, and $\bar{X}_t$ or $\bar{X}_t^{\gamma}$ instead of $\bar{X}_t^{\gamma,x_0}$ to alleviate the notations. We now mimic \eqref{art2:eq:ergod} with the Euler scheme to approximate the target measure $\pi$. Thus, consider the following occupation measure (for background see \cite{art:talay1990second}), for $N \in \mathbb{N}$
\begin{equation*}
\nu_N^\gamma (f) := \frac{1}{N } \sum_{i=0}^{N-1} f \left( \bar{X}_{i\gamma}\right).
\end{equation*}
\subsection{Multilevel methods}
Multilevel methods introduced by M. Giles in \cite{art:giles2008multilevel}. These methods, initially used for the approximation of $\mathbb{E}[f(X_T)]$, are now widely exploited in many settings. The rough idea is the following: assume that the target $\mathbb{E}[X]$ is the expectation of a random variable that cannot be sampled (with a reasonable cost) and consider a family of random variables $(X_j)_{j}$ approximating $X$, with a cost of simulation and a precision which typically increases with $j$. The principle of the multilevel is to stack correcting layers with a low variance to a coarse approximation $X_0$ of the target. More precisely, writing
\begin{equation}\label{art2:eq:telescopHeuri}
\mathbb{E} [X_J] = \underbrace{\mathbb{E} [X_0]}_{\text{Coarse}} + \sum_{j=1}^J \underbrace{\mathbb{E} [X_j-X_{j-1}]}_{\text{Correcting layer}},
\end{equation}
the multilevel method consists in building a procedure based on the addition of Monte-Carlo approximations of $\mathbb{E}[X_0]$ and of $\mathbb{E}[X_j-X_{j-1}]$, $j=1,\ldots,J$. Then, if the random variables $X_j-X_{j-1}$ have low variance, the approximation of $\mathbb{E}[X_j-X_{j-1}]$ requires few simulations and, in view of \eqref{art2:eq:telescopHeuri}, we can obtain a procedure which has the bias related to $X_J$ but with a cost which may be much less than the one generated by a standard Monte-Carlo method applied to estimate $ \mathbb{E} [X_J]$.
\bigskip \newline In the discretization setting, the family of random variables $(X_j)_{j}$ is a sequence of Euler schemes $(\bar{X}^{\gamma_j})_{j}$ where $(\gamma_j)_j$ is a family of decreasing time steps\footnote{Decreasing according the levels but constant for each layer.}. Following the heuristic \eqref{art2:eq:telescopHeuri}, the (independent) correcting layers are built by coupling of Euler schemes with steps $\gamma_{j-1}$ and $\gamma_j$. Note that in view of the simulation of the (synchronous) coupling, we need $\gamma_{j-1}$ to be a multiple of $\gamma_j$ (in this paper, we will assume that $\gamma_j=\gamma_0 2^{-j}$).
\bigskip Multilevel methods have been already studied in the literature for the approximation of the invariant distribution of the Langevin diffusion. In \cite{art:gilesmajkameteuszSzpruch}, the authors take advantage of the convergence in distribution to equilibrium. Thus, the classical Monte-Carlo point of view is adopted: the approximation of $\pi(f)$ is obtained by sampling a large number of Euler schemes for each level. In \cite{art:pagespanloup2018weighted}\footnote{This paper is written in the multiplicative setting with a so-called Multilevel-Romberg point of view.} and \cite{art:egeapanloup2021multilevel}, the point of view is to take advantage of the convergence of the occupation measure. Thus, each level is based on only one path of the Euler scheme or of the couple of Euler schemes whose length decreases (since the variance of the correcting layers decreases) and discretization step increases. All these papers show that, in the uniformly strongly convex setting, the invariant distribution can be approximated (along Lipschitz continuous functions) with a precision $\varepsilon$ (in a $L^2$-sense) using a Multilevel procedure whose complexity is of order $\varepsilon^{-2}$ or $\varepsilon^{-2}\log^p(\varepsilon)$ with $p\in[1,3]$. Moreover, in \cite{art:egeapanloup2021multilevel}, a particular attention is paid to the dependency in the dimension. In this case, it is shown that one can build a multilevel procedure that produces an $\varepsilon$-approximation of the target for a complexity cost proportional to $d \varepsilon^{-2}$ (with an explicit expression of the dependence in the Lipschitz constant $L$ and the contraction parameter $\alpha$).
\bigskip The more involved weakly convex case seems to be less explored in the multilevel paradigm but, in view of applications (for instance for Bayesian Lasso), it is natural to ask about the robustness of these methods when one relaxes the contraction assumption.
\subsection{Contributions and plan of the paper}
The main goal of this paper is thus to extend the multilevel Langevin algorithm for the Gibbs sampling to the weakly convex setting, and if possible to obtain some quantitative bounds for the complexity related to the computation of an $\varepsilon$-approximation of the target (see Section \ref{art2:notations} for a definition of $\varepsilon$-approximation).
\bigskip We first investigate the \textit{penalized multilevel} method: in the continuity of \cite{art:DalalyanKaragulyan} and \cite{art:DalalyanRiouKaragulyan}, we build a multilevel procedure based on the following observation: consider a new equation with another potential $U_\a(x):= U(x)+\frac{\a}{2}|x|^2$, this new equation has an invariant distribution named $\pi_\a$ which converges to $\pi$ when $\a$ tends to $0$ in Wasserstein metric. The idea is that this new invariant distribution is easiest to sample because of the uniform convexity of the potential $U_\a$. In Section \ref{art2:sec:penalised}, Theorem \ref{art2:theo:PenaMain} combines the benefits of the penalized approach and of the multilevel methods. For a Lipschitz-continuous function $f:\mathbb{R}^d \rightarrow \mathbb{R}$ and a ${\cal C}^2$-potential $U$, the multilevel procedure performs an $\varepsilon$-approximation of $\pi(f)$ with a complexity cost proportional to $\pi(|.|^2)^{3}d \varepsilon^{-5}$. As in \cite{art:DalalyanRiouKaragulyan}, our result depends on the generally unknown constant $\pi(|.|^2)$ which is at least proportional to $d$ (see Remark \ref{rem:dalriou} for details and comparisons with \cite{art:DalalyanRiouKaragulyan}).
\bigskip Because of the above remarks, we chose in a second part to try to develop some tools which tackle the weakly convex setting from a dynamic point of view and which can improve the complexity in terms of $\varepsilon$. More precisely, in the spirit of \cite{art:gadat2020cost}, we study an intermediary framework (called weakly parametric convex setting in the sequel). We assume that the eigenvalues of the Hessian matrix of $U$ vanish when $|x|$ goes to $+\infty$, but with a rate controlled by the function $x\mapsto U^{-r}(x)$ with $r \in [0,1)$ (see Assumption $(\mathbf{H_{r}^1})$). The parameter $r$ characterizes the ``lack of strong convexity'', the case $r=1$ referring to the ``Laplace case''\footnote{The ``Laplace case'' will refer to the setting where the potential has a flat gradient. The simplest example is $U(x)=|x|$. In this case, the invariant distribution is a Laplace distribution. That is the reason why we use this terminology.} whereas $r=0$ corresponds to the uniformly convex setting. When one assumes such an assumption, one can get some bounds for the exponential moments of the Euler scheme (at the price of technicalities). One is also able to preserve some \textit{confluence} properties, \emph{i.e.} two paths of the Euler scheme have a tendency to stick at infinity. Finally, in this setting, it is also possible to control the distance between diffusion paths and the related Euler schemes. These three ingredients (obtained with a lower quality than in the strongly convex setting) allow us to tackle the multilevel procedure in this framework.
\bigskip \newline The related main contribution is Theorem \ref{art2:theo:maintheo2}. In this result, we provide a series of statements under different sets of assumptions: when $U$ is only ${\cal C}^2$ or when $U$ is ${\cal C}^3$. Under $(\mathbf{H_{r}^1})$ only or under $(\mathbf{H_{r}^1})$ and $(\mathbf{H_{r}^2})$, where $(\mathbf{H_{r}^2})$ denotes an additional assumption which requires the highest eigenvalue to be also controlled by the function $x\mapsto U^{-r}(x)$ (we could roughly say that under $(\mathbf{H_{r}^1})$ and $(\mathbf{H_{r}^2})$, the potential is \textit{uniformly} weakly convex in the sense that ``the decrease of the contraction is uniform''). In each statement, we provide a multilevel procedure adapted to the assumptions. The related complexity is exhibited in terms of $d$, $\varepsilon$ but also in terms of the contraction parameter and the Lipschitz constant. Without going too much into the details, when $U$ is only ${\cal C}^2$, the complexity is of the order $\varepsilon^{-3}$ whereas when $U$ is ${\cal C}^3$, we can obtain a rate of the order $\varepsilon^{-2-\rho}$ for any $\rho>0$ and thus approach the ``optimal'' complexity $\varepsilon^{-2}$. Now, in terms of the dimension, when only $(\mathbf{H_{r}^1})$ holds the dependence in the dimension of the complexity is bounded by
$d^{\frac{\frac{3}{2}+(\frac{9}{2}+\delta) r}{1-r}}$ whereas when $(\mathbf{H_{r}^1})$ and $(\mathbf{H_{r}^2})$ hold, we obtain in $d^{{\frac{3}{2}+(\frac{9}{2}+\delta) r}}$ when $U$ is ${\cal C}^2$
and $d^{1+\frac{\rho}{2}+4r}$ for any $\rho>0$ when $U$ is ${\cal C}^3$. This means that when $U$ is ${\cal C}^3$ and $(\mathbf{H_{r}^1})$ and $(\mathbf{H_{r}^2})$ hold, the complexity is of the order $\varepsilon^{-2-\rho} d^{1+\frac{\rho}{2}+4r}$ for any $\rho>0$. With respect to the paper \cite{art:gadat2020cost}, our multilevel procedure improves the dependence in $\varepsilon$ and is most comparable in terms of the dimension. Note that when only $(\mathbf{H_{r}^1})$ holds, the dependency in the dimension dramatically increases on $r$. Whereas, when the potential is \textit{uniformly weakly} convex, the dependence in the dimension does not explode when $r\rightarrow 1$ (see Theorem \ref{rem:theoparam} for more details).
\bigskip \newline \textbf{Plan of the paper.} As detailed in the previous paragraphs, Sections \ref{art2:sec:penalised} and \ref{art2:sec:weaklyconvex} are respectively devoted to the statement of the main theorems for the penalized multilevel and in the parametric weakly convex case. Then, Section \ref{art2:sec:proofs} is dedicated to the proof of the first main theorem (Theorem \ref{art2:theo:PenaMain}). In Proposition \ref{art2:prop:alphaBias}, we obtain a Wasserstein bound related to the bias induced by the penalization on the invariant distribution. The proof of Theorem \ref{art2:theo:PenaMain} is then an adaptation of \cite[Theo 2.2]{art:egeapanloup2021multilevel}. From Section \ref{art2:sec:prelimtool}, we focus on the proof of Theorem \ref{art2:theo:maintheo2}. In Section \ref{art2:sec:prelimtool}, we prove some preliminary results on the diffusion and its Euler scheme under $(\mathbf{H_{r}^1})$: we begin with some controls of the exponential moments (Proposition \ref{art2:prop:exponentcontrol} and Proposition \ref{art2:prop:Eulexponentcontrol}) which in turn lead to some bounds of the polynomial moments (Proposition \ref{art2:prop:moment}). In this section, we also show that the discretization error can be controlled in long time (Proposition \ref{art2:prop:HypH2}) and finally obtain an integrable rate of convergence to equilibrium for the Euler scheme (Proposition \ref{art2:prop:HypH1}). With the help of these fundamental bricks, in Section \ref{art2:sec:maintheo2}, we obtain some bounds on the bias (Proposition \ref{art2:prop:HypH3}) and of the variance of the procedure (Proposition \ref{art2:varcontr}) which in turn allow us to finally provide the proof of Theorem \ref{art2:theo:PenaMain} in Theorem \ref{art2:theo:maintheo2}.
\subsection{Design of the algorithm}
We now build the multilevel procedure. Let $x\in \mathbb{R}^d$ be the initialization of the procedure, $J \in \mathbb{N}$ be a number of levels, $\left(\gamma_j\right)_{0 \le j \le J}$ be a sequence of time steps, $\left(T_j\right)_{0 \le j \le J}$ be a sequence of final times. Define by $ \mathcal{Y}(J,\left(\gamma_j\right)_j,\tau,\left(T_j\right)_j,x,.)$ the \textit{multilevel occupation measure} : for all $f : \mathbb{R}^d \rightarrow \mathbb{R}$,
\begin{equation}\label{art2:eq:estimat}
\begin{split}
\mathcal{Y}(J,\l(\gamma_j\r)_j,\tau,\l(T_j\r)_j,x,\a,f) &:= \frac{1}{T_0-\tau}\int_{\tau}^{T_0} f(\bar{X}_{\underline{s}_{\gamma_0}}^{\gamma_0,x}) \mathrm{d}s
\\ &+ \sum_{j=1}^J \frac{1}{T_j-\tau}\int_\tau^{T_j} f(\bar{X}_{\underline{s}_{\gamma_{j-1}}}^{\gamma_{j},x}) -f(\bar{X}_{\underline{s}_{\gamma_{j-1}}}^{\gamma_{j-1},x}) \mathrm{d}s,
\end{split}
\end{equation}
where trajectories on each level are coupled with the same Brownian motion. To ease notation, we simplify $\mathcal{Y}(J,\left(\gamma_j\right)_j,\tau,\left(T_j\right)_j,x,f)$ by $\mathcal{Y}(f)$. The parameter $\tau \ge 0$ is the time where we begin the average. Indeed, this ``warm-start trick'' may improve the precision of the estimation in some cases (we refer to \cite{art:egeapanloup2021multilevel} for more details). But, it could be equal to $0$ when the gain is non-efficient, for example in the second part of our main result.
\subsection{Notations}\label{art2:notations}
The usual scalar product on $\mathbb{R}^d$ and the induced Euclidean norm are respectively denoted by $\langle \cdot, \cdot\rangle$ and $|\cdot|$. The set $\mathcal{M}_{d,d}$ refers to the set of $d \times d$-real matrices, we denote by $\|\cdot\|$ the operator norm associated with the Euclidean norm. For a symmetric matrix $A$, we denote respectively by $\underline{\lambda}_A$ and $\bar{\lambda}_A$ its lowest and highest eigenvalues. The Frobenius norm for $A \in {\cal M}_{d,d}$ is denoted by $\|A\|_F$.\\
\noindent The Lipschitz constant of a given (Lipschitz) function $f:\mathbb{R}^d\rightarrow\mathbb{R}$ is denoted by $[f]_1$.
A function $f:\mathbb{R}^d\rightarrow\mathbb{R}$ is ${\cal C}^k$, $k\in\mathbb{N}$, if all its partial derivatives are well-defined and continuous up to order $k$. The gradient and Hessian matrix of $f$ are respectively denoted by $\nabla f$ and $D^2f$. The probability space is denoted by $(\Omega,{\mathcal{ F}},\mathbb{P})$. The Laplace operator is denoted by $\Delta$: $\Delta f=\sum_{i=1}^d \partial^2_{i,i} f$. The $L^p$-norm on $(\Omega,{\cal F},\mathbb{P})$ is denoted by $\|\cdot\|_p$. For two probability measures $\mu$ and $\nu$, we define the Wasserstein distance of order $p$ by
\begin{equation*}
\mathcal{W}_p(\mu,\nu) = \inf_{\zeta \in \Pi(\mu,\nu)} \left(\int_{\mathbb{R}^d} |x-y|^p \mathrm{d}\zeta(x,y) \right)^{\frac{1}{p}},
\end{equation*}
where $\Pi(\mu,\nu)$ is the set of couplings of $\mu$ and $\nu$.\\
\begin{itemize}
\item{} \textbf{$\lesssim_{P}$ and $\lesssim_{uc}$}: For two positive real numbers $a$ and $b$ and a set of parameters $P$, one writes $a\lesssim_{P} b$ if $a\le C_P b$ where $C_P$ is a positive constant which only depends on the parameters $P$. When $a\le C b$ where $C$ is a universal constant, we write $a\lesssim_{uc} b$.
\item{} \textbf{$\varepsilon$-approximation}: We say that ${\cal Y}$ is an $\varepsilon$-approximation (or more precisely an $\varepsilon$-approximation of $a$ for the $L^2$-norm) if $\|{\cal Y}-a\|_2=\mathbb{E}[|{\cal Y}-a|^2]^{\frac{1}{2}}\le \varepsilon$. Equivalently, ${\cal Y}$ is said to be an $\varepsilon$-approximation of $a$ if the related Mean-Squared Error (MSE) is lower than $\varepsilon^2$.
\item{} \textbf{Complexity/$\varepsilon$-complexity}: For a random variable ${\cal Y}$ built with some iterations of a standard Euler scheme, we denote by ${\cal C}(\cal Y)$, the number of iterations of the Euler scheme which is needed to compute ${\cal Y}$. For instance, ${\cal C}(\bar{X}_{n\gamma}^{\gamma})=n$. We sometimes call $\varepsilon$-complexity of the algorithm, the complexity related to the algorithm which produces an $\varepsilon$-approximation.
\end{itemize}
\section{Main results}
\subsection{The penalized approach}\label{art2:sec:penalised}
In this section, we develop a \textit{penalized multilevel method} to sample a non-strongly log-concave probability distribution $\pi$. The idea is based on \cite{art:DalalyanRiouKaragulyan} and \cite{art:karagulyan2020penalized} where the authors consider the potential $U^\a(x) := U(x)+\frac{\a}{2} |x|^2$ with $\a>0$ which is called \textit{penalized} version of $U$. We here assume that $U$ satisfies the following assumption:
\bigskip \newline $\mathbf{WC_L}$: $U$ is a non-zero ${\cal C}^2$-function and there exists $L > 0$ such that
\begin{equation}\label{art2:hyp:hessianpot}
\forall x\in \mathbb{R}^d, \qquad 0 \preccurlyeq D^2U(x) \preccurlyeq L \mathrm{Id}_{\mathbb{R}^d},
\end{equation}
the inequalities being taken in the sense of symmetric matrices. Denote by $\pi_{\a}$ the invariant measure of the diffusion process $\left(X_t^\a\right)_{t \ge 0}$ solution of the stochastic differential equation
\begin{equation}\label{art2:eq:SDEr}
\mathrm{d} X_t^{\a} =- \nabla U^\a(X_t^\a) \mathrm{d}t + \sigma \mathrm{d}B_t.
\end{equation}
It appears that $\pi_\a$ satisfies the Bakry-\'Emery criterion thus we can apply our multilevel method that requires strong convexity to approximate $\pi_\a$. But our target is $\pi$ then we have to control the distance (in a Wasserstein sense) between $\pi$ and $\pi_\a$. To this end, the results of \cite{art:bolley2005weighted} and \cite{art:DalalyanRiouKaragulyan} ensure the convergence of $\pi_\a$ when $\a$ goes to $0$ with a bound of the Kullback-Leibler divergence. This leads to the following theorem:
\begin{theoreme}\label{art2:theo:PenaMain} Assume that $\mathbf{WC_L}$ hold. Let $\varepsilon>0$, let $f : \mathbb{R}^d \rightarrow \mathbb{R}$ be a Lipschitz function and $x \in \mathbb{R}^d$. For $\varepsilon \ge 0$ let
\begin{equation} \label{art2:eq:parameterPenawithoual}
\begin{split}
J_\varepsilon&= \lceil 2\log_2(\sigma^2 d m_4^{1/2} \varepsilon^{-2})\rceil,\quad T_j= \sigma^2 d\log(\gamma_0^{-1})m_4 \varepsilon^{-5} J_\varepsilon^2 2^{-J},
\\& \qquad j\in\{0,\ldots,J_\varepsilon\}, \quad \gamma_0=\varepsilon m_4^{-1/2} L^{-2},
\end{split}
\end{equation}
with $m_4=\mathbb{E}_{\pi}[|\cdot|^4]$. Then we have
\begin{equation}
\| \mathcal{Y}(f) - \pi(f) \|_2^2 \le \varepsilon,
\end{equation}
with a complexity satisfying
\begin{equation}\label{art2:eq:complexitePenawithoutal}
{{\cal C}({\cal Y})\le \frac{1}{3}\log\l(\gamma_0\r)m_4^{3/2} L^2\sigma^2 d\varepsilon^{-5}\left\lceil\log_2\l(\frac{1}{2}\sigma^2d m_4^{1/2} \varepsilon^{-3}\r)\right\rceil^3.}
\end{equation}
\end{theoreme}
\begin{rem}\label{rem:dalriou} By Borell's inequality (see \emph{e.g.} \cite{alonso-bastero}), $m_4\lesssim_{uc} \pi(|.|^2)^2$ for any log-concave probability. This means that, up to logarithmic terms, the complexity is controlled by $\pi(|.|^2)^3d \varepsilon^{-5}$. When $\pi$ is an \textit{isotropic probability}, $\pi(|.|^2)\lesssim_{uc} d$ (see \emph{e.g.} \cite{alonso-bastero}\footnote{Note that the control of $\pi(|.|^2)$ is strongly linked to the so-called \textit{KLS-conjecture}.}) and this implies that the complexity is of the order $d^4 \varepsilon^{-5}$.
\smallskip \newline Let us now compare with \cite{art:DalalyanRiouKaragulyan}: note that in this paper, the cost is not explicitly written. As usual in the Monte-Carlo literature, the authors control the number of iterations of the Euler scheme which is necessary to draw a random variable whose distance to the target is lower than $\varepsilon$ instead of giving the real cost. In the Langevin Monte-Carlo case, when $U$ is ${\cal C}^2$, they then obtain a number of iterations which is, up to logarithmic terms, of order $\pi(|.|^2)d\varepsilon^{-4}$. Normalizing $\varepsilon$ (\emph{i.e.} replacing $\varepsilon$ by $\varepsilon/\sqrt{\pi(|.|^2)}$) leads to a number of iterations of order $\pi(|.|^2)^{3} d\varepsilon^{-4}$. But to compare with our work, we need to include the Monte-Carlo cost, \emph{i.e.} to the number of simulations which is necessary to make the variance lower than $\varepsilon^{2}$. Then, we have to multiply the previous number of iterations by ${\rm Var}_\pi(f) \varepsilon^{-2}$ which can be reasonably bounded by $\pi(|.|^2)\varepsilon^{-2}$(when $f$ is Lipschitz). This means that the complexity of the penalized Langevin Monte-Carlo in \cite{art:DalalyanRiouKaragulyan} is of the order $\pi(|.|^2)^{4} d\varepsilon^{-6}$. In consequence, the multilevel method allows us to improve the result of \cite{art:DalalyanRiouKaragulyan}. Note that the authors also provide other algorithms such as the Kinetic-LMC where the bound in $\varepsilon$ is improved (it seems that our result meets the complexity given for this algorithm).
\end{rem}
\noindent \textbf{About decreasing penalization.}In the above result, we propose a Multilevel strategy based on a fixed penalization. A natural question arises: could we take advantage of the Multilevel strategy by keeping the same penalization for the highest level and progressively reducing it on the lower layers? Indeed, this is precisely what we do with the discretization bias, thus we can wonder about the effect of such a strategy for the penalization: to this end, let us introduce a decreasing sequence $(\a_j)_{0\le j \le J }$ of penalization levels such that the bias induced by the distance between $\pi_{{\a}_J}$ and $\pi$ is small with respect to the required precision $\varepsilon$. More specifically, we want to replace \eqref{art2:eq:telescopHeuri} with the following telescoping series
\begin{equation*}
\pi^{\gamma_J}_{{\a}_J}= \pi^{\gamma_0}_{{\a}_0} + \sum_{j=1}^J \pi^{\gamma_{j}}_{{\a}_j} - \pi^{\gamma_{j-1}}_{\a_{j-1}}.
\end{equation*}
However, the above decomposition requires several longtime bounds on the underlying dynamics to be an efficient multilevel procedure. In particular, for the control of the variance generated by each level, we need to control the pathwise distance between two paths of the dynamics related to penalization levels ${\a}_j$ and ${\a}_{j-1}$. But, oppositely to the constant penalization case where we can obtain some \emph{confluence} properties, we can observe that for two trajectories computed with two different degrees of penalization $\a$ and $\tilde{\alpha}$, there is a ``lack of confluence" quantified by the following inequality,
\begin{equation*}
\mathbb{E} \left[ \left|X_t^{x, \a}-X_t^{y,\tilde{\alpha}}\right|^2 \right] \le e^{-2 \a t}|x-y|^2 + \frac{ (\a-\tilde{\alpha})d\sigma^2}{\tilde{\alpha}},
\end{equation*}
where the bound dramatically depends on $\a$. In this inequality we voluntary treated the continuous case to ease the readability but a discrete analogous result can be shown. We refer to the section \ref{art2:sec:proofs} to get the proof of this result. This inequality is certainly related to the shape of the penalization sequence $(\a_j)_{0\le j \le J }$. In fact, this result must be considered in addition of the error induced by the difference between two Euler schemes with different time steps. Up to an universal constant we have (see \cite[Prop 5.1]{art:egeapanloup2021multilevel})
\begin{equation*}
\sup_{t \ge 0} \mathbb{E} \left[ \left|\bar{X}_{\underline{t}}^{x, \gamma}-\bar{X}_{\underline{t}}^{x,\gamma/2}\right|^2 \right] \le \frac{\gamma d \sigma^2}{\alpha^2}.
\end{equation*}
Then, the additional variance generated by the decrease of the penalization does not have an impact on the results, we have to impose that
$ \displaystyle{\frac{\gamma d \sigma^2}{\alpha_j^2}}\lesssim \displaystyle{ \frac{ (\a_{j}-\a_{j-1})d\sigma^2}{\a_{j-1}}}$. In particular, the sequence $(\a_j)_{0\le j \le J }$ cannot be too decreasing. Going further into the computations, it seems that we cannot expect a significant gain with this approach.
\subsection{Parametric weakly convex setting}\label{art2:sec:weaklyconvex}
The purpose of this section is to study the non-penalized multilevel procedure in the weakly convex setting. Instead of penalizing the dynamics, it is actually natural to ask about the robustness of the ``standard'' multilevel method in this case.
To answer this question, we have to prove a series of properties in the spirit of the assumptions $\mathbf{H_i}$ ($i \in \{1,2,3,4\}$) in \cite{art:egeapanloup2021multilevel}. These assumptions include the convergence to equilibrium of the Euler scheme with a quantitative rate, the long-time control of the $L^2$-distance between the Euler scheme and the diffusion, the control of the Wasserstein distance between $\pi^\gamma$ and $\pi$ and the control of the moments. Some of these properties (especially the long-time control of the $L^2$-distance) seem hard to check in a general convex setting. We thus propose to work in the parametric weakly convex setting used in \cite{art:egeapanloup2021multilevel} by introducing $(\mathbf{H_{r}^1})$ (see below) where we assume that the contraction vanishes at $\infty$ but with a rate controlled by $U^{-r}$.
\\
\noindent Let us now introduce our assumptions depending on a parameter $r\in[0,1)$ :\\
\noindent $(\mathbf{H_{r}^1}):$ The potential $U$ is a positive $C^2$-convex function with a unique minimum $x^\star$ such that $U(x^\star)=1$. $\nabla U$ is $L$-Lipschitz {with $L\ge1$\footnote{The fact that $L$ is greater than $1$ is clearly not fundamental but allows to simplify the usually technical expressions which appear in the sequel.}}. The function $x \mapsto \underline{\lambda}_{D^2 U (x)}$ is positive and there exist $L$ and $\underline{c} > 0$ such that,
\begin{equation*}
\forall x \in \mathbb{R}^d, \quad \underline{\lambda}_{D^2 U (x)} \ge \underline{c} U^{-r}(x).
\end{equation*}
The lower-bound can be seen as the ``lack of uniform strong convexity'' for the potential. Indeed, if $r=0$ we recover the strong convexity and $r=1$ corresponds to the weakest convexity case where the gradient is flat at infinity. \\
\noindent The fact that $\nabla U$ is $L$-Lipschitz implies that $x \mapsto \bar{\lambda}_{D^2 U (x)}$ is upper-bounded by $L$. In order to improve the dependence in the dimension, we also introduce an additional assumption that deals with the case where the largest eigenvalue decreases at infinity with an intensity that is of the same order as the lowest eigenvalue:\\
\noindent $(\mathbf{H_{r}^2}):$ There exists a positive $\bar{c}$ such that for all $x\in\mathbb{R}^d$, $ \bar{\lambda}_{D^2 U (x)} \le \bar{c} U^{-r}(x).$\\
\noindent For instance, it can be checked that the function $x\mapsto (1+|x|^2)^p$ with $p\in(1/2,1]$ satisfies $(\mathbf{H_{r}^1})$ and $(\mathbf{H_{r}^2})$ with $r=\frac{1-p}{p}$, $\underline{c}=2p(2p-1)$ and $\bar{c}=2p$. \\
\noindent Let us finally define the couple $(\gamma^\star,\bar{\Psi})$ by:
\begin{equation}\label{art2:def:gamzero}
(\gamma^\star,\bar{\Psi})=\begin{cases}
\left(\frac{1}{4L},(1+\sigma^2)\left(dL+\left(1+\frac{dL}{\underline{c}}\right)^{\frac{1}{1-r}}\right)\right)&\textnormal{if only $(\mathbf{H_{r}^1})$ holds}\\
\left(\frac{1-r}{4 (\bar{c}\vee L)}, c_r\frac{ d(1+\sigma^2)(\bar{c}\vee L)}{\underline{c}}\right)& \textnormal{if $(\mathbf{H_{r}^1})$ and $(\mathbf{H_{r}^2})$ hold true},
\end{cases}
\end{equation}
where $c_r$ is a constant which only depends on $r$. $\gamma^\star$ will denote the largest value for $\gamma_0$ whereas, $\bar{\Psi}$ controls the moments of $U(\bar{X}_{n\gamma})$ (see Proposition \ref{art2:prop:moment} for details). It is worth noting that on the one hand, $\gamma^\star$ does not depend on $d$ and on the other hand, that
$\bar{\Psi} \propto d^{\frac{1}{1-r}}$ when only $(\mathbf{H_{r}^1})$ holds and $\bar{\Psi} \propto d$ when $(\mathbf{H_{r}^1})$ and $(\mathbf{H_{r}^2})$ hold true. This means that in the first case, the dependence in $d$ dramatically increases with $r$ whereas in the second case, it does not depend on $r$. In the next result, the reader will have to keep in mind that
the definition of these parameters depends on the assumptions. In particular, even if $(\mathbf{H_{r}^2})$ does not appear in the statement, it is hidden in the value of the parameters $\gamma^\star$ and $\bar{\Psi}$.\\
We are now ready to state our main theorem in this setting:
\begin{theoreme}\label{art2:theo:maintheo2}
Assume $(\mathbf{H_{r}^1})$ and let $x \in \mathbb{R}^d$ such that $U(x)\lesssim_{r} \bar{\Psi}$, $\gamma_0 \in (0,\gamma^\star]$, $\delta \in (0,1/4]$ and let $f$ be a Lipschitz-continuous function.
For an integer $J\ge 1$, set $\forall j \in \{1,\dots,J\}$, $\gamma_j= \gamma_0 2^{-j}$ and $T_j= T_0 2^{-(1-\rho)j}$ with $\rho\in[1/2,1)$.
Let $\varepsilon>0$.
\begin{itemize}
\item[(i)]
Set $\rho=1/2$,
\begin{equation*}
J =\lceil \log_2(\frac{L}{\underline{c}^{\frac{2}{1-\delta}}\wedge\underline{c}}\bar{\Psi}^{1+(3+\delta)r} \gamma_0 \varepsilon^{-2})\rceil, \quad \textnormal{and}\quad T_0\coloneqq (\underline{c}^{-\frac{3}{4}}\vee \underline{c}^{-\frac{5}{2}-\delta}) \bar{\Psi}^{\frac{3}{2}+(\frac{9}{2}+\delta) r}\varepsilon^{-2}.
\end{equation*}
Then, for $\delta$ small enough,
\begin{equation}\label{art2:eq:pluspetitqueepsilon}
\| \mathcal{Y}(J,\l(\gamma_j\r)_j,\tau,\l(T_j\r)_j,f)- \pi (f) \|_2 \lesssim_{r,\delta} \varepsilon,
\end{equation}
with a complexity cost,
\begin{equation}\label{art2:eq:complex2}
\mathcal{C}(\mathcal{Y})\le\sqrt{L}(\underline{c}^{-\frac{5}{4}}\wedge \underline{c}^{-{\frac{7}{2}}-\delta})\bar{\Psi}^{\frac{3}{2}+(\frac{9}{2}+\delta)r} \varepsilon^{-3}.
\end{equation}
\item[(ii)] Assume that $U$ is a ${\cal C}^3$-function with $\sup_{x\in\mathbb{R}^d} \sum_{i=1}^d |\Delta (\partial_i U)|^2\lesssim_{r} \sigma^{-4} L^3 \bar{\Psi}$. Set
\begin{equation*}
J=\lceil \log_2\left(\underline{c}^{-\frac{2}{1-\delta}} L^{{3}}\bar{\Psi}^{1+\frac{2r}{1-\delta}} \gamma_0 \varepsilon^{-1}\right)\rceil\quad \textnormal{and}\quad T_0=
L^{\frac{\rho}{2}} \left(\underline{c}^{-(\frac{5}{4}-\rho)\wedge(3\rho)+\delta}\vee \underline{c}^{-\frac{5}{2}-\delta}\right)\bar{\Psi}^{1+(4-2\rho+\delta) r} \varepsilon^{-2}.
\end{equation*}
Then, for $\delta$ small enough, $\| \mathcal{Y}(J,\l(\gamma_j\r)_j,\tau,\l(T_j\r)_j,f)- \pi (f) \|_2 \lesssim_{r,\delta,\rho} \varepsilon,$
with
$$\mathcal{C}(\mathcal{Y})\le \gamma_0^{-\rho} L^{2\rho} \left(\underline{c}^{\frac{5}{4}\wedge(2\rho)+\delta}\vee \underline{c}^{-\frac{5}{2}-\delta}\right)\bar{\Psi}^{1+\frac{\rho}{2}+(4-\rho+\delta) r} \varepsilon^{-2-\rho}.$$
\end{itemize}
\end{theoreme}
\begin{rem} \label{rem:theoparam}
This technical result deserves several comments:\\
$\rhd$ \textit{Complexity in terms of $\varepsilon$.} If we only consider the dependence in $\varepsilon$, we obtain $\varepsilon^{-3}$ when $U$ is only ${\cal C}^2$ and $\varepsilon^{-2-\rho}$ for any $\rho>0$ when $U$ is ${\cal C}^3$ and an additional (but reasonable\footnote{See Remark \ref{art2:rem:laplacienU} for details on this assumption}) assumption on $\Delta(\nabla U)$ is satisfied. We can thus theoretically approach the complexity in $\varepsilon^{-2}$. However, it is worth noting that the non-explicit constants depending on $\rho$ and $\delta$ go to $\infty$ (independently of the other parameters) when $\rho$ and $\delta$ go to $0$. The fact that we ``only'' obtain a complexity in $\varepsilon^{-3}$ when $U$ is ${\cal C}^2$ is due to the fact that in this case, our bound of the $1$-Wasserstein distance between $\pi^\gamma$ and $\pi$ is of the order $\sqrt{\gamma}$. When $U$ is ${\cal C}^3$, the bound on the $1$-Wasserstein distance between $\pi^\gamma$ and $\pi$ is of order $\delta$. This allows us to clearly improve the complexity but it can be noted that we do not retrieve the $\varepsilon^{-2}$-bound of the uniformly convex case. This is due to the rate of convergence to equilibrium. Actually, our rate is polynomial and not exponential, which in turn, implies a ``slight cost'' on the dependence in $\varepsilon$. In fact, we could get some (sub)-exponential rates but without controlling the dependence in the dimension which is of first importance for applications. \\
$\rhd$ \textit{Complexity in terms of the dimension.} The dependence in the dimension strongly varies with the assumptions. In the ``worst'' case where $U$ is only ${\cal C}^2$ and only $(\mathbf{H_{r}^1})$ holds, the complexity is of the order $d^{\frac{\frac{3}{2}+(\frac{9}{2}+\delta) r}{1-r}}$. Unfortunately, when $r$ is close to $1$, this means that this dependence seriously worsens. We retrieve this same phenomenon when $U$ is ${\cal C}^3$ and only $(\mathbf{H_{r}^1})$ holds but with a better bound of the order $d^{\frac{{1+\frac{\rho}{2}+(4-\rho+\delta) r}}{1-r}}$ for any positive $\rho$ and $\delta$. This bad behavior when $r$ goes to $1$ is due to the fact that when only $(\mathbf{H_{r}^1})$ holds, the bounds on the exponential moments of $\sup_{t\ge 0}\mathbb{E}[e^{U(\bar{X}_{\underline{t})}}]$ are of the order $\exp({d^{\frac{1}{1-r}}})$. Introducing $(\mathbf{H_{r}^2})$ dramatically improves this exponential bound since in this case, we are able to prove that this is of the order $e^d$ (this implies that $\sup_{t\ge 0}\mathbb{E}[U^p(\bar{X}_{\underline{t}})]$ is of the order $d^p$, see Propositions \ref{art2:prop:moment} and \ref{art2:prop:Eulexponentcontrol} for details). It is worth noting that in this case, the dependence in the dimension does not explode when $r$ goes to $1$ being of the order $d^{{1+\frac{\rho}{2}+(4-\rho+\delta) r}}$ for any $\rho>0$. Remark that when $r=0$, we formally approach the rate of the uniformly convex case in $d\varepsilon^{-2}$ obtained in \cite{art:egeapanloup2021multilevel}. \\
$\rhd$ \textit{Comparison with the literature:} In this setting, the only paper which we may reasonably compare with is \cite{art:gadat2020cost} since we use similar assumptions. Compared with this paper, our multilevel procedure certainly improves the dependence in $\varepsilon$, replacing $\varepsilon^{-4}$ by $\varepsilon^{-3}$ when $U$ is ${\cal C}^2$ and $\varepsilon^{-3}$ by $\varepsilon^{-2-\rho}$ when $U$ is ${\cal C}^3$. In terms of the dimension, our approach slightly increases the dependence on the dimension. For instance, when $(\mathbf{H_{r}^1})$ and $(\mathbf{H_{r}^2})$ hold, \cite{art:gadat2020cost} obtain a bound in $d^{1+4r}$ when $U$ is ${\cal C}^2$ or ${\cal C}^3$. We here retrieve a dependence which is somewhat similar when $U$ is ${\cal C}^3$ but when $U$ is ${\cal C}^2$, our bound in $d^{\frac{3}{2}+(\frac{9}{2}+\delta) r}$ is clearly worse.\\
$\rhd$ \textit{About the parameters.} In applications, the dependence in the parameters, $L$, $\underline{c}$, and $\bar{c}$ may be of importance (think for instance to applications to Bayesian estimation where these parameters can strongly depend on the number of observations). This is why here, we chose to keep all these dependencies in the main result even if it sometimes adds many technicalities in the proof.
\end{rem}
\section{Proof of Theorem \ref{art2:theo:PenaMain}} \label{art2:sec:proofs}
This section is devoted to the proof of the first main result. We first quantify the bias induced by the approximation of $\pi$ by $\pi_\a$. To this end, we use the Talagrand concentration inequality that estimates the Wasserstein distance between these two measures by their Kullback-Leibler divergence.
\begin{prop} \label{art2:prop:Pinsker}
Assume that $\mathbf{WC_L}$ hold. Then for all $\a \ge 0$, there is a universal constant $C$ such that
\begin{equation*}
\mathcal{W}_1 (\pi,\pi_\a) \le C \sqrt{D_{\mathrm{KL}} (\pi| \pi_\a )} .
\end{equation*}
\end{prop}
We refer to \cite[Cor 2.4]{art:bolley2005weighted} to find a proof of this result. In addition, in \cite{art:DalalyanRiouKaragulyan} the authors show that $C \le 2\mathbb{E}_{\pi}[|X|^2]$ (page 24).
It remains to compute the Kullback Leibler divergence of $\pi$ from $\pi_\a$, to bound the bias induced by the penalization.
\begin{prop}\label{art2:prop:alphaBias}
Assume that $\mathbf{WC_L}$ hold. Then for all $\a \ge 0$,
\begin{equation*}
\mathcal{W}_1 (\pi, \pi_\alpha) \le \frac{\alpha}{2\sqrt{2}} \mathbb{E}_{\pi} [|.|^4]^{1/2}.
\end{equation*}
\end{prop}
\begin{proof}
The Kullback-Leibler divergence is defined by
\begin{equation*}
D_{\mathrm{KL}} (\pi| \pi_\a ) = \int_{\mathbb{R}^d} \mathrm{log} \left( \frac{\mathrm{d}\pi}{\mathrm{d}\pi_\a}(x)\right) \pi(\mathrm{d}x).
\end{equation*}
By definition of $\pi$ and $\pi_\a$,
\begin{equation*}
\begin{split}
D_{\mathrm{KL}} (\pi| \pi_\a ) &= \int_{\mathbb{R}^d} \mathrm{log} \left( \frac{Z_\a}{Z} \right) + \frac{\a}{2} |x|^2 \pi(\mathrm{d}x)
\\ & = \mathrm{log} \left( \mathbb{E}_{\pi} \left[ e^{-\frac{\alpha}{2}|.|^2} \right] \right) + \frac{\alpha}{2} \mathbb{E}_{\pi} \left[|.|^2\right].
\end{split}
\end{equation*}
Using the inequality $e^{-x} \le 1-x+\frac{x^2}{2}$ for $x \ge 0$, this leads to,
\begin{equation*}
D_{\mathrm{KL}} (\pi| \pi_\a ) \le \mathrm{log} \left( 1 - \mathbb{E}_{\pi} \left[ \frac{\a}{2} |.|^2 - \frac{\alpha^2}{8}|.|^4 \right] \right) + \frac{\a}{2} \mathbb{E}_{\pi} \left[|.|^2\right],
\end{equation*}
and by the inequality $\mathrm{log}(1-x) \le -x$ for $x \le 1$ we get,
\begin{equation*}
D_{\mathrm{KL}}(\pi|\pi_\a) \le \frac{\a^2}{8} \mathbb{E}_{\pi} [|.|^4].
\end{equation*}
Proposition \ref{art2:prop:Pinsker} for $\pi_\a$ implies the result.
\end{proof}
Now we switch to the proof of the main theorem. With the two previous propositions, we control the bias induced by the penalization, then it remains to compute the error and the complexity of a multilevel procedure in a uniformly convex setting. To this end, we use \cite[Theorem 2.2]{art:egeapanloup2021multilevel} which gives parameters to perform an $\varepsilon$-approximation of the invariant distribution with an explicit complexity in terms of the parameters, especially in terms of the \textit{contraction parameter}. Here, this is exactly our penalization parameter $\a$ and we will thus optimize its choice in the proof.
\begin{proof}\textit{of Theorem \ref{art2:theo:PenaMain}}
Let $\varepsilon$ be a positive number and $f:\mathbb{R}^d\rightarrow\mathbb{R}$ be a Lipschitz continuous function. By the bias/variance decomposition, triangular inequality and the Monge-Kantorovich duality, we have
\begin{equation}\label{art2:eq:decomperr}
\begin{split}
\| \mathcal{Y}(f) - \pi(f) \|_2^2 & \le \mathbb{E} \left[ \left|\mathcal{Y}(f) - \pi(f)\right|\right]^2 + \mathrm{Var}( \mathcal{Y}(f))
\\ & \le 2|\pi_{\a}(f)- \pi(f)|^2 + 2 \mathbb{E} [\mathcal{Y}(f) - \pi_{\a}(f)]^2 + \mathrm{Var}( \mathcal{Y}(f)))
\\ & \lesssim_{uc} \underbrace{\mathcal{W}_1^2(\pi,\pi_\a)}_{P_1} + \underbrace{ \mathbb{E} [\mathcal{Y}(f) - \pi_{\a}(f)]^2 + \mathrm{Var}( \mathcal{Y}(f))}_{P_2}.
\end{split}
\end{equation}
The second term denoted by $P_2$ is the mean squared error of a Multilevel procedure for the approximation of $\pi_\a$. This penalized measure is invariant for the diffusion process defined with the potential $U_\a$. By assumption $\mathbf{WC_L}$, $U_\a$ satisfies the following property
\noindent \\ $\mathbf{\mathbf{C_\a}}$: For all $x,y \in \mathbb{R}^d$,
\begin{equation*}
\langle \nabla U^{\a}(x)-\nablaU^{\a}(y),x-y\rangle \ge \a |x-y|^2.
\end{equation*}
\cite[Theorem 2.2]{art:egeapanloup2021multilevel} ensures that with $\alpha/L^2\le 1$, $\sigma^2 \alpha^{-1} d\ge 1$, $|U^{\a} (x_0)|^2\lesssim_{uc} \sigma^2 \alpha d$ and the following parameters\footnote{To ease notation we have voluntarily omitted an assumption about $\varepsilon$: we have to consider $\varepsilon$ small enough, we refer to \cite{art:egeapanloup2021multilevel} to get more precision. }:
\begin{equation} \label{art2:eq:parameterPena}
\begin{split}
J_\varepsilon&= \lceil 2\log_2(\sigma^2 d \a^{-1} \varepsilon^{-1})\rceil,\quad T_j= \frac{\sigma^2 d \log(\gamma_0^{-1})}{\a^2} \varepsilon^{-2} J_\varepsilon^2 2^{-j},
\\& \qquad r\in\{0,\ldots,J_\varepsilon\}, \quad \gamma_0=\a/(2L^2),
\end{split}
\end{equation}
we have
\begin{equation*}
P_2 \le \varepsilon,
\end{equation*}
with a complexity satisfying
\begin{equation}\label{art2:eq:complexitePena}
{\cal C}({\cal Y})\le 5\log\left(\frac{2 L^2}{\a}\right)\frac{L^2}{2\a^3}\sigma^2 d\varepsilon^{-2}\left\lceil\log_2\left( \frac{\sigma^2d}{\a} \varepsilon^{-2}\right)\right\rceil^3.
\end{equation}
It remains to calibrate the penalization parameter $\a$. Proposition \ref{art2:prop:alphaBias} implies
\begin{equation*}
P_1 \lesssim_{uc} \frac{\a^2}{4} \mathbb{E}_{\pi} [|.|^4],
\end{equation*}
so that $P_1 \le \varepsilon^2$ for
\begin{equation*}
\a = \frac{2 \varepsilon}{ \mathbb{E}_{\pi} [|.|^4]^{1/2}}.
\end{equation*}
Putting $\a$ in \eqref{art2:eq:parameterPena} and \eqref{art2:eq:complexitePena},
\begin{equation*}
{\cal C}({\cal Y})\le \frac{5d\varepsilon^{-5}}{16}\log\left( \mathbb{E}_{\pi} [|.|^4]^{1/2} L^2 \varepsilon^{-1}\right)\mathbb{E}_{\pi} [|.|^4]^{3/2} L^2\sigma^2 \left\lceil\log_2\left(\frac{1}{2} \mathbb{E}_{\pi} [|.|^4]^{1/2}\sigma^2d \varepsilon^{-3}\right)\right\rceil^3.
\end{equation*}
\end{proof}
\subsection*{Precisions about the ``decreasing penalization":}
For $\a>\tilde{\alpha}>0$ and $x,y \in \mathbb{R}^d$ consider the couple $(X_t^{x,\a},X_t^{x,\tilde{\alpha}})_{t\ge0}$ defined by
\begin{equation}
\begin{cases}
&X_t^{x,\a}=x+\int_0^t \nabla U_{\a}(X_s^{x,\a}) \mathrm{d}s+\sigma B_t\\
&X_t^{y, \tilde{\alpha}}=y+\int_0^t \nabla U_{\tilde{\alpha}}(X_s^{y,\tilde{\alpha}}) \mathrm{d}s+\sigma B_t.
\end{cases}
\end{equation}
\begin{prop}
For all $t>0$ we have
\begin{equation*}
\mathbb{E} \left[ \left|X_t^{x, \a}-X_t^{y,\tilde{\alpha}}\right|^2 \right] \le e^{-2 \a t}|x-y|^2 + \frac{ (\a-\tilde{\alpha})d\sigma^2}{\tilde{\alpha}}.
\end{equation*}
\end{prop}
\begin{proof}
By the Itô formula we have,
\begin{equation*}
\begin{split}
e^{2 \a t}\left|X_t^{x, \a}-X_t^{y,\tilde{\alpha}}\right|^2=|x-y|^2 & + \int_0^t 2 e^{2 \a s} \left\langle \b_{\a} \left(X_s^{x, \a}\right)-\b_{\tilde{\alpha}} \left(X_s^{y,\tilde{\alpha}}\right),X_s^{y,\tilde{\alpha}} -X_s^{x, \a}\right\rangle \mathrm{d}s
\\ & + \int_0^t 2 \a e^{2 \a s}\left|X_s^{x, \a}-X_s^{y,\tilde{\alpha}}\right|^2 \mathrm{d}s,
\end{split}
\end{equation*}
We now use the fact that for all $ x\in \mathbb{R}$ : $\b_{\tilde{\alpha}}(x)=\b_{\a}(x) + \left(\tilde{\alpha} - \a \right)|x|$ which yields
\begin{equation*}
\begin{split}
e^{2 \a t}\left|X_t^{x, \a}-X_t^{y,\tilde{\alpha}}\right|^2=|x-y|^2 & +\int_0^t 2 e^{2 \a s} \left\langle \b_\a\left(X_s^{x, \a}\right)-\b_\a\left(X_s^{y,\tilde{\alpha}}\right),X_s^{y,\tilde{\alpha}} -X_s^{x, \a}\right\rangle \mathrm{d}s
\\ & + \int_0^t 2(\a-\tilde{\alpha}) e^{2 \a s} \left\langle X_s^{y,\tilde{\alpha}},X_s^{y,\tilde{\alpha}} -X_s^{x, \a}\right\rangle \mathrm{d}s
\\ & + \int_0^t 2 \a e^{2 \a s}\left|X_s^{x, \a}-X_s^{y,\tilde{\alpha}}\right|^2 \mathrm{d}s.
\end{split}
\end{equation*}
The strong convexity property of $U_\a$: $\langle \b_\a(x)-\b_\a(y),x-y\rangle \le \a |x-y|^2$ implies
\begin{equation*}
\begin{split}
e^{2 \a t}\mathbb{E} \left[ \left|X_t^{x, \a}-X_t^{y,\tilde{\alpha}}\right|^2 \right]& \le |x-y|^2 +\int_0^t 2(\a-\tilde{\alpha}) e^{2 \a s} \left\langle X_s^{y,\tilde{\alpha}},X_s^{y,\tilde{\alpha}} -X_s^{x, \a}\right\rangle \mathrm{d}s
\\ & \le |x-y|^2 +\int_0^t 2(\a-\tilde{\alpha}) e^{2 \a s} \mathbb{E} \left[ \left| X_s^{y,\tilde{\alpha}}\right|^2 \right]\mathrm{d}s.
\end{split}
\end{equation*}
Up to an universal constant, the moment of order two of the diffusion process under the strong convexity hypothesis are bounded by $\frac{\sigma^2 d }{\tilde{\alpha}}$ (see \cite[Lem 5.1]{art:egeapanloup2021multilevel}), we get
\begin{equation*}
\mathbb{E} \left[ \left|X_t^{x, \a}-X_t^{y,\tilde{\alpha}}\right|^2 \right] \le e^{-2 \a t}|x-y|^2 + \frac{ 2(\a-\tilde{\alpha})d\sigma^2}{\tilde{\alpha}}\int_0^t e^{2\a (s-t)}\mathrm{d}s.
\end{equation*}
The result follows.
\end{proof}
\section{Preliminary bounds under \texorpdfstring{$(\mathbf{H_{r}^1}) $ and $(\mathbf{H_{r}^2})$}{Hr}:}\label{art2:sec:prelimtool}
From now, we switch to the proof of the second part of the main results \textit{i.e.} we consider the weakly convex case under the parametric assumptions $(\mathbf{H_{r}^1})$ and $(\mathbf{H_{r}^2})$. As mentioned before, these hypotheses deal with the behavior of the lowest and highest eigenvalues of the Hessian matrix of $U$. In some sense, $(\mathbf{H_{r}^1})$ quantifies the strict convexity of the potential which in turn implies the contraction of the dynamics. Note that such an assumption also appears in \cite{cattiaux:hal-02486264} where the authors obtain exponential rates to equilibrium under this parametric assumption.
\bigskip \newline In this preliminary section we state a series of results related to the diffusion and its Euler scheme under Assumption $(\mathbf{H_{r}^1})$. For the upper-bounds of the eigenvalues of $D^2 U$, we distinguish two cases: the first one where we assume that we have a uniform upper-bound by $L$ (in others words that $\nabla U$ is $L$-Lipschitz) and the second one, we add Assumption $(\mathbf{H_{r}^2})$ where the largest eigenvalues also decrease at infinity with a rate which is comparable to the one of the lowest eigenvalues.
In fact, in the second case, we will see that we are able to preserve a dependency of the moments in the dimension which is linear, whereas, without this assumption, the dependency is $O(d^{\frac{1}{1-r}})$.
\bigskip \newline In the second part we state a result about the longtime pathwise control of the distance between the diffusion and its Euler scheme. Third, we study the convergence to equilibrium for the Euler scheme. Finally, we quantify the bias induced by the discretization with some results on the $1$-Wasserstein distance between $\pi$ and $\pi^\gamma$ (the invariant measure of the Euler scheme).
\subsection{Bounds on the exponential moment}
In order to study the confluence between the continuous time process and its Euler scheme, let us start this section by a control of the moment of the continuous time process and the discrete time when the potential $U$ is supposed convex. We first state a result on the control of the exponential moment of the continuous time process.
\begin{prop} \label{art2:prop:exponentcontrol}
For all $x \in \mathbb{R}^d$ and $t>0$,
\begin{equation*}
\sup_{t\ge0} \mathbb{E}_x \left[e^{ \frac{U\left(X_t\right)}{\sigma^2}} \right] \le e^{\frac{U(x)}{\sigma^2}}+ \begin{cases} e^{\frac{1}{\sigma^2} \left(1+\frac{ dL}{2\underline{c}}\right)^{\frac{1}{1-r}}}&\textnormal{under $(\mathbf{H_{r}^1})$}\\
\frac{d L}{2\underline{c}} e^{\frac{1}{\sigma^2} (4^{\frac{1}{1-r}}\vee \frac{d \bar{c}}{\underline{c}})}
&\textnormal{under $(\mathbf{H_{r}^1})$ and $(\mathbf{H_{r}^2})$.}
\end{cases}
\end{equation*}
\end{prop}
We preface the proof by a technical lemma.
\begin{lem} \label{art2:lem:compactset}
Let $\theta \in (0,\frac{2}{\sigma^2})$ and $M>0$, then
\begin{equation*}
{\cal C}_M := \left\{x \in \mathbb{R}^d ; (1-\theta)\left|\nabla U(x)\right|^2- \frac{\sigma^2}{2} \theta \Delta U(x) \le M \right\}\subset\left\{x\in\mathbb{R}^d, U(x)\le K_M\right\},
\end{equation*}
where,
\begin{equation*}
K_M=\begin{cases}\left(1+\frac{ (2M+\theta \sigma^2 dL)}{2\underline{c}(2-\theta\sigma^2)}\right)^{\frac{1}{1-r}} &\textnormal{under $(\mathbf{H_{r}^1})$}\\
\max\left(\left(\frac{M(1-r)}{(4-2\theta\sigma^2)\underline{c}}\vee 4\right)^{\frac{1}{1-r}},\frac{2\sigma^2 d\theta \bar{c}}{\underline{c}(2-\theta\sigma^2)}\right) &\textnormal{under $(\mathbf{H_{r}^1})$ and $(\mathbf{H_{r}^2})$.}
\end{cases}
\end{equation*}
In particular, ${\cal C}_M$ is a compact set (since it is included in a level set of a coercive function).
\end{lem}
\begin{proof}
Denote by $y$ the solution of the ordinary differential equation $y'(t)=-\nabla U (y(t))$ starting from $y(0)=x$. Define the function $f: t \mapsto |\nabla U(y(t))|^2 $ we have by chain rule for all $t\in \mathbb{R}^+$. By $(\mathbf{H_{r}^1})$,
\begin{equation*}
\begin{split}
\frac{\mathrm{d}}{\mathrm{d}t}f(t) &= 2 \left\langle y''(t), y'(t) \right\rangle
\\ & = 2\left\langle D^2 U(y(t))y'(t), y'(t) \right\rangle
\\ & \ge 2\underline{c} \left\langle U^{-r}(y(t))y'(t), y'(t) \right\rangle = \frac{2 \underline{c}}{1-r} \frac{\mathrm{d}}{\mathrm{d}t} \big( U^{1-r}(y(t)) \big),
\end{split}
\end{equation*}
Since $\displaystyle{\lim_{t \rightarrow + \infty} y(t)=x^\star}$, we get by integration
\begin{equation}\label{art2:eq:minogradU}
|\nabla U(x)|^2 \ge \frac{2\underline{c}}{1-r}\left( U^{1-r}(x) - U^{1-r}(x^\star) \right).
\end{equation}
Therefore,
\begin{equation*}
\left(1-\frac{\theta\sigma^2}{2}\right)\left|\nabla U(x)\right|^2- \frac{\sigma^2\theta}{2} \Delta U(x) \ge \frac{2(1-\theta)\underline{c}}{1-r}\left( U^{1-r}(x) - U^{1-r}(x^\star) \right) - \frac{\sigma^2\theta}{2} \Delta U(x).
\end{equation*}
Since $ \Delta U={\rm Tr}(D^2 U)\le d\bar{\lambda}_{D^2 U}\le d L$ (where $\bar{\lambda}_A$ stands for the largest eigenvalue of symmetric matrix $A$) and $U(x^\star)=1$, it follows that
\begin{equation*}
\left(1-\frac{\theta\sigma^2}{2}\right)\left|\nabla U(x)\right|^2- \frac{\sigma^2}{2} \theta \Delta U(x)\ge \frac{2\left(1-\frac{\theta\sigma^2}{2}\right){\underline{c}}}{1-r}(U^{1-r}(x)-U^{1-r}(x^\star))-\frac{\theta \sigma^2 dL}{2},
\end{equation*}
so that
\begin{equation}
{\cal C}_M\subset \left\{x\in\mathbb{R}^d; U(x)\le \left(1+\frac{2M+\theta \sigma^2 dL}{2\underline{c}(2-{\theta\sigma^2}) }\right)^{\frac{1}{1-r}}\right\}.
\end{equation}
If we now consider the case where $(\mathbf{H_{r}^2})$ also holds, we use that $ \Delta U\le\underline{c} U^{-r}(x)$ to obtain:
\begin{equation*}
\begin{split}
\left(1-\frac{\theta\sigma^2}{2}\right)\left|\nabla U(x)\right|^2- \frac{\sigma^2}{2} \theta \Delta U(x)
\ \ge\frac{(2-\theta\sigma^2)\underline{c}}{1-r} U^{1-r}(x) \left( 1-U^{r-1}(x)- \frac{(1-r) \sigma^2 \theta \bar{c}d}{2\underline{c}(2-\theta\sigma^2})U(x)\right).
\end{split}
\end{equation*}
To ensure that the right-hand member is lower-bounded by $M$, it is enough to ensure that
\begin{equation*}
\frac{(2-\theta\sigma^2)\underline{c}}{1-r} U^{1-r}(x)>\frac{M}{2},\quad U^{r-1}(x)\le \frac{1}{4}\quad \textnormal{and} \quad \frac{ \sigma^2 \theta \bar{c} d}{\underline{c}(4-2\theta\sigma^2})U(x) \le \frac{1}{4}.
\end{equation*}
This concludes the proof.
\end{proof}
\begin{proof}\textit{(of Proposition \ref{art2:prop:exponentcontrol})}
Let $\theta \in (0,1)$, (to be choosen latter) and for all $x \in \mathbb{R}$ define,
\begin{equation} \label{art2:eq:LyapFunction}
f_{\theta}(x):=e^{\theta U(x)},
\end{equation}
show that $f_{\theta}$ is a Lyapunov function for the dynamic $\mathcal{L}$ :
\begin{equation*}
\begin{split}
\mathcal{L}f_{\theta}(x) & = -\theta f_{\theta}(x)\left(\left(1-\frac{\theta\sigma^2}{2}\right)\left|\nabla U(x)\right|^2- \frac{ \theta \sigma^2}{2} \Delta U(x) \right)
\\ & \le - \theta M f_\theta(x) -\theta f_{\theta}(x)\left(\left(1-\frac{\theta\sigma^2}{2}\right)\left|\nabla U(x)\right|^2- \frac{ \theta \sigma^2}{2} \Delta U(x) \right) 1_{{\cal C}_M}(x),
\end{split}
\end{equation*}
where ${\cal C}_M$ is defined in Lemma \ref{art2:lem:compactset}. In the proof of this lemma we showed that ${\cal C}_M$ is included in a level set $L_{K_M} = \{x \in \mathbb{R}^d \mid U(x) \le K_M \}$. Thus,
\begin{equation*}
\begin{split}
-\theta f_{\theta}(x)\left(\left(1-\frac{\theta\sigma^2}{2}\right)\left|\nabla U(x)\right|^2- \frac{\sigma^2}{2} \theta \Delta U(x) \right) 1_{{\cal C}_M}(x) & \le \frac{ \theta^2 \sigma^2}{2} f_{\theta}(x) \Delta U(x) 1_{L_{K_M}}
\\ & \le \frac{ \theta^2 \sigma^2 d L }{2} e^{\theta K_M}1_{L_{K_M}}.
\end{split}
\end{equation*}
Finally $f_\theta$ is a Lyapunov function for the dynamics: \textit{i.e.} for all $x\in \mathbb{R}^d$,
\begin{equation*}
\mathcal{L}f_{\theta}(x) \le - \theta M f_\theta(x) + \frac{ \theta^2 \sigma^2 d L }{2} e^{\theta K_M} 1_{L_{K_M}}(x),
\end{equation*}
Hence by a Gronwall argument we get,
\begin{equation*}
P_t f_\theta(x) \le e^{-\theta M t} f_\theta(x) + \frac{ \theta \sigma^2 d L}{2M} e^{\theta K_M}.
\end{equation*}
Choosing $\theta=\frac{1}{\sigma^2}$, $M=\frac{\theta\sigma^2 dL}{2} $ under $(\mathbf{H_{r}^1})$ and $M=\frac{(4-2\theta\sigma^2)\underline{c}}{1-r} $ under $(\mathbf{H_{r}^1})$ and $(\mathbf{H_{r}^2})$ leads to the result.
\end{proof}
\noindent We now state an analogous result for the Euler scheme.
\begin{prop} \label{art2:prop:Eulexponentcontrol}
\begin{itemize}
\item[$(i)$] Assume $(\mathbf{H_{r}^1})$. Then, if $\gamma \in (0,\frac{1}{4L}]$ and $\theta \in [0,\frac{1}{8\sigma^2}]$,
\begin{equation*}
\sup_{n\ge0} \mathbb{E}_x \left[ e^{ \theta U \left(\bar{X}_{n\gamma}\right)} \right] \le e^{\thetaU(x)} + e^{\theta\left(1+\frac{8dL}{\underline{c}}\right)^{\frac{1}{1-r}} +\theta dL}{\cosh}(\theta\gamma d L).
\end{equation*}
\item[$(ii)$] Assume $(\mathbf{H_{r}^1})$ and $(\mathbf{H_{r}^2})$. If $\gamma \in (0,\frac{1-r}{4\bar{c}\vee L}]$ and $\theta \in [0,\frac{1}{8\sigma^2}\wedge 1]$, then for all $x \in \mathbb{R}^d$ and $ n \in \mathbb{N}$ we have
\begin{equation*}
\sup_{n\ge0} \mathbb{E}_x \left[ e^{ \theta U \left(\bar{X}_{n\gamma}\right)} \right] \le e^{\thetaU(x)} + c e^{c_r\frac{\theta d(1+\sigma^2)\bar{c}}{\underline{c}}},
\end{equation*}
where $c$ denotes a constant independent of the parameters and $c_r$ a constant which only depends on $r$.
\end{itemize}
\end{prop}
The proof of this proposition is postponed in \cref{art2:appen:appendixA}.
\begin{rem} The reader will find more explicit (but more technical) bounds in the proof of the second case. It is worth noting that we can preserve a condition on $\gamma$ does not depend on $d$ (as in the strongly convex setting). This is of first importance in our multilevel setting where it is much more efficient if the rough layers of the method can be implemented with step sequences with large sizes. The proof is very close to \cite{art:gadat2020cost} but the bounds are refined. In particular, compared to this paper, we precisely do not require that the step size decreases with $d$.
\end{rem}
Thanks to the two previous results we are now able to control the moment of the continuous time and the discrete time processes.
\begin{prop}\label{art2:prop:moment}
Assume $(\mathbf{H_{r}^1})$.
\newline (i) For all $x \in \mathbb{R}^d$, $p\ge 0 $,
\begin{equation*}
\sup_{t\ge0} \mathbb{E}_x \left[U^p\left( X_t\right)\right] \lesssim_{p} \left(U(x)+ \Psi\right)^p.
\end{equation*}
where
\begin{equation*}
\Psi=\begin{cases}
(1+\sigma^2)\left(1+\frac{dL}{\underline{c}}\right)^{\frac{1}{1-r}}&\textnormal{under $(\mathbf{H_{r}^1})$ only}\\
4^{\frac{1}{1-r}}\vee \frac{(1+\sigma^2) (\bar{c}\vee L)}{\underline{c}}d& \textnormal{under $(\mathbf{H_{r}^1})$ and $(\mathbf{H_{r}^2})$},
\end{cases}
\end{equation*}
\newline (ii) Let $\gamma \in (0,\gamma^\star]$ with $\gamma^\star$ defined by \eqref{art2:def:gamzero}. Then,
\begin{equation*}
\sup_{n\ge0} \mathbb{E}_x \left[U^p\left( \bar{X}_{n\gamma}\right)\right] \lesssim_{p} (U(x)+ \bar{\Psi})^p
\end{equation*}
where $\bar{\Psi}$ is defined by \eqref{art2:def:gamzero}
\newline (iii) In particular,
\begin{equation}\label{art2:eq:boundfacileatransporter}
\max\left(\sup_{t\ge0} \mathbb{E}_x \left[U^p\left( X_t\right)\right],\sup_{t\ge0,\gamma\in(0,\gamma^\star]}\mathbb{E}_x \left[U^p\left( \bar{X}_{\underline{t}}\right)\right]\right)\lesssim_{r,p} (U(x)+ \bar{\Psi})^p.
\end{equation}
and
\begin{equation}\label{art2:eq:boundfacileinvariant}
\max\left(\int U^p(x)\pi(dx),\sup_{\gamma\in(0,\gamma^\star]}\int U^p(x)\pi^\gamma(dx)\right)\lesssim_{r,p} \bar{\Psi}^p.
\end{equation}
\end{prop}
\begin{proof}
The proof similar to \cite[Prop B.4]{art:gadat2020cost} is postponed in \cref{art2:appen:appendixB}.
\end{proof}
\begin{rem} In order to avoid the distinction between cases in all the proofs, we choose to adopt only one notation for $\Psi$ and $\bar{\Psi}$ but the reader has to keep in mind that the definition of these quantities depends on the fact that $(\mathbf{H_{r}^2})$ is satisfied or not. Let us also recall that
the notation $\lesssim_{r,p}$ means that the constant only depends on $r$ and $p$. These constants are certainly locally bounded: for any compact subset $K$ of $[0,1)\times [1,+\infty)$, there exists a universal constant $c$ such that for any $(r,p)\in K$, the underlying constant $c_{r,p}$ related to $\lesssim_{r,p}$ is bounded by $c$. Finally, note that we chose to keep all the dependencies in the other parameters.
\end{rem}
\subsection{Longtime strong discretization error under \texorpdfstring{$(\mathbf{H_{r}^1}) $}{Hr}:}
The following proposition studies the $L^2$-error induced by the discretization of the SDE under $(\mathbf{H_{r}^1}) $. The notations $\gamma^\star$ and $\bar{\Psi}$ come from Proposition
\ref{art2:prop:moment}.
\begin{prop}\label{art2:prop:HypH2} Assume $(\mathbf{H_{r}^1})$ and let $x\in \mathbb{R}^d$, $\gamma \in (0,\gamma^\star]$ and $\delta \in (0,1)$. Then,
\begin{equation*}
\sup_{t\ge0}\mathbb{E}_x\left[|X_t - \bar{X}_t|^2\right]\lesssim_{r}
\frac{L\gamma}{\underline{c}\delta} \left(U(x)+\bar{\Psi}\right)^{1+2r} \left(\Gamma \left( \frac{1}{\delta}\right) + \frac{(U(x)+\bar{\Psi})^{r+\frac{2\delta r}{1-\delta}}}{\underline{c}^{1+\frac{2\delta}{1-\delta}}} \right)
\end{equation*}
where $\Gamma$ is the gamma function.
\end{prop}
\begin{rem}The control of the $L^2$-distance between the diffusion and its Euler scheme is a fundamental property for the efficiency of the multilevel method. Actually, it allows us to control the variance of each level. The fact that we are able to obtain such a property in this (semi)-weakly convex setting is new.
\end{rem}
We start with two technical lemmas.
\begin{lem}\label{art2:lem:momenttoU}
Assume $(\mathbf{H_{r}^1})$ then for all $x\in \mathbb{R}^d$ we have
\begin{equation*}
|x-x^\star|^2 \le \frac{U^{1+r}(x)- U^{1+r}(x^\star)}{\underline{c} (1+r)}.
\end{equation*}
\end{lem}
\begin{proof}
First, one can check that for all $x\in \mathbb{R}^d$ and for all eigenvalue of the Hessian we have
\begin{equation}\label{art2:eq:eigenvalue}
\lambda_{D^2U^{1+r}(x)} \ge (1+r)U^r(x) \underline{\lambda}_{D^2U(x)} \ge (1+r)\underline{c},
\end{equation}
where in the last inequality we have used assumption $(\mathbf{H_{r}^1})$. By the Taylor formula,
\begin{equation*}
U^{1+r}(x) -U^{1+r}(x^\star) = \langle \nablaU^{1+r} (x^\star),x-x^\star \rangle + \int_0^1 \langle D^2 U^{1+r}(\xi_{\theta})(x-x^\star),x-x^\star \rangle \mathrm{d}\theta,
\end{equation*}
where $\xi_\theta = \lambda x + (1-\lambda)x^{*} $. By \eqref{art2:eq:eigenvalue} and the fact that $\nablaU^{1+r} (x^\star)=0$, we get
\begin{equation*}
U^{1+r}(x) -U^{1+r}(x^\star) \ge (1+r)\underline{c} \int_0^1 |x-x^\star|^2 \mathrm{d}\theta.
\end{equation*}
This concludes the proof.
\end{proof}
The next lemma is a bound on the moments of the increment of the Euler scheme (with the notations $\gamma^\star$ and $\bar{\Psi}$ introduced in Proposition \ref{art2:prop:moment}).
\begin{lem}\label{art2:lem:EulIncr}Assume $(\mathbf{H_{r}^1})$ with $r\in[0,1)$. Let $\gamma \in (0,\gamma^\star]$. Then for all $t>0$ and $k\in \mathbb{N}^*$,
\begin{equation*}
\mathbb{E}_x \left[\left| \bar{X}_t - \bar{X}_{\underline{t}}\right|^{2k} \right]^{1/k} \lesssim_{r,k} L^2 (t-\underline{t})^2\left(U(x)+\bar{\Psi}\right)^{1+r} +(t-\underline{t})d\sigma^2.
\end{equation*}
\end{lem}
\begin{proof}
By the definition of the Euler scheme we have
\begin{equation*}
\begin{split}
\mathbb{E}_x \left[\left| \bar{X}_t - \bar{X}_{\underline{t}}\right|^{2k} \right]^{1/k} &= \mathbb{E}_x \left[\left(\left|- (t-\underline{t})\nablaU \left(\bar{X}_{\underline{t}}\right)+\sigma(B_t-B_{\underline{t}})\right|^{2}\right)^k \right]^{1/k}
\\ & \le \mathbb{E}_x \left[\left(2 (t-\underline{t})^2 \left|\nablaU \left(\bar{X}_{\underline{t}}\right)\right|^{2}+2\sigma^2\left|B_t-B_{\underline{t}}\right|^{2}\right)^k \right]^{1/k}.
\end{split}
\end{equation*}
Then, by Minkowski inequality,
\begin{equation*}
\begin{split}
\mathbb{E}_x \left[\left| \bar{X}_t - \bar{X}_{\underline{t}}\right|^{2k} \right]^{1/k} & \le 2 (t-\underline{t})^2 \mathbb{E}_x \left[\left|\nablaU \left(\bar{X}_{\underline{t}}\right)\right|^{2k}\right]^{1/k}+2\sigma^2\mathbb{E}_x \left[\left|B_t-B_{\underline{t}}\right|^{2k} \right]^{1/k}
\\ & \le 2 L^2 (t-\underline{t})^2 \mathbb{E}_x \left[\left|\bar{X}_{\underline{t}}-x^\star\right|^{2k}\right]^{1/k}+2 (t-\underline{t})\sigma^2\mathbb{E}_x \left[|Z|^{2k} \right]^{1/k},
\end{split}
\end{equation*}
where in the last line we used the $L$-Lipschitz continuous property of $\nabla U$ and the fact that $B_t-B_{\underline{t}} \sim \sqrt{t-\underline{t}}Z$ with $Z \sim \mathcal{N}(0,\mathrm{Id})$. Finally, by Lemma \ref{art2:lem:momenttoU} and Proposition \ref{art2:prop:moment}, we obtain
\begin{align*}
\mathbb{E}_x \left[\left| \bar{X}_t - \bar{X}_{\underline{t}}\right|^{2k} \right]^{1/k}& \le \frac{2L^2 (t-\underline{t})^2}{1+r} \sup_{n\ge0} \mathbb{E}[\left(U^{1+r}(\bar{X}_{n\gamma})\right)^{2k}]^{\frac{1}{2k}}
+ 2 d \sigma^2 \left(\frac{(2k)!}{2^k k!}\right)^{1/k} (t-\underline{t})\\
& \lesssim_{r,k} L^2 (t-\underline{t})^2\left(U(x)+\bar{\Psi}\right)^{1+r} +(t-\underline{t})d\sigma^2.
\end{align*}
\end{proof}
\noindent We are now ready to prove Proposition \ref{art2:prop:HypH2}.
\begin{proof}\textit{(of Proposition \ref{art2:prop:HypH2})}
Let $x \in \mathbb{R}^d$ and consider the following process in $\mathbb{R}^d\times\mathbb{R}^d$,
\begin{equation*}
\begin{cases}
&X_t=x-\int_0^t \nablaU(X_s) \mathrm{d}s+\sigma B_t\\
&\bar{X}_t=\bar{X}_{\underline{t}}-(t-\underline{t}) \nablaU(\bar{X}_{\underline{t}})+\sigma \left(B_t-B_{\underline{t}}\right) ,
\end{cases}
\end{equation*}
with $\bar{X}_0=x.$ Denote by $F_x(t):=|X_t - \bar{X}_t|^2$. By the Lebesgue differentiability theorem,
\begin{equation*}
\begin{split}
F'_x(t)&=2\left\langle X_t-\bar{X}_t, \nabla U(\bar{X}_{\underline{t}})-\nablaU(X_t)\right\rangle \\
&=2\underbrace{ \left\langle X_t-\bar{X}_t, \nablaU(\bar{X}_t)-\nabla U(X_t)\right\rangle}_{E_1}+2\underbrace{ \left\langle X_t-\bar{X}_t, \nablaU(\bar{X}_{\underline{t}})-\nablaU(\bar{X}_t)\right\rangle}_{E_2}.
\end{split}
\end{equation*}
To control $E_1$ we use a Taylor expansion and obtain,
\begin{equation*}
\begin{split}
E_1 &= -2\left\langle X_t-\bar{X}_t, \int_0^1 D^2U\left(\lambda X_t + (1-\lambda)\bar{X}_t\right)(X_t-\bar{X}_t) \mathrm{d}\lambda\right\rangle \\
& = -2\int_0^1 \left\langle X_t-\bar{X}_t, D^2U\left(\lambda X_t + (1-\lambda)\bar{X}_t\right)(X_t-\bar{X}_t)\right\rangle \mathrm{d}\lambda.
\end{split}
\end{equation*}
Since $D^2U$ is a symmetric matrix,
\begin{equation*}
E_1 \le -2 \xi_t \left| X_t-\bar{X}_t \right|^2,
\end{equation*}
where $\xi_t = \int_0^1 \underline{\lambda}_{D^2U\left(\lambda X_t + (1-\lambda)\bar{X}_t\right)} \mathrm{d}\lambda$. For the second term $E_2$, using the inequality $\langle a,b \rangle \le \frac{\xi_t}{2}|a|^2+\frac{1}{2 \xi_t}|b|^2$ and the fact that $\nabla U$ is $L$-Lipschitz, we get
\begin{equation*}
E_2 \le \xi_t \left| X_t-\bar{X}_t \right|^2 + \frac{L^2}{\xi_t} \left| \bar{X}_t -\bar{X}_{\underline{t}} \right|^2.
\end{equation*}
Thus,
\begin{equation*}
F_x'(t) \le - \xi_t F_x(t) + \frac{L^2}{\xi_t} \left| \bar{X}_t -\bar{X}_{\underline{t}} \right|^2,
\end{equation*}
and a Gronwall argument leads to
\begin{equation*}
F_x(t) \le \int_0^t \frac{L^2}{\xi_s} \left| \bar{X}_s -\bar{X}_{\underline{s}} \right|^2 e^{-\int_s^t \xi_u \mathrm{d}u}\mathrm{d}s .
\end{equation*}
Taking the expectation, the Fubini's theorem implies
\begin{equation*}
\mathbb{E}_x\left[|X_t - \bar{X}_t|^2\right] \le \int_0^t \underbrace{\mathbb{E}_x \left[ \frac{1}{\xi_s} \left| \bar{X}_s -\bar{X}_{\underline{s}} \right|^2 e^{-\int_s^t \xi_u \mathrm{d}u}\right]}_{A_s} \mathrm{d}s.
\end{equation*}
Now let $\phi$ be a real non negative function, $\phi: t \mapsto \phi(t)$
\begin{equation} \label{art2:eq:integ}
\begin{split}
A_s & \le \mathbb{E}_x \left[ \frac{1}{\xi_s} \left| \bar{X}_s -\bar{X}_{\underline{s}} \right|^2e^{- \int_s^t \xi_u\mathrm{d}u} 1_{\{ \int_s^t \xi_u \mathrm{d}u > \phi(s) \}}\right]
\\ &+ \mathbb{E}_x \left[ \frac{1}{\xi_s} \left| \bar{X}_s -\bar{X}_{\underline{s}} \right|^2e^{- \int_s^t \xi_u\mathrm{d}u} 1_{\{ \int_s^t \xi_u \mathrm{d}u \le \phi(s) \}}\right]
\\ & \le \underbrace{ \mathbb{E}_x \left[ \frac{1}{\xi_s} \left| \bar{X}_s -\bar{X}_{\underline{s}} \right|^2 \right]e^{- \phi(s)}}_{A_s^{(1)}}+ \underbrace{ \mathbb{E}_x \left[\frac{1}{\xi_s} \left| \bar{X}_s -\bar{X}_{\underline{s}} \right|^2 1_{\{ \int_s^t \xi_u \mathrm{d}u \le \phi(s) \}}\right]}_{A_s^{(2)}},
\end{split}
\end{equation}
Using $(\mathbf{H_{r}^1})$, the convexity of $x \mapsto U(x)$ and $t \mapsto t^{-r}$ and the Jensen inequality, we have
\begin{equation*}
\begin{split}
\xi_t=\int_0^1 \underline{\lambda}_{D^2U\left(\lambda X_t + (1-\lambda)\bar{X}_t\right)} \mathrm{d}\lambda & \ge \underline{c}\int_0^1 U^{-r} \left(\lambda X_t + (1-\lambda)\bar{X}_t\right) \mathrm{d}\lambda
\\ & \ge \underline{c} \int_0^1 \left( \lambda U \left( X_t\right) +(1-\lambda) U \left(\bar{X}_t\right)\right)^{-r} \mathrm{d}\lambda
\\ & \ge \underline{c}\left( \int_0^1 \lambda U \left( X_t\right) +(1-\lambda) U \left(\bar{X}_t\right) \mathrm{d}\lambda \right)^{-r}
\\ & \ge 2^{r}\underline{c}\left( U \left( X_t\right) +U \left(\bar{X}_t\right) \right)^{-r}.
\end{split}
\end{equation*}
Thus by the inequality $(a+b)^{p} \le a^p + b^p$ for $a,b \ge 0$ and $p \in [0,1]$,
\begin{equation}\label{art2:ineq:invxi}
\begin{split}
\xi_t^{-1} \le \frac{1}{2^{r}\underline{c}}\left(U \left(X_t\right) + U\left(\bar{X}_t\right) \right)^{ r} \le \frac{1}{2^{r}\underline{c}}\left(U^r\left(X_t\right) + U^r \left(\bar{X}_t\right)\right).
\end{split}
\end{equation}
By Cauchy-Schwarz inequality, Proposition \ref{art2:prop:moment}\emph{(iii)} and Lemma \ref{art2:lem:EulIncr} we get
\begin{align}
A_s^{(1)} & \le \mathbb{E}[\xi_s^{-2}]^{\frac{1}{2}} \mathbb{E}[|\bar{X}_s-\bar{X}_{\underline{s}}|^4]^{\frac{1}{2}} e^{- \phi(s)}\nonumber \\
& \lesssim_{r} e^{- \phi(s)} \underline{c}^{-1}\left(U(x)+\bar{\Psi}\right)^{r} \left[ L^2 (s-\underline{s})^2\left(U(x)+\bar{\Psi}\right)^{1+r} +(s-\underline{s})d\sigma^2\right]\nonumber\\
&\lesssim_{r} e^{- \phi(s)} \frac{L\gamma}{\underline{c}} \left(U(x)+\bar{\Psi}\right)^{1+2r},\label{art2:eq:aunes}
\end{align}
where in the last line, we used that $d\sigma^2\le \bar{\Psi}$ and $\gamma L\le 1$.
For the second term use Cauchy-Schwarz inequality,
\begin{equation*}
\begin{split}
A_s^{(2)} & \le \mathbb{E}_x \left[\left(\frac{1}{\xi_s} \left| \bar{X}_s -\bar{X}_{\underline{s}} \right|^2 \right)^2\right]^{1/2} \Pr_x \left( \int_s^t \xi_u \mathrm{d}u \le \phi(s) \right)^{1/2}
\\ & \le \mathbb{E}_x \left[\frac{1}{\xi_s^4} \right]^{1/4}\mathbb{E}_x \left[\left| \bar{X}_s -\bar{X}_{\underline{s}} \right|^8\right]^{1/4} \Pr_x \left( \int_s^t \xi_u \mathrm{d}u \le \phi(s) \right)^{1/2}.
\end{split}
\end{equation*}
With the help of Inequality \eqref{art2:ineq:invxi} and Proposition \ref{art2:prop:moment}, we have
\begin{equation*}
A_s^{(2)} \lesssim_{r} \underline{c}^{-1}\left(U(x)+\bar{\Psi}\right)^{r} \left[ L^2 (s-\underline{s})^2\left(U(x)+\bar{\Psi}\right)^{1+r} +(s-\underline{s})d\sigma^2\right] \Pr_x \left( \int_s^t \xi_u \mathrm{d}u \le \phi(s) \right)^{1/2}.
\end{equation*}
For the third term of this product, let $\kappa$ be a positive number and use Markov inequality:
\begin{equation*}
\begin{split}
\Pr_x \left( \int_s^t \xi_u \mathrm{d}u \le \phi(s) \right)^{1/2} & \le \Pr_x \left( \left( \int_s^t \xi_u \mathrm{d}u\right)^{-\kappa} \ge \phi^{\kappa}(s)\right)^{1/2}
\\ & \le \phi^{\kappa/2}(s) \mathbb{E} \left[ \left( \int_s^t \xi_u \mathrm{d}u\right)^{-\kappa} \right]^{1/2}.
\end{split}
\end{equation*}
The function $x \mapsto x^{-\kappa}$ being convex on $(0,+\infty)$ it follow from Jensen inequality that
\begin{equation*}
\Pr_x \left( \int_s^t \xi_u \mathrm{d}u \le \phi(s) \right)^{1/2} \le \phi^{\kappa/2}(s) (t-s)^{-\kappa/2} \mathbb{E} \left[ \frac{1}{t-s} \int_s^t \xi_u^{-\kappa} \mathrm{d}u \right]^{1/2}.
\end{equation*}
Using again inequality \eqref{art2:ineq:invxi} and Proposition \ref{art2:prop:moment},
\begin{equation*}
\begin{split}
\Pr_x \left( \int_s^t \xi_u \mathrm{d}u \le \phi(s) \right)^{1/2} & \lesssim_r \underline{c}^{-\frac{\kappa}{2}}\left(\frac{ \phi(s)}{t-s} \right)^{\kappa/2} \sup_{u \in [s,t]} \mathbb{E} \left[ U^{ \kappa r}\left( X_t\right) + U^{ \kappa r}\left(\bar{X}_t\right)\right]^{1/2}
\\ & \lesssim_r \underline{c}^{-\frac{\kappa}{2}}\left(\frac{ \phi(s)}{t-s} \right)^{\kappa/2} \left(U(x)+\bar{\Psi}\right)^{\frac{\kappa r}{2}} .
\end{split}
\end{equation*}
Finally, we get
\begin{equation*}
A_s^{(2)} \lesssim_r \underline{c}^{-1-\frac{\kappa}{2}}\left(U(x)+\bar{\Psi}\right)^{r+\frac{\kappa r}{2}} \left[ L^2 (s-\underline{s})^2 \left(U(x)+\bar{\Psi}\right)^{1+r} +(s-\underline{s})d\sigma^2\right] \left(\frac{ \phi(s)}{t-s} \right)^{\kappa/2} .
\end{equation*}
Now let $\phi(s)=\frac{t-s}{(t+1-s)^{1-\delta}}$ with $a>0$, $\delta \in (0,1)$ and $\kappa=\frac{2(1+\delta)}{1-\delta}$ we have
\begin{equation*}
A_s^{(2)} \lesssim_{r} \underline{c}^{-2-\frac{2\delta}{1-\delta}}\left[L^2 (s-\underline{s})^2 (U(x)+\bar{\Psi})^{1+3r+\frac{2\delta r}{1-\delta}}+(s-\underline{s}){d\sigma^2}(U(x)+\bar{\Psi})^{2r+\frac{2\delta r}{1-\delta}}\right] \left(\frac{1}{t+1-s} \right)^{1+\delta}.
\end{equation*}
As a consequence, since $\gamma L\le 1$ and $d\sigma^2 \le {\bar{\Psi}}$, we obtain
\begin{equation*}
A_s^{(2)} \lesssim_{r} \frac{L \gamma}{\underline{c}^{2+\frac{2\delta}{1-\delta}}} (U(x)+\bar{\Psi})^{1+3r+\frac{2\delta r}{1-\delta}}\left(\frac{1}{t+1-s} \right)^{1+\delta}.
\end{equation*}
Back to \eqref{art2:eq:integ}, we deduce from \eqref{art2:eq:aunes} and from the above inequality that
\begin{align*}
\mathbb{E}_x\Big[|X_t &- \bar{X}_t|^2\Big] \le \int_0^t A_s^{(1)} \mathrm{d}s + \int_0^t A_s^{(2)} \mathrm{d}s \\
& \lesssim_{r} \frac{L\gamma}{\underline{c}} \left(U(x)+\bar{\Psi}\right)^{1+2r} \left(\int_0^t e^{-\frac{ u}{(u+1)^{1-\delta}}} \mathrm{d}u +
\frac{(U(x)+\bar{\Psi})^{r+\frac{2\delta r}{1-\delta}}}{\underline{c}^{1+\frac{2\delta}{1-\delta}}}\int_0^t \left( \frac{1}{1+u}\right)^{1+ \delta} \mathrm{d}u \right) \\
& \lesssim_{r} \frac{L\gamma}{\underline{c}} \left(U(x)+\bar{\Psi}\right)^{1+2r}
\left( \int_0^t e^{-u^{\delta}} \mathrm{d}u + \frac{(U(x)+\bar{\Psi})^{r+\frac{2\delta r}{1-\delta}}}{\delta \underline{c}^{1+\frac{2\delta}{1-\delta}}}\right) \\
& \lesssim_{r} \frac{L\gamma}{\underline{c}\delta} \left(U(x)+\bar{\Psi}\right)^{1+2r} \left( \Gamma \left( \frac{1}{\delta}\right) + \frac{(U(x)+\bar{\Psi})^{r+\frac{2\delta r}{1-\delta}}}{\underline{c}^{1+\frac{2\delta}{1-\delta}}} \right)
\end{align*}
The result follows.
\end{proof}
\subsection{Convergence to equilibrium for the Euler Scheme under \texorpdfstring{$(\mathbf{H_{r}^1}) $}{Hr}:}
We now proceed to establish the weak error between the discrete semi group and its invariant measure denoted by $\pi^\gamma$. The proof of this result is based on the control of the so-called tangent process $\bar{T}_t^x:= \nabla X^x_t$.
\begin{prop} \label{art2:prop:HypH1} Assume $(\mathbf{H_{r}^1})$ and let $x \in \mathbb{R}^d$, $\gamma \in (0,\gamma^\star]$. Let $\kappa>0$ and $\phi:\mathbb{R}_+\rightarrow\mathbb{R}$ be a positive function. Assume that $U(x)\lesssim_r\bar{\Psi}$. Then, for any $r\in[0,1)$, there exists a constant $c_{r,\kappa}$ (depending only on $r$ and $\kappa$) such that for all $n>0$, for all Lipschitz continuous function $f:\mathbb{R}^d\rightarrow\mathbb{R}$,
\begin{equation*}
\left| \mathbb{E}_x \left[ f(\bar{X}_{n \gamma}) \right] - \pi^{\gamma}(f) \right| \le c_{r,\kappa} [f]_1 h_{\phi,\kappa}(n),
\end{equation*}
where
\begin{equation*}
h_{\phi,\kappa}(n) = \underline{c}^{-\frac{1}{2}}\bar{\Psi}^{\frac{1+r}{2}} e^{- \phi(n)}+\underline{c}^{-\frac{\kappa+1}{2}}\bar{\Psi}^{\frac{1+r(1+\kappa)}{2}}\left( \frac{\phi(n)}{n\gamma} \right)^{\frac{\kappa}{2}},
\end{equation*}
with $\gamma^\star$ and $\bar{\Psi}$ defined in Proposition \ref{art2:prop:moment}.
\end{prop}
\begin{rem} \label{art2:rem:summablefunction}
$\rhd$ In order to alleviate the purpose, the result is stated under the assumption that the initial condition $x$ satisfies $U(x)\lesssim_r\bar{\Psi}$ but the reader will find some bounds without this assumption in the proofs.
\bigskip \newline \noindent $\rhd$ The function $h_{\phi,\kappa}$ plays the role of convergence rate to equilibrium. In this setting where the Hessian is not lower-bounded, we adopt a strategy which consists in separating the space into two parts. In the first one, we assume that we have some good contraction properties parametrized by the function $\phi$ and in the other one, we try simply to control the probability that such a good contraction does not occur. This leads to a balance between two terms depending on $\phi$ and $\kappa$. In the following, we will choose $\phi$ and $\kappa$ in order that $h_{\phi,\kappa}$ is summable with the smallest impact on the dependence in the dimension.
\bigskip \newline \noindent Note that in \cite{cattiaux:hal-02486264}, some exponential rates are exhibited under similar assumptions in the continuous case (with the help of concentration inequalities). However, this exponential rate depends on some constants whose control seems to be difficult to obtain (typically, when the starting distribution is absolutely continuous with respect to the invariant distribution, the constants involve the $L^2$-moment of the related density). Probably, some ideas could be adapted to the Euler scheme (starting from a deterministic point) but with technicalities that seem to carry us too far for this paper.
\end{rem}
We preface the proof of Proposition \ref{art2:prop:HypH1} by a lemma about the shape of the \textit{first variation process} of the continuous time Euler scheme, $\bar{T}_t^x= \nabla_x \bar{X}^x_t$.
\begin{lem}\label{art2:lem:TDP}
For all $n \in \mathbb{N}$, $x\in \mathbb{R}^d$ and $\gamma\in [0,1)$,
\begin{equation*}
\bar{T}_{(n+1)\gamma}^x = \prod_{i=0}^n \left(\mathrm{Id}_{\mathbb{R}^d}- \gamma D^2 U \left( \bar{X}_{i\gamma}^x \right)\right).
\end{equation*}
\end{lem}
\begin{proof}\textit{(of Lemma \ref{art2:lem:TDP})}
First, observe that for all $n \in \mathbb{N}$,
\begin{equation*}
\frac{\bar{T}_{(n+1)\gamma}^x-\bar{T}_{n\gamma}^x}{\gamma} = \nabla_x \left(\frac{\bar{X}_{(n+1)\gamma}^x-\bar{X}_{n\gamma}^x}{\gamma} \right),
\end{equation*}
and by the definition of the Euler scheme and the chain rule,
\begin{equation*}
\begin{split}
\frac{\bar{T}_{(n+1)\gamma}^x-\bar{T}_{n\gamma}^x}{\gamma} &= \nabla_x \left(-\nabla_x U \left(\bar{X}_{n\gamma}^x\right) + \sigma \left( B_{(n+1)\gamma}- B_{(n-1)\gamma} \right) \right) \\
& = - D^2 U \left(\bar{X}_{n\gamma}^x\right) \bar{T}_{n\gamma}^x.
\end{split}
\end{equation*}
Then we get,
\begin{equation*}
\bar{T}_{(n+1)\gamma}^x=\bar{T}_{n\gamma}^x \left(\mathrm{Id}_{\mathbb{R}^d} - \gamma D^2 U \left(\bar{X}_{n\gamma}^x\right) \right),
\end{equation*}
and the proof follows by a simple induction.
\end{proof}
Consider two paths defined with the same Brownian motion and different starting points: $x,y \in \mathbb{R}^d$. The following proposition shows that there is a pathwise confluence, \textit{i.e.} the two trajectories get closer when $n$ goes to infinity.
\begin{prop}\label{art2:prop:discretconfl}
Assume $(\mathbf{H_{r}^1})$ and let $x,y \in \mathbb{R}^d$, $\gamma \in (0,\gamma^\star]$, $\kappa>0$. Let $\phi:\mathbb{R}_+\rightarrow\mathbb{R}$ be a positive function. Then,
\begin{equation*}
\sup_{n\ge0} \mathbb{E}_{(x,y)} \left[ \left|\bar{X}_{n\gamma}^x-\bar{X}_{n\gamma}^y \right|^2 \right] \lesssim_{r,\kappa} \left| x-y \right|^2 h_{\phi,\kappa,x,y}(n),
\end{equation*}
where,
\begin{equation*}
h_{\phi,\kappa,x,y}(n) = e^{-2\phi(n)} + \left( \frac{\phi(n)}{n\gamma} \right)^{\kappa}\underline{c}^{-\kappa} (U(x)+U(y)+\bar{\Psi})^{\kappa r},\end{equation*}
with $\gamma^\star$ and $\bar{\Psi}$ given by Proposition \ref{art2:prop:moment}.
\end{prop}
\begin{rem} \label{art2:rem:exponenrates} In the sequel, this property is typically applied with a polynomial function $\phi$ which leads to polynomial rates to equilibrium. It is worth noting that the proof could be adapted to provide exponential rates (the idea would be to consider an exponentially decreasing convex function instead of $x\mapsto x^{-\kappa}$ in the proof below). However, with our method, such rates would lead to exponential dependence in the dimension. This is why we do not give such bounds here.
\end{rem}
\begin{proof}\textit{(of Proposition \ref{art2:prop:discretconfl})}
For $x,y \in \mathbb{R}^d$ and $n\in \mathbb{N}$, let us start by a Taylor expansion of the function $x \mapsto \bar{X}_{n\gamma}^x $,
\begin{equation*}
\begin{split}
\left|\bar{X}_{n\gamma}^x-\bar{X}_{n\gamma}^y \right|^2 & = \left| \int_0^1 \bar{T}_{n\gamma}^{\lambda x + (1-\lambda)y}(x-y) \mathrm{d}\lambda \right|^2
\\ & = \left| x-y \right|^2 \left| \int_0^1 \bar{T}_{n\gamma}^{\lambda x + (1-\lambda)y} \mathrm{d}\lambda \frac{x-y}{\left| x-y \right|} \right|^2
\\ & \le \left| x-y \right|^2 \left\| \int_0^1 \bar{T}_{n\gamma}^{\lambda x + (1-\lambda)y} \mathrm{d}\lambda \right\|^2,
\end{split}
\end{equation*}
where $\|.\|$ is the operator norm associated with the Euclidean norm. By Jensen inequality and Lemma \ref{art2:lem:TDP},
\begin{equation*}
\begin{split}
\left|\bar{X}_{n\gamma}^x-\bar{X}_{n\gamma}^y \right|^2 & \le \left| x-y \right|^2 \int_0^1 \left\|\prod_{i=0}^{n-1} \mathrm{Id}_{\mathbb{R}^d}- \gamma D^2 U \left( \bar{X}_{i\gamma}^{\lambda x + (1-\lambda)y} \right) \right\|^2 \mathrm{d}\lambda
\\ & \le \left| x-y \right|^2 \int_0^1 \prod_{i=0}^{n-1} \left\| \mathrm{Id}_{\mathbb{R}^d}- \gamma D^2 U \left( \bar{X}_{i\gamma}^{\lambda x + (1-\lambda)y} \right) \right\|^2 \mathrm{d}\lambda.
\end{split}
\end{equation*}
The operator norm associated with the Euclidean norm of a symmetric matrix is equal to its spectral radius, so we get
\begin{equation*}
\begin{split}
\left|\bar{X}_{n\gamma}^x-\bar{X}_{n\gamma}^y \right|^2 & \le \left| x-y \right|^2 \int_0^1 \prod_{i=0}^{n-1}\left( 1 - \gamma \un{\lambda}_{D^2 U\left(\bar{X}_{i\gamma}^{\lambda x - (1-\lambda)y}\right)}\right)^2 \mathrm{d}\lambda
\\ & \le \left| x-y \right|^2 \int_0^1 e^{-2\gamma \sum_{i=0}^{n-1} \un{\lambda}_{D^2 U\left(\bar{X}_{i\gamma}^{\lambda x - (1-\lambda)y}\right)} } \mathrm{d}\lambda,
\end{split}
\end{equation*}
and
\begin{equation} \label{art2:eq:confluencediscrete}
\mathbb{E}_{(x,y)} \left[ \left|\bar{X}_{n\gamma}^x-\bar{X}_{n\gamma}^y \right|^2 \right] \le \left| x-y \right|^2 \int_0^1 \mathbb{E}_{(x,y)} \left[ e^{-2\xi_n (\lambda)} \right] \mathrm{d}\lambda,
\end{equation}
with,
\begin{equation*}
\begin{split}
\xi_n (\lambda)& := \gamma \sum_{i=0}^{n-1} \un{\lambda}_{D^2 U\left(\bar{X}_{i\gamma}^{\lambda x - (1-\lambda)y}\right)} = \int_0^{n\gamma}\un{\lambda}_{D^2 U\left(\bar{X}_{\underline{u}}^{\lambda x - (1-\lambda)y}\right)} \mathrm{d}u.
\end{split}
\end{equation*}
For a given real non negative function $\phi: \mathbb{R} \to \mathbb{R} $ we have
\begin{equation*}
\begin{split}
\mathbb{E}_{(x,y)} \left[ e^{-2 \xi_n (\lambda)} \right] & \le \mathbb{E}_{(x,y)} \left[ e^{-2 \xi_n} 1_{\{ \xi_{n} > \phi(n) \}}\right] + \mathbb{E}_{(x,y)} \left[ e^{-2 \xi_n} 1_{\{ \xi_{n} \le \phi(n) \}}\right]
\\ & \le e^{-2 \phi(n)}+ \mathbb{E}_{(x,y)} \left[ 1_{\{ \xi_{n} \le \phi(n) \}}\right]
\\ & \le e^{-2 \phi(n)}+ \Pr_{(x,y)} \left( \int_0^{n\gamma}\un{\lambda}_{D^2 U\left(\bar{X}_{\underline{u}}^{\lambda x - (1-\lambda)y}\right)} \mathrm{d}u \le \phi(n) \right).
\end{split}
\end{equation*}
For a positive number $\kappa$ we have
\begin{equation*}
\mathbb{E}_{(x,y)} \left[ e^{-2 \xi_n (\lambda)} \right] \le e^{-2 \phi(n)} + \Pr_{(x,y)} \left( \left( \int_0^{n\gamma}\un{\lambda}_{D^2 U\left(\bar{X}_{\underline{u}}^{\lambda x - (1-\lambda)y}\right)} \mathrm{d}u\right)^{-\kappa} \ge \phi^{-\kappa}(n) \right),
\end{equation*}
and using the Markov inequality,
\begin{equation*}
\mathbb{E}_{(x,y)} \left[ e^{-2 \xi_n (\lambda)} \right]
\le e^{-2 \phi(n)} + \phi^{\kappa}(n) \mathbb{E}_{(x,y)} \left[ \left( \int_0^{n\gamma}\un{\lambda}_{D^2 U\left(\bar{X}_{\underline{u}}^{\lambda x - (1-\lambda)y}\right)} \mathrm{d}u\right)^{-\kappa} \right].
\end{equation*}
The function $x \mapsto x^{-\kappa}$ is convex on $(0,+\infty)$ then by Jensen inequality,
\begin{equation*}
\begin{split}
\mathbb{E}_{(x,y)} \left[ e^{-2 \xi_n (\lambda)} \right] &\le e^{-2 \phi(n)} + \phi^{\kappa}(n) (n\gamma)^{-\kappa} \mathbb{E}_{(x,y)} \left[ \frac{1}{n\gamma} \int_0^{n\gamma} \un{\lambda}_{D^2 U\left(\bar{X}_{\underline{u}}^{\lambda x - (1-\lambda)y}\right)}^{-\kappa} \mathrm{d}u \right]
\\ & \le e^{-2\phi(n)} + \left( \frac{\phi(n)}{ n\gamma } \right)^{\kappa} \sup_{k \in \{0,\dots,n-1\}} \mathbb{E} \left[ \un{\lambda}_{D^2 U\left(\bar{X}_{k\gamma}^{\lambda x - (1-\lambda)y}\right)}^{-\kappa} \right].
\end{split}
\end{equation*}
Observe that assumption $(\mathbf{H_{r}^1})$ implies,
\begin{equation*}
\mathbb{E}_{(x,y)} \left[ e^{-2 \xi_n (\lambda)} \right] \le e^{-2\phi(n)} + \left( \frac{\phi(n)}{n\gamma} \right)^{\kappa} \sup_{k \in \{0,\dots,n-1\}} \mathbb{E} \left[\underline{c}^{-\kappa}U^{\kappa r}\left(\bar{X}_{k \gamma}^{\lambda x + (1-\lambda)y}\right)\right].
\end{equation*}
By Proposition \ref{art2:prop:moment} and the convexity of $U$, this implies that for any $\gamma\in(0,\gamma^\star]$,
\begin{equation*}
\mathbb{E}_{(x,y)} \left[ e^{-2 \xi_n (\lambda)} \right] \lesssim_{r,\kappa} e^{-2\phi(n)} + \left( \frac{\phi(n)}{n\gamma} \right)^{\kappa}\underline{c}^{-\kappa} (U(x)+U(y)+\bar{\Psi})^{\kappa r}.
\end{equation*}
\end{proof}
Thanks to this confluence property we are now able to prove the convergence to equilibrium of the Euler scheme and to give the rate of this convergence.
\begin{proof}\textit{(of Proposition \ref{art2:prop:HypH1})}
Since $\pi^\gamma$ is invariant for $\left(\bar{X}_{n\gamma}\right)_{n \in \mathbb{N}}$ we deduce from Fubini's Theorem and Jensen inequality that
\begin{equation*}
\begin{split}
\left| \mathbb{E}_x \left[ f(\bar{X}_{n \gamma}) \right] - \pi^{\gamma}(f) \right|^2 &= \left| \int_{\mathbb{R}^d} \mathbb{E}_{(x,y)} \left[f(\bar{X}_{n \gamma}^x) - f(\bar{X}_{n \gamma}^y) \right] \pi^{\gamma}(\mathrm{d}y) \right|^2
\\ & \le \int_{\mathbb{R}^d} \mathbb{E}_{(x,y)} \left[ \left| f(\bar{X}_{n \gamma}^x) - f(\bar{X}_{n \gamma}^y)\right|^2 \right] \pi^{\gamma}(\mathrm{d}y).
\end{split}
\end{equation*}
The Lipschitz property of $f$ implies that
\begin{equation*}
\left| \mathbb{E}_x \left[ f(\bar{X}_{n \gamma}) \right] - \pi^{\gamma}(f) \right|^2 \le[f]_1^2 \int_{\mathbb{R}^d} \mathbb{E}_{(x,y)} \left[ \left| \bar{X}_{n \gamma}^x - \bar{X}_{n \gamma}^y\right|^2 \right] \pi^{\gamma}(\mathrm{d}y),
\end{equation*}
where $[f]_1$ is the Lipschitz constant of $f$. Proposition \ref{art2:prop:discretconfl} implies
\begin{equation}\label{art2:eq:jlskjdl}
\left| \mathbb{E}_x \left[ f(\bar{X}_{n \gamma}) \right] - \pi^{\gamma}(f) \right|^2 \lesssim_{r,\kappa} [f]_1^2 \int_{\mathbb{R}^d} \left| x-y \right|^2 h_{\phi,\kappa,x,y}(n)\pi^{\gamma}(\mathrm{d}y).
\end{equation}
By Lemma \ref{art2:lem:momenttoU},
\begin{equation*}
|x-y|^2\le 2 (|x-x^\star|^2+|y-y^\star|^2)\le \frac{2}{\underline{c}(1+r)}\left(U^{1+r}(x)+U^{1+r}(y)\right).
\end{equation*}
With the help of the Young inequality, we also have
\begin{equation*}
|x-y|^2(U(x)+U(y))^{r\kappa}\lesssim_{r,\kappa} \underline{c}^{-1} \left(U^{1+r(1+\kappa)}(x)+U^{1+r(1+\kappa)}(y)\right).
\end{equation*}
Plugging these controls into \eqref{art2:eq:jlskjdl} yields
\begin{equation}\label{art2:eq:vitessexxx}
\begin{split}
\left| \mathbb{E}_x \left[ f(\bar{X}_{n \gamma}) \right] - \pi^{\gamma}(f) \right|^2 & \lesssim_{r,\kappa} \frac{[f]_1^2}{\underline{c}} \left(U^{1+r}(x)+\pi^\gamma(U^{1+r})\right)e^{-2 \phi(n)}\\
&+ [f]_1^2\underline{c}^{-\kappa-1}\left( U^{1+r(1+\kappa)}(x)+\pi^\gamma(U^{1+r(1+\kappa)})\right)\left( \frac{\phi(n)}{n\gamma} \right)^{\kappa} .\\
&+ [f]_1^2\underline{c}^{-\kappa-1} \bar{\Psi}^{r\kappa}\left( U^{1+r}(x)+\pi^\gamma(U^{1+r})\right)\left( \frac{\phi(n)}{n\gamma} \right)^{\kappa} .
\end{split}
\end{equation}
To conclude, we now use the bound \eqref{art2:eq:boundfacileinvariant} of Proposition \ref{art2:prop:moment}\emph{(iii)} and the assumption $U(x)\lesssim_r \bar{\Psi}$.
\end{proof}
\subsection{Bias induced by the discretization under \texorpdfstring{$(\mathbf{H_{r}^1}) $}{Hr}:}\label{art2:sec:weakerror}
We now need to provide estimates of ${\cal W}_1(\pi,\pi^\gamma)$. We provide two results:
Lemma \ref{art2:lem:firstresult} where we directly derive from Proposition \ref{art2:prop:HypH2} a bound in $O(\sqrt{\gamma})$ which ``only'' requires the potential $U$ to be ${\cal C}^2$. However, such a bound has a serious impact on the dependency in $\varepsilon$ of the complexity. Thus, we propose a second result when $U$ is ${\cal C}^3$ where we recover a bound in $O({\gamma})$.
\subsubsection{A first bound in \texorpdfstring{$O(\sqrt{\gamma})$}{O(sqrt(gamma))}}
As mentioned before, a first estimate can be directly deduced from Proposition \ref{art2:prop:HypH2}. Actually, since in this result, the $L^2$-error between the process and its discretization is controlled uniformly in time, this leads to a similar bound for $ {\cal W}_1(\pi,\pi^\gamma) $ by letting $t$ go to $\infty$. More precisely,
\begin{lem}\label{art2:lem:firstresult}
Assume $(\mathbf{H_{r}^1})$. Let $\gamma\in(0,\gamma^\star]$. Then,
\begin{equation}
{\cal W}_1(\pi,\pi^\gamma)^2 \lesssim_{r,\delta}\frac{L\gamma}{\underline{c}^{\frac{2}{1-\delta}}\wedge\underline{c}}\bar{\Psi}^{1+3r+\frac{2\delta r}{1-\delta}}.
\end{equation}
\end{lem}
\begin{proof}
Owing to the stationarity of $\pi$, we have for every $n\ge0$,
\begin{equation*}
{\cal W}_1(\pi,\pi^\gamma)\le {\cal W}_1(\pi P_{n\gamma},\pi \bar{P}_{n\gamma})+{\cal W}_1(\pi \bar{P}_{n\gamma}, \pi^\gamma),
\end{equation*}
so that
\begin{equation*}
{\cal W}_1(\pi,\pi^\gamma)\le \limsup_{n\rightarrow+\infty} {\cal W}_1(\pi P_{n\gamma},\pi \bar{P}_{n\gamma}),
\end{equation*}
since ${\cal W}_1(\pi \bar{P}_{n\gamma}, \pi^\gamma)\xrightarrow{n\rightarrow+\infty}0$ by
Proposition \ref{art2:prop:HypH1} (more precisely, this property can be deduced from an integration of \eqref{art2:eq:vitessexxx} with respect to $\pi$ and from the fact that $\pi (U^p)<+\infty$ for any $p$ by Proposition \ref{art2:prop:moment}).
\newline Now, integrating with respect to $\pi$ the bound of Proposition \ref{art2:prop:HypH2} and using that $\pi(U^p)\lesssim_{r} \bar{\Psi}^p$ by Proposition \ref{art2:prop:moment} leads to the result.
\end{proof}
\subsubsection{A second bound in \texorpdfstring{$O(\gamma)$}{O(gamma)}}
Even if the above bound is quite explicit in terms of its dependency with respect to $L$, $\underline{c}$, $\bar{c}$ and $d$, the fact that it is in $O(\sqrt{\gamma})$ dramatically impacts the complexity in terms of $\varepsilon$ (at least).
\newline In fact, it is possible to get a $1$-Wasserstein error of the order $\gamma$ by using a combination of the control of the rate of convergence to equilibrium of the continuous process and of the finite-time weak error (between the process and its discretization). Such a strategy is used in several papers : in \cite{art:PagesPanloup2022unajusted}, this idea is developed in a multiplicative setting with a so-called ``domino'' approach for the control of the $1$-Wasserstein and $TV$ distances between the process and its discretization, uniformly in time. For the control of ${\cal W}_1(\pi,\pi^\gamma)$ itself, our approach follows \cite{durmus2021asymptotic} which provides a series of bounds in many models and sets of assumptions which are mainly based on the following principle (see Lemma 1 of \cite{durmus2021asymptotic}). Taking advantage of the stationarity of $\pi^\gamma$, for any $p\ge1$, for any $t>0$,
\begin{equation*}
{\cal W}_p(\pi,\pi^\gamma)\le {\cal W}_p(\pi,\pi^\gamma P_{\underline{t}} )+{\cal W}_p(\pi^\gamma {P}_{\underline{t}},\pi^\gamma \bar{P}_{\underline{t}}),
\end{equation*}
so that if we assume that
\begin{equation*}
{\cal W}_p(\pi,\pi^\gamma P_{t} )\le \varepsilon_1(t){\cal W}_p(\pi,\pi^\gamma)\quad\textnormal{and}\quad {\cal W}_p(\pi^\gamma {P}_{\underline{t}},\pi^\gamma \bar{P}_{\underline{t}})\le \varepsilon_2(\underline{t}),
\end{equation*}
then,
\begin{equation}\label{art2:eq:ebdur}
{\cal W}_p(\pi,\pi^\gamma)\le \inf\left\{\frac{\varepsilon_2(\underline{t})}{1-\varepsilon_1(\underline{t})}, t>0\right\}.
\end{equation}
We thus propose to estimate $\varepsilon_1(t)$ and $\varepsilon_2(t)$ under Assumption $(\mathbf{H_{r}^1})$ (with or without $(\mathbf{H_{r}^2})$). This is the purpose of Lemmas \ref{art2:lem:varepsilonn} and
\ref{art2:lem:psin} respectively. These two estimates lead to the following proposition
\begin{prop}\label{art2:prop:HypH3}
Assume $(\mathbf{H_{r}^1})$ and let $\delta \in (0,1)$. Assume that $U$ is ${\cal C}^3$ with $\|\Delta (\nabla U)\|_{2,\infty}^2\lesssim_{r} \sigma^{-4} L^3 \bar{\Psi}$ (with $\|\Delta (\nabla U)\|_{2,\infty}$ defined in Lemma \ref{art2:lem:psin}). Then, a constant $c_{r,\delta}$ (depending only on $r$ and $\delta$) exists such that for all $\gamma \in (0,\gamma^\star]$,
\begin{equation*}
{\cal W}_2 (\pi,\pi^\gamma)\le c_{r,\delta} \left( \underline{c}^{-\frac{1}{1-\delta}} L^{\frac{3}{2}}\bar{\Psi}^{\frac{1}{2}+\frac{r}{1-\delta}}\right) \gamma.
\end{equation*}
\end{prop}
\begin{rem} $\rhd$ Note that this result is clearly in the spirit of \cite[Theorem 6]{durmus2021asymptotic}. However, there are several differences. First, we need here to adapt our proof to a setting where we only have polynomial convergence to equilibrium (instead of exponential convergence). Second, under our assumptions on $U$ which are more restrictive than the one of \cite[Theorem 6]{durmus2021asymptotic}, we can improve the constants (and in particular avoid some exponential dependence in the Lipschitz constant $L$).\\
$\rhd$ Compared with Lemma \ref{art2:lem:firstresult} , this result improves the dependence in $\gamma$ but it is worth noting that the bound is also better with respect to $\bar{\Psi}$ (and thus to the dimension).
\end{rem}
\begin{proof} With the notations of \cite[Theorem 6]{durmus2021asymptotic}, let $t_{\rm rel}=\inf\{t> 0, \varepsilon_1(\underline{t})\le 1/\sqrt{2}\}$. Using Lemma \ref{art2:lem:varepsilonn},
one can upper-bound $t_{\rm rel}$ by asking the two right-hand terms of \eqref{art2:eq:weppigamma} to be bounded by $1/4$. With $\phi(t)= t^\delta$ with $\delta \in(0,1)$, this leads to:
\begin{equation*}
t_{\rm rel}\lesssim_{r} \max\{(\log 4)^\delta, (4{\underline{c}^{-1}})^{\frac{1}{1-\delta}}\bar{\Psi}^{\frac{r}{1-\delta}}\}.
\end{equation*}
By \eqref{art2:eq:ebdur} and Lemma \ref{art2:lem:psin}\emph{(ii)}, we get for any $\lambda\in(0,1]$,
\begin{equation*}
{\cal W}_2(\pi,\pi^\gamma)\le c_r \frac{\gamma}{\lambda} \sqrt{e^{\lambda t_{\rm rel}}L^3 \bar{\Psi}}.
\end{equation*}
Let $\lambda= t_{\rm rel}^{-1}$. In this case, we obtain
\begin{equation*}
{\cal W}_2 (\pi,\pi^\gamma)\le c_r \gamma t_{\rm rel} \sqrt{L^3 \bar{\Psi}}\lesssim_{r,\delta} \gamma \underline{c}^{-\frac{1}{1-\delta}} L^{\frac{3}{2}} \bar{\Psi}^{\frac{1}{2}+\frac{r}{1-\delta}}.
\end{equation*}
\end{proof}
\begin{lem}\label{art2:lem:varepsilonn} Assume $(\mathbf{H_{r}^1})$. Then, for any $\gamma\in(0,\gamma^\star]$ and for any positive function $\phi:\mathbb{R}_+\rightarrow \mathbb{R}$,
\begin{equation}\label{art2:eq:weppigamma}
{\cal W}_2^2(\pi,\pi^\gamma P_t)\lesssim_{r} {\cal W}_2^2(\pi,\pi^\gamma)\left(e^{-2\phi(t)} + \frac{\phi(t)}{ \underline{c} t} \bar{\Psi}^{ r} \right).
\end{equation}
\end{lem}
\begin{rem} $\rhd$ The proof of \emph{(i)} is mostly a continuous-time version of the one of Proposition \ref{art2:prop:discretconfl}.
\bigskip \newline $\rhd$ As mentioned in Remark \ref{art2:rem:exponenrates}, the proof can be adapted to provide exponential rates but unfortunately our method would lead to exponential dependence in the dimension. For this section, the lack of exponential rate does not have a serious impact on the bounds.
\newline Nevertheless, if we needed to improve our bounds, an idea would be to apply \cite[Theorem 5.6]{cattiaux:hal-02486264}. In this result, the authors provide exponential rates under assumptions which are similar to ours. However the related constant depends on the density of the semi-group and it would be necessary to be able to control it with respect to the parameters of the model.
\end{rem}
\begin{proof} Denoting by $T^x_t=\partial_x X_t^x$, the first variation process related to $(X_t^x)_{t\ge0}$, we have:
\begin{equation*}
X_t^y-X_t^x=\int_0^1 T^{x+\lambda(y-x)}_t (y-x) d\lambda.
\end{equation*}
Thus,
\begin{equation*}
|X_t^y-X_t^x|^2\le \int_0^1 \|T^{x+\lambda(y-x)}_t \|^2 |y-x|^2 d\lambda,
\end{equation*}
where $\|.\|$ stands for the operator norm associated with the Euclidean norm.
Since $T^x$ is the solution to $dT_t^x=-D^2 U(X_t^x) T_t^x dt$ with $T_0^x=I_d$,
one easily checks that
\begin{equation*}
\|T^{x+\lambda(y-x)}_t \|^2\le e^{-2\int_0^t \underline{\lambda}_{U}(X_s^{x+\lambda(y-x)}) {\rm{d}}s}.
\end{equation*}
Thus,
\begin{equation*}
|X_t^y-X_t^x|^2\le |x-y|^2 \int_0^1 e^{-2\int_0^t \underline{\lambda}_{U}(X_s^{x+\lambda(y-x)}) {\rm{d}}s} d\lambda.
\end{equation*}
Following the arguments of Proposition \ref{art2:prop:discretconfl}, we get for any positive function $\phi$,
\begin{equation*}
\begin{split}
\mathbb{E} \left[ e^{-2\int_0^t \underline{\lambda}_{U}(X_s^{x+\lambda(y-x)}) {\rm{d}}s} \right] &\le e^{-2 \phi(t)} + \frac{\phi(t)}{t } \mathbb{E} \left[ \frac{1}{t} \int_0^{t} \un{\lambda}_{D^2 U\left(X_{\underline{u}}^{x+\lambda(y-x)}\right)}^{-1} \mathrm{d}u \right]
\\ & \le e^{-2\phi(t)} + \frac{\phi(t)}{ t} \sup_{s\in[0,t]\}} \mathbb{E} \left[ \un{\lambda}_{D^2 U\left(X_{s}^{x+\lambda(y-x)}\right)}^{-1} \right].
\end{split}
\end{equation*}
By $(\mathbf{H_{r}^1})$, one deduces that
\begin{align*}
\mathbb{E}[|X_t^y-X_t^x|^2]&\le |x-y|^2 \left(e^{-2\phi(t)} +\frac{\phi(t)}{ \underline{c}t} \sup_{t\ge0]\}} \mathbb{E} \left[ U^{r}\left(X_{t}^{(1-\lambda) x +\lambda y}\right) \right]\right)\\
&\lesssim_{uc} |x-y|^2 \left(e^{-2\phi(t)} + \frac{\phi(t)}{ \underline{c} t} (U^{ r}(x)+U^{ r} (y)+\Psi^{ r}) \right),
\end{align*}
where in the second line, we used Proposition \ref{art2:prop:moment} and the convexity of $U$.
\newline Let now $\nu$ be a coupling of $\pi$ and $\pi^\gamma$. We have
\begin{equation*}
{\cal W}_2^2(\pi,\pi^\gamma P_t)\le \int |x-y|^2(\nu(dx,dy)\left(e^{-2\phi(t)} + \frac{\phi(t)}{\underline{c} t} (\pi(U^{ r})+\pi^\gamma(U^{ r})+\Psi^{r} )\right).
\end{equation*}
Taking the infimum over the set of couplings $\nu$ of $\pi$ and $\pi^\gamma$ and using again Proposition \ref{art2:prop:moment}, this yields
\begin{equation*}
{\cal W}_2^2(\pi,\pi^\gamma P_t)\lesssim_{r} {\cal W}_2^2(\pi,\pi^\gamma)\left(e^{-2\phi(t)} + \frac{\phi(t)}{ \underline{c} t} \bar{\Psi}^{ r} \right)
\end{equation*}
\end{proof}
\begin{lem}\label{art2:lem:psin}
$(i)$ Let $U$ be a ${\cal C}^3$-convex function. Then,
\begin{equation*}
\mathbb{E}[|X_\gamma^x-\bar{X}_\gamma^y|^2]\le |x-y|^2 e^{\lambda \gamma} +\mathfrak{c}_{\gamma,\lambda}(x,y)\gamma^3.
\end{equation*}
with
\begin{equation*}
\mathfrak{c}_{\gamma,\lambda}(x,y)= \frac{e^{\lambda\gamma}}{6\lambda} \left(L^2 |\nabla U(y)|^2+{\sigma^4} \|\Delta (\nabla U)\|_{2,\infty}^2 + 2\lambda\sigma L \|D^2 U\|_{2,\infty} \left(\sqrt{\gamma}\right)\right)
\end{equation*}
where
\begin{equation*}
\|D^2 U\|_{2,\infty}=\sup_{x\in\mathbb{R}^d} \|D^2 U\|_F, \quad \|\Delta (\nabla U)\|_{2,\infty}=\sup_{x\in\mathbb{R}^d} \sum_{i=1}^d |\Delta \partial_i U|^2,
\end{equation*}
and $S(x,\gamma )=\sup_{u\in[0,\gamma]} \mathbb{E}[|b(X_u^x)|^2] ^{\frac{1}{2}}$.
\bigskip \newline $(ii)$ Let $(\mathbf{H_{r}^1})$ hold. Assume that $U$ is ${\cal C}^3$ with $\|\Delta (\nabla U)\|_{2,\infty}^2\lesssim_{r} \sigma^{-4} L^3 \bar{\Psi}$. Then, a constant $c_r$ exists such that for all $\gamma\in(0,\gamma^\star]$, for all $\lambda\in(0,1]$,
\begin{equation*}
{\cal W}_2^2(\pi^\gamma P_{n\gamma},\pi^\gamma)\le c_r \frac{\gamma^2}{\lambda^2} e^{\lambda (n+1)\gamma}L^3 \bar{\Psi}.
\end{equation*}
\end{lem}
\begin{rem}\label{art2:rem:laplacienU}The assumption on $\Delta (\nabla U)$ is calibrated to control its contribution by $L^3 \bar{\Psi}$. That simplifies the purpose and we could keep its specific contribution at the price of technicalities. However, this assumption is not really restrictive: denoting by $A(x)$ the $d\times d$-matrix defined by
$A_{i,j}(x)=D^3_{i,j,j} U(x)$. One easily checks that
\begin{equation*}
\|\Delta (\nabla U)\|_{2,\infty}^2=\sup_{x\in\mathbb{R}^d} \|A(x)\|_F^2\le d\sup_{x\in\mathbb{R}^d} \bar{\lambda}_{A(x)},
\end{equation*}
the second inequality coming from a classical inequality related to the Frobenius norm. Since
$L\ge \sup_{x\in\mathbb{R}^d} \bar{\lambda}_{D^2 U(x)}$, the assumption is for instance true if
\begin{equation*}
\sup_{x\in\mathbb{R}^d} \bar{\lambda}_{A(x)}\le \sigma^{-2} \sup_{x\in\mathbb{R}^d} (\bar{\lambda}_{D^2 U(x)})^4 \frac{\bar{\Psi}}{\sigma^2 d L}.
\end{equation*}
To conclude, note that $\frac{\bar{\Psi}}{\sigma^2 d L}$ is well controlled: for instance, under $(\mathbf{H_{r}^1})$ and $(\mathbf{H_{r}^2})$,
\begin{equation*}
\frac{\bar{\Psi}}{\sigma^2 d L}\lesssim_r \underline{c}^{-1}\left(\frac{\bar{c}}{L}\vee 1 \right).
\end{equation*}
\end{rem}
\begin{rem} The calibration of the parameter $\lambda$ is of first importance in the proof of Proposition \ref{art2:prop:HypH3} in order to avoid exponential dependence in the dimension.
\end{rem}
\begin{proof} The proof is an adaptation of Lemma 5.2 and Proposition 5.3 of \cite{art:egeapanloup2021multilevel} but with the viewpoint that $U$ is only convex. More precisely, we start with a one-step control of the error between the diffusion and its Euler scheme by setting
\begin{equation*}
F_{x,y}(t)=\frac{1}{2}\mathbb{E}[|X_t^x-\bar{X}_t^y|^2], \quad t\in[0,\gamma].
\end{equation*}
Then, setting $b=-\nabla U$, we have
\begin{equation} \label{art2:eq:basiceq}
\begin{split}
F'_{x,y}(t)&=\mathbb{E}\left[\langle X_t^x-\bar{X}_t^y, {{b}}(X_t^x)-{{b}}(y)\rangle\right]\\
&=\mathbb{E}\left[\langle X_t^x-\bar{X}_t^y, {{b}}(X_t^x)-{{b}}(\bar{X}_t^y)\rangle\right]+\mathbb{E}\left[\langle X_t^x-\bar{X}_t^y, {{b}}(\bar{X}_t^y)-{{b}}(y)\rangle\right]\nonumber \\
&\le \mathbb{E}\left[\langle X_t^x-\bar{X}_t^y, {{b}}(\bar{X}_t^y)-{{b}}(y)\rangle\right],
\end{split}
\end{equation}
where in the last line we used the convexity of $U$ which involves that
\begin{equation*}
\langle \nabla U(x)-\nabla U(y),x-y\rangle \ge 0.
\end{equation*}
We then write
\begin{align}
\mathbb{E}\left[\langle X_t^x-\bar{X}_t^y, {{b}}(\bar{X}_t^y)-{{b}}(y)\rangle\right]&= \mathbb{E}\left[\langle X_t^x-\bar{X}_t^y, {{b}}(\bar{X}_t^y)-b(y+\sigma B_t)\rangle\right]\label{art2:eq:gghbis}\\
&+\mathbb{E}\left[\langle X_t^x-\bar{X}_t^y,b(y+\sigma B_t)-b(y)\rangle\right].\label{art2:eq:ggh2bis}
\end{align}
Let $\lambda>0$. For the right-hand side of \eqref{art2:eq:gghbis}, we use the elementary inequality, ${|uv|\le \frac{\lambda}{2}|u|^2+\frac{1}{2\lambda}|v|^2}$ to obtain
\begin{equation}\label{art2:eq:part111bis}
\mathbb{E}\left[\langle X_t^x-\bar{X}_t^y, {{b}}(\bar{X}_t^y)-b(y+\sigma B_t)\rangle\right]\le \frac{\lambda}{2} F_{x,y}(t)+\frac{t^2}{2\lambda} L^2 |b(y)|^2.
\end{equation}
For \eqref{art2:eq:ggh2bis}, the It\^o formula applied to $b_i=-\partial_i U$ leads to
\begin{equation*}
b_i(y+\sigma B_t)-b_i(y)=\sigma^2 \int_0^t \Delta b_i (y+\sigma B_s) ds+\sigma\int_0^t \langle\nabla b_i(y+\sigma B_s),dB_s\rangle.
\end{equation*}
On the one hand, setting $\Delta b=(\Delta b_i)_{i=1}^d$,
\begin{equation*}
\mathbb{E}\left[\langle X_t^x-\bar{X}_t^y,\sigma^2 \int_0^t \Delta b (y+\sigma B_s) ds\rangle \right]\le \frac{\lambda}{2} F_{x,y}(t)+ \frac{\sigma^4}{{2\lambda}} t^2 \|\Delta b\|_{2,\infty}^2,
\end{equation*}
where
\begin{equation*}
\|\Delta b\|_{2,\infty}^2=\sup_{x\in\mathbb{R}^d} \sum_{i=1}^d |\Delta b_i(x)|^2.
\end{equation*}
On the other hand, using the fact that ${\cal M}$ defined by ${\cal M}_t=\int_0^t \langle \nabla b(y+\sigma B_s), dB_s\rangle$ is a martingale (we refer to \cite{art:egeapanloup2021multilevel} for the details), we get
\begin{equation*}
|\mathbb{E}\left[\langle X_t^x-\bar{X}_t^y,\sigma \int_0^t \langle\nabla b(y+\sigma B_s),dB_s\rangle\rangle\right]|\le \sigma L \|\nabla b\|_{2,\infty} \left(s^{\frac{3}{2}} S(x,\gamma) +\sigma s \sqrt{ d}\right),
\end{equation*}
where $S(x,\gamma )=\sup_{u\in[0,\gamma]} \mathbb{E}[|b(X_u^x)|^2] ^{\frac{1}{2}}$ and
\begin{equation*}
\|\nabla b\|_{2,\infty} =\sup_{x\in\mathbb{R}^d} \|\nabla b(x)\|_F.
\end{equation*}
Finally, from what precedes, we deduce that
\begin{equation*}
F'_{x,y}(t)\le \lambda F_{x,y}(t)+\frac{t^2}{{2\lambda}}\left(L^2 |b(y)|^2+{\sigma^4} \|\Delta b\|_{2,\infty}^2
+ 2\lambdaL \|\nabla b\|_{2,\infty} \left(\sqrt{t} S(x,\gamma) +\sigma \sqrt{ d}\right)\right).
\end{equation*}
A standard Gronwall argument then leads to
\begin{align*}
\mathbb{E}[|X_\gamma^x-\bar{X}_\gamma^y|^2]&\le |x-y|^2 e^{\lambda \gamma} \\
& +\int_0^{\gamma} s^2 e^{\lambda(\gamma -s)} \mathrm{d}s \left(L^2 |b(y)|^2+{\sigma^4} \|\Delta b\|_{2,\infty}^2
+ 2\lambda\sigma L \|\nabla b\|_{2,\infty} \left(\sqrt{t} S(x,\gamma) +\sigma \sqrt{ d}\right)\right)\\
&\le |x-y|^2 e^{\lambda \gamma} +\gamma^3 \mathfrak{c}_{\gamma,\lambda}(x,y)
\end{align*}
with
\begin{equation*}
\mathfrak{c}_{\gamma,\lambda}(x,y)= \frac{e^{\lambda\gamma}}{6\lambda} \left(L^2 |b(y)|^2+{\sigma^4} \|\Delta b\|_{2,\infty}^2
+ 2\lambda\sigma L \|\nabla b\|_{2,\infty} \left(\sqrt{\gamma} S(x,\gamma) +\sigma \sqrt{ d}\right)\right).
\end{equation*}
$(ii)$ Iterating the above inequality, we obtain for each $n\ge 1$,
\begin{align*}
\mathbb{E}[|X_{n\gamma}^x-\bar{X}_{n\gamma}^x|^2]\le \gamma^3 \sum_{k=0}^{n-1}\mathbb{E}[\mathfrak{c}_{\gamma,\lambda}(X_{k\gamma}^x,\bar{X}_{k\gamma}^x))] e^{\lambda (n-k)\gamma}
\end{align*}
Integrating the initial condition with respect to $\pi^\gamma$, we get
\begin{align*}
&{\cal W}_2(\pi^\gamma P_{n\gamma},\pi^\gamma)\le \gamma^2 e^{\lambda n\gamma} \sup_{k\ge 0} \int\mathbb{E}[\mathfrak{c}_{\gamma,\lambda}(X_{k\gamma}^x,\bar{X}_{k\gamma}^x))]\pi^\gamma(dx)\\
&\le \frac{\gamma^2}{\lambda^2} e^{\lambda (n+1)\gamma} \left(L^2 \pi^\gamma(|b|^2)+{\sigma^4} \|\Delta b\|_{2,\infty}^2
+ 2\lambda\sigma L \|\nabla b\|_{2,\infty} \left(\sqrt{\gamma} \sup_{n\ge 0} \int \mathbb{E}[S(X_{n\gamma}^x,\gamma)]\pi^\gamma(dx) +\sigma \sqrt{ d}\right)\right),
\end{align*}
where in the second line, we used the stationarity property of $\pi^\gamma$. Now, under $(\mathbf{H_{r}^1})$, $|b|^2=|\nabla U|^2\le 2 L U$ (with the same idea than one which leads to \eqref{art2:eq:minogradU}) so that by Proposition \ref{art2:prop:moment}$(iii)$, $\pi^\gamma(|b|^2)\lesssim_{r} L\bar{\Psi}$.
On the other hand, by the It\^o formula and the fact that $\Delta U\le d L$,
\begin{equation*}
\mathbb{E}[U(X_t^x)]\le U(x)+\sigma^2 \int_0^t\mathbb{E}[\Delta U(X_s^x)] ds \le U(x)+\sigma^2 d L,
\end{equation*}
so that
\begin{equation*}
S(x,\gamma)\le \sqrt{2L} \sup_{u\in[0,\gamma]} \mathbb{E}[U(X_u^x)] ^{\frac{1}{2}}\le \sqrt{2L U(x)}+\sigma \sqrt{2d}L.
\end{equation*}
Again, with the help of Proposition \ref{art2:prop:moment}\emph{(iii)},
\begin{equation*}
\sup_{n\ge 0} \int \mathbb{E}[S(X_{n\gamma}^x,\gamma)]\pi^\gamma(dx)\lesssim_{r} \sqrt{L}\bar{\Psi}^\frac{1}{2}+\sigma \sqrt{d}L.
\end{equation*}
Thus, using that $\gamma\le L^{-1}$,
\begin{equation*}
{\cal W}_2(\pi^\gamma P_{n\gamma},\pi^\gamma)\lesssim_{r}\frac{\gamma^2}{\lambda^2} e^{\lambda (n+1)\gamma} \left(L^3 \bar{\Psi}+{\sigma^4} \|\Delta b\|_{2,\infty}^2+\sigma \lambda \|\nabla b\|_{2,\infty}(\sqrt{\bar{\Psi}}+\sigma \sqrt{d L})\right).
\end{equation*}
Since for a symmetric $d\times d$-matrix $A$, $\|A\|_F\le \sqrt{d}\bar{\lambda}_A$, one deduces that
$\|\nabla b\|_{2,\infty}=\|D^2 U\|_{2,\infty}\le \sqrt{d} L$. It easily follows that $\sigma\lambda L \|\nabla b\|_{2,\infty}(\sqrt{\bar{\Psi}}+\sigma \sqrt{d L})\le L^3\bar{\Psi}$ (using that $L\ge1$ and $\bar{\Psi}\ge d$). The result follows.
\end{proof}
\section{Proof of Theorem \ref{art2:theo:maintheo2}}\label{art2:sec:maintheo2}
Following the bias-variance decomposition of the MSE:
\begin{equation*}
\| \mathcal{Y}(f) - \pi(f) \|_2^2 \le \left[\mathbb{E} [\mathcal{Y}(f)] - \pi(f)\right]^2 + \mathrm{Var}( \mathcal{Y}(f)),
\end{equation*}
we successively study the bias and the variance contributions and end the section by the proof of
Theorem \ref{art2:theo:maintheo2}.
\subsection{Step 1: Bias of the procedure}
{In the sequel, $\mathcal{Y}(J,\left(\gamma_j\right)_j,\tau,\left(T_j\right)_j,f)$ is usually written $\mathcal{Y}$ for the sake of simplicity.} We start with a telescopic-type decomposition:
\begin{equation}\label{art2:eq:telescop}
\begin{split}
\mathcal{Y}(f)-\pi(f) &= \frac{1}{T_0}\int_0^{T_0} f(\bar{X}_{\underline{s}_{\gamma_0}}^{\gamma_0,x_0}) - \pi^{\gamma_0} (f) \mathrm{d}s
\\ & + \sum_{j=1}^J \left(\frac{1}{T_j}\int_0^{T_j}f(\bar{X}_{\underline{s}_{\gamma_{j-1}}}^{\gamma_j,x_0})- \pi^{\gamma_j} (f) \mathrm{d}s-\frac{1}{T_j}\int_0^{T_j}f(\bar{X}_{\underline{s}_{\gamma_{j-1}}}^{\gamma_{j-1},x_0})-\pi^{\gamma_{j-1}} (f) \mathrm{d}s \right)
\\ & + \pi^{\gamma_J}(f)-\pi(f).
\end{split}
\end{equation}
Let us now study the bias generated by the first and second terms of the right-hand side of \eqref{art2:eq:telescop}.
\begin{lem}\label{art2:couchcontr}
Assume $(\mathbf{H_{r}^1})$ and $\gamma_0 \in (0,\gamma^\star]$. Let $x \in \mathbb{R}^d$ such that $U(x)\lesssim_r \bar{\Psi}$. Then, for any $r\in[0,1)$ and $\delta\in(0,\frac{1}{2}]$, there exists a constant $c_{r,\delta}$ (depending only on $r$ and $\delta$) such that for all $T\ge 1$, for all Lipschitz continuous function $f:\mathbb{R}^d\rightarrow\mathbb{R}$,
\begin{equation*}
\left| \frac{1}{T}\int_0^{T} \mathbb{E}_{x}[f(\bar{X}_{\underline{s}}^{\gamma,x_0})] - \pi^{\gamma} (f) \mathrm{d}s \right| ^2 \le c_{r,\delta} \frac{[f]_1^2 \mathfrak{C}^{(1)}_{\mathrm{bias}}}{T^2}
\quad\textnormal{with}\quad
\mathfrak{C}^{(1)}_{\mathrm{bias}}=\left(\underline{c}^{-\frac{1}{2}}\vee\underline{c}^{-3-\frac{4\delta}{1-\delta}}\right)\bar{\Psi}^{{1+3r}+\frac{4\delta r}{1-\delta}}.
\end{equation*}\end{lem}
\begin{proof}
Let us apply Proposition \ref{art2:prop:HypH1} with $\phi(n)=\frac{n\gamma}{(n\gamma+1)^{1-\delta}}$ and $\kappa=\frac{2(1+\delta)}{1-\delta}=2+\frac{4\delta}{1-\delta}$ for $\delta\in(0,1/2]$. Using that
$\phi(n)\ge (n\gamma)^\delta$, we have
\begin{equation*}
\begin{split}
\left| \frac{1}{T}\int_0^{T} \mathbb{E}_{x}[f(\bar{X}_{\underline{s}}^{\gamma,x_0})] - \pi^{\gamma} (f) \mathrm{d}s \right| &\lesssim_{r,\kappa} \frac{[f]_1}{T} \int_0^T {h_{\phi,\kappa}(\underline{s})} \mathrm{d}s\\
\\ & \lesssim_{r} \frac{[f]_1 }{T}\left( \underline{c}^{-\frac{1}{2}}\bar{\Psi}^{\frac{1+r}{2}} \int_0^T e^{- \underline{t}^{\delta}}dt+\underline{c}^{-\frac{3}{2}-\frac{2\delta}{1-\delta}}\bar{\Psi}^{\frac{1+3r}{2}+\frac{2\delta r}{1-\delta}}\int_0^T\left(1+\underline{t} \right)^{-1-\delta} \mathrm{d}t\right)\\
&\lesssim_r \frac{[f]_1 }{\delta T} \left(\Gamma\left(\frac{1}{\delta}\right)\underline{c}^{-\frac{1}{2}}\bar{\Psi}^{\frac{1+r}{2}}+\underline{c}^{-\frac{3}{2}-\frac{2\delta}{1-\delta}}\bar{\Psi}^{\frac{1+3r}{2}+\frac{2\delta r}{1-\delta}}\right),
\end{split}
\end{equation*}
where in the last line, we used standard arguments of comparisons between series and integrals.
The result follows.
\end{proof}
We are now ready to state a proposition about the control of the bias of the procedure.
\begin{prop}\label{art2:prop:biascontr}
Assume $(\mathbf{H_{r}^1})$ and $\gamma_0 \in (0,\gamma^\star]$. Let $x \in \mathbb{R}^d$ such that $U(x)\lesssim_r \bar{\Psi}$. Let $\delta \in (0,\frac{1}{2}]$ and let $f$ be a continuous Lipschitz function with $[f]_1= 1$. Let $J \in\mathbb{N}^*$. Then,
\newline (i)
\begin{equation*}
\left| \mathbb{E}_{x}[\mathcal{Y}(f)]-\pi(f) \right|^2 \lesssim_{r,\delta}
\mathfrak{C}^{(1)}_{\mathrm{bias}}
\sum_{j=0}^J \frac{1}{T_j^2}+ \mathfrak{C^{(2,1)}_{\mathrm{bias}}} \gamma_J \quad\textnormal{with}\quad\mathfrak{C^{(2,1)}_{\mathrm{bias}}}=\frac{L}{\underline{c}^{\frac{2}{1-\delta}}\wedge\underline{c}}\bar{\Psi}^{1+3r+\frac{2\delta r}{1-\delta}}.
\end{equation*}
\noindent (ii) If the assumptions of Proposition \ref{art2:prop:HypH3} are fulfilled,
\begin{equation*}
\left| \mathbb{E}_{x}[\mathcal{Y}(f)]-\pi(f) \right|^2 \lesssim_{r,\delta}
\mathfrak{C}^{(1)}_{\mathrm{bias}}
\sum_{j=0}^J \frac{1}{T_j^2}+ \mathfrak{C^{(2,2)}_{\mathrm{bias}}} \gamma_J^2 \quad\textnormal{with}\quad\mathfrak{C^{(2,2)}_{\mathrm{bias}}}=\underline{c}^{-\frac{2}{1-\delta}} L^{{3}}\bar{\Psi}^{1+\frac{2r}{1-\delta}}.
\end{equation*}
\end{prop}
\begin{proof}
\emph{(i)} Taking the expectation in \eqref{art2:eq:telescop}, we obtain:
\begin{equation*}
\begin{split}
\left| \mathbb{E}_x[\mathcal{Y}(f)-\pi(f)] \right| &\le \left|\frac{1}{T_0}\int_0^{T_0}\mathbb{E}_{x_0}\left[ f(\bar{X}_{\underline{s}_{\gamma_0}}^{\gamma_0,x_0})\right]- \pi^{\gamma_0} (f) \mathrm{d}s \right|
+ \sum_{j=1}^J \frac{1}{T_j}\left|\int_0^{T_j}\mathbb{E}\left[f(\bar{X}_{\underline{s}_{\gamma_{j-1}}}^{\gamma_j,x_0})\right]- \pi^{\gamma_j} (f) \mathrm{d}s\right|
\\&+ \sum_{j=1}^J\frac{1}{T_j}\left|\int_0^{T_j}\mathbb{E}\left[ f(\bar{X}_{\underline{s}_{\gamma_{j-1}}}^{\gamma_{j-1},x_0})\right]-\pi^{\gamma_{j-1}} (f) \mathrm{d}s\right|
+ \left| \pi^{\gamma_J}(f)-\pi(f) \right|.
\end{split}
\end{equation*}
For the three first terms, we apply Lemma \ref{art2:couchcontr} and for the last one, Lemma \ref{art2:lem:firstresult}. The result follows.
\bigskip \newline \emph{(ii)} It is the same proof using Proposition \ref{art2:prop:HypH3} to control the last term (instead of
Lemma \ref{art2:lem:firstresult}).
\end{proof}
\subsection{Step 2 : Control of the variance}
Now we have to control the variance of our estimator. Owing to the independency between the layers,
\begin{equation}\label{art2:eq:decompofvar}
\mathrm{Var}(\mathcal{Y}(J,\left(\gamma_j\right)_j,\left(T_j\right)_j,f)) = \mathrm{Var}\left(\frac{1}{T_0}\int_{0}^{T_0} f(\bar{X}_{\underline{s}_{\gamma_0}}^{\gamma_0,x}) \mathrm{d}s\right) + \sum_{j=1}^J \mathrm{Var}\left( \frac{1}{T_j} \int_0^{T_j}G_{s}^{\gamma_{j}}\mathrm{d}s\right),
\end{equation}
where for some given $\gamma>0$ and $s>0$,
\begin{equation*}
\begin{split}
G_{s}^\gamma &= f\left(\bar{X}_{\underline{s}_\gamma}^{\frac{\gamma}{2},x}\right) - f\left(\bar{X}_{\underline{s}_\gamma}^{\gamma,x}\right).
\end{split}
\end{equation*}
Before going further, let us recall that in order that the multilevel method be efficient, the correcting layers must have a small variance. In the long-time setting, this involves to be able to control the $L^2$-distance between couplings of Euler schemes with steps $\gamma$ and $\gamma/2$. By Proposition \ref{art2:prop:HypH2}, this is still possible under $(\mathbf{H_{r}^1})$, and such a property allows to obtain the following result:
\begin{lem}\label{art2:lem:propvarlvl}
Assume $(\mathbf{H_{r}^1})$ and $\gamma_0 \in (0,\gamma^\star]$. Let $x \in \mathbb{R}^d$ such that $U(x)\lesssim_r \bar{\Psi}$. Let $\delta \in (0,\frac{1}{2}]$ and $\kappa>\frac{2}{1-\delta}$. Let $f$ be a continuous Lipschitz function.Then, for all $T>0$,
\begin{equation*}
\mathrm{Var} \left( \frac{1}{T} \int_0^{T} G_s^\gamma \mathrm{d}s \right) \lesssim_{r,\delta,\kappa} [f]_1^2 \frac{\mathfrak{C_{\rm var}^{(1)}}(\kappa,\delta) \gamma^{1-\frac{1}{\kappa(1-\delta)}}}{T},
\end{equation*}
where
\begin{equation*}
\mathfrak{C_{\rm var}^{(1)}}(\kappa,\delta)= L^{1-\frac{1}{\kappa(1-\delta)}} (\underline{c}^{-\frac{3-\delta}{\kappa(1-\delta)^2}}\vee \underline{c}^{-\frac{3+\kappa-(\kappa+1)\delta}{\kappa(1-\delta)^2}})\bar{\Psi}^{1+\left(4-\frac{2}{\kappa}\right)r+\varepsilon(\delta,\kappa) r}\;\textnormal{with}\;
\varepsilon(\delta,\kappa)=\left(3-\frac{2}{\kappa}-\frac{2}{\kappa(1-\delta)}\right) \frac{\delta}{1-\delta}.
\end{equation*}
\end{lem}
\begin{rem} In the uniformly convex case, the variance is controlled by
$\frac{\gamma \log(1/\gamma)}{T}$ whereas, here, we are only able to obtain $\frac{\gamma^{1-\frac{1}{\kappa(1-\delta)}}}{T}$. This difference is due to the lack of exponential convergence to equilibrium under our assumptions. Note that if we leave $\kappa$ go to $\infty$, we are moving ever closer to the uniformly convex bound. However, the constant depends on $\kappa$ and explodes when $\kappa\rightarrow+\infty$. The interesting point is that the exponent of $\bar{\Psi}$ remains bounded when $\kappa\rightarrow+\infty$, which means that the dependence in the dimension is slightly impacted by the choice of $\kappa$.
\end{rem}
\begin{proof}
A standard computation shows that
\begin{equation*}
\mathrm{Var} \left( \frac{1}{T} \int_0^{T} G_s^\gamma \mathrm{d}s \right) \le \frac{2}{T^2} \int_0^{T} \int_u^{T} \mathrm{Cov} \left(G_s^\gamma,G_u^\gamma\right) \mathrm{d}s \mathrm{d}u.
\end{equation*}
First, at the price of replacing $f$ by $f/[f]_1$,we can assume in the sequel that $[f]_1\le 1$. Then,
\begin{equation*}
|G_{s}^\gamma|\le |\bar{X}_{\underline{s}_\gamma}^{\frac{\gamma}{2},x}-X_{\underline{s}_\gamma}|+|\bar{X}_{\underline{s}_\gamma}^{\gamma,x}-X_{\underline{s}_\gamma}|.
\end{equation*}
By Proposition \ref{art2:prop:HypH2} and the fact that $U(x)\lesssim_{uc} \bar{\Psi}$, we deduce that for every $\delta\in(0,1]$,
\begin{equation}\label{art2:controlgupas}
\mathbb{E}[ |G_u^\gamma|^2]\lesssim_{r,\delta} \frac{L\gamma}{\underline{c}^{\frac{2}{1-\delta}}}\bar{\Psi}^{1+3r+\frac{2\delta r}{1-\delta}}.
\end{equation}
This yields a first bound for $\mathrm{Cov} \left(G_s^\gamma,G_u^\gamma\right)$:
\begin{equation*}
\mathrm{Cov} \left(G_s^\gamma,G_u^\gamma\right)\le \mathbb{E}[ |G_s^\gamma|^2]^{\frac{1}{2}}\mathbb{E}[ |G_u^\gamma|^2]^{\frac{1}{2}}\lesssim_{r,\delta} \frac{L\gamma}{\underline{c}^{\frac{2}{1-\delta}}}\bar{\Psi}^{1+3r+\frac{2\delta r}{1-\delta}}.
\end{equation*}
Hence, for any $t_0>0$,
\begin{equation}\label{art2:eq:covgsgtbound1}
\int_u^{u+t_0} \mathrm{Cov} \left(G_s^\gamma,G_u^\gamma\right) \mathrm{d}s\lesssim_{r,\delta} \frac{L\gamma t_0}{\underline{c}^{\frac{2}{1-\delta}}}\bar{\Psi}^{1+3r+\frac{2\delta r}{1-\delta}}.
\end{equation}
We now want to take advantage of the convergence to equilibrium to get a second bound when $s-u\ge t_0$: since $G_u^{\gamma}$ is ${\cal F}_{\underline{u}_{\gamma}}$-measurable, we have for any $s\ge \underline{u}_{\gamma}$,
\begin{equation*}
\mathbb{E}\left[G_s^{\gamma} G_u^{\gamma}\right]=\mathbb{E}[\mathbb{E}[G_s^\gamma|\mathcal{F}_{\underline{u}_{\gamma}}]G_u^\gamma].
\end{equation*}
Setting $ \mathfrak{F}(\gamma,t,x)=\mathbb{E}[f(\bar{X}_t^{\gamma,x})]-\pi^\gamma(f)$, we deduce from the Markov property that
\begin{equation*}
\mathbb{E}[G_s^\gamma|\mathcal{F}_{\underline{u}_{\gamma}}]= \mathfrak{F}\left(\frac{\gamma}{2}, \underline{s}_\gamma-\underline{u}_\gamma, \bar{X}_{\underline{u}_{\gamma}}^{\frac{\gamma}{2},x}\right)-\mathfrak{F}\left({\gamma}, \underline{s}_\gamma-\underline{u}_\gamma, \bar{X}_{\underline{u}_{\gamma}}^{\gamma,x}\right) +(\pi^\gamma-\pi^\frac{\gamma}{2})(f),
\end{equation*}
and hence,
\begin{align*}
\mathbb{E}\left[G_s^{\gamma} G_u^{\gamma}\right]= \mathbb{E} \left[\left(\mathfrak{F}\left(\frac{\gamma}{2}, \underline{s}_\gamma-\underline{u}_\gamma, \bar{X}_{\underline{u}_{\gamma}}^{\frac{\gamma}{2},x}\right)-\mathfrak{F}\left({\gamma}, \underline{s}_\gamma-\underline{u}_\gamma, \bar{X}_{\underline{u}_{\gamma}}^{\gamma,x}\right)\right)G_u^\gamma \right]+(\pi^\gamma-\pi^\frac{\gamma}{2})(f)\mathbb{E}[G_u^\gamma].
\end{align*}
On the other hand,
\begin{align*}
\mathbb{E}[G_s^{\gamma}] \mathbb{E}[G_u^{\gamma}]= \mathbb{E} \left[\left(\mathfrak{F}\left(\frac{\gamma}{2}, \underline{s}_\gamma, \bar{X}_{\underline{u}_{\gamma}}^{\frac{\gamma}{2},x}\right)-\mathfrak{F}\left({\gamma}, \underline{s}_\gamma, \bar{X}_{\underline{u}_{\gamma}}^{\gamma,x}\right)\right)\right]\mathbb{E}\left[G_u^\gamma \right]+(\pi^\gamma-\pi^\frac{\gamma}{2})(f)\mathbb{E}[G_u^\gamma].
\end{align*}
As a consequence,
\begin{align}
\mathrm{Cov} \left(G_s^\gamma,G_u^\gamma\right)&= \mathbb{E} \left[\left(\mathfrak{F}\left(\frac{\gamma}{2}, \underline{s}_\gamma-\underline{u}_\gamma, \bar{X}_{\underline{u}_{\gamma}}^{\frac{\gamma}{2},x}\right)-\mathfrak{F}\left({\gamma}, \underline{s}_\gamma-\underline{u}_\gamma, \bar{X}_{\underline{u}_{\gamma}}^{\gamma,x}\right)\right)G_u^\gamma \right]\label{art2:eq:gupasun}\\
&-\mathbb{E} \left[\left(\mathfrak{F}\left(\frac{\gamma}{2}, \underline{s}_\gamma, x\right)-\mathfrak{F}\left({\gamma}, \underline{s}_\gamma, x\right)\right)\right]\mathbb{E}\left[G_u^\gamma \right].\label{art2:eq:gupasdeux}
\end{align}
Let us study the two right-hand members successively.
For \eqref{art2:eq:gupasun}, the Cauchy-Schwarz inequality and \eqref{art2:controlgupas} yield:
\begin{align*}
\mathbb{E} \Big[\Big(\mathfrak{F} &\left(\frac{\gamma}{2}, \underline{s}_\gamma -\underline{u}_\gamma , \bar{X}_{\underline{u}_{\gamma}}^{\frac{\gamma}{2},x}\right)-\mathfrak{F}\left({\gamma}, \underline{s}_\gamma-\underline{u}_\gamma, \bar{X}_{\underline{u}_{\gamma}}^{\gamma,x}\right)\Big)G_u^\gamma \Big]\\
&\lesssim_{r,\delta} \left(\left\|\mathfrak{F}\left(\frac{\gamma}{2}, \underline{s}_\gamma-\underline{u}_\gamma, \bar{X}_{\underline{u}_{\gamma}}^{\frac{\gamma}{2},x}\right)\right\|_2+\left\|
\mathfrak{F}\left({\gamma}, \underline{s}_\gamma-\underline{u}_\gamma, \bar{X}_{\underline{u}_{\gamma}}^{\gamma,x}\right)\right\|_2\right)\sqrt{\frac{L\gamma}{\underline{c}^{\frac{2}{1-\delta}}}\bar{\Psi}^{1+3r+\frac{2\delta r}{1-\delta}}}.
\end{align*}
By Proposition \ref{art2:prop:HypH1} or more precisely by \eqref{art2:eq:vitessexxx} combined with Proposition \ref{art2:prop:moment}\emph{(iii)}\footnote{\label{art2:footnoteone} In fact, Proposition \ref{art2:prop:HypH1} is written under the assumption $U(x)\lesssim \bar{\Psi}$ but here, we need to integrate with respect to the initial condition. To extend to this setting, the idea is to start from \eqref{art2:eq:vitessexxx} and to use the bounds of Proposition \ref{art2:prop:moment}\emph{(iii)}. This allows us to retrieve controls which are similar to Proposition \ref{art2:prop:HypH1}.} applied with $\phi(n)=(n\gamma)^{\delta}$,
\begin{equation*}
\begin{split}
\left\|\mathfrak{F}\left(\frac{\gamma}{2}, \underline{s}_\gamma-\underline{u}_\gamma, \bar{X}_{\underline{u}_{\gamma}}^{\frac{\gamma}{2},x}\right)\right\|_2+\left\|
\mathfrak{F}\left({\gamma}, \underline{s}_\gamma-\underline{u}_\gamma, \bar{X}_{\underline{u}_{\gamma}}^{\gamma,x}\right)\right\|_2 \lesssim_{r,\delta,\kappa} \underline{c}^{-\frac{1}{2}} \bar{\Psi}^{\frac{1+r}{2}} e^{-(\underline{s}_\gamma-\underline{u}_\gamma)^\delta} &
\\ +\underline{c}^{-\frac{\kappa+1}{2}}\bar{\Psi}^{\frac{1+r(1+\kappa)}{2}} (\underline{s}_\gamma-\underline{u}_\gamma)^{-\frac{\kappa(1-\delta)}{2}}.
\end{split}
\end{equation*}
Now, let us remark that if $s-u\ge t_0$, $t_0\ge 2$ and $\gamma\in(0,\gamma^\star]$, then $\underline{s}_\gamma-\underline{u}_\gamma\ge 1$ (since $\gamma^\star\le 1$). Noting that for any $t\ge 1$, $e^{-t^\delta}\lesssim_{\delta} t^{-\frac{\kappa(1-\delta)}{2}}$ and that for any $\kappa>\frac{2}{1-\delta}$,
\begin{equation*}
\int_{(u+t_0)\wedge T}^T(\underline{s}_\gamma-\underline{u}_\gamma)^{-\frac{\kappa(1-\delta)}{2}} \mathrm{d}s\le \int_{u+t_0}^{+\infty} (s-u-\gamma)^{-\frac{\kappa(1-\delta)}{2}} \mathrm{d}s\le \frac{(t_0-\gamma)^{1-\frac{\kappa(1-\delta)}{2}}}{\frac{\kappa(1-\delta)}{2}-1}\lesssim_{\kappa,\delta} \left(\frac{t_0}{2}\right)^{1-\frac{\kappa(1-\delta)}{2}}
\end{equation*}
we deduce that for any $t_0\ge 2$ and for any $\gamma\in(0,\gamma^\star]$
\begin{align*}
\int_{(u+t_0)\wedge T}^T \mathbb{E} \Big[\Big(\mathfrak{F}\left(\frac{\gamma}{2}, \underline{s}_\gamma-\underline{u}_\gamma, \bar{X}_{\underline{u}_{\gamma}}^{\frac{\gamma}{2},x}\right)-\mathfrak{F}\left({\gamma}, \underline{s}_\gamma-\underline{u}_\gamma, \bar{X}_{\underline{u}_{\gamma}}^{\gamma,x}\right)\Big)G_u^\gamma \Big] \mathrm{d}s\lesssim_{r,\delta,\kappa} \frac{(\gamma L)^{\frac{1}{2}} \bar{\Psi}^{1+2r+\frac{\delta r}{1-\delta}+\frac{\kappa r}{2}}}{\underline{c}^{\frac{3}{2}+\frac{\delta}{1-\delta}}\wedge \underline{c}^{\frac{3+\kappa}{2}+\frac{\delta}{1-\delta}}} \left(\frac{t_0}{2}\right)^{1-\frac{\kappa(1-\delta)}{2}}.
\end{align*}
For \eqref{art2:eq:gupasdeux}, using that $\underline{s}_\gamma\ge \underline{s}_\gamma-\underline{u}_\gamma$, we remark that we can obtain the same bound so that:
\begin{equation*}
\int_{(u+t_0)\wedge T}^T \mathrm{Cov} \left(G_s^\gamma,G_u^\gamma\right)\mathrm{d}s \lesssim_{r,\delta,\kappa} \frac{(\gamma L)^{\frac{1}{2}} }{\underline{c}^{\frac{3}{2}+\frac{\delta}{1-\delta}}\wedge \underline{c}^{\frac{3+\kappa}{2}+\frac{\delta}{1-\delta}}}\bar{\Psi}^{1+2r+\frac{\delta r}{1-\delta}+\frac{\kappa r}{2}}\left(\frac{t_0}{2}\right)^{1-\frac{\kappa(1-\delta)}{2}}.
\end{equation*}
In view of the above bound and of the one obtained in \eqref{art2:eq:covgsgtbound1}, we now optimize the choice of $t_0$ by taking $t_0$ solution to:
\begin{equation*}
\frac{(\gamma L)^{\frac{1}{2}} }{\underline{c}^{\frac{3}{2}+\frac{\delta}{1-\delta}}\wedge \underline{c}^{\frac{3+\kappa}{2}+\frac{\delta}{1-\delta}}} \bar{\Psi}^{1+2r+\frac{\delta r}{1-\delta}+\frac{\kappa r}{2}}\left(\frac{t_0}{2}\right)^{1-\frac{\kappa(1-\delta)}{2}}=
\frac{L\gamma t_0}{\underline{c}^{\frac{2}{1-\delta}}}\bar{\Psi}^{1+3r+\frac{2\delta r}{1-\delta}}.
\end{equation*}
\emph{i.e.},
\begin{equation*}
t_0=2^{1-\frac{2}{\kappa(1-\delta)}}(\underline{c}^{\frac{1}{2}+\frac{\delta}{1-\delta}}\vee \underline{c}^{\frac{1-\kappa}{2}+\frac{\delta}{1-\delta}})^{\frac{2}{\kappa(1-\delta)}} (L\gamma)^{-\frac{1}{\kappa(1-\delta)}}\bar{\Psi}^{(1-\frac{2}{\kappa}) \frac{r}{1-\delta}-\frac{2\delta}{\kappa(1-\delta)^2}r}.
\end{equation*}
Plugging this value of $t_0$ into \eqref{art2:eq:covgsgtbound1}, this leads to: for any $\kappa>\frac{2}{1-\delta}$,
\begin{equation*}
\int_u^T \mathrm{Cov} \left(G_s^\gamma,G_u^\gamma\right) \mathrm{d}s\lesssim_{r,\delta,\kappa} (\gamma L)^{1-\frac{1}{\kappa(1-\delta)}} (\underline{c}^{-\frac{3}{2}-\frac{\delta}{1-\delta}}\vee \underline{c}^{-\frac{3+\kappa}{2}-\frac{\delta}{1-\delta}})^{\frac{2}{\kappa(1-\delta)}}
\bar{\Psi}^{1+\left(4-\frac{2}{\kappa}\right)r+\varepsilon(\delta,\kappa) r},
\end{equation*}
where $\varepsilon(\delta,\kappa)=\left(3-\frac{2}{\kappa}-\frac{2}{\kappa(1-\delta)}\right) \frac{\delta}{1-\delta}$. The result follows.
\end{proof}
In the next proposition, we are now able to work on the variance of the multilevel procedure.
\begin{prop}\label{art2:varcontr}
Assume $(\mathbf{H_{r}^1})$ and $\gamma_0 \in (0,\gamma^\star]$. Let $x \in \mathbb{R}^d$ such that $U(x)\lesssim_r \bar{\Psi}$. Let $\delta \in (0,\frac{1}{2}]$ and $\kappa>\frac{2}{1-\delta}$. Let $f$ be a continuous Lipschitz function.Then,
\begin{equation*}
\mathrm{Var}(\mathcal{Y}(J,\left(\gamma_j\right)_j,\left(T_j\right)_j,f)) \lesssim_{r,\delta,\kappa}
[f]_1^2 \left(\mathfrak{C_{\rm var}^{(1)}}(\kappa,\delta)\sum_{j=1}^J \frac{\gamma^{1-\frac{1}{\kappa(1-\delta)}}_j}{T_j}+
\frac{\mathfrak{C_{\rm var}^{(2)}}}{T_0}\right),
\end{equation*}
where $\mathfrak{C_{\rm var}^{(1)}}(\kappa,\delta)$ is defined in Lemma \ref{art2:lem:propvarlvl} and $\mathfrak{C_{\rm var}^{(2)}}= \left(\underline{c}^{-\frac{3}{2}}\vee \underline{c}^{-\frac{5}{2}-\frac{2\delta}{1-\delta}}\right) \bar{\Psi}^{1+2r+\frac{2\delta r}{1-\delta}}$.
\end{prop}
\begin{proof}
We assume (without loss of generality) that $[f]_1=1$ and $f(x^\star)=0$. In view of the decomposition \eqref{art2:eq:decompofvar}, we apply Lemma \ref{art2:lem:propvarlvl} for each level $j\in\{1,\ldots,J\}$ with $T=T_j$ and $\gamma=\gamma_{j-1}$. We obtain for any $\delta\in(0,1/2]$ and $\kappa>2/(1-\delta)$,
\begin{equation*}
\mathrm{Var}(\mathcal{Y}(J,\left(\gamma_j\right)_j,\left(T_j\right)_j,f)) \le \mathrm{Var}\left(\frac{1}{T_0}\int_{0}^{T_0} f(\bar{X}_{\underline{s}_{\gamma_0}}^{\gamma_0,x_0}) \mathrm{d}s\right) +\mathfrak{C_{\rm var}^{(1)}}(\kappa,\delta)\sum_{j=1}^J \frac{\gamma^{1-\frac{1}{\kappa(1-\delta)}}_j}{T_j}.
\end{equation*}
It remains to control the first term, \emph{i.e.} the variance related to the first level. We use similar arguments as in the proof of Lemma
\ref{art2:lem:propvarlvl} (see in particular \eqref{art2:eq:gupasun} and what follows). First, one can check that for every $0\le u\le s\le T$,
\begin{align*}
\mathrm{Cov} \left(f(\bar{X}_{\underline{u}_{\gamma_0}}^{\gamma_0,x_0}),f(\bar{X}_{\underline{s}_{\gamma_0}}^{\gamma_0,x_0})\right)&= \mathbb{E} \left[\mathfrak{F}\left(\gamma_0, \underline{s}_{\gamma_0}-\underline{u}_{\gamma_0}, \bar{X}_{\underline{u}_{\gamma_0}}^{\gamma_0,x}\right)f(\bar{X}_{\underline{u}_{\gamma_0}}^{\gamma_0,x_0}) \right]-\mathbb{E} \left[\mathfrak{F}\left(\gamma_0, \underline{s}_{\gamma_0}, x\right)\right]\mathbb{E}\left[f(\bar{X}_{\underline{u}_{\gamma_0}}^{\gamma_0,x_0}) \right]\\
&\le \left(\left\|\mathfrak{F}\left(\gamma_0, \underline{s}_{\gamma_0}-\underline{u}_{\gamma_0}, \bar{X}_{\underline{u}_{\gamma_0}}^{\gamma_0,x}\right)\right\|_2+\left\|\mathfrak{F}\left(\gamma_0, \underline{s}_{\gamma_0}, x\right)\right\|_2\right) \left\|f\left(\bar{X}_{\underline{u}_{\gamma_0}}^{\gamma_0,x_0}\right)\right\|_2,
\end{align*}
by Cauchy-Schwarz inequality. Then, by Proposition \ref{art2:prop:HypH1} (and footnote \ref{art2:footnoteone}) applied with $\phi(n)=n\gamma$ and $\kappa=\frac{2(1+\delta)}{1-\delta}$, one deduces that (we leave the details to the reader),
\begin{align*}
\int_u^T \mathrm{Cov} \left(f(\bar{X}_{\underline{u}_{\gamma_0}}^{\gamma_0,x_0}),f(\bar{X}_{\underline{s}_{\gamma_0}}^{\gamma_0,x_0})\right) ds&\lesssim_{r,\delta}
\left(\underline{c}^{-\frac{1}{2}}\vee \underline{c}^{-\frac{3}{2}-\frac{2\delta}{1-\delta}} \right)\bar{\Psi}^{\frac{1+3r}{2}+\frac{2\delta r}{1-\delta}} \times \sup_{u\ge0} \left\|f\left(\bar{X}_{\underline{u}_{\gamma_0}}^{\gamma_0,x_0}\right)\right\|_2 \\
&\lesssim_{r,\delta} \left(\underline{c}^{-\frac{3}{2}}\vee \underline{c}^{-\frac{5}{2}-\frac{2\delta}{1-\delta}}\right) \bar{\Psi}^{1+2r+\frac{2\delta r}{1-\delta}},
\end{align*}
where in the second line, we used Proposition \ref{art2:prop:moment}\emph{(iii)} and the fact that (by Lemma \ref{art2:lem:momenttoU})
\begin{equation*}
|f(x)|^2=|f(x)-f(x^\star)|^2\le |x-x^\star|^2\le \frac{U^{1+r}(x)}{\underline{c}}.
\end{equation*}
\end{proof}
\subsection{Step 3 : Proof of Theorem \ref{art2:theo:maintheo2}}
Back to the bias-variance decomposition, we deduce from Proposition \ref{art2:prop:biascontr} and Proposition \ref{art2:varcontr} that, up to a constant depending on $\kappa$, $\delta$ and $r$, the MSE is lower than $\varepsilon^2$ if the following conditions are satisfied (with $\delta\in(0,1/2]$ and $\kappa>2/(1-\delta)$):
\begin{equation}\label{art2:eq:troisconditions}
\begin{cases}
\displaystyle{\emph{(i.a)}: \quad \mathfrak{C}^{(1)}_{\mathrm{bias}}
\sum_{j=0}^J \frac{1}{T_j^2}\le \varepsilon^2,}&\displaystyle{\emph{(i.b)}: \quad \mathfrak{C^{(2,1)}_{\mathrm{bias}}} \gamma_J\le \varepsilon^2 \quad \textnormal{or} \quad \displaystyle{(i.b)'}:\quad \mathfrak{C^{(2,2)}_{\mathrm{bias}}} \gamma_J^2\le \varepsilon^2, }\\
\displaystyle{\emph{(ii.a)}:\quad \frac{ \mathfrak{C_{\rm var}^{(2)}}}{T_0}\le \varepsilon^2,}&
\displaystyle{ \emph{(ii.b)}: \quad \mathfrak{C_{\rm var}^{(1)}}(\kappa,\delta)\sum_{j=1}^J \frac{\gamma^{1-\frac{1}{\kappa(1-\delta)}}_j}{T_j}\le \varepsilon^2.}
\end{cases}
\end{equation}
Note that \emph{(i.b)} corresponds to Proposition \ref{art2:prop:biascontr}\emph{(i)} whereas \emph{(i.b)}' corresponds to Proposition \ref{art2:prop:biascontr}\emph{(ii)}. Let us assume that
\begin{equation*}
\gamma_j=\gamma_0 2^{-j}\quad \textnormal{and}\quad T_j=T_0 2^{-(1-\rho) j},
\end{equation*}
where $\rho\ge 1/2$, $\gamma_0$ and $T_0$ are positive numbers which will calibrated further. With these choices, the above conditions read (up to universal constants):
\begin{equation}\label{art2:eq:troisconditionsbis}
\begin{cases}
\displaystyle{\emph{(i.a)}: \quad T_0 \ge 2^{(1-\rho) J} \sqrt{\mathfrak{C^{(1)}_{\mathrm{bias}}}}\varepsilon^{-1}} &\displaystyle{\emph{(i.b)}: 2^J\ge \mathfrak{C^{(2,1)}_{\mathrm{bias}}} \gamma_0 \varepsilon^{-2} \quad \textnormal{or} \quad \displaystyle{(i.b)'}:\quad 2^J\ge \sqrt{\mathfrak{C^{(2,2)}_{\mathrm{bias}}}} \gamma_0 \varepsilon^{-1},}\\
\displaystyle{\emph{(ii.a)}:\quad {T_0}\ge \mathfrak{C_{\rm var}^{(2)}}\varepsilon^{-2},}& \displaystyle{ \emph{(ii.b)}: \quad T_0\ge \mathfrak{C_{\rm var}^{(1)}}(\kappa,\delta){\gamma_0^{1-\frac{1}{\kappa(1-\delta)}}\sum_{j=1}^J 2^{(-\rho+\frac{1}{\kappa(1-\delta)})j} \varepsilon^{-2}.}}
\end{cases}
\end{equation}
\noindent \textit{Proof of Theorem \ref{art2:theo:maintheo2}} \emph{(i)} In this case, we have to calibrate the parameters according to \emph{(i.a)}, \emph{(i.b)}, \emph{(ii.a)} and \emph{(ii.b)} are satisfied. First, for \emph{(i.b)}, we need $ 2^{ J}\ge \mathfrak{C^{(2,1)}_{\mathrm{bias}}} \gamma_0 \varepsilon^{-2}$ so we can set:
\begin{equation*}
J=\left\lceil \log_2\left(\mathfrak{C^{(2,1)}_{\mathrm{bias}}} \gamma_0 \varepsilon^{-2}\right)\right\rceil.
\end{equation*}
Then, set $\rho=1/2$. With the above value of $J$ and the condition $\kappa>2/(1-\delta)$,
\begin{equation*}
2^{\frac{J}{2}}\le\sqrt{2\gamma_0\mathfrak{C^{(2,1)}_{\mathrm{bias}}}}\varepsilon^{-1}\quad\textnormal{and}\quad \sum_{j=1}^J 2^{(-\frac{1}{2}+\frac{1}{\kappa(1-\delta)})j}\lesssim_{\kappa,\delta} 1.
\end{equation*}
Hence, \emph{(i.a)}, \emph{(ii.a)}, \emph{(ii.b)} are satisfied (up to a constant depending on $\kappa$, $\delta$ and $r$ only) if
\begin{equation*}
T_0\ge \mathfrak{C} \varepsilon^{-2} \quad\textnormal{with}\quad\mathfrak{C}=\max\left(\sqrt{\gamma_0\mathfrak{C^{(1)}_{\mathrm{bias}}}\mathfrak{C^{(2,1)}_{\mathrm{bias}}}},\mathfrak{C_{\rm var}^{(2)}},\mathfrak{C_{\rm var}^{(1)}}(\kappa,\delta)\gamma_0^{1-\frac{1}{\kappa(1-\delta)}}\right).
\end{equation*}
The complexity of the procedure is then:
\begin{equation*}
{\cal C}({\cal Y})=2\left(\frac{T_0}{\gamma_0}+\sum_{j=1}^J \frac{T_j}{\gamma_j}\right)\lesssim \frac{T_0}{\gamma_0} 2^{\frac{J}{2}}\lesssim_{r,\delta,\kappa} \frac{\mathfrak{C} \sqrt{\mathfrak{C^{(2,1)}_{\mathrm{bias}}}}\varepsilon^{-3}}{\sqrt{\gamma_0}}.
\end{equation*}
To deduce the result, it is now enough to remark that with $\kappa=\frac{2(1+\delta)}{1-\delta}$, $\gamma_0\le 1/(4L)$,
\begin{equation*}
\mathfrak{C}\le(\underline{c}^{-\frac{3}{4}}\vee \underline{c}^{-\frac{5}{2}-\frac{2\delta}{1-\delta}})\bar{\Psi}^{1+3r+\frac{3\delta r}{1-\delta}}\quad \textnormal{so that}\quad \mathfrak{C}\sqrt{\mathfrak{C^{(2,1)}_{\mathrm{bias}}}}\le \sqrt{L}(\underline{c}^{-\frac{5}{4}}\wedge \underline{c}^{-{\frac{7}{2}}-\frac{3\delta}{1-\delta}})\bar{\Psi}^{\frac{3}{2}+\frac{9}{2}r+\frac{4\delta r}{1-\delta}},
\end{equation*}
as soon as $\delta <1/3$. The main result then follows from a change of variable replacing $\delta$ by $\tilde{\delta}= a\delta$ with $a$ small enough.
\bigskip \newline \emph{(ii)} In this case, we have to calibrate the parameters according to \emph{(i.a)}, \emph{(i.b)}', \emph{(ii.a)} and \emph{(ii.b)}. First, for \emph{(i.b)}', we need $ 2^{ J}\ge \sqrt{\mathfrak{C^{(2,2)}_{\mathrm{bias}}}} \gamma_0 \varepsilon^{-1}$ so we can set:
\begin{equation*}
J=\left \lceil \log_2\left(\sqrt{\mathfrak{C^{(2,2)}_{\mathrm{bias}}}} \gamma_0 \varepsilon^{-1}\right)\right \rceil.
\end{equation*}
Then, if $\rho>\frac{1}{\kappa(1-\delta)}$, \emph{(i.a)}, \emph{(ii.a)}, \emph{(ii.b)} are satisfied (up to a constant depending on $\kappa$, $\delta$ and $r$ only) if
\begin{equation*}
T_0\ge \widetilde{\mathfrak{C}} \varepsilon^{-2} \quad\textnormal{with}\quad \widetilde{\mathfrak{C}}=\max\left(\sqrt{\mathfrak{C^{(1)}_{\mathrm{bias}}}}\left(\gamma_0^2\mathfrak{C^{(2,2)}_{\mathrm{bias}}}\right)^{\frac{1-\rho}{2}},\mathfrak{C_{\rm var}^{(2)}},\mathfrak{C_{\rm var}^{(1)}}(\kappa,\delta)\gamma_0^{1-\frac{1}{\kappa(1-\delta)}}\right),
\end{equation*}
and the complexity of the procedure satisfies:
\begin{equation*}
{\cal C}({\cal Y})\lesssim \frac{T_0}{\gamma_0} 2^{\rho{J}} \lesssim_{r,\delta,\kappa,\rho} \frac{\widetilde{\mathfrak{C}} (\mathfrak{C^{(2,2)}_{\mathrm{bias}}})^{\frac{\rho}{2}}\varepsilon^{-2-\rho}}{\gamma_0^{\rho}}.
\end{equation*}
Set $\kappa=\frac{1+\delta}{\rho(1-\delta)}=\frac{1}{\rho}+\frac{2\delta}{\rho(1-\delta)}$. For $\delta$ small enough and $\gamma_0\le 1/L$,
\begin{equation*}
\gamma_0^{1-\frac{1}{\kappa(1-\delta)}}\mathfrak{C_{\rm var}^{(1)}}(\kappa,\delta)\le (\underline{c}^{-3\rho(1-\frac{\delta}{1-\delta})}\vee \underline{c}^{-(1+3\rho)-5\delta})\bar{\Psi}^{1+ (4-2\rho) r+\frac{4\delta r}{1-\delta}}.
\end{equation*}
Using that $\gamma_0\le 1/(4L)$,
\begin{equation*}
\sqrt{\mathfrak{C^{(1)}_{\mathrm{bias}}}}\left(\gamma_0^2\mathfrak{C^{(2,2)}_{\mathrm{bias}}}\right)^{\frac{1-\rho}{2}}\le L^{\frac{\rho}{2}}(\underline{c}^{-\frac{1}{4}-\frac{1-\rho}{1-\delta}}\vee \underline{c}^{-\frac{3}{2}-\frac{1-\rho}{1-\delta}-\frac{2\delta}{1-\delta}}) \bar{\Psi}^{1-\frac{\rho}{2}+\frac{5-2\rho}{2} r+(3-\rho)\frac{\delta r}{1-\delta}}.
\end{equation*}
Using that $\rho\le 1/2$, this yields (at the price of replacing $\delta$ by $a\delta$ for $a$ small enough)
\begin{equation*}
\widetilde{\mathfrak{C}}\le L^{\frac{\rho}{2}} \left(\underline{c}^{-(\frac{5}{4}-\rho)\wedge(3\rho)+\delta}\vee \underline{c}^{-\frac{5}{2}-\delta}\right)\bar{\Psi}^{1+(4-2\rho+\delta) r}.
\end{equation*}
The result follows.
\section*{Acknowledgements} The present author is deeply grateful to F.Panloup for his numerous suggestions and discussions that improved the quality of this paper. The author also thanks the SIRIC ILIAD Nantes-Angers program supported by the French National Cancer Institute (INCA-DGOS-Inserm 12558 grant) for funding M. Eg\'ea's Ph.D. thesis.
\bibliographystyle{alpha}
|
1,116,691,499,295 | arxiv | \section{Introduction}
Let $\mathbf{x}$ be an infinite word, i.e., a sequence of letters belonging to a finite alphabet. Imagine a window of size~$n$ moving along $\mathbf{x}$. Such a reading frame permits to detect all factors of length~$n$ occurring in~$\mathbf{x}$. For instance, the factor complexity function of $\mathbf{x}$ mapping $n\in\mathbb{N}$ to the number of distinct factors of length~$n$ is extensively studied in combinatorics on words. Now let $n,\ell$ be such that $n\ge \ell$. Assume that within the sliding window, we only focus on its first and last $\ell$ symbols. Otherwise stated, for a factor $uyv$ of length $n$, we only consider its borders $u$ and $v$ of length~$\ell$.
\begin{figure}[h!t]
\begin{center}
\begin{tikzpicture}
\node [fit={(4.5,1) (7,1.5)}, inner sep=0pt, draw=black, thick] (x) {};
\node [fit={(3,1) (4.5,1.5)}, inner sep=0pt, draw=black, thick, fill=gray!30] (u) {};
\node [fit={(7,1) (8.5,1.5)}, inner sep=0pt, draw=black, thick, fill=gray!30] (v) {};
\draw [thick, decoration={brace, mirror, raise=0.3cm}, decorate] (u.west) -- node[below=.4cm] {$u$} (x) {};
\draw [thick, decoration={brace, mirror, raise=0.3cm}, decorate] (x.east) -- node[below=.4cm] {$v$} (v.east) ;
\draw [thick, decoration={brace, mirror, raise=0.3cm}, decorate] (x.west) -- node[below=.4cm] {$y$} (v.west) ;
\draw[<->] (3,1.7) -- (8.5,1.7) ;
\node [fit={(5,1.8)}] (t) {$n$} ;
\draw[<->] (7,1.8) -- (8.5,1.8) ;
\node [fit={(7.75,1.82)}] (t) {$\ell$} ;
\draw[<->] (3,1.8) -- (4.5,1.8) ;
\node [fit={(3.75,1.82)}] (t) {$\ell$} ;
\foreach \r in {1,...,40}{
\node[circle,fill=black,inner sep=0pt,minimum size=3pt] (a) at (.125+.25*\r,1.25) {};
}
\end{tikzpicture}
\end{center}
\caption{A sliding window where we focus on two regions of a fixed length.}
\label{fig:frame}
\end{figure}
For any given window length~$n$, we would like to determine what are the pairs of length-$\ell$ borders that may occur. This leads to the following definition, where, to simplify notation, we consider borders of factors of length $n + \ell$ rather than $n$.
\begin{definition}
Let $\ell\in\mathbb{N}_{>0}$ and $\infw{x} \in A^{\mathbb{N}}$. For $n\ge \ell$, we define the
\emph{$n$th boundary set} by
\begin{equation*}
\partial_{\mathbf{x},\ell}[n] := \{(u,v)\in A^\ell\times A^\ell\mid uyv \text{ is a factor of }\mathbf{x} \text{ for some } y\in A^{n-\ell}\}
\end{equation*}
and call the sequence $\partial_{\mathbf{x},\ell} := (\partial_{\mathbf{x},\ell}[n])_{n\ge \ell}$ the {\em $\ell$-boundary sequence} of $\mathbf{x}$. When $\ell=1$, we write $\partial_{\mathbf{x},1}=\partial_\mathbf{x}$ and simply talk about the {\em boundary sequence}.
\end{definition}
The $\ell$-boundary sequence takes values in $2^{A^\ell\times A^\ell}$, and hence itself can be seen as an infinite word over a finite alphabet. We give an introductory example.
\begin{example}\label{exa:intro}
Consider the Fibonacci word $\mathbf{f}=0100101001\cdots$; the fixed point of the morphism $0\mapsto 01, 1\mapsto 0$.
We have $\partial_{\mathbf{f}} = a\, b\, b\, a\, b\, b\, b\, b\, a\, b\, b\, a\, b\, b\, b\, b\, a\, b\, b\, b\, b\, a\, b\, b\, a\, b\, b\, b\, b\cdots$,
where $a:=\{(0,0), (0,1), (1,0)\}$ and $b:=\{0,1\} \times \{0,1\}$.
For instance, $\partial_{\mathbf{f}}[1]=a$ because the length-$2$ factors of $\mathbf{f}$ are $00, 01, 10$, while $\partial_{\mathbf{f}}[2]=b$ because its length-$3$ factors are of the form $0\_0, 0 \_ 1, 1 \_ 0, 1 \_ 1$ (they are in fact $010, 001, 100, 101$).
The $2$-boundary sequence starts with
\[
\partial_{\mathbf{f},2} = a_0\ a_1\ a_2 \ a_3\ a_4\ a_5\ a_1\ a_2\ a_3\ a_1\ a_2\ a_3\ a_4\ a_5\ a_1\ a_2\ a_3\ a_4\ a_5\ a_1\ a_2\ a_3\ a_1\ a_2\ a_3\cdots
\]
where
\begin{align*}
a_0 &:= \{(00,10),(01,00),(01,01),(10,01),(10,10)\}, \\
a_1 &:= \{(00,00),(00,01),(01,01),(01,10),(10,00),(10,10)\}, \\
a_2 &:= \{(00,01),(00,10),(01,00),(01,10),(10,00),(10,01)\}, \\
a_3 &:= \{(00,00),(00,10),(01,00),(01,01),(10,01),(10,10)\}, \\
a_4 &:= \{(00,01),(01,01),(01,10),(10,00),(10,01),(10,10)\}, \\
a_5 &:= \{(00,10),(01,00),(01,01),(01,10),(10,01),(10,10)\}.
\end{align*}
The first element $\partial_{\mathbf{f},2}[2]=a_0$ is peculiar; it
corresponds exactly to the five length-$4$ factors occurring in
$\mathbf{f}$. Our \cref{prop:first-letter} shows that $a_0$ appears only once in $\partial_{\infw{f}}$. Then, e.g.,
$\partial_{\mathbf{f},2}[3] = a_1$ because the length-$5$ factors of
$\mathbf{f}$ are of the form $00\_00$, $00\_01$, $01\_01$, $01\_10$,
$10\_00$ and $10\_10$ (the factors are $00100$, $00101$, $01001$, $10100$, $10010$, and $01010$). For
length-$6$ factors, note that two are of the form $10u01$ for some
$u\in\{0,1\} \times \{0,1\}$. For $i\geq 1$, the letter $a_i$ appears infinitely often
in $\partial_{\mathbf{f},2}$: see \cref{the:sturmian}.
\end{example}
\subsection{Motivation and related work}
In combinatorics on words, borders and boundary sets are related to important concepts.
For instance, a word $v$ is {\em bordered} if there
exist $u,x,y$ such that $v=ux=yu$ and $0<|u|<|v|$. One reason to
study bordered words is Duval's theorem: for a sufficiently long word $v$, the
maximum length of unbordered factors of $v$ is equal to the
period of $v$ \cite{Duval}.
In formal language theory, a language $L$ is
{\em locally $\ell$-testable} (LT) if the membership of a word
$w$ in $L$ only depends on the prefix, suffix and factors of
length~$\ell$ of $w$. In \cite{Place}, the authors consider the
so-called {\em separating problem} of languages by LT languages;
they utilize {\em $\ell$-profiles} of a word, which can again be
related to boundary sets. Let us also mention that, in bioinformatics and computational biology, one of the aims is to reconstruct sequences from subsequences \cite{Margaritis}. To determine DNA segments by bottom-up analysis, {\em paired-end} sequencing is used. In this case both ends of DNA fragments of known length are sequenced. See, for instance, \cite{Fullwood}. This is quite similar to the theoretical concept we discuss here.
The notion of a ($1$-)boundary sequence was introduced by Chen and Wen in
\cite{ChenWen} and was further studied in \cite{Guo}, where it is
shown that the boundary sequence of a \emph{$k$-automatic}
word (in the sense of Allouche and Shallit \cite{AS}: see \cref{def:Saut-kaut}) is $k$-automatic.
It is well-known that a $k$-automatic word $\infw{x}$ is
\emph{morphic}, i.e., there exist morphisms $f\colon A \to A^*$
and $g\colon A \to B$ and a letter $a \in A$ such that $\infw{x} = g(f^{\omega}(a))$,
where $f^{\omega}(a) = \lim_{n\to \infty} f^n(a)$. However, $k$-automatic words (with $k$ ranging over the integers) do not capture all morphic words: a well-known
characterization of $k$-automatic words is given by Cobham~\cite{Cobham1972uniform} (the generating morphism $f$ maps each letter to a length-$k$ word).
This paper is driven by the natural
question whether, in general, the $\ell$-boundary sequence of a morphic word is morphic.
In case such generating morphisms can be constructed, we have at our disposal a simple algorithm providing the set of length-$\ell$ borders in factors of all lengths.
We briefly present several situations in which the notion of
boundary sets is explicitly or implicitly used. In
\cite[Thm.~4]{CurrieHOR2019squarefree}, the authors study the boundary sequence to exhibit a squarefree word for which each subsequence arising from an arithmetic progression contains a square.
Boundary sets play an important role in the study of so-called \emph{$k$-abelian} and \emph{$k$-binomial complexities} of infinite words (for definitions, see \cite{Rigo2017}).
For instance, computing the $2$-binomial complexity of generalized Thue--Morse words \cite{lu20212binomial} requires inspecting pairs of prefixes and suffixes of factors, which is again related to the boundary sequence when these prefixes and suffixes have equal length. The $k$-binomial complexities of images of binary words under powers of the
Thue--Morse morphism are studied in \cite{RigoSW2022binomial}; there some
general properties of boundary sequences of binary words are required.
Moreover, if $\partial_{\mathbf{x}}$ is automatic, then the
abelian complexity of the image of $\mathbf{x}$ under a
so-called Parikh-constant morphism is automatic \cite{ChenWen}.
Guo, L\"u, and Wen combine this result with theirs in \cite{Guo} to
establish a large family of infinite words with automatic abelian
complexity.
Let $k\ge 1$. We let $\equiv_k$ denote the $k$-abelian
equivalence, i.e., $u \equiv_k v$ if the words $u$ and $v$ share the same set of factors of length at most $k$ with the same multiplicities \cite{KSZ}. For $u$ and $v$ equal length factors
of a Sturmian word $\mathbf{s}$, we have $u\equiv_k v$ if and only if they share a common prefix and a
common suffix of length $\min\{|u|, k-1\}$ and $u\equiv_1 v$ \cite[Prop.~2.8]{KSZ}.
Under the assumption that the largest power of a letter appearing in
$\infw{s}$ is less than $2k-2$,
the requirement $u \equiv_1 v$ in the previous result may be
omitted \cite[Thm.~3.6]{PeltoWhiteland} (compare to \cref{prop:first-letter}). Thus the quotient of the set of factors of
length $n$ occurring in a Sturmian word by the
relation~$\equiv_k$ is completely determined by
$\partial_{\mathbf{s},k-1}[n-k+1]$ for large enough~$k$
(depending on $\mathbf{s}$). Other families of words with
$k$-abelian equivalence determined by the boundary sets are given in \cite[Prop.~4.2]{PeltoWhiteland}.
\subsection{Our contributions}
Up to our knowledge, we are the first to propose a systematic study of the
$\ell$-boundary sequences of infinite words. It is therefore
natural to consider the notion on well-known classes of words.
In this paper, we consider morphic words and Sturmian words.
Any morphic word is $S$-automatic for some abstract numeration system $S$ \cite{Maes}.
With \cref{the:kernel}, we prove that for a large class of numeration systems $U$, if $\mathbf{x}$ is a $U$-automatic word, then the boundary sequence $\partial_\mathbf{x}$ is again $U$-automatic. Our approach generalizes the arguments provided by \cite{Guo}. Considering exotic numeration systems allows a better understanding of underlying mechanisms, which do not arise in the ordinary integer base systems. In particular, we deal with addition within the numeration system; in integer base systems, the carry propagation is easy to handle (by a two-state finite automaton). Our arguments apply to so-called \emph{addable} numeration systems (see \cref{def:addable}).
As an alternative, we observe that the B\"uchi--Bruy\`ere theorem
\cite{BruyereHansel} can be extended to addable positional numeration
systems $U$ (\cref{thm:logic}). The $U$-automaticity of the
$\ell$-boundary sequence then follows from the fact that it
is definable by a first-order formula of the structure $\langle \mathbb{N},+ \rangle$
extended with a unary function relating an integer with the least
element of $U$ properly appearing in its representation.
This alternative proof however hides the important details that
might help identifying the technical limits of the result: not all morphic words allow an addable system to work with.
However, the finiteness of a suitable kernel captures all morphic words (see \cref{thm:Ukernel}).
Also, one practical difficulty
when one wants to use automatic provers (such as Walnut
\cite{Walnut}) is to be
able to provide the relevant automaton for addition.
To identify the contours of our result, we also discuss the case where $\mathbf{x}$ is $U$-automatic and $\partial_\mathbf{x}$ is not $U$-automatic. To construct such examples, we have to consider non-addable numeration systems in \cref{ss:ce}.
We then turn to the other class of words under study. Letting
$\mathbf{s}$ be a Sturmian word with slope $\alpha$, with \cref{the:sturmian} we show that the $\ell$-boundary sequence of $\mathbf{s}$ is obtained through a sliding block code
from the \emph{characteristic Sturmian word of slope $\alpha$} (see \cref{sec:Sturmian} for a definition) up to the first letter.
This result
holds even for non-morphic Sturmian words, so for an arbitrary irrational $\alpha$. Where the techniques used in the first part of the paper have an automata-theoretic flavor, the second part relies on the geometric characterization of Sturmian words as codings of rotations.
We provide another description of the $\ell$-boundary sequence of a
Sturmian word as the morphic image of some characteristic Sturmian
word in \cref{prop:another-charact}. We remark that it is unclear to
us whether some of the results in \cref{sec:Sturmian} can be proved
automatically using the very recent tool developed in \cite{HieronymiMOSSS2022decidability}.
\section{Preliminaries}
Throughout this paper we let $A$ denote a finite alphabet. Then $A^n$ denotes the set of length-$n$ words. For an infinite word $\mathbf{x}$, we let $\mathbf{x}[n]$ denote its $n$th letter, for all $n\ge 0$. For general references on numeration systems, see \cite{Fraenkel} and \cite[Chap.~1--3]{cant2010}. We assume that the reader has some knowledge in automata theory. For a reference see \cite{ShallitAutomata} or \cite[Chap.~1]{RigoBook}.
\subsection{Numeration systems and automatic words}
Let $U=(U_n)_{n\ge 0}$ be an increasing sequence of integers such that $U_0=1$. Any integer $n$ can be decomposed (not necessarily uniquely) as $n=\sum_{i=0}^t c_i\, U_i$ with non-negative integer coefficients $c_i$. The finite word $c_t\cdots c_0\in\mathbb{N}^*$ is a {\em $U$-representation} of $n$. If this representation is computed greedily \cite{Fraenkel,RigoBook}, then for all $j\le t$ we have $\sum_{i=0}^j c_i\, U_i<U_{j+1}$ and $\rep_U(n)=c_t\cdots c_0$ is said to be the {\em greedy} (or {\em normal}) $U$-representation of $n$. By convention, the greedy representation of $0$ is the empty word~$\varepsilon$ and the greedy representation of $n>0$ starts with a non-zero digit. An extra condition on the boundedness of $\sup_{i\ge 0} (U_{i+1}/U_i)$ implies that the digit-set for greedy representations is finite. For any $c_t\cdots c_0\in\mathbb{N}^*$, we let $\val_U(c_t\cdots c_0)$ denote the integer $\sum_{i=0}^t c_i\, U_i$. A sequence $U$ satisfying all the above conditions is said to define a {\em positional numeration system}.
For the following, we refer to the terminology in \cite{peltomaki2021} (addable systems are called regular in \cite{Shallit2022logical}).
\begin{definition}\label{def:addable}
A positional numeration system $U$ with digit-set $A$ is {\em addable} if the following \emph{graph of addition}, denoted by $\mathcal{L}_+$, is regular:
\[
\left\{ \smatrix{ u \\ v \\ w} \in (0^*\rep_U(\mathbb{N}))^3 \cap (A \times A \times A)^*
\mid \val_U(u) + \val_U(v) = \val_U(w) \right\} \setminus \smatrix{ 0 \\ 0 \\ 0} (A \times A \times A)^*.
\]
\end{definition}
Notice that words in $\rep_U(\mathbb{N})$ do not start with $0$; however, when dealing with tuples of such words, shorter $U$-representations are padded with leading zeroes to get words of equal length (so they can be processed by an automaton over
tuples of letters).
For general references about automatic words and abstract numeration systems, see \cite{AS} and \cite{Maes} or \cite[Chap.~3]{cant2010}. An \emph{abstract numeration system}
is a triple $S = (L,A,<)$ with $L$ an infinite regular language over the totally ordered alphabet $A$ (with $<$). Genealogically (i.e., radix or length-lexicographic) ordering $L$ gives a one-to-one correspondence $\rep_S$ between $\mathbb{N}$ and $L$;
the $S$-representation of $n$ is the $(n+1)$st word of $L$, and the inverse map is
denoted by $\val_S$.
A \emph{deterministic finite automaton with output} (DFAO) $\mathcal{A}$ is a
DFA (with state set~$Q$) equipped with a mapping
$\tau\colon Q \to A$ (with $A$ an alphabet). The output $\mathcal{A}(w)$ of
$\mathcal{A}$ on a word $w$ is $\tau(q)$, where $q$
is the state reached by reading $w$ from the initial state.
\begin{definition}\label{def:Saut-kaut}
An infinite word $\mathbf{x}$ is {\em $S$-automatic} if there exists a DFAO $\mathcal{A}$ such that $\mathbf{x}[n]=\mathcal{A}(\rep_S(n))$.
In particular, for $k\ge 2$ an integer, if $\mathcal{A}$ is fed with the genealogically ordered language $L= \{\varepsilon\} \cup \{1,\ldots,k-1\}\{0,\ldots,k-1\}^*$, then $\mathbf{x}$ is said to be {\em $k$-automatic}.
\end{definition}
We introduce abstract numeration systems due to the following theorem:
\begin{theorem}[\cite{Maes}]
\label{thm:morphicIsAutomatic}
A word $\infw{x}$ is morphic if and only if it is $S$-automatic
for some abstract numeration system $S$.
\end{theorem}
Fix $s \in A^*$. For a word $\mathbf{x}$, define the subsequence $\mathbf{x} \circ s$ by
$(\mathbf{x}\circ s)[n]:=
\mathbf{x}[\val_S(p_{s,n}\, s)]$,
where $p_{s,n}$ is the $n$th word in the genealogically ordered language $Ls^{-1}=\{u\in A^*\mid us\in L\}$. The \emph{$S$-kernel} of the word $\mathbf{x}$ is defined as the set of words $\{\mathbf{x}\circ s \mid s\in A^*\}$. The following theorem is critical to our arguments. Details are given in \cite[Prop.~3.4.12--16]{cant2010}.
\begin{theorem}[\cite{Maes}]\label{thm:Ukernel}
A word $\mathbf{x}$ is $S$-automatic if and only if its $S$-kernel is finite.
\end{theorem}
\begin{example}
Consider the Fibonacci numeration system based on the sequence $(F_n)_{n\ge 0}$ with $F_0 = 1$, $F_1 = 2$,
and $F_{n+1} = F_n + F_{n-1}$ for $n\geq 1$.
The first few terms of the associated subsequences $\mu_s:\mathbb{N}\to\mathbb{N}$, such that $(\mathbf{x}\circ s)[n]=\mathbf{x}[\mu_s(n)]$, are given in \cref{tab:fib-ker}.
One simply computes the numerical value of all the Fibonacci representations with the suffix $s$.
\end{example}
\begin{table}[h!t]
\centering
\begin{tabular}{r|lr|l}
$s$ & $(\mu_s(n))_{n\ge 0}$ & $s$ & $(\mu_s(n))_{n\ge 0}$ \\
\hline
$\varepsilon$ & 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, \ldots & 01& 4, 6, 9, 12, 14, 17, 19, 22, 25, \ldots \\
0& 2, 3, 5, 7, 8, 10, 11, 13, 15, \ldots & 00& 3, 5, 8, 11, 13, 16, 18, 21, 24, \ldots\\
1 & 1, 4, 6, 9, 12, 14, 17, 19, 22, 25, \ldots & 10& 2, 7, 10, 15, 20, 23, 28, 31, 36, 41, \ldots \\
\end{tabular}
\caption{The first few terms of some subsequences $\mu_s$
for the Fibonacci numeration system.}
\label{tab:fib-ker}
\end{table}
Notice that some kernel elements $\mathbf{x}\circ s$ may be finite; more precisely, this occurs exactly when the language $L s^{-1}$ is finite.
Our reasoning will not be affected by such particular cases, and we let the reader adapt it to such situations.
\underline{\smash{General assumptions}}.\label{assumptions}
From now on we let $U$ be a positional numeration system with digit-set $A$ such
that $\rep_U(\mathbb{N})$ is regular, and thus it is an abstract
numeration system $(\rep_U(\mathbb{N}),A,<)$ for the natural ordering of the digits; this is because for all $m,n\in\mathbb{N}$, $m < n$ if and only if
$\rep_U(m) \gen \rep_U(n)$ for the genealogical order.
In \cref{sec:addable}, we further require $U$ to be addable. All these assumptions are shared by many systems. For instance, classical integer base systems or the Fibonacci numeration system have all the assumed properties including being addable. For the latter system, the minimal automaton (reading most significant digits first) of $\mathcal{L}_+$ has $17$ states (its transition table is given in \cite{MousaviShallit}). The one reading least significant digits first has $22$ states. The largest known family of positional systems with all these properties (addable with $\rep_U(\mathbb{N})$ being regular) is the one of those based on a linear recurrence sequence whose characteristic polynomial is the minimal polynomial of a Pisot number \cite{BruyereHansel,RigoBook}.
\subsection{Link with first-order logic}\label{ss:logic}
The result stated below is nothing else but the B\"uchi--Bruy\`ere theorem \cite[Thm.~16]{BruyereHansel}, originally given in the context of Pisot numeration systems.
Its proof can be straightforwardly adapted to the case where $U$ is addable, as the key ingredients are that
$\rep_U(\mathbb{N})$ and the graph of addition $\mathcal{L}_+$ are regular languages.
For a positional numeration system $U$, we define the map $V_U$ by $V_U(n)=U_j$ whenever $n\in\mathbb{N}$ and $\rep_U(n)=c_t\cdots c_j 0^j$ and $c_j\neq 0$.
\begin{theorem}\label{thm:logic}
Let $U$ be an addable positional numeration system. A word $\mathbf{x}$ over $B$ is $U$-automatic if and only if for each symbol $b\in B$, the set $\{n\ge 0\mid \mathbf{x}[n]=b\}$ can be defined by a first-order formula $\varphi_b(n)$ of the structure $\langle\mathbb{N},+,V_U\rangle$.
\end{theorem}
As already mentioned in \cite{Guo}, the boundary sequence of a $k$-automatic
word~$\mathbf{x}$ may be defined by means of a first-order formula and
therefore automaticity readily follows. This extends to addable systems: let $\mathbf{x}\in B^\mathbb{N}$ be $U$-automatic
for an addable system $U$. The above theorem implies that, for all $b\in B$,
we have a formula $\varphi_b(n)$ which is true if and only if
$\mathbf{x}[n]=b$. We have
$(u_1\cdots u_\ell,v_1\cdots v_\ell)\in\partial_{\mathbf{x},\ell}[m]$ if and
only if
\[
(\exists i)\bigwedge_{j=1}^{\ell}\varphi_{u_j}(i+j-1)\wedge \bigwedge_{j=1}^{\ell}\varphi_{v_j}(i+m + j-1).
\]
For each subset $R$ of $A^{\ell} \times A^{\ell}$ there is thus a formula $\psi_R(m)$ which is true if and only if $\partial_{\mathbf{x},\ell}[m] = R$.
We may now apply \cref{thm:logic} to conclude that
$\partial_{\mathbf{x},\ell}$ is $U$-automatic. We remark that a
similar proof of our \cref{the:kernel} as sketched above is
given in a forthcoming book of Shallit \cite{Shallit2022logical}. Also,
this proof works for $U$ not necessarily positional using
\cite[Lem.~37 and Thm.~55]{CharlierCS2022regular}. In particular, they show the following: Let $\infw{x}$ be an $S$-automatic sequence, where $S$ is an addable abstract numeration system. Any first-order formula involving the predicates defined by the letters of $\infw{x}$ leads to an automaton accepting the $S$-representations of integers for which the formula holds.
\section{On the boundary sequences of automatic words}\label{sec:automatic}
In this section we provide the first of our main contributions, an alternative proof (not relying on \cref{thm:logic}) to the fact that
a $U$-automatic word has a $U$-automatic boundary sequence whenever $U$ is addable and satisfies the assumptions laid down in \cref{assumptions}. We then
show that this result does not necessarily hold for non-addable $U$.
\subsection{Addable systems: automatic boundary sequences}\label{sec:addable}
For the sake of presentation, we only consider the case of the $1$-boundary sequence.
Our proof provides a precise description of a set containing the
$U$-kernel of $\partial_\mathbf{x}$ in terms of three equivalence
relations based on the kernel of~$\mathbf{x}$, the graph of addition,
and the numeration language; see~\eqref{eq:new-equiv-rel}. This set is finite, and so \cref{thm:morphicIsAutomatic} gives the claim.
In particular, one is the Myhill--Nerode congruence associated with the graph of addition since we have to consider the elements $\mathbf{x}[i]$ and $\mathbf{x}[i+m]$ for some $m>0$.
For $\ell>1$, the only technical difference is that we have to consider longer factors $\mathbf{x}[i] \cdots \mathbf{x}[i+\ell-1]$ and $\mathbf{x}[i+m] \cdots \mathbf{x}[i+m+\ell-1]$.
\begin{theorem}\label{the:kernel}
Let $U$ be an addable numeration system with digit-set $A$ and $\mathbf{x}$ be a $U$-automatic word. The boundary sequence $\partial_\mathbf{x}$ is $U$-automatic.
\end{theorem}
\begin{proof}
Thanks to \cref{thm:Ukernel}, the $U$-kernel of $\mathbf{x}$ is finite, say of cardinality $m$.
Moreover, since $L=\rep_U(\mathbb{N})$ and $\mathcal{L}_+$ are regular, the following two sets of languages are finite by the Myhill--Nerode theorem \cite[Sec.~3.9]{ShallitAutomata}, say of cardinality
$k$ and $\ell$, respectively:
\[
\{Ls^{-1}\mid s\in A^*\} \quad \text{and} \quad
\Big\{\mathcal{L}_+ \smatrix{s \\ t \\ r}^{-1} \mathrel{\Big|} \smatrix{s \\ t \\ r} \in (A \times A \times A)^* \Big\}.
\]
Let $\partial_\mathbf{x}$ be the boundary sequence of $\mathbf{x}$. An element of the $U$-kernel of $\partial_\mathbf{x}$ is given by
$
\partial_\mathbf{x}\circ s =
\partial_\mathbf{x}[\val_U(p_{s,0}s)]\
\partial_\mathbf{x}[\val_U(p_{s,1}s)]\
\partial_\mathbf{x}[\val_U(p_{s,2}s)]\cdots
$
where $p_{s,n}$ is the $n$th word in the language $Ls^{-1}$, $n\ge 0$.
Let us inspect the $n$th term of such an element of the kernel: it is precisely the set
\begin{equation}
\label{eq:kernelElement}
\partial_\mathbf{x}[\val_U(p_{s,n}s)]=\{(\mathbf{x}[i],\, \mathbf{x}[i+\val_U(p_{s,n}s)]) \mid i\ge 0\}
\end{equation}
of pairs of letters.
Let $t$, $r$ be length-$|s|$ suffixes of words in $L$ for which
$\mathcal{L}_+(s,t,r)^{-1}$ is non-empty. There exist words $w$, $x$, $y$ such
that $ws$, $xt$, $yr \in 0^*L$ and $\val_U(ws)+\val_U(xt)=\val_U(yr)$. We
let $\mathcal{P}(s)$ denote the set of such pairs $(t,r)\in (A\times A)^{|s|}$.
Now partition \eqref{eq:kernelElement} depending on the suffixes of length $|s|$ of
$\rep_U(i)$ and $\rep_U(i+\val_U(p_{s,n}s))$: we may write
\[
\partial_\mathbf{x}[\val_U(p_{s,n}s)] = \bigcup_{(t,r)\in \mathcal{P}(s)}\left\{
(\mathbf{x}[\val_U(xt)],\, \mathbf{x}[\val_U(yr)])
\mathrel{\Big|} \smatrix{w \\ x \\ y} \in \mathcal{L}_+\smatrix{s \\ t \\ r}^{-1} \wedge w\in 0^*p_{s,n}
\right\}.
\]
Roughly speaking, we look at all pairs of positions such that the first one is represented by a word ending with $t$, the second position is a shift of the first one by $\val_U(p_{s,n}s)$ and is represented by a word ending with $r$.
For convenience, we set
$
L(s,t,r,n):=\mathcal{L}_+ \smatrix{s \\ t \\ r}^{-1} \cap \left( 0^*p_{s,n} \times A^* \times A^* \right)$ for all $n\ge 0$.
Note that if $\mathcal{L}_+(s,t,r)^{-1}= \mathcal{L}_+(s',t',r')^{-1}$ and $Ls^{-1}=Ls'^{-1}$ then, for all~$n$, $L(s,t,r,n)=L(s',t',r',n)$. Indeed, the second condition means that $p_{s,n}=p_{s',n}$ for all $n$.
\medskip
\noindent
$\bullet$ Ordering $L(s,t,r,n)$.
\noindent
For each $w$, $x$ of the same length, there is at most one $y$ not starting with $0$ such that $(w,x,y)$ belongs to $\mathcal{L}_+(s,t,r)^{-1}$. Similarly if $y$ does not start with $0$, for each $w\in A^{|y|}$ (resp., $x\in A^{|y|}$) there is at most one $x$ (resp., $w$) such that $(w,x,y)$ belongs to $\mathcal{L}_+(s,t,r)^{-1}$.
Now let $(w,x,y)$ and $(w',x',y')$ in $L(s,t,r,n)$. We will always assume (this is not a restriction) that triplets do not start with $(0,0,0)$ --- otherwise, different triplets may have the same numerical value. Note that $\val_U(x)<\val_U(x')$ if and only if $\val_U(y)<\val_U(y')$.
Indeed, $w,w'$ both belong to $0^*p_{s,n}$ thus $\val_U(ws)=\val_U(p_{s,n} s)=\val_U(w's)$. We have
\[
\val_U(yr)-\val_U(xt)=\val_U(ws)=\val_U(w's)=\val_U(y'r)-\val_U(x't),
\]
so, since $U$ is positional,
$\val_U(y0^{|r|})-\val_U(y'0^{|r|})=\val_U(x0^{|r|})-\val_U(x'0^{|r|})$.
Without loss of generality, we may assume that $x$ and $x'$ (resp., $y$ and $y'$) have the same length (indeed one can pad shorter representations with leading $0$'s). Then the above equation means that $x$ is lexicographically less than $x'$ if and only if the same holds for $y$ and $y'$. We can thus order $L(s,t,r,n)$ by the numerical value of the second component of an element, and therefore the $j$th element of $L(s,t,r,n)$ is well-defined.
\medskip
\noindent
$\bullet$ Defining two subsequences by the maps $\lambda_{s,t,r,n}:\mathbb{N}\to\mathbb{N}$ and $\mu_{s,t,r,n}:\mathbb{N}\to\mathbb{N}$.
\noindent
Let $(w_j,x_j,y_j)$ be the $j$th element in $L(s,t,r,n)$ with $j\ge 0$. After removing the leading $0$'s, the word
$x_j$ belongs to $Lt^{-1}\cup\{\varepsilon\}$, which can also be ordered by genealogical order. We let $\lambda_{s,t,r,n}(j)$ denote the index (i.e., position counting from $0$) of $x_j$ within this language. Similarly, the word
$y_j$ belongs to $Lr^{-1}\cup\{\varepsilon\}$ and has an index $\mu_{s,t,r,n}(j)$ within this language.
Note that if $\mathcal{L}_+(s,t,r)^{-1}= \mathcal{L}_+(s',t',r')^{-1}$,
$Ls^{-1}=Ls'^{-1}$, and $Lt^{-1}=Lt'^{-1}$ then, for all~$n$, the maps
$\lambda_{s,t,r,n}$ and $\lambda_{s',t',r',n}$ are the same. Indeed, the
first two conditions imply that $L(s,t,r,n)=L(s',t',r',n)$. Similarly, if
$\mathcal{L}_+(s,t,r)^{-1}= \mathcal{L}_+(s',t',r')^{-1}$,
$Ls^{-1}=Ls'^{-1}$, and $Lr^{-1}=Lr'^{-1}$ then, for all~$n$, the maps
$\mu_{s,t,r,n}$ and $\mu_{s',t',r',n}$ are the same.
We now obtain
\begin{align*}
\partial_\mathbf{x}[\val_U(p_{s,n}s)]
&=\!
\bigcup_{(t,r)\in \mathcal{P}(s)} \!\! \left\{
( \mathbf{x}[\val_U(xt)],\, \mathbf{x}[\val_U(yr)] )
\mathrel{\Big|} \smatrix{w \\ x \\ y}\in \mathcal{L}_+ \smatrix{s \\ t \\ r}^{-1} \wedge w\in 0^*p_{s,n}
\right\}\\
&= \! \bigcup_{(t,r)\in \mathcal{P}(s)} \!\! \left\{
( \mathbf{x}[\val_U(x_jt)],\, \mathbf{x}[\val_U(y_jr)] )
\mathrel{\Big|} \smatrix{w_j \\ x_j \\ y_j} \in L(s,t,r,n), j\ge 0
\right\}\\
&= \! \bigcup_{(t,r)\in \mathcal{P}(s)} \!\! \left\{
( (\mathbf{x}\circ t)[\lambda_{s,t,r,n}(j)], (\mathbf{x}\circ r)[\mu_{s,t,r,n}(j)] )
\mid j\ge 0 \right\}.
\end{align*}
Let us define an equivalence relation on triplets by $(s,t,r)\sim(s',t',r')$ if and only if all the following hold:
\begin{align}\label{eq:new-equiv-rel}
\mathcal{L}_+\smatrix{ s \\ t \\ r}^{\!\!\!-1}
= \mathcal{L}_+ \smatrix{s' \\ t' \\ r'}^{\!\!\!-1}\!\!\!,\quad
& Ls^{-1} = Ls'^{-1}, \quad
Lt^{-1} = Lt'^{-1}, \quad
Lr^{-1}=Lr'^{-1}, \\
&\mathbf{x} \circ t = \mathbf{x} \circ t', \quad \text{ and } \quad
\mathbf{x}\circ r = \mathbf{x}\circ r'. \nonumber
\end{align}
Since we have regular languages and the kernel of $\mathbf{x}$ is finite by assumption, this relation has a finite index (bounded by $\ell k^3 m^2$). Given $s$, the set $\{(s,t,r)\mid (t,r)\in\mathcal{P}(s)\}$ can be replaced by a set $\Lambda(s)$ of representatives of the equivalence classes for $\sim$. Since $\sim$ has a finite index, there are finitely many possible subsets of the form $\Lambda(s)$. So, we can write
\[
\partial_\mathbf{x}[\val_U(p_{s,n}s)] =
\bigcup_{(b,c,a)\in \Lambda(s)}
\left\{
((\mathbf{x}\circ c)[\lambda_{b,c,a,n}(j)], (\mathbf{x}\circ a)[\mu_{b,c,a,n}(j)])
\mid j\ge 0 \right\}.
\]
Now if $s$ and $s'$ are such that $Ls^{-1}=Ls'^{-1}$ and $\Lambda(s)=\Lambda(s')$, then $\partial_\mathbf{x}\circ s = \partial_\mathbf{x}\circ s'$. This proves that the kernel of $\partial_\mathbf{x}$ is finite (of size bounded by $k\cdot 2^{\ell k^3 m^2}$).
\end{proof}
\subsection{Non-addable systems: counterexamples}\label{ss:ce}
Our aim is to show that the boundary sequence of a $U$-automatic word is not always $U$-automatic.
We give two such examples.
The numeration system defined first is a variant of the base-$2$ system.
\begin{example}\label{exa:ce1}
Take the numeration system $(U_n)_{n\ge 0}$ defined by $U_n=2^{n+1}-1$ for all $n\ge 0$. We have $0^*\rep_U(\mathbb{N})=(0+1)^*(\varepsilon+20^*)$.
Consider the characteristic word $\mathbf{u}$ of $U$, i.e., $\mathbf{u}[n]=1$ if and only if $n\in\{U_j\mid j\ge 0\}$. The boundary sequence $\partial_\mathbf{u}$ starts with
\[
a\, b\, a\, b\, a\, b\, a\, b\, a\, a\, a\, b\, a\, b\, a\, b\, a\, a\, a\, a\, a\, a\, a\, b\, a\, a\, a\, b\, a\, b\, a\, b\, a\, a\, a\, a\, a\, a\, a\, a\, a\, a\, a
\, a\, a\, a\, a \, b\, a\, a\, a
\cdots
\]
where $a:=\{(0,0),(0,1),(1,0)\}$ and $b:=\{0,1\} \times \{0,1\}$. \end{example}
One can show that the language $\{\rep_U(n) \colon \partial_{\infw{u}}[n]= b\}$
is not regular (\cref{app:counterexamples}), hence:
\begin{restatable}{proposition}{restU}
\label{prop:example_U}
Let $U=(2^{n+1}-1)_{n\ge 0}$. The word $\mathbf{u}$ from \cref{exa:ce1} is $U$-automatic but its boundary sequence $\partial_\mathbf{u}$ is not $U$-automatic.
\end{restatable}
As a consequence of the previous proposition and \cref{the:kernel}, $U$ is non-addable.
\begin{remark}\label{rk:u-partialu-2aut}
One may notice that both $\mathbf{u}$ and
$\partial_\mathbf{u}$ are $2$-automatic: this follows by \cref{thm:logic}
from the set $X:= \{ U_{m+r}-U_m\mid m \ge 0, r>0 \}$ (which equals $\{n \in \mathbb{N} \colon \partial_{\infw{u}}[n] = b\}$) being $2$-definable by the formula $\varphi(n) := (\exists x) \, (\exists y)\,
(x<y \wedge V_2(x)=x \wedge V_2(y)=y \wedge n=y-x)$,
where $V_2(y)$ is the smallest power of $2$ occurring with a non-zero coefficient in the binary expansion of $y$.
\end{remark}
In view of the above remark, \cref{exa:ce1} could be considered as unsatisfactory. We now make use of a similar strategy but with a more complicated numeration system, for which we do not know any analogue of \cref{rk:u-partialu-2aut}. To this end, consider the non-addable numeration system from \cite[Ex.~3]{Frougny1997} or
\cite[Ex.~2]{MassuirPeltomakiRigo} defined by
\begin{equation}\label{eq:Un}
V_0 = 1,\ V_1 = 4,\ V_2 = 15,\ V_3 = 54\quad \text{ and }\quad V_n = 3V_{n-1} + 2V_{n-2} + 3V_{n-4}, \quad \forall \, n\ge 4.
\end{equation}
\begin{example}\label{exa:ce2}
Consider the characteristic word $\mathbf{v}$ of $V$, i.e., $\mathbf{v}[n]=1$ if and only if $n\in\{V_j\mid j\ge 0\}$. This word is trivially $V$-automatic.
The boundary sequence $\partial_\mathbf{v}$ starts with
\[
a\, a\, b\, a\, a\, a\, a\, a\, a\, a\, b\, a\, a\, b\, a\, a\, a\, a\, a\, a\, a\, a\, a\, a\, a\, a\, a\, a\, a\, a\, a\, a\, a\, a\, a\, a\, a\, a\, b\, a\, a\, a\, a\, a\, a\, a\, a\, a\, a\, b \cdots
\]
where again $a:=\{(0,0),(0,1),(1,0)\}$ and $b:=\{0,1\}\times \{0,1\}$.
\end{example}
Similar to the above, $\{\rep_V(n) \colon \partial_{\infw{v}}[n] = b\}$ is not regular (\cref{app:counterexamples}), whence
\begin{restatable}{proposition}{restV}
\label{prop:example_V}
Let $V$ be the numeration system given by \eqref{eq:Un}. The word $\mathbf{v}$ from \cref{exa:ce2} is $V$-automatic but its boundary sequence $\partial_\mathbf{v}$ is not $V$-automatic.
\end{restatable}
\begin{remark}
We do not know whether $\mathbf{v}$ and $\partial_{\mathbf{v}}$ are both $V'$-automatic for some numeration system $V'$.
\end{remark}
\section{The extended boundary sequences of Sturmian words}
\label{sec:Sturmian}
We give two descriptions of the $\ell$-boundary sequences of
Sturmian words (\cref{the:sturmian,prop:another-charact}) and discuss some of their word combinatorial properties.
We first recap minimal background on Sturmian words seen as codings of rotations.
For a general reference, see \cite[\S 2]{Lothaire}.
Let $\alpha$, $\rho \in \mathbb{T}:= [0,1)$ with $\alpha$ irrational. Define the \emph{rotation} of the $1$-dimensional torus $R_{\alpha}\colon \mathbb{T} \to \mathbb{T}$ by $R_{\alpha}(x) = \{x + \alpha\}$,
where $\{\,\cdot\, \}$ denotes the fractional part. Let
$I_0 = [0,1-\alpha)$ (or $I_0 = (0,1-\alpha]$) and
$I_1 = \mathbb{T}\setminus I_0$.
(The endpoints of $I_0$ will not matter in the forthcoming arguments.)
Define the coding $\nu \colon \mathbb{T} \to \{0,1\}$ by $\nu(x) = 0$ if $x \in I_0$, otherwise $\nu(x) = 1$. We define the word
$\mathbf{s}_{\alpha,\rho}$ by
$\mathbf{s}_{\alpha,\rho}[n] = \nu(R_{\alpha}^n(\rho))$, for all $n\ge 0$.
We call $\alpha$ the {\em slope} and $\rho$ the {\em intercept} of $\mathbf{s}_{\alpha,\rho}$.
The \emph{characteristic Sturmian word of slope $\alpha$} is $\mathbf{s}_{\alpha,\alpha}$.
\subsection{A description of the extended boundary sequence}\label{subsec:first}
In the following, a \emph{sliding block code of length $r$} is a mapping
$\mathfrak{B}\colon A^{\mathbb{N}} \to B^{\mathbb{N}}$ defined by
$\mathfrak{B}(\infw{x})[n] = \mathcal{B}(\infw{x}[n]\cdots \infw{x}[n + r-1])$
for all $n\ge 0$ and some $\mathcal{B} \colon A^{r} \to B$. Let $T \colon A^{\mathbb{N}} \to A^{\mathbb{N}}$
denote the shift map $Tx_0x_1x_2\cdots = x_1x_2\cdots$.
\begin{restatable}{theorem}{sturmianFirst}
\label{the:sturmian}
For a Sturmian word $\infw{s}$ of slope $\alpha$ (and intercept $\rho$) and $\ell \geq 1$, the (shifted) $\ell$-boundary sequence $T\partial_{\infw{s},\ell}$
is obtained by a sliding block code of length $2\ell$ applied to
the characteristic Sturmian word of slope $\alpha$.
\end{restatable}
To prove the theorem we develop the required machinery. For a word~$u=u_0\cdots u_{\ell-1}$, we let $I_u= \bigcap_{i=0}^{\ell-1}R_\alpha^{-i} (I_{u_i})$.
It is well-known that $u$ occurs at position $i$ in $\mathbf{s}_{\alpha,\rho}$ if and only if $R_{\alpha}^{i}(\rho) \in I_u$. These intervals of factors
of length $\ell$ can also be described as follows: order the set
$\bigr\{ \{-j\alpha\} \bigr\}_{j = 0}^{\ell}$
as $0 = i_0 <i_1<i_2<\cdots < i_\ell$.
For convenience, we set $i_{\ell+1}=1$.
If the $\ell+1$ factors of length~$\ell$ of the Sturmian word
$\mathbf{s}_{\alpha,\rho}$ are lexicographically ordered as $w_0<w_1<\cdots <w_\ell$, then
$I_{w_j}=[i_j,i_{j+1})$ for each $j\in\{0,\ldots,\ell\}$. From the
following claim (whose proof is in \cref{app:firstSturmian}) it is evident that the intercept $\rho$ plays no further role in our considerations. (This also follows from the fact that two Sturmian words have the same set of factors if and only if they have the same slope.)
\begin{restatable}{claim}{Claim}
Let $n\ge\ell$ and $u$, $v$ be length-$\ell$ factors of $\mathbf{s}_{\alpha,\rho}$. Then $ (u,v)\in \partial_{\mathbf{x},\ell}[n]$ if and only if
$R_\alpha^n(I_u) \cap I_v\neq\emptyset$.
\end{restatable}
The endpoints of $I_u$ are of the form $i_j$ and $i_{j+1}$ for some $j\in\{0,\ldots,\ell\}$. Hence, for $n \geq \ell$,
the set of pairs belonging to $\partial_{\mathbf{x},\ell}[n]$ is determined by the positions of the rotated endpoints $R_\alpha^n(i_j)$ within the intervals $I_{w_k}$.
Notice that each rotated endpoint $R_{\alpha}^n(i_j)$ always
lies in the interior of some $I_{w_k}$ whenever $n > \ell$. When $n = \ell$,
we have $R_{\alpha}^n(\{-\ell\alpha\}) = 0$, which is an endpoint of
one of the intervals $I_{w_k}$.
For the time being we assume $n > \ell$, and return to the
case $n=\ell$ in \cref{prop:first-letter}.
Now, for example, if $R_\alpha^{n}(i_j) \in I_{w_k}$ then we have
$(w_j,w_k)$, $(w_{j-1},w_k) \in \partial_{\mathbf{x},\ell}[n]$
(if $j=0$, $w_{j-1}$ is replaced with $w_\ell$). Determining the boundary sets can be quite an intricate exercise; see \cref{exa:ffib}.
An alternative to considering the positions of the points $R_\alpha^{n}(i_j)$
within the intervals $I_{w_k}$ is to consider the positions
of the points $R_\alpha^{n}(\{-j \alpha\})$ within the intervals
$I_{w_k}$---the only difference is the order of enumeration.
For each $n > \ell$, there is a map $\sigma = \sigma_n \in T_{\ell}$,
where $T_{\ell}$ is the set of mappings from $\{0,\ldots,\ell\}$
to itself, such that
\begin{equation}\label{eq:realizable-constellations}
R_{\alpha}^n(\{-j\alpha\}) \in I_{w_{\sigma(j)}} \quad \forall\, j\in \{0,\ldots,\ell\}.
\end{equation}
The realizable such configurations in \eqref{eq:realizable-constellations} are called \emph{constellations}.
These points, when ordered according to the
$i_j$'s, determine the boundary set
$\partial_{\infw{s},\ell}[n]$ as described above.
See \cref{exa:ffib} (and \cref{exa:intricated}) for an illustration of the construction.
\begin{definition}\label{def:r}
Let $\sigma \in T_{\ell}$ be such that
\eqref{eq:realizable-constellations} holds for some $n\in \mathbb{N}$. We
define $\partial_{\sigma}\in 2^{A^\ell\times A^\ell}$ as the
boundary set corresponding to any constellation inducing
$\sigma$.
\end{definition}
It is now evident that if $\sigma_n = \sigma_m =: \sigma$, then $\partial_{\infw{s},\ell}[n] = \partial_{\sigma} = \partial_{\infw{s},\ell}[m]$.
\begin{example}\label{exa:ffib}
The Fibonacci word $\mathbf{f}$ is $\mathbf{s}_{\alpha,\alpha}$ for $\alpha=(3-\sqrt{5})/2\simeq 0.382$. In \cref{fig:rotations}, the outer circle shows the partition with the interval $I_{w_0}$, \ldots, $I_{w_\ell}$ and the inner circle shows the positions of the points $R_\alpha^n(\{-j\alpha\})$ for $\ell=4$ and $n=17$.
The corresponding words $w_0,\ldots,w_\ell$ are written next to their interval.
Here $\sigma_n$ is defined by
$(0,1,2,3,4) \mapsto (2,0,3,1,4)$. For any constellation inducing $\sigma_n$, we see the pairs belonging to $\partial_{\sigma_n} = \partial_{\mathbf{f},4}[17]$ from \cref{fig:rotations}: the inner intervals (obtained from the outer intervals by applying $R_\alpha^{17}$) give the prefix matching the suffix of the overlapping outer intervals, in clockwise order:
\[
\smatrix{0010 \\0101}, \smatrix{0010 \\ 1001}, \smatrix{0100 \\1001}, \smatrix{0100\\1010}, \smatrix{0101 \\ 1010}, \smatrix{0101\\ 0010}, \smatrix{1001 \\ 0010}, \smatrix{1001\\0100}, \smatrix{1010 \\ 0100}, \smatrix{1010 \\ 0101}.
\]
Coming back to the introductory \cref{exa:intro}, the five sets $a_1$, \ldots, $a_5$ correspond to the situations depicted from left to right in \cref{fig:exa2fib}. For instance, in the fourth picture, we understand why $10$ is a prefix belonging to three pairs in $a_4$: the red inner interval intersects the three outer intervals of the partition. The situation is similar in the fifth picture where $01$ is the prefix of three pairs in $a_5$. It is however not the case with the first three sets/pictures.
\begin{figure}
\centering
\begin{minipage}{.39\textwidth}
\centering
\includegraphics[width=5.4cm]{./figure1.pdf}
\captionof{figure}{A constellation for\\ $\alpha=(3-\sqrt{5})/2$, $\ell=4$ and $n=17$.}
\label{fig:rotations}
\end{minipage}%
\begin{minipage}{.61\textwidth}
\centering
\includegraphics[height=5.35cm]{./figure2.pdf}
\captionof{figure}{Some constellations for $\alpha=(3-\sqrt{5})/2$ and $\ell=2$ inducing the five maps $\sigma_n$ sending $(0,1,2)$, resp., to\\
$(0,2,1)$, $(1,0,2)$, $(2,1,0)$, $(1,2,1)$, $(2,1,2)$.}
\label{fig:exa2fib}
\end{minipage}
\end{figure}
\end{example}
\begin{remark}\label{rem:necessary}
It is possible that $\partial_{\sigma} = \partial_{\sigma'}$ for distinct maps $\sigma$, $\sigma' \in T_{\ell}$. Indeed,
for the Fibonacci word and $\ell=1$, we have equality for the identity mapping $\text{id}$ and $\sigma \colon (0,1) \mapsto (1,0)$;
in this case $\partial_{\text{id}} = \partial_{\sigma} =
\{0,1\}\times \{0,1\}$.
So two constellations inducing different maps in $T_{\ell}$ lead to
the same set of boundary pairs. (See however \cref{lem:distinctSets}.)
\end{remark}
\begin{definition}
Let $\infw{r}$ be the rotation word defined by $\infw{r}[n] = \eta(R_{\alpha}^n(\alpha))$ for all $n\ge 0$, where $\eta \colon \mathbb{T} \to \{0,\ldots,\ell\}$ is defined by $\eta(x) = j$ when $x \in I_{w_j}$ (recall $I_{w_i}$ corresponds to the $i$th factor of length $\ell$).
\end{definition}
We have that $\infw{r}[n] = j$ if and only if the characteristic Sturmian word $\infw{s}_{\alpha,\alpha}$ has the length-$\ell$ factor $w_j$
occurring at position $n$.
\begin{proof}[Proof sketch of \cref{the:sturmian}]
It can be shown that the boundary sequence of $\infw{s}$ can be obtained from $\infw{r}$ constructed above
using a sliding block code of length $\ell + 1$ (see \cref{app:firstSturmian}). Notice that by definition $\infw{r}$ is obtained by a sliding block code of length $\ell$ of the characteristic Sturmian word $\infw{s}_{\alpha,\alpha}$. The claim
follows as composing sliding block codes of length
$r$ and $r'$, respectively, yields a sliding block code of length $r + r' - 1$.
\end{proof}
\begin{example}\label{exa:fib}
We apply \cref{the:sturmian} to the Fibonacci word $\infw{f}$. Take $\alpha=(3-\sqrt{5})/2$, $\ell=1$, $I_0=[0,1-\alpha)$ and $I_1=[1-\alpha,1)$. Then the rotation word $\infw{r}$ associated with the partition
$\{I_0,I_1\}$, slope $\alpha$, and intercept $\alpha$ is $\infw{s}_{\alpha,\alpha}$ by definition, which happens to be the Fibonacci word $\infw{f}$.
We have
$
\mathbf{f}= 0\ 1\ 0\ 0\ 1\ 0\ 1\ 0\ 0\ 1\ 0\ 0\ 1\ 0\ 1\ 0\ 0\ 1\ 0\ 1\ 0\ 0\ 1\ 0\ 0\ 1\ 0\ 1\ 0\ 0\ 1 \cdots.
$
Recall from the construction that the length-$2$ factors of the rotation
word determine the boundary sets. The three length-$2$ factors of $\infw{f}$ are $01$, $10$, and $00$ occurring at positions $m = 0$, $1$, and $2$,
respectively. We get the three maps $\sigma_{m+2}\in T_1$ defined by $(0,1) \mapsto (1,0)$,
$(0,1) \mapsto (0,1)$, and $(0,1) \mapsto (0,0)$, respectively. We deduce that an occurrence of $01$ or $10$ corresponds to the boundary set
$b := \{0,1\} \times \{0,1\}$, and $00$ to
$a := \{(0,0),(0,1),(1,0)\}$.
We may therefore define $\mathcal{B} \colon 01, 10 \mapsto b$, $00 \mapsto a$ and the associated sliding block code $\mathfrak{B}$ of length~$2$;
applying $\mathfrak{B}$ to $\infw{f}$, we get
\begin{align*}
& \mathfrak{B} ( (01)(10)(00)(01)(10)(01)(10)(00)(01)(10)(00)(01)(10)(01)(10)(00)(01)(10)(01) \cdots )\\
&= \hskip6pt b \hskip14pt b \hskip13pt a \hskip13pt b \hskip13pt
b \hskip14pt b \hskip13pt b \hskip13pt a \hskip13pt b \hskip14pt b \hskip13pt a \hskip13pt b \hskip13pt b \hskip14pt b \hskip13pt b \hskip13pt
a \hskip13pt b \hskip14pt b \hskip14pt b \cdots,
\end{align*}
which indeed gives back \cref{exa:intro} after prepending the letter $a$.
\end{example}
We next discuss the first element
$\partial_{\mathbf{s},\ell}[\ell]$
of the (extended) boundary sequence.
Notice that the set is in one-to-one correspondence with the factors
of length $2\ell$, and thus has cardinality $2 \ell + 1$.
The points $\{-j\alpha\}$ and $R_{\alpha}^{\ell}(\{-j\alpha\})$, $j \in\{0,\ldots,\ell\}$, on the torus
still determine the boundary set, but notice that there are only $2\ell + 1$
distinct pairs. The following proposition describes rather precisely
how the first element appears in the boundary sequence. The proof is given in \cref{app:firstSturmian}.
\begin{restatable}{proposition}{firstLetter}
\label{prop:first-letter}
For a Sturmian word $\infw{s}$, the boundary set
$\partial_{\infw{s},\ell}[\ell]$ appears infinitely often in
$\partial_{\infw{s},\ell}$ if and only if
$0^{2\ell}$ or $1^{2\ell}$ appears in $\infw{s}$. Otherwise it appears exactly once.
\end{restatable}
Notice that either $00$ or $11$ appears in a Sturmian word $\infw{s}$, so the above implies that the
first letter of the ($1$-)boundary sequence~$\partial_{\infw{s}}$
always appears infinitely often in the sequence.
Returning to \cref{exa:intro}, since $0^4$ does not appear in the Fibonacci word, the letter $a_0$ appears only once in $\partial_{\infw{f},2}$.
We conclude with the immediate corollary of \cref{the:sturmian,prop:first-letter}; here we say that a word
$\infw{w}$ is \emph{uniformly recurrent} if each of its factors occurs infinitely often within bounded gaps (the distance between two consecutive occurrences depends on the factor). It is known that, e.g., Sturmian words are uniformly recurrent.
\begin{corollary}
For any Sturmian word $\infw{s}$, the shifted sequence $T\partial_{\infw{s},\ell}[n]$ is uniformly recurrent. The sequence $\partial_{\infw{s},\ell}$ is uniformly recurrent if and only if $0^{2\ell}$ or $1^{2\ell}$ appears in $\infw{s}$.
\end{corollary}
\subsection{Another description of the extended boundary sequence}\label{subsec:second}
We give another description of the $\ell$-boundary sequences of Sturmian words when $\ell \geq 2$.
For any irrational number $\alpha \in (0,1)$ there is a unique infinite continued fraction expansion
\[
\alpha = [0; a_1 , a_2, a_3, \ldots] := \dfrac{1}{a_1 + \dfrac{1}{a_2 + \dfrac{1}{a_3 + \ldots}}},
\]
where $a_n \geq 1$ are integers for all $n\geq 1$. Then the
characteristic Sturmian word $\infw{s}_{\alpha,\alpha}$ of slope
$\alpha$ equals $\lim_{k\to\infty} S_k$,
where $S_{-1} = 1$, $S_0 = 0$, $S_1 = S_0^{a_1-1}S_{-1}$, and $S_{k+1} = S_{k}^{a_{k+1}}S_{k-1}$ for all $k\geq 1$ \cite[Chap.~9]{AS}.
The main result of this part is the following (proof in \cref{app:secondSturmian}, see also \cref{ex:morphicImage,ex:morphicImagePi}).
\begin{restatable}{proposition}{otherSturmian}
\label{prop:another-charact}
Let $\infw{s}$ be a Sturmian word of slope $\alpha = [0;a_1+1,a_2,\ldots]$.
For each $\ell \geq 2$, there exists $k_{\ell} \in \mathbb{N}$ such that for any $k \geq k_{\ell}$ there is a morphism $h_{k,\ell}$ such
that $T\partial_{\mathbf{s},\ell} = h_{k,\ell}(\infw{s}_{\beta_k,\beta_k})$, where
$\beta_k = [0;a_{k+1}+1,a_{k+2},\ldots]$.
\end{restatable}
\begin{example}
Take the slope $\alpha=(3-\sqrt{5})/2$; its continued fraction expansion is $[0;2,1,1,1,\ldots]$.
Using the previous notation, $S_{-1} = 1$, $S_0 = 0$, and $S_{k+1} = S_k S_{k-1}$ for all $k\geq 0$.
Then the sequence $(S_k)_{k\geq 0}$ converges to the Fibonacci word; the first few words in the sequence $(S_k)_{k\ge 0}$ are $0, 01, 010, 01001, 01001010$.
Now for any $\ell \geq 2$, the above proposition thus gives that
$\partial_{\infw{f},\ell}$ is the morphic image of the characteristic
Sturmian word of slope
$\beta = \alpha$. In other words, the $\ell$-boundary sequence is
always a morphic image of $\infw{f}$.
\end{example}
We generalize the last observation made in the above example.
\begin{restatable}{corollary}{morphicSturmian}
\label{cor:morphicSturmian}
Let $\mathbf{s}$ be a Sturmian word with quadratic slope.
Then $\partial_{\mathbf{s},\ell}$ is morphic. In particular,
the $\ell$-boundary sequence of a Sturmian word fixed by a non-trivial morphism is morphic.
\end{restatable}
\begin{proof}
A remarkable result of Yasutomi \cite{Yasutomi1999sturmian} (see also \cite{BertheEIR2007substitution}), characterizing those Sturmian words that
are fixed by some non-trivial morphism,
implies that if a Sturmian word of slope~$\alpha$ is fixed by a
non-trivial morphism, then so is the characteristic Sturmian word of
slope~$\alpha$. Furthermore, the slope is characterized by the property that $\alpha = [0;1,a_2,\overline{a_3,\ldots,a_r}]$ with $a_r \geq a_2$
or $\alpha = [0;1+a_1,\overline{a_2,\ldots,a_r}]$ with
$a_r \geq a_1 \geq 1$ \cite{CrispMPS1993substitution} (see also
\cite[Thm.~2.3.25]{Lothaire}). Here $\overline{x_1,\ldots,x_t}$
indicates the periodic tail of the infinite continued fraction
expansion. As $\alpha$ is quadratic, it has an eventually periodic
continued fraction expansion. There thus exist arbitrarily large $k$ for
which $\beta = [0;a_k+1,a_{k+1},\ldots]$ gives a characteristic Sturmian word of slope $\beta$ which is the fixed point of a non-trivial morphism (it is of the latter form). \cref{prop:another-charact} then posits that
$T\partial_{\infw{s},\ell}$ is the morphic image of this word, and the claim follows (because prepending the letter $\partial_{\infw{s},\ell}[\ell]$ preserves morphicity \cite[Thm.~7.6.3]{AS}).
\end{proof}
Notice that given the morphism fixing a Sturmian word
$\infw{s}$, one can compute (the continued fraction expansion of) the quadratic slope (and intercept) of $\infw{s}$ \cite{TanW2003invertible,PengT2011Sturmian,LepsovaPS2022faithful}.
The above corollary has an alternative proof via the logical
approach as well. For the definitions of notions that follow, we
refer to the cited papers. From the work of Hieronymi and Terry
\cite{HieronymiT2018Ostrowski}, it is known that addition in
the \emph{Ostrowski-numeration system} based on an irrational
quadratic number $\alpha$ is recognizable by a finite
automaton. This motivated Baranwal, Schaeffer, and Shallit to
introduce \emph{Ostrowski-automatic sequences}
in \cite{BaranwalSS2021Ostrowski}. For example, they showed
that the characteristic Sturmian word of slope $\alpha$ is
Ostrowski $\alpha$-automatic. Since the numeration system is
addable, the above corollary follows by the same arguments as
in \cref{ss:logic}.
\subsection{Factor complexities of the extended boundary sequences}
\label{subsec:complexities}
\begin{definition}
A word over an alphabet $A$ is of {\em minimal complexity}
if its factor complexity is $n + |A|-1$ for all $n \geq 1$.
\end{definition}
Minimal complexity words can be seen as a generalization of
Sturmian words to larger alphabets: if a word (containing all
letters of $A$) has less than $n + |A| - 1$ factors of length
$n$ for some $n$, then it is ultimately periodic. Otherwise it
is aperiodic (a consequence of the Morse--Hedlund theorem).
See \cite{FerencziM1997transcendence,Cassaigne1997sequences,Didier1999characterisation} for characterizations and generalizations.
The following proposition is almost immediate after the key \cref{lem:distinctSets} (both proofs can be found in \cref{app:complexities}).
\begin{restatable}{proposition}{minimalComplexity}
\label{pro:minimal-complexity}
Let $\ell \geq 2$. The $\ell$-boundary sequence of a Sturmian
word is a minimal complexity word (of complexity $n \mapsto n + 2\ell$, $n\geq 1$).
\end{restatable}
\begin{restatable}{lemma}{distinctConstellations}
\label{lem:distinctSets}
Let $\sigma$ and $\sigma' \in T_{\ell}$, $\ell \geq 2$, be distinct mappings both satisfying \eqref{eq:realizable-constellations} (for different $n$).
Then $\partial_{\sigma} \neq \partial_{\sigma'}$.
\end{restatable}
We conclude with a formula for the factor complexity of the
$1$-boundary sequence of Sturmian words (proof in \cref{app:complexities}).
\begin{restatable}{proposition}{boundaryComplexity}
Let $r$ be the maximal integer such that
$(01)^r$ appears in the Sturmian word $\infw{s}$.
The boundary sequence
$\partial_{\infw{s}}$ has factor complexity
\begin{equation*}
n \mapsto
\begin{cases}
n+1, &\text{if }n < 2r;\\
n+2, & \text{ otherwise}.
\end{cases}
\end{equation*}
\end{restatable}
As an immediate corollary, we see that the $\ell$-boundary
sequence is aperiodic for all $\ell \geq 1$.
|
1,116,691,499,296 | arxiv | \section{Introduction}\label{s:intro}
Let $f: {\mathbb C}^n \to {\mathbb C}$ be a holomorphic function germ. What can be said about functions whose Milnor fibre $F$ has the property $b_i(F) = 0$ for all $i \ge 1$ ? If $F$ is connected then $f$ is non-singular and equivalent to a linear function by A'Campo's trace formula. The remaining question: \emph{What happens if $F$ is non-connected~?} is only relevant for non-reduced plane curve singularities.
This question is related to a recent paper \cite{HM}. That paper contains a statement about the so-called Bobadilla conjectures \cite{Bo} in case of plane curves. The invariant $\beta= 0$, used by Massey \cite{Ma} should imply that the singular set of $f$ is a smooth line.
In this note we give a short topological proof of a stronger statement.
\begin{proposition} \label{p:main}
If the first Betti number of the Milnor fibre of a plane curve singularity is zero,
then the defining function is equivalent to $x^r$.
\end{proposition}
\begin{corollary} \label{c:line}
In the above case the singular set is a smooth line and the system of transversal singularities is trivial.
\end{corollary}
\section{Non-reduced plane curves}
Non-isolated plane curve singularities have been thoroughly studied by Rob Schrauwen in his dissertation \cite{Sch1}. Main parts of it are published as \cite{Sch2} and \cite{Sch3}. The above Proposition \ref{p:main} is an easy consequence of his work.
\medskip
We can assume that $f = f_1^{m_1}. \cdots . f_r^{m_r}$
(partition in powers of reduced irreducible components).
\begin{lemma}
Let $d = \gcd (m_1,\cdots,m_r)$
\begin{itemize}
\item[(a.)] $F$ has $d$ components, each diffeomorphic to the Milnor fibre $G$ of
$g = g_1^{\frac{m_1}{d}}. \cdots . g_r^{\frac{m_r}{d}}$. The Milnor monodromy of $f$ permutes these components,
\item[(b.)] if $d=1$ then $F$ is connected.
\end{itemize}
\end{lemma}
\begin{proof}
(a.) Since $f = g^d$ the fibre $F$ consists of $d$ copies of $G$.\\
(b.) We recall here the reasoning from \cite{Sch1}. Deform the reduced factors $f_i$ into $\hat{f}_i$ such that
the product $ \hat{f}_1. \cdots .\hat{f}_r = 0 $ contains the maximal number of double points (cf. Figure \ref{f:Qpoints}). This is called a network deformation by Schrauwen. The corresponding deformation $\hat{f}$ of $f$ near such a point has local equation are of the form $x^p y^q = 0$ (point of type $D[p,q]$).
\begin{figure}[htbp]
\centering
\includegraphics[width=6cm]{DoublePointDeformation}
\caption{Deformation to maximal number of double points.}
\label{f:Qpoints}
\end{figure}
Near every branch $\hat{f}_i=0$ the Milnor fibre is a $m_i$-sheeted covering of the zero-locus, except in the $D[p,q]$-points. We construct the Milnor fibre $F$ of $f$ starting with $S=\sum m_i$ copies of the affine line ${\mathbb A}$.
Cover the {\it i}th branch with $m_i$ copies of ${\mathbb A}$ and delete $(p+q)$ small discs around the $D[p,q]$-points. Glue in the holes $\gcd (p,q)$ small annuli (the Milnor fibres of $D[p,q]$). The resulting space is the Milnor fibre $F$ of $f$.
A hyperplane section of a generic at a generic point of $\hat{f}_i=0$ defines a transversal Milnor fibre $F'_1$.
Start now the construction of $F$ from $F'_1$, which consists of $m_1$ cyclic ordered points.
As soon as $f_1=0$ intersects $f_k=0$ it connects the sheets of $f_1=0$ modulo $m_k$. Since $\gcd (m_1,\cdots,m_r) = 1$ we connect all sheets.
\end{proof}
\begin{proofp}
If $b_1(F)=0$, then also $b_1(G)=0$. The Milnor monodromy has trace$(T_g)$ = 1. According to
A'Campo's observation \cite{AC} $g$ is regular: $g=x$.
It follows that $f = x^r$.
\end{proofp}
\section{Relation to Bobadilla's question}
We consider first in any dimension $f: {\mathbb C}^{n+1} \to {\mathbb C}$ with a 1-dimensional singular set, see especially the 1991-paper \cite{Si} for definitions, notations and statements.
We focus on the group $H_n (F,F^{\pitchfork})$
which occurs in two exact sequences on p. 468 of \cite{Si}:
\[ 0 \rightarrow H_{n-1}^{\mbox{\tiny f}}(F) \rightarrow H_{n-1}( F^{\pitchfork})
\rightarrow H_n(F) \oplus H_{n-1}^{\mbox{\tiny t}} (F)
\rightarrow 0 \]
\[ 0 \rightarrow H_{n}(F) \rightarrow H_n (F,F^{\pitchfork})
\rightarrow H_{n-1} (F^{\pitchfork})
\rightarrow H_{n-1} (F)
\rightarrow 0 \]
Here $F^{\pitchfork}$ is the disjoint union of the transversal Milnor fibres $F^{\pitchfork}_i$, one for each irreducible branch of the 1-dimensional singular set.\footnote{ $F^{\pitchfork}$ was originally denoted by $F'$. In the second sequence a misprint $n$ in the third term has been changed to $n-1$.}
Note that $ H_{n}(F) ,\; H_n (F,F^{\pitchfork})$ and $H_{n-1}( F^{\pitchfork})$ are free groups. $H_{n-1}(F)$ can have torsion,
we denote its free part by $H_{n-1}(F)^{\mbox{\tiny f}}$ and its torsion part by $H_{n-1}(F)^{\mbox{\tiny t}}$. All homologies here are taken over ${\mathbb Z}$, but also other coefficents are allowed.
From both sequences it follows that the $\beta$-invariant, introduced in \cite{Ma} has a 25 years history, since is nothing else than:
\[ \mathop{{\rm{dim}}} H_n(F,F^{\pitchfork}) = b_n - b_{n-1} + \sum \mu_i^{\pitchfork} : = \beta \]
From this definition is immediately clear that $\beta \ge 0$ and that $\beta$ is topological. The topological defintion has as direct consequence:
\begin{proposition}\label{beta=0}
Let $f: {\mathbb C}^{n+1} \to {\mathbb C}$ with a 1-dimensional singular set, then:
\[ \beta = 0 \; \; \Leftrightarrow \; \; \chi(F) = 1 + (-1)^n \sum \mu_i^{\pitchfork} \; \; \Leftrightarrow \; \;
H_n(F,{\mathbb Z}) = 0 \; \mbox{\rm and} \; H_{n-1}(F,{\mathbb Z}) = {\mathbb Z}^{\sum \mu_i} \]
\end{proposition}
The original Bobadilla conjecture C \cite{Bo} was in \cite{Ma} generalized to the reducible case as follows: \emph{Does $\beta = 0$ imply that the singular set is smooth?} As consequence of our main Proposition \ref{p:main} we have:
\begin{corollary}
In the curve case $\beta = 0$ implies that the singular set is smooth; and that the function is equivalent to $x^r$.
\end{corollary}
\begin{remark}
In \cite{HM} the first part of this corollary was obtained with the help of L\^e numbers.
\end{remark}
\begin{remark}
From the definition $\beta = H_n (F,F^{\pitchfork})$ follow direct and short proofs of several statements from \cite{Ma}.
\end{remark}
An other consequence from \cite{Si} is the composition of surjections:
\[ H_{n-1}(F^{\pitchfork})= \oplus {\mathbb Z}^{\mu_i} \twoheadrightarrow H_{n-1}(\partial_2 F) = \oplus \frac{{\mathbb Z}^{\mu_i}}{A_i-I} \twoheadrightarrow H_{n-1}(F) \]
From this follows:
\begin{proposition} \label{beta=bound}
If $\mathop{{\rm{dim}}} H_{n-1}(F) = \sum \mu_i$ (upper bound) then
\begin{itemize}
\item[a.] $ H_{n-1}(\partial_2 F)$ and $ H_{n-1}(F)$ are free and isomorphic to $ {\mathbb Z}^{\sum \mu_i}$.
\item[b.] All transversal monodromies $A_i$ are the identity.
\end{itemize}
\end{proposition}
The second part of \cite{Ma} contains an elegant statement about $\beta=1$ via the A'Campo trace formula.
Also the reduction of the generalized Bobadilla conjecture to the (irreducible) Bobadilla conjecture.
As final remark: The great work (the irreducible case) has still has to be done!
Together with the L\^e-conjecture this seems to be an important question in the theory of hypersurfaces 1-dimensional singular sets.\\
\section*{References}
|
1,116,691,499,297 | arxiv | \section{Introduction}\label{intro}
The Brian\c con-Skoda theorem, \cite{brianconskoda}, states that for any
ideal $\mathfrak{a} \subset \O_{\C^n,0}$ generated by $m$ germs, we have the inclusion
$\overline{\mathfrak{a}^{\min(m,n)+l-1}}\subset\mathfrak{a}^l$, where $\overline{I}$
denotes the integral closure of $I$.
The generalization to an arbitrary regular Noetherian ring was proven
algebraically in \cite{lipmansathaye}, and for rational singularities in \cite{aberbachhuneke:frational}.
Huneke, \cite{huneke:uniform}, showed that for a quite general Noetherian reduced local ring $S$,
there is an integer $N$ such that $\overline{\mathfrak{a}^{N+l-1}}\subset\mathfrak{a}^l$,
for all ideals $\mathfrak{a}$ and $l\geq 1$. In particular this applies when $S=\O_{V,x}$,
the local ring of holomorphic functions of a germ of a reduced analytic space $V$.
This case of Huneke's theorem was recently reproven analytically, \cite{ass:bsvar}.
Let $|\mathfrak{a}|^2 = \sum_1^m |a_i|^2$, which up to constants does not depend on the choice of the generators $a_i$.
Since a function $\phi$ in
$\overline{\mathfrak{a}^M}$ is characterized by the property that $|\phi|\leq C|\mathfrak{a}|^M$,
\cite{lejeunejalaberttessier}, an equivalent formulation of the theorem is that
$\phi$ belongs to $\mathfrak{a}^l$ whenever $|\phi|\leq C|\mathfrak{a}|^{N+l-1}$ (on $V$).
We will consider a germ of an analytic space, i.e. a pair $(Z,\O_{Z,x})$, or just $Z$ for brevity,
of a germ of an analytic variety $Z$ at a point $x$ and its local ring $\O_{Z,x} = \O_{\C^n,0} / \mathcal{J}$
for some $\mathcal{J} \subset \O_{\C^n,0}$ such that $Z(\mathcal{J}) = Z$.
The reduced space $Z_{red}$ has the same
underlying variety $Z$, but the structure ring is $\O_{Z,x}/\sqrt{0} = \O_{\C^n,0} / \sqrt{\mathcal{J}}$.
We assume throughout this paper that $\O_{Z,x}$ is Cohen-Macaulay, which in particular gives that $\mathcal{J}$
has pure dimension.
The aim of this paper is to find an appropriate generalization of the Brian\c con-Skoda
theorem to this setting -- when $S=\O_{Z,x}$.
Now that we have dropped the assumption that $Z$ is reduced, the situation becomes different;
the integral closure of any ideal contains the nilradical $\sqrt{0}$ by definition, so
$\overline{\mathfrak{a}^N}\subset\mathfrak{a}$ can only hold if
$\sqrt{0} \subset \mathfrak{a}$, or equivalently, if each element of $\O_{Z,x}$ that vanishes on $Z$ belongs to $\mathfrak{a}$.
Clearly this does not hold for any $\mathfrak{a}$. In the following example we consider the most simple non-reduced
space. It will nevertheless help illustrate some general notions, and also our main result, Theorem \ref{bs}.
\begin{example}\label{enkeltex}
Consider the analytic space $Z$ whose underlying space $Z_{red}$ is
$\left\{w=0\right\}=\mathbb{C}^{n-1}\subset \mathbb{C}^n$, such that $\mathcal{O}_Z = \mathbb{C}[[z_1,\dots z_{n-1},w]]/w^k$, $k>2$.
The nilradical is $(w)$, and is not contained in $\mathfrak{a}=(w^2)$. It may be helpful to think of
the space $Z$ as $\mathbb{C}^{n-1}$ with an extra infinitesimal direction transversal to $Z$,
and its structure sheaf being the $k$:th order Taylor expansions in that direction. For each $f\in\O_{Z,x}$ we have
\begin{align*}
f(z,w) &= \sum_{i=0}^{k-1} \frac{\partial^i f}{\partial w^i}(z,0) w^i, \quad \text{and} \\
\O_{Z,x} &\simeq \bigoplus_0^{k-1} \mathcal{O}_{Z_{red},x}.
\end{align*}
Although the function $w$ is identically zero on $Z$ (and therefore it most definitely satisfies
$|w| \leq C|\mathfrak{a}|^M$ for any $M$), but the element $w$ does not belong to $\mathfrak{a} = (w^2)$.
Since $Z$ is non-reduced, evaluating $w$, or any other element, as a function on $Z$ does not give enough
information to determine ideal memebership. We also have to take into account the transversal derivatives.
\end{example}
A germ of a holomorphic differential operator $L$ is called
\textit{Noetherian} with respect to an ideal $\mathcal{J} \subset \O_{\C^n,0}$
if $L\phi \in \sqrt{\mathcal{J}}$ for all $\phi \in \mathcal{J}$.
We say that $L_1, \dots ,L_M$ is a \textit{defining} set of Noetherian operators for $\mathcal{J}$,
if $\phi \in \mathcal{J}$ if and only if $L_1 \phi, \dots, L_M \phi \in \sqrt{\mathcal{J}}$.
The existence of a defining set for any ideal $\mathcal{J}$ is due to Ehrenpreis \cite{ehrenpreis}
and Palamodov \cite{palamodov}, see also
\cite{bjork:abel}, \cite{hormander:scv} and \cite{oberst}. In the example above, a defining set
is $1, \partial/\partial w, \dots, \partial^{k-1}/\partial w^{k-1}$.
If $L$ is Noetherian with respect to $\mathcal{J}$, then $L \psi$ is a well-defined function on
$Z_{red}$ for any $\psi \in \O_{Z,x}$, and $L$ induces an intrinsic mapping
$L: \O_{Z,x} \to \mathcal{O}_{Z_{red},x}$. Let $\mathcal{N}(Z)$ be the set of all such mappings;
this set does not depend on the choice of local embedding of $Z$.
If $(L_i)$ is a defining set for $\mathcal{J}$, then by definition any element $\phi\in\O_{Z,x}$ is determined
uniquely by the tuple of functions $(L_i \phi)$ on $Z_{red}$, cf. Example~\ref{enkeltex}.
This fact indicates that it is natural to impose size conditions on the whole set $(L_i \phi)$ to generalize the
Brian\c con-Skoda theorem:
\begin{theorem}\label{bs}
Let $Z$ be a germ of an analytic space such that $\O_{Z,x}$ is Cohen-Macaulay.
Then there exists an integer $N$ and operators $L_1,\dots,L_M \in \mathcal{N}(Z)$ such that
for all ideals $\mathfrak{a}\subset \O_{Z,x}$ and all $l\geq 1$,
\begin{align} \label{premise}
|L_j \phi|\leq C |\mathfrak{a}|^{N+l-1}, \quad 1 \leq j \leq M,
\end{align}
implies that $\phi\in\mathfrak{a}^l$.
\end{theorem}
\begin{remark}
Let $\mu_0$ be the maximal order of $L_j$ for $1 \leq j \leq M$. If $\phi \in \mathfrak{a}^{l+\mu_0}$,
then $L_j\phi \in \mathfrak{a}^l$, so for $N = -\mu_0+1$ and $l >>0$, the inequalities \eqref{premise}
are necessary conditions for $\phi$ belonging to $\mathfrak{a}^l$.
Note also that the special case $\mathfrak{a} = (0)$ of the theorem means that $\mathcal{L}_1, \dots, \mathcal{L}_M$
is a defining set of Noetherian operators for $\mathcal{J}$, if $\mathcal{L}_j$ is any representative for
$L_j$ in the ambient space.
\end{remark}
We will now prove Theorem~\ref{bs} by elementary means in the case of Example~\ref{enkeltex}:
\begin{example}\label{bevisatexempel}
As we saw in the previous example,
a set of defining differential operators for the ideal $(w^k)$ consists of $L_j = \partial^j/\partial w^j$,
$0\leq j \leq k-1$. We assume that
\begin{align}\label{smpremise}
|L_j \phi| \leq |\mathfrak{a}|^{r +(k-1)-j},
\end{align}
for all $z\in \mathbb{C}^{n-1}, w=0$, where
$r=\min(n-1,m)$. We will allow ourselves to abuse notation; for example, we will write simply $\mathfrak{a}$ when we
actually are referring to some element that belongs to $\mathfrak{a}$.
Using Brian\c con-Skoda for $\mathbb{C}^{n-1}$ we get
\begin{align}\label{phipartials}
\frac{\partial^j\phi}{\partial w^j} = \mathfrak{a}^{k-j} + k_j w, \quad k_j \in \O_{\C^n,0}.
\end{align}
We will show that
\begin{align} \label{induktion}
\phi = \sum_{i=0}^p w^i \mathfrak{a}^{k-i} + g_p w^{p+1}, \quad g_p \in \O_{\C^n,0}
\end{align}
holds for $p \leq k-1$. For $p=k-1$ it implies that $\phi \in \mathfrak{a}$ and for $p=0$ it reduces to
\eqref{phipartials} with $j=0$. Assume that \eqref{induktion} holds for some $p < k-1$.
Let us differentiate \eqref{induktion} $p+1$ times with respect to $w$, and compare the result with
\eqref{phipartials} for $j=p+1$.
This gives
\begin{align*}
g_p \in \sum_{i=0}^p w^i \mathfrak{a}^{k-i-p-1} + (w).
\end{align*}
Now we substitute this back into \eqref{induktion}, and get
\begin{align*}
\phi \in \sum_{i=0}^{p+1}w^i \mathfrak{a}^{k-i} + (w^{p+2}).
\end{align*}
By induction \eqref{induktion} holds also for $p=k-1$.
This proves the theorem for $l=1$, and the same argument works for all $l$.
Moreover we get that $N=r+k-1$ works in Theorem~\ref{bs}. This is optimal as the
following example shows, i.e. $N=r+k-1$ is the Brian\c con-Skoda number for this particular
analytic space.
We get of course back the usual Brian\c con-Skoda number $N=r$ in the reduced case $k=1$.
\end{example}
\begin{example}\label{optimalt}
To show that the example above is optimal, we need to find $\phi_p$, for each $0\leq p \leq k-1$,
such that $\phi_p \notin \mathfrak{a}$
and $|\partial^j_w \phi_p| \leq |\mathfrak{a}|^{r +(k-1)-j}$ holds for $j \neq p$ and
$|\partial^p_w \phi_p| \leq |\mathfrak{a}|^{r +(k-1)-p-1}$. Take $n=2$ and $\mathfrak{a}=(z+w)$. Then $r=1$, since $n-1=m=1$.
A suitable choice is now $\phi_p = w^p z^{k-1-p}$. This does not belong to $\mathfrak{a}$, because if it did
we would have $w^p z^{k-1-p} = (z+w)(a_0(z) + \dots + a_{k-1}(z)w^{k-1})$, which would give
$a_0=\dots=a_{p-1}=0$ and $za_p = z^{k-1-p}$, $za_{p+1}=-a_p$, $za_{p+2}=-a_{p+1}$, etc, so
$a_{k-1}=\pm 1/z$. This is a contradiction since $a_{k-1}$ is holomorphic at $0\in \mathbb{C}^n$.
\end{example}
In Section \ref{smooth} we will apply the proof of the main theorem, with a minor modification,
to the case where $Z_{red}$ is smooth. We then
recover the optimal result as in Examples \ref{bevisatexempel} and \ref{optimalt} when $\mathcal{J} = (w^k)$.
Although the formulation of Theorem \ref{bs} is intrinsic, we will choose an embedding and
work in the ambient space exclusively. If
$\phi\in\O_{\C^n,0}$ annihilates a certain vector-valued residue current, $R^{a^l}\wedge R^Z$, then it turns out that it
belongs to $\mathfrak{a}^l+\mathcal{J}$ (i.e. the image of $\phi$ in $\O_{Z,x}$ belongs to $\mathfrak{a}^l$);
this follows by solving a certain sequence of $\overline{\partial}$-equations. Alternatively, one
can also use a division formula to obtain an explicit integral representation of the membership.
This way of proving ideal membership is described in \cite{ass:bsvar} and goes back to \cite{andersson:residueideals}
and \cite{andersson:strong}.
We conclude that the proof is reduced to showing that
$\phi R^{a^l}\wedge R^Z = 0$ whenever $\phi$ satisfies \eqref{premise}. The vanishing of this
current is proved in Section \ref{proof}.
As a preparation, we begin to discuss Coleff-Herrera currents and the properties of $R^{a^l}$, $R^Z$ and their
product $R^{a^l}\wedge R^Z$.
\section{Coleff-Herrera currents}\label{coleffherrera}
Assume that $Z$ is a germ of an analytic variety of pure codimension $p$ in $\mathbb{C}^n$.
\begin{definition}
A Coleff-Herrera current on $Z$ is a $\overline{\partial}$-closed current $\mu$ of bidegree $(0,p)$ with support on $Z$
that is annihilated by $\overline{\phi}$ if $\phi \in \O_{\C^n,0}$ vanishes on Z. One also requires that $\mu$
satisfies the standard extension property ($\text{SEP}$), that is,
$\mu = \lim_{\varepsilon \to 0} \chi(|h|^2/\varepsilon^2)\mu$,
if $h \in \O_{\C^n,0}$ does not vanish identically on any component of $Z$, and $\chi$ is
a smooth cut-off function approximating $1_{[1,\infty)}$.
\end{definition}
The set of all Coleff-Herrera currents on $Z$ is an $\mathcal{O}$-module which we denote by $\mathcal{CH}_Z$.
It is easy to see for any $\mu \in \mathcal{CH}_{Z,x}$, that $\ann \mu$ is a pure-dimensional ideal whose associated
primes correspond to components of $Z$. There is a direct way, due to J-E. Bj\"{o}rk \cite{bjork:abel},
to obtain a defining set $L_1 \dots L_\nu$ for the ideal $\mathcal{J}=\ann \mu$. Furthermore, the operators $L_j$ are related to $\mu$ by the formula \eqref{phimu}.
To obtain the formula \eqref{phimu}, we need to consider
\begin{proof}[Bj\"ork's proof]
Let $\zeta$ denote the first $n-p$ coordinates in $\mathbb{C}^n$ and $\eta$ the $p$ last.
It follows from the local parametrization theorem one can find holomorphic functions
$f_1 \dots f_p$ forming a complete intersection, such that $Z$ is a union of a number of
irreducible components of $V_f = \{f_1=\dots=f_p=0\}$.
Moreover,
\begin{align}
z &=\zeta \notag\\
w &=f(\zeta,\eta)\label{coordinates}
\end{align}
are local holomorphic coordinates outside the hypersurface $W$ defined by
\begin{align}\label{detdef}
h:=\det \frac{\partial f}{\partial \eta}.
\end{align}
By possibly rotating the coordinates $(\zeta,\eta)$, we can make sure $h$ will not vanish identically on
any component of $Z$. Since $\mu$ is a Coleff-Herrera current on the complete intersection $V_f$,
we get by Theorem 4.2 in \cite{andersson:ch},
\begin{align}\label{chproduct}
\mu = A\left[\overline{\partial} \frac 1{f_1^{1+M_1}} \wedge \dots \wedge \overline{\partial} \frac 1{f_p^{1+M_p}} \right]
\end{align}
for some integers $M_i$ and strongly holomorphic function $A$.
A basic fact is that for a $(n-p,n-p)$ test form $\xi$,
\begin{align*}
d\eta\wedge\left[\overline{\partial} \frac 1{\eta_1^{1+M_1}} \wedge \dots \wedge \overline{\partial} \frac 1{\eta_p^{1+M_p}}\right].\xi
= \int_{\eta=0} \partial_{\eta_1}^{M_1} \dots \partial_{\eta_p}^{M_p} \xi,
\end{align*}
where the derivative symbols refer to Lie derivatives.
Now let $d\eta \wedge \xi$ be an arbitrary $(n,n-p)$ test form supported outside
$W$ so that $(z,w)$ are coordinates on its support.
Then, since $d\eta \wedge \xi = \frac 1h dw \wedge \xi$, the Leibniz rule gives that
\begin{align}\label{ch}
\mu.d\eta \wedge \xi = \int_{w=0} \sum_{\alpha_j \leq M_j}
\frac {c_\alpha}{h} \partial_w^{M-\alpha}(A) \partial_w^{\alpha}\xi,
\end{align}
where $M=(M_1,\dots,M_p)$.
We now want to express $\phi\mu.d\eta\wedge\xi$ in terms of derivatives with respect to the variables
$\eta_i$ instead of $w_i$. By the chain rule,
\begin{align*}
\frac{\partial}{\partial w_j} = \frac 1h \sum_k \gamma_{jk}
\frac{\partial}{\partial \eta_k}.
\end{align*}
Combining this with \eqref{ch}, we thus get operators $Q_\alpha$ so that
\begin{align}\label{phimu}
\phi\mu.d\eta\wedge\xi &= \int_{w=0} \frac 1h \sum_{\alpha_j \leq M_j}
Q_\alpha (\phi) \partial_w^\alpha (\xi) = \notag\\
&= \int_Z \sum_{\alpha_j \leq M}\frac 1{h^{N_0}} L_\alpha (\phi) K_\alpha (\xi),
\end{align}
where $N_0$ is some (large) integer and $L_\alpha = h^{N_0} Q_\alpha$ and
$K_\alpha = h^{N_0} \partial^\alpha_w$ are differential operators
with respect to the variables $\eta$ that are holomorphic across $W$.
Clearly, the values of $\partial_w^\alpha \xi$ can be
prescribed on $\{w=0\}$. Therefore $\phi \mu = 0$ on $Z \setminus W$
if and only if $L_\alpha (\phi) = 0$ on $Z \setminus W$ for all $\alpha\leq M$, but by continuity and $\text{SEP}$,
these relations hold if and only if they hold across $W$.
\end{proof}
Let $\mathcal{J}\subset \O_{\C^n,0}$, assume that $\O_{\C^n,0}/\mathcal{J}$ is Cohen-Macaulay, and let $Z=Z(\mathcal{J})$.
It is well-known, that there is a finite set $\mu_1, \dots \mu_{\nu_0} \in \mathcal{CH}_Z$
such that $\mathcal{J} = \bigcap \ann \mu_j$.
Hence we get a defining set for $\mathcal{J}$ by taking the union of the defining sets for $\ann \mu_j$.
In the next section, we will use the construction in \cite{anderssonwulcan:prescribed} of a set $\{\mu_j\}$ associated
to $\mathcal{J}$ as above. It is proved in \cite{andersson:chduality} that this set actually generates the $\mathcal{O}$-module
of all $\mu\in\mathcal{CH}_Z$ such that $\mathcal{J} \mu = 0$.
Finally, we extend, for later convenience, the formula \eqref{phimu} across $W$,
i.e. so that the support of $\xi$ may intersect $W$. To simplify the argument we assume, without loss of generality,
that $\phi = 1$. Then \eqref{phimu} agrees with
\begin{align}\label{utanphi}
\mu.d\eta\wedge\xi = \lim_{\varepsilon \to 0}\int_Z \chi({|h|}^2/\varepsilon^2)
\frac{\mathcal{L}(\xi)}{h^{N_0}},
\end{align}
for some holomorphic differential operator $\mathcal{L}$, since the cut-off function makes no difference when $\xi$ has support in $Z \setminus W$ and $\varepsilon$ is sufficiently small.
We proceed to show that \eqref{utanphi} continues to hold for general $\xi$.
The right hand side of \eqref{utanphi} defines a current, say $\tau$, which must equal $\mu$ if it too has $\text{SEP}$
with respect to $h$, so what we need to show is
\begin{align*}
\lim_{\delta\to 0}\chi({|h|}^2/\delta^2)\tau - \tau = 0.
\end{align*}
By expanding
\begin{align}\label{extending}
\chi({|h|}^2/\delta^2)\tau.\xi=\int_Z \frac{\mathcal{L}(\chi({|h|}^2/\delta^2) \xi)}{h^{N_0}}
\end{align}
we get one term when all derivatives of $\mathcal{L}$ hit $\xi$,
and clearly this term is precisely $\tau.\xi$ in the limit. We will now explain why all other contributions vanish;
all these terms contain derivatives of $\chi({|h|}^2/\delta^2)$ as a factor, and
such a factor can be written as a sum of terms
\begin{align*}
\chi^{(k)} \left(\frac{{|h|}^2}{\delta^2}\right) \cdot \left(\frac{{|h|}^2}{\delta^2}\right)^k
\frac{\sigma}{h^\kappa},
\end{align*}
for some integers $k$ and $\kappa$ and a smooth function $\sigma$.
Let us define $\chi_0 (x) := x^k \chi^{(k)}(x)$. This function is identically zero in some neighbourhoods of
$0$ and infinity. After applying the following lemma to $\chi_0$, we are done, since $\chi_0(\infty)=0$.
\begin{lemma}
If $\tilde{\chi}$ is any bounded function on $[0,\infty)$ that vanishes identically near $0$ and
has a limit at $\infty$, then
\begin{align}\label{chiinf}
\lim_{\delta \to 0} \tilde{\chi}({|h|}^2/\delta^2) [\frac 1f]\wedge[Z] =
\tilde{\chi}(\infty)[\frac 1f]\wedge[Z]
\end{align}
for any $h,f \in \O_{\C^n,0}$ that are generically non-vanishing on $Z$.
\end{lemma}
For a proof see \cite{bjorksamuelsson:reg}, Lemma 2.
\section{Residue currents associated to ideals}\label{currents}
A summary of the machinery of residue currents needed to prove the Brian\c con-Skoda theorem on a singular variety,
or in the present setting, appears in \cite{ass:bsvar}, Section 2. For the convenience of the reader,
we give here an even shorter summary.
In \cite{anderssonwulcan:prescribed} a method was presented for constructing currents $R$ and $U$
associated to any generically exact complex of hermitian vector bundles $(E_k)$ with maps $(f_k)$.
These currents take values in the vector bundle $\End_E := \End (\bigoplus E_k)$.
We define $f=\oplus f_j$ and $\nabla_f = f - \overline{\partial}$. The latter is an operator acting on $\End_E$-valued forms and currents,
and $U$ is related to $R$ by $\nabla_f U = 1 - R$. Since $\nabla_f^2 = 0$, it therefore follows that
that $\nabla_f R = 0$.
We restrict our attention to the case $\rank E_0 = 1$ and define the ideal $\mathcal{J} = Im(\mathcal{O}(E_1) \to \mathcal{O}(E_0))$.
Then $\ann R \subset \mathcal{J}$, but in general the inclusion is proper.
However, a main result of \cite{anderssonwulcan:prescribed}
is that if $\mathcal{O}(E_k)$ with maps $(f_k)$ is a resolution of $\mathcal{O}_{\mathbb{C}^n}/\mathcal{J}$, that is,
an exact complex of sheaves, then $\ann R = \mathcal{J}$.
Now suppose that we start with any ideal $\mathcal{J} \subset\O_{\C^n,0}$
with the assumptions mentioned before, that is, $\mathcal{J}$ has pure codimension $p$ and $\O_{\C^n,0} / \mathcal{J}$ is Cohen-Macaulay.
We denote by $R^Z$ the current obtained from a resolution so that $\ann R^Z = \mathcal{J}$. We have a
decomposition $R^Z = \sum_{k=0}^N R^Z_k$ so that $R^Z_k$ has bidegree $(0,k)$. The class of so called
\textit{pseudomeromorphic} currents $\mathcal{PM}$ was introduced in \cite{anderssonwulcan:decomp}.
Since $R^Z_k\in \mathcal{PM}$ has support on $Z=Z(\mathcal{J})$, which has codimension $p$, $R^Z_k$ must vanish for $k < p$ by
Corollary 2.4 in \cite{anderssonwulcan:decomp}, so $R^Z=R^Z_p + R^Z_{p+1} + \dots$.
The same corollary implies that $R^Z$ has $\text{SEP}$. The Cohen-Macaulay condition means that there is a resolution of
length $p$ of $\mathcal{O}_{\mathbb{C}^n}/\mathcal{J}$, so $R^Z_k$ vanishes also for $k > p$, and thus $R^Z=R^Z_p$.
Now $\nabla_f R^Z=0$ implies that $\overline{\partial} R^Z=0$ by a consideration of bidegrees.
By Proposition 2.3 in \cite{anderssonwulcan:decomp}, $\overline{I(Z)} = \overline {\sqrt{\mathcal{J}}}$ annihilates $R^Z$.
Collecting all information so far, we get that $R^Z$ is a tuple of Coleff-Herrera currents, since its components satisfy the
three required conditions. Thus $\mathcal{J} = \ann R^Z = \bigcap \ann \mu_j$, where $\mu_j$ are the
components of $R^Z$, so by the proof of Bj\"ork's theorem in Section \ref{coleffherrera}, we have a defining set
of operators for $\mathcal{J}$, such that the current $R^Z$ can be represented as a prinicipal value integral on $Z$ in terms
of these defining operators, \eqref{phimu}.
Next we construct currents $R^{a^l}$ and $U^{a^l}$ associated to a complex that is the Koszul complex for $l=1$
and a slight modification otherwise. The current $R^{a^l}$ is of Bochner-Martinelli type and
$\ann R^{a^l} \subset \mathfrak{a}^l$,
but in general we do not have equality. The restriction $u^{a^l}$ of $U^{a^l}$ to the complement of $Z(\mathfrak{a})$ is displayed in \eqref{Ual}, and $R^{a^l}$
is obtained as the limit of $R_\delta^{a^l} = \left(1 - \chi^a_\delta\right) + \overline{\partial} \chi^a_\delta \wedge u^{a^l}$,
where $\chi^a_\delta=\chi(|\mathfrak{a}|^2/\delta)$. We want to form the product of the currents $R$ and $R^{a^l}$, which
corresponds to restricting $R^{a^l}$ to $Z$ in a certain way. We thus define
\begin{align*}
R^{a^l} \wedge R^Z = \lim_{\delta \to 0} R_\delta^{a^l} \wedge R^Z.
\end{align*}
This product takes values in the product of the two complexes of the two factors, see \cite{ass:bsvar}.
It follows by Proposition 2.2 in \cite{ass:bsvar} that $\phi R^{a^l} \wedge R^Z = 0$ for $\phi \in \O_{\C^n,0}$
implies that $\phi \in \mathcal{J} + \mathfrak{a}^l$, that is, the image of $\phi$ in $\O_{Z,x}$ belongs to $\mathfrak{a}^l$.
Although $R^{a^l} \wedge R^Z$ is a current in $\mathbb{C}^n$, $\phi R^{a^l} \wedge R^Z$ depends only on the image of $\phi$.
The proof of Theorem \ref{bs} is reduced to
showing that indeed $\phi R^{a^l} \wedge R^Z = 0$ if $\phi$ satisfies \eqref{premise}, which we now begin to prove.
\section{Proof of Theorem 1.2}\label{proof}
Since $R^Z$ is, under our assumptions, a tuple of Coleff-Herrera currents, we can use \eqref{phimu}
with $\xi=\omega\wedge R^{a^l}_\delta$ to calculate $\phi R^{a^l}_\delta\wedge\mu.\omega$, where $\mu$ is a component
of $R^Z$ and $\omega$ is a test form.
We now choose a resolution $X \overset{\pi}{\to} Z$ such that $X$ is smooth and $\pi^* h$ is locally a monomial.
Thus to show that $\phi R^{a^l} \wedge R^Z=0$, it suffices by linearity to show that
\begin{align}\label{suffices}
I_\delta := \int_X \frac{\sigma}{s_1^{1+n_1}\cdot\dots\cdot s_{n-p}^{1+n_{n-p}}}\wedge
\pi^*{\left(L_i \phi\right)}\pi^*\left(\partial^\alpha_{\eta}R^{a^l}_\delta\right) \to 0,
\end{align}
where $\sigma$ is a smooth form with compact support.
This integral is really a principal value integral with a smooth cut-off function as in \eqref{utanphi}.
Recall that $|\alpha| \leq |M|=\sum M_i$, see \eqref{chproduct}.
We want to integrate \eqref{suffices} by parts.
The reciprocal of the monomial occuring in this integral is just a tensor product of one variable
distributions and
\begin{align*}
\frac{\partial}{\partial z}\left[\frac{1}{z^m}\right] = -m \left[\frac{1}{z^{m+1}}\right], \quad m \geq 1.
\end{align*}
This yields indeed that
\begin{align}\label{ideltaparts}
I_\delta = \int_X \frac{ds_1 \wedge\dots\wedge ds_{n-p}}{s_1\cdot\dots\cdot s_{n-p}}\wedge
\partial_s^{(n_1,\dots,n_{n-p})}\left(\sigma\wedge\pi^*{\left(L_i\phi\right)}\pi^*\left(\partial^\alpha_{\eta}
R^{a^l}_\delta\right)\right).
\end{align}
We now extend $\pi$ to a resolution $X' \overset{\gamma}{\to} X \overset{\pi}{\to} Z$
that principalizes $\mathfrak{a}$, and we call the generator in $\mathcal{O}_{X'}$ for $a_0$, i.e.
$\gamma^*\pi^* a_j = a_0 a'_j$ and $|\gamma^*\pi^* \mathfrak{a}| \sim |a_0|$. Note that the form
$\frac{ds_1\wedge\dots\wedge ds_{n-p}}{s_1\cdot\dots\cdot s_{n-p}}$
becomes a sum of similar forms when we pull it back to $X'$. If we show, for all integers $k_j \leq n_j$, $1\leq j \leq n-p$,
that the form
\begin{align}\label{remains}
\gamma^* \left[\partial_s^{(k_1,\dots,k_{n-p})}\left(\pi^*{\left(L_i\phi\right)}\pi^*\left(\partial^\alpha_{\eta}
R^{a^l}_\delta\right)\right)\right]
\end{align}
is bounded on $X'$ by a constant independent of $\delta$,
then an application of dominated convergence gives that $I_\delta \to 0$,
and thereby concludes the proof. We assume that $k_j=n_j$ for $1\leq j \leq n-p$. It will become apparent that this is
the worst case.
The philosophy will be to write the expression in \eqref{remains} as a product of
factors of three types; factors whose modulii are equivalent to $|a_0|$, or to $|a_0|^{-1}$, and remaining factors, which
we require to be bounded. We count the number of factors of the first type
minus the number of factors of the second type, that is, the number of zeroes minus the number of singularities.
Let us call this difference \textit{homogeneity}. Since the factors of the third type may also be of the
first type, the homogeneity depends on our factorization. However, we only want to find one factorization such that
the homogeneity is non-negative, because then \eqref{remains} is indeed bounded. Nothing will be lost if
we instead get a finite number of terms which can be factorized in this way.
We examine first the factor $L_i\phi$. Then we consider the second factor, $\partial^\alpha_{\eta}R^{a^l}_\delta$,
and add the two results.
Since $X$ is a manifold, the usual Brian\c con-Skoda theorem and \eqref{premise} gives that
$\pi^* \left(L_i \phi\right) \in {(\pi^* \mathfrak{a})}^{N-\varrho+l}$, where $\varrho=\min(m,\dim Z)$.
The conclusion is that the term in \eqref{remains} for which no derivatives with respect to $s$ hit
$\pi^*{\left(L_i\phi\right)}$ gives a contribution of $N-\varrho+l$ to the homogeneity. A term for which $\pi^*{\left(L_i\phi\right)}$ is hit by $k$ derivatives gives a contribution reduced by $k$, since $\partial \mathfrak{a}^{M+1} \subset \mathfrak{a}^M$ for any $M$.
We proceed now with the second factor. For simplicity we begin with the case $(n_1,\dots,n_{n-p}) = 0$, so that no derivatives appear with respect to the coordinates $s$. We will then consider the general case and see
that the derivatives $\partial_s$ have the same effect on homogeniety as $\partial^\alpha_{\eta}$ -- namely each
derivative decreases the homogeneity by one.
From \cite{andersson:bsexplicit} we know that
\begin{align}\label{Ual}
u^{a^l}=\sum_{\substack{1 \leq k \leq \varrho\\\beta_1 + \dots + \beta_p=k-1}}\left(
\frac{\bigwedge_{i=1}^l (\sum_{j=1}^m \bar{a}_j e^i_j) \wedge \bigwedge_{i=1}^l
(\sum_{j=1}^m \overline{\partial} \bar{a}_j\wedge e^i_j)^{\beta_i}}{|\mathfrak{a}|^{2(k+l-1)}}\right),
\end{align}
where $e^i_*$ are frames of trivial bundles, and $a_j$, $1\leq j \leq m$, are generators for $\mathfrak{a}$.
For simplicity we consider only the most singular term
of $R^{a^l}_\delta = \left( 1- \chi^a_\delta \right) - \overline{\partial}\chi^a_\delta\wedge u^{a^l}$, that is,
the term of $\overline{\partial}\chi^a_\delta\wedge u^{a^l}$ for which $k=\varrho$ above.
All other terms can be treated similarily, but easier.
We denote the most singular term by $\nu_\delta$. Using the notation from Example \ref{bevisatexempel}, we get
\begin{align}\label{nudelta}
\nu_\delta=\chi'\left(\frac{|\mathfrak{a}|^2}{\delta^2}\right)
\left(\frac{\mathfrak{a}\overline{\partial}\overline{\mathfrak{a}}}{\delta^2}\right)
\wedge \sum_{|\beta|=\varrho-1}
\frac{\theta_\beta (a)}{|\mathfrak{a}|^{2(\varrho+l-1)}},
\end{align}
where
\begin{align*}
\theta_\beta (a) = \bigwedge_{i=1}^l (\sum_{j=1}^m \bar{a}_j e^i_j) \wedge \bigwedge_{i=1}^l
(\sum_{j=1}^m \overline{\partial} \bar{a}_j\wedge e^i_j)^{\beta_i}.
\end{align*}
\begin{lemma}\label{thetalemma}
The homogeneity of $\gamma^*\pi^*\theta_\beta(a)$ is $\varrho+l-1$.
\begin{proof}
Recall that $\gamma^*\pi^* a_j = a_0 a'_j$.
All terms for which $\overline{\partial}$ hits $\overline{a_0}$ vanish since they contain the wedge product of
$(\sum_{j=1}^m \overline{a'_j} e^i_j)$ with itself, that is, the square of a $1$-form. Thus all
remaining terms are divisible by $a_0^{\rho + l -1}$.
\end{proof}
\end{lemma}
By differentiating \eqref{nudelta}, one sees that $\partial^\alpha_\eta \nu_\delta$ is a sum of terms like
\begin{align}\label{sumofterms}
\sigma \chi^{(1+|\alpha_1|)}\left(\frac{|\mathfrak{a}|^2}{\delta^2}\right)
\left(\frac{\mathfrak{a}^{1-|\epsilon|+|\alpha_1|}}{\delta^{2|\alpha_1|+2}}\right)
\wedge \sum_{|\beta|=\varrho-1}
\frac{\overline{\mathfrak{a}}^{|\alpha_2|}\theta_\beta (a)}{|\mathfrak{a}|^{2(\varrho+l-1)+2|\alpha_2|}},
\end{align}
where $\sigma$ is a smooth function, and $\epsilon+\alpha_1+\alpha_2$ are multi-indices such that $\epsilon+\alpha_1+\alpha_2 \leq \alpha$.
It is readily seen that the homogeneity of \eqref{sumofterms} only depends on $|\epsilon|+|\alpha_1|+|\alpha_2|$
(for $l$ and $\rho$ fixed), except if $1 - |\epsilon| + |\alpha_1|$ is negative, but this only improves the estimate.
For the worst term it is therefore $-l-\varrho - |\alpha|$, due to Lemma \ref{thetalemma}.
In other words, no matter which terms the derivatives hit, each one of them reduce the homogeneity by one (or zero).
It remains to show that the homogeneity
of $\gamma^* \partial^{\alpha_0}_s \pi^* \nu_{\delta}$ drops by at most $|\alpha_0|$, for any $\alpha_0$.
This is however quite immediate;
we apply $\pi^*$ to \eqref{sumofterms} and then differentiate with respect to $s$. The ideal $\mathfrak{a}$ is then replaced
by $\pi^* \mathfrak{a}$ and since $\pi^* \theta_\beta (a)$ is anti-holomorphic, the differentiation follows exactly the same
pattern as when we differentiated with respect to $\eta$ -- the derivatives
can hit $\chi^{(k)}$, or a power of $\pi^* \mathfrak{a}$ or $|\pi^* \mathfrak{a}|^2$, or a smooth function, so we already know that each derivative decreases the homogeneity by no more than one.
The total homogeneity of \eqref{remains} is the sum of $N-\varrho+l$ and $-l-\varrho - |\alpha|$ less the number of derivatives with respect both to $s$ and to $\eta$, that is $N - 2\varrho - |\alpha| - \sum_1^{n-p} n_j$. This is non-negative if $N\geq 2\min(m,\dim Z) + |\alpha| + \sum_1^{n-p} n_j$, so any such integer $N$ has the desired property of Theorem \ref{bs}, where $|\alpha|$ is
maximized over the components of $R^Z$ and $\sum_1^{n-p} n_j$ is maximized over an open covering of $\pi^{-1}(0)$ (recall that these numbers are constructed locally in the resolution).
\section{An improvement for smooth analytic spaces}\label{smooth}
We return now to the case where $Z_{red}$ is smooth. We then get a bound for $N$ and the proof also
simplifies significantly. Recall from Section \ref{currents} that $R^Z=(\mu_1,\dots,\mu_\nu)$ is tuple of Coleff-Herrera
currents, and each of its components therefore
gives rise to holomorphic differential operators $L_\alpha$, and all of these together form a defining set for $\mathcal{J}$ and satisfy \eqref{phimu}.
Note however that, since $Z_{red}$ is smooth,
we can take $w=\eta$ in \eqref{coordinates} so that $h=1$ in \eqref{detdef}. For each differential operator $L$ we denote by $\order(L)$ its order as a differential operator in the ambient space. This gives also a well-defined notion of order for the induced intrinsic mapping in $\mathcal{N}(Z)$.
\begin{theorem}\label{bssmooth}
Let $Z$ be a germ of an analytic space such that $Z_{red}$ is pure-dimensional and smooth
(which implies that $Z$ is Cohen-Macaulay). Let $L_1 \dots L_M$ be the Noetherian operators obtained from
$R$ as before and let $d$ be the maximal order of these operators.
Then if $l\geq 1$, and $\mathfrak{a} \subset \O_{Z,x}$ can be generated by $m$ elements,
\begin{align} \label{premisesmooth}
|L_j \phi|\leq C |\mathfrak{a}|^{\min(m,\dim Z)+d-\order(L_j)+l-1}, \quad 1 \leq j \leq M
\end{align}
implies that $\phi\in\mathfrak{a}^l$.
\end{theorem}
Note that we have already seen \eqref{premisesmooth} in a slightly different appearance in \eqref{smpremise}.
\begin{proof}
We can assume that $Z_{red} = \mathbb{C}^{n-p} \subset \mathbb{C}^n$, and we call the last $p$ coordinates
$w_1, \dots, w_p$. Clearly these functions form a complete intersection, so if $\mu$ is
a component of $R^Z$, we have
\begin{align*}
\mu = A\left[\overline{\partial} \frac 1{w_1^{1+M_1}} \wedge\dots\wedge
\overline{\partial} \frac 1{w_p^{1+M_p}} \right],
\end{align*}
for some holomorphic function $A$, cf. \eqref{chproduct}. We choose $A$ so that
the numbers $M_i$ are minimal. Then the distribution order of $\mu$ is precisely $\sum M_i$,
which, as we soon shall see, is also the highest order of the operators obtained from $\mu$.
By Example 1 in \cite{andersson:chduality},
the components of $R^Z$ generate the $\mathcal{O}$-module of Coleff-Herrera
currents on $Z$ which are annihilated by $\mathcal{J}$. Hence, $d$ is an invariant of $Z$.
We proceed as in the general case to construct
a set of defining Noetherian operators for the ideal $\mathcal{J}$ corresponding to $Z$. Since
\begin{align*}
\mu.dw \wedge \xi = \int_{w=0} \sum_{\alpha \leq M}
\partial_w^{M-\alpha}(A) \partial_w^{\alpha}\xi,
\end{align*}
where $M=(M_1,\dots,M_p)$, we get that
\begin{align}\label{smoothphimu}
\phi\mu.dw\wedge\xi &= \int_{w=0} \sum_{\alpha \leq M}
L_\alpha (\phi) \partial_w^\alpha \xi,
\end{align}
and
\begin{align*}
L_\alpha = \sum_{\alpha\leq\gamma\leq M}C_{(\alpha,\gamma)}
\partial_w^{M-\gamma}(A) \partial_w^{\gamma-\alpha},
\end{align*}
for some combinatorial constants $C_{(\alpha,\gamma)}$.
Clearly $\order(L_\alpha) = |M|-|\alpha|$ and $L_0$ has the maximal order $\order(L_0) = |M|$.
As before, we need to substitue $\xi=\omega\wedge R^{a^l}_\delta$ into \eqref{smoothphimu}.
By principalizing $\mathfrak{a}$ as before, it follows that the limit of \eqref{smoothphimu} is zero when $\delta$ goes to zero, if the integrand can be factored in to two parts,
one that is integrable, and one that is bounded. To see that the second factor is bounded
we count its homogeneity as before.
In the proof of the general case we used that
\begin{align}\label{dbarchi}
\overline{\partial} \chi\left(\frac{|\mathfrak{a}|^2}{\delta^2}\right) =
\chi'\left(\frac{|\mathfrak{a}|^2}{\delta^2}\right)
\left(\frac{\mathfrak{a}\overline{\partial}\overline{\mathfrak{a}}}{\delta^2}\right),
\end{align}
which was counted as a term of homogeneity $-1$. It is however well-known that $\partial a_0/|a_0|$
is integrable, so taking it as the first factor, it no longer counts into the homogeneity.
Thus, the homogeneity contribution of \eqref{dbarchi} is now improved to $0$.
The most singular term of $\partial_w^\alpha (R^{a^l}_\delta)$ has thus homogeneity $1-\varrho-l-|\alpha|$
(one unit more than in the non-smooth case). Since
$1-\varrho-l-|\alpha|\geq 1-\varrho-l-d+\order(L_\alpha)$,
the homogeneity is non-negative if $|L_\alpha(\phi)|\leq C|\mathfrak{a}|^{\varrho+d-\order(L_\alpha)+l-1}$,
so we are done.
\end{proof}
\begin{remark}
In the non-smooth case the form in \eqref{dbarchi} has to be included into the bounded factor, since in
that case we already had the form $ds/s$ occupying the integrable factor.
Note also that we avoided using Hironaka desingularization and applying the classical Brian\c con-Skoda
theorem in the resolution, which previously cost us $\varrho -1$ units of homogeneity.
\end{remark}
|
1,116,691,499,298 | arxiv | \section{Introduction}
Large scale hypertexts, such as the World Wide Web, are well-known examples of scale-free networks in the real-world \cite{pietro}
and the Barab\'asi--Albert model \cite{ba} is certainly one of their most investigated descriptors.
When the interest focuses on the popularity of a webpage, its connectivity (i.e.\ the number of its in-links)
is usually considered. On the other hand, the knowledge of in- and out-links becomes necessary to deal with the topology of the network.
Both approaches exist in the literature for the so called \textit{preferential attachment model} \cite{ba}, in which
new webpages are constantly added to the network
with a given probability and the number of in-links of each webpage grows proportionally to the number of in-links already possessed
by that webpage. It turns out that with these assumptions, the limiting distribution of the number
of in-links for a webpage chosen at random exhibits a power-law tail (see \cite{ba} for a more in-depth explanation).
It is pointed out in \cite{born} that the Barab\'asi--Albert model
for in-links
can be classified as a specific case of the so-called
Yule--Simon model \cite{simon} which in turn is related to the well-known Yule model for macroevolution \cite{yule}
(do not confuse it with the \emph{Yule process} which is a homogeneous linear birth process).
Specifically, Yule--Simon and Yule models are asymptotically equivalent \cite{willis}.
The Yule model appears recurrently with different names in relation to different applications \cite{willis}.
Despite the long history of the Yule model, it maintains a relevant role in a variety of applications
\cite{doro}; variants of it have been studied in the past \cite{doro2,new} and are part of very recent research \cite{pnas,plos}.
The aim of this paper is to study formally the consequences of detachment of in-links.
This is accomplished by considering the Yule model and replacing the linear birth process governing
the growth of in-links with a linear birth-death process (following \cite{reed}, where the model was first introduced
within the context of macroevolution).
It turns out that the introduction of the possibility of detachment of in-links in the Yule model still leads to an
analytically tractable model and thus still permits to obtain exact results.
In Section \ref{inaina} we determine the analytical form of the distribution of the number of in-links to a webpage.
Our results include the probabilities
of any finite number of in-links not being limited to the asymptotic
limits, as happens in the literature.
In fact, the analysis of the role of the detachment becomes possible for any range of the number of in-links.
The availability of the in-links distribution allows us the comparison of its features with those of the World Wide Web in 2012
(Section \ref{ddataa}).
Further, the moments are calculated explicitly and their expressions are
presented in Section \ref{mome}.
In Section \ref{fuego} we compare the generalized model and the classical Yule model.
\section{Classical model}
The construction based on Yule model \cite{yule} proceeds as follows.
Consider an initial webpage present at time $t=0$ with a single in-link.
The number of in-links increases proportionally to the number of existing in-links with
an in-link rate constant $\lambda>0$.
The number of webpages develops independently as a linear birth process
of webpage rate constant $\beta>0$.
\begin{figure}
\centering
\includegraphics[scale=.6]{p0.pdf}
\caption{\label{figure1}An example of a realization of the generalized Yule model with
$\beta=0.1$, $\lambda=1.1$, $\mu=1$ (supercritical case). The evolution of the number of in-links in different webpages is
highlighted with different colors. The instants when new webpages are introduced are indicated with colored dots
while the moments when webpages disappear due to the removal of the last in-link
are denoted by colored squares.}
\end{figure}
Let $N_\beta(t)$ and $N_\lambda(t)$ be the counting processes reflecting the number of webpages and of in-links of a given webpage,
respectively, observed at time $t$.
The assumption that the process $N_\beta(t)$ starts
at time zero with one initial webpage simplifies the notation but
has no effects on the asymptotic properties of the results.
Recall that the probability distribution $\mathbb{P} ( N_\lambda(t) = n )$, $n \ge 1$, of
a linear birth process is geometric \cite{bailey},
\begin{align}
\label{bai}
\mathbb{P} ( N_\lambda(t) = n ) = e^{-\lambda t} (1-e^{-\lambda t})^{n-1}, \quad t \ge 0.
\end{align}
In the above formula, as removal of in-links is not permitted, $n$ cannot reach zero.
Referring to
the results obtained in \cite{neuts,crump,feigin,puri}, by conditioning
on the number of webpages present at time $t$, we obtain that the
random instants at which appearance of new webpages occurs are distributed as the order statistics
of independent and identically distributed random variables $T$ with probability distribution function
\begin{align}
\label{distr}
\mathbb{P} ( T \le \tau ) = \frac{e^{\beta \tau}-1}{e^{\beta t}-1}, \qquad 0 \le \tau \le t.
\end{align}
Let $\mathcal{N}^Y_t$ denote the size of a webpage chosen uniformly at random at time $t$ and call $\mathcal{N}^Y
= \lim_{t \to \infty}\mathcal{N}^Y_t$.
Plainly,
\begin{align*}
\mathbb{P} (\mathcal{N}^Y_t = n ) = \mathbb{E}_T \mathbb{P} ( N_\lambda(t) = n | N_\lambda(T) = 1 ),
\end{align*}
meaning that the selected webpage was introduced at a random time $T$ with
distribution \eqref{distr}. Note that the age $t-T$ of the randomly selected webpage appeared at time $T$ is
distributed as a truncated exponential random variable.
Hence, using \eqref{bai} we have
\begin{align}
\label{lenovo}
\mathbb{P} (\mathcal{N}^Y_t = n )
= \frac{\beta}{1-e^{-\beta t}} \int_0^t e^{-\beta y} e^{-\lambda y} (1-e^{-\lambda y})^{n-1} \mathrm dy,
\end{align}
and in the limit for $t \to \infty$ we obtain
\begin{align}
\label{two}
\mathbb{P} ( \mathcal{N}^Y = n ) & = \int_0^\infty \beta e^{-\beta y} e^{-\lambda y}
(1-e^{-\lambda y})^{n-1} \mathrm dy.
\end{align}
Expression \eqref{two} is simple
to evaluate (see for example \cite{yule}),
\begin{align}
\label{yule}
\mathbb{P}(\mathcal{N}^Y=n) = \frac{\beta}{\lambda} \frac{\Gamma(n)\Gamma
\left( 1+ \frac{\beta}{\lambda} \right)}{\Gamma \left( n+1+\frac{\beta}{\lambda} \right)}.
\end{align}
Distribution \eqref{yule} is known as Yule--Simon distribution \cite{bala}.
\section{Generalized model: basic properties}
\label{inaina}
As stressed above, we are concerned with the limiting behavior of a stochastic process in which the
appearance (observation) of new webpages (each with only one in-link) is controlled by a linear birth process $N_\beta(t)$
of intensity $\beta>0$.
Now the evolution of the number of in-links for each webpage is governed by
independent (of one another and of $N_\beta(t)$) linear birth-death processes $N_{\lambda,\mu}(t)$ with birth and death intensities
$\lambda>0$ and $\mu>0$, respectively (see also the description of this model in
\cite{reed}). A schematic example of the model can be seen in Fig.\ \ref{figure1}.
Note how, with that particular choice of the parameters, many webpages may lose all of their links in a short time.
However, a few of them develop fast gaining a considerable number of in-links in a relatively short timespan.
To derive the asymptotic properties of the model,
we proceed by following the same lines leading to formulae \eqref{two} and \eqref{yule}. To compute the distribution
of the number of in-links $\mathcal{N}$ of a webpage chosen uniformly at random,
we first write the analogue of \eqref{lenovo} randomizing by the time $T$ of first
appearance of a webpage and then we let $t \to \infty$.
Since in this generalized model $\mathcal{N}$ can be zero with a positive probability,
in the following, we distinguish the case $\mathbb{P}(\mathcal{N}=0)$ from $\mathbb{P}(\mathcal{N}=n)$, $n \ge 1$.
For the sake of brevity we will omit $n \ge 1$ in formulae.
As we are dealing with birth-death processes for the evolution of the number of in-links,
we must separate the three cases $\lambda=\mu$ (critical), $\lambda<\mu$ (subcritical), and $\lambda>\mu$ (supercritical)
in which the processes behave rather differently (e.g.\ \cite{bailey}, Section 8.6).
\subsection{Critical regime}
Let us start with the case $\lambda=\mu$. Considering that for $n \ge 0$,
\begin{align}
\label{su}
\mathbb{P}(\mathcal{N}=n) = \int_0^\infty \beta e^{-\beta t} \mathbb{P} (N_{\lambda,\mu}(t) = n) \, \mathrm dt,
\end{align}
we have
\begin{align*}
\mathbb{P}(\mathcal{N}=0) = \int_0^\infty \beta e^{-\beta t} \frac{\lambda t}{1+\lambda t} \mathrm dt
\end{align*}
and
\begin{align*}
\mathbb{P}(\mathcal{N}=n)
= \int_0^\infty \beta e^{-\beta t} \frac{(\lambda t)^{n-1}}{(1+\lambda t)^{n+1}} \, \mathrm dt.
\end{align*}
Exploiting the integral representation of the confluent hypergeometric function (see \cite{abramowitz}, Section 13)
\begin{align}
\label{pdf}
U(a,b,z) = \frac{1}{\Gamma(a)} \int_0^\infty e^{-zy} y^{a-1} (1+y)^{b-a-1} \, \mathrm dy
\end{align}
and the transformation formula 13.1.29 of \cite{abramowitz},
the above expressions can be written as
\begin{align}
\mathbb{P}(\mathcal{N}=0) = U(1,0,\beta/\lambda),
\end{align}
and
\begin{align}
\label{ext}
\mathbb{P}(\mathcal{N}=n) =
(\beta/\lambda) \, \Gamma(n) \, U(n,0,\beta/\lambda).
\end{align}
Fig.\ \ref{figure2b}(b) shows the probability mass function of $\mathcal{N}$ in the specific critical case $\lambda=\mu=1/2$.
Notice the fast decay of the tail.
\begin{figure}
\centering
\subfigure[]{\includegraphics[scale=0.75]{p2.pdf}}
\subfigure[]{\includegraphics[scale=0.75]{p1.pdf}}
\subfigure[]{\includegraphics[scale=0.75]{p3.pdf}}
\caption{\label{figure2b}
Distribution of the number of in-links for different values of the webpage rate constant, $\beta=\{1,0.25,0.5,0.1\}$.
(a) Supercritical case, $\lambda=1/2$, $\mu=1/4$; the tails decay as power-laws (see the discussion in Section \ref{fuego}).
(b) Critical case, $\lambda=\mu=1/2$;
we can see that the tails decay faster than a power-law.
(c) Subcritical case, $\lambda=1/4$, $\mu=1/2$; note that
the webpage rate constant $\beta$ plays almost no role.}
\end{figure}
\subsection{Non-critical regimes}
Let us move now to the cases in which the linear birth-death process governing the appearance of in-links is supercritical or subcritical,
i.e.,\ $\lambda>\mu$ or $\lambda<\mu$. For notational purposes let us define the functions (with $x,y,b,\nu>0$)
\begin{align*}
& r_b(x,y) = c_b(x,y)
{}_2 F_1 \left( 1,\frac{b}{x-y}; 2+\frac{b}{x-y}; \frac{y}{x} \right), \\
& q_b^\nu(x,y) = d_b^\nu(x,y) {}_2 F_1 \left( \nu+1, 1+\frac{b}{x-y};
\nu+1+\frac{b}{x-y}; \frac{y}{x} \right),
\end{align*}
where ${}_2 F_1$ is the Gauss hypergeometric function and where
\begin{align*}
& c_b(x,y) = \frac{b}{x-y} \Gamma\left( \frac{b}{x-y} \right) \Bigr/ \Gamma
\left( 2+\frac{b}{x-y}\right), \\
& d_b^\nu(x,y) = \frac{b (x-y)}{x^2} \Gamma\left(1+\frac{b}{x-y}\right)\Gamma(\nu) \Bigr/
\Gamma\left(\nu+1+\frac{b}{x-y}\right).
\end{align*}
By exploiting the integral representation
\begin{align}
\label{mattarello}
{}_2 F_1 (a,b;c;z) = \frac{\Gamma(c)}{\Gamma(b) \Gamma(c-b)} \int_0^1 y^{b-1} \frac{(1-y)^{c-b-1}}{
(1-yz)^{a}} \, \mathrm dy,
\end{align}
(see \cite{abramowitz}, page 558) considering \eqref{su}
and after tedious calculations we obtain that in the supercritical case ($\lambda>\mu$)
\begin{align}
\mathbb{P}(\mathcal{N}=0) = (\mu / \lambda) r_\beta(\lambda,\mu),
\end{align}
and
\begin{align}
\label{dec}
\mathbb{P}(\mathcal{N}=n) =
q_\beta^n (\lambda,\mu),
\end{align}
while in the subcritical case ($\lambda<\mu$)
\begin{align}
\mathbb{P}(\mathcal{N}=0) = r_\beta(\mu,\lambda),
\end{align}
and
\begin{align}
\label{dec2}
\mathbb{P}(\mathcal{N}=n) =
(\lambda/\mu)^{n+1} q_\beta^n (\mu,\lambda).
\end{align}
The examples of the limiting distribution are depicted in Fig.\ \ref{figure2b}(a,c).
Notably, in the supercritical case a power-law behavior is present, while in the subcritical case the tail of the distribution
decays more rapidly. Moreover, if $\lambda < \mu$, the subcritical character of the process governing the arrival of in-links
dominates the dynamics of the whole system and the parameter $\beta$ plays an almost irrelevant role.
Fig. \ref{graph1} presents two examples of the $\mathbb{P}(\mathcal{N}=0)$,
as a function of the webpage rate constant $\beta$, in the three cases, supercritical, critical, and subcritical.
The insets highlight the behaviors
of the relative distances between the three curves with respect to different choices of the parameters $\lambda$ and $\mu$.
We retrieve the Yule--Simon distribution \eqref{yule} for fixed $\lambda>0$ and $\mu \to 0$ (classical Yule model)
from \eqref{dec} by recalling
that ${}_2 F_1 (a,b;c;0) = 1$. Clearly, in this case
$\mathbb{P}(\mathcal{N}=0) = 0$.
Manipulation of the probabilities is made simple by recalling that Gauss Hypergeometric functions admit a huge varieties of
different representations and relations (see e.g.\ \cite{MR2360010}, Sections 9.1, 9.2,
\cite{spanier}, Chapter 60, or \cite{MR2723248}, Chapters 13, 15). For example, when $\lambda>\mu$ and $n \ge 1$, by considering the representation
15.2.1 of \cite{MR2723248} we have that $\mathbb{P}(\mathcal{N}=n)$ can be written also as the absolutely convergent series
\begin{align}
\mathbb{P}(\mathcal{N}=n) = \frac{\beta(\lambda-\mu)}{n\lambda^2} \sum_{r=0}^\infty
\frac{\Gamma(n+1+r)\Gamma\left(1+r+\frac{\beta}{\lambda-\mu}\right)}{
r! \Gamma\left(n+1+\frac{\beta}{\lambda-\mu}+r\right)} \left(\frac{\mu}{\lambda}\right)^r,
\end{align}
or by means of generalized Wright functions ${}_2\Psi_1$ as
\begin{equation*}
\mathbb{P}(\mathcal{N}=n) = \frac{\beta(\lambda-\mu)}{n\lambda^2} {}_2\Psi_1
\left[ \left.
\begin{array}{l}
(n+1,1)\left(1+\frac{\beta}{\lambda-\mu},1\right) \\
\left( n+1+\frac{\beta}{\lambda-\mu},1 \right)
\end{array}
\right| \frac{\mu}{\lambda}
\right].
\end{equation*}
The representation of the probabilities by means of the power series expansion of the Gauss hypergeometric function provides a simple and efficient
means for its computation (see \cite{temme}, Section 2.3.1.
Similar representations hold for all the other cases $\lambda=\mu$ and $\lambda < \mu$.
\section{Generalized model: moments}
\label{mome}
To complete the analysis of the model we calculate the moments of $\mathcal{N}$.
In order to derive the expectation $\mathbb{E} \, \mathcal{N}$ it is
useful to realize that it can be written as
\begin{align*}
\mathbb{E} \, \mathcal{N} = \int_0^\infty \beta e^{-\beta t} \mathbb{E} N_{\lambda,\mu}(t) \, \mathrm dt.
\end{align*}
Therefore we immediately have
\begin{align}
\label{media}
\mathbb{E} \, \mathcal{N} & = \beta \int_0^\infty e^{-\beta t} e^{(\lambda-\mu)t} \mathrm dt
= \frac{\beta}{\beta - (\lambda-\mu)}.
\end{align}
The above integral converges for $\mu>\lambda$ and, if $\lambda > \mu$, for $\beta > (\lambda-\mu)$,
otherwise $\mathbb{E} \, \mathcal{N} = \infty$.
It holds $\mathbb{E} \, \mathcal{N} = 1$ for $\lambda=\mu$.
To derive the variance, we first determine the second moment.
By recalling that for the classical linear birth-death process
\begin{align*}
\mathbb{E}\left[ N_{\lambda,\mu}(t) \right]^2 = \frac{2\lambda}{\lambda-\mu} e^{2(\lambda-\mu)t}
- \frac{\lambda+\mu}{\lambda-\mu}
e^{(\lambda-\mu)t}, \qquad \lambda \ne \mu,
\end{align*}
we calculate the second moment of $\mathcal{N}$ as
\begin{align*}
\mathbb{E}\,\mathcal{N}^2 = \frac{2\lambda}{\lambda-\mu} \frac{\beta}{\beta-2(\lambda-\mu)}
-\frac{\lambda+\mu}{\lambda-\mu} \frac{\beta}{\beta-(\lambda-\mu)},
\end{align*}
and therefore
\begin{align}
\label{var}
\mathbb{V}\text{ar} \, \mathcal{N} = \frac{\frac{2 \lambda\beta}{\beta-2(\lambda-\mu)}
-\frac{(\lambda+\mu)\beta}{\beta-(\lambda-\mu)}}{\lambda-\mu} - \left( \frac{\beta}{\beta-(\lambda-\mu)} \right)^2.
\end{align}
The integral giving rise to the above expressions converges
if $\beta/(\lambda-\mu)>2$, otherwise $\mathbb{V}\text{ar}\, \mathcal{N}=\infty$.
For $\mu=\lambda$, as $\mathbb{E}\left[ N_{\lambda,\mu}(t) \right]^2 = 2\lambda t +1$,
\begin{align*}
\mathbb{E} \, \mathcal{N}^2 = 1+2 \lambda \beta \int_0^\infty e^{-\beta t} t \, \mathrm dt
= 1+2 (\lambda/\beta).
\end{align*}
It immediately follows that
\begin{align}
\mathbb{V}\text{ar} \, \mathcal{N} = 2(\lambda/\beta).
\end{align}
\begin{figure}
\centering
\subfigure[]{\includegraphics[scale=0.75]{g1.pdf}}
\subfigure[]{\includegraphics[scale=0.85]{g2.pdf}}
\caption{\label{graph1}Probability of webpage ``death'' $\mathbb{P}(\mathcal{N}=0)$, in dependency on webpage rate constant $\beta$
in critical ((a): $\lambda=\mu=5$; (b): $\lambda=\mu=50$), supercritical ((a): $\lambda=10$, $\mu=5$;
(b): $\lambda=50$, $\mu=25$), and subcritical ((a): $\lambda=5$, $\mu=10$; (b): $\lambda=50$, $\mu=100$) regimes.
In the insets, relative distance between supercritical and critical (solid line), and subcritical and critical
regimes (dashed line) with respect to the critical regime.}
\end{figure}
Higher-order moments can be derived from the probability generating function
$\mathcal{G}(u) = \sum_{n=0}^\infty u^n \mathbb{P}(\mathcal{N}=n)$. This is determined
by similar reasoning as before but this time a different special function is involved.
In particular the so-called Appell hypergeometric function (see \cite{erdelyi}, Chapter 5) will be used,
\begin{align}
\label{appell}
& F_1(a;b_1,b_2;c;z_1,z_2) \\ & = \frac{\Gamma(c)}{\Gamma(a)\Gamma(c-a)} \int_0^1 \frac{y^{a-1} (1-y)^{c-a-1}}{
(1-y z_1)^{b_1} (1-yz_2)^{b_2}} \, \mathrm dy. \notag
\end{align}
Let $G(u,t)$ be the probability generating function of the linear birth-death process $N_{\lambda,\mu}(t)$.
The probability generating function of $\mathcal{N}$ can be determined by using
$\mathcal{G}(u)=\int_0^\infty \beta e^{-\beta t}G(u,t) \, \mathrm dt$,
and \eqref{appell}. After some computations, for $\lambda > \mu > 0$, we get
\begin{align}
\mathcal{G}(u) = (\mu/\lambda) f_\beta \left( \lambda,\mu,\frac{\lambda u-\mu}{\mu(u-1)}, \frac{\lambda u-\mu}{
\lambda(u-1)} \right), \quad |u|\le 1,
\end{align}
where
\begin{align*}
f_b(x,y,z,s) = F_1 \left( \frac{b}{x-y};-1,1;1+\frac{b}{x-y};z,s \right).
\end{align*}
For the subcritical case, i.e.\ when $\lambda<\mu$, we obtain
\begin{align}
\mathcal{G}(u) = f_\beta \left( \mu,\lambda, \frac{\mu(u-1)}{\lambda u-\mu}, \frac{\lambda(u-1)}{\lambda u-\mu} \right),
\quad |u|\le 1.
\end{align}
The critical case $\lambda=\mu$ is simpler to cope with and results in a form which involves a difference of confluent hypergeometric
functions. In particular we have
\begin{align}
\mathcal{G}(u) = \frac{\beta \, u \, U\left(1,1,\frac{\beta}{\lambda(u-1)}\right)}{\lambda(u-1)}
- U\left( 1,0,\frac{\beta}{\lambda(u-1)} \right).
\end{align}
\section{Comparison with real data}
\label{ddataa}
\begin{figure}
\centering
\includegraphics[scale=0.85]{mnb}
\caption{\label{data}The generalized Yule model compared to the empirical probability mass function
for the number of in-links in the World Wide Web ($n \ge 1$). Data are taken from
Web Data Commons, University of Manheim. It can be seen that the model (in red)
fits the data even for small values of $n$ (with parameters $(\lambda;\mu;\beta) = (4;3.8;0.272)$).
In the inset, the empirical distribution function calculated on the data. Note the importance of the small values of $n$ as
the cumulative distribution function immediately saturates.}
\end{figure}
As it is well known, data from World Wide Web show a scale free behavior. The proposed generalized model exhibits
this feature in the supercritical regime.
Having the complete distribution of in-links we are able to compare our model predictions against real data.
The most recent available data on in-links of World Wide Web we found were collected in 2012 (Web Data Commons, University of
Manheim\footnote{http://webdatacommons.org/hyperlinkgraph/}). In Fig.~\ref{data} we show the excellent performance
of the proposed model with respect to these data. Note the goodness of fit also in the case of a few in-links.
Furthermore, let us stress that most of the probability concentrates on points corresponding to a very few in-links
(the cumulative distribution function at $n=20$ is already $0.9$) and this tells us that it is essential to derive closed form
representations for these probabilities.
\section{Generalized model: effect of link removal}
\label{fuego}
In the previous sections we presented a generalization of Yule model characterized by the introduction
of the possibility of link detachment. This resulting model
is more complex and one may wonder about the necessity of this added complexity.
In the supercritical case, an intuition could suggest to replace the proposed model with a simple Yule model characterized by a decreased
intensity $\delta = \lambda-\mu$ for the links generation. We show here that this intuition is misleading.
The most evident difference is that only in the proposed model the event $\{\mathcal{N}=0\}$ has a positive probability.
A further evidence is highlighted by reparametrizing
formula \eqref{dec} for $n \ge 1$ setting $\delta=\lambda-\mu$, $\lambda=\mu+\delta$, resulting in
\begin{align*}
\mathbb{P} (\mathcal{N}=n) = {} & \frac{\beta \delta \Gamma \left( 1+\frac{\beta}{\delta} \right) \Gamma(n)
}{(\mu+\delta)^2\Gamma\left( n+1+\frac{\beta}{\delta} \right)} \\
& \times {}_2 F_1 \left(n+1,1+\frac{\beta}{\delta};n+1+\frac{\beta}{\delta};\frac{\mu}{\mu+\delta}\right), \notag
\end{align*}
which can be written as
\begin{align}
\label{petr}
\mathbb{P} (\mathcal{N}=n) = {} & \frac{\beta \Gamma \left( 1+\frac{\beta}{\delta} \right) \Gamma(n)
}{\delta \Gamma\left( n+1+\frac{\beta}{\delta} \right)} \left( \frac{\delta}{\mu+\delta} \right)^{1-\frac{\beta}{\delta}} \\
& \times {}_2 F_1 \left(\frac{\beta}{\delta},1+\frac{\beta}{\delta};n+1+\frac{\beta}{\delta};-\frac{\mu}{\delta}\right). \notag
\end{align}
The last step comes from formula (1.3) of \cite{olde}.
Note that the first factor of \eqref{petr} coincides with the analogous probability for the
Yule model of parameter $\delta$. However, the second and third factors depend both on $\delta$ and $\mu$.
The same feature characterizes the variance \eqref{var} while the mean \eqref{media} depends only on $\delta$.
Figs.~\ref{new} and \ref{cinque} show the shape of the distribution of $\mathcal{N}$ for different values of the death intensity
$\mu$. We can see in Fig.\ \ref{new} how the initial part of the distribution, as $\mu$ becomes larger and larger, is
more and more distant from a pure power-law behavior.
An interesting property concerns the asymptotics of $\mathbb{P}(\mathcal{N}=n)$ for large values of $n$, i.e.\ the tail
of the distribution. To investigate this aspect let us consider the asymptotics of the ratio
\begin{align}
\frac{\mathbb{P}(\mathcal{N}=n)}{\mathbb{P}(\mathcal{N}^Y=n)}
= \left( \frac{\delta}{\mu+\delta} \right)^{1-\frac{\beta}{\delta}}
{}_2 F_1 \left( \frac{\beta}{\delta},1+\frac{\beta}{\delta};n+1+\frac{\beta}{\delta};-\frac{\mu}{\delta} \right), \notag
\end{align}
where $\mathbb{P}(\mathcal{N}^Y=n)$ represents the probability of interest, \eqref{yule},
for a classical Yule model of parameters $(\beta,\delta)$.
When $n$ is large we have (see \cite{spanier}, formulae 60:9:3, page 605, and 48:6:1, page 474)
\begin{align}
\frac{\mathbb{P}(\mathcal{N}=n)}{\mathbb{P}(\mathcal{N}^Y=n)} \sim
\left( \frac{\delta}{\mu+\delta} \right)^{1-\frac{\beta}{\delta}}
- \left( \frac{\delta}{\mu+\delta} \right)^{1-\frac{\beta}{\delta}}
\frac{\frac{\beta}{\delta}\left( 1+\frac{\beta}{\delta} \right)\left( 1+\frac{\mu}{\delta} \right)}{
n+1+\frac{\beta}{\delta}} + \mathcal{O}\left( n^{-2} \right). \notag
\end{align}
The reader should realize that the first order asymptotics of the distribution of $\mathcal{N}$ and $\mathcal{N}^Y$
are characterized by the same power-law exponent. However, it holds
\begin{align*}
\left( \frac{\delta}{\mu+\delta} \right)^{1-\frac{\beta}{\delta}} \ne 1
\end{align*}
if $\mu>0$ (i.e.\ if removal of in-links is introduced) and $\beta\ne\delta$. Furthermore
the second order contribution to the asymptotics depends upon $\mu$.
The inset of Fig.~\ref{new} compares the distribution of $\mathcal{N}$ with its asymptotics.
The asymptotics of $\mathbb{P}(\mathcal{N}=n)$ can be exploited to estimate the value of $\beta/\delta$ and $\mu/\delta$.
Indeed, using exact asymptotics of hypergeometric functions,
we consider the dominant term of $\mathbb{P}(\mathcal{N}=n)$ as
\begin{align*}
\mathbb{P}(\mathcal{N}=n) \sim n^{-1-\frac{\beta}{\delta}} \Gamma\left( 1+\frac{\beta}{\delta}
\right) \frac{\beta}{\delta}
\left( \frac{\delta}{\mu+\delta} \right)^{1-\frac{\beta}{\delta}} + \mathcal{O}\left( n^{-2-\beta/\delta} \right)
\end{align*}
(see also \cite{reed} for a different derivation).
Taking the logarithm and setting $y=\log n$ we get
\begin{align}
\label{e130}
\log \, & \mathbb{P}(\mathcal{N}=n) \sim -\left( 1+\frac{\beta}{\delta} \right) y \\
& + \left( 1-\frac{\beta}{\delta} \right) \log \left( \frac{1}{1+\mu/\delta} \right)
+ \log \Gamma\left( 1+\frac{\beta}{\delta}
\right) + \mathcal{O}\left( n^{-1} \right). \notag
\end{align}
An estimate of the slope and intercept of the straight-line in \eqref{e130} allows the estimation
of $\beta/\delta$ and $\mu/\delta$. Hence, from data a possible non-zero value of $\mu$ can be detected.
Similarly, statistical test on the intercept and the constant term of the regression equation can be performed with
classical regression methods, determining the p-value of the considered tests.
These test can then be adapted to perform test on $\mu/\delta$ and $\beta/\delta$.
We do not detail further this fact because other statistical features could be investigated and this will be the subject of a future work.
Another possibility to obtain from data information on the value of the
death intensity $\mu$ would be for example by means of the behavior of the distribution of $\mathcal{N}$ for small values
of $n$.
\section{Discussion}
Traditional models disregard the detachment phenomenon but it may play a relevant role
in network dynamics.
As it is remarked in \cite{barab}, the observed growth of the World Wide Web
measured as the number of connected webpages in time is exponential.
This suggests that the assumption of the Yule process for the appearing of new webpages is reasonable.
However, the original model assumes that the in-links exist forever. To avoid this hypothesis
we generalized the model by using linear birth-death processes in substitution of
classical birth processes for in-links dynamics.
The proposed model admits the presence of \emph{lost} webpages, that is of webpages
with zero in-links. The growth of the number of in-links per webpage is due to a linear birth
phenomenon and therefore webpages with zero in-links are lost forever.
This feature of the model may be questionable and further modifications can be proposed.
The introduction of removal was already considered in models motivated by different applications
\cite{pnas,plos,reed,biol,patzkowsky,shi}
but here we obtain closed form results not yet appeared in the literature. In \cite{pnas,plos},
the framework is that of macroevolutionary dynamics. Two different models are described and analyzed:
the SEO model (speciation-extinction-origination model) and the BDM model (birth-death-mutation model) also studied in \cite{biol}.
However, the microscopic dynamics of our model is different from all these models.
In the SEO model, the origination rate for new families (new genera) is proportional to the number of species already present in the
system while the BDM model is a discrete-time generalization of the Yule--Simon model allowing for deaths.
The model discussed in \cite{patzkowsky} considers an origination rate for families which depends upon the number of species already
present. Although an analytical treatment is performed, the overall behavior of the model is evaluated trough Monte-Carlo
simulations.
The framework of \cite{shi} is that of network growth. The model introduced and analyzed is
a generalization of the Barab\'asi--Albert model in which removal of links is permitted. The model implements the
usual preferential attachment for the link addition while an anti-preferential detachment is used for the link removal.
The analysis is then performed by means of mean-field theory.
Other papers studied growth models allowing also negative fluctuations \cite{gabaix,saichev,pop}.
These are works considering growth models with continuous state-space. They are considerably different from our
work in the motivations, in the implemented models and also in the mathematical treatment.
Finally, in the model treated in \cite{reed} (see also the references therein) the process governing the origination
of new families is independent of that describing the growth of the number of species per family.
Despite the different motivations and framework, the model considered in that paper coincides with that shown in the present paper.
However, while \cite{reed} implements numerical and asymptotic methods, we arrive at closed form analytical results.
We explicitely underline that differently from the Barab\'asi--Albert model, the Yule model disregards out-links.
Their introduction would request further hypotheses on the model that are not necessary when the focus is on the popularity of a webpage,
a quantity of interest for many applications.
The results obtained in the present paper are rather general and therefore relevant to many different fields.
Nonetheless, we wanted to stress the fact that network growth models allowing fluctuations in the number of in-links can
be successfully modeled by means of suitable generalizations of the Yule model.
\section{Concluding remarks}
\begin{figure}
\centering
\includegraphics[scale=1]{p4.pdf}
\caption{\label{new}The probability distribution $\mathbb{P}(\mathcal{N}=n)$, $n \ge 1$, for different values of
removal rate $\mu$
(note that $\mu = 0$ corresponds to Yule model) and
$\beta=\delta=1$. Note that, with the choice $\beta=\delta$,
the tail behavior for different values of $\mu$ can be compared more easily (see also Fig. \ref{cinque}
for an example of $\beta\ne \delta$).
In the inset (solid line) the theoretical distribution
$\mathbb{P}(\mathcal{N}=n)$ for $\mu=80$, $\delta=1$, and (dashed line) its
asymptotic behavior.}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=1]{p5.pdf}
\caption{\label{cinque}The probability distribution $\mathbb{P}(\mathcal{N}=n)$, $n \ge 1$, for different values of $\mu$
and $(\beta,\delta)=(10,1)$. Notice that, since $\delta=1$, the underlying behavior is supercritical but
$\lambda$ increases together with $\mu$.}
\end{figure}
The main advantage of the investigated model lies in its mathematically tractability.
We are able to present closed form expressions
for the limiting distribution of the number of in-links per webpage. Its tail behavior is
also analytically determined.
The use of these formulae permits to recognize features not exhibited by the classical Yule model and to reproduce the observed
behavior of in-links of World Wide Web.
The main difference with the classical Yule model is that it did not allow the extinction of the in-links
to a specific webpage. The presence of a non zero probability for $\{\mathcal{N}=0\}$ implies
a decrease of the probabilities of $\{\mathcal{N}=n\}$, $n \ge 1$, and
the probability of having a positive number of in-links is strictly smaller
than the analogous quantity of the Yule model.
This is also well illustrated by the fact that for the classical model always holds
$\mathbb{E}\,\mathcal{N}^Y>1$ while this is not valid in the current version of the model.
The comparison of the classical and generalized versions has been performed in the supercritical case, i.e.\ in the most relevant instance.
The first-order tail behavior of the two models is similar but with a different proportionality
coefficient. Differences arise in the second-order term.
This is our first attempt to deal with the detachment issue and more realistic models can be considered.
Growth phenomena admitting saturation or with more general nonlinear evolution rates are possible extensions.
For example, a different situation arises if for a lost webpage there is a (possibly very small) probability of
reobtaining an in-link. This circumstance would correspond to a birth-death process with immigration
governing the increase of the number of in-links.
Similarly, detachment can happen with higher probability when the number of in-links
is low.
A model with removal of webpages in spite of existing in-links was considered in \cite{fenner} and could imply
a variant of the studied model.
We finally underline that our study is motivated by Web applications but scale-free networks are of interest in a
variety of different contexts.
Power-laws are the common feature of these models \cite{newman,barba,mit,hughes}.
Social networks, macroevolutionary trees, scientific collaborations,
words occurrences, neural networks, are some of examples of networks studied in the literature
\cite{simon,pnas,plos,reed,patzkowsky,noh,grabowski,lansky}.
\ack{The authors have been supported by project AMALFI (Universit\`{a} di Torino/Compagnia di San Paolo).
Petr Lansky has been supported by Institute of Physiology RVO67985823 and the Centre for
Neuroscience P304/12/G069.
The authors would like to thank the anonymous referees for their useful comments and criticisms which improved the
quality of the paper.}
\section*{References}
|
1,116,691,499,299 | arxiv |
\section{Introduction}
\label{sec:intro}
Randomised Controlled Trials (RCTs) are the gold-standard study design for evaluating the effectiveness and safety of new interventions.
Despite their successful history of demonstrating the benefits
and uncovering the risks of new treatments,
they face substantial challenges.
Recent evidence has shown that the success rate of RCTs is low~\citep{wong2019estimation}.
This low success rate and the increasing cost of conducting RCTs is resulting in large real-term increases in the cost of drug development~\citep{dimasi2016innovation}.
Since the sample size of a trial is a key determinant of the chances of detecting a treatment effect (if it is present) and
an important cost factor,
the choice of an adequate sample size enables more
economic drug development.
Purely economic arguments would suggest a utility based approach
as discussed in~\citet{lindley-1997}, for example.
However, specifying utility functions, particularly in investigator-sponsored studies concerned with non-drug interventions,
can be hard in practice.
These practical problems, the desire to maintain minimal ethical standards, as well as recommendations of health authority guidelines result in the vast majority of confirmatory clinical trials deriving their sample size based on desired conditional type~I and type~II error rates.
For instance, a randomised clinical trial with an unnecessarily large sample size (`overpowered') is unethical if the treatment shows a substantial effect and the consequences of being randomised to the control arm are severe.
Thus the issue of selecting an appropriate sample size is a vitally important part of conducting a clinical trial.
The traditional approach to determining the required sample size for a trial is to choose a point alternative and
derive a sample size such that the probability to reject the null hypothesis exceeds a certain threshold (typically 80\% or 90\%) while maintaining a certain maximal type~I error rate (typically 2.5\% one-sided).
The maximal type~I error rate is usually realised at the boundary of the null hypothesis and thus available immediately.
The type~II error rate, however, critically depends on the choice of the (point) alternative.
There are at least two ways of justifying the choice of point alternative.
The first is based on a relevance argument, which requires the specification of a minimal clinically relevant difference (MCID).
Since the probability to reject the null hypothesis is typically monotonic in the effect size, determining the sample size such that the desired probability to reject the null is exceeded at the MCID implies that the power for all other relevant differences will be even larger.
Guidance has recently been published on choosing the MCID~\citep{cook2018delta2}, but making this choice may still be difficult in practice.
The second perspective is based on \textit{a~priori} considerations about the likelihood of the treatment effect.
Here, an \textit{a~priori} likely effect size is used as the point alternative.
This implies that the resulting sample size might be too small to detect smaller but still relevant differences reliably.
On the other hand, the potential savings in terms of sample size might outweigh the risk of ending up with an underpowered study.
The core difference between these approaches is that basing the required sample size on the MCID might be ineffective if prior evidence for a larger effect is available, but the MCID is chosen based on \emph{relevance} arguments and is thus not subject to uncertainty.
In contrast, choosing the point alternative based on considerations about the relative \textit{a~priori} likelihood of effect sizes implies that there is an inherent uncertainty about the effect size that can be expected -- otherwise no trial would be needed in the first place.
Depending on the study objective, the sample size can also be
derived based on entirely different considerations.
For example, studies that aim to establish new diagnostic tools or biomarkers would rather target a certain width of the confidence interval for the AUC~\citep{obuchowski1998}.
Similarly, studies aimed at estimating population parameters with sufficient precision should target the standard error of the estimate, rather than deriving a sample size based on power arguments~\citep{Thompson2012,grouin-2007}.
These approaches are beyond the scope of this manuscript and we will only discuss sample size derivation based on error rate considerations.
To make things more tangible, consider the case of a simple
one-stage, one-arm $Z$-test.
Let $X_i$, $i=1,\ldots,n$, be $iid$ observations with mean $\theta$ and known standard deviation $\sigma$.
Under suitable regularity conditions, their mean is asymptotically normal and
{\small $Z_n := \sqrt{n}\,\big(\overline{X} - \theta_0\big)/\sigma\stackrel{\cdot}{\sim}\mathcal{N}\big(\theta_n,1\big)$} where $\theta_n := \sqrt{n}(\theta - \theta_0)/\sigma$.
Assume that interest lies in testing the null hypothesis $\mathcal{H}_0:\theta \leq \theta_0 = 0$ at a one-sided significance level of $\alpha$.
\newcommand{z_{1-\alpha}}{z_{1-\alpha}}
The critical value for rejecting $\mathcal{H}_0$ is given by the $(1-\alpha)$-quantile of the standard normal distribution, $z_{1-\alpha}$, and is independent of $n$.
\newcommand{\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}}{\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}}
The probability of rejecting the null hypothesis for given $n$ and $\theta$ is
\begin{align}
\label{eq:prob-to-reject}
\Pr_{\theta}[\,Z_n > z_{1-\alpha}\,] = 1 - \Phi\big(z_{1-\alpha} - \theta_n\big) = \Phi\big(\theta_n - z_{1-\alpha}\big),
\end{align}
where $\Phi$ is the cumulative distribution function (CDF) of the standard normal distribution.
Often, $\Pr_{\theta}[\,Z_n > z_{1-\alpha}\,]$ is seen as a function of $\theta$ and termed the `power function'.
This terminology may lead to confusion when considering parameter values $\theta\leq\theta_0$ and $\theta\geq\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}$, since the probability
to reject the null hypothesis corresponds to the type~I error rate in the former case and classical `power' in the latter.
For the sake of clarity we will thus use the neutral term `probability to reject'.
\newcommand{\theta_{\scaleto{\operatorname{alt}}{3.5pt}}}{\theta_{\scaleto{\operatorname{alt}}{3.5pt}}}
Assume that a point alternative $\theta_{\scaleto{\operatorname{alt}}{3.5pt}}>\theta_0$ is given.
A sample size can then be chosen as the smallest sample size that results in a probability to reject of at least $1-\beta$ at $\theta_{\scaleto{\operatorname{alt}}{3.5pt}}$
\begin{align}
n^*_{\theta_{\scaleto{\operatorname{alt}}{3.5pt}}} :=\quad &\argmin{n}\ &&n \nonumber\\
&\operatorname{subject\ to:\ } &&\Pr_{\theta_{\scaleto{\operatorname{alt}}{3.5pt}}}[\,Z_n > z_{1-\alpha}\,] \geq 1 - \beta \ .
\label{eq:simle-pwr-cnstr}
\end{align}
Since $\Pr_{\theta}[\,Z_n > z_{1-\alpha}\,]$ is monotone in $\theta$, the probability to reject the null hypothesis for $\theta>\theta_{\scaleto{\operatorname{alt}}{3.5pt}}$ is also greater than $1-\beta$.
Thus, if $\theta_{\scaleto{\operatorname{alt}}{3.5pt}}=\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}$, the null hypothesis can be rejected for all clinically relevant effect sizes with a probability of at least $1-\beta$.
This approach requires no assumptions about the \textit{a priori} likelihood of the value of $\theta$ but only
about $\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}$ and the desired minimal power level
(see also~\cite[Section 3]{chuang2011} and~\citealp{chuang-2006}).
However, the required sample size increases quickly as $\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}$ approaches $\theta_0$.
In almost all practically relevant situations, a maximal feasible sample size is given (e.g., due to budget constraints), which might not be sufficient to achieve the desired power if $\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}$ is close to $\theta_0$.
This problem might arise when considering overall survival in oncology trials, for example.
However, it may then be hard to justify a value for $\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}$ much larger than $\theta_0$ since almost any improvement in overall survival can be considered relevant
\footnote{It is important to distinguish between the (unknown) true effect size and the observed effect size.
It might still be reasonable to additionally require a certain deviation of the observed effect from the null, e.g.~a hazard ratio of less than 0.85.}.
The problem becomes even more pressing if the null hypothesis is defined as $\mathcal{H}_0':\theta \leq \theta_{\scaleto{\operatorname{MCID}}{3.5pt}}$, i.e.,
if the primary study objective is to demonstrate a clinically important effect.
In either case it is clearly impossible to derive a feasible sample size based on the minimal clinically important difference alone~\citep{chuang2011}.
Formulating a principled approach to eliciting a sample size in situations like these is difficult, and in practice trialists may resort to back-calculating an effect size in order to achieve the desired power given the maximum feasible sample size~\citep{lenth2001,lan-2012,grouin-2007}.
One way of justifying a smaller sample size is to consider a point alternative $\theta_{\scaleto{\operatorname{alt}}{3.5pt}}>\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}$ based on \textit{a~priori} likelihood arguments: if there is prior evidence for effect sizes larger than $\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}$, determining the sample size based on $\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}$ might well be inefficient and lead to an unnecessarily large trial.
Therefore, planning of the required sample size is often based on a single point alternative $\theta_{\scaleto{\operatorname{alt}}{3.5pt}}$, $\theta_{\scaleto{\operatorname{alt}}{3.5pt}} > \theta_{\scaleto{\operatorname{MCID}}{3.5pt}}$.
Yet, this pragmatic approach is unsatisfactory in that it ignores any uncertainty about the effect
size~\citep{lenth2001}.
In the following, we review approaches to sample size derivation that do account for \textit{a~priori} uncertainty via a prior density for the effect size.
We propose a framework encompassing the most relevant
quantities discussed in this context, give precise definitions of the terms, and highlight connections between individual items.
Where necessary, we refine existing definitions to improve overall consistency.
Note that we exclusively focus on what is usually termed
a `hybrid' Bayesian-frequentist approach
to sample size derivation~\citep{spiegelhalter-2004}.
This means that, although, Bayesian arguments are used to derive
a sample size under uncertainty about the true effect size,
the final analysis is strictly frequentist.
After introducing the individual concepts in detail,
a structured overview of all quantities considered is provided in Figure~\ref{fig:diagram}.
We then present a review of the literature on the subject, showcasing the confusing diversity of terminology used in the field and relating our definitions back to the existing literature.
Finally, we present some numerical examples and conclude with a discussion.
\section{A Bayesian argument for the choice of the point alternative}
\label{sec:quantile-approach}
\newcommand{\operatorname{RPR}}{\operatorname{RPR}}
One way of incorporating planning uncertainty is to make assumptions about the relative \textit{a~priori} likelihood of the unknown effect size.
This approach can be formalised within a Bayesian framework by seeing the true effect size $\theta$ as the realisation of a random
variable $\Theta$ with prior density $\varphi(\theta)$.
This means that the CDF of
$\Theta$ is given by $\Pr_{}[\,\Theta\leq x\,] = \int_0^x \varphi(\theta) \operatorname{d} \theta$.
At the planning stage, the probability to reject the null hypothesis is then given by the random variable $\operatorname{RPR}(n)$, the `random probability to reject':
\begin{align}
\operatorname{RPR}(n):=\Pr_{\Theta}[\,Z_n > z_{1-\alpha}\,]\ .
\end{align}
We explicitly denote this quantity as `random' to emphasise the distinction between the
(conditional on $\Theta=\theta$) probability to reject given in equation~\eqref{eq:prob-to-reject} and the
unconditional `random' probability to reject.
The variation of the random variable $\operatorname{RPR}(n)$ reflects the \textit{a~priori} uncertainty about the unknown underlying effect size that is encoded in the prior density $\varphi(\cdot)$ of the random variable $\Theta$.
We prefer the term `random probability to reject' over `random power' since $\operatorname{RPR}(n)$ is unconditional on the effect size, and consequently does not distinguish between rejections under the null hypothesis and under relevant effect sizes.
\newcommand{\rpower}[1]{\operatorname{RPow}({ #1 })}
Instead, we define the conditional random variable `random power' as
\begin{align}
\rpower{n}:= &\ \Pr_{\, \Theta\geq\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}\, }[\,Z_n > z_{1-\alpha}\,] \\ =& \ \operatorname{RPR}(n) \mid \Theta\geq\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}.
\end{align}
This definition more closely resembles the concept of frequentist power since it conditions on a relevant effect size.
Determining the required sample size based on a point alternative as outlined in the introduction evaluates the probability to reject the null hypothesis solely on $\theta_{\scaleto{\operatorname{alt}}{3.5pt}}$.
This can be understood as conditioning the random probability to reject, or the random power, on $\Theta=\theta_{\scaleto{\operatorname{alt}}{3.5pt}}$, i.e., to consider $\big(\operatorname{RPR}(n) \mid \Theta = \theta_{\scaleto{\operatorname{alt}}{3.5pt}} \big) = \big(\rpower{n} \mid \Theta = \theta_{\scaleto{\operatorname{alt}}{3.5pt}} \big)$.
Due to conditioning on a single parameter value $\theta_{\scaleto{\operatorname{alt}}{3.5pt}}\geq\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}$, under any prior density with $\varphi(\theta_{\scaleto{\operatorname{alt}}{3.5pt}})>0$, both random variables (almost surely) reduce to the deterministic expression $\Pr_{\theta_{\scaleto{\operatorname{alt}}{3.5pt}}}[\,Z_n > z_{1-\alpha}\,]$ which is often termed `power' in a frequentist context.
Basing the sample size derivation on this quantity means that the probability to reject the null hypothesis for relevant values of $\theta\geq\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}, \theta\neq\theta_{\scaleto{\operatorname{alt}}{3.5pt}}$ is completely ignored and the \textit{a~priori} evidence encoded in $\varphi(\cdot)$ is not used.
\citet{spiegelhalter1986} have pointed out that a power constraint for sample size derivation could be computed based on \textit{``[...] a somewhat arbitrarily chosen location parameter of the [prior] distribution (for example the mean, the median or the 70th percentile).''}
Using a location parameter of the unconditional prior distribution of $\Theta$, however, might lead to situations where no sample size can be determined if the location parameter lies within the null hypothesis.
Here, we follow a similar idea but motivate the choice of location parameter in terms of the \textit{a~priori} distribution of random power.
To this end, let
\newcommand{\quant}[1]{{\operatorname{Q}_{ #1 }}}
\begin{align}
\quant{ 1-\gamma }[\ \rpower{n} \,] \ := \
\inf_{x}\ \Pr_{\varphi(\cdot)}\big[\, \rpower{n} \geq x \,\big] \geq \gamma\
\end{align}
be the $(1-\gamma)$-quantile of the random power ($\rpower{n}$).
Furthermore, let
\begin{align}
\varphi(\theta\,|\,\Theta\geq \theta_{\scaleto{\operatorname{MCID}}{3.5pt}}):= \frac{\displaystyle \boldsymbol{1}_{\theta\geq \theta_{\scaleto{\operatorname{MCID}}{3.5pt}}} \ \varphi(\theta)}{\displaystyle \int_{\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}}^\infty \varphi(x)\ \operatorname{d} x } \label{eq:conditional-prior-density}
\end{align}
be the conditional prior density of $\big(\Theta \mid \Theta \geq \theta_{\scaleto{\operatorname{MCID}}{3.5pt}} \big)$.
We choose to make the dependency of \begin{align}
\Pr_{\varphi(\cdot)}\big[\, \rpower{n} \geq x \,\big] = \int_{-\infty}^{\infty} \varphi(\theta\,|\,\Theta\geq \theta_{\scaleto{\operatorname{MCID}}{3.5pt}})\, \Pr_{ \theta}[\,Z_n > z_{1-\alpha}\,] \operatorname{d}\theta
\end{align}
on the prior density explicit by using the index `$\varphi(\cdot)$' since the random parameter $\Theta$ does not appear directly in the description of the event `$\rpower{n} \geq x$'.
Whenever $\Theta$ appears explicitly, we omit the index since the dependency on~$\varphi(\cdot)$ is then clear from the context.
The expression ${\Pr_{\varphi(\cdot)}\big[\, \rpower{n} \geq x \,\big]}$ is a real number in ${[\,0, 1\,]}$ and thus different from $\operatorname{RPR}(n)=\Pr_{\Theta}[\,Z_n > z_{1-\alpha}\,]$ which is a random variable.
If a sample size was then chosen such that $\quant{1-\gamma}[\,\rpower{n}\,]\geq 1 - \beta$, the \textit{a~priori} probability of exceeding a probability to reject of $1-\beta$ given a relevant effect would be, by definition, at least $\gamma$.
The required sample size for this approach is the solution of
\begin{align}
\argmin{n}\quad &n \nonumber\\
\operatorname{subject\ to:\ } \quad &\quant{1-\gamma}[\,\rpower{n}\,] \geq 1 - \beta \ .
\end{align}
Since $\Pr_{\theta}[\,Z_n > z_{1-\alpha}\,]$ is monotonic in $\theta$, this problem is equivalent to solving
\begin{align}
n_{\gamma}^* \quad := \quad & \argmin{n} \quad &&n \nonumber\\
&\operatorname{subject\ to:\ } \quad &&\Pr_{\ \quant{1 - \gamma}[\, \Theta \geq \theta_{\scaleto{\operatorname{MCID}}{3.5pt}} \,]}\big[\,Z_n > z_{1-\alpha}\,\big] \geq 1 - \beta \ ,
\end{align}
where $\quant{1 - \gamma}[\, \Theta \geq \theta_{\scaleto{\operatorname{MCID}}{3.5pt}} \,]$ is the $(1-\gamma)$-quantile of the prior distribution of the random variable $\Theta$ conditional on a relevant effect.
Consequently, this `prior quantile approach' can be used with any existing frequentist sample size derivation formula.
It~is merely a formal Bayesian justification for determining the sample size of a trial based on a point alternative ${\theta_{\scaleto{\operatorname{alt}}{3.5pt}}:=\quant{1 - \gamma}[\, \Theta \geq \theta_{\scaleto{\operatorname{MCID}}{3.5pt}} \,]\geq\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}}$.
The prior quantile approach reduces to powering on $\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}$ whenever the target power needs to be met with absolute certainty for all relevant effect sizes ($\gamma=1$).
One may thus see the prior quantile approach as a principled relaxation of powering on $\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}$.
The approach differs from Spiegelhalter and Freedman's suggestions in two key aspects.
Firstly, the point alternative naturally emerges as a quantile of the prior \emph{conditional on a relevant effect} by imposing a lower boundary on the \textit{a~priori} probability to undershoot the target power.
This intuitively makes sense since a large probability to reject is only desirable when the underlying $\theta$ is relevant.
This also ensures that $\quant{1 - \gamma}[\, \Theta \geq \theta_{\scaleto{\operatorname{MCID}}{3.5pt}} \,]>\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}\geq\theta_0$ for $\gamma<1$ and thus guarantees a finite sample size irrespective of the choice of prior.
Secondly, to guarantee a more than 50\% chance of exceeding the target power, the conditional prior quantile will typically be chosen smaller (i.e.,~$\gamma>0.5$) than the conditional median ($\gamma=0.5$) which was discussed by Spiegelhalter and Freedman.
\section{Probability of success and expected power}
\label{sec:pos}
\newcommand{\operatorname{PoS}}{\operatorname{PoS}}
Spiegelhalter and Freedman also proposed the use of the \textit{``probability of concluding that the new treatment is superior and of this being correct''} ($P_{Ss}$ in their notation) to derive a required sample size~\citep{spiegelhalter1986}.
The quantity has also been referred to as `prior adjusted power'~\citep{spiegelhalter-2004,shao-2008}.
This definition of probability of success is also discussed in \citet{liu-2010} and \citet{ciarleglio-2015}.
In the situation at hand, it reduces to
\begin{align}
\operatorname{PoS}(n) :=&\ \Pr_{}[\,Z_n > z_{1-\alpha}, \Theta \geq \theta_{\scaleto{\operatorname{MCID}}{3.5pt}}\,] \\
=&\ \int_{\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}}^\infty \int_{z_{1-\alpha}}^\infty \phi(z - \theta_n)\ \varphi(\theta)\ \operatorname{d}z\operatorname{d}\theta \ ,
\end{align}
where $\phi$ is the PDF of the standard normal distribution.
Here, we are slightly more general than previous authors in that
we allow $\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}>0$ and use a tighter definition of `success': a trial is only successful if the null hypothesis is rejected \emph{and} the effect is relevant.
Whenever $\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}=0$ this coincides with the definitions used previously in the literature.
The definition of $\operatorname{PoS}(n)$ critically relies on what is being considered a `success'.
The original proposal of Spiegelhalter and Freedman only considers a significant study result a success if the underlying effect is also non-null (i.e., the joint probability of non-null \emph{and} detection).
In more recent publications, a majority of authors tend to follow
O'Hagan~\textit{et~al.} who consider a slightly
different definition of the probability of success by integrating the probability to reject over the entire parameter range~\citep{o2001bayesian,ohagan-2005}
and term this `assurance'.
For a more comprehensive overview of the terms used in
the literature, see Section~\ref{sec:literature}.
The alternative definition for probability of success introduced by O'Hagan~\textit{et~al.}
corresponds to the marginal probability of rejecting the null hypothesis irrespective of the corresponding parameter value
\begin{align}
\operatorname{PoS}'(n) :=&\ \Pr_{}[\,Z_n > z_{1-\alpha}\,] \\
=&\ \int_{-\infty}^\infty \int_{z_{1-\alpha}}^\infty \phi(z - \theta_n)\ \varphi(\theta)\ \operatorname{d}z\operatorname{d}\theta\\
=&\ \operatorname{PoS}(n) + \underbrace{\Pr_{}[\,Z_n > z_{1-\alpha}, 0<\Theta<\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}\,]}_{\text{probability of rejection and irrelevant effect}} + \underbrace{\Pr_{}[\,Z_n > z_{1-\alpha}, \Theta \leq 0\,]}_{\text{probability of a type~I error}} \ .
\label{eq:pos-prime-decomposition}
\end{align}
The decomposition in equation~\eqref{eq:pos-prime-decomposition} makes it clear that the implicit definition of `success' underlying $\operatorname{PoS}'(n)$ is at least questionable \citep{liu-2010}.
The marginal probability of rejecting the null hypothesis includes rejections under irrelevant or even null values of $\theta$, and is thus inflated by type~I errors and rejections under irrelevant values of~$\theta$.
This issue was first raised by \citet{spiegelhalter-2004}
for simple (point) null and alternative hypotheses.
The degree to which $\operatorname{PoS}(n)$ and $\operatorname{PoS}'(n)$ differ numerically is discussed in more detail in Section~\ref{sec:results:pos-for-sample-size-determination}.
Which definition of `success' is preferred mainly depends on perspective: a short-term oriented pharmaceutical company might just be interested in rejecting the null hypothesis to monetise a new drug - irrespective of it actually showing a relevant effect. This view would then correspond to $\operatorname{PoS}'(n)$.
Regulators and companies worried about the longer-term consequences of potentially having to retract ineffective drugs,
might tend towards the joint probability of correctly rejecting the null, i.e.,~$\operatorname{PoS}(n)$.
We take the latter perspective and focus on~$\operatorname{PoS}(n)$.
$\operatorname{PoS}(n)$ is an unconditional quantity and must therefore implicitly depend on the \textit{a~priori} probability of a relevant effect.
To see this, consider the following decomposition
\begin{align}
\operatorname{PoS}(n) &=
\Pr_{}[\,Z_n > z_{1-\alpha}, \Theta \geq \theta_{\scaleto{\operatorname{MCID}}{3.5pt}}\,] \\
&= \int_{\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}}^\infty \Pr_\theta[\,Z_n > z_{1-\alpha}\,]\ \varphi(\theta)\ \operatorname{d} \theta \label{eq:pos-integral} \\
&= \Pr_{ }[\,Z_n > z_{1-\alpha} \mid \Theta \geq \theta_{\scaleto{\operatorname{MCID}}{3.5pt}}\,] \, \Pr_{}[\,\Theta \geq \theta_{\scaleto{\operatorname{MCID}}{3.5pt}}\,] \\
&= \underbrace{\operatorname{E}\big[\, \Pr_{\, \Theta \geq \theta_{\scaleto{\operatorname{MCID}}{3.5pt}}}[\,Z_n > z_{1-\alpha}\,] \,\big]}_{= \ \operatorname{E}[\,\rpower{n}\,]\ =:\ \operatorname{EP}(n)} \,
\Pr_{}[\,\Theta \geq \theta_{\scaleto{\operatorname{MCID}}{3.5pt}}\,] \ .
\label{eq:pos-and-ep}
\end{align}
This means that the probability of success can be expressed as the product of the `expected power', $\operatorname{EP}(n)$, and the \textit{a~priori} probability of a relevant effect (see again \citet{spiegelhalter-2004} for the situation with point hypotheses).
Expected power was implicitly mentioned in \citet{spiegelhalter1986} ($P_{Ss}/P_{\cdot s}$ in their notation) as a way to characterise the properties of a design.
The use of expected power as a means to derive the required sample size of a design under uncertainty was then proposed in \citet{brown1987projection} by solving
\begin{align}
{n_{\scalebox{.5}{EP}}^*} :=\quad &\argmin{n} &&n \nonumber\\
&\operatorname{subject\ to:\ } &&\operatorname{EP}(n) \geq 1 - \beta \ . \label{eq:ep-problem}
\end{align}
Since the power function is monotonically increasing in $\theta$, expected power is strictly larger than power at the minimal relevant value whenever $\Pr[\,\Theta>\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}\,] > 0 $.
This implies that a constraint on expected power instead of a constraint on the probability to reject the null hypothesis at $\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}$ is less restrictive, and consequently~${n_{\scalebox{.5}{EP}}^*}<{n^*_{\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}}}$.
The terms `expected power' and `probability of success' are sometimes used interchangeably in the literature (see Section~\ref{sec:literature}).
In the following, we take a closer look at their connection to clarify their characteristic differences.
Expected power is merely a weighted average of the probability to reject in the relevance region $\theta\geq\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}$, where the weight function is given by the conditional prior density $\varphi(\cdot\,|\,\Theta\geq \theta_{\scaleto{\operatorname{MCID}}{3.5pt}})$ defined in equation~\eqref{eq:conditional-prior-density}
\begin{align}
\operatorname{EP}(n) =&\ \operatorname{E}\big[\, \Pr_{\Theta}[\,Z_n >z_{1-\alpha}\,] \mid \Theta > \theta_{\scaleto{\operatorname{MCID}}{3.5pt}}\,\big] \\
=& \int_{\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}}^\infty \Pr_\theta[\,Z_n > z_{1-\alpha}\,]\ \varphi(\theta\,|\,\Theta\geq \theta_{\scaleto{\operatorname{MCID}}{3.5pt}})\ \operatorname{d} \theta \ .
\label{eq:ep-integral}
\end{align}
$\operatorname{PoS}(n)$, on the other hand, integrates the probability to reject over the same region using the unconditional prior density (see equations~\eqref{eq:pos-integral} and \eqref{eq:ep-integral}).
Thus, in~contrast to $\operatorname{PoS}(n)$, expected power does not depend on the \textit{a~priori} probability of a relevant effect size but only on the relative magnitude of the prior density (`\textit{a~priori} likelihood') of relevant parameter values.
Since the conditional prior density differs from the unconditional one only by normalisation via the \textit{a~priori} probability of a relevant effect,
it follows from equation~\eqref{eq:pos-and-ep} that $\operatorname{EP}(n)$ and $\operatorname{PoS}(n)$ differ only by the constant factor $ \Pr_{}[\,\Theta \geq \theta_{\scaleto{\operatorname{MCID}}{3.5pt}}\,]$.
Consequentially, any constraint on probability of success can be reformulated as a constraint on expected power and \textit{vice~versa}
\begin{align}
\operatorname{PoS}(n)\geq 1 - \beta
\quad \Leftrightarrow \quad
\operatorname{EP}(n)\geq (1 - \beta)\,/\,\Pr_{ }[\,\Theta \geq \theta_{\scaleto{\operatorname{MCID}}{3.5pt}}\,] \ .
\label{eq:pos-to-ep-cnstr}
\end{align}
Furthermore, since $\operatorname{PoS}(n) = \operatorname{EP}(n)\Pr_{}[\,\Theta \geq \theta_{\scaleto{\operatorname{MCID}}{3.5pt}}\,]$ and $\operatorname{EP}(n)\leq 1$, $\operatorname{PoS}(n)$ can never exceed the \textit{a~priori} probability of a relevant effect, $\Pr_{}[\,\Theta \geq \theta_{\scaleto{\operatorname{MCID}}{3.5pt}}\,]$.
This implies that the usual conventions on the choice of $\beta$
as the maximal type~II error rate for a point alternative
cannot be meaningful in terms of the unconditional $\operatorname{PoS}(n)$, since the maximum attainable probability of success is the situation-specific \textit{a~priori} probability of a relevant effect.
The need to recalibrate typical benchmark thresholds when considering probability of success was previously discussed in the literature.
For instance, \citet{o2001bayesian} states that
``[t]he assurance figure is often much lower [than the power], because there is an appreciable prior probability that the treatment difference is less than $\delta^*$'', where in their notation, $\delta^*$ corresponds to $\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}$ in our notation.
A similar argument is put forward in \citet[Section~2]{rufibach_15} for $\operatorname{PoS}'(n)$.
The key issue is thus whether one is interested in the joint probability of rejecting the null hypothesis \emph{and}
the effect being relevant,~$\operatorname{PoS}(n)$, or
the conditional probability of the rejecting the null hypothesis \emph{given} a relevant effect,~$\operatorname{EP}(n)$.
While the interpretation of both quantities is different, in any particular situation, they only differ by a constant factor.
\subsection{Expected power versus quantile-based approach for sample size derivation}
Since expected power and probability of success are proportional,
it suffices to compare expected power and the quantile-based approach outlined in Section~\ref{sec:quantile-approach} with respect to sample size derivation.
Consider
${\theta'' := \quant{1 - \gamma}[\, \Theta \geq \theta_{\scaleto{\operatorname{MCID}}{3.5pt}} \,]}$ and an arbitrary but fixed parameter value $\theta'>\theta'' >\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}$.
Clearly, under the quantile-based approach, the rejection probability at any $\theta'> \theta''$ does not contribute towards the fulfilment of the power constraint since the probability to reject is only evaluated at
${\theta''}$.
For expected power, however, the total functional derivative with respect to changes in the probability to reject at $\theta'$ and $\theta''$ is
\begin{align}
\operatorname{d} \operatorname{EP}(n) &=
\frac{\partial\operatorname{EP}(n)}{\partial\Pr_{\theta''}[\,Z_n > z_{1-\alpha}\,]}\operatorname{d}\Pr_{\theta''}[\,Z_n > z_{1-\alpha}\,] +
\frac{\partial\operatorname{EP}(n)}{\partial\Pr_{\theta'}[\,Z_n > z_{1-\alpha}\,]}\operatorname{d} \Pr_{\theta'}[\,Z_n > z_{1-\alpha}\,] \\[1em]
&= \varphi(\theta''\mid\theta \geq \theta_{\scaleto{\operatorname{MCID}}{3.5pt}})\operatorname{d} \Pr_{\theta''}[\,Z_n > z_{1-\alpha}\,] +
\varphi(\theta'\mid\theta \geq \theta_{\scaleto{\operatorname{MCID}}{3.5pt}})\operatorname{d} \Pr_{\theta'}[\,Z_n > z_{1-\alpha}\,] \ .
\end{align}
Keeping expected power constant, i.e., setting ${\operatorname{d} \operatorname{EP}(n)=0}$ and solving for $\operatorname{d}\Pr_{\theta'}[\,Z_n > z_{1-\alpha}\,]$ yields
\begin{align}
0 &= \varphi(\theta''\mid\Theta \geq \theta_{\scaleto{\operatorname{MCID}}{3.5pt}})\operatorname{d} \Pr_{\theta''}[\,Z_n > z_{1-\alpha}\,] +
\varphi(\theta'\mid\Theta \geq \theta_{\scaleto{\operatorname{MCID}}{3.5pt}})\operatorname{d} \Pr_{\theta'}[\,Z_n > z_{1-\alpha}\,] \\[1em]
\Leftrightarrow \operatorname{d} \Pr_{\theta'}[\,Z_n > z_{1-\alpha}\,] &= -\frac{\varphi(\theta'\mid\Theta \geq \theta_{\scaleto{\operatorname{MCID}}{3.5pt}})}{\varphi(\theta''\mid\Theta \geq \theta_{\scaleto{\operatorname{MCID}}{3.5pt}})} \operatorname{d} \Pr_{\theta''}[\,Z_n > z_{1-\alpha}\,] \\[1em]
\Leftrightarrow \operatorname{d} \Pr_{\theta'}[\,Z_n > z_{1-\alpha}\,] &= -\frac{\varphi(\theta')}{\varphi(\theta'')} \operatorname{d} \Pr_{\theta''}[\,Z_n > z_{1-\alpha}\,]
\ .
\end{align}
A reduction in the probability to reject at $\theta''$ by one percentage point can thus be compensated by an increase in the probability to reject at $\theta'$ by~${\varphi(\theta')\,/\,\varphi(\theta'')}$ percentage points.
This demonstrates that the core difference between the prior quantile-based approach and the expected power approach is whether or not a trade-off between power at different parameter values is deemed permissible (expected power) or not (quantile-based approach).
A structured overview of the terms introduced so far and the respective connections between them is given in Figure~\ref{fig:diagram}.
\section{Connection to utility maximisation}
\label{sec:utility}
In a regulatory environment, and most scientific fields,
the choice of the significance level, $\alpha$, is a pre-determined
quality criterion.
In the life sciences a one-sided $\alpha$ of 2.5\% is common.
Yet, the exact choice of the threshold $1-\beta$ is much more arbitrary.
In clinical trials, $1-\beta=0.9$ or $1-\beta=0.8$ are common choices when
a classical sample size derivation is conducted.
From the previous section it is already clear that a
generic threshold for $1-\beta$ that is independent of the specific context of a trial only makes sense with conditional approaches like the (conditional prior) quantile approach or when using $\operatorname{EP}(n)$ to derive a required sample size.
In principle, the unconditional $\operatorname{PoS}(n)$ should be easier to interpret
by non-statisticians.
Equation~\eqref{eq:pos-to-ep-cnstr} allows the transformation of an $\operatorname{EP}(n)$-based
sample size derivation, which can readily use any of the established values for $1-\beta$, into a $\operatorname{PoS}(n)$-based sample size derivation by
re-calibrating the threshold with the proportionality factor linking $\operatorname{EP}(n)$ and $\operatorname{PoS}(n)$.
This only transforms the conditional criteria (minimum $\operatorname{EP}(n)$)
of the classical sample size derivation to the unconditional domain (minimum $\operatorname{PoS}(n)$) without affecting the derived sample size in any way.
For instance, if $1-\beta=0.8$ and $\Pr[\,\Theta\geq\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}\,] = 0.683$,
the transformed threshold for $\operatorname{PoS}(n)$ would be~$0.5464$.
In making the assumptions underlying the sample size derivation more transparent by formulating them in terms of unconditional probabilities, there is a need to explain to practitioners why the threshold
now differs from study to study.
The Bayesian view and the prior density $\varphi(\cdot)$ give a natural
answer to this issue of trial-specific thresholds via
the concept of utility maximisation or maximal expected utility (MEU).
An in-depth discussion of the MEU concept is beyond the scope of
this manuscript and we refer the reader to, for example, ~\citet{lindley-1997}.
We merely want to highlight the fact that the choice of the constraint threshold $1-\beta$ can be justified by making the link to MEU principles.
To this end we consider a particularly simple utility function.
Assume that the maximal type~I error rate is still to be controlled at level $\alpha$.
For sake of simplicity, further assume that a \emph{correct}
rejection of the null hypothesis yields an expected return of $\lambda$.
Here the return is given in terms of the average per-patient costs within the trial.
Ignoring fixed costs, the expected trial utility (in units of average per-patient costs) is then given by
\begin{align}
U(n) := \lambda\,\operatorname{PoS}(n) - n
\label{eq:utility-function}
\end{align}
and the utility-maximising sample size is $n_{U}^*(\lambda) := \argmax{n} U(n)$.
Obviously, the same sample size would be obtained by solving problem \eqref{eq:ep-problem}, given
\begin{align}
1-\beta = \operatorname{EP}\big( n_{U}^*(\lambda) \big) = \Pr_{ }[\, \Theta \geq \theta_{\scaleto{\operatorname{MCID}}{3.5pt}} \,] \, \operatorname{PoS}\big(n_{U}^*(\lambda)\big) \ .
\end{align}
The right hand side, $\Pr_{}[\, \Theta \geq \theta_{\scaleto{\operatorname{MCID}}{3.5pt}} \,] \, \operatorname{PoS}\big(n_{U}^*(\lambda)\big)$, is the utility-maximising expected power threshold given the utility parameter~$\lambda$.
Similarly, one could start with $n_{\operatorname{EP}}^*$ for a given $\beta$ and derive the corresponding $\lambda$ such that $n_{U}^*(\lambda)=n_{\operatorname{EP}}^*$.
This value of $\lambda$ would then correspond to the implied expected reward upon successful rejection of the null for given $\beta$.
Under the assumption of a utility function of the form (\ref{eq:utility-function}), $\lambda$ and $\beta$ can thus be matched such that the corresponding utility maximisation problem and the constraint minimisation of the sample size under a power constraint both lead to the same required sample size.
Consequently, practitioners are free to either define an expected return upon successful rejection, $\lambda$, or a threshold on the minimal expected power, $1-\beta$.
While it is theoretically attractive to derive the sample size directly based on a utility function, an informed choice of $\lambda$ is often hard to justify in practice.
In these situations one may instead reverse the perspective and determine the value of $\lambda$ under which the utility-maximising design would coincide with the design obtained under a standard (expected) power threshold of, say, 80\% or 90\%.
The implied reward parameter might then be used to communicate the consequences of different choices of power thresholds to decision makers and to inform the final choice of $1-\beta$.
This approach can, of course, be generalised to more detailed utility functions.
Note, however, that for utility functions with more than one free parameter there is no longer a one-to-one correspondence between power level and utility parameters.
Rather, for a given power level, there will be a level-set of values for the utility parameters that match the specified power.
We give a practical example of this process in Section~\ref{sec:example:clinical-trial}.
\begin{figure}
\centering
\resizebox{0.925\textwidth}{!}{\input{diagram}}
\caption{Structured overview of all quantities related to `power' that are introduced in Sections~\ref{sec:intro} to~\ref{sec:pos}. The symbols used in the text (\textit{Sym.}), their exact definitions (\textit{Def.}), and verbal interpretation (\textit{Int.}) are summarised in the respective boxes. The relationships between the individual quantities are given as labelled arrows.
For an overview of previous mentions and synonyms used in the literature, see Table~\ref{tab:literature-overview}.}
\label{fig:diagram}
\end{figure}
\begin{center}
\renewcommand{\arraystretch}{1.5}
\begin{longtable}{p{3cm}p{4cm}p{8cm}}
\caption{%
Selected publications on `hybrid' sample size derivation based on error rates. Structured by concepts as defined in Figure~\protect\ref{fig:diagram}.
}%
\label{tab:literature-overview}\\
\textbf{Concept} & \textbf{References} & \textbf{Notes} \\
\hline
\endfirsthead
\input{literature-review-table}
\end{longtable}
\end{center}
\section{Literature review of terminology}
\label{sec:literature}
A structured overview of the literature on `hybrid' Bayesian sample size derivation in the context of clinical trials is given
in Table~\ref{tab:literature-overview}.
The table relates publications in the field to the terms defined
in Figure~\ref{fig:diagram}.
Publications with a similar take on the matter are grouped.
In the following, we highlight a few particularly interesting contributions and how they relate to the definitions used in this manuscript.
The majority of the manuscripts only consider
the marginal probability to reject $\mathcal{H}_0$ ($\operatorname{PoS}'(n)$).
Many publications refer to \citet{o2001bayesian} or \citet{ohagan-2005},
where this quantity was introduced as `assurance'.
The range of names for what we call the `marginal probability to reject $\mathcal{H}_0$' is, however, quite diverse:
`assurance', `probability of success',
`predictive probability of success', `average probability of success', `probability of statistical success', `probability of study success', `predictive power', `predictive frequentist power', `expected power', `average power', `strength', `extended Bayesian expected power 1', and `hybrid Neyman-Pearson-Bayesian probability'.
However, only a handful of authors elaborate on the intricacies of defining what exactly constitutes a `success' and
whether to consider an unconditional measure of success or to condition on the presence of a relevant effect for sample size derivation \citep{spiegelhalter1986,brown1987projection,shao-2008,liu-2010,ciarleglio-2015}.
Most publications fail to define explicitly what exactly constitutes a `success'.
Yet, the use of $\operatorname{PoS}'(n)$ implies that rejection of the null hypothesis,
irrespective of its truth, must be considered a success.
Our analysis in Section~\ref{sec:results:pos-for-sample-size-determination}
confirms the statement in \citet{spiegelhalter-2004} that
$\operatorname{PoS}'(n)$ can be used as a practical approximation to
$\operatorname{PoS}(n)$ in many situations.
The exact definition of `probability of success' becomes more
interesting when allowing for $\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}>\theta_0$,
a potential extension rarely considered in the literature
(see, e.g.,~\citet{brown1987projection} for the binary case).
We revisit the distinction between $\operatorname{PoS}'(n)$ and $\operatorname{PoS}(n)$
in a concrete example in Section~\ref{sec:results:pos-for-sample-size-determination}.
The exact choice of wording should not be given too much weight.
However, we feel that any notion of power in the `hybrid' Bayesian/frequentist setting should be \emph{conditional} on a relevant effect (or at least a non-null effect) to preserve the conditional nature of the purely frequentist power.
Using the term `power' to refer to a joint probability like
the `expected power' of
\citet{brown1987projection} and \citet{ciarleglio-2015} (our $\operatorname{PoS}(n)$) or
the `average/expected power' of
\citet{spiegelhalter-2004} (our $\operatorname{PoS}'(n)$) is potentially misleading.
Others suggest `conditional expected power' for $\operatorname{EP}(n)$ to distinguish it from `expected power' (our $\operatorname{PoS}'(n)$)~\citep{brown1987projection,ciarleglio-2015}.
This wording, however, may lead to confusion when also considering interim analyses where `conditional power' is a well-established term for the probability of rejecting the null hypothesis given $\theta_{\scaleto{\operatorname{alt}}{3.5pt}}$ and partially observed data~\citep{bauer-2016}.
A particularly interesting publication is
\citet{liu-2010}.
They extend hybrid sample size derivation in the normal case to also incorporate uncertainty about the
variance and clearly distinguish between
$\operatorname{PoS}'(n)$ = `extended Bayesian expected power 1',
$\operatorname{PoS}(n)$ = `extended Bayesian expected power 2',
and $\operatorname{EP}(n)$ = `extended Bayesian expected power 3'.
Apart from nomenclature, our definitions of these three quantities
only differ in that they assume the standard deviation to be
fixed and the fact that we accommodate the optional notion of a relevant effect via $\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}$.
The former makes explicit formulas more manageable,
the latter is important to keep sample sizes small in situations with vague or conservative prior information but substantial relevance thresholds.
\citet{liu-2010} and \citet{rufibach_15} are also the only
publications we found that study the distribution of the quantities that are averaged over ($\operatorname{RPR}(n)$ and $\rpower{n}$ in our notation, see Figures~\ref{fig:power-distribution} and~\ref{fig:power-distribution-quantile-approach}).
In \citet{ciarleglio-2015}, the distinction between all three
quantities is also made explicit (`expected power' is our $\operatorname{PoS}'(n)$, `prior-adjusted power' is our $\operatorname{PoS}(n)$, and `conditional expected power' is our $\operatorname{EP}(n)$).
\section{Prior elicitation}
A major issue in modelling uncertainty and computing sample size via prior densities is the elicitation of an adequate prior.
At first glance, non-informative or `objective' priors seem to be a viable choice.
As illustrated in \citet{rufibach_15}, the prior crucially impacts the properties and interpretability of $\operatorname{PoS}'$ and likewise any other quantity depending on $\varphi(\cdot)$ in Figure~\ref{fig:diagram}, so careful selection is paramount.
Often in clinical research, there is no direct prior knowledge on the effect size of interest, e.g., overall survival in a phase III trial, as no randomised trials comparing these same treatments using the same endpoint have been run previously.
Researchers are then often tempted to use a {\it vague} prior, typically a normal prior with large variance, as, e.g., advocated in \cite{saint-hilary-2019}.
Assuming a non-informative, improper prior for $\Theta$ would imply that arbitrarily large effect sizes are just as likely as small ones.
Yet, in clinical trials, the standardised effect size rarely exceeds 0.5 \citep{lamberink2018}.
We thus illustrate the characteristics of the different approaches to defining power constraints under uncertainty using a convenient truncated Gaussian prior.
The truncated Gaussian is conjugate to a Gaussian likelihood and allows us to restrict the plausible range of effect sizes to, e.g., liberally $[-1, 1]$.
Also, the truncated Gaussian is the maximum entropy distribution on the truncation interval, given mean and variance which can be interpreted as a `least-informative' property under constraints on the first two moments.
Alternatively, a trial designer can formally elicit a prior on the effect size of interest.
Kinnersley and Day describe a structured approach to the elicitation of expert beliefs in a clinical trial context based on the SHELF framework~\citep{kinnersley_13,shelf}.
\citet{dallow_18} discusses how SHELF is routinely used within a pharmaceutical company to define prior distributions that are used in conjunction with calculation of probability of success and to inform internal decision making at key project milestones.
Formal and informal prior elicitation is also discussed in~\citet{spiegelhalter-2004}.
\section{Results}
\subsection{Comparison of required sample sizes for various prior choices}
Let $\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}=0.1$ and let the maximal feasible sample size be 1000.
Figure~\ref{fig:n-vs-prior-parameters} shows the required sample sizes under the expected power, the probability of success, and the quantile approach (for $\gamma = 0.5, 0.9$).
We use $\alpha = 0.025$ and $\beta = 0.2$.
\begin{figure}[htbp]
\centering
\includegraphics[width=\textwidth]{%
figures/fig2-required-sample-size-comparison}
\caption{Required sample size plotted against prior parameters (Normal truncated to [-0.3, 0.7], with varying mean and standard deviation); $\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}=0.1$; EP = Expected Power, PoS = Probability of Success, quantile = quantile approach with $\gamma=0.5$ and $\gamma=0.9$, respectively.}
\label{fig:n-vs-prior-parameters}
\end{figure}
The patterns of required sample sizes are qualitatively different between the three approaches.
For probability of success, large prior uncertainty implies low \textit{a~priori} probability of a relevant effect and thus the required sample sizes explode for large prior standard deviations (in relation to the prior mean).
For very large standard deviations, the constraint on probability of success becomes infeasible.
The expected power criterion leads to a completely different sample size pattern.
Since expected power is defined conditional on a relevant effect, large prior uncertainty increases the weight in the upper tails of the power curve where power quickly approaches one.
Consequently, for small prior means, larger uncertainty decreases the required sample size.
For large prior means, however, smaller prior uncertainty leads to smaller sample sizes since again more weight is concentrated in the tails of the power curve.
The characteristics of the prior-quantile approach
very much depend on the choice of $\gamma$.
When using the conditional prior median ($\gamma=0.5$) the approach is qualitatively similar to the expected power approach.
This is due to the fact that computing power on the conditional median of the prior is close to computing power on the conditional prior mean.
Since the power function is locally linear around the centre of mass of the conditional prior, this approximates computing expected power by interchanging forming the expected value and computing power (i.e., first average the prior and then compute power or average over power with weights given by the conditional prior).
For a stricter criterion ($\gamma=0.9$) the required
sample sizes are much larger.
This is due to the fact that the quantile approach
does not allow a trade-off between power in the upper tails of the power curve and regions with low power. Higher uncertainty then decreases the $(1-\gamma)$-quantile towards the minimal relevant effect and thus increases the required sample size.
\subsection{Probability of success as the basis for sample size derivation}
\label{sec:results:pos-for-sample-size-determination}
In theory, one might be inclined to derive a sample size based on the probability of success instead of using expected power.
Consider a situation in which the \textit{a~priori} probability of $\Theta \geq \theta_{\scaleto{\operatorname{MCID}}{3.5pt}}$ is $0.51$.
The probability of success is then only $41\%$ (for 80\% expected power) or $46\%$ (for 90\% expected power).
A sponsor might want to increase these relatively low unconditional success probabilities by deriving a sample size based on a minimal $\operatorname{PoS}(n)$ of $1-\beta$ instead.
The choice of $1-\beta$ is limited by the \textit{a~priori}
probability of a relevant effect (0.51 in this case).
Using equation~\eqref{eq:pos-to-ep-cnstr} a minimal probability of success of 0.5 is equivalent to requiring an expected power of more than 98\%.
In essence, the attempt to increase $\operatorname{PoS}(n)$ via a more stringent threshold on $\operatorname{EP}(n)$ implies that low \textit{a~priori} chances of success are to be offset with an almost certain detection ($\operatorname{EP}(n)\approx 1$) in the unlikely event of an effect actually being present.
The ethical implication of this approach to sample size derivation is that an extremely large number of patients would be exposed to treatment although the sponsor more or less expects it to be ineffective.
This example demonstrates that the (situation agnostic) conventions on power levels cannot be transferred to thresholds for probability of success without adjustment for the situation-specific \text{a~priori} probability of a relevant effect.
It thus seems much easier to directly impose a constraint on expected power, which implicitly adjusts for the \textit{a~prior} probability of a relevant effect via equation~\eqref{eq:pos-to-ep-cnstr}.
To investigate the difference between the probability of success, $\operatorname{PoS}(n)$, and the marginal probability of rejecting the null hypothesis, $\operatorname{PoS}'(n)$, Figure~\ref{fig:pos-components} visualises the proportion of the individual components of $\operatorname{PoS}'(n)$ for varying prior standard deviation and prior means.
The sample size is fixed at $n=150$, $\theta_0=0$, the maximal type~I error rate is $\alpha=0.025$, and the minimal clinically important difference is $\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}=0.1$.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{%
figures/fig3-pos-prime-composition}
\caption{Components of $\operatorname{PoS}'(n)$ for $n=150$, $\theta_0=0$, $\alpha=0.25$, $\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}=0.1$ and varying prior mean and standard deviation;
numbers correspond to overall $\operatorname{PoS}'(n)$;
proportions in individual pie charts correspond to:
A = probability to reject and null effect (type~I error),
B = probability to reject and irrelevant but non-null effect,
C = probability to reject and relevant effect ($\operatorname{PoS}$).}
\label{fig:pos-components}
\end{figure}
Evidently, the contribution of type~I errors (component `A' in Figure~\ref{fig:pos-components}) to $\operatorname{PoS}'(n)$ is mostly negligible unless the prior is sharply peaked at an effect size slightly smaller than the null.
The \textit{a~priori} probability of a relevant effect size is close to zero in these cases and so is $\operatorname{PoS}'(n)$.
For the more practically relevant scenarios with prior mean greater than $\theta_0=0$, the contribution of the average type~I error rate to $\operatorname{PoS}'(n)$ is almost negligible.
Still, if $\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}>\theta_0$, $\operatorname{PoS}'(n)$ might be inflated substantially by rejections under parameter values that are non-null but also not clinically relevant.
This phenomenon evidently depends on the magnitude of $\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}$; the more of the prior mass concentrated in $[\,0,\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}\,]$, the larger the contribution towards $\operatorname{PoS}'(n)$.
If $\theta_{\scaleto{\operatorname{MCID}}{3.5pt}} = 0$, the numeric difference between $\operatorname{PoS}$ and $\operatorname{PoS}'$ is negligible since the maximal type~I error rate is controlled at level $\alpha$ and the power curve quickly approaches zero on the interior of the null hypothesis.
This was already pointed out by \citet{spiegelhalter-2004},
who argue that $\operatorname{PoS}'(n)$ can be used as approximation to $\operatorname{PoS}(n)$ in many practically relevant situations.
\subsection{Distribution of random power under a constraint on expected power}
To further investigate the properties of the random variable $\Pr_{\Theta}[\,Z_n>z_{1-\alpha}\,]$, we consider three example prior configurations with means $-0.25, 0.3, 0.5$ and standard deviations $0.4, 0.125, 0.05$ respectively.
The corresponding sample sizes to reach an expected power of at least 80\% are 854, 126, and 32.
Figure~\ref{fig:power-distribution} shows the unconditional and conditional (on a relevant effect) priors, the corresponding probability of rejecting the null as a function of $\theta$, and histograms of the distributions of random power ($\rpower{n}$), and the unconditional probability to reject the null hypothesis ($\operatorname{RPR}(n)$).
\begin{figure}
\centering
\includegraphics[width=\textwidth]{%
figures/fig4-power-distribution-ep-approach}
\caption{%
Top: conditional and unconditional prior PDF and power curves corresponding to $n_{\protect\scalebox{.5}{EP}}^*$ for $1-\beta=0.8$.
Bottom: histogram of the random power (conditional on relevant effect) and the random probability to reject (unconditional); vertical lines indicate 80\% power and the numbers are the respective (conditional) probabilities to exceed a probability to reject of 80\%.
}
\label{fig:power-distribution}
\end{figure}
Clearly, the distributions of the conditional and unconditional rejection probabilities (random power and random probability to reject, respectively) are qualitatively very different in the three situations.
In the first case (mean $-0.25$, standard deviation 0.4), the prior mass is mostly concentrated on the null hypothesis and the normalising factor that links the unconditional and the conditional prior is clearly noticeable.
The conditional prior then assigns most weight to values of $\theta$ close to the relevance threshold leading to
a large required sample size.
The large sample size then implies a steep power curve and a distribution of $\rpower{n}$ that is highly right-skewed towards~1 since $\rpower{n}$ is conditional on $\Theta>\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}$.
If the unconditional distribution of the rejection probability is considered instead ($\operatorname{RPR}(n)$), the characteristic u-shape discussed in \citet{rufibach_15} is recovered.
For the intermediate setting (mean 0.3, standard deviation 0.125), most prior mass is already concentrated on relevant values.
The difference between conditional and unconditional prior is less pronounced (the normalising factor is closer to~1) and even the unconditional distribution of the rejection probability is no longer u-shaped.
Finally, in the last setting (mean 0.5, standard deviation 0.05), $\theta$ is almost certainly highly relevant.
Since the normalising factor is thus close to~1, there is no discernible difference between the conditional and the unconditional prior densities.
Not surprisingly, the assumption of a highly relevant effect with high certainty only requires a small sample size (32).
This leads to a relatively flat curve of the rejection probability and to a peaked distribution of $\rpower{n}$ and $\operatorname{RPR}(n)$.
Flat power curves and high certainty about the effect size tend to result in peaked distributions of $\rpower{n}$ and $\operatorname{RPR}(n)$ because the power curve is almost linear in the region of the parameter where the prior mass is concentrated.
The distribution of $\rpower{n}$ and $\operatorname{RPR}(n)$ is thus well approximated by a linear transformation of the (conditional) prior, which is a peaked truncated normal distribution.
Since conditioning has almost no effect, the unconditional distribution of the probability to reject is the same in this case.
Interestingly, both settings with higher \textit{a~priori} uncertainty lead to a high chance of exceeding a power of 80\%.
This is due to the fact that the rare occurrence of very low rejection probabilities needs to be compensated to achieve an overall expected power of 80\%.
\subsection{Distribution of random power under quantile approach}
To compare the results in the previous section with the prior quantile-based approach, we consider the intermediate example with prior mean 0.3 and prior standard deviation 0.2 again.
For this situation, the required sample sizes under $\gamma=0.5,0.9$ and target power $0.7$ or $0.8$, the corresponding curves of the rejection probability, and histograms of the distribution of the rejection probability are given in Figure~\ref{fig:power-distribution-quantile-approach}.
\begin{figure}
\centering
\includegraphics[width=.99\textwidth]{%
figures/fig5-power-distribution-quantile-approach}
\caption{%
Top: conditional and unconditional prior PDF for a truncated normal prior on $[-0.3, 0.7]$ with mean 0.3, standard deviation 0.2 and
power curves corresponding to $n_{\gamma}^*$ for $1-\beta=0.8$ or $0.7$ and $\protect\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}=0.1$;
Bottom: histogram of the random power (conditional on relevant effect) and the random probability to reject (unconditional); vertical lines mark 80\% power and the number are the respective (conditional) probabilities to exceed a probability to reject of either 70\% or 80\%.
}
\label{fig:power-distribution-quantile-approach}
\end{figure}
The required sample sizes depend heavily on the choice of $\gamma$.
The crucial difference between the quantile-based and the expected power approach is that for the prior quantile approach the exact distribution of power below the target value of 80\% is irrelevant; only the total mass of the distribution below this critical point matters.
This means that the sample size for the $\gamma=0.5$ cases are substantially lower than the corresponding sample size derived from an expected power constraint.
The flip side of this ignorance about \emph{the exact amount} by which the target power is undershot is that there is a relatively high chance of ending up with a severely underpowered study in these cases.
Increasing the certainty to exceed a power of 80\% or 70\% by setting $\gamma=0.9$, however, leads to substantially larger required sample sizes than under the expected power approach.
The example demonstrates the problems arising from having to specify both $\gamma$ and $1-\beta$.
While this allows more fine-grained control over the distribution of the (conditional) rejection probability, there seems to be no canonical choice for $\gamma$, which is critical in determining the required sample size.
\subsection{A clinical trial example}
\label{sec:example:clinical-trial}
To make things more tangible, consider the case of a clinical trial designed to demonstrate superiority of an intervention over a historical control group with respect to the endpoint of overall survival.
To stay within the framework of (approximately) normally distributed test statistics, we assume that effect sizes are given on a standardised log hazard scale, i.e., $\theta=0$ corresponds to no difference in overall survival and $\theta>0$ to superiority of the intervention group.
Assume that the prior for the treatment effect of the intervention is given by a truncated Normal distribution on $[\,-0.3, 0.7\,]$ with mean $0.2$ and standard deviation $0.2$ (pre-truncation).
The minimal clinically relevant difference is set to $\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}=0.05$.
This setting corresponds to an \textit{a~priori} probability of a relevant effect of approximately $0.86$.
Figure~\ref{fig:example} shows the (conditional) prior density, the curves of the rejection probability corresponding to the required sample sizes derived from constraints on a minimal probability to reject of $1-\beta=0.8$ at $\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}$ (MCID), at ${\quant{0.5}[\,\Theta\geq\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}\,]\approx0.26}$ (quantile, 0.5), at ${\quant{0.9}[\,\Theta\geq\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}\,]\approx0.10}$ (quantile, 0.9), or a minimal expected power of $1-\beta=0.8$ (EP).
\begin{figure}
\centering
\includegraphics[width=\textwidth]{%
figures/fig6-clinical-trial-example}
\caption{Left panel: prior PDF and conditional prior PDF (on $\Theta>\theta_{\scaleto{\operatorname{MCID}}{3.5pt}} = 0.05$); middle panel: probability to reject the null hypothesis as function of $\theta$ for the expected power design (EP), the design powered for $\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}$ (MCID), the design based on power at the conditional prior median (quantile 0.5),
and the design using the conditional 0.1-quantile, i.e. $\gamma=0.9$ (quantile 0.9); right panel: CDF of random power (probability to reject given $\Theta>\theta_{\scaleto{\operatorname{MCID}}{3.5pt}} = 0.05$) for the four different design choices.}
\label{fig:example}
\end{figure}
In this case the MCID criterion requires $n=3140$.
The quantile approach (with $\gamma=0.9$) already reduces this to $n=834$ while still maintaining an \textit{a~priori} chance of 90\% to exceed the target power of 80\%.
The quantile approach with $\gamma=0.5$ results in the lowest sample size of $n=120$ at the cost of only having a 50\% chance to exceed the target power of 80\%.
The EP approach is more liberal than the quantile approach ($\gamma=0.9$) with $n=218$ but still guarantees a chance of exceeding the target power of roughly 75\%.
A sample size based on $\operatorname{PoS}(n)\geq1-\beta=0.8$ cannot be derived in this example since the \textit{a~priori} probability of a relevant value is lower than $0.8$.
The large spread between the derived sample sizes shows how sensitive the
the required sample size is to the changes in the power constraint.
Clearly, the MCID approach is highly ineffective, as accepting a small chance to undershoot the target power with the quantile approach ($\gamma=0.9$) reduces the required sample size from $n=3140$ to roughly a quarter ($n=834$).
At the other extreme, constraining power only on the conditional prior median (quantile approach, $\gamma=0.5$) leads to a rather unattractive \textit{a~priori} distribution of the random power:
by definition, the probability to exceed a rejection probability of $0.8$ is still $0.5$ but the \textit{a~priori} chance of ending up with a severely underpowered study is non-negligible.
These considerations leave the trial-sponsor with essentially two options.
Either a range of scenarios for the quantile approach with values of $\gamma$ between $0.5$ and $0.9$ could be discussed in more detail and a decision on the exact value of $\gamma$ could be reached by considering the corresponding distributions of $\rpower{n}$, or the intermediate EP approach could be used.
We assume that the trial-sponsor accepts the implicit trade-off inherent to expected power and decides to base the sample size derivation on the EP approach.
The required sample size for an (expected) power of 80\% is then $n=218$.
Note that this still means that there is a roughly one-in-five \textit{a~priori} probability to end up in a situation with less than 50\% power (see Figure~\ref{fig:example}, CDF panel).
In a situation where $1-\beta=0.8$ is not set in stone, further insights might be gained by making the link to
utility maximisation explicit.
In a first step, we will assume that the sponsor has no way of quantising the reward parameter $\lambda$ directly.
One may then guide decision making by computing the values of $\lambda$ that lead to the same required sample size for a range of values of $\beta$.
Figure~\ref{fig:matched-reward} shows this `implied reward' as function of the minimal expected power constraint.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{figures/fig7-matched-reward}
\caption{%
Utility-maximising implied reward $\lambda$ for varying expected power levels in the situation discussed in Section~\ref{sec:example:clinical-trial}.
}
\label{fig:matched-reward}
\end{figure}
An expected power of $0.8$ is thus ideal in this situation if the expected reward upon successful (i.e., the effect is indeed relevant) rejection of the null hypothesis is approximately $1732$ times the average per-patient costs within the planned trial.
Using the curve depicted in Figure~\ref{fig:matched-reward}, a discussion with the trial sponsor about the plausibility of certain reward levels can be started.
Usually the average per-patient costs are well-known in advance, so that the scale can even be transformed to monetary units, e.g., \$US.
Assume to this end, that for the particular study at hand, the expected average per-patient costs are $30\,000$ \$US.
Then, the sample size corresponding to an expected power of $0.8$ is maximising utility if the expected reward is $30\,000\cdot1732=51.96\cdot 10^6$ \$US.
The utility-maximising reward for an expected power of $0.9$ would be approximately $6006$, i.e., $180.18\cdot 10^6$ \$US.
Even without committing to a fixed value of $\lambda$, these considerations can be used to guide the decision as to which of the `standard' power levels (0.8 or 0.9) might be more appropriate in the situation at hand.
Of course, one might also directly optimise utility if the reward upon successful rejection of the null hypothesis can be specified.
To that end, assume that a reward of $100\cdot 10^6$ \$US is expected.
Under the same assumption about average per-patient costs, this translates to $\lambda\approx3333$.
The utility-maximising sample size is then $n=329$ and the corresponding utility-maximising expected power is $0.86$.
\section{Discussion}
The concept of `hybrid' sample size derivations based on
Bayesian priors for planning and the design's frequentist error rate properties is well-established in the literature on clinical trial design.
Nevertheless, the substantial variation in the terminology used and small differences in the exact definition of the terms used can be confusing.
We have tried our best to formulate a consistent naming scheme,
to be explicit about the exact definitions,
highlight connections between the different quantities
(see Figure~\ref{fig:diagram}),
and to relate back to previous authors (see Table~\ref{tab:literature-overview}).
Any naming scheme necessarily has a subjective element to it
and ours is by no means exempt from this problem (see also \url{https://xkcd.com/927/}).
We do hope, however, that our review encourages a clearer separation
between terminology for joint probabilities (avoiding the use of the word `power') and for probabilities that condition on the presence of an effect (`power' seems more appropriate here), as well as a more transparent distinction between arguments based on \textit{a~priori} likelihood of effects and their relevance.
We also strongly believe that an explicit definition (in formulae)
of any quantities used should be given when discussing
the subject.
Merely referring to terms like `expected power' or
`probability of success' are too ambiguous given their inconsistent use in the literature.
Often, the main argument for a `hybrid' approach to sample size derivation is the fact that the uncertainty about the true underlying effect can be incorporated in the planning of a design.
This is certainly a major advantage but it is equally important that
the `hybrid' approach allows a very natural distinction between
arguments relating to the (relative) \textit{a~priori} likelihood of different parameter values (encoded in the prior density)
and relevance arguments (encoded in the choice of $\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}$).
The fact that these two components can be represented naturally within the `hybrid' approach has the potential to make sample size derivation much more transparent.
The `hybrid' quantity considered most commonly in the literature is the marginal probability to reject $\mathcal{H}_0$.
Often, it is not clear whether the authors are aware of the fact that this quantity includes the error of rejecting the null hypothesis incorrectly, i.e. when $\theta<\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}$.
In many practical situations this problem is numerically negligible and $\operatorname{PoS}'(n)\approx\operatorname{PoS}(n)$, i.e.,~the marginal probability to reject the null hypothesis is approximately the same
as the joint probability of a non-null effect and the rejection of the null hypothesis.
If, however, the definition of `success' also takes into account a non-trivial relevance threshold $\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}>\theta_0$,
the distinction becomes more important in practice.
Given the great emphasis on strict type~I error rate control in the clinical trials community it seems at least strange to implicitly consider type~I errors as `successful' trial outcomes.
Beyond these principled considerations,
a practical advantage of $\operatorname{PoS}(n)$ over $\operatorname{PoS}'(n)$ is the
direct and simple connection to $\operatorname{EP}(n) = \operatorname{PoS}(n) / \Pr[\,\Theta\geq\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}\,]$.
While $\operatorname{EP}(n)$ is independent of the \textit{a~priori} probability
of a relevant effect and only depends on the relative \textit{a~priori} likelihood of different effects through the
conditional prior, $\operatorname{PoS}(n)$ does directly depend on $\Pr[\,\Theta\geq\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}\,]$.
Although \citet{spiegelhalter-2004} see this as a disadvantage of $\operatorname{EP}(n)$, it is actually a necessary property to use it for sample size derivation without re-calibrating the conventional values for $1-\beta$ (see also \citealp{brown1987projection}).
If one tried to derive a sample size such that $\operatorname{PoS}(n)=0.8$ this would be impossible for situations with $\Pr[\,\Theta\geq\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}\,]<0.8$.
In a situation where $\Pr[\,\Theta\geq\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}\,]$ exceeds 0.8 only slightly, the expected power ($\operatorname{EP}(n)$) would have to be close to 1 to compensate for the \textit{a~priori} probability of a relevant effect.
In essence, one would thus increase the sample size in situations where the efficacy of the new treatment is still uncertain.
This would put more study participants at risk just to make sure that the treatment effect is detected almost certainly if it is indeed present.
The use of $\operatorname{PoS}(n)$ for sample size derivation thus only makes sense in a setting where the threshold $1 - \beta$ is adapted to the \textit{a~priori} probability of a relevant effect.
The simplest way to do so is by using $\operatorname{PoS}(n) \geq \Pr[\,\Theta\geq\theta_{\scaleto{\operatorname{MCID}}{3.5pt}}\,]\,(1 - \beta)$ which is, however, entirely equivalent to $\operatorname{EP}(n) \geq 1 - \beta$.
Another option to derive situation-specific thresholds is via utility maximisation, and $\operatorname{PoS}(n)$ is a key term in the simple expected utility function proposed in Section~\ref{sec:utility}.
Ultimately, $\operatorname{PoS}(n)$ and $\operatorname{EP}(n)$ can be used interchangeably once the prior distribution is fixed as long as the respective multiplicative factor is taken into account.
The main advantage of $\operatorname{PoS}(n)$ is that it is an unconditional probability
which might be easier to interpret by practitioners, while $\operatorname{EP}(n)$ can be readily used in conjunction with an already established power threshold in a research field.
A slightly different concept to sample size derivation via expected power is what we call the `quantile approach'.
This approach uses a different functional of the probability to reject the null hypothesis given a relevant effect.
Instead of the mean, we propose to use a $\gamma$ quantile of this distribution.
Compared to expected power, this allows direct control of
the left-tail of the \textit{a~priori} distribution of the probability to reject the null hypothesis given a relevant effect.
This can be desirable since a sample size derived via a threshold for expected power might still lead to a substantial chance of ending up with an underpowered study.
This can be avoided with the quantile approach and a higher value for $\gamma$ (see Figure~\ref{fig:power-distribution-quantile-approach}).
The quantile approach is also relatively easy to implement in practice, since it is just a Bayesian justification for powering on a point alternative.
This flexibility comes at the price of having to specify an additional parameter, $\gamma$ (the acceptable risk of ending up with an underpowered study).
Theoretically, both expected power and the prior quantile approach are perfectly viable to determine a sample size.
Whichever approach is preferred, it is certainly advisable to
not only plot the corresponding power curves but also the resulting distribution of $\rpower{n}$ (see Figure~\ref{fig:power-distribution}).
In essence, the problem of defining a `hybrid' power constraint boils down to finding a summary functional of the power curve that reflects the planning objectives.
Ideally, one would like to control the \textit{a~priori} distribution of $\rpower{n}$ such that it is sharply peaked around a certain target value avoiding both over- and underpowered studies.
Yet, controlling both location (e.g.,~mean) and spread (e.g.,~standard deviation) of the distribution of $\rpower{n}$ is
impossible.
A second constraint on the standard deviation of $\rpower{n}$
in addition to the mean constraint (expected power)
would led to an over-determined problem since there is only one free parameter,~$n$.
To increase expected power, the sample size must be increased.
The standard deviation of $\rpower{n}$, however, decreases as the sample size is lowered since this flattens the power curve of the resulting test (the standard deviation would be 0 if the power curve was constant).
Both conflicting objectives (high expected power, low standard deviation of power) are thus not fulfillable at the same time.
Finally, it should be stressed again that the key frequentist property of strict type~I error rate control of the designs are not affected by the fact that the arguments for calculating a required sample size are Bayesian.
In fact, at no point, the Bayes theorem is invoked (i.e.~the posterior distribution of the effect size is not required).
The Bayesian perspective is merely a principled and insightful way of specifying a weight function (prior density) that can then be used to guide the choice of the power level of the design, or as \citet{brown1987projection} put it: ``This proposed use of Bayesian methods should not be criticised by frequentists in that these methods do not replace any current statistical techniques, but instead offer additional guidance where current practice is mute''.
\subsection*{Supplemental Materials}
The code required to reproduce the figures is available at \url{https://github.com/kkmann/sample-size-calculation-under-uncertainty}.
A permanent backup of the exact version of the repository used for this manuscript is available under the digital object identifier
\href{https://doi.org/10.5281/zenodo.3899943}{10.5281/zenodo.3899943} (release 0.2.1).
An interactive version of the repository at the time of publication is
hosted at \url{https://mybinder.org/v2/gh/kkmann/sample-size-calculation-under-uncertainty/0.2.1?urlpath=lab/tree/notebooks/figures-for-manuscript.ipynb} using Binder~\citep{jupyter-2018}.
A simple shiny app implementing the sample size calculation procedures is
available at \url{https://mybinder.org/v2/gh/kkmann/sample-size-calculation-under-uncertainty/0.2.1?urlpath=shiny/apps/sample-size-calculation-under-uncertainty/}.
\subsection*{Funding}
DSR was funded by the Biometrika Trust and the Medical Research Council (MC\_UU\_00002/6).
\subsection*{Conflicts of interest}
None to declare.
|
1,116,691,499,300 | arxiv | \section{Introduction}
Lattice QCD aims at describing hadrons and interactions between them based on first-principle Quantum ChromoDynamics. It is based on the evaluation of correlation functions by means of Feynman path integral of QCD action on discretized and finite Euclidean space-time. I review recent results on spectrum for the exotic and conventional mesons.
\section{ The discrete spectrum from lattice and information encoded in it}
The physics information on a meson (below, near or above threshold) is commonly extracted from the discrete energy spectrum in lattice QCD.
The physical system for given quantum numbers is created from the vacuum $|\Omega\rangle$ using interpolator ${\cal O}_j^\dagger$ at time $t\!=\!0$ and the system propagates for time $t$ before being annihilated by ${\cal O}_i$. To study a meson state with given $J^P$ one can use ${\cal O}\simeq \bar q \Gamma q,~$ $(\bar q \Gamma_1 q)_{\vec p_1}(\bar q \Gamma_2 q)_{\vec p_2},~$$[\bar q \Gamma_1 \bar q][q\Gamma_2 q]$ with desired quantum numbers. After the spectral decomposition the correlators are expressed in terms of the energies $E_n$ of eigenstates $|n\rangle$ and their overlaps $Z_j^n$
\begin{equation}
\label{C}
C_{ij}(t)= \langle \Omega|{\cal O}_i (t) {\cal O}_j^\dagger (0)|\Omega \rangle=\sum_{n}Z_i^nZ_j^{n*}~e^{-E_n t}~,\qquad Z_j^n\equiv \langle \Omega|{\cal O}_j|n\rangle~.
\end{equation}
The most widely used method to extract the discrete spectrum $E_n$ and overlaps from the correlation matrix $C_{ij}(t)$ is the generalised
eigenvalue method $C(t)u^{(n)}(t)=\lambda^{(n)}(t)C(t_0)u^{(n)}(t)$ \cite{Michael:1985ne,Blossier:2009kd}.
The energies $E_n$ are extracted from the exponential
behavior of the eigenvalues $\lambda^{(n)}(t)\propto e^{-E_n t}$ at large $t$.
All physical eigenstates with given quantum numbers appear as energy levels in principle. These can be "one-meson" states (for example $\chi_{c1}$ in $1^{++}$ charmonium channel), "two-meson" states (for example $D\bar D^*$) and the multi-meson states (for example $J/\psi \pi\pi$). In reality the eigenstates are mixtures of these Fock components. Three- and more-meson states have never been taken into account in the study of the meson spectroscopy yet. The major step during the preceding years came from treating two-meson states rigorously.
These have discrete spectrum due to periodic boundary condition on finite lattice. If the two mesons do not interact, then momenta of each meson is $\vec{p}= \!\tfrac{2\pi}{L}\vec{N}$ with $\vec{N}\in {N}^3$, and the energies of $M_1(\vec p)M_2(-\vec p)$ are $E^{n.i.}=E_1(p)+E_2(p)$ with $E_{1,2}(p)=(m_{1,2}^2+p^2)^{1/2}$. The energies $E_n$ extracted from the lattice are slightly shifted in presence of the interaction and the shift provides rigorous information on the scattering matrix, as discussed bellow.
In experiment, two-meson states correspond to the two-meson decay products
with a continuous energy spectrum.
The mass of a hadron well below strong decay threshold is simply $m=E\vert_{\vec {p}=0}$. The masses of resonances and near-threshold bound-states have to be inferred from the infinite-volume scattering matrix of the one-channel (elastic) or multiple-channel (inelastic) scattering. The bound states correspond to the poles of scattering matrix on the real axis below threshold, while the resonances masses and widths are extracted from the Breit-Wigner type fits of the corresponding cross-section or phase shift. The bound states and narrow resonances typically manifest themselves as levels that appear in addition to the expected discrete two-meson levels.
The most widely used approach for extracting the infinite-volume scattering matrix from the finite volume $E_n$ is based on L\"usher's seminal work and its generalizations. In the case of elastic scattering between two hadrons with zero-total momenta, the energy $E_n=(m_1^2+p^2)^{1/2}+(m_2^2+p^2)^{1/2}$ renders momenta $p$ of each meson in the region outside the interaction. The infinite volume phase shift at that energy is given by the L\"uscher's relation $\delta_l(p)=\mathrm{atan}[\sqrt{\pi} p L/2Z_{00}(1;(pL/2\pi)^2)]$ if the partial wave $l$ dominates the scattering \cite{Luscher:1991cf}.
This is a favorable case where one equation determines one unknown $\delta_l(E_n)$ for each energy level $E_n$. The generalizations of this relation to multiple partial waves, non-zero total momenta, twisted boundary conditions, coupled-channel scattering and three-particle systems have also been derived in a series of papers recently. For each energy level $E_n$ this generally leads to one (determinant) equation with several unknown $\delta_l^{a}(E_n^{cm})$ and the rigorous extraction becomes much more challenging. In this case the analysis may relay on certain parametrizations of the scattering matrix as a function of $E^{cm}$, which may render otherwise unsolvable problem tractable. It is encouraging that the Hadron Spectrum Collaboration presented the first simulation of two-coupled channel system $K\pi-K\eta$ and extracted the poles corresponding to strange mesons relying on the parametrization of the scattering matrix \cite{Dudek:2014qha}.
The overlaps $Z_i^n=\langle \Omega|{\cal O}_i|n\rangle$ provide wealth of information about the composition of each lattice eigenstate $|n\rangle$. This information has been used so far mostly as a qualitative guidance on the importance of various Fock components. It remains an open question how to use this rich source of information to rigorously extract physics information on the physical states (especially for smeared quarks) and analytic considerations in this direction may prove fruitful.
\section{Mesons well below threshold}
Well below strong decay threshold there are no multi-hadron states, and the mass of a single hadron is extracted from $m=E\vert_{\mathbf{P}=0}$ extrapolated to $L\to \infty$, $a\to 0$ and $m_q\to m_q^{phys}$. Particular care has to be taken concerning discretization errors related to heavy quarks and complementary methods lead to compatible results in the continuum limit.
Many precision results are available for a number of years. The continuum and chiral extrapolations of low-lying charmonia were, for example, addressed by Fermilab/MILC \cite{Mohler:lat14} and HPQCD/MILC \cite{Galloway:lat14} collaborations recently. The resulting splittings between ground-state masses in different channels as well as spin-averaged masses of $2S$ and $1S$ charmonia are in good agreement with experiment. In all simulations of charmonia and other hidden charm channels reported here, the charm-quark annihilation contribution is omitted (while possible Wick contractions with $u/d/s$ annihilation are taken into account) and it is indeed OZI suppressed in experiment. The rigorous treatment of charm annihilation presents an unsolved problem due to the mixing with a number of light hadron channels and the noise in the disconnected diagrams.
The $\eta$ and $\eta'$ can strongly decay only to the three-meson states, therefore they are very narrow and can be treated using standard technique to a good approximation. Their masses as well as the flavour mixing angle were determined as a function of $m_\pi$ by ETMC collaboration \cite{Michael:2013gka}, recovering experimental values in the chiral limit.
\section{Excited mesons within a single-hadron approach}
The great majority of hadrons lie near or above strong decay threshold. Yet most of them have been treated until recently based on a single-hadron approximation. For meson states this entails (i) using only quark-antiquark interpolating fields ${\cal O}\simeq \bar qq$ for mesons, (ii) assuming that all energy levels correspond to ``one-particle'' states and (iii) that the mass of the excited resonance equals $m\!=\!E$. These are strong assumptions for the resonances, which are not asymptotic states. The approach also ignores the effect of the threshold on near-threshold states.
The most extensive light isoscalar \cite{Dudek:2013yja}, $D$, $D_s$ \cite{Moir:2013ub} and $\bar cc$ \cite{Liu:2012ze} spectrum was extracted by the Hadron Spectrum Collaboration (HSC) on $N_f\!=\!2\!+\!1$ anisotropic configurations with $m_\pi\simeq 400~$MeV. The mixing angle between $(\bar uu+\bar dd)/\sqrt{2}$ and $\bar ss$ components for isoscalar mesons mass was also calculated and the mixing is found small for most of the states \cite{Dudek:2013yja}. The continuum $J^{PC}$ was reliably identified using advanced spin-identification method. An impressive number of excited states was extracted in each channel with a good accuracy in spite of the disconnected contribution for isoscalars. States are identified with members of $\bar qq$ multiplets $nS$, $nP$, $nD$ and $nF$ based on overlaps $\langle O_i|n\rangle$, where interpolators are chosen to resemble multiplet members.
There are several remaining states which are identified as hybrids: they do not fit $\bar qq$ multiplets and show strong overlap with ${\cal O}\simeq \bar q F_{\mu\nu}q$.
\section{Near-threshold mesons (beyond single-hadron approach)}
Most of the exciting states found by experiments are located near thresholds, for example $X(3872)$, $Z_c^+(3900)$, $Z_b^+(10610)$, $Z_b^+(10650)$, $D_s^0(2317)$ and $\Lambda(1405)$. The quarkonium-like states, which lie near threshold and above threshold, are listed in Tables 10 and 12 of a review by Brambilla et al. \cite{Brambilla:2014jmp}. Identifying whether these states arise from QCD or not, and what is their nature, presents an exciting and important challenge to the lattice community.
Indeed most of the effort in the hadron spectroscopy during past few years went in going beyond the single-hadron approximation and taking into account two-hadron eigenstates rigorously. Note that majority of the studies focus on the (elastic) energy region near threshold, where the methods may be tractable at present, but one can not expect spectra of highly excited multiplets from rigorous approach soon.
\\
\underline{$\mathbf{Z_c^+}$}: Several charged-charmonia with quark content $\bar cc\bar d u$ were discovered recently in experiment. Most notably these are $Z^+(4430)$ with $J^P=1^+$ discovered by Belle and confirmed by LHCb, and $Z_c^+(3900)$ with unknown $J^P$ discovered by BESIII and confirmed by Belle and CLEOc \cite{Brambilla:2014jmp}. In is important to note that $Z_c^+(3900)$ was found in $J/\psi\,\pi$ invariant mass only through $e^+e^- \to Y(4460)\to (J/\psi\, \pi^+)\pi^- $. No resonant structure in $J/\psi\, \pi^+$ was seen in $\bar B^0\to (J/\psi\, \pi^+) K^-$ by BELLE \cite{Chilikin:2014bkk}, in $\bar B^0 \to (J/\psi \pi^+) \pi^-$ by LHCb \cite{Aaij:2014siy} or in $\gamma p\to (J/\psi \,\pi^+) n$ by COMPASS \cite{Adolph:2014hba}. This might indicate that the peak seen in $Y(4460)$ decay might not be of dynamical origin \cite{Chen:2013coa,Swanson:2014tra}.
The first search for $Z_c^+(3900)$ on the lattice considered $J/\psi\, \pi$ and $D\,\bar D^*$ scattering and no $Z_c^+$ candidate was found \cite{Prelovsek:2013xba}. The $D\bar D^*$ scattering was considered in \cite{Chen:2014afa} and the authors conclude that they do not find the state either.
The most extensive lattice search for $Z_c^+$ with mass below $4.2~$GeV in the channel $I^G(J^{PC})=1^+(1^{+-})$ is performed in \cite{Prelovsek:2014v2}. The major challenge is presented by the two-meson states $J/\psi\, \pi$, $\psi_{2S}\,\pi$, $\psi_{1D}\,\pi$, $D\bar D^*$, $D^*\bar D^*$, $\eta_c\,\rho$ that are inevitably present in this channel in addition to potential $Z_c^+$ candidates. The spectrum of eigenstates is extracted using a number of meson-meson and diquark-antidiquark interpolating fields. All the expected two-meson states are found but no additional candidate for $Z_c^+$ \cite{Prelovsek:2014v2}. It is also illustrated how a simulation incorporating low-lying two-mesons states seems to render a $Z_c^+$ candidate \cite{Prelovsek:2014v1}, which is however not robust after further two-meson states around $4.2~$GeV are implemented \cite{Prelovsek:2014v2}. It is concluded that the experimental $Z_c^+$ candidates with $I^G(J^{PC})=1^+(1^{+-})$ and a mass below $4.07~$GeV are either very broad or most likely not dominated by the $[\bar c\bar d]_{3_c}[cu]_{\bar 3_c}$ Fock component.
\\
\underline{\bf $\mathbf{D_{s0}^*}$, $\mathbf{D_{s1}}$}: The quark models expected $D_{s0}^*(2317)$ and $D_{s1}(2460)$ above $DK$ and $D^*K$ thresholds, but they were experimentally found slightly below them. The first lattice QCD simulations that take the effect of these thresholds into account used $DK$ and $D^*K$ interpolating fields in addition to the $\bar sc$ \cite{Mohler:2013rwa,Lang:2014yfa}. The position of thresholds is almost physical in this $N_f\!=\!2+1$ simulation with nearly physical $m_\pi\simeq 156~$MeV. The $D^{(*)}K$ phase shift is extracted from each energy level and then parametrized in the region close to threshold using effective range formula. The large negative scattering length is an indication for the presence of the bound states. The effective range expansion renders the position of the poles in $S\propto (\cot\delta -i)^{-1}$ related to $D_{s0}^*(2317)$ and $D_{s1}(2460) $ close to the experimental masses. The summary of the resulting $D_s$ spectrum for these two states as well as other $D_s$ states in summarized in Fig. 9 of \cite{Lang:2014yfa} for two values of pion masses.
\\
\underline{$\mathbf{X(3872)}$}: A candidate for the charmonium(like) state $X(3872)$ is found $11\pm 7~$MeV below the $D\bar D^*$ threshold for $J^{PC}\!=\!1^{++}$, $I\!=\!0$, $N_f\!=\!2$ and $m_\pi\!\simeq\! 266~$MeV \cite{Prelovsek:2013cra}. This is the first lattice simulation that establishes a candidate for $X(3872)$ in addition to $\chi_{c1}$ and the nearby scattering states $D\bar D^*$ and $J/\psi\,\omega$. The large and negative $a_0\!=\!-1.7\pm 0.4~$fm for $D\bar D^*$ scattering is one indication for a shallow bound state $X(3872)$. The mass of $X$ is determined from the position of the pole in $S$ matrix which is obtained by interpolating $D\bar D^*$ scattering phase shift near threshold. The established $X(3872)$ has a large overlap with $\bar cc$ as well as $D\bar D^*$ interpolating fields \cite{Prelovsek:2013cra}. The single-hadron approach using just $\bar cc$ interpolators renders only one level near $DD^*$ threshold just like in previous simulations. In this case one can not reliably establish whether this level is related to $X(3872)$ or $D(0)\bar D^*(0)$.
In the $I\!=\!1$ channel, only the $D\bar D^*$ and $J/\psi\,\rho$ scattering states are found, and no candidate for $X(3872)$ \cite{Prelovsek:2013cra}. This is in agreement with a popular interpretation that $X(3872)$ is dominantly $I\!=\!0$, while its small $I\!=\!1$ component arises solely from the isospin breaking and is therefore absent in the simulation with $m_u\!=\!m_d$.
\section{Rigorous treatment of hadronic resonances}
The rigorous treatment of a resonance in an elastic channel $M_1M_2$ amounts to determination of the discrete spectrum including two-meson states, determination of the scattering phase shift from each energy level and making a Breit-Wigner type fit of the phase shift as described in Section 2. The only hadron resonance studied in this way until recently is $\pi\pi\to\rho\to \pi\pi$, which has been simulated by a number of lattice collaborations until now (see for example \cite{Dudek:2012xn}). In the following I summarize results for other channels, where only pioneering steps have been made. \\
\underline{$\mathbf{K^*}$, $\mathbf{\kappa}$, $\mathbf{K_0^*}$ {\bf and } $\mathbf{K_2}$}: $K^*$ mesons and in particular the $K^*(892)$ were frequently addressed in lattice simulations, but always ignoring that the $K^*(892)$ decays strongly. The simulation \cite{Prelovsek:2013ela} presents the first extraction of the masses and widths for the $K^*$ resonances by simulating $K\pi$ scattering in $p$-wave with $I=1/2$.
A Breit-Wigner fit of the phase renders a $K^*(892)$ resonance mass $m^{lat}=891\pm 14~$MeV and the $K^*(892)\to K\pi$ coupling $g^{lat}=5.7\pm 1.6$ compared to the experimental values $m^{exp}\approx 892$ MeV and $g^{exp}=5.72\pm 0.06$, where $g$ parametrizes the $K^*\to K\pi$ width. Mixing of $s$ and $p$-wave is taken into account
when extracting the phase shift around the $K^*(1410)$ and $K_2^*(1430)$ resonances.
This gives an estimate of the $K^*(1410)$ resonance mass $m^{lat}=1.33\pm 0.02 ~$GeV compared to $m^{exp}=1.414\pm 0.0015~$GeV assuming the experimental $K^*(1410)\to K\pi$ coupling.
The first simulation of two-coupled channel system $K\pi-K\eta$ was presented just around the time of this meeting \cite{Dudek:2014qha}.
The scattering matrix in complex plane was parametrized and the parameters were extracted using the fit to the finite volume spectrum via L\"uscher-type method. The poles corresponding $\kappa$ and $K^*(892)$ are found below $K\pi$ threshold for the employed $m_\pi\simeq 400~$MeV, while $K_0^*(1430)$ and $K_2$ are found as resonances above threshold.
\\
\underline{$\mathbf{D_{0}^*}$ {\bf and} $\mathbf{D_{1}}$}: The first rigorous simulation of a hadronic resonance that contains charm quarks addresses the broad scalar $D_0^*(2400)$ and the axial $D_1(2430)$ charmed-light mesons, which appear in $D\pi$ and $D^*\pi$ scattering \cite{Mohler:2012na}. The simulation is done for $N_f=2$ and $m_\pi\simeq 266~$MeV. The resonance parameters are obtained using a Breit-Wigner fit to the elastic phase shifts. The resulting $D_0^*(2400)$ mass is $351\pm 21~$MeV above the spin-average $\tfrac{1}{4}(m_D+3m_{D^*})$, in agreement with the experimental value of $347\pm 29~$MeV above. The resulting $D_0^*\to D\pi$ coupling $g^{lat}=2.55\pm 0.21~$GeV is close to the experimental value $g^{exp}\le1.92\pm 0.14~$GeV, where $g$ parametrizes the width $\Gamma\equiv g^2p^*/s$. The results for $D_1(2430)$ are also found close to the experimental values; these are obtained by appealing to the heavy quark limit, where the neighboring resonance $D_1(2420)$ is narrow.
The charmed scalar meson puzzle wonders why the strange $D_{s0}^*(2317)$ and the non-strange $D_0^*(2400)$ charmed scalar mesons have a mass within 1 MeV of each other experimentally, while one would naively expect a larger slitting $O(m_s)$. The question is whether this near degeneracy is due to the strange meson being pushed down or the non-strange one being pushed up. This puzzle can be addressed by considering the lattice results for $D_0^*(2400)$ \cite{Mohler:2012na}, which is found as a resonance in $D\pi$, and $D_{s0}^*(2317)$ \cite{Mohler:2013rwa}, which is found as a pole below $DK$. Both masses are found close to experiment. This favors the interpretation that the near degeneracy is a consequence of strange meson being pushed down due to $DK$ threshold. On the other hand, the interpretation that $D_0^*(2400)$ is pushed up due to tetra quark Fock component $\bar u\bar ss c$ is disfavored since $N_f=2$ simulation \cite{Mohler:2012na} renders its mass close to the experiment without any strange content in valence or sea sectors. \\
\underline{$\mathbf{a_1}$ {\bf and} $\mathbf{b_1}$}: The light axial-vector resonances $a_1(1260)$ and $b_1(1235)$ are explored for $N_f\!=\!2$ by simulating the corresponding scattering channels $\rho \pi$ and $\omega\pi$ \cite{Lang:2014tia}.
Interpolating fields $\bar qq$ and $\rho\pi$ or $\omega\pi$ are used to extract the $s$-wave phase shifts for the first time. It is assumed that $\rho$ and $\omega$ are stable, which is justified in the energy region of interest for the employed parameters $m_\pi\simeq 266~$MeV and $L\simeq 2~$fm.
A Breit-Wigner fit of the phase shift gives the $a_1(1260)$ resonance mass $m_{a_1}^{\textrm{res}}=1.435(53)(^{+0}_{-109})~$GeV compared to $m_{a_1}^{\textrm{exp}}=1.230(40)~$GeV. The $a_1$ width $\Gamma_{a_1}(s)\equiv g^2 p/s$ is parametrized in terms of the coupling, which results in $g_{a_1\rho\pi}=1.71(39)~$GeV compared to $g_{a_1\rho\pi}^{\textrm{exp}}=1.35(30)~$GeV derived from experiment.
\section{Conclusions}
I have reviewed recent lattice results for conventional and exotic mesons. A number of precise results for states well below strong decay threshold are available for a number of years and there is impressive agreement with experiment. I reported on the first rigorous simulations aimed at near-threshold states $Z_c^+$, $X(3872)$, $D_{s0}^*(2317)$ as well as light, strange and charmed resonances.
\begin{acknowledgement}
I would like to thank C.B. Lang, L. Leskovec, D. Mohler and R. Woloshyn for the pleasure of collaborating on the described topics. This work is supported by ARRS project number N1-0020 and FWF project number I1313-N27.
\end{acknowledgement}
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1,116,691,499,301 | arxiv | \section{#1}}
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\title{{\boldmath Gauge theory of Lorentz group as a source of the dynamical electroweak symmetry breaking}\\
{\small \it dedicated to the memory of D.I.Diakonov}}
\author{M.A.Zubkov}
\affiliation{Institute for Theoretical and Experimental Physics\\B.Cherjemushkinskaya - 25, Moscow, Russia}
\emailAdd{[email protected]}
\abstract{
We consider the gauge theory of Lorentz group coupled in a nonminimal way to fermions.
We suggest the hypothesis that the given theory may
exist in the phase with broken chiral symmetry and without confinement. The lattice discretization of the model is described. This unusual strongly coupled theory may appear to be the source of the dynamical electroweak symmetry breaking. Namely, in this theory all existing fermions interact with the $SO(3,1)$ gauge field. In the absence of the other interactions the chiral condensate may appear and all fermionic excitations may acquire equal masses. Small corrections to the gap equations due to the other interactions may cause the appearance of the observed hierarchy of masses.
}
\begin{document}
\maketitle
\flushbottom
\section{Introduction}
The scalar excitation recently found at $125$ GeV is now interpreted as the Higgs boson \cite{CMSHiggs, ATLASHiggs}. The existence of the other Higgs bosons with the same (or, larger) production cross - sections is excluded within the wide ranges of masses approximately from $130$ GeV to $550$ GeV \cite{CMSexc,ATLASexc}. However, this does not mean that the existence of scalar particles within these ranges of masses is excluded completely. The particles with smaller values of production cross - sections are allowed. In \cite{VZ2013,VZ2012,VZ2013_2} it was suggested that such particles may appear. It was supposed that these particles together with the $125$ GeV Higgs may be composed of known Standard Model fermions due to the unknown strong interaction between them (with the scale $\Lambda$ above $1$ TeV). This unknown interaction if it exists should have very specific properties that make it different from the conventional technicolor (TC) interactions \cite{Technicolor,Technicolor_,Technicolor__} (for a recent alternative to the technicolor see, for example, \cite{Simonov}).
First, these interactions cannot be confining since otherwise they would confine quarks and leptons to the extremely small regions of space $\sim 1/\Lambda$, so that all strong and weak interaction physics would be missed. At the same time these interactions should provide the spontaneous chiral symmetry breaking needed to make W and Z bosons massive. Some models with such properties were already discussed in the framework of the topcolor \cite{topcolor1,topcolor,Marciano:1989xd,cveticreview,Miransky,modern_topcolor,Simmons,Hill:1996te,Buchalla:1995dp}. For the review of the conventional technicolor see \cite{Weinberg:1979bn,Susskind:1978ms,Simmons,Chivukula:1990bc}. The topcolor assisted technicolor (combines both technicolor and topcolor ingredients) was considered in \cite{Hill:1994hp, LaneEichten, Popovic:1998vb, Braam:2007pm,Chivukula:2011ag}.
The models based on the extended color sector were considered in \cite{top_coloron,top_coloron1}. For the top - seesaw see \cite{topseesaw}.
In \cite{Simonov,VZ2013,VZ2012} the scenario was suggested according to which the chiral condensates appear corresponding to some of the Standard Model fermion fields (not necessarily of the top quark only). This pattern provides both masses for the W, Z bosons and for the fermionic particles.
The considerations of \cite{VZ2013,VZ2012,VZ2013_2} were based on the NJL approximation. Due to the non - renormalizability it is to be considered as the phenomenological model with the finite ultraviolet cutoff $\Lambda$. In the majority of papers on the NJL approximation only the leading order in $1/N$ expansion was considered. The contributions of higher loops, however, may be strong. Actually, nonperturbative lattice results and higher orders of the perturbation theory demonstrate that all dimensional parameters of the relativistic NJL models are of the order of the cutoff \cite{cveticreview,cvetic,latticeNJL}. Moreover, the higher loop corrections may destroy the dynamical electroweak symmetry breaking (DEWSB) just like the quadratic divergent contributions to the Higgs boson mass destroy the DEWSB in the Standard Model. That's why the problem of higher order corrections in the NJL models is related to the hierarchy problem of the Standard Model.
In \cite{VZ2012} it was proposed to look at the effective four - fermion (NJL) approximation as at the effective low energy theory, in which only the one - loop results have sense. The higher loops are simply disregarded.
This approach is similar to the consideration of the hydrodynamics with the corrections due to phonon loops disregarded.
In quantum hydrodynamics there exist the divergent contributions to physical quantities (say, to the vacuum energy). If these divergences are omitted, the classical hydrodynamics appears as a low energy approximation to quantum theory. Why is this so? The quantum hydrodynamics is an effective low energy theory with the cutoff $\Lambda$. The complete theory contains the trans - $\Lambda$ degrees of freedom, whose contribution to vacuum energy exactly cancels the divergent loop phonon contributions of quantum hydrodynamics. This occurs due to the thermodynamical stability of vacuum \cite{quantum_hydro}. It was also suggested that the similar mechanism may be responsible for the cancellation of the ultraviolet divergences in quantum gravity as well as for the cancellation of the dangerous quadratically divergent contribution to the Higgs boson mass in the Standard Model \cite{hydro_gravity}.
It is worth mentioning that in QCD \cite{NJLQCD} and in Technicolor \cite{Simmons} usually only the one - loop NJL results are taken into account. In many papers on the NJL models of top quark condensation (TC) \cite{Miransky,topcolor1} the cutoff was assumed to be many orders of magnitude larger than the electroweak scale, but the consideration was limited to one loop. Next to leading approximation in TC was considered in \cite{cvetic,cveticreview}. In the Extended Technicolor (ETC) \cite{Simmons} the effect of the ETC gauge group is taken into account through the effective four - fermion term. Loop contributions would give values of masses $\sim \Lambda^2_{ETC}, \Lambda_{ETC} \gg M_T$, but these contributions are usually omitted.
In \cite{VZ2012} it was proposed that in the models of the dynamical electroweak symmetry breaking with composite Higgs bosons there exist the contributions of the microscopic theory due to the trans - $\Lambda$ degrees of freedom that cancel the dominant higher loop divergences of the low energy NJL effective theory. Therefore, the one - loop results dominate. In the present paper we follow this proposition. Sure, this is a kind of the fine - tuning that is often considered as unnatural. But, is this possible to avoid the fine - tuning while providing the fermion masses from less than $1$ eV for neutrino to about $174$ GeV for the top - quark (the difference is eleven orders of magnitude)?
It was proposed \cite{VZ2012,VZ2013,VZ2013_2} that in the zero order approximation all fermions have the same masses. Small corrections to the gap equations are able to provide considerable difference between the quark masses. This is how the hierarchy of masses appears. Unfortunately, in addition, in this approach numerous light higgs bosons appear. Some physics is to be added to \cite{VZ2012,VZ2013,VZ2013_2} in order to provide extra light higgs bosons with the masses on the order of the top quark mass (see discussion below in section 5). The given pattern prompts that the mentioned unknown forces binding quarks are to be universal, i.e. they have to act in the same way on all fermions. We know only one kind of forces that act in the same way on all particles: the gravitational forces. At the same time, the scale of the Riemannian gravity is the Plank mass that is many orders of magnitude larger than the expected scale for these interactions. However, one may consider quantum gravity with torsion. Its dynamical variables are the vierbein (that is related to Riemannian gravity) and the $SO(3,1)$ connection that gives rise to torsion. With the vierbein frozen we come to the gauge theory of Lorentz group \cite{Minkowski,Minkowski_}. At the present moment we cannot point out any particular quantum gravity theory that gives rise to the dynamical torsion with the scale much smaller than the scale of the fluctuating metric. (The former scale is not fixed but is assumed to be between $10^3$ TeV and $10^{15}$ GeV while the latter scale is assumed to be given by the Plank mass $\approx 10^{19}$ GeV.)
In this paper we suggest the possibility that the gauge theory of Lorentz group binds quark and lepton fields giving rise to the dynamical electroweak symmetry breaking in the spirit of the models of top - quark condensation \cite{topcolor1, topcolor} and the models considered in \cite{Simonov}. This theory is strongly coupled but it has the properties that make it different from the QCD - like technicolor theories \cite{Technicolor,Technicolor_,Technicolor__}. Therefore, we expect that it may provide chiral symmetry breaking without the confinement.
One of the characteristic features of this theory is that the term linear in curvature appears similar to the Palatini action of the Poincare gravity. When the other terms for the gauge field are neglected, this term gives rise to the four - fermion interactions. In the context of Poincare gravity (i.e. in the presence of nontrivial vierbein) this was considered in
\cite{Freidel:2005sn,Randono:2005up,Mercuri:2006um,Alexandrov}. The induced four -
fermion interactions were considered in
\cite{Xue,Alexander1,Alexander2} as a source of the fermion condensation used,
mainly, in a cosmological background. In our previous paper \cite{Z2010_3} we
have suggested that nonminimal coupling \cite{Shapiro} of fermion fields to torsion (when Parity is broken by the torsion action)
may provide the condensation of the fermion bilinears composed of the right - handed fermions,
and cause the Dynamical
Electroweak Symmetry Breaking (DEWSB) if the technifermions are placed in the right - handed spinors while the Standard Model fermions are placed in the left - handed spinors.
In the present paper we do not consider such a pattern and do not introduce the technifermions at all.
It is worth mentioning that the most general form of the nonminimal coupling of fermions to gauge field of Lorentz group (the gravity with torsion)
was suggested only recently in \cite{Alexandrov}. Later it was used, for example, in \cite{Diakonov}. It is important that dealing with the Lorentz gauge theory we cannot consider only the minimal coupling because the nonminimal interaction between fermions and the gauge group is generalted dynamically. Therefore, following \cite{Diakonov} we consider the nonminimal coupling from the very beginning.
It is worth mentioning that the parity violation appears in the model of the fermions coupled to gravity with torsion in two cases:
if there is the nonminimal parity breaking coupling of fermions to gravity (this was considered, for example, in \cite{Alexandrov}) and if the pure gravity action contains the parity - breaking terms (for example, in \cite{Freidel:2005sn} the Holst action is considered that breaks parity).
In both cases there take place effects of atomic parity violation.
However, in the model considered in the present paper neither of the two mentioned above sources of
parity violation appear:
the considered non-minimal coupling of fermions to gravity does not break parity, while the Holst term of the gravity action is absent; also
there are no parity breaking terms of the action quadratic in curvature.
Strongly coupled theories are to be investigated using nonperturbative methods. The most powerful nonperturbative method is the numerical simulation of the model in lattice discretization. Therefore, we consider the lattice discretization. In its essence our discretization is similar to that of suggested in \cite{Z2006} for the Poincare gravity, and in \cite{Diakonov} for the theory of quantum gravity suggested by D.I.Diakonov.
The paper is organized as follows. In the 2-nd section we consider fermion
fields coupled to the gauge theory of the Lorentz group. In the 3-rd section we consider the action
for the gauge field and derive the four - fermion interactions that appear after the integration over
torsion. In the $4$ - th section we apply NJL technique to the four -
fermion interactions of fermions and describe how the chiral symmetry
breaking occurs. In the $5$ - the section we consider the model that appears when small perturbations to the exchange by the Lorentz group gauge bosons are taken into account. In section $6$ we introduce the lattice discretization needed for the numerical investigation of the model. In section $7$ we end with the conclusions.
\section{Fermions coupled to the gauge field of Lorentz group}
We consider the CP - invariant action of a massless Dirac spinor coupled to the $SO(3,1)$ gauge field in
the form \cite{Alexandrov}:
\begin{eqnarray}
S_f & = & \frac{i}{2}\int \{ \bar{\psi} \gamma^{\mu} \zeta
D_{\mu} \psi - [D_{\mu}\bar{\psi}]\bar{\zeta}
\gamma^{\mu}\psi \} d^4 x \label{Sf}
\end{eqnarray}
Here $\zeta=\eta + i\theta$ is the coupling constant,
the covariant derivative is $D_{\mu} =
\partial_{\mu} + \frac{1}{4}C_{..\mu}^{ab} \gamma_{[a}\gamma_{b]}$;
$\gamma_{[a}\gamma_{b]} =
\frac{1}{2}(\gamma_{a}\gamma_{b}-\gamma_{b}\gamma_{a})$. The spin
connection is denoted by $C_{\mu}$. It is related to Affine connection $\Gamma^{i}_{jk}$
and torsion $T^a_{.\mu \nu}$ as follows:
\begin{eqnarray}
\Gamma^{a}_{\mu
\nu} &=& C^{a}_{ . \mu\nu}\nonumber\\
T^a_{.\mu \nu} & = & C^{a}_{ .
[\mu\nu]}
\end{eqnarray}
Indices are lowered and lifted via metric $g$ of Minkowsky space as
usual.
The symmetry group of the theory is the group of local Lorentz transformations \cite{Minkowski,Minkowski_}. The infinitesimal transformations are given by
\begin{eqnarray}
x^i &\rightarrow &\Lambda^i_{.j} (x) x^j = x^i + \epsilon^{i}_{j}(x) x^j, \quad \epsilon(x) \in so(3,1)\nonumber\\
\psi(x) &\rightarrow& \tilde{\Lambda}\psi(\Lambda^i_{.j} (x) x^j)=(1-\frac{1}{4} \epsilon^{ab} \gamma_{[a}\gamma_{b]})\psi(\Lambda^i_{.j} (x) x^j),\nonumber\\
\partial_j+\frac{1}{4}C_j^{ab}(x)\gamma_{[a}\gamma_{b]}& \rightarrow &\Lambda_{.j}^{k}\tilde{\Lambda} \Bigl(\partial_k+ C^{{a}{b}}_{\bar{k}}(\Lambda^i_{.j} (x) x^j)\gamma_{[a}\gamma_{b]}\Bigr)\tilde{\Lambda}^{-1}
\end{eqnarray}
Another way to look at this symmetry is to restore the field of the vierbein that corresponds to vanishing Riemannian curvature. The symmetry group of this system is $Diff \otimes SO(3,1)_{local}$. Fixing the trivial value of the vierbein $e^a_k = \delta^a_k$ we break this symmetry: $Diff \otimes SO(3,1)_{local} \rightarrow SO(3,1)_{local}$. This breakdown is accompanied by distinguishing local Lorentz transformations out of the general coordinate transformations.
Eq. (\ref{Sf}) can be rewritten as follows:
\begin{eqnarray}
S_f & = & \frac{1}{2}\int \{i\bar{\psi} \gamma^{\mu}\zeta\nabla_{\mu} \psi - i[\nabla_{\mu}\bar{\psi}]\bar{\zeta} \gamma^{\mu}\psi\nonumber\\&& + \frac{i}{4}
C_{abc}\bar{\psi}[\{ \gamma^{[a}\gamma^{b]},\gamma^c\}\eta - i[
\gamma^{[a}\gamma^{b]},\gamma^c] \theta] \psi\} d^4 x \label{Sf1}
\end{eqnarray}
Here $\nabla$ is the usual derivative. Next, we
obtain:
\begin{eqnarray}
S_f & = & \frac{1}{2}\int \{i\bar{\psi} \gamma^{\mu}{\zeta}\nabla_{\mu} \psi -
i[\nabla_{\mu}\bar{\psi}]\bar{\zeta}\gamma^{\mu}\psi\nonumber\\&& - \frac{1}{4}
C_{abc}\bar{\psi}[-2\epsilon^{abcd}\gamma^5 \gamma_d \eta +4
\eta^{c[a} \gamma^{b]} \theta ] \psi\} d^4 x \label{Sf2}
\end{eqnarray}
Now let us introduce the irreducible components of torsion:
\begin{eqnarray}
S^i& =& \epsilon^{jkli}T_{jkl}\nonumber\\
T_i& =& T^j_{.ij}\nonumber\\
T_{ijk}& =& \frac{1}{3}(T_j \eta_{ik}-T_k\eta_{ij}) -
\frac{1}{6}\epsilon_{ijkl}S^l + q_{ijk}
\end{eqnarray}
In terms of $S$ and $T$ (\ref{Sf2}) can be rewritten as:
\begin{eqnarray}
S_f & = & \frac{1}{2}\int \{i\bar{\psi} \gamma^{\mu}\eta
\nabla_{\mu} \psi - i[\nabla_{\mu}\bar{\psi}]{\eta} \gamma^{\mu}\psi\nonumber\\&& +
\frac{1}{4}\bar{\psi}[\gamma^5 \gamma_d \eta S^d - 4\theta T^b \gamma_{b}] \psi\} d^4 x \label{Sf22}
\end{eqnarray}
One can see that $\theta$ disappears from the first term with usual derivatives. However, it remains in the second term. It is worth mentioning that in the theory with dynamical field $C^{ab}_{..\mu}$ it is not possible to consider only the minimal coupling with $\theta = 0$ because the second term with nonzero $\theta$ is to appear dynamically.
\section{Gauge field action}
Let us consider the action of the form $S_g = S_T + S_G$ with
\begin{equation}
S_T = -M_T^2 \int G d^4x \label{ST}
\end{equation}
and
\begin{eqnarray}
S_G &=& \beta_1 \int G^{abcd}G_{abcd}d^4x+\beta_2 \int
G^{abcd}G_{cdab}d^4x\nonumber\\&& +\beta_3 \int G^{ab}G_{ab}d^4x+\beta_4 \int
G^{ab}G_{ba}d^4x\nonumber\\&& +\beta_5 \int G^2d^4x+\beta_6 \int
A^2 d^4x\label{ST2}
\end{eqnarray}
with coupling constants $\beta_{1,2,3,4,5,6}$. Here $G^{ab}_{..\mu\nu} = [D_{\mu},D_{\nu}]$ is the $SO(3,1)$ curvature,
$G^{abcd}=\delta^c_{\mu}\delta^d_{\nu}G^{ab}_{\mu\nu}$, $G^{ac}=G^{abc}_{...b}$, $G =
G^a_a$,
\begin{equation}
A = \epsilon^{abcd} G_{abcd} \label{A}
\end{equation}
Actually,
Eq. (\ref{ST2}) is the most general quadratic in curvature action that does
not break Parity \cite{Diakonov2}. The topological charge may be composed of different terms of Eq. (\ref{ST2}):
\begin{equation}
{\cal Q} \sim \int d^4x (R^2 - 4 R^{ab} R_{ab} + R^{abcd}R_{cdab})
\end{equation}
Therefore, up to the the topological term we have five independent terms quadratic in curvature.
$S_T$ may be represented in terms of torsion
\cite{Mercuri:2006um}:
\begin{eqnarray}
S_T = M_T^2 \int \{\frac{2}{3}T^2 - \frac{1}{24}S^2 \}d^4x + \tilde{S}
\end{eqnarray}
Here $\tilde{S}$ depends on $q$ but does not depend on $S, T$.
Let us suppose for a moment that it is possible to neglect $S_G$ in some approximation. Then the integration over torsion degrees of freedom can be performed for the system that consists of the Dirac fermion
coupled to the gauge field. The result of this integration is \cite{Alexandrov,Freidel:2005sn}:
\begin{eqnarray}
S_{eff}& = & \frac{1}{2}\int \{i\bar{\psi} \gamma^{\mu} \eta
\nabla_{\mu} \psi - i[\nabla_{\mu}\bar{\psi}] \eta \gamma^{\mu}\psi\}d^4x\nonumber\\&& -
\frac{3}{32M_T^2} \int
\{V^2 \theta^2 - A^2\eta^2 \} d^4x \label{F42}
\end{eqnarray}
Here we have defined:
\begin{eqnarray}
V_{\mu} & = & \bar{\psi} \gamma_{\mu} \psi \nonumber\\A_{\mu} & = &
\bar{\psi}\gamma^5 \gamma_{\mu} \psi
\end{eqnarray}
The next step is the consideration of the fermions coupled to the gauge field of Lorentz group
with action $S_T+S_G$. This theory may appear to be renormalizable due to the presence of terms quadratic
in curvature \cite{Sesgin,Elizalde}. Our supposition
is that the theory contains the scale $\Lambda_{\chi} \sim M_T$. The effective charges entering the term of the action with the
derivative of torsion depend on the ratio $\epsilon/\Lambda_{\chi}$, where $\epsilon$ is the energy scale of the considered physical process.
We suppose that under certain circumstances running coupling constants $\beta_i(\epsilon/\Lambda_{\chi})$ are decreased with the decrease of $\epsilon$. This does not contradict to the asymptotic freedom ($\beta_i(\epsilon/\Lambda_{\chi}) \rightarrow \infty $ at $\epsilon \rightarrow \infty$) that is often considered as the condition for the self - consistency of the gauge theory. However, due to the existence of $S_T$ in the action we do not expect confinement, and at small enough energies $\epsilon << \Lambda_{\chi}$ it is possible to neglect $S_G$. The theory with the low energy effective action Eq. (\ref{F42}) has the natural cutoff $\Lambda_{\chi}$. In the next section we shall demonstrate that in this theory the dynamical breakdown of chiral symmetry is possible.
\section{Chiral symmetry breaking }
As it was mentioned in the Introduction we restrict ourselves with the one - loop analysis of the effective NJL model derived in the previous secton. This is based on the assumption that due to a certain symmetry the dominant divergencies that appear at more than one loop are exactly cancelled by contribution of the trans - $\Lambda_{\chi}$ degrees of freedom of the microscopic theory. (The microscopic theory in our case is the complete gauge theory of the Lorentz group or the Poincare gravity that contains it, or some other unified theory. As a result of the cancellation the one - loop results dominate and we are able to use the one - loop gap equation for the consideration of the chiral symmetry breaking.
Let us
apply Fierz transformation to the four fermion term of (\ref{F42}):
\begin{eqnarray}
S_{4} &=& \frac{3}{32M_T^2}\int \{ -\theta^2(\bar{\psi}^a\gamma^i
\psi^a)(\bar{\psi}^b\gamma_i \psi^b)\} d^4 x
+ \eta^2(\bar{\psi}^a\gamma^i \gamma^5
\psi^a)(\bar{\psi}^b\gamma_i \gamma^5 \psi^b)\} d^4 x\nonumber\\
&=& \frac{3}{32M_T^2}\int\{4 (\eta^2+\theta^2)
(\bar{\psi}^a_{L}\psi^b_{R})(\bar{\psi}^b_{R} \psi^a_{L})
+(\eta^2-\theta^2)[(\bar{\psi}^a_{L}\gamma_i\psi^b_{L})(\bar{\psi}^b_{L}\gamma^i
\psi^a_{t,L})\nonumber\\&&+(L\leftarrow \rightarrow R)]\} d^4 x \label{S4}
\end{eqnarray}
In this form the action is equal to that of the extended NJL model (see Eq. (4), Eq. (5), Eq. (6) of \cite{ENJL}). In
the absence of the other gauge fields the $SU({ N})_L\otimes
SU({ N})_R$ symmetry is broken down to $SU({N})_V$
(here ${ N}$ is the total number of fermions). We suppose that the Electroweak interactions act
as a perturbation.
For the purpose of the further consideration we
denote $G_S = \frac{3 (\theta^2+\eta^2)
\Lambda^2_{\chi}}{64 M_T^2\pi^2}$; $G_V = \frac{\theta^2-\eta^2}{4(\theta^2+\eta^2))}G_S$. Here
$\Lambda_{\chi}$ is the cutoff that is now the physical parameter of the model.
We also denote $g_s =
\frac{4\pi^2G_S}{\Lambda_{\chi}^2}=\frac{3(\theta^2+\eta^2)}{16 M_T^2}$.
Next, the auxiliary fields $H$, $L_i$, and $R_i$ are introduced and the new
action for $\psi$ has the form:
\begin{eqnarray}
S_{4,t} &=& \int\{ -(\bar{\psi}^a_{t, L}H^+_{ab} \psi^b_{R} + (h.c.)) -
\frac{8 M_T^2}{3 (\theta^2+\eta^2)} \, H_{ab}^+H_{ab}\}d^4x \nonumber\\&&
+\int\{(\bar{\psi}^a_{t,L}\gamma^i L^{ab}_i\psi^b_{t,L})
-\frac{32M_T^2}{3(\theta^2-\eta^2)} {\rm Tr}\,L^i L_{i} +(L\leftarrow \rightarrow R)\}
d^4 x\label{eff}
\end{eqnarray}
Integrating out fermion fields we arrive at the effective action for the
mentioned auxiliary fields (and the source currents for fermion bilinears). The
resulting effective action receives its minimum at $H = m {\bf 1}$, where
$m$ plays the role of the fermion mass (equal for all fermions).
We apply the following regularization: $\frac{1}{-p^2+m^2} \rightarrow \int_{\frac{1}{\Lambda_{\chi}^2}}^{\infty} d\tau
e^{-\tau (-p^2+m^2)}$.
With this regularization the expression for the condensate of $\psi$ is:
\begin{eqnarray}
<\bar{\psi}\psi> &=& i \int\frac{d^4p}{(2\pi)^4}\frac{1}{p\gamma - m}
=-\frac{1}{16\pi^2}4m^3
\Gamma(-1,\frac{m^2}{\Lambda_{\chi}^2})\label{GM}
\end{eqnarray}
Here $\Gamma(n,x) = \int_x^{\infty}\frac{dz}{z}e^{-z}z^{n}$. The gap equation
is $m = -g_s<\bar{\psi}\psi>$, that is
\begin{equation}
m = G_S m
\{\exp(-\frac{m^2}{\Lambda_{\chi}^2})-\frac{m^2}{\Lambda_{\chi}^2}
\Gamma(0,\frac{m^2}{\Lambda_{\chi}^2})\} \approx G_S m (1 - \frac{m^2}{\Lambda^2}{\rm log} \frac{\Lambda_{\chi}^2}{m^2}), \quad (\Lambda_{\chi} \gg m) \label{gap}
\end{equation}
There exists the critical value of
$G_S$: at $G_S > 1$ the gap equation has the nonzero solution for $m$ while
for $G_S < 1$ it has not. This means that in this approximation the
condensation of fermions occurs at
\begin{equation}
M_T < M_T^{\rm critical} = \sqrt{3 (\theta^2+\eta^2) }\frac{\Lambda_{\chi}}{8\pi}\sim
\Lambda_{\chi}\label{cond}
\end{equation}
The analogue of the technipion decay constant $F_T$ in the given approximation is:
\begin{equation}
F^2_T = \frac{m^2}{8\pi^2}
\Gamma(0,\frac{m^2}{\Lambda_{\chi}^2}) \approx \frac{m^2}{8\pi^2} \, {\rm log}\, \frac{\Lambda_{\chi}^2}{m^2}, \quad (\Lambda_{\chi} \gg m) \label{FT}
\end{equation}
In order to have appropriate values of $W$ and $Z$ - boson masses we need
$F_T\sim 250/\sqrt{N}$ Gev, where $N=24$ is the total number of fermions. It is given by
$N = (3 \, {\rm generations}) \times (3\, {\rm colors} + 1 \, {\rm lepton}) \times (2\, {\rm fermions}\, {\rm in}\, SU(2) \, {\rm doublets}) = 3\times 4\times 2 = 24$. Rough estimate via Eq. (\ref{FT}) with $m = 174$ GeV (top quark mass) gives the value $\Lambda_{\chi}\sim 5$ TeV. At the same time if only one $t$ quark contributes to the formation of the gauge boson masses we would have $\Lambda_{\chi}\sim 10^{15}$ GeV.
This, probably, means that Eq. (\ref{FT}) cannot be used directly and should be a subject of a subsequent change.
In a realistic theory the fermions should have different masses due to the corrections to the gap equations caused by the other interactions (see below). In order to avoid the constraints on the FCNC we need $\Lambda_{\chi} \ge 1000$ TeV.
In the absence of SM interactions the relative orientation of the SM gauge
group $G_W = SU(3)\otimes SU(2)\otimes U(1)$ and $SU({ N})_V$ from
$SU({ N})_L\otimes SU({ N})_R \rightarrow SU({ N})_V$ is
irrelevant. However, when the SM interactions are turned on, the effective
potential due to exchange by SM gauge bosons depends on this relative
orientation. Minimum of the potential is achieved in the true vacuum state and
defines the pattern of the breakdown of $G_W$. This process is known as the vacuum
alignment. In \cite{Align, Align1} this process was considered for the conventional technicolor. It was shown that under very natural suppositions $G_W$
is broken in a minimal way. This means that the subgroups of $G_W$ are not
broken unless they should. We suppose, that the same arguments work also for the dynamical theory of Lorentz group as a source of the dynamical electroweak symmetry breaking. The form of the condensate requires that $SU(2)$ and
$U(1)$ subgroups are broken. That's why in the complete analogy with $SU(N_{TC})$
Farhi - Susskind model the Electroweak group in our case is broken correctly while
the $SU(3)$ group remains unbroken.
\section{Taking into account small perturbations}
\label{NJL}
Let us briefly mention what happens to the gap equation Eq. (\ref{gap}) when the corrections are taken into account due to the Standard Model gauge fields and, due to some other unknown interactions. The four - fermion interaction term of Eq. (\ref{S4}) is modified in this case. To demonstrate, how this may work, let us consider the toy model that involves only $t$ and $b$ - quarks, and ignore the color. Correspondingly, we have the left - handed doublet $\chi_{A, L}^T = (t_L,b_L)$ and the right - handed doublet $\chi_{A, R}^T = (t_R,b_R)$. The unperturbed action has the form
\begin{eqnarray}
S & = & \int d^4x \Bigl(\bar{\chi}[ i \nabla \gamma ]\chi + \frac{8\pi^2}{\Lambda_{\chi}^2} (\bar{\chi}_{A,L}
\chi^{B}_R)(\bar{\chi}_{\bar{B}, R} {\chi}^{{A}}_{L}) I_B^{\bar{B}}\Bigr) \label{toy0}
\end{eqnarray}
with $I_B^{\bar{B}} = \delta_B^{\bar{B}}(1+y)$. The four - fermion terms similar to the second term of Eq. (\ref{S4}) with the product of two vector - like fermion bilinears do not contribute to the one - loop gap equation. Therefore, we omit these terms. The one - loop gap equation gives equal masses of top and bottom quarks $M = M_t = M_b$:
\begin{equation}
\frac{M^2}{\Lambda^2}{\rm log} \frac{\Lambda^2}{M^2} = y\label{gap00}
\end{equation}
Now let us consider as a perturbation the gauge field $B$ interacting with the right - handed top - quark only.
We imply that the corresponding $U(1)$ symmetry is broken spontaneously, and $B$ receives mass $M_B$ much larger than $\Lambda_{\chi}$. The corrections to Eq. (\ref{toy0}) due to the exchange by $B$ give the modification of Eq. (\ref{toy0}) with $I_B^{\bar{B}} = {\rm diag} \, \Bigl(1+y_t, 1+y_b\Bigr)$, where $y_b \approx y$, while $|y_t - y_b| \sim \frac{\Lambda^2_{\chi}}{M_B^2}$.
This toy pattern demonstrates how perturbations may affect action of Eq. (\ref{S4}). Next, we consider the NJL model of general type that generalizes the model with action Eq. (\ref{toy0}) and that involves all $6$ quarks and all $6$ leptons (neutrino is supposed to be of Dirac type). The particular form of the four - fermion action is obtained under the supposition that the tensor of coupling constants standing in front of the four - fermion
term is factorized and under the supposition that lepton number originates from the fourth color. At the present moment we do not intend to consider this model as realistic. It is rather an example of how this may work and does not exhaust all possible
deformations of the theory by small perturbations. However, in principle, with certain updates this model may pretend on the description of the TeV - scale physics. (For the consideration of points, where it is to be updated see the discussion below.) The action of this NJL model is suggested in \cite{VZ2013_2} and generalizes the action suggested in \cite{Miransky} considered in details in \cite{VZ2012,VZ2013}:
\begin{eqnarray}
S & = & \int d^4x \Bigl(\bar{\chi}[ i \nabla \gamma ]\chi + \frac{8\pi^2}{\Lambda^2} (\bar{\chi}_{k, \alpha A,L} \chi^{l, \beta, B}_R)(\bar{\chi}_{\bar{l},\bar{{\beta}}
\bar{B}, R} {\chi}^{\bar{k},\bar{\alpha} {A}}_{L}) W^k_{\bar{k}} W_l^{\bar{l}} L_{\bar{\alpha}}^{\alpha}
R_{\beta}^{\bar{\beta}} I_B^{\bar{B}}\Bigr) \label{Stopcolor_}
\end{eqnarray}
Here $\chi_{k, \alpha A}^T = \{(u_k,d_k); (c_k,s_k); (t_k,b_k)\}$ for $k = 1,2,3$ is the set of the quark doublets
with the generation index $\alpha$ while $\chi_{4, \alpha A}^T = \{(\nu_e, e); (\nu_{\mu},\mu); (\nu_{\tau},\tau)\}$ is the set of the lepton doublets. $\Lambda$ is the dimensional parameter.
Hermitian matrices $L,R,I,W$ contain dimensionless coupling constants. The form of action Eq. (\ref{Stopcolor_}) with $W = {\rm diag} \, (1+\frac{1}{2}W_{e\mu\tau},1,1,1)$ is fixed by the requirement that there is the $SU(3)\otimes SU(2) \otimes U(1)$ symmetry.
We imply that all eigenvalues of matrices $L,R,I$ are close to each
other.
If all quarks and leptons would experience only the interaction via the gauge bosons of Lorentz group, then in this action the eigenvalues of $L,R,I$ are all equal to each other, and $W_{e\mu\tau}=0$. In this case we would come to the scalar - scalar interaction term of Eq. (\ref{S4}).
Any small corrections to this equality gives the eigenvalues of $L,R,I$ that only
slightly deviate from each other, and the value of $W_{e\mu\tau}$ that only slightly deviates from $0$. (After the suitable rescaling $\Lambda$ plays the role of the cutoff $\Lambda_{\chi}$,
while the eigenvalues of $L,R,I$ are all close to $1$.)
The difference between the model with action of Eq. (\ref{Stopcolor_}) and the model of \cite{VZ2012,VZ2013} is that in the present model the leptons are included and the color indexes (the fourth colour is the lepton number) are contracted in the different way.
The basis of observed quarks corresponds to the diagonal form of $L,R,I$. We
denote
$L = {\rm diag}(1+L_{ude},1+L_{cs\mu},1+L_{tb\tau})$, $R = {\rm
diag}(1+R_{ude},1+R_{cs\mu},1+R_{tb\tau})$,
$ I = {\rm diag}(1+I_{up},1+I_{down})$, and
\begin{eqnarray}
&&y_u = L_{ude} +R_{ude}+ I_{up}, \quad y_d = L_{ude}+ R_{ude}
+I_{down},\nonumber\\&& ...
\nonumber\\ &&
y_{ud} = L_{ude} +R_{ude}+ I_{down}, \quad y_{du} = L_{ude} +R_{ude}
+I_{up},\nonumber\\
&& ... \nonumber\\
&&y_{\nu_e} = L_{ude} +R_{ude}+ I_{up} + W_{e\mu\tau}, \quad y_{e} = L_{ude}+ R_{ude}
+I_{down} + W_{e\mu\tau},\nonumber\\&& ...
\nonumber\\ &&
y_{\nu_e e} = L_{ude} +R_{ude}+ I_{down}+W_{e\mu\tau}, \quad y_{e \nu_e} = L_{ude} +R_{ude}
+I_{up}+W_{e\mu\tau},\nonumber\\
&& ...
\label{parameters}
\end{eqnarray}
These coupling constants satisfy the relation $y_{q_1q_2}+y_{q_1q_2} =
y_{q_1}+y_{q_2}$.
As it was mentioned above, it is implied that $|y_{q}|, |y_{q_1 q_2}| << 1$.
Bosonic spectrum of this model is formally given by the expressions for the bosonic spectrum of the model suggested in \cite{Miransky} calculated in one - loop approximation in
\cite{VZ2012,VZ2013}. It is implied that in vacuum the composite scalar
fields $h_q = \bar{q}q$ are condensed for all fermions $q=u,d,c,s,t,b,e,\mu,\tau, \nu_e,\nu_{\mu}, \nu_{\tau}$. The
induced fermion masses $M_q$ are related to the coupling constants $y_q$, $\Lambda$ as
\begin{equation}
\frac{M_q^2}{\Lambda^2}{\rm log} \frac{\Lambda^2}{M_q^2} = y_q\label{gap2}
\end{equation}
(Here it is implied that $\Lambda \gg M_q$.)
Instead of Eq. (\ref{FT}) we have in each of the $q\bar{q}$ channel:
\begin{equation}
F^2_T = \frac{M_q^2}{8\pi^2} \, {\rm log}\, \frac{\Lambda^2}{M_q^2}\label{FT2}
\end{equation}
The $t$ quark dominates. Therefore, in order to have appropriate values of $W$ and $Z$ - boson masses we need
$F_T\sim 250/\sqrt{N}$ Gev, where $N=3$ is the number of colours. This gives $\Lambda = \Lambda_{\chi}\sim 10^{15}$ GeV.
There are two excitations in each $q\bar{q}$ channel:
\begin{equation}
M^P_{q\bar{q}} = 0; \quad M^S_{q\bar{q}} = 2 M_q
\end{equation}
and four excitations (i.e. two doubly degenerated excitations) in each
$q_1\bar{q}_2$ channel. (Pairings of leptons and quarks are also allowed and give the colored scalar fields.) We denote the masses $M^{\pm}_{q_1q_2},
M^{\pm}_{q_2q_1}$. They are given by
\begin{eqnarray}
M_{q_1 {q}_2}^2 &= &M_{q_1}^2 + M_{q_2}^2 \pm
\sqrt{(M_{q_2}^2 - M_{q_1}^2)^2\zeta_{q_1q_2}^2+ 4 M_{q_1}^2M_{q_2}^2}
\end{eqnarray}
with $ \zeta_{q_1 q_2} = \frac{2 y_{q_1q_2}- y_{q_2} - y_{q_1}}{y_{q_2}-y_{q_1}} =
\zeta_{q_2 q_1}$.
It is worth mentioning that each of the scalar quark - antiquark bosons carries two color indexes. In the absence of the $SU(3)$ gauge field each of these channels represents the degenerate nonet. When the color interactions are turned on we are left with the singlet and octet states. Traceless octet states as well as the color scalar excitations of the quark - lepton channels cannot exist as distinct particles due to color confinement.
Instead of the trivial Nambu sum rule of the simplest models of top - quark condensation $M_H = 2 M_t$ we have the sum rule \cite{VZ2013}:
\begin{eqnarray}
&& [M^{+}_{q_1\bar{q}_2}]^2 + [M^{-}_{q_1\bar{q}_2}]^2 +
[M^{+}_{q_2\bar{q}_1}]^2 + [M^{-}_{q_2\bar{q}_1}]^2\approx 4 [M_{q_1}^2 +
M_{q_2}^2], \quad (q_1\ne q_2); \nonumber\\
&& [M^{P}_{q\bar{q}}]^2 + [M^{S}_{q\bar{q}}]^2 \approx 4 M_{q}^2\label{NSRR}
\end{eqnarray}
In the case when the t - quark contributes to the formation of the given
scalar excitation,
its mass dominates, and in each channel ($t\bar{t}, t\bar{c}$, ...) we come
to the relation $\sum M_{H, i}^2 \approx 4 M^2_t$,
where the sum is over the scalar excitations in the given channel.
We already do not have the only gap equation. Instead, for each fermion particle its own gap equation (\ref{gap2}) appears.
It is important, that although the corrections to the eigenvalues of $L,R,I,W$ are small, this does not mean that the corrections to the masses are small. Instead, the large difference between masses may appear in this way. We imply that this is provided by the fine - tuning that is responsible for the cancellation of dominant higher - loop divergences in the NJL model. As it was mentioned in the Introduction, this mechanism may be similar to the mechanism that provides the cancellation of the divergent contributions to vacuum energy (due to sound waves) in quantum hydrodynamics.
It is worth mentioning that among the
mentioned Higgs bosons there are 24 Goldstone bosons that are exactly
massless (in the channels $t(1\pm \gamma^5)\bar b, t \gamma^5\bar{t},
c(1\pm\gamma^5)\bar{s}, c\gamma^5\bar{c}$, $u(1\pm \gamma^5)\bar{d},
u\gamma^5\bar{u},b\gamma^5\bar{b}$, $s\gamma^5\bar{s}$, $d\gamma^5\bar{d}$ and $\nu_e(1\pm \gamma^5)\bar e, \nu_e \gamma^5\bar{\nu_e}, \nu_{\mu}(1\pm \gamma^5)\bar \mu, \nu_{\mu} \gamma^5\bar{\nu_{\mu}}, \nu_{\tau}(1\pm \gamma^5)\bar \tau, \nu_{\tau} \gamma^5\bar{\nu_{\tau}}, {\tau} \gamma^5\bar{{\tau}}, {\mu} \gamma^5\bar{{\mu}}, e \gamma^5\bar{e}$).
There are Higgs bosons with the masses of the order of the t-quark mass ($
t(1\pm \gamma^5)\bar b, t \bar{t}, t(1\pm\gamma^5)\bar{s},
t\gamma^5\bar{c}, t(1\pm \gamma^5)\bar{d}, t\gamma^5\bar{u}$ and
$t(1\pm \gamma^5)\bar e, t(1\pm\gamma^5)\bar{\mu},
t(1\pm\gamma^5)\bar{\tau}, t(1\pm \gamma^5)\bar{\nu_e}, t\gamma^5\bar{\nu_{\mu}}, t\gamma^5\bar{\nu_{\tau}}$). The other
Higgs bosons have masses much smaller than the t - quark mass.
If we want to heave realistic model, extra light Higgs bosons should be provided with the masses of
the order of $M_t$. In principle, this may be achieved if the new gauge symmetries are added, that are spontaneously broken. Then the extra light Higgs bosons may become massive via the Higgs mechanism. The consideration of such a completion of the theory is out of the scope of the present paper.
In principle, all neutral Higgs bosons $h^{a}, a = t \bar{t},b \bar{b}, \bar{\tau} \tau, \bar{\nu}_{\tau} \nu_{\tau} ... $ are coupled to the fields of the Standard Model in a similar way. However, already at the tree level the corresponding coupling constants are different for different Higgs bosons.
At the tree level in the effective decay lagrangian \cite{status} the following terms dominate:
\begin{eqnarray}
\label{eq:1}
L_{eff} &= &
{2 m_W^2 \over v} h^{(\bar{t}t)} \, W_\mu^+ W_\mu^- + {m_Z^2 \over v} h^{(\bar{t}t)} \, Z_\mu Z_\mu
+ c^{}_{g} {\alpha_s \over 12 \pi v} h^{(\bar{t}t)} \, G_{\mu \nu}^a G_{\mu \nu}^a \nonumber \\&& + c^{}_{\gamma} { \alpha \over \pi v} h^{(\bar{t}t)} \, A_{\mu \nu} A_{\mu \nu} - {m_b \over v^{(\bar{b}b)} } h^{(\bar{b}b)} \,\bar b b - {m_c \over v^{(\bar{c}c)}} h^{(\bar{c}c)} \, \bar c c \ - {m_{\tau} \over v^{(\bar{\tau}\tau)} } h^{(\bar{\tau}\tau)} \,\bar \tau \tau .
\end{eqnarray}
We denote by $v^{(a)}$ the vacuum average of the scalar boson in the $a$ - th channel; $v \approx v^{(\bar{t}t)} \approx 245\, {\rm GeV}$ is the vacuum average of the scalar field in the $\bar{t}t$ channel, that dominates the formation of the gauge boson masses.
One can see, that at the tree level in the channels $h \rightarrow gg, \gamma \gamma, ZZ, WW$ the decay of the $\bar{t}t$ Higgs boson dominates. The contributions of the decays of the other Higgs bosons are suppressed. The top quark has been integrated out in Eq.~\eqref{eq:1} and its effects are included in the effective couplings $c_g$ and $c_\gamma$ that are given by $c_{g} \simeq 1.03\,, c_{\gamma} = (2/9)c_{g} \simeq 0.23$ (see \cite{status}).
One can see, that in the processes like $p p \rightarrow h^{(a)} \rightarrow \bar{c}c, \bar{b}b, \bar{\tau} \tau$ (that may be observed at the LHC) the scalar states $h^{(a)}$ with $a=\bar{c}c$, $\bar{t}{t}$, $\bar{\tau}\tau$ correspondingly dominate at the tree level.
From this consideration it also follows that the cross - sections $\sigma^{(a)}$ of the processes like $p p \rightarrow h^{(a)} \rightarrow WW, ZZ, gg, \gamma \gamma$ for the $\bar{t} t $ Higgs bosons are much larger than for the other Higgs Bosons and are close to that of the Standard Model.
This means, in particular, that the scalar boson of this model in the $\bar{t}t$ channel with mass $\approx 2 M_t$ is excluded by the LHC data. Therefore, some additional physics is necessary that either suppresses the corresponding cross - section or makes this state much heavier.
The decays of the other neutral Higgs bosons to $ZZ, WW, \gamma \gamma, gg$ are suppressed compared to that of $\bar{t}t$. Therefore, these scalar states are not excluded by the LHC data.
\section{Lattice discretization}
We suggest the geometrical pattern of manifestly gauge invariant discretization that is similar to that of the Regge calculus or the pattern suggested in \cite{Z2006} for the discretization of Poincare quantum gravity. It is implied that the Wick rotation is performed. Therefore, we deal with the Euclidean theory and with the gauge group $SO(4)$ instead of $SO(3,1)$. We use the hypercubic form of the lattice cells. We approximate the given manyfold with the $SO(4)$ curvature by the set of the adjacent hypercubes of the same size. It is implied that the Lorentz connection is localized on the sides of these hypercubes. As for the fermion field we consider the approximation, in which inside each hypercube the fermion field is constant. As a result, the lattice sites of the given regularization are placed in the middles of the hypercubes. The $SO(4)$ connection is attached to links. The curvature is localized on the plaquettes. Its values in the adjacent hypercubes are related by the gauge transformation.
The model to be discretized has the action $S = S_f + S_T + S_G$, where the different terms are
given by Eq. (\ref{Sf}), Eq. (\ref{ST}), Eq. (\ref{ST2}).
The lattice discretized action for the fermions is similar to that of suggested in \cite{Diakonov} (we performed the rotation to the Euclidean signature of space - time):
\begin{eqnarray}
S_f & = & \sum_{xy} \{ {\psi}^+_{L,x} H^{xy}_L \psi_{L,y} + \psi^+_{R,x} H^{xy}_R \psi_{R,y} \}, \label{Sfl}
\end{eqnarray}
where
\begin{eqnarray}
H_L & = & \sum_k\Bigl( \eta\{ U^+_{R,yx} \bar{\tau}^k \delta_{x-e_k,y} - \bar{\tau}^k U_{L,xy} \delta_{x+e_k,y} \}\nonumber\\
&&-i\theta\{ U^+_{R,yx} \bar{\tau}^k \delta_{x-e_k,y} + \bar{\tau}^k U_{L,xy} \delta_{x+e_k,y} - 2 \bar{\tau}^k \delta_{x,y} \}\Bigr)\nonumber\\
H_R & = & \sum_k \Bigl(\eta\{ U^+_{L,yx} {\tau}^k \delta_{x-e_k,y} - {\tau}^kU_{R,xy} \delta_{x+e_k,y} \} \nonumber\\
&&-i\theta\{ U^+_{L,yx} {\tau}^k \delta_{x-e_k,y} + {\tau}^k U_{R,xy} \delta_{x+e_k,y} - 2 {\tau}^k \delta_{x,y} \}\Bigr)\nonumber\\
&& \tau^4 = \bar{\tau}^4 = -i, \quad \tau^a = -\bar{\tau}^a = \sigma^a (a=1,2,3) \label{Sfl_}
\end{eqnarray}
Here the link matrix is
\begin{eqnarray}
U_{yx} &=& \left(\begin{array}{cc} U_{L,yx} & 0 \\
0 & U_{R,yx} \end{array}\right), \quad U_L,U_R \in SU(2)
\end{eqnarray}
The action is (formally) invariant under the gauge transformation
\begin{eqnarray}
\psi_x & \rightarrow & G_x \psi_x, \quad \psi_x^+ \rightarrow \psi_x^+\Gamma^4 G_x^+ \Gamma^4, \quad U_{xy} \rightarrow G_x U_{xy} G^+_y, \quad \tau^k \rightarrow G_L \tau^k G^+_R, \quad \bar{\tau}^k \rightarrow G_R \bar{\tau}^k G^+_L \nonumber\\
G &=& \left(\begin{array}{cc} G_L & 0 \\
0 & G_R \end{array}\right), \quad G_L,G_R \in SU(2), \quad \Gamma^4 = \left(\begin{array}{cc} 0 & 1 \\
1 & 0 \end{array}\right)
\end{eqnarray}
This symmetry is broken by the lattice discretization, and is restored in the continuum limit, when the invariance under local $SO(4)$ coordinate transformations comes back. (The appropriate transformations of the coordinates compensate the transformation of $\tau$ and $\bar{\tau}$.)
The fermion action can also be rewritten in the following way:
\begin{eqnarray}
S_f & = & \sum_{xy} {\psi}^+_{x} \Gamma^4 {\cal H}^{xy}_L \psi_{y}, \nonumber\\
{\cal H} &=& \left(\begin{array}{cc} 0 & H_R \\
H_L & 0 \end{array}\right), \quad H_L^+ = H_R \label{HH}
\end{eqnarray}
Remarkably, for $\theta\ne 0$ in the absence of the gauge field the given discretization gives no doublers just like the Wilson formulation of lattice fermions.
Next, let us introduce the definition of lattice $SO(4)$ curvature. It corresponds to the closed path along the boundary of the given plaquette started from the given lattice site $x$. Therefore, it is marked by the position of the lattice site $x$ and the couple of the directions in space - time $n,j$.
\begin{eqnarray}
{\cal G}_{x,n,j}^{4,k} & = & - {\cal G}_{x,n,j}^{k,4} = i \,{\rm sign}(n)\, {\rm sign}(j) \, \Bigl( {\rm Tr}\, ( U_{L,x,n,j} - U^+_{L,x,n,j}) \sigma^k - {\rm Tr}\, ( U_{R,x,n,j} - U^+_{R,x,n,j}) \sigma^k \Bigr) \nonumber\\
{\cal G}_{x,n,j}^{k,l} & = & i\, {\rm sign}(n)\, {\rm sign}(j) \, \epsilon^{klm} \Bigl( {\rm Tr}\, ( U_{L,x,n,j} - U^+_{L,x,n,j}) \sigma^m + {\rm Tr}\, ( U_{R,x,n,j} - U^+_{R,x,n,j}) \sigma^m \Bigr)
\end{eqnarray}
Here $n,j = \pm 1,\pm 2,\pm 3,\pm 4$. Positive sign corresponds to the positive direction while negative sign corresponds to the negative direction in $4D$ space - time. The plaquette variables obtained by the product of link matrices along the boundary of the plaquette located in $n,j$ plane (starting from the point $x$) are denoted by $U_{R,x,n,j}$ and $U_{R,x,n,j}$. Contraction of indices results in the definition of lattice Ricci tensor $\cal R$ and the lattice scalar curvature $\cal S$. Both these quantities are also attached to the closed paths around the boundaries of the plaquettes and are marked by $x, n, j$.
\begin{eqnarray}
{\cal R}_{x,n,j}^{k} & = & {\cal G}_{x,n,j}^{k,|n|}, \quad {\cal S}_{x,n,j} = {\cal R}_{x,n,j}^{|j|}
\end{eqnarray}
For the part of the action linear in curvature we have:
\begin{eqnarray}
S_T & = & -\kappa \sum_{x} \sum_{n,j=\pm 1,\pm 2, \pm 3, \pm 4} {\cal S}_{x, n, j}
\end{eqnarray}
Analogue of quantity $A$ from Eq. (\ref{A}) is given by
\begin{eqnarray}
{\cal A}_{x,n,j} & = & \sum_{kl}\epsilon^{|n||j|kl} {\cal G}_{x,n,j}^{k,l}
\end{eqnarray}
Contraction of indices in quadratic expressions results in the following combinations:
\begin{eqnarray}
Q_{x}^{(1)} & = & \sum_{k,l, n, j} {\cal G}_{x,n,j}^{|k|,|l|}{\cal G}_{x,n,j}^{|k|,|l|} \nonumber\\
Q^{(2)}_x &=& \sum_{k,l, n, j} {\cal G}_{x,n,j}^{|k|,|l|}{\cal G}_{x,k,l}^{|n|,|j|}\nonumber\\
Q^{(3)}_x &=& \sum_{k, n, j} {\cal R}_{x,n,j}^{|k|}{\cal R}_{x,n,j}^{|k|}\nonumber\\
Q^{(4)}_x &=& \sum_{k, n, j} {\cal R}_{x,n,j}^{|k|}{\cal R}_{x,n,k}^{|j|}\nonumber\\
Q^{(5)}_x &=& \sum_{n, j} {\cal S}_{x,n,j}{\cal S}_{x,n,j}\nonumber\\
Q^{(6)}_x &=& \sum_{ n, j} {\cal A}_{x,n,j}{\cal A}_{x,n,j}
\end{eqnarray}
(The sum is over positive and negative values of $n,j,k,l$.)
Finally, the term in the action quadratic in curvature has the form
\begin{eqnarray}
S_G & = & \sum_{x}\sum_{i=1}^6 \beta_i Q^{(i)}_x
\end{eqnarray}
Our main expectation is that in this theory the chiral symmetry is broken dynamically while the confinement is absent.
The main quantities to be measured are the static potential extracted from the Polyakov loops and the chiral condensate.
The chiral condensate $\chi$ may be calculated using the Banks - Casher relation
\begin{eqnarray}
\chi & = & - \frac{\pi}{V} \langle \nu(0)\rangle,
\end{eqnarray}
where $\nu(\lambda)$ is the density of eigenvalues of the operator $\cal H$ given by Eq. (\ref{HH}), $V$ is the $4D$ volume.
The potential $V_{L\bar{L}}(R)=V_{R\bar{R}}(R)$ between the static (either left - handed or right - handed) fermion and anti - fermion is defined through the relation
\begin{eqnarray}
&&{\rm exp}( - V_{L\bar{L}}(|x-y|) T) = \langle {\cal P}_L(x) {\cal P}_R(y) \rangle, \nonumber\\
&& {\cal P}_L(x) = {\rm Tr} \Bigl(\Pi_{K = 0, ... , T-1}U_{L, x + Ke_4, x+(K+1)e_4}\Bigr) , \nonumber\\
&& {\cal P}_R(y) = {\rm Tr} \Bigl(\Pi_{K = 0, ... , T-1}U_{R, x + Ke_4, x+(K+1)e_4}\Bigr)
\end{eqnarray}
Here $x_4 = y_4 = 0$, while $T$ is the lattice extent in time direction.
The potential $V_{L\bar{R}}(R)=V_{R\bar{L}}(R)$ between the static fermion and anti - fermion of different chiralities is given by \begin{equation}
{\rm exp}( - V_{L\bar{R}}(|x-y|) T) = \langle {\cal P}_L(x) {\cal P}_L(y)\rangle = \langle {\cal P}_R(x) {\cal P}_R(y)\rangle
\end{equation}
\section{Conclusions}
In this paper we considered the fermions coupled in a nonminimal way to the gauge theory of Lorentz group. First, we have derived the effective four - fermion low energy theory (the NJL approximation). The derivation was based on the assumption that the couplings $\beta_i$ entering Eq. (\ref{ST2}) are asymptotic free. As a result the term of the gauge field action squared in curvature can be neglected at small enough energies.
In our analysis of the effective NJL low energy theory we follow the proposition of \cite{VZ2012} that the microscopic theory is organized in such a way that the contributions from the trans - $\Lambda_{\chi}$ degrees of freedom cancel the dominant divergences at more than one loop existing in the NJL model. (The unified theory that generalizes the gauge theory of Lorentz group may play the role of such a microscopic theory. We even do not exclude that the gauge theory of Lorentz group itself may play such a role and provide mechanism for this cancellation.) This pattern is similar to that of quantum hydrodynamics \cite{quantum_hydro} and was proposed for the solution of the hierarchy problem and the problem of vacuum energy in \cite{hydro_gravity}.
The one - loop analysis demonstrates that the effective NJL model may provide the chiral symmetry breaking. This results in the dynamical electroweak symmetry breaking and in the appearance of the masses for W,Z bosons, as well as the masses of all existing fermions.
Thus we have suggested the possible new scenario of the dynamical electroweak symmetry breaking. In this scenario Lorentz group plays the role of the technicolor group. But there are no technifermions. The Higgs bosons are composed directly of the existing Standard Model fermions. We have demonstrated that the chiral symmetry breaking may take place in the given model in the approximation when the squared curvature gauge field action $S_G$ is neglected. In the same approximation the confinement is absent. This indicates that in the complete theory with $S_G$ taken into account the chiral symmetry breaking may take place in the absence of confinement. If this pattern indeed takes place, the chiral condensate provides masses for the electroweak gauge bosons. In the zero order approximation all fermions have equal masses. The fermionic excitations are not confined and may exist as distinct particles. We suggest that the observed massive Standard Model fermions may appear in this way.
As it was mentioned above, in the zero order approximation all fermion masses are equal to each other. The standard model gauge fields (and, probably, some other unknown fields) interact with the fermions and provide small corrections to the gap equations. In \cite{VZ2013_2} the model was suggested that appears as a result of the deformation of the model, in which the fermions experience only the interaction with the gauge bosons of the Lorentz group. This model is considered in details in section 5 of the present paper. The spectrum of bosonic and fermionic excitations in this model is given by the expressions similar to that of \cite{VZ2012,VZ2013}. We observe, that in this case small corrections to the original model may result in the considerable difference between the fermion masses. This is provided by the mentioned above fine tuning that allows to disregard the higher loops in the NJL approximation. Although the fine tuning of this type is often thought of as unnatural, we do not think it is possible to avoid it in the model that pretends to explain the appearance of the fermion masses from less than $1$ eV for neutrino to above $100$ GeV for the top - quark. The models considered in \cite{VZ2012,VZ2013,VZ2013_2} suffer from various problems that at the present moment do not allow to consider them as realistic. First of all, in the strongly coupled model of this type the spectrum of bosonic excitations contains scalar excitations with small masses (much smaller than the electroweak scale). One of the possible solutions of this problem was sketched in \cite{VZ2013}. Namely, these light scalar bosons may appear to be the Goldstone bosons eaten by some gauge fields during the spontaneous breakdown of the additional gauge symmetries. Besides, as it was mentioned in \cite{VZ2013_2}, the scalar state of the $\bar{t}t$ channel is excluded by the present LHC data. So, the mechanism should be added that either suppresses the cross section with the creation of this state, or increases its mass considerably. It is worth mentioning that the model of \cite{VZ2013_2} considered in section 5 of the present paper does not exhaust all possible deformations of the original theory, in which the fermions interact with the gauge field of the Lorentz group. That's why, it could be that more realistic deformations may appear.
We have suggested the lattice discretization of the model, in which fermions interact with the gauge bosons of the Lorentz group. On the lattice the gauge invariance is lost. It is supposed to be restored in the continuum limit. Lattice Dirac operator has several peculiar properties. In the presence of nonminimal interactions the fermion doublers disappear like in the model with Wilson fermions. The Dirac operator is Hermitian, and the Banks - Casher relation can be used in order to check the appearance of the chiral condensate.
\section*{Acknowledgements}
Numerous discussions with G.E.Volovik are kindly acknowledged. The author is greatful to Yu.A.Simonov for the discussion that stimulated him to consider the unusual scenario for the dynamical electroweak symmetry breaking. The author is especially indebted to D.I.Diakonov for the continuous support during the last two years and for the discussions of issues related to the fermions coupled to gravitational fields. Also the author is greatful to B.Svetitsky for the discussion of the lattice discretization and to V.I.Zakharov for the discussion of the chiral symmetry breaking in the considered model. This work was partly supported by RFBR grant 11-02-01227, by the
Federal Special-Purpose Programme 'Human Capital' of the Russian Ministry of
Science and Education, by Federal Special-Purpose Programme 07.514.12.4028.
|
1,116,691,499,302 | arxiv | \section*{Supplemental Material}
\section{Renormalization group analysis}
First we consider the easy-plane CP$^1$ model with a soft constraint and without any additional gauge terms,
\begin{eqnarray}
\label{eq:CP1soft}
\mathcal{L}&=&\sum_{a=1,2}|(\partial_\mu-ia_\mu)z_a|^2+ \frac{K}{2}(|z_1|^2- |z_2|^2)^2\nonumber\\
&+&m_0^2(|z_1|^2+|z_2|^2)+\frac{u}{2}(|z_1|^2+|z_2|^2)^2.
\end{eqnarray}
We define the couplings $U=u+K$ and $V=u-K$ and derive the renormalized coupings at one-loop order,
\begin{equation}
U_r=U-(5U^2+V^2)I(m^2),
\end{equation}
\begin{equation}
V_r=V-2(V^2+2UV)I(m^2),
\end{equation}
where,
\begin{equation}
I(m^2)=\int_p\frac{1}{(p^2+m^2)^2}=\frac{m^{d-4}}{(4\pi)^{d/2}}\Gamma\left(2-
\frac{d}{2}\right),
\end{equation}
with $m$ being the renormalized mass, and we have introduced the notation $\int_{p} = \int \frac{d^3p}{(2 \pi)^3}$.
Since the CS is absent, we are generalizing the
calculation to $d$ dimensions.
We define the dimensionless renormalized couplings by $\hat{U}=m^{d-4}U_r$ and
$\hat{V}=m^{d-4}V_r$, along with the rescalings $\hat{U}\to\hat{U}/c_d$,
$\hat{V}\to\hat{V}/c_d$, where $c_d=(4-d)(4\pi)^{-d/2}\Gamma(2-d/2)$, so that the
RG $\beta$ functions, $\beta_{\hat{U}}=md\hat{U}/dm$ and $\beta_{\hat{V}}=md\hat{V}/dm$
are obtained,
\begin{equation}
\label{Eq:beta-U}
\beta_{\hat{U}}=m\frac{d\hat{U}}{dm}=-(4-d)\hat{U}+5\hat{U}^2+\hat{V}^2,
\end{equation}
\begin{equation}
\label{Eq:beta-V}
\beta_{\hat{V}}=m\frac{d\hat{V}}{dm}=-(4-d)\hat{V}+2(\hat{V}^2+2\hat{U}\hat{V}).
\end{equation}
We obtain three fixed points, namely, the Gaussian fixed point, $(\hat{U}_G,\hat{V}_G)=(0,0)$,
the $XY$ fixed point, $(\hat{U}_{\rm XY},\hat{V}_{\rm XY})=(\epsilon/5,0)$, and the
easy-plane anisotropy fixed point, $(\hat{U}_*,\hat{V}_*)=(\epsilon/6)(1,1)$,
where $\epsilon=4-d$. The
$XY$ fixed point is stable for $\hat{V}=0$, but becomes unstable for any small $\hat{V}$.
Note that this fixed point actually corresponds $\hat{U}=2\hat{K}$, so this yields
a fixed point $\hat{K}_*=\epsilon/10$ associated to the $O(2)\times O(2)$ symmetry.
The fixed point $(\hat{U}_*,\hat{V}_*)$
actually corresponds to vanishing anisotropy, i.e., $\hat{K}=0$, since $\hat{U}_*=\hat{V}_*$,
implying $\hat{u}_*=\epsilon/6$. Hence, this fixed point governs the
$O(4)$ universality class. This fixed point becomes IR unstable only in a region
where $K<0$, which would correspond to easy-axis rather than easy-plane anisotropy.
Next, we consider the coupling to the gauge field $a_\mu$. We include CS and
Maxwell terms,
\begin{equation}
\mathcal{L}_{\rm gauge}=\frac{1}{2e^2}(\epsilon_{\mu\nu\lambda}\partial_\nu a_\lambda)^2+i\frac{\kappa}{2}\epsilon_{\mu\nu\lambda}a_\mu\partial_\nu a_\lambda.
\end{equation}
The gauge field propagator in the absence of interactions is given in the Landau gauge by,
\begin{equation}
D_{\mu\nu}(p)=\frac{1}{p^2+M^2}\left(
\delta_{\mu\nu}-\frac{p_\mu p_\nu}{p^2}-M\epsilon_{\mu\nu\lambda}\frac{p_\lambda}{p^2}\right),
\end{equation}
where $M=e^2\kappa$, and where we used the rescaling $a_\mu\to ea_\mu$.
Since the CS term is defined in three spacetime dimensions, the renormalized couplings $U_r$ and $V_r$ will be calculated for fixed dimensionality $d=3$ \cite{Parisi}. This has the drawback of making $\epsilon=4-d$ as a control parameter in principle unavailable to us (see section \ref{sec:control} for a more thorough discussion on this point). Instead, we generalize the easy-plane model to a system featuring a global $O(N)\times O(N)$ symmetry with $N$ even and explicitly consider the large $N$ limit. This is achieved by considering $N/2$ complex fields $z_{1a}$ and $z_{2a}$ ($a=1,\dots,N/2$). In this case the special case where the anisotropy is absent (i.e., $K=0$) will correspond to an $O(2N)$ global symmetry.
The coupling to a dynamical gauge field will cause $U_r$ and $V_r$ to
receive a contribution from the
diagram at Fig. \ref{Fig:self-energy} through the square of the wave function
renormalization. This diagram is the only one giving a momentum dependent contribution to the total self-energy at one-loop. Its explicit expression is given by,
\begin{figure}[h]
\includegraphics[width=4cm]{graph4p-2-1.pdf}
\caption{Scalar field self-energy}
\label{Fig:self-energy}
\end{figure}
\begin{eqnarray}
\Sigma(p)&=&-e^2\int\frac{d^3k}{(2\pi)^3}\frac{(2p_\mu-k_\mu)(2p_\nu-k_\nu)}{(p-k)^2+m^2}D_{\mu\nu}(k)
\nonumber\\
&=&4e^2\left\{\int\frac{d^3k}{(2\pi)^3}\frac{(k\cdot p)^2}{k^2[(p-k)^2+m^2](k^2+M^2)}
\right.\nonumber\\
&-&\left.\int\frac{d^3k}{(2\pi)^3}\frac{1}{[(p-k)^2+m^2](k^2+M^2)}
\right\}.
\end{eqnarray}
In the small external momentum $|p|$ limit, we obtain,
\begin{equation}
\Sigma(p)=-\frac{2e^2}{3\pi}\frac{p^2}{|m|+|M|}+\mathcal{O}(p^4).
\end{equation}
Another diagram contributing to both
$U_r$ and $V_r$ is shown in Fig. \ref{Fig:photon4p-loop}. However, the
latter vanishes at zero external momenta (see the Appendix A in Ref. \cite{Conformality_loss}).
\begin{figure}[h]
\includegraphics[width=4cm]{photon-fish.pdf}
\caption{One-loop photon diagram contributing to the couplings $U_r$ and $V_r$.}
\label{Fig:photon4p-loop}
\end{figure}
Therefore, we obtain the wave function renormalization,
\begin{equation}
Z=1+\frac{2}{3\pi}\frac{e^2}{m+e^2|\kappa|}
\end{equation}
and dimensionless renormalized couplings,
\begin{eqnarray}
\label{Eq:U-exp}
\hat{U}&=&Z^2\left[\frac{U}{m}-\frac{(N+8)U^2+NV^2}{16\pi m^2}\right]
\nonumber\\
&\approx&\left[\frac{U}{m}-\frac{e^2}{2\pi m|\kappa|}+\frac{4e^2U}{3\pi m(m+e^2|\kappa|)}
\right.\nonumber\\
&-&\left.\frac{(N+8)U^2+NV^2}{16\pi m^2}\right],
\end{eqnarray}
\begin{eqnarray}
\label{Eq:V-exp}
\hat{V}&=&Z^2\left[\frac{V}{m}-\frac{2V^2+(N+2)UV}{8\pi m^2}\right]
\nonumber\\
&\approx&\left[\frac{V}{m}-\frac{e^2}{2\pi m|\kappa|}+\frac{4e^2V}{3\pi m(m+e^2|\kappa|)}
\right.\nonumber\\
&-&\left.\frac{2V^2+(N+2)UV}{8\pi m^2}\right],
\end{eqnarray}
where now the dimensionality is fixed to $d=3$.
We define an additional dimensionless coupling, $\hat{e}^2=e_r^2/m$, where $e_r^2$ is the
renormalized gauge coupling which is calculated at one-loop order by considering the vacuum polarization diagram of Fig. \ref{Fig:poldiag}.
\begin{figure}[h]
\includegraphics[width=4cm]{graph4p-1-1.pdf}
\caption{Vacuum polarization diagram. The wiggles represent the gauge field $a_\mu$. The continuous line can be either fermion or
boson propagators.}
\label{Fig:poldiag}
\end{figure}
The new RG $\beta$ functions are given by,
\begin{equation}
\label{Eq:beta-U-1}
\beta_{\hat{U}}=-\left[1+\frac{4\hat{e}^2}{3\pi(1+\hat{e}^2|\kappa|)^2}\right]\hat{U}+
\frac{(N+8)\hat{U}^2+N\hat{V}^2}{16\pi},
\end{equation}
\begin{equation}
\label{Eq:beta-V-1}
\beta_{\hat{V}}=-\left[1+\frac{4\hat{e}^2}{3\pi(1+\hat{e}^2|\kappa|)^2}-
\frac{2\hat{V}+(N+2)\hat{U}}{8\pi}\right]\hat{V},
\end{equation}
along with the $\beta$ function,
\begin{equation}
\label{Eq:betae2}
\beta_{\hat{e}^2}=-\hat{e}^2+\frac{N\hat{e}^4}{24\pi}.
\end{equation}
On the other hand, the non-renormalization of the CS term \cite{COLEMAN1985184} implies,
\begin{equation}
\beta_\kappa=\left(\frac{N\hat{e}^2}{24\pi}-1\right)\kappa.
\end{equation}
Therefore, at the charged fixed point an arbitrary value of
$\kappa$ is allowed, leading to a critical behavior featuring continuously varying
critical exponents as a function of $\kappa$. The vanishing of $\beta_{\hat{e}^2}$ at the IR stable fixed point
(i.e., $\hat{e}^2_*\neq 0$), automatically implies the vanishing of $\beta_\kappa$ for
arbitrary $\kappa$. Thus,
the fixed point structure of the $\beta$ functions (\ref{Eq:beta-U-1}) and
(\ref{Eq:beta-V-1}) at the IR stable fixed point $\hat{e}^2_*=24\pi/N$ is similar to the one
of the $\beta$ functions for the charge neutral system given by Eqs. (\ref{Eq:beta-U}) and
(\ref{Eq:beta-V}). Plugging the fixed point $e_*^2$ into Eqs. (\ref{Eq:beta-U-1}) and (\ref{Eq:beta-V-1}) we find that these $\beta$ functions have two nontrivial fixed points, namely,
\begin{equation}
(\hat{U}_*,\hat{V}_*)=\left(\frac{16\pi[32N+(N+12n)^2]}{(N+8)(N+12n)^2},0\right),
\end{equation}
corresponding to the $O(N)\times O(N)$ symmetry regime,
while the $O(2N)$ symmetric case,
\begin{equation}
\hat{U}_*=\hat{V}_*=\frac{8\pi[32N+(N+12n)^2]}{(N+4)(N+12n)^2},
\end{equation}
where we have assumed a level $n$ CS term, $\kappa=n/(2\pi)$.
An additional fixed point $(\hat{U}_1,\hat{V}_1)$ corresponding to a regime where $K>u$ is obtained for $N>4$ (recall that $N$ is even), where,
\begin{equation}
\hat{U}_1=\frac{8\pi[32N+(N+12n)^2]}{(N^2+8)(N+12n)^2},
\end{equation}
\begin{equation}
\hat{V}_1=(4-N)\hat{U}_1.
\end{equation}
Note that for $N=2$ the fixed point $(\hat{U}_1,\hat{V}_1)$ coincides with the $O(4)$ symmetric one.
The flow diagram in terms of the original couplings $u$ and $K$ is shown in Fig. \ref{Fig:RG-flow}. Interestingly, we see that the $O(4)$ symmetric fixed point occurring for a vanishing anisotropy is IR stable. This implies that for the CS CP$^1$ theory a deconfined critical point occurs in the more symmetric case. The anisotropy fixed point is stable along the line $\hat{u}=\hat{K}$, corresponding to the case of a scalar self-coupling interaction of the form, $|z_1|^4+|z_2|^4$.
Hence, we arrive at two non-trivial fixed points that govern second-order
phase transitions. Clearly, a new universality
class emerges, since the critical exponents will depend on the CS level.
\begin{figure}[t]
\begin{center}
\includegraphics[width=9cm]{uKflow-1.pdf}
\caption{The RG flow for the $\beta$ functions of the dimensionless couplings $\hat{u}$ and $\hat{K}$. The $O(4)$ symmetric fixed point corrsponding to vanishing anisotropy is IR stable, while the anisotropy fixed point corresponding to an $O(2)\times O(2)$ symmetry is stable along the line $\hat{u}=\hat{K}$.}
\label{Fig:RG-flow}
\end{center}
\end{figure}
For instance, for either the $O(2)\times O(2)$ or $O(4)$
symmetric fixed points, we obtain,
\begin{equation}
\frac{1}{\nu}=2-\frac{(\mathcal{N}+2)\hat{U}_*}{16\pi}+\frac{2\hat{e}^2_*}{3\pi(1+\hat{e}_*^2|\kappa|)^2},
\end{equation}
where $\mathcal{N}=2$ and $\mathcal{N}=4$ correspond to the $O(2)\times O(2)$ and $O(4)$ symmetries, respectively.
Hence,
\begin{equation}
\frac{1}{\nu^{O(2)\times O(2)}}=\frac{8}{5}\left[1+\frac{1}{(1+6n)^2}\right],
\end{equation}
\begin{equation}
\frac{1}{\nu^{O(4)}}=\frac{3}{2}.
\end{equation}
Thus, for a level 1 CS term this yields $\nu^{O(2)\times O(2)}=49/80\approx 0.613$. This is nearly the same as
the one-loop value $\nu=5/8$ of the $XY$ universality class.
For the $O(4)$ symmetric criticality we obtain a larger
correlation length critical exponent, $\nu^{O(4)}=2/3$, which at this order is independent of the CS level. Interestingly,
the same value is obtained for the limit case of a neutral system.
Finally, we would like to calculate the anomalous dimension $\eta_N$ of the critical
magnetization correlation function.
Using the relation,
\begin{equation}
\sigma^a_{\alpha\beta}\sigma^a_{\gamma\delta}=2\delta_{\alpha\delta}\delta_{\beta\gamma}
-\delta_{\alpha\beta}\delta_{\gamma\delta},
\end{equation}
we obtain that,
\begin{eqnarray}
\label{Eq:nn-correlation}
\mathcal{G}(x)=\langle\vec{n}(x)\cdot\vec{n}(0)\rangle&=&2\langle z_\alpha^*(x)z_\beta(x)z_\alpha(0)
z_\beta^*(0)\rangle
\nonumber\\
&-&\langle|z_\alpha(x)|^2|z_\beta(0)|^2\rangle.
\end{eqnarray}
Note that in the hard constraint case the second term in the equation above is unity.
The calculation of $\eta_N$ amounts to finding the anomalous dimension of the
operator $z_\alpha^*(x)z_\beta(x)$ \cite{zinn2002quantum}. The anomalous
dimension of this operator is one of the eigenvalues occurring in the matrix,
\begin{equation}
\llbracket\eta_2\rrbracket=-\mathcal{N}\frac{\hat{U}_*}{16\pi}\llbracket P\rrbracket
-\left[\frac{\hat{U}_*}{8\pi}-\frac{2\hat{e}_*^2}{3\pi(1+\hat{e}_*^2|\kappa|)^2}
\right]\llbracket I\rrbracket,
\end{equation}
where $\mathcal{N}=2$ corresponds to the $O(2)\times O(2)$ symmetric case and $\mathcal{N}=4$ to the
$O(4)$ symmetric one. The matrix elements of $\llbracket P\rrbracket$ and
$\llbracket I\rrbracket$ are given by,
\begin{equation}
\llbracket I\rrbracket_{\alpha\beta,\gamma\delta}=\frac{1}{2}(\delta_{\alpha\gamma}\delta_{\beta\delta}+\delta_{\alpha\delta}\delta_{\beta\gamma}),
\end{equation}
\begin{equation}
\llbracket P\rrbracket_{\alpha\beta,\gamma\delta}=\frac{1}{\mathcal{N}}\delta_{\alpha\beta}\delta_{\gamma\delta},
\end{equation}
where the indices run from 1 to 4. The trace of $\llbracket\eta_2\rrbracket$ yields
$\eta_2\delta_{\alpha\beta}$, which corresponds to insertions of $|z_\alpha|^2$. In
other words, we recover the formula for $1/\nu$ via the well known relation
$\eta_2=1/\nu-2$. The insertion of the operator $z_\alpha^*(x)z_\beta(x)$ with
$\alpha\neq \beta$ corresponds to the zero eigenvalue of $\llbracket P\rrbracket$, and so
we obtain,
\begin{equation}
\widetilde{\eta}_2=-\frac{\hat{U}_*}{8\pi}+\frac{2\hat{e}_*^2}{3\pi(1+\hat{e}_*^2|\kappa|)^2},
\end{equation}
which is related to $\eta_N$ by the formula $\eta_N=1-2\widetilde{\eta}_2$. In the
$O(2)\times O(2)$ symmetric case we obtain,
\begin{equation}
\eta_N^{O(2)\times O(2)}=\frac{1}{5}\left[7-\frac{48}{(1+6n)^2}\right],
\end{equation}
and similarly for $O(4)$ symmetry,
\begin{equation}
\eta_N^{O(4)}=\frac{4}{3}\left[1-\frac{56}{5(1+6n)^2}\right].
\end{equation}
For a level 1 CS term we obtain, $ \eta_N^{O(2)\times O(2)}=59/49\approx 1.2$ and
$\eta_N^{O(4)}=164/147\approx 1.12$.
\section{Controlling the RG analysis}
\label{sec:control}
The advantage of the $\epsilon$-expansion is that it allows for a reliable expansion parameter in perturbation theory. Of course, one then is ultimately interested in the $\epsilon=1$ case, and several mathematical techniques have been used in the past to show that the perturbation series actually converge \cite{zinn2002quantum}. There is also the fixed dimension approach by Parisi \cite{Parisi}, but this typically applies to scalar field theories without the coupling to a gauge field. The difficulty can be seen in our case, where for $N=2$, which is the case we are interested in, the RG fixed point $\hat{e}^2_*=12\pi$ is too large. We cannot use the $\epsilon$-expansion to obtain $\hat{e}^2_*\sim\mathcal{O}(\epsilon)$ in this case because the CS term imposes a fixed dimension $d=3$ from the outset. Furthermore, thanks to the CS term the Feynman diagram of Fig. \ref{Fig:photon4p-loop} vanishes. This diagram is a known obstruction towards reaching a fixed point for the scalar couplings, since it leads to a large contribution in the $\beta$ functions, even within the $\epsilon$-expansion. Only for $N$ sufficiently large the theory can become critical for a nonzero gauge coupling \cite{Hikami}. Hence, fixed dimensionality $d=3$ is a desirable feature in our case.
Introducing a larger global symmetry group at fixed dimension $d=3$ provides a way to control the perturbation expansion, since all fixed points behave as $\sim\mathcal{O}(1/N)$ for $N$ large. However, let us give an additional argument that even though we take $N=2$ at the end, the results for universal quantities can be relied upon.
The main source of difficulty for $N=2$ is the fixed point value for the gauge coupling, which is $\hat{e}^2_*=12\pi$ in this case. However, quite generally, the renormalization of the gauge coupling follows from the self-energy of the gauge field propagator, which is obtained from the vacuum polarization diagram of Fig. \ref{Fig:poldiag}. This leads to the effective Maxwell Lagrangian,
\begin{equation}
\label{Eq:Maxwell-eff}
\mathcal{L}_{M}=\frac{1}{2e^2}\left[1+\Pi(0)\right](\epsilon_{\mu\nu\lambda}\partial_\nu a_\lambda)^2,
\end{equation}
where $\Pi(0)=Ne^2/(24\pi m)$ is the vacuum polarization at $p=0$.
From this we read off the dimensionless renormalized gauge coupling,
\begin{eqnarray}
\label{Eq:e2-eff}
\hat{e}^2&=&\frac{e^2m^{-1}}{1+\Pi(0)}
=\frac{e^2m^{-1}}{1+\frac{Ne^2}{24\pi m}}.
\end{eqnarray}
Thus, the $\beta$ function of Eq. (\ref{Eq:betae2}) is easily obtained by simple differentiation, $\beta_{\hat{e}^2}=md\hat{e}^2/dm$ without any need of a series expansion. Furthermore, we note the following two important facts. First, the result obtained in Eq. (\ref{Eq:e2-eff}) is the same as the one obtained within an $1/N$ expansion. Second, the fixed point follows from Eq. (\ref{Eq:e2-eff}) in two different ways, namely, either by directly letting $m\to 0$, or by taking the limit where the bare (dimensionful) gauge coupling $e^2\to\infty$. The latter limit highlights the strong coupling character of the theory at $d=3$. Furthermore, this is the regime of interest to us in the duality analysis.
As far as the couplings $\hat{U}$ and $\hat{V}$ are concerned, $\hat{e}^2$ enters only via the wavefunction renormalization, since the diagram of Fig. \ref{Fig:photon4p-loop} vanishes. Since the CS mass also depends on $e^2$, the perturbative results in Eqs. (\ref{Eq:U-exp}) and (\ref{Eq:V-exp}) are not jeopardized by the strong-coupling character of the gauge coupling.
Finally, we could, somewhat artificially, make an $\epsilon$-expansion analysis in which we compute Feynman diagrams in $d$ dimensions for the cases where $\epsilon_{\mu\nu\lambda}$ does not play any role, while still keeping $d=3$ in the diagram of Fig. \ref{Fig:photon4p-loop}, since in this case $\epsilon_{\mu\nu\lambda}$ plays a crucial role. It is worth to carry out this calculation as well, in order to clearly show the need of the fixed dimension approach in this case. In fact, we will show below that while fixed points exist as before, they lead to unphysical values of the critical exponent $\nu$ in the $O(2)\times O(2)$ invariant case.
Most of what we need for this calculation is already available, since we have discussed the $d$-dimensional example in absence of the gauge coupling earlier in the previous section. It remains to discuss the changes in the diagram of Fig. \ref{Fig:self-energy}. We have,
\begin{eqnarray}
\Sigma(p)&=&\frac{4e^2m^{d-4}}{(4\pi)^{d/2}}\left(1-\frac{1}{d}\right)\Gamma\left(1-\frac{d}{2}\right)
\nonumber\\
&\times&\frac{1-(m/M)^{d-2}}{1-M^2/m^2}p^2+\mathcal{O}(p^4),
\end{eqnarray}
which upon expanding around $d=4$ yields the wavefunction renormalization at one-loop order,
\begin{equation}
Z=1+\frac{3\hat{e}^2}{8\pi^2\epsilon}.
\end{equation}
Hence, the $\beta$ functions become for $N=2$,
\begin{equation}
\label{Eq:beta-U-2}
\beta_{\hat{U}}=-(\epsilon+6\hat{e}^2)\hat{U}+5\hat{U}^2+\hat{V}^2,
\end{equation}
\begin{equation}
\label{Eq:beta-V-2}
\beta_{\hat{V}}=-(\epsilon+6\hat{e}^2)\hat{V}+2(\hat{V}^2+2\hat{U}\hat{V}),
\end{equation}
\begin{equation}
\label{Eq:beta-e2}
\beta_{\hat{e}^2}=-\epsilon\hat{e}^2+\frac{\hat{e}^4}{3},
\end{equation}
\begin{equation}
\beta_\kappa=\left(\frac{\hat{e}^4}{3}-\epsilon\right)\kappa,
\end{equation}
where we have performed a rescaling similar to the one described above Eqs. (\ref{Eq:beta-U}) and (\ref{Eq:beta-V}).
Now, if we consider the correlation length exponent for the $O(2)\times O(2)$ case, we obtain to order $\epsilon$,
\begin{equation}
\label{Eq:nu-unphys}
\nu_{O(2)\times O(2)}=\frac{1}{2+8\epsilon/3}\approx\frac{1}{2}-\frac{2\epsilon}{3},
\end{equation}
and we see after setting $\epsilon=1$ at the end that $\nu_{O(2)\times O(2)}<0$ and therefore unphysical. Even if one does not completely adhere to the $\epsilon$-expansion and use Eq. (\ref{Eq:nu-unphys}) without making the expansion, a result smaller than 1/2 is obtained after setting $\epsilon=1$. This is also unphysical, since the critical exponent $\nu$ should be larger than or equal to its mean-field value for a local field theory of this type.
\section{Self-duality}
Let us introduce a change of the variables for the gauge fields, $b_{+\mu} = (b_{1\mu}+b_{2\mu})/2$, $b_{-\mu} = (b_{1\mu}-b_{2\mu})/2$. Then, the Eq. (13) of the paper takes the form,
\begin{eqnarray}
\label{Eq:Seff-dual}
\mathcal{L}_{\text{dual}}&=&g[(\epsilon_{\mu\nu\lambda}\partial_\nu b_{+ \lambda})^2 +(\epsilon_{\mu\nu\lambda}\partial_\nu b_{- \lambda})^2 ]\nonumber\\
&+&i2\pi (w_{1\mu} +w_{2\mu}) b_{+\mu}+ i2\pi (w_{1\mu} -w_{2\mu}) b_{-\mu}\nonumber\\&-&\frac{i}{2\kappa}\epsilon_{\mu\nu\lambda}
b_{+\mu} \partial_\nu b_{+\lambda}.
\end{eqnarray}
To integrate out the gauge fields, we need to find the propagator $\widetilde{D}_{\mu \nu} $ which is the inverse of the tensor,
\begin{equation}
M_{\mu \nu} =2 g\left(p^{2} \delta_{\mu \nu}-p_{\mu} p_{\nu}\right)+\frac{2 }{\alpha} p_{\mu} p_{\nu}
+\frac{4}{\kappa} \varepsilon_{\mu \nu \lambda} p_{\lambda},
\end{equation}
with a gauge fixing $\alpha$.
After a straightforward calculation one obtains the propagator $\widetilde{D}_{\mu \nu}$ in the momentum space,
\begin{equation}
\widetilde{D}_{\mu \nu}(p)=\frac{1}{2\left(g^{2} \kappa^{2} p^{2}+4\right)}\left(\delta_{\mu \nu}-2\frac{\varepsilon_{\mu \nu \lambda}p_{\lambda}}{\kappa g p^{2}}\right),
\label{eq:propagator}
\end{equation}
where the Landau gauge ($\alpha =0$) was used and we have dropped a longitudinal part $\sim p_\mu p_\nu$, since the zero divergence constraint of the vortex loop variables causes such a term to give a vanishing contribution. Therefore, the effective action for vortex fields in the momentum space is
\begin{eqnarray}
\label{eq:sdual-momentum-dmunu}
S_{\text {dual }}&=&2 \pi^{2} g \kappa^{2} \int_{p} D_{\mu \nu}(p)(w_{1 \mu}+w_{2 \mu} )(p)(w_{1 \mu}+w_{2 \mu})(-p) \nonumber\\
&+&\frac{\pi^{2}}{g} \int_{p} \frac{(w_{1 \mu}-w_{2 \mu} )(p)(w_{1 \mu}-w_{2 \mu})(-p)}{p^{2}}.
\end{eqnarray}
And in the coordinate space one obtains,
\begin{eqnarray}
\label{eq:sdual-coor-dmunu}
S_{\text {dual }}&=&2 \pi^{2} g \kappa^{2} \int_{x} \int_{x'} D_{\mu \nu}(x-x')(w_{1 \mu}+w_{2 \mu} )(w_{1 \mu}'+w_{2 \mu}')\nonumber\\
&+&\frac{\pi^{2}}{g} \int_{x} \int_{x'}\frac{(w_{1 \mu}-w_{2 \mu} )(w_{1 \mu}'-w_{2 \mu}')}{|x-x'|}.
\end{eqnarray}
We perform explicit calculations with the propagator Eq. \eqref{eq:propagator}, the first term in Eq. \eqref{eq:sdual-momentum-dmunu} becomes
\begin{eqnarray}
&&2 \pi^{2} g \kappa^{2} \int_{p} D_{\mu \nu}(p)(w_{1 \mu}+w_{2 \mu} )(p)(w_{1 \mu}+w_{2 \mu})(-p)\nonumber\\&=& 2 \pi^{2} g \kappa^{2} \left( \int_{p} \frac{(w_{1 \mu}+w_{2 \mu} )(p)(w_{1 \mu}+w_{2 \mu})(-p)}{2\left(g^{2} \kappa^{2} p^{2}+4\right)}\right.\nonumber\\&-&\left. \int_{p} \frac{\varepsilon_{\mu \nu \lambda}p_{\lambda}(w_{1 \mu}+w_{2 \mu} )(p)(w_{1 \nu}+w_{2 \nu})(-p)}{\kappa g p^2(g^{2} \kappa^{2} p^{2}+4)}\right),
\end{eqnarray}
In the case of $g^2p^2\ll 1$, the last line of the expression simplifies. Let us take a closer look at the integral not taking coefficients into account,
\begin{eqnarray}
&& \int_{p} \frac{\varepsilon_{\mu \nu \lambda}p_{\lambda}(w_{1 \mu}+w_{2 \mu} )(p)(w_{1 \nu}+w_{2 \nu})(-p)}{ p^2}\nonumber\\&=& \int_{p}\ \frac{\varepsilon_{\mu \nu \lambda} p_{\lambda}}{p^{2}} \int_{x} \int_{y} e^{i p\cdot(y-x)} (w_{1\mu}+w_{2\mu})(x) (w_{1\nu}+w_{2\nu})(y)\nonumber\\&=&
\frac{i}{4\pi} \int_{p} \int_{z} \frac{\varepsilon_{\mu \nu \lambda} z_{\lambda} e^{-i p \cdot y}}{|z|^{3}} \nonumber \\ &\times&\int_{x} \int_{y} e^{i p \cdot(y - x)}(w_{1\mu}+w_{2\mu})(x) (w_{1\nu}+w_{2\nu})(y),
\end{eqnarray}
where we used a Fourier transform and an exponential representation of the $\delta$-function. Further calculations lead to
\begin{eqnarray}
&&\frac{i}{4\pi} \varepsilon_{\mu \nu \lambda} \int_{x} \int_{y} \frac{\left(x_{\lambda}-y_{\lambda}\right)}{|x-y|^{3}}(w_{1\mu}+w_{2\mu})(x) (w_{1\nu}+w_{2\nu})(y) \nonumber \\&=&-\frac{i}{4\pi} \int_{x} \int_{y}\frac{\left(x_{\alpha}-y_{\alpha}\right)}{|x-y|^{3}} \partial_{\alpha} (v_{1\beta}+v_{2\beta})(x) (w_{1\beta}+w_{2\beta})(y)\nonumber \\&=& i \int d^3x (v_{1\beta}+v_{2\beta}) (w_{1\beta}+w_{2\beta}).
\end{eqnarray}
Finally, the Eq. \eqref{eq:sdual-coor-dmunu} takes the form,
\begin{eqnarray}
\label{eq:sdual}
S_{\text {dual }}&=& \frac{\pi}{4 g} \int_{x} \int_{x'} e^{-\frac{2|x-x'|}{g\kappa}}\frac{(w_{1 \mu}+w_{2 \mu} )(x)(w_{1 \mu}+w_{2 \mu})(x')}{ |x-x'|} \nonumber\\
&+&\frac{\pi^{2}}{g} \int_{x} \int_{x'}\frac{(w_{1 \mu}-w_{2 \mu} )(w_{1 \mu}'-w_{2 \mu}')}{|x-x'|}\nonumber\\
&+&\frac{i\pi^{2}\kappa }{2} \int d^3x (v_{1\beta}+v_{2\beta}) (w_{1\beta}+w_{2\beta}).
\end{eqnarray}
\section{Flux attachment bosonization duality}
In this section we conciser the bosonization duality that we obtain in the scope of a duality web approach \cite{Karch_PhysRevX.6.031043,SEIBERG2016395}. To do so, we first need to write down the field theory for the dual bosonic system obtained in Eq. (13) of the main body of the paper. To this end we introduce complex scalar fields $\phi_I$, $I=1,2$ yielding a second-quantized representation for the ensemble of vortex loops \cite{kleinert1989gauge}. This yields the Lagrangian,
\begin{eqnarray}
\label{Eq:Dual-L}
\mathcal{L}_{dual}&=&\sum_{I=1,2}\left[|(\partial_\mu-ib_{I\mu})\phi_I|^2+m^2|\phi_I|^2+\frac{\lambda}{2}|\phi_I|^4
\right]\nonumber\\
&-&\frac{i}{8\pi^2\kappa}\epsilon_{\mu\nu\lambda}(b_{1\mu}+b_{2\mu})\partial_\nu (b_{1\lambda}+ b_{2\lambda}),
\end{eqnarray}
where we have rescaled gauge fields $b_{I\mu}\to b_{I\mu}/(2\pi)$, thus assigning a unit charge to both $\phi_1$ and $\phi_2$. Following the technique employed in Ref. \cite{Karch_PhysRevX.6.031043}, we awoke the bosonization conjectures,
\begin{equation}
\label{Eq:Boson+flux}
Z_{f Q E D}[A] e^{\frac{1}{2}S_{CS}[A]}=Z_{b Q E D+flux}[A],
\end{equation}
\begin{equation}
\label{Eq:Fermion+flux}
Z_{f Q E D+flux}[A]=Z_{b Q E D}[A] e^{-S_{CS}[A]},
\end{equation}
where $S_{CS}[A]$ is the action for a level 1 CS term, $A_\mu$ is the background field. The fermionic and bosonic partition functions are, respectively,
\begin{eqnarray}
\label{Eq:ZfQED}
Z_{fQED}[A]&=&\int\mathcal{D}\bar{\psi}\mathcal{D}\psi e^{-S_{fQED}[A]},
\nonumber\\
S_{fQED}[A]&=&\int d^3x\bar{\psi}(\slashchar{\partial}-i\slashchar{A})\psi ,
\end{eqnarray}
\begin{eqnarray}
\label{Eq:ZbQED}
Z_{bQED}[A]&=&\int\mathcal{D}\phi^*\mathcal{D}\phi e^{-S_{bQED}[A]},
\nonumber\\
S_{bQED}[A]&=&\int d^3x\left[|(\partial_\mu-iA_\mu)\phi|^2+m^2|\phi|^2+\frac{\lambda}{2}|\phi|^4
\right],\nonumber\\
\end{eqnarray}
while the flux attachment operates as follows,
\begin{eqnarray}
\label{Eq:ZfQED+flux}
Z_{fQED+flux}[A]&=&\int\mathcal{D}a_\mu\mathcal{D}\bar{\psi}\mathcal{D}\psi e^{-S_{fQED+flux}[A]},
\nonumber\\
S_{fQED+flux}[A]&=&S_{fQED}[a] -\frac{1}{2}S_{CS}[a] - S_{BF}[a;A]
\nonumber\\
\end{eqnarray}
\begin{eqnarray}
\label{Eq:ZbQED+flux}
Z_{bQED+flux}[A]&=&\int\mathcal{D}a_\mu\mathcal{D}\phi^*\mathcal{D}\phi e^{-S_{bQED+flux}[A]},
\nonumber\\
S_{bQED+flux}[A]&=&S_{bQED}[a] + S_{CS}[a] + S_{BF}[a;A]\nonumber\\
\end{eqnarray}
where the BF term is given by,
\begin{equation}
S_{BF}[a;A]=\frac{i}{2\pi}\int d^3 xa_\mu\epsilon_{\mu\nu\lambda}\partial_\nu A_\lambda.
\end{equation}
Now that we have recalled the basic flux attachment dualities (\ref{Eq:Boson+flux}) and (\ref{Eq:Fermion+flux}), we can derive the duality described in the main text by multiplying both these relations together,
\begin{eqnarray}
\label{eq:zmult}
&Z&_{f Q E D}[A] e^{\frac{1}{2}S_{CS}[A]} Z_{f Q E D+flux}[A]\nonumber\\=&Z&_{b Q E D+flux}[A]Z_{b Q E D}[A] e^{-S_{CS[A]}},
\end{eqnarray}
After we promote the background field $A_\mu$ to a dynamical field $b_\mu$, the left-hand side of the Eq. \eqref{eq:zmult} takes the form,
\begin{eqnarray}
&&\int\prod_{I=1,2}\mathcal{D}b_{I\mu}\mathcal{D}\bar{\psi}_I\mathcal{D}\psi_I e^{-S},
\nonumber\\
S&=&\int d^3x\left\{\sum_{I=1,2}\left[ \bar{\psi}_I(\slashchar{\partial}-i\slashchar{b}_I)\psi_I-\frac{i}{8\pi}b_{I\mu}\epsilon_{\mu\nu\lambda}\partial_\nu b_{I\lambda}\right] \right.\nonumber\\
&-& \left.\frac{i}{2\pi}b_{1\mu}\epsilon_{\mu\nu\lambda}\partial_\nu b_{2\lambda}\right\}.\nonumber\\
\end{eqnarray}
Integrating out $\psi_2$ generates a level 1/2 CS term with a minus sign. We can integrate out the dynamical field $b_{2\mu}$, which enforces $b_{1\mu}=b_\mu$. Eventually, we can write down the left-hand side of the Eq. \eqref{eq:zmult},
\begin{eqnarray}
&&\int\mathcal{D}b_{\mu}\mathcal{D}\bar{\psi}\mathcal{D}\psi e^{-\int d^3x\left[\bar{\psi}(\slashchar{\partial}-i\slashchar{b})\psi + \frac{i}{8\pi}b_{\mu}\epsilon_{\mu\nu\lambda}\partial_\nu b_{\lambda}\right]},
\end{eqnarray}
where we have set $\psi_{1}\equiv \psi$.
Now, let us write explicitly the right-hand side of the Eq. \eqref{eq:zmult},
\begin{eqnarray}
&&\int\prod_{I=1,2}\mathcal{D}b_{I\mu}\mathcal{D}\phi^*_I\mathcal{D}\phi_I e^{-S},
\nonumber\\
S&=&\int d^3x\left\{\sum_{I=1,2}\left[ |(\partial_\mu-ib_{I\mu})\phi_I|^2+…
\right] \right.\nonumber\\
&+& \left.
\frac{i}{4\pi}\epsilon_{\mu\nu\lambda}(b_{1\mu}+b_{2\mu})\partial_\nu (b_{1\lambda}+ b_{2\lambda})
\right\},
\nonumber\\
\end{eqnarray}
where the ellipsis represent scalar field self-interactions. This is precisely the time-reversal transformed version of the dual Lagrangian (\ref{Eq:Dual-L}) for $\kappa=1/(2\pi)$. Therefore, we have obtained that our derivation is consistent with the bosonization duality performed via flux attachments to fermions and bosons.
\end{document}
|
1,116,691,499,303 | arxiv | \section{Introduction}
Nearly two decades ago, Katz, Lebowitz and Spohn introduced a seemingly
trivial generalization of the Ising model: a lattice gas driven far from
equilibrium by an external ``electric'' field, $E$ \cite{KLS}. The
motivations are twofold: the statistical mechanics of non-equilibrium
systems and the physics of fast ionic conductors \cite{FIC}. Since then,
this model has provided a rich variety of surprises, showing how our
intuition built on the foundations of equilibrium statistical mechanics
fails when applied to non-equilibrium stationary states. Though some of its
unusual properties are now understood, it continues to offer new surprises
\cite{DL17}. In this paper, we report recent simulation results which may be
signals of a novel ``phase''. If this new state is established, it would
lead to a resolution of a long standing controversy concerning the critical
properties. In the next section, we briefly describe the model and the
disagreement between Marro, et. al. \cite{Marro} and the Leung-Wang team\cite
{KTLJSW}. Since the methods used in these two approaches are quite distinct,
we explore, in Section III, the possibility that the discrepancies may lie
in the existence of different types of ordering. In particular we introduce
a new parameter which distinguishes the two low temperature ``phases''. The
systems used in our simulations are simple generalizations of the standard
model -- to include anisotropic interparticle {\em interactions} -- so that
the presence of the novel phase is more pronounced even for small lattices.
The final section is devoted to some concluding remarks.
\section{The driven lattice gas and its critical properties}
The driven lattice gas is based on the Ising model \cite{Ising,YangLee} with
attractive nearest-neighbor interactions. In the spin language, it is a
ferromagnetic Ising model with {\em biased} spin-exchange \cite{Kawasaki}
dynamics. Since both the Onsager solution \cite{Onsager} and most
simulations concern a system in two dimensions ($d=2$), we give a brief
description of that model only. Each of the $L_{x}\times L_{y}$ sites of a
square lattice (on a torus, i.e., with periodic boundary conditions: PBC)
may be occupied by a particle or left vacant. A configuration of our system
is specified by the occupation numbers $\{n_{i}\}$, where $i$ is a site
label and $n$ is either 1 or 0. The interparticle attraction is modeled by
the usual Hamiltonian: ${\cal H}=-4J\sum_{<i,j>}n_{i}n_{j}$, where $<i,j>$
are nearest-neighbor sites and $J>0$. In thermal equilibrium, a half filled
system undergoes a second order phase transition, in the thermodynamic
limit, at the Onsager temperature $T_{O}=(2.2692..)J/k_{B}$. To perform
Monte Carlo simulations for this system, particles are allowed to hop to
vacant nearest neighbor sites with probability $\min [1,e^{-\Delta {\cal H}%
/k_{B}T}]$, where $\Delta {\cal H}$ is the change in ${\cal H}$ after the
particle-hole exchange. The deceptively simple modification introduced in
\cite{KLS} is to bias the hops along, say, the $y$ axis, so that the new
rates are $\min [1,e^{-(\Delta {\cal H+}E\Delta y)/k_{B}T}]$. Locally, the
effect of $E$ is identical to that due to gravity. However, due to the PBC,
this modification cannot be accommodated by a (single-valued) Hamiltonian.
Instead, the system settles into a {\em non-equilibrium} steady state with a
non-vanishing global particle current. For temperatures above some finite
critical $T_{c}$, the particle density in this state is homogeneous, while
below $T_{c}$, the system displays phase segregation. Superficially, the
driven system behaves like the equilibrium Ising model. With deeper probing,
dramatic differences surface. For example, there is no question that its
critical properties fall {\em outside} the Ising universality class. For
recent reviews of this and other differences, we refer the interested reader
to \cite{DL17}.
Focusing on critical properties here, let us review a long standing
discrepancy between two sets of Monte Carlo results. While one set \cite
{KTLJSW} shows data entirely consistent with field theoretic renormalization
group predictions \cite{JSLC}, another group \cite{Marro} finds very
different results, for which no viable theoretical analysis exists \cite
{PsFT}. Technically, there are two serious differences between the
simulation methods. In this section, we conjecture that {\em different types
of ordering} may be responsible for the discrepancies. In view of the
presence of ``shape dependent'' thermodynamics in the {\em disordered} phase
of this system \cite{A+E}, we believe that the scenario pictured below is
quite plausible. Let us begin with a brief review of the controversy.
Consider the (untruncated!) two point correlation function (in spin
language): $G(x,y)\equiv \left\langle \left[ 2n(x,y)-1\right] \left[
2n(0,0)-1\right] \right\rangle $, and its Fourier transform: $S(k,p)\equiv
\sum_{x.y}G(x,y)e^{2\pi i\left( kx/L_x+py/L_y\right) }$, the structure
factor. Due to the conserved dynamics, $S(0,0)\equiv 0$, so that non-trivial
large distance properties are found in, e.g., $S(1,0)$ and $S(0,1)$. In
equilibrium systems with square geometries ($L_x=L_y=L$), isotropy dictates
that $S(1,0)=S(0,1)$, which approaches the static susceptibility in the
thermodynamic limit for $T>$ $T_O$. As $T\rightarrow $ $T_O,$ this quantity
diverges (as $L^{\gamma /\nu }$ for finite systems at $T_O$). Below $T_O$,
ordering is displayed as phase segregation, into a strip (of, say, a
particle rich region) aligned with either the $x$- (or $y$-) axis, so that $%
S(0,1)$ (or $S(1,0)$) $\rightarrow L^2$. For the {\em driven} system,
simulations \cite{KLS} first revealed a discontinuity singularity ($%
S(1,0)>S(0,1)$) in the disordered phase and showed further that, at low $T$,
phase segregation occurs {\em only} into ``vertical'' strips ($%
S(1,0)=O(L^2), $ $S(0,1)=O(1)$). Since these structure factors are related
to (the inverse of) diffusion co-efficients in a continuum theory, these
observations formed the basis for the study of a model where, first, the
diffusion co-efficients are anisotropic, and second, more importantly, {\em %
only one} of these co-efficients vanishes as $T\rightarrow $ $T_c$ \cite
{JSLC}. This approach is also consistent with subsequent simulations \cite
{Newton} showing that, while $S(1,0)$ diverges as $T\rightarrow $ $T_c$, $%
S(0,1)$ remains $O(1)$ throughout the transition. A serious consequence of
this ``strongly anisotropic'' behavior is that parallel and transverse
lengths and momenta scale with different powers ($p_y\sim p_x^{1+\Delta }$, $%
\Delta =1$ in mean field theory) in the critical region. Analogues of this
unusual property exist in equilibrium systems \cite{anisoEQ}. When
fluctuations are taken into account via renormalization group methods \cite
{JSLC}, the upper critical dimension ($d_c$) is found to be $5$, while
anomalous scaling modifies $\Delta $ to $2$ in $d=2.$ A more subtle result
is that one set of exponents retains the classical values.
Now, in order to /measure/ critical properties through simulations, with
some accuracy, it is crucial to exploit finite size scaling methods \cite
{FSS}. Systems with various $L$'s are used, and the quality of data collapse
serves as an indicator of the presence of scaling. Given the highly
anisotropic scaling of lengths, the analysis would be much simplified if
samples obeying $L_{y}\sim L_{x}^{3}$ are used. This route is pioneered and
followed in one of the studies \cite{KTLJSW}. More importantly, the order
parameter used is closely related to $S(1,0)$, so that it is sensitive to
the presence of a {\em single} domain (i.e., one vertical strip, regardless
of the system size). Also measured are cumulant ratios associated with $%
S(1,0)$. {\em All} data thus obtained display good collapse, using a set of
exponents that is entirely consistent with those predicted by the
renormalization group \cite{JSLC}. In another set of studies \cite{Marro},
only square samples are used. Furthermore, the order parameter used is
\begin{equation}
m\equiv \sqrt{\left\langle M_{y}^{2}\right\rangle -\left\langle
M_{x}^{2}\right\rangle } \label{m}
\end{equation}
where $M_{x(y)}^{2}\equiv L^{-2}\sum_{y(x)}\left[ \sum_{x(y)}\left\{
2n_{i}-1\right\} \right] ^{2}$. Note that, {\em if} the ordered phase
consists of a single vertical strip, then $m$ provides information about its
average density. However, configurations with vertically aligned {\em %
multiple strips }also contribute to $m.$ The conclusion is that, e.g., $%
\beta \simeq \frac{1}{4}$, contradicting the field theory results \cite
{Marro}.
\section{A possible novel phase}
It should be quite alarming that two sets of simulations lead to drastically
different conclusions, unless we allow the possibility that the ``ordered''
states in the two studies are {\em not} the same. Considering $m$ is {\em not%
} sensitive to the presence of multiple strips, we conjecture the following.
\begin{itemize}
\item In the thermodynamic limit, the low temperature state in systems with
square geometry ($L_x=L_y$) is characterized by multiple strips, with a
nontrivial distributions of widths.
\end{itemize}
In other words, we believe that there is no long range order
(which would lead to a single-strip state) while strip-strip correlations
are controlled by a finite correlation length. For short, we will use a more
picturesque term: ``stringy phase.''
\begin{figure}[tbph]
\hspace*{2.5cm}
\epsfxsize=5cm \epsfbox{fig1aa.eps}
\vspace*{-6.68cm}
\end{figure}
\begin{figure}[tbph]
\hspace*{8cm}
\epsfxsize=5cm \epsfbox{fig1bb.eps}
\vspace*{-1.8cm}
\caption{A typical ``stringy'' configuration vs. a typical single strip one.
(See text for system parameters.)}
\end{figure}
Significantly, the existence and
behavior of such configurations were first noted in \cite{Marro}. Indeed,
these authors wrote: ``The system was never seen to escape from (these
multiple-strip) states... and we had to manipulate some system
configurations for $L=50$ to create artificial one-strip states...''. For $%
L\geq 100$ systems, they were ``unable to observe the decay (of these
multiple-strip states) toward one-strip states...'' The conclusion was that
multiple states are ``metastable'' with exceedingly long life times.
Another source for our conjecture comes from simulation studies of a driven
lattice gas with anisotropic interparticle {\em interactions} ($J_x\neq J_y$)
\cite{LBS}. Though typical configurations are disordered/ordered(single strip)
for high/low temperatures, there is a significant range of $T$ where the
system appears to be ``stringy.'' In Fig.1, we show such a configuration at
$T=0.80$, found in a $90\times 90$ lattice with $3J_x=J_y/3=J$, driven
with
saturation $E$. (Here, as in the following, $T$ is given in units of $T_O$.)
In contrast, the single strip configuration was obtained with $T=0.70$.
To be more precise,
the ``stringy'' states may be characterized by two properties: (a) the
densities in each column (i.e., along the drive) are bi-modally distributed,
yet (b) $S(1,0)$ remains small \cite{LBSMS}. An example is provided in Fig.2.
While the column densities begin to develop a
double peak around $T\sim 0.90$, the structure factor displays an
inflection
point at $T\sim 0.70$ (which is also where the {\em fluctuations} in
$S(1,0)$
reach a maximum). Note that bi-modal column densities correspond to $m>0$.
Yet, in this range ($0.75\lesssim T\lesssim 0.95$), the typical configurations
are far from being single-stripped. To show that multiple-strip states are
not simply metastable, we ran simulations starting from a completely ordered
(single-strip) state. Unlike in the previous study \cite{Marro}, we {\em do }%
observe the split-up into multiple strip
states\footnote{Figures in the Marro studies
(e.g., Fig 2.7 in \cite{MDbook}) show typical
configurations of $T$ well below the transition value. It is possible that
the failure to observe break-up/re-merging is due to $T$ being too low (for
the sizes studied).}.
In Fig.3, we show a time trace of 3500 measurements
(700K MCS) of $S(k,0)$, $k=1,2,3$, which are sensitive to the presence of
$k$-strip domains. Note that $S(1,0)$ is often smaller than $S(2,0)$ and
sometimes even smaller than $S(3,0)$, as shown in the inset. As a check, we
made three separate runs of this length and observed the same type of
behavior. Our conclusion is that, despite being started in a completely
ordered state, the system evolves towards a ``stringy'' state. In other
words, the probability to find the system in multiple strip configurations
can be considerable.
\begin{figure}[tbph]
\hspace*{-1.2cm} \epsfxsize=17cm \epsfbox{fig2.eps} \vspace*{-6.5cm}
\caption{(a) Histogram for column densities at $T=0.80$[$-$],
$0.90$[$\bullet $],
$2.0$[$+$], and (b) $S(1,0)$ vs. $T$ in a $30\times 30$ lattice with
$3J_x=J_y/3=J$, driven with saturation $E$.}
\end{figure}
\begin{figure}[tbph]
\hspace*{-1.2cm}
\epsfxsize=17cm \epsfbox{fig3.eps} \vspace*{-8cm}
\caption{Time trace of $S(k,0)$, $k=1$ [solid line], $2$ [$\bullet $], $3$,
[dashed line], for a $60\times 60$ lattice at $T=0.85$, driven with
saturation $E$. Inset: magnified view of a portion.}
\end{figure}
In order to make quantitative statements about the ``stringy state'', we
introduce the ratio:
\begin{equation}
{\cal R}\equiv L^{d}\frac{G(0,L/2)}{S(1,0)} \label{R}
\end{equation}
for a system in $d$ dimensions. The numerator is a measure of ``long range
order'' along the $y$-axis, while the denominator detects the presence of a
single strip. In particular, we are interested in the behavior of ${\cal R}$%
, as $L\rightarrow \infty $, when the system is in various states.
Far in the disordered phase, the two-point correlation decays as $r^{-d}$
\cite{LRC}. Thus, $G(0,L/2)\rightarrow O(L^{-d})$, while $S(1,0)$ remains $%
O(1)$. On the other hand, deep in the ordered phase $G\rightarrow O(1)$
while $S\rightarrow O(L^d)$. As a result, we expect
\begin{equation}
{\cal R}\rightarrow O(1) \label{far-from Tc}
\end{equation}
for both high and low temperatures. In between, a finite system makes a
transition, either through a ``stringy phase'' or directly into the ordinary
ordered state (as in the equilibrium case). In the latter scenario, the
results of the renormalization group analysis \cite{JSLC} should hold: $%
G(0,r)\rightarrow r^{-[d-2+\Delta ]/(1+\Delta )}$ and $S(k,0)\rightarrow
k^{-2}$. Thus, for simulations in $d=2$, we conclude that ${\cal R}$ {\em %
decreases} with $L:$
\begin{equation}
{\cal R}\rightarrow L^{-2/3}. \label{DDS-Tc}
\end{equation}
In sharp contrast, for a{\em \ }stringy state, ordering has set in along $y$
(so that $G(0,L/2)\rightarrow O(1)$) but complete phase segregation in $x$
is yet to take hold (so that $S(1,0)\rightarrow O(1)$). The consequence is
an {\em increasing }${\cal R}:$
\begin{equation}
{\cal R}\rightarrow L^{d}. \label{stringy}
\end{equation}
which is a function of $L.$ The drastic differences between these asymptotic
properties (\ref{far-from Tc}, and \ref{DDS-Tc} vs. \ref{stringy}) should be
detectable through simulations. As a final remark, we note that, for the
{\em equilibrium} Ising model, ${\cal R}$ decreases exponentially as $%
L^{2}e^{-L/\xi }$ for $T>T_{c}$, while for $T\lesssim T_{c}$, ${\cal R}$ is $%
O(1)$.
Turning to Monte Carlo data, we find that the evidence in favor of stringy
states is quite strong, at least for the $L$'s we can access. In particular,
using models with $J_{x}=J/3$ and $J_{y}=3J$ under saturation drive, we
collected data from a number of systems with $L\in \left[ 10,90\right] $.
The runs are up to 800K MCS long, with both random and ordered initial
conditions. In Fig.4, we show a plot of ${\cal R}$ vs. $L^{2}$, for various
temperatures. For $T=0.85$ and $0.90$, this ratio appears to be
proportional
to $L^{2}$. Meanwhile, its behavior is consistent with ${\cal R}\rightarrow
O(1)$ at both higher and lower $T$'s. We believe that these results support
the presence of a ``stringy phase''.
\begin{figure}[htbp]
\hspace*{-1.5cm} \epsfxsize=17cm \epsfbox{fig4.eps}
\vspace*{-14cm}
\caption{Correlation-Structure Factor ratio ${\cal R}$ for systems with $%
\alpha =3$ and saturation drive, plotted against $L^2.$ The legend refers to
the temperatures of the various runs.}
\end{figure}
Of course, this ``phase'' may be just a mirage in the landscape of a giant
crossover. In the absence of analytic results, we can never be certain of
its existence, since all simulations are based on finite $L$. Nevertheless,
if future simulations continue to support ${\cal R}\propto O(L^2)$, then it
behooves us find a sound analytic basis for such a state. To conclude this
section, we conjecture further. As $L$ increases further, the stringy phase
will persist at lower and lower temperatures, so that it is the {\em only}
low temperature state, for {\em square} systems, in the limit $L\rightarrow
\infty $. However, if the thermodynamic limit is approached through systems
obeying $L_y\propto L_x^3$, the low temperature phase will consist of the
ordinary, {\em single-domain} state. If this conjecture turns out to be
true, then an avenue is opened for the resolution of the controversy over
critical properties.
\section{Concluding remarks}
In this short paper, we explored the possibility that the low temperature
phase of the driven lattice gas depends on the geometry of the system. For
{\em square} systems, we conjecture that a new, ``stringy'' phase is the
only ordered state in the limit $L\rightarrow \infty $. On the other hand,
if an Ising-like, ordered state with a single domain is desired, then the
``unusual'' thermodynamic limit, with {\em rectangular }systems obeying $%
L_{\Vert }\propto L_{\bot }^3$, must be used. For finite square geometries,
we have some Monte Carlo evidence for the presence of a ``stringy'' state:
multiple strips of high/low density regions. The system appears to be
``ordered'' in the direction parallel to the drive, but the strips are of
varying widths and multiplicities. Specifically, as $T$ is lowered, the
disordered, homogeneous phase gives way to this multi-strip state for a
sizable range of $T$, before settling into the fully phase segregated,
single domain state at lower temperatures.
Many open questions, besides those mentioned above, naturally arise. More
quantitative analyses can be carried out, in order to better characterize
this ``stringy'' state. For example, we could measure the distribution of
strip-widths, or compile histograms for both the structure factors and the
correlations. Once a clearer picture emerges, assuming such a state still
exists, we should attempt to develop a reliable field theory. Certainly, the
order parameter of such a theory, $m$ in Eqn. (\ref{m}), is necessarily
quite different from the one for an equilibrium Ising lattice gas.
Hopefully, renormalization group methods can be applied, a nontrivial fixed
point can be obtained, and its associated critical properties can be
computed. Until then, to compare field-theoretic predictions with simulation
data from the Marro studies makes as much sense as ``comparing apples with
oranges.''
\section{Acknowledgments}
It is a pleasure to dedicate this article to Joel Lebowitz, a co-author of
this model. Over the years, we have benefitted from numerous conversations
with him, not to mention all the encouragements when we run into apparent
dead-ends. Also, we are grateful to J. Hager for help with simulations,
as well as G.L. Eyink and U.C. T\"{a}uber for illuminating
discussions. This research was supported in part by a grant from US National
Science Foundation through the Division of Materials Research and a
supplement from the Research Experience for Undergraduates program.
|
1,116,691,499,304 | arxiv | \section{Introduction. Theoretical framework}
To date understanding is that at the filling factor
$\nu=1/3$, $1/5$ the Laughlin
variational wave function \cite{laughlin1983} gives the best
known so far analytical approximation of exact many-body ground-state
wave function. At $\nu=1/3$, there were many attempts
to obtain lower energy for ground-state than Laughlin's
\cite{laughlin1983} incompressible liquid state, e.g., see references in
\cite{balev2006}. Here we will start with the
same many-electron Hamiltonian,
$\hat{H}(\textbf{r}_{1},\ldots,\textbf{r}_{N})$, for
2DES of $N$ electrons as in \cite{laughlin1983,morf1986},
only we adopt the Landau gauge for vector potencial
$\textbf{A}=-By\hat{\textbf{x}}$. We assume that $N$ electrons
are localized in MR of $(z=0)-$plane;
$N/L_{x}L_{y}=\nu/2\pi \ell_{0}^{2}$,
where $\nu=1/m$,
$\ell_{0}=\sqrt{\hbar c/|e|B}$.
As in \cite{laughlin1983,morf1986}, the classical
model of neutralizing ion jellium background (IJB) is used,
cf. \cite{balev2006}. Eigenstates
$\Psi(\textbf{r}_{1},\ldots,\textbf{r}_{N})$ of the Hamiltonian
$\hat{H}(\textbf{r}_{1},\ldots,\textbf{r}_{N})$ and their energies
are determined by
\begin{equation}
\hat{H}(\textbf{r}_{1},\ldots,\textbf{r}_{N})
\Psi(\textbf{r}_{1},\ldots,\textbf{r}_{N})=E_{N}
\Psi(\textbf{r}_{1},\ldots,\textbf{r}_{N}) .
\label{1}
\end{equation}%
We can assume that 2DES, with IJB, is located within the ribbon of
width $L_{y}$ bent into loop of radius $L_{x}/2\pi$.
Then Born-Carman periodic boundary conditions
$\textbf{r}_{i} \pm L_{x}\hat{\textbf{x}}=\textbf{r}_{i}$ are
holded, where $i=1,\dots,N$.
It is seen that the area of MR per electron, $L_{x}L_{y}/N=
(L_{x}^{\square})^{2}$, where $L_{x}^{\square}=\sqrt{2 m \pi}\ell_{0}$.
Then the strip of the width $L_{x}^{\square}$, along $x$-direction,
and of the length $L_{y}$ contains $\tilde{N}=L_{y}/L_{x}^{\square}$
of the (square) unit cells,
$L_{x}^{\square} \times L_{x}^{\square}$. The integer number
of such strips within MR is
given as $n_{xs}^{\max}=L_{x}/L_{x}^{\square}=N/\tilde{N}$;
for definiteness, odd (as $\tilde{N}$). Now we assume
that the ground-state and, at the least, the lowest excited-states of
Eq. (\ref{1}) correspond to partial crystal-like correlation order
among $N$ electrons of MR as
\begin{eqnarray}
&&\textbf{r}_{1+k \tilde{N}}=\textbf{r}_{1}+k L_{x}^{\square}
\hat{\textbf{x}}, \;\;\;
\textbf{r}_{2+k \tilde{N}}=\textbf{r}_{2}+k L_{x}^{\square}
\hat{\textbf{x}},
\notag \\
&& \cdots, \;\;\;
\textbf{r}_{\tilde{N}+k \tilde{N}}=
\textbf{r}_{\tilde{N}}+k L_{x}^{\square} \hat{\textbf{x}} ,
\label{2}
\end{eqnarray}%
where $k=0, \pm 1,\ldots,\pm (n_{xs}^{\max}-1)/2$.
Then Hamiltonian in Eq. (\ref{1}) becomes:
i) dependent only on $\tilde{N}$ $\mathbf{r}_{i}=(x_{i},y_{i})$
and ii) periodic, with period $L_{x}^{\square}$,
on any $x_{i}$; $i=1,\ldots,\tilde{N}$.
For the wave function in Eq. (\ref{1}): the property (i)
obviously holds and it is natural to assume that the property (ii)
is valid as well. Then
the study within MR of Eq. (\ref{1}), for 2DES of $N$ electrons,
can be reduced to the treatment of the
Schr\"{o}dinger equation for 2DES of $\tilde{N}$ ``compound'' electrons
within the main strip (MS) $L_{x}^{\square}n_{xs}^{MS}
>x_{i}>L_{x}^{\square}(n_{xs}^{MS}-1)$, cf. Ref. \cite{balev2006}, as
\begin{equation}
\hat{H}_{\tilde{N}}(\textbf{r}_{1},\ldots,\textbf{r}_{\tilde{N}})
\Psi_{\tilde{N}}=E_{\tilde{N}}
\Psi_{\tilde{N}}(\textbf{r}_{1},\ldots,\textbf{r}_{\tilde{N}}) ,
\label{3}
\end{equation}%
where, for definiteness, we assume $n_{xs}^{MS}=0$.
In Eq. (\ref{3})
\begin{equation}
\hat{H}_{\tilde{N}}=\hat{H}_{0}+V_{ee}+V_{eb}+V_{bb} ,
\label{4}
\end{equation}%
where the kinetic energy term
\begin{equation}
\hat{H}_{0}=\sum_{i=1}^{\tilde{N}}\hat{h}_{0i}=\frac{1}{2m^{\ast }}%
\sum_{i=1}^{\tilde{N}}[\hat{\mathbf{p}}_{i}-\frac{e}{c}\mathbf{A}(\mathbf{r}%
_{i})]^{2},
\label{5}
\end{equation}%
here $\hat{\mathbf{p}}=-i\hbar \mathbf{\nabla }$, $m^{\ast }$ the
electron effective mass.
In Eq. (\ref{4}) the
electron-electron potential, cf. \cite{yoshioka1983}
\begin{eqnarray}
V_{ee} &=&\frac{1}{2}\sum_{i=1}^{\tilde{N}}\sum_{j=1,j\neq i}^{\tilde{N}%
}\sum_{k=-N_{C}}^{N_{C}}\frac{e^{2}}{\varepsilon |\mathbf{r}_{i}-\mathbf{r}%
_{j}-kL_{x}^{\square }\hat{\mathbf{x}}|} \notag \\
&&+\tilde{N}\sum_{k=1}^{N_{C}}\frac{e^{2}}{\varepsilon L_{x}^{\square }k},
\label{6}
\end{eqnarray}%
physical results will not depend on $N_{C}\rightarrow \infty $.
The
electron-IJB interaction potential, cf. \cite{morf1986},
\begin{equation}
V_{eb}=-\sum_{i=1}^{\tilde{N}}\int_{MR}d\mathbf{R}
\frac{e^{2}n_{b}(\mathbf{R}%
)}{\varepsilon |\mathbf{r}_{i}-\mathbf{R}|},
\label{7}
\end{equation}%
and IJB-IJB interaction term
\begin{equation}
V_{bb}=\frac{1}{2}\int_{MS}d\mathbf{R}\int_{MR}d\mathbf{R}^{\prime }\frac{%
e^{2}n_{b}(\mathbf{R})n_{b}(\mathbf{R}^{\prime })}{\varepsilon |\mathbf{R}-%
\mathbf{R}^{\prime }|}, \label{8}
\end{equation}%
where the subscript MR (MS) shows that integration is carried out
over MR (MS); $n_{b}(\mathbf{R})=const=n_{b}$ and
$\int_{MS}d\mathbf{R}n_{b}=\tilde{N}$.
Notice $\langle \Psi _{\tilde{N}}| \Psi_{\tilde{N}}\rangle=1$, where
integrations are over the MS, and
$\epsilon=E_{\tilde{N}}/\tilde{N}=E_{N}/N$ is the total energy
per electron.
At $\nu=1/m$ we assume ground-state
$\Psi _{\tilde{N}}^{(m)}(\mathbf{r}_{1},\ldots ,\mathbf{r}_{\tilde{N}})$
of Eq. (\ref{3}) as \cite{balev2006}
\begin{equation}\Psi _{\tilde{N}}^{(m)}=
\sum_{n=-\ell }^{\ell }C_{n}(m)\Psi _{%
\tilde{N}}^{n,(m)}(\mathbf{r}_{1},\ldots , \mathbf{r}_{\tilde{N}}) ,
\label{9}
\end{equation}%
where $|C_{n}(m)|^{2}=1/m$ and
\begin{equation}
\Psi _{\tilde{N}}^{n,(m)}=\frac{1}{\sqrt{\tilde{N}!}}%
\begin{vmatrix}
\varphi _{k_{x1}^{(n)}}^{(m)}(\mathbf{r}_{1}) & \cdots & \varphi
_{k_{x1}^{(n)}}^{(m)}(\mathbf{r}_{\tilde{N}}) \\
\varphi _{k_{x2}^{(n)}}^{(m)}(\mathbf{r}_{1}) & \cdots & \varphi
_{k_{x2}^{(n)}}^{(m)}(\mathbf{r}_{\tilde{N}}) \\
\vdots & \ddots & \vdots \\
\varphi _{k_{x\tilde{N}}^{(n)}}^{(m)}(\mathbf{r}_{1}) & \cdots & \varphi
_{k_{x\tilde{N}}^{(n)}}^{(m)}(\mathbf{r}_{\tilde{N}})%
\end{vmatrix}%
, \label{20a}
\end{equation}%
is
the Slater determinant
of single-electron wave functions (orthonormal within MS;
$y_{0}(k_{x})=\ell_{0}^{2}k_{x}$ )
\begin{equation}
\varphi _{k_{xi}^{(n)}}^{(m)}(\mathbf{r})=
e^{ik_{xi}^{(n)} x} \Psi_{0}(y-y_{0}(k_{xi}^{(n)})) /\sqrt{%
L_{x}^{\square}},
\label{10}
\end{equation}%
where $\Psi_{0}(y)$ is the harmonic oscillator function,
$i=1,2,\ldots,\tilde{N}$ is the number of a unit cell within MS,
\begin{equation}
k_{xi}^{(n)}=\frac{2\pi \; m}{L_{x}^{\square}} \left[%
n_{ys}^{(i)} + \frac{n}{m} \right] , \label{11}
\end{equation}%
here $n_{ys}^{(i)}=0, \pm 1, \dots,\pm(\tilde{N}-1)/2$,
the "set" number $n=0,\ldots,\pm \ell$;
$\langle \Psi _{\tilde{N}}^{k,(m)}|
\Psi _{\tilde{N}}^{n,(m)}\rangle =\delta _{k,n}$.
Notice, for the ground-state
Eq. (\ref{9}) electron density \cite{balev2006},
\begin{equation}
n(y)=\frac{m^{-1}}{2\pi \ell _{0}^{2}}[1+2\sum_{k=1}^{\infty }
e^{-\frac{\pi mk^{2}}{2}}\cos (\frac{\sqrt{2\pi m}}{\ell _{0}}ky)],
\label{12}
\end{equation}
depends only on $y$, periodically; $L_{x}^{\square}$ is the period.
It is readily seen that the periodicity property (ii) is valid as
for ground-state wave function, Eq. (\ref{9}), so for
$\Psi _{\tilde{N}}^{n,(m)}(\mathbf{r}_{1},\ldots ,\mathbf{r}_{\tilde{N}})$.
\section{Results and Discussion}
Using Eq. (\ref{4}) we obtain
the energy $\epsilon^{(m)}=\hbar \omega_{c}/2+
(e^{2}/\varepsilon \ell_{0}) U(m)$ of the ground-state
Eq. (\ref{9}) as $<\Psi _{\tilde{N}}^{(m)}|\hat{H}_{\tilde{N}}|
\Psi _{\tilde{N}}^{(m)}>/\tilde{N}$, where $U(m)$ is given by
analytical expressions,
deduced explicitly in
\cite{balev2006}. After simple numerical
calculations, we obtain: $U(3)=-0.42854$, $U(5)=-0.33885$,
and $U(1)=-0.66510$. So $U(3)$, $U(5)$, and $U(1)$ there are
substantially lower than pertinent total lowering at $\nu=1/3$,
$1/5$, and $1$ for the Laughlin's variational wave function
\cite{laughlin1983}: $-0.4156\pm 0.0012$ ($-0.410 \pm 0.001$
is obtained in Ref. \cite{morf1986}, by fixing relevant
numerical study of \cite{laughlin1983}), $-0.3340\pm0.0028$, and
$-\sqrt{\pi/8} \approx -0.6267$.
We assume, for $m \geq 3$, the lowest excited state
(the compound exciton)
of the ground-state Eq. (\ref{9}) as\cite{balev2006}
\begin{equation}
\Psi_{\tilde{N};(m)}^{i_{0},j_{0};\tilde{n}}= \sum_{n=-\ell}^{\ell }
C_{n}(m) \Phi_{\tilde{N},(m);n}^{i_{0},j_{0};\tilde{n}} (\mathbf{r}%
_{1},\ldots,\mathbf{r}_{\tilde{N}}) ,
\label{13}
\end{equation}
where
the excited ``partial'' many-electron wave function
$\Phi_{\tilde{N},(m);n}^{i_{0},j_{0};\tilde{n}}$ it follows
from the ground-state ``partial'' many-electron wave function
$\Psi _{\tilde{N}}^{n,(m)}$
after changing in its $\tilde{N}-$dimensional Slater determinant
of the $i_{0}-$th row $\varphi_{k_{xi_{0}}^{(n)}}^{(m)}(\mathbf{r}_{1}),
\varphi_{k_{xi_{0}}^{(n)}}^{(m)}(\mathbf{r}_{2}),\cdots,
\varphi_{k_{xi_{0}}^{(n)}}^{(m)}(\mathbf{r}_{\tilde{N}})$ by
the row
\begin{equation}
\varphi _{k_{xj_{0}}^{(n+\tilde{n})}}^{(m)}(\mathbf{r}_{1}),
\varphi_{k_{xj_{0}}^{(n+\tilde{n})}}^{(m)}(\mathbf{r}_{2}),
\cdots,\varphi_{k_{xj_{0}}^{(n+\tilde{n})}}^{(m)}(\mathbf{r}_{\tilde{N}}),
\label{14}
\end{equation}%
where $\tilde{n}=\pm 1,\ldots, \pm \ell$,
the implicit spin wave function
is omitted.
We assume
that the $i_{0}$-th unit cell defined by $n_{ys}^{(i_{0})}$ (where $m$
quasihole excitations appear that constitute the compound hole) as well as
the $j_{0}-$th unit cell (where $m$ quasielectron excitations are
mainly localized that constitute the excited
compound electron) there are well inside of MS. For the compound
exciton Eq. (\ref{13}) the spin of the compound electron is
not changed in a process of the excitation; excited states Eq. (\ref{13})
are orthonormal and they are orthogonal to the ground-state Eq. (\ref{9}).
The total charge of any quasielectron (quasihole) within MS
is given as $e/m$ ($-e/m$). Due to periodic boundary conditions
(in particular, the periodicity of single-electron wave functions)
it is clear that the charge density of the compound exciton has its
``images'' along $x$, outside MS, with the period $L_{x}^{\square}$.
The study shows that at $m \geq 3$ the lowest excited state,
Eq. (\ref{13}), of the compound exciton
(here it defines the activation gap
$E_{ac}^{(m)}$) is obtained for $n_{ys}^{(j_{0})}=
n_{ys}^{(i_{0})}$ and $\tilde{n}=\pm 1$.
We obtain that dimensionless activation gap
of the compound exciton $\Delta_{ac}^{(m)}=E_{ac}^{(m)}/(e^{2}/\varepsilon
\ell_{0}) $ is given, for $m=3, \; 5$, as
$\Delta_{ac}^{(3)} \approx 0.1016$,
$\Delta_{ac}^{(5)} \approx 0.0257$. Further, the study
shows \cite{balev2006}
that at $m=1$ the compound spin-exciton (with the change of the spin
of the excited compound electron) gives the activation gap as
$E_{ac}^{(1)}=|g_{0}| \mu_{B}B+1.1843 e^{2}/\varepsilon \ell_{0}$,
where many-body contribution is a bit smaller than relevant result
of Hartree-Fock approximation $\sqrt{\pi/2}(e^{2}/\varepsilon \ell_{0}$);
$g_{0}$ is the bare g-factor.
Assuming that the Fermi level is located within the finite energy
gap between the ground-state, Eq. (\ref{9}), and excited-states,
we calculate \cite{balev2006} (in particular, from
the Kubo formula) that the Hall conductance
$\sigma_{H}=-\sigma_{xy}=e^{2}/(2m\pi\hbar)$. I.e., for $m=3, 5, \dots$
the ground-state Eq. (\ref{9}) corresponds to the fractional Hall
effect, $\nu=1/m$. As the activation gap is experimentally observable
from the activation behavior of the direct current
magnetotransport coefficients, it is
given by the excitation gap of relevant compound exciton,
at $m \geq 3$, and by the gap of pertinent
compound spin-exciton, at $m=1$.
Point out that present study has some important similarities with
well known theory by Yoshioka, Halperin and Lee \cite{yoshioka1983}
of ground-state of the $\nu=1/3$ fractional quantum Hall
effect (for related recent works see, e.g.,
\cite{tapash1990}), using periodic boundary condition along $x$ and $y$.
In particular, the Hamiltonian of \cite{yoshioka1983}, for $n$ electrons
in the rectangular cell $b \times a$, can be related with the
Hamiltonian, Eq. (\ref{4}),
$\hat{H}_{\tilde{N}}(\textbf{r}_{1},\ldots,\textbf{r}_{\tilde{N}})$
for $\tilde{N} \equiv n \rightarrow \infty$ electrons within MS. If to take
into account that the role of $b$ ($a$) now is played by $L_{x}^{\square}$
($L_{y}$), etc.; some minor differences are related with another form
of the Landau gauge in Ref. \cite{yoshioka1983}.
Present study shows that proper periodic boundary condition
can be totally relevant to symmetry, periodicity, correlations, and etc.
properties of a sought state; i.e., it will not
lead to any oversimplification.
|
1,116,691,499,305 | arxiv | \section{Introduction}
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{img/scheme.png}
\caption{MisConv{} redefines the first convolutional layer of existing CNN to process incomplete images. First, it estimates a distribution of possible replacements for an incomplete image using a Mixture of Factor Analyzers. Next, the convolutional layer computes the expected value of the convolutional operator taken over the MFA distribution. This mechanism takes the uncertainty of imputation into account and can be implemented efficiently using matrix operations.
}
\label{fig:scheme}
\end{figure*}
Convolutional neural networks (CNNs) present state-of-the-art performance on various image processing tasks, such as classification, segmentation, object detection \cite{krizhevsky2012imagenet, karpathy2014large, ciresan2011flexible}. Nevertheless, standard CNNs cannot directly process images, which contain missing regions. Since missing data are very common in medical diagnosis, self-driving vehicles or robots, and other computer vision applications, generalizing CNNs to this case is of great importance.
The typical strategy for using neural networks with incomplete images relies on replacing absent pixels with the most probable candidates in the initial stage \cite{cheng2019novel, li2019misgan}. While imputation-based techniques can be combined with various machine learning models, they do not take into account the uncertainty of estimation. In particular, more than one replacement usually conforms to a given image, but only a single one is selected in the imputation stage\footnote{In the case of multiple imputations, a selected number of candidates are considered.}. The use of such a hard decision may be destructive because important information about data distribution is lost during this process.
To overcome the aforementioned problems, many classical machine learning models, such as logistic regression or SVM, were adapted to learn from missing data directly \cite{chechik2008max, dekel2010learning, globerson2006nightmare}. They either ignore missing values or integrate the probability distribution of missing data to make a final prediction. Although these approaches do not substitute missing values with a single point estimate, the works on adapting neural networks, including CNNs, to the case of incomplete images are limited \cite{smieja2018processing, liu2018partialinpainting}.
In this paper, we propose a general mechanism, MisConv{}, for adapting CNN architectures to incomplete images. MisConv{} does not rely on a single imputation, but takes the uncertainty contained in missing pixels into account. Our idea is to model the probability distribution of missing values and calculate the expected value of the convolution operator taken over all possible replacements -- see Figure \ref{fig:scheme} for the illustration. Making use of the modified Mixture of Factor Analyzers (MFA) \cite{richardson2018gans, przewiezlikowskiestimatingICONIP}, we model the distribution of missing pixels with sufficient accuracy as well as we are able to find an analytical formula for the expected value of convolution. Although related approaches have been considered before for shallow models, such as logistic regression \cite{williams2005incomplete} and kernel methods \cite{smieja2019generalized}, or fully connected neural networks \cite{smieja2018processing}, there are no works on applying similar ideas to CNNs.
MisConv{} can be applied to various CNN architectures and requires only a modification of the first convolutional layer. The whole framework is realized by matrix operations and classical convolutions. In consequence, the complexity of MisConv{} is comparable to typical CNNs operating on complete images. The proposed approach is experimentally verified on various image processing tasks, including image classification, reconstruction, and generation. We demonstrate that MisConv{} compares favorably with state-of-the-art techniques used for processing incomplete images, which confirms its practical usefulness.
Our contributions can be summarized as follows:
\begin{compactitem}
\item We introduce a general mechanism for adapting CNNs to the case of incomplete data, which takes into account the uncertainty contained in missing values.
\item We show that MisConv{} can be easily implemented using typical matrix operations, which makes it extremely efficient in practice.
\item Experimental results demonstrate that MisConv{} can be successfully applied to various tasks and CNN architectures, achieving state-of-the-art performance.
\end{compactitem}
\section{Related work}
\paragraph{Imputation-based techniques} One of the most common approaches for applying machine learning models to the case of missing data relies on completing absent attributes and next using a given model on complete inputs. Missing attributes may be replaced using simple statistical techniques, such as mean value or k-nn imputation \cite{cheng2019novel}, or by applying more advanced machine learning methods \cite{sharpe1995dealing, sovilj2016extreme}. Although iterative regression approaches \cite{buuren2011micemice, azur2011multiple} are frequently used for small data, they are computationally too demanding for large high dimensional data such as images. The authors of \cite{gondara2017multiple} used denoising autoencoders for multiple imputations. It was shown that zero imputation can also obtain competitive results if only the rate of missingness is carefully taken into account. There are also many works that apply deep learning methods for image inpainting \cite{li2017generative,pathak2016context, yu2018generative, smieja2020iterative}, but they require complete data for training, which is in contradiction with many real data sets (such as medical ones).
\paragraph{Adaptation of shallow models} Replacing or deleting missing values is not always necessary when the task is to perform a prediction, e.g. classification, regression, etc. \cite{ipsen2020deal}. Given a rough estimation of missing data density by GMM (Gaussian mixture model) \cite{ghahramani1994supervised, tresp1994training}, one can adapt logistic regression \cite{williams2005incomplete} or kernel methods \cite{smola2005kernel, williams2005analytical, smieja2019generalized, mesquita2019gaussian} to operate on incomplete data directly. The authors of \cite{dick2008learning, liao2007quadratically} estimated the parameters of the probability model and the classifier jointly. Decision function can also be learned based on the visible inputs alone \cite{dekel2010learning, globerson2006nightmare}, see \cite{chechik2008max, xia2017adjusted} for SVM and random forest. The authors of \cite{hazan2015classification} designed an algorithm for kernel classification under the low-rank assumption, while Goldberg et. al. \cite{goldberg2010transduction} used matrix completion strategy to solve the missing data problem. Pelckmans et. al. \cite{pelckmans2005handling} modeled the expected risk under the uncertainty of the predicted outputs. Khosravi et al. embedded logistic regression classifier by computing the expected prediction on a given feature distribution \cite{khosravi2019expect}.
\paragraph{Adaptation of deep learning models}
Partial convolution is a general technique that redefines convolutional layers in CNNs and deals with arbitrary missing patterns in images \cite{liu2018partialinpainting}. Danel et al. generalized GCNs (graph convolutional networks) to operate on incomplete images and show that this method works well on small images \cite{danel2020processing}. The authors of \cite{ghorbani2018embedding}, proposed a trainable embedding of incomplete inputs, which can be used as an input layer to a given neural network. Mesquita et al. adapted neural networks with random weights to the case of incomplete data by estimating the weights of the output layer as a function of the uncertainty of the missing data \cite{mesquita2019artificial}. The paper \cite{bengio1996recurrent} used recurrent neural networks with feedback into the input units, which fills absent attributes for the sole purpose of minimizing a learning criterion. By applying the rough set theory, the authors of \cite{nowicki2016novel} presented a feedforward neural network that gives an imprecise answer as the result of input data imperfection. Goodfellow et. al. \cite{goodfellow2013multi} introduced the multi-prediction deep Boltzmann machine, which is capable of solving different inference problems, including classification with missing inputs.
Many recent works focus on imputation strategies. Yoon et al. adapted conditional GANs to fill missing values \cite{yoon2018gain}. In this approach, discriminator is trained to classify which components were observed and which have been imputed. Mattei and Frellsen introduced MIWAE technique, which allows for training autoencoders on missing data by approximating the maximum likelihood of only the observed portion of the data \cite{mattei2019miwae}. The authors of \cite{li2019acflow} proposed a flow-based model (ACFlow), which gives an explicit formula for a conditional density function of missing values.
\'Smieja et al. defined a generalized neuron's response, which computes the expected activation for missing data points, and applied this mechanism for fully connected networks \cite{smieja2018processing}. However, this work ignores the case of CNNs and does not show how to compute the expected activation in this situation. Moreover, modeling high dimensional data, such as images, using Gaussian mixture models (GMMs) requires special attention. On the one hand, the number of parameters of Gaussian distribution with full covariance matrix grows quadratically with data dimension, while on the other hand, restricting to diagonal covariance is insufficient for modeling images \cite{richardson2018gans}. In this paper, we show how to efficiently apply a similar probabilistic approach for CNNs and image data.
\section{Convolutional layer for missing data}
\paragraph{Model overview} We consider the problem of processing missing data by convolutional neural networks (CNNs). Typical CNNs can only process complete images, in which values of all pixels are known. For incomplete images, the convolutional operation is not defined and there appears a question of how to harness the power of CNNs in this case.
We tackle this problem using a probabilistic approach, where missing data is represented by probability distributions. Such a representation takes into account the uncertainty of missing data estimation, which is more reliable than using a single or multiple imputation. As a probabilistic model, we use a Gaussian density represented in the form of the Mixture of Factor Analyzers (MFA), which was successfully used to model image distributions~\cite{richardson2018gans}. Moreover, its Gaussian form allows for performing an analytical calculation on this random vector.
To process the probabilistic representation of missing data, we adapt the convolution operator. Here, we introduce a convolution that is a generalization of typical convolution, which computes the mean activation (expected value) taken over all possible replacements. Formally, if $\mathbf{Z}$ is a random vector representing incomplete image $\mathbf{x}$, $\mathbf{M}$ is linear convolution and $f$ is the activity function, our convolutional layer computes:
$$
f(\mathbf{M} \mathbf{x}) := \mathbb{E}[f(\mathbf{M} \mathbf{Z})].
$$
Since the expected value transforms random variables to numeric values, our convolution has to be applied only in the first layer. As a benefit, the rest of the architecture can be left unchanged, which is very important in practice, because typical neural networks can be directly reused in our framework.
\paragraph{Probabilistic representation of incomplete images} A missing data point is denoted by $\mathbf{x} = (\mathbf{x}_o, \mathbf{x}_m) \in \mathbb{R}^n$, where $\mathbf{x}_o \in \mathbb{R}^{d}$ represents pixels with known values, while $\mathbf{x}_m \in \mathbb{R}^{n-d}$ denotes absent pixels. The set of indices (pixels) with missing values at sample $\mathbf{x}$ is denoted $\mathcal{J} \subset \{1,\ldots,n\}$. Formally $\mathcal{J} = \mathcal{J}_\mathbf{x}$, which means that the missing pattern can be different for every instance $\mathbf{x}$, but we drop the subscript $\mathbf{x}$ to simplify notation.
To account for uncertainty contained in missing pixels, we employ a conditional probability distribution $p_{\mathbf{x}_m|\mathbf{x}_o}$. Making use of conditional density, we can cover a wider spectrum of possible replacements than using point estimates.
While the conditional density $p_{\mathbf{x}_m|\mathbf{x}_o}$ is defined on $(n-d)$-th dimensional space, we make its natural extension to the whole $\mathbb{R}^n$ space, by putting
$$
P_{\mathbf{x}_m|\mathbf{x}_o}(\mathbf{t}) = \left\{
\begin{array}{ll}
p_{\mathbf{x}_m|\mathbf{x}_o}(\mathbf{t}_{\mathcal{J}'}), & \text{ if } \mathbf{t}_{\mathcal{J}'} = \mathbf{x}_o,\\
0, & \text{ otherwise},
\end{array}
\right.
$$
where $\mathbf{t}_{\mathcal{J}'}$ denotes the restriction of $\mathbf{t} \in \mathbb{R}^n$ to the observed pixels $\mathcal{J}' = \{1,\ldots,n\} \setminus \mathcal{J}$.
As a parametric model of conditional density $p_{\mathbf{x}_m|\mathbf{x}_o}$, we select the Mixture of Factor Analyzers (MFA), a type of Gaussian mixture model (GMM) \cite{ghahramani1996algorithm}. Let us recall that a single component of MFA, called Factor Analyzer (FA), has a Gaussian distribution with a covariance matrix spanned on a low dimensional space, which drastically reduces the number of model parameters. In particular, the memory and complexity of estimating MFA grow linearly with data dimension (not quadratically as in the standard GMM). Moreover, the Gaussian form of MFA allows us to perform analytical calculations on the conditional density. In the case of deep generative models, such as GANs \cite{Goodfellow}, VAEs \cite{VAE} or INFs \cite{dinh2014nice}, analytical calculation is difficult or even impossible and sampling remains as the only possible strategy. Finally, it was recently shown that MFA is competitive to simple versions of GANs in estimating a density of high dimensional data such as images \cite{richardson2018gans}. Since we do not need sharp, realistic replacements, but only a rough estimation of possible values, MFA suits our needs perfectly.
Formally, a single Factor Analyzer (FA) defined in $\mathbb{R}^n$ is described by the mean vector $\bm{\mu} \in \mathbb{R}^n$ and the covariance matrix $\mathbf{\Sigma} = \mathbf{A} \mathbf{A}^T + \mathbf{D}$, where $\mathbf{A}_{n \times l}$ is a low rank factor loading matrix composed of $l$ vectors $\mathbf{a}_1,\ldots,\mathbf{a}_l \in \mathbb{R}^n$, such that $l \ll n$, and $\mathbf{D} = \mathbf{D}_{n \times n} = \mathrm{diag}(\mathbf{d})$ is a diagonal matrix representing noise defined by $\mathbf{d} \in \mathbb{R}^n$. Formally, FA is modeled as a random vector defined by:
\begin{equation} \label{eq:wl}
\mathbf{Z}=\bm{\mu}+\sqrt{\mathbf{d}} \odot\mathbf{X}+\sum_{j=1}^l Y_j \cdot \mathbf{a}_j,
\end{equation}
where $\mathbf{X} \sim N(0,\mathbf{I})$, $Y_j \sim N(0,1)$ are independent, $\sqrt{\mathbf{d}}$ denotes element-wise square root of vector $\mathbf{d}$, and $\mathbf{a} \odot \mathbf{b}$ stands for element-wise multiplication of vectors $\mathbf{a}$ and $\mathbf{b}$.
\paragraph{Linear transformation of missing data}
We consider a random vector $\mathbf{Z}$ with MFA distribution $P_\mathbf{Z}$ representing a missing data point $\mathbf{x}=(\mathbf{x}_o,\mathbf{x}_m)$. The introduced convolutional layer allows us to transform a probabilistic MFA representation $\mathbf{Z}$ into a numeric output vector. Our idea is to calculate the expected value of the convolution applied to $\mathbf{Z}$. Below, we present an analytical formula for this transformation.
Let $\mathbf{M}$ be a linear convolution operator (without applying nonlinear activity function), which produces a random vector $\mathbf{M} \mathbf{Z}$ when applied to $\mathbf{Z}$. First, we show that if $\mathbf{Z}$ is a Factor Analyzer then $\mathbf{M} \mathbf{Z}$ is also a Factor Analyzer.
\begin{theorem}\label{thm:rv}
Let $\mathbf{Z}$ be a random vector with FA distribution defined by $\mathbf{Z}=\bm{\mu}+\sqrt{\mathbf{d}} \odot \mathbf{X}+\sum_{j=1}^l Y_j \cdot \mathbf{a}_j$. Then the random vector $\mathbf{M} \mathbf{Z}$ has Factor Analyzer distribution with mean and variance given by
\begin{equation}\label{eq:conv}
\begin{array}{l}
\mathbb{E}[\mathbf{M} \mathbf{Z}] = \mathbf{M} \bm{\mu}\\[0.8ex]
\mathbb{V}[\mathbf{M} \mathbf{Z}] = \mathrm{diag}(\mathbf{M} \mathbf{d}) + \sum_{j=1}^l (\mathbf{M} \mathbf{a}_j) \cdot (\mathbf{M} \mathbf{a}_j)^T
\end{array}
\end{equation}
\end{theorem}
\begin{proof}
Observe that $\mathbf{M} \mathbf{Z}$ produces Gaussian distribution, because $\mathbf{M}$ is a linear operator. Its parameters can be calculated as follows:
$$
\begin{array}{ll}
\mathbb{E}[\mathbf{M} \mathbf{Z}] & =\mathbf{M} \mathbb{E}[\mathbf{Z}]=\mathbf{M} \bm{\mu},\\[0.8ex]
\mathbb{V}[\mathbf{M} \mathbf{Z}] & =\mathbb{V}[\mathbf{M}(\sqrt{\mathbf{d}}.\mathbf{X})]+\sum_{j=1}^l \mathbb{V}[\mathbf{M}(Y_j \cdot \mathbf{a}_j)]\\
& =\mathrm{diag}(\mathbf{M} \mathbf{d}) +\sum_{j=1}^l \mathbb{V}[Y_i \cdot \mathbf{M} \mathbf{a}_j]\\[0.8ex]
&=\mathrm{diag}(\mathbf{M} \mathbf{d}) +\sum_{j=1}^l (\mathbf{M} \mathbf{a}_j) \cdot (\mathbf{M} \mathbf{a}_j)^T.
\end{array}
$$
\end{proof}
If $\mathbf{Z}$ is the MFA then we need to apply the convolution operator to every component according to Theorem \ref{thm:rv} (the mixture proportions remains the same).
\paragraph{Expected activation}
The next step is the application of the activation function to every coordinate of the feature map produced by $\mathbf{M} \mathbf{Z}$. Thus, we ignore the correlations between coordinates and take into account only the elements on the diagonal of the covariance matrix. The diagonal of $\mathbf{M} \mathbf{Z}$ from Theorem \ref{thm:rv} is given by:
\begin{equation}\label{eq:mar}
\mathbf{M} \mathbf{d}+\sum_{j=1}^l (\mathbf{M} \mathbf{a}_j) \odot (\mathbf{M} \mathbf{a}_j).
\end{equation}
Due to the above marginalization, it is sufficient to consider 1-dimensional MFA representing the distribution of $\mathbf{M} \mathbf{Z}$ on each coordinate of the feature map. Since we deal with random vectors, we calculate the expected value of the activation function. We restrict our attention to ReLU activation because it is commonly used in CNNs.
\begin{theorem}
Let $P = \sum_{i=1}^k p_i N(m_i, \sigma_i^2)$ be 1-dimensional Gaussian density. The expected value of ReLU applied to random variable with density $P$ equals:
\begin{multline}\label{eq:expected}
\mathbb{E}[\mathrm{ReLU}(P)] = \\
\frac{1}{2}\sum_{i=1}^k p_i \left(m_i +
\frac{\sigma_i}{2\sqrt{2 \pi}} \exp(-\frac{m_i^2}{2\sigma_i^2}) + m_i \cdot \mathrm{erf}(\frac{m_i}{\sigma_i\sqrt{2}})\right),
\end{multline}
where $\mathrm{erf}(z) = \frac{2}{\sqrt{\pi}} \int_0^z \exp(-t^2)dt$ is the error function.
\end{theorem}
\begin{proof}
First, observe that:
\begin{align*}
\mathbb{E}[\mathrm{ReLU}(P)] = \int_\mathbb{R} \mathrm{ReLU}(x) \sum_i N(m_i,\sigma_i^2)(x) dx \\
= \sum_i p_i \int_0^\infty x N(m_i,\sigma_i^2)(x)dx.
\end{align*}
Finally, the last integral can be calculated as follows:
\begin{multline*}
\int_0^\infty x N(m,\sigma^2)(x)dx =\\
\frac{1}{2}\left(m + \frac{\sigma}{2\sqrt{2\pi}} \exp(-\frac{m^2}{2\sigma^2}) +m \cdot \mathrm{erf}(\frac{m}{\sigma \sqrt{2}})\right).
\end{multline*}
\end{proof}
\paragraph{Practical realization}
MisConv{} allows us to efficiently adapt typical CNNs to the case of incomplete data by redefining the first hidden layer. The whole procedure is summarized in Algorithm 1.
\begin{figure*}
{\bf Algorithm 1:}
\medskip
\hrule
\medskip
\begin{algorithmic}[1]
\STATE \textbf{INPUT:}
\INDSTATE $\mathbf{x} = (\mathbf{x}_o,\mathbf{x}_m)$ -- incomplete image\\
\STATE \textbf{OUTPUT:}\\
\INDSTATE $\mathrm{ReLU}(\mathbf{M} \mathbf{x}))$ -- transformation of $\mathbf{x}$ by the 1st hidden layer\\
\STATE \textbf{1st HIDDEN LAYER}\\
\INDSTATE Compute a density $F_\mathbf{Z} = \sum_{i=1}^k p_i N(\bm{\mu}_i, \mathbf{A}_i^T \mathbf{A}_i + \mathrm{diag}(\mathbf{d}_i))$ of MFA representation $\mathbf{Z}$ of $\mathbf{x}$.
\INDSTATE Compute a distribution $F_{\mathbf{M} \mathbf{Z}} = \sum_{i=1}^k p_i N(\mathbf{m}_i,\mathrm{diag}(\bm{\sigma}_i))$ of $\mathbf{M} \mathbf{Z}$ on every pixel where:
\INDSTATE[2] $\mathbf{m}_i = (m_i^1,\ldots,m_i^n) = \mathbf{M} \bm{\mu}_i$
\INDSTATE[2] $\bm{\sigma}_i = (\sigma_i^1,\ldots,\sigma_i^n) = \mathbf{M} \mathbf{d}+\sum_{j=1}^l (\mathbf{M} \mathbf{a}_j) \odot (\mathbf{M} \mathbf{a}_j)$
\INDSTATE Compute the expected ReLU activation of $\sum_{i=1}^k p_i N(m_i^j,(\sigma_i^j)^2)$ on every pixel $j$ using \eqref{eq:expected}
\end{algorithmic}
\medskip
\hrule
\end{figure*}
As can be seen, MisConv{} can be implemented with the use of typical matrix operations and classical convolutions. The first layer applies a convolution operator $\mathbf{M}$ to the mean vector $\bm{\mu}$ of every FA as well as to each column of matrix $\mathbf{A}$ and noise vector $\mathbf{d}$ defining the covariance matrix. Since MisConv{} involves only the modification in the first layer, the remaining part of the CNN architecture can be left unchanged.
We experimentally verify that the computational overhead introduced by this modification is small and has little effect on the overall network performance, see the supplementary materials.
Incomplete image $\mathbf{x}$ can be represented by MFA in two ways. One way is to estimate MFA from an incomplete data set and compute conditional MFA for every image $\mathbf{x}$ \cite{richardson2018gans}. An alternative option relies on direct estimation of conditional MFA using density network \cite{przewiezlikowskiestimatingICONIP, bishop1994mixture}. In the latter case, we construct a neural network that takes incomplete image $\mathbf{x}$ and returns the parameters of a conditional MFA. Such an inpainting network is trained using log-likelihood loss. Experimental evaluation, presented in the following section, demonstrates that this approach leads to better results, see Section~\ref{sec:an}.
\section{Experiments}
\label{sec:experiments}
We evaluate MisConv{} on three machine learning problems: classification, image generation and image reconstruction\footnote{The code implementing our technique is added to the supplemental material and will be made publicly available when the review period ends.}.
\paragraph{Experimental setting} If not stated otherwise, every model is trained and tested on missing data. To generate missing regions, we remove a square patch covering 1/4 area of the image. The location of the patch is uniformly sampled for each example. In the supplementary materials, we also consider more challenging tasks, where larger portions of images are missing.
MisConv{} is parametrized by MFA, which builds a probabilistic representation of incomplete images. For this purpose, we use DMFA (Deep Mixture of Factor Analyzers) \cite{przewiezlikowskiestimatingICML, przewiezlikowskiestimatingICONIP} -- a variant of density network \cite{bishop1994mixture}, which returns the parameters of conditional FA for the incomplete image. It should be emphasized that in contrast to \cite{przewiezlikowskiestimatingICONIP}, our DMFA is also trained on missing data, see supplementary material for details.
As our main baseline, we consider Partial Convolution ({\bf PC}) \cite{liu2018partialinpainting}, where the convolution is masked and renormalized to be conditioned on only the valid pixels. To the best of our knowledge, it is the only publicly available method for adapting CNNs to the case of missing data. Additionally, we consider various imputation strategies, which modify input images, but leave CNN architecture unchanged:
\begin{compactitem}
\item {\bf ACFlow}: missing features are replaced with imputations produced by Arbitrary Conditional Flow \cite{li2019acflow}. This is a recent generative model, which allows for sampling from conditional distribution of missing values. In coherence with the authors of this method, we utilize ACFlow models trained on complete data (other models considered here, including MisConv{}, are trained only on incomplete data).
\item {\bf zero}: missing values are replaced with zeros.
\item {\bf mask}: the results of zero imputation is concatenated with a mask vector indicating missing pixels.
\item {\bf k-NN}: missing pixels are replaced with mean values of those pixels computed from the $k$ nearest training samples (we use $k=5$). Because of the computational complexity of k-NN, we use a random subset of 5000 training samples for finding neighbors.
\end{compactitem}
We also report the performance of CNN trained and tested on complete datasets (no missing values), which is denoted by {\bf GT} (ground-truth). For a comparison, we only considered methods with publicly available codebases suitable for processing of images and thus many methods described in the related work section \cite{yoon2018gain, mattei2019miwae} have not been taken into account.
\paragraph{Image classification}
For the classification task, we consider three popular datasets: MNIST, SVHN, and CIFAR-10. We use a classifier built from blocks of Convolution-ReLU-Batch-Normalization-Dropout layers \cite{ioffe2015batchnorm}, followed by two fully-connected layers with a ReLU nonlinearity between them. Detailed parameters of classifiers used for each dataset can be found in the supplemental material.
Classification accuracies reported in Table \ref{tab:results-accuracy} clearly indicate that MisConv{} obtains better results than competitive methods. While the advantage of MisConv{} over second-best mask imputation on MNIST is small, the disproportion on SVHN is large. It is evident that recent methods, such as ACFlow and PC, are unstable and frequently give lower scores than MisConv{} and a simple zero imputation strategy.
In Table \ref{tab:results-accuracy-full-images} we also report the accuracies achieved by testing the above classifiers on complete images (without missing values).
Although most results are fairly similar on the simplest case of MNIST, experiments on more challenging SVHN and CIFAR10 datasets show that the classifier equipped with MisConv{} shows the strongest ability to adapt to the complete data.
\begin{table}[]
\centering
\caption{Classification accuracy when training and testing on incomplete images.} \label{tab:results-accuracy}
\begin{tabular}{cccc}
\hline
& MNIST & SVHN & CIFAR10 \\ \hline
zero & 0.910 & 0.679 & 0.709 \\
mask & 0.925 & 0.493 & 0.563 \\
k-NN & 0.875 & 0.523 & 0.660 \\
PC & 0.920 & 0.376 & 0.522 \\
ACFlow & 0.908 & 0.666 & 0.671 \\
MisConv{} & {\bf 0.931} & {\bf 0.757} & {\bf 0.722} \\
GT & 0.990 & 0.889 & 0.811 \\
\hline
\end{tabular}
\end{table}
\begin{table}[]
\centering
\caption{Classification accuracy when training on incomplete images and testing on complete images.}
\label{tab:results-accuracy-full-images}
\begin{tabular}{cccc}
\hline
& MNIST & SVHN & CIFAR10 \\
\hline
zero & 0.984 & 0.843 & 0.761 \\
mask & 0.986 & 0.827 & 0.752 \\
k-NN & 0.957 & 0.805 & 0.742 \\
PC & {\bf 0.987} & 0.759 & 0.731 \\
ACFlow & 0.983 & 0.845 & 0.746 \\
MisConv{} & 0.986 & {\bf 0.860} & {\bf 0.769} \\
GT & 0.990 & 0.889 & 0.811 \\
\hline
\end{tabular}
\end{table}
\begin{table}[]
\centering
\caption{Frechet Inception Distance (FID) between true images and images generated by WAE (lower is better).}
\label{tab:results-fid}
\begin{tabular}{cccc}
\hline
& MNIST & SVHN & CelebA \\
\hline
zero & 9.72 & 21.97 & 56.62 \\
mask & 10.96 & 24.36 & 59.45 \\
k-NN & 9.46 & 19.90 & 61.48 \\
PC & 9.75 & 22.61 & 64.92 \\
ACFlow & {\bf 9.08} & 21.49 & 59.04 \\
MisConv{} & 9.60 & {\bf 19.53} & {\bf 56.60} \\
GT & 6.45 & 19.65 & 51.75 \\
\hline
\end{tabular}
\end{table}
\begin{figure*}[]
\centering
\begin{subfigure}{.4\textwidth}
\centering
\includegraphics[width=0.95\linewidth]{img/gen_mnist}
\label{fig:gen-mnist}
\end{subfigure}%
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=0.95\linewidth]{img/recon_mnist}
\label{fig:recon-mnist}
\end{subfigure}\\
\begin{subfigure}{.4\textwidth}
\centering
\includegraphics[width=0.95\linewidth]{img/gen_svhn}
\label{fig:gen-svhn}
\end{subfigure}%
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=0.95\linewidth]{img/recon_svhn}
\label{fig:recon-svhn}
\end{subfigure}\\
\begin{subfigure}{.4\textwidth}
\centering
\includegraphics[width=0.95\linewidth]{img/gen_celeba}
\label{fig:gen-celeba}
\end{subfigure}%
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=0.95\linewidth]{img/recon_celeba}
\label{fig:recon-celeba}
\end{subfigure}%
\caption{Samples (right) and reconstructions (left) produced by WAE models.}
\label{fig:wae}
\end{figure*}
\begin{table}[]
\centering
\caption{Structural similarities (SSIM) between true and reconstructed images (higher is better).}
\label{tab:results-ssim}
\begin{tabular}{cccc}
\hline
& MNIST & SVHN & CelebA \\
\hline
zero & 0.696 & 0.777 & 0.795 \\
mask & 0.658 & 0.775 & 0.802 \\
k-NN & 0.770 & 0.771 & 0.798 \\
PC & 0.662 & 0.774 & 0.786 \\
ACFlow & 0.628 & 0.787 & 0.805 \\
MisConv{} & {\bf 0.803} & {\bf 0.792} & {\bf 0.812} \\
GT & 0.944 & 0.811 & 0.832 \\
\hline
\end{tabular}
\end{table}
\begin{table}[]
\centering
\caption{Peak signal-noise ratio (PSNR) between true and reconstructed images (higher is better).}
\label{tab:results-psnr}
\begin{tabular}{cccc}
\hline
& MNIST & SVHN & CelebA \\
\hline
zero & 15.05 & 23.81 & 22.74 \\
mask & 14.26 & 23.64 & 22.94 \\
k-NN & 16.01 & 23.62 & 22.74 \\
PC & 14.15 & 23.74 & 22.64 \\
ACFlow & 14.91 & 24.09 & 23.14 \\
MisConv{} & {\bf 16.27} & {\bf 24.25} & {\bf 23.40} \\
GT & 21.48 & 24.70 & 24.02 \\ \hline
\end{tabular}
\end{table}
\paragraph{Image generation}
In the second experiment, we use Wasserstein Auto-Encoder (WAE) \cite{WAE} for the image generation task. During training, the WAE reconstruction loss function is evaluated only on the observed pixels. We consider three datasets: MNIST, SVHN, and CelebA. In the case of CelebA dataset, we employ a common preprocessing step -- cropping out the background part of the image and resizing the remaining part to the size of $64 \times 64$. The models were assessed using Frechet Inception Distance (FID) \cite{heusel2018fid, Seitzer2020FID} calculated between the set of ground-truth images and images sampled from WAE models. Due to the large volatility of FID scores, we decided to report the best (lowest) value across the whole training process.
The architecture of the WAE decoder and encoder is based on the DCGAN network \cite{radford2015unsupervised} -- namely, the encoder and decoder are built from repeated blocks of Convolution-ReLU-Batch-Normalization layers, which gradually downsample and then upsample the feature maps. We use fully connected layers to map the encoded feature maps into the latent dimension and a fully connected network to discriminate between random samples and features of real images in the latent space. Details of the architectures can be found in the supplemental material.
As shown in Table \ref{tab:results-fid}, MisConv{} outperforms competitive methods on two more challenging datasets, SVHN and CelebA. In the case of SVHN, the FID score obtained by MisConv{} is even slightly better than the one of the original model trained on complete images. It means that MisConv{} is capable of producing images of similar quality to standard WAE. While ACFlow performs very well on MNIST, its performance drops significantly on other datasets. PC, our main baseline, again gives unsatisfactory results. For the illustration, we present sample images generated by WAE models, see Figure \ref{fig:wae} (left).
\paragraph{Image reconstruction}
To complement the previous experiment, we also benchmark MisConv{} in the task of image reconstruction. For this purpose, we use the same architecture and datasets as before and report structural similarity (SSIM) and peak signal-noise ratio (PSNR) between the ground-truth and reconstructed images \cite{wang2004ssim}.
As can be seen in Tables \ref{tab:results-ssim} and \ref{tab:results-psnr}, MisConv{} obtained the best scores on all datasets in terms of both metrics. It is difficult to identify the second-best method because their performances vary across datasets. While k-NN performs quite well on MNIST, it is worse than ACFlow on SVHN and CelebA. In Figure \ref{fig:wae} (right), we show sample reconstructions.
\begin{table}[]
\centering
\caption{Classification accuracy of MisConv{} using SMFA and DMFA.}\label{tab:mfa-ablation}
\begin{tabular}{lcc}
\hline
& SMFA & DMFA\\
\hline
MNIST & 0.922 & {\bf 0.931}\\
SVHN & 0.554 & {\bf 0.757}\\
CIFAR-10 & 0.640 & {\bf 0.722}\\
\hline
\end{tabular}
\end{table}
\begin{table}[]
\centering
\caption{Quality of SMFA and DMFA on SVHN dataset.}\label{tab:dmfa-vs-mfa}
\begin{tabular}{lcccc}
\hline
& NLL & MSE & PSNR & SSIM \\ \hline
SMFA & -1127.71 & 5.81 & 26.90 & 0.86 \\
DMFA & {\bf -1584.08} & {\bf 5.34} & {\bf 30.14} & {\bf 0.94} \\ \hline \\
\end{tabular}
\end{table}
\section{Analysis of MisConv{}} \label{sec:an}
In this section, we experimentally analyze MisConv{}. First, we examine the influence of MFA representation on the performance of MisConv{}. Next, we show that calculating the expected value of the convolution layer applied to MFA is essentially better strategy than using imputation taken from analogical MFA model. We restrict our attention to the classification problem.
\paragraph{MFA representation} MisConv{} relies on applying the expected value of the convolution operator applied to the MFA representation of the incomplete image. To investigate the influence of MFA parametrization, we consider its two variants. In the first one, the MFA is estimated from the whole dataset, and next its conditional distribution is calculated for each incomplete image. This classical approach, termed here {\bf SMFA} (shallow MFA), was recently explored in \cite{richardson2018gans}. In the second variant, conditional MFA is directly estimated for each incomplete image using density network \cite{przewiezlikowskiestimatingICONIP}. The latter option, called {\bf DMFA} (deep MFA), was used in the previous experiments.
As can be seen in Table \ref{tab:mfa-ablation}, MisConv{} with DMFA is consistently better than with SMFA. To find the reason behind this behavior, we inspect both models regardless on MisConv{} (we restrict our attention to the SVHN dataset). Table \ref{tab:dmfa-vs-mfa} presents the Negative Log-Likelihood (NLL) and mean squared error (MSE) of the most probable imputation inferred by the DMFA and SMFA, as well as SSIM and PSNR between the original image and the image with missing pixels replaced by the imputation. Higher quality of DMFA results in a better description of missing values, which impacts the performance of MisConv{}.
\paragraph{Comparison with MFA imputation} There is a question whether taking the expected value in MisConv{} is a better strategy than using a single imputation from an analogical MFA model. To verify this aspect, we compare MisConv{} with typical CNN applied to the imputation produced by the MFA. It is evident from Table \ref{tab:mis-vs-impute} that taking the whole distribution of missing values is more profitable than using only a point estimate.
We additionally compare the outputs of the initial classifier layer produced by those two variants, see Figure \ref{fig:misconv-filters}. It can be seen that MisConv{} is able to differentiate between the observed and missing pixels. This leads us to hypothesize that MisConv{} allows for processing the signal from the observed and missing regions with varying levels of trust, treating the latter with lower confidence.
\begin{table}[t]
\centering
\caption{Comparing MisConv{} with MFA imputation in terms of classification accuracy.}\label{tab:mis-vs-impute}
\begin{tabular}{lcc}
\hline
& MFA imputation & MisConv{}\\
\hline
MNIST & 0.923 & {\bf 0.931}\\
SVHN & 0.720 & {\bf 0.757}\\
CIFAR-10 & 0.716 & {\bf 0.722}\\
\hline
\end{tabular}
\end{table}
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{img/misconv_filters.png}
\caption{Outputs of the first convolutional layer for MFA imputaton and MisConv{}. }\label{fig:misconv-filters}
\end{figure}
\section{Conclusion}
We have presented MisConv{} -- a generalization of CNNs, capable of processing incomplete images. By taking the expected value over all possible replacements after the first convolutional layer, MisConv{} allows for exploring the uncertainty contained in missing pixels. Making use of MFA as the missing data representation, we were able to efficiently implement the whole framework using matrix operations without the need of sampling. Experimental results confirm that MisConv{} obtains better results across various tasks and datasets than the recent adaptation of CNNs to the case of missing data as well as typical imputation strategies.
\section*{Acknowledgement}
The research of M. Przewięźlikowski was funded by the Priority Research Area DigiWorld under
the program Excellence Initiative -– Research University at the Jagiellonian
University in Kraków. The research of M. \'Smieja was funded by the Priority Research Area SciMat under the program Excellence Initiative -- Research University at the Jagiellonian University in Kraków. The research of \L{}. Struski was supported by the Foundation for Polish Science co-financed by the European
Union under the European Regional Development Fund
in the POIR.04.04.00-00-14DE/18-00 project carried out
within the Team-Net programme. The research of J. Tabor was supported by the
National Science Centre (Poland), grant no. 2018/31/B/ST6/00993. For the purpose of Open Access, the authors have applied a CC-BY public copyright licence to any Author Accepted Manuscript (AAM) version arising from this submission.
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1,116,691,499,306 | arxiv | \section{\label{sec:intro}Introduction}
Dispersive shock waves~(DSWs) are widespread nonlinear phenomena in physics, from hydrodynamics~\cite{1966Peregrine,1988Smyth,2016Maiden} to acoustics~\cite{1970Taylor}, from Bose-Einstein condensates~\cite{2004Damski,2004Kamchatnov,2004Perez,2005Simula,2006Hoefer,2008Chang} to plasma physics~\cite{2008Romagnani}, and from quantum liquids~\cite{2006Bettelheim} to optics~\cite{1967Akhmanov,1989Rothenberg,2007El,2007HoeferPhysD,2007Ghofraniha,2007Wan,2008Conti,2012Ghofraniha,2013Garnier,2013Gentilini,2014Gentilini,2014Smith1,2015GentiliniPRA,2015GentiliniSciRep,2015Xu,2016Braidotti,2016Wetzel,2016XuMussot,2016XuGarnier,2017Zannotti,2019Gautam,2019Marcucci,2019MarcucciReview}.
In optics and photonics, when light propagates in a medium whose refractive index depending on the beam intensity, (e.g., light experiences the Kerr effect~\cite{2008Boyd}), the interplay between diffraction (or dispersion) and nonlinearity can lead to steep gradients in the phase profile, and, in some cases, to a wave breaking~\cite{2019MarcucciReview}.
Such discontinuity is regularized by very rapid oscillations, both in phase chirp and in intensity outlines, called undular bores~\cite{2007Wan}.
If we slightly mutate this third-order nonlinearity, making it nonlocal (or noninstantaneous), then the phenomenology changes very little. When the phase chirp reaches the discontinuity and starts to oscillate, the intensity does not develop undular bores, but rather the annular collapse singularity~(ACS) (or M-shaped singularity if the beam is $1+1$-dimensional)~\cite{2015Xu,2016Braidotti,2018Xu,2019MarcucciReview}. Spatial collapse-DSWs occur in thermal media, where the radiation-matter interaction is led by a thermo-optic effect, and the refractive index perturbation depends on the whole intensity profile~\cite{2008Boyd,2019MarcucciReview}.
Theoretically, such modification to the Kerr nonlinearity has significant consequences. Laser beam propagation in a standard Kerr medium is ruled by the nonlinear Schr\"odinger equation~(NLSE), exactly solvable by the inverse scattering transform method~\cite{1967Gardner,1972Zacharov,2015Fibich}. However, the NLSE with a nonlocal potential cannot be solved by inverse scattering transform (despite some recent progress in two-dimensional~(2D) media~\cite{2017Hokiris,2019Hokiris}), but only through the Whitham modulation and the hydrodynamic approximation~\cite{1999Whitham}. Moreover, the dynamics beyond the shock appears to be intrinsically irreversible, as recently shown by applying the time asymmetric quantum mechanics~\cite{1981Bohm,1989Bohm,1998Bohm,1999Bohm,2002Chruscinski,2002Delamadrid,2003Gadella,2003Chruscinski,2004Chruscinski,2004Civitarese,2016Celeghini,2016Marcucci} to the description of DSWs in highly nonlocal approximation~\cite{2015GentiliniPRA,2015GentiliniSciRep,2016Braidotti,2017Marcucci,2019MarcucciReview}.
In highly nonlocal Kerr media, the ACS is modeled by the simplest Hamiltonian of time asymmetric quantum mechanics: the reversed harmonic oscillator~(RHO). This Hamiltonian has a real complete continuous spectrum, which corresponds to a basis of eigenfunctions that have simple poles in their analytical prolongations to the complex plane. Since the RHO is a harmonic oscillator~(HO) with a pure imaginary frequency, starting from the HO complete point spectrum, it turns out that the RHO point spectrum is the set of the mentioned simple poles, and the related eigenfunctions are non-normalizable eigenvectors belonging to a rigged Hilbert space, called Gamow vectors~(GVs)~\cite{1928GamowDE,1928GamowENG}.
Light propagation beyond the collapse is then expressed as a superposition of GVs, which exponentially decay with quantized decay rates. Such laser beam evolution is the outcome of a phenomenon, the shock, that is intrinsically irreversible: in the absence of absorption and interaction with an external thermal bath, the dynamics cannot be inverted, i.e., it is time asymmetric.
Can this theoretical model be used to describe much more complex scenarios? To answer this question, we need to access regimes with much stronger nonlinearity
In recent experiments, it is showed that M-Cresol/Nylon solutions exhibit an isotropic giant self-defocusing nonlinearity, tunable by varying the nylon concentration~\cite{2014Smith}.
M-Cresol/Nylon is a thermal chemical mixture, consisting of an organic solvent (m-cresol) and a synthetic polymeric solute (nylon).
The nonlinear Kerr coefficient $n_2$ of pure m-cresol is $-9\times10^{-8} cm^2/W$, but it was found that, in such mixtures, $n_2=-1.6\times10^{-5} cm^2/W$ for a nylon mass concentration of $3.5\%$, higher than other thermal nonlinear materials in which ACSs have been observed~\cite{2014Smith1,2019MarcucciReview}.
In this Letter, we report on our theoretical discovery and experimental evidence of optical DSWs with an \textit{anisotropic zero-singularity}~(ZS) (i.e., a gap in the intensity profile along only one direction) in M-Cresol/Nylon solutions. Fixing $z$ as the longitudinal and $x,y$ as the transverse directions, we consider an initial beam which is even in the $y$ direction, and odd along the $x$ direction. This initial condition causes a new phenomenon: the shock does develop an annular collapse, but around the ZS it presents an abrupt intensity discontinuity.
We theoretically analyze this anisotropic wave breaking. We model the beam propagation beyond the shock point by time asymmetric quantum mechanics and uncover the mechanism that determines how such an abrupt intensity discontinuity is generated. We numerically simulate these results and find remarkable agreement between experimental outcomes and theoretical predictions.
For a laser beam propagating in a thermal medium with refractive index $n=n_0+\Delta n[|A|^2](\mathbf{R})$, where $\mathbf{R}=(\mathbf{R_{\perp}},Z)=(X,Y,Z)$, the NLSE describes the evolution of the envelope $A(\mathbf{R})$ of the monochromatic field $\mathbf{E}(\mathbf{R})=\hat{\mathbf{E}}_0 A(\mathbf{R})e^{\imath k Z}$, and it reads as follows:
\begin{equation}
2\imath k \partial_Z A+\nabla_{\mathbf{R_{\perp}}}^2 A+2k^2 \frac{\Delta n[|A|^2]}{n_0} A=-\imath\frac{k}{ L_{loss}}A,
\label{eq:NLSE}
\end{equation}
where $\nabla_{\mathbf{R_{\perp}}}^2=\partial_X^2+\partial_Y^2$, $k=\frac{2\pi n_0}{\lambda}$ is the wavenumber, $\lambda$ is the wavelength, and $L_{loss}$ is the linear loss length. By defining $I=|A|^2$ the intensity, $\bar{P}(Z)=\int\int\mathrm{d}\mathbf{R_{\perp}}I(\mathbf{R})$ the power, $L_d=k W_0^2$ the diffraction length, with $W_0$ the initial beam waist, and $\alpha=\frac{L_d}{L_{loss}}$, it turns out that $\bar{P}$ is not conserved only if $\alpha\neq0$. Indeed, if $\alpha\sim0$, then $\partial_Z \bar{P}\sim0$~\cite{1991Lisak}.
In low absorption regime, the refractive index perturbation in Eq.~(\ref{eq:NLSE}) is~\cite{2019MarcucciReview}
\begin{equation}
\Delta n[|A|^2](\mathbf{R_{\perp}})=n_2 \int\int\mathrm{d}\mathbf{R_{\perp}'}K(\mathbf{R_{\perp}}-\mathbf{R_{\perp}'})I(\mathbf{R_{\perp}'}),
\label{eq:n1}
\end{equation}
with $n_2$ the Kerr coefficient and $K(\mathbf{R_{\perp}})$ is the kernel function describing the nonlocal nonlinearity, normalized such that $\int\int\mathrm{d}\mathbf{R_{\perp}}K(\mathbf{R_{\perp}})=1$.
For $K(\mathbf{R_{\perp}})=\delta(\mathbf{R_{\perp}})$ we attain the well-known local Kerr effect, i.e., $n=n_0+n_2I$~\cite{2008Boyd}. In our nonlocal case, where the laser beam produces a thermo-optic effect that generates an isotropic variation of the medium density distribution, the response function is
\begin{equation}
K(X,Y)=\tilde{K}(X)\tilde{K}(Y),\;\; \tilde{K}(X)=\frac{e^{-\frac{|X|}{L_{nloc}}}}{2L_{nloc}}.
\label{eq:kernel}
\end{equation}
Here $L_{nloc}$ is the nonlocality length~\cite{1997Snyder,2003Conti,2007Minovich,2007Ghofraniha,2019Marcucci}.
We rescale Eq.~(\ref{eq:NLSE}), with $\alpha\sim0$, by defining the dimensionless variables $x=X/W_0$, $y=Y/W_0$ and $z=Z/L_d$, and obtain
\begin{equation}
\imath \partial_z \psi+\frac{1}{2}\nabla_{\mathbf{r_{\perp}}}^2\psi+\chi P K_0*|\psi|^2 \psi=0,
\label{eq:NLSEnorm}
\end{equation}
where $\mathbf{r}=(\mathbf{r_{\perp}},z)=(x,y,z)$, $\nabla_{\mathbf{r_{\perp}}}^2=\partial_x^2+\partial_y^2$, $\psi(\mathbf{r})=\frac{W_0}{\sqrt{\bar{P}}}A(\mathbf{R})$, $\chi=\frac{n_2}{|n_2|}$ and $P=\frac{\bar{P}}{P_{REF}}$ with $P_{REF}=\frac{\lambda^2}{4\pi^2 n_0 |n_2|}$.
The asterisk in Eq. (\ref{eq:NLSEnorm}) stands for the convolution product, while $K_0(x,y)=\tilde{K_0}(x)\tilde{K_0}(y)$ with $\tilde{K_0}(x)=W_0 \tilde{K}(X)=\frac{e^{-\frac{|x|}{\sigma}}}{2\sigma}$, $\tilde{K_0}(y)$ of the same form, and $\sigma=\frac{L_{nloc}}{W_0}$ the nonlocality degree.
In highly nonlocal approximation ($\sigma>>1$), once the initial conditions are fixed, $|\psi|^2$ mimics a delta function (or a narrow superposition of delta functions), and the nonlocal potential looses its $I$-dependence, becoming a simple function of the transverse coordinates~\cite{1997Snyder,2012Folli}:
\begin{equation}
\begin{array}{l}
K_0*|\psi|^2\simeq \kappa(\mathbf{r_{\perp}})\simeq\kappa(\mathbf{0})+\left(\partial_x\kappa|_{\mathbf{r_{\perp}}=\mathbf{0}}\right)x+\left(\partial_y\kappa|_{\mathbf{r_{\perp}}=\mathbf{0}}\right)y+\\ \\
+\frac{1}{2}\left(\partial_x^2\kappa|_{\mathbf{r_{\perp}}=\mathbf{0}}\right)x^2+\left(\partial_x\partial_y\kappa|_{\mathbf{r_{\perp}}=\mathbf{0}}\right)xy+\frac{1}{2}\left(\partial_y^2\kappa|_{\mathbf{r_{\perp}}=\mathbf{0}}\right)y^2,
\end{array}
\label{eq:nonlocpot}
\end{equation}
after a Taylor second-order expansion.
This approximation maps the NLSE in Eq.~(\ref{eq:NLSEnorm}) into a linear Schr\"odinger equation $\imath \partial_z\psi(\mathbf{r})=\hat{H}(\mathbf{p_{\perp}},\mathbf{r_{\perp}})\psi(\mathbf{r})$, with $\hat{H}(\mathbf{p_{\perp}},\mathbf{r_{\perp}})=\frac{1}{2}\hat{\mathbf{p_{\perp}}}^2+\hat V(\mathbf{r_{\perp}})$ the Hamiltonian, $\mathbf{\hat p_{\perp}}=(\hat p_x, \hat p_y)=(-\imath\partial_x, -\imath\partial_y)$ the transverse momentum and $\hat V(\mathbf{r_{\perp}})=-\chi P\kappa(\mathbf{r_{\perp}})\mathbb{1}$ the multiplicative potential ($\mathbb{1}$ is the identity operator). Let us consider the initial condition
\begin{equation}
\psi_{\mathrm{ISO}}(\mathbf{r_{\perp}})=\psi_{even}(x)\psi_{even}(y),\;\;\psi_{even}(x)=\frac 1{\sqrt[4]{\pi}}e^{-\frac{x^2}2},
\label{eq:isoini}
\end{equation}
and $\psi_{even}(y)$ of the same form.
The shape of $\kappa(\mathbf{r_{\perp}})$ depends on $\psi_{\mathrm{ISO}}(\mathbf{r_{\perp}})$. Indeed, since $\psi_{\mathrm{ISO}}$ is an even, separable function, all the first derivatives in Eq.~(\ref{eq:isoini}) vanish, hence $\kappa(\mathbf{r_{\perp}})=\kappa_0^2-\frac{1}{2}\kappa_2^2\left|\mathbf{r_{\perp}}\right|^2$, where $\kappa_0^2=\frac1{4\sigma^2}$ and $\kappa_2^2=\frac{1}{2\sqrt{\pi}\sigma^3}$.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=\linewidth]{figISO.eps}
\end{center}
\caption{Solution of the NLSE~(\ref{eq:NLSEnorm}) with initial condition~(\ref{eq:isoini}), for $P=4\times10^6$ and $\sigma=120$, in arbitrary units: (a) shows the intensity transverse profile at $z=0$, (b) exhibits the symmetric response function derived from Eq.~(\ref{eq:isoini}), and (c) reports the intensity longitudinal outline, here on the plane $(x,z)$, equal to one on the plane $(y,z)$.
\label{fig:iso}}
\end{figure}
In the defocusing case ($n_2<0$), the transverse profile of the solution of Eq.~(\ref{eq:NLSEnorm}) with initial condition~(\ref{eq:isoini}) is shown in Fig.~\ref{fig:iso}(a). Figure~\ref{fig:iso}(b) exhibits the central part of the symmetric response function $K_0(x,y)$, while the longitudinal profile on $x,z$ (same of $y,z$) is reported in Fig.~\ref{fig:iso}(c). The corresponding Hamiltonian reads
$\hat{H}=P\kappa_0^2 +\hat{H}_{\mathrm{RHO}}(p_x,x)+\hat{H}_{\mathrm{RHO}}(p_y,y)$,
where
\begin{equation}
\hat{H}_{\mathrm{RHO}}(p_x,x)=\frac 12\hat{p_x}^2-\frac{\gamma^2}{2}\hat{x}^2
\label{eq:RHO}
\end{equation}
is the 1D-RHO Hamiltonian of frequency $\gamma=\sqrt{P}\kappa_2$.
Once moved to $\phi(\mathbf{r})=e^{\imath P\kappa_0^2}\psi(\mathbf{r})$, the Schr\"odinger equation becomes
$\imath \partial_z\phi(\mathbf{r})=\left[\hat{H}_{\mathrm{RHO}}(p_x,x)+\hat{H}_{\mathrm{RHO}}(p_y,y)\right]\phi(\mathbf{r})$, which is completely separable.
In bra-ket notation
\begin{equation}
\begin{array}{l}
\imath \frac{\mathrm{d}}{\mathrm{d}z}|\phi(z)\rangle=\hat{H}_{\mathrm{ISO}}(\mathbf{p_{\perp}},\mathbf{r_{\perp}})|\phi(z)\rangle,\\ \\
\hat{H}_{\mathrm{ISO}}(\mathbf{p_{\perp}},\mathbf{r_{\perp}})=\hat{H}_{\mathrm{RHO}}(p_x,x)\otimes\mathbb{1}_y+\mathbb{1}_x\otimes\hat{H}_{\mathrm{RHO}}(p_y,y),\\ \\
|\phi(z)\rangle=|\phi_{even}(z)\rangle_x\otimes|\phi_{even}(z)\rangle_y,
\end{array}
\label{eq:iso}
\end{equation}
with $\otimes$ the tensorial product, no more explicitly written hereafter. The solution of Eq.~(\ref{eq:iso}) lives in a tensorial product between two 1D rigged Hilbert spaces. Indeed, if we consider the evolution operator $\hat{U}(z)=e^{-\imath\hat{H}z}$ such that $|\phi(z)\rangle=\hat{U}(z)|\phi(0)\rangle$, for Eq.~(\ref{eq:iso}) $|\phi(z)\rangle=e^{-\imath\hat{H}_{\mathrm{RHO}}z}|\psi_{even}\rangle_xe^{-\imath\hat{H}_{\mathrm{RHO}}z}|\psi_{even}\rangle_y$. The representation of $|\phi_{even}(z)\rangle_{x,y}=e^{-\imath\hat{H}_{\mathrm{RHO}}z}|\psi_{even}\rangle_{x,y}$ in terms of GVs was already studied and was also already demonstrated to describe 1D DSWs in thermal media~\cite{2015GentiliniPRA,2015GentiliniSciRep,2016Braidotti,2017Marcucci,2019MarcucciReview}. It is $|\phi_{even}(z)\rangle_{x,y}=|\phi_N^G(z)\rangle+|\phi_N^{BG}(z)\rangle$, with
\begin{equation}
|\phi_N^G(z)\rangle=\sum_{n=0}^N e^{-\frac{\gamma}2(2n+1)z}|\mathfrak{f}_n^-\rangle\langle \mathfrak{f}_n^+|\psi_{even}\rangle
\label{eq:GV}
\end{equation}
the decaying superposition of Gamow states $|\mathfrak{f}_n^-\rangle$, corresponding to the energy levels $E_n^{RHO}=\imath\frac{\gamma}2(2n+1)$, and $|\phi_N^{BG}(z)\rangle$ the background function, both belonging to the same 1D rigged Hilbert space.
Our experiments are performed in an isotropic medium with asymmetric initial conditions for the beam, designed through a phase mask, to attain an anisotropic light propagation. The setup is illustrated in Fig.~\ref{fig:exp}(a). A laser beam with wavelength $\lambda=532$nm passes through two collimating lenses (L1 and L2), and then through a beam splitter (BS1), which divides the beam into two arms, one used for the nonlinear experiment, and the other for getting a reference beam for interference measurements. The beam outcoming from the first arm is transformed into an asymmetric input by a phase mask, and then is focused (via L3) onto the facet of a $2$mm-long cuvette, which contains a M-Cresol/Nylon solution (with $3.5\%$ Nylon). The output is imaged (via L4 and BS2) onto a CCD-camera.
Figure~\ref{fig:exp}(b) reports the input beam (zoom-in intensity and phase patterns at initial power $\bar{P}=2$mW and waist $W_0=15.8\mu$m) and its outputs at different initial powers. The input beam presents a phase discontinuity of $\pi$ along $x=0$.
The output beam exhibits diffraction at a low power, but evolves into an overall shock pattern with two parts at high powers. It's important to note that, while the whole pattern expands, the gap between two parts remains constant. This represents the first realization of what we define as \textit{anisotropic DSWs}: ACSs with an initial ZS, which generates two barriers of light intensity around a constant gap in the middle of the beam. Despite the medium isotropy, the oddity of the initial condition generates an anisotropic final transverse profile.
\begin{figure}[h!]
\begin{center}
\textbf{(a)
\raisebox{-1\height}{\includegraphics[width=0.95\linewidth]{Setup.eps}}
\\\vspace{3mm}
\textbf{(b)}\raisebox{-0.85\height}{\includegraphics[width=\linewidth]{anisotropic.png}}
\end{center}
\caption{
(a) Experimental setup. A $\lambda=532$nm CW-laser beam is collimated through two lenses (L1 and L2). A beam splitter (BS1) divides the beam in two arms. the first is made asymmetric by a phase mask and propagates in a $2$mm-long cuvette filled with M-Cresol/Nylon $3.5\%$-solution. The second is a reference beam for interference measurements. The output is imaged (via L4 and BS2) onto a CCD-camera.
(b) Input and outputs observed. The phase mask in (a) generates a $\pi$ discontinuity in the input phase, here reported with initial power $\bar{P}=2$mW and waist $W_0=15.8\mu$m, together with the intensity profile. Several output at different initial powers are shown. The low power cannot distinguish nonlinear effects from diffraction, but the higher the power is, the stronger the nonlinear effects are, leading to the formation of anisotropic DSWs.
\label{fig:exp}}
\end{figure}
We model the initial asymmetric beam-shape as follows:
\begin{equation}
\psi_{\mathrm{ANI}}(\mathbf{r_{\perp}})=\psi_{odd}(x)\psi_{even}(y),\;\;\psi_{odd}(x)=-\frac{\sqrt{2}}{\sqrt[4]{\pi}}xe^{-\frac{x^2}2},
\label{eq:aniini}
\end{equation}
$\psi_{even}(y)=\frac 1{\sqrt[4]{\pi}}e^{-\frac{y^2}2}$ as in Eq.~(\ref{eq:isoini}).
In this case, Eq.~(\ref{eq:nonlocpot}) is reduced to $\kappa(\mathbf{r_{\perp}})=\kappa_0^2+\frac{1}{2}\kappa_1^2 x^2-\frac{1}{2}\kappa_2^2 y^2$, with $\kappa_0^2=\frac1{4\sigma^2}$, $\kappa_1^2=\frac{1}{4\sigma^4}$ and $\kappa_2^2=\frac{1}{2\sqrt{\pi}\sigma^3}$.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=\linewidth]{figANI.eps}
\end{center}
\caption{
Solution of the NLSE~(\ref{eq:NLSEnorm}) with initial condition~(\ref{eq:aniini}), for $P=4\times10^6$ and $\sigma=120$, in arbitrary units: (a) shows the intensity transverse profile at $z=1$, (b) exhibits the asymmetric response function derived from Eq.~(\ref{eq:isoini}), and (c) reports the intensity longitudinal outline on the plane $(x,z)$, with the zero-singularity.
\label{fig:ani}}
\end{figure}
The anisotropy appears evident: not only the initial condition presents a ZS, but also the response function has two different behaviors along $x,y$ directions. Numerical simulations are illustrated in Fig.~\ref{fig:ani}. Figure~\ref{fig:ani}(a) shows the anisotropic DSWs, solution of the NLSE~(\ref{eq:NLSEnorm}) with initial condition~(\ref{eq:aniini}). Figure~\ref{fig:ani}(b) gives numerical proof of the response function anisotropy: the $(x,y)$-plane origin corresponds to a saddle point, with a locally increasing profile along $x>0$, $y<0$ and a locally decreasing outline along $x<0$, $y>0$. Figure~\ref{fig:ani}(c) reports the intensity-ZS in a neighborhood of $x=0$ during propagation.
The presence of the saddle point in the response function has direct consequences through highly nonlocal approximation in mapping the NLSE in the quantum-like linear Schr\"odinger equation. From the expression of $\kappa(\mathbf{r_{\perp}})$ above, for $\phi(\mathbf{r})=e^{\imath P\kappa_0^2}\psi(\mathbf{r})$ we obtain
\begin{equation}
\begin{array}{l}
\imath \frac{\mathrm{d}}{\mathrm{d}z}|\phi(z)\rangle=\hat{H}_{\mathrm{ANI}}(\mathbf{p_{\perp}},\mathbf{r_{\perp}})|\phi(z)\rangle,\\ \\
\hat{H}_{\mathrm{ANI}}(\mathbf{p_{\perp}},\mathbf{r_{\perp}})=\hat{H}_{\mathrm{HO}}(p_x,x)\mathbb{1}_y+\mathbb{1}_x\hat{H}_{\mathrm{RHO}}(p_y,y),\\ \\
|\phi(z)\rangle=|\phi_{odd}(z)\rangle_x|\phi_{even}(z)\rangle_y,
\end{array}
\label{eq:ani}
\end{equation}
where $\hat{H}_{\mathrm{HO}}(p_x,x)=\frac 12\hat{p_x}^2+\frac{\omega^2}{2}\hat{x}^2$ is the one-dimensional harmonic oscillator Hamiltonian with $\omega=\sqrt{P}\kappa_{odd}$, and $\hat{H}_{\mathrm{RHO}}$ is the one-dimensional RHO Hamiltonian in Eq.~(\ref{eq:RHO}).
The solution of Eq.~(\ref{eq:ani}) is the tensorial product of $|\phi_{odd}(z)\rangle_{x}=\sum_{n=0}^{+\infty} e^{\imath\frac{\omega}2(2n+1)z}|\Psi_n^{HO}\rangle\langle\Psi_n^{HO}|\psi_{odd}\rangle$, where $|\Psi_n^{HO}\rangle$ are $\hat{H}_{\mathrm{HO}}$-eigenstates corresponding to the energy levels $E_n^{HO}=\frac{\omega}2(2n+1)$~\cite{2016Marcucci}, and $|\phi_{even}(z)\rangle_{y}=|\phi_N^G(z)\rangle+|\phi_N^{BG}(z)\rangle$, explicitly written in Eq.~(\ref{eq:GV}).
\begin{figure}[h!]
\begin{center}
\includegraphics[width=\linewidth]{GVSign.eps}
\end{center}
\caption{
GVs signature. From Fig.~\ref{fig:ani}, in the same conditions, (a) is the $y,z$ profile at fixed $x=2.29$. Intensity along the pink line, i.e., $|\phi(x=2.29,y=2.29,z)|^2=\langle\phi_{even}(z)|\phi_{even}(z)\rangle_{y}=\sum_{n=0}^Ne^{-\Gamma_n z}\left|\langle\mathfrak{f}_n^+|\phi_{even}\rangle\right|^2$, is in (b) [$\Gamma_n=\gamma(2n+1)$ quantized decay rates]. The continuous lines represent the two first exponential functions of the summation above that fit the decaying part: the red line is the fundamental GV, with decay rate $\Gamma_0$, and the blue line is the first excited GV, with decay rate $\Gamma_2$.
\label{fig:GV}}
\end{figure}
Evidence of the presence of GVs is given in Fig.~\ref{fig:GV}. By defining $\Gamma_n=\gamma(2n+1)$, we look for the first two quantized decay rates $\Gamma_{0,2}$ (the even Gaussian initial function only leads to even energy levels) in the longitudinal propagation in $y$-direction. Indeed, if one computes the intensity of the $y$-part, one finds $\langle\phi_{even}(z)|\phi_{even}(z)\rangle_{y}\stackrel{N>>0}{\simeq}\langle\phi_N^G(z)|\phi_N^G(z)\rangle=\sum_{n=0}^Ne^{-\Gamma_n z}\left|\langle\mathfrak{f}_n^+|\phi_{even}\rangle\right|^2$. Figure~\ref{fig:GV}(a) shows the theoretical section of the nonlinear sample where we seek decaying states. We fix $x=2.29$, a little distant from the shock-gap, and report the corresponding intensity in $y,z$ plane. The pink line is equivalent to $x=y=2.29$. Figure~\ref{fig:GV}(b) exhibits $|\phi(x=2.29,y=2.29,z)|^2$, exponentially decaying. Two exponential fits demonstrate the GV occurrence: the fundamental Gamow state represents the plateau with decay rate $\Gamma_0=1.51$, whereas the first excited one interpolates the peak, with decay rate $\Gamma_2=7.51$.
We stress that the rule $\frac{\Gamma_2}{\Gamma_0}=5$ is respected.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=\linewidth]{GapExp.eps}
\end{center}
\caption{
Observation of the shock-gap versus the initial power $\bar{P}$. The red line shows the theoretical fit with a functions $\propto\frac{1}{\sqrt{\bar{P}}}$.
\label{fig:gap}}
\end{figure}
The analysis of the barriers, due to the HO component, and the corresponding shock-gap is also examinated. Since $\hat{H}_{\mathrm{HO}}$ has potential $\hat{V}_{\mathrm{HO}}(x)=\frac{\omega^2}2\hat{x}^2$, we expect a shock-gap with the same behavior of the potential width $\Delta x\propto\frac1{\sqrt{\omega}}=\frac{2\sigma^2}{\sqrt{P}}$, that is, a shock-gap that varies among different experiments by changing the initial power but, in a single observation with a given $P$, does not change by varying $z$ [as shown in Fig.~\ref{fig:ani}(c)].
This is experimentally proven and reported in Fig.~\ref{fig:gap}, which shows measurements of the shock-gap at variance of initial power $\bar{P}$. A theoretical fit with a function $\propto\frac{1}{\sqrt{\bar{P}}}$ is drawn in the red line. The agreement between observations and numerical simulations confirms the theoretical statement.
We have proved that the interplay of a trapping (harmonic oscillator) and an antitrapping (reversed harmonic oscillator) potential generates a novel kind of dispersive shock waves, with the simultaneous presence of annular collapse singularities and a shock-gap enclosed by very intense light barriers.
The use of a thermal medium with a giant Kerr coefficient, the M-Cresol/Nylon solution with $3.5\%$ of nylon concentration, let us access an extremely-nonlinear highly-nonlocal regime and perform accurate experiments with negligible loss.
We modeled the outcoming dynamics through an advanced theoretical description in rigged Hilbert spaces, by means of time asymmetric quantum mechanics, proving its intrinsic irreversibility.
Our results not only confirm previous studies on the giant nonlinear response of M-Cresol/Nylon, but also disclose fundamental insights on propagation of dispersive shock waves with a singular initial intensity profile.
We believe that this work can be a further step towards a complete description of optical nonlinear phenomena, where inverse scattering transform, Whitham modulation, hydrodynamic approximation and time asymmetric quantum mechanics cooperate in establishing one uniform theory of dispersive shock waves.
We are pleased to acknowledge support from the QuantERA ERA-NET Co-fund 731473 (Project QUOMPLEX), H2020 project grant number 820392, Sapienza Ateneo, PRIN 2015 NEMO, PRIN 2017 PELM, Joint Bilateral Scientic Cooperation CNR-Italy/RFBR-Russia 2018-2020, the NSF Award DMR-1308084, and the National Key R\&D Program of China (2017YFA0303800).
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1,116,691,499,307 | arxiv | \section{\@startsection{section}{1}{\z@}{-3.5ex plus -1ex minus
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\def\vr#1,#2;{\vrule height#1pt depth#2pt}
\def\blb#1#2#3#4#5
{\hbox{\ifnum#2=0\hskip11.5pt
\else\ifnum#2=1\hr13,5;\hskip-1.5pt
\else\ifnum#2=2\hr13.5,6.5;\hskip-13.5pt
\hr13.5,3.5;\hskip-2pt
\else\ifnum#2=3\hr13.7,8;\hskip-13.7pt
\hr13,5;\hskip-13pt
\hr13.7,2;\hskip-2.2pt\fi\fi\fi\fi
$#1$
\ifnum#4=0
\else\ifnum#4=1\hskip-9.2pt\vr22,-9;\hskip8.8pt
\else\ifnum#4=2\hskip-10.9pt\vr22,-8.75;\hskip3pt
\vr22,-8.75;\hskip7.1pt
\else\ifnum#4=3\hskip-12.6pt\vr22,-8.5;\hskip3pt
\vr22,-9;\hskip3pt
\vr22,-8.5;\hskip5.4pt\fi\fi\fi\fi
\ifnum#5=0
\else\ifnum#5=1\hskip-9.2pt\vr1,12;\hskip8.8pt
\else\ifnum#5=2\hskip-10.9pt\vr1.25,12;\hskip3pt
\vr1.25,12;\hskip7.1pt
\else\ifnum#5=3\hskip-12.6pt\vr1.5,12;\hskip3pt
\vr1,12;\hskip3pt
\vr1.5,12;\hskip5.4pt\fi\fi\fi\fi
\ifnum#3=0\hskip8pt
\else\ifnum#3=1\hskip-5pt\hr13,5;
\else\ifnum#3=2\hskip-5.5pt\hr13.5,6.5;
\hskip-13.5pt\hr13.5,3.5;
\else\ifnum#3=3\hskip-5.7pt\hr13.7,8;
\hskip-13pt\hr13,5;
\hskip-13.7pt\hr13.7,2;\fi\fi\fi\fi
}}
\def\blob#1#2#3#4#5#6#7{\hbox
{$\displaystyle\mathop{\blb#1#2#3#4#5 }_{#6}\sp{#7}$}}
\def\up#1#2{\dimen1=33pt\multiply\dimen1by#1\hbox{\raise\dimen1\rlap{#2}}}
\def\uph#1#2{\dimen1=17.5pt\multiply\dimen1by#1\hbox{\raise\dimen1\rlap{#2}}}
\def\dn#1#2{\dimen1=33pt\multiply\dimen1by#1\hbox{\lower\dimen1\rlap{#2}}}
\def\dnh#1#2{\dimen1=17.5pt\multiply\dimen1by#1\hbox{\lower\dimen1\rlap{#2}}}
\def\rlbl#1{\kern-8pt\raise3pt\hbox{$\scriptstyle #1$}}
\def\llbl#1{\raise3pt\llap{\hbox{$\scriptstyle #1$\kern-8pt}}}
\def\elbl#1{\kern3pt\lower4.5pt\hbox{$\scriptstyle #1$}}
\def\lelbl#1{\rlap{\hbox{\kern-9pt\raise2.5pt\hbox{{$\scriptstyle #1$}}}}}
\def\wht#1#2#3#4{\blob\circ#1#2#3#4{}{}}
\def\blk#1#2#3#4{\blob\bullet#1#2#3#4{}{}}
\def\whtd#1#2#3#4#5{\blob\circ#1#2#3#4{#5}{}}
\def\blkd#1#2#3#4#5{\blob\bullet#1#2#3#4{#5}{}}
\def\whtu#1#2#3#4#5{\blob\circ#1#2#3#4{}{#5}}
\def\blku#1#2#3#4#5{\blob\bullet#1#2#3#4{}{#5}}
\def\whtr#1#2#3#4#5{\blob\circ#1#2#3#4{}{}\rlbl{#5}}
\def\blkr#1#2#3#4#5{\blob\bullet#1#2#3#4{}{}\rlbl{#5}}
\def\whtl#1#2#3#4#5{\llbl{#5}\blob\circ#1#2#3#4{}{}}
\def\blkl#1#2#3#4#5{\llbl{#5}\blob\bullet#1#2#3#4{}{}}
\def\rwng{\hbox{$\vbox{\offinterlineskip{
\hbox{\phantom{}\kern6pt{$\circ$}}\kern-2.5pt\hbox{$\Biggr/$}\kern-0.5pt
\hbox{\phantom{}\kern-5pt$\circ$}\kern-3.0pt\hbox{$\Biggr\backslash$}
\kern-1.5pt\hbox{\phantom{}\kern6pt{$\circ$}} }}$}}
\def\hbox{$\vbox{\offinterlineskip{ \hbox{$\circ${\hbox{$\vbox{\offinterlineskip{ \hbox{$\circ$}
\kern-3.0pt\hbox{\phantom{}\kern6.0pt{$\Biggr\backslash$}}
\kern-0.5pt\hbox{\phantom{}\kern11pt{$\circ$}}\kern-3.5pt
\hbox{\phantom{}\kern5.0pt {$\Biggr/$}}\kern-1.0pt\hbox{$\circ$} }}$}}
\def\drwng#1#2#3{\hbox{$\vcenter{ \offinterlineskip{
\hbox{\phantom{}\kern7pt{$\circ^{\elbl{#3}}$}}
\kern-2.5pt\hbox{$\Biggr/$}\kern-0.5pt
\hbox{\phantom{}\kern-5pt$\circ^{ \elbl{#1}}$}
\kern-3.0pt\hbox{$\Biggr\backslash$}
\kern-1.5pt\hbox{\phantom{}\kern7pt{$\circ^{\elbl{#2}}$}} } }$}}
\def\drwngt#1#2#3{\hbox{$\vcenter{ \offinterlineskip{
\hbox{\phantom{}\kern7pt{$\bullet^{\elbl{#3}}$}}
\kern-2.5pt\hbox{$\Biggr/$}\kern-0.5pt
\hbox{\phantom{}\kern-5pt$\circ^{ \elbl{#1}}$}
\kern-3.0pt\hbox{$\Biggr\backslash$}
\kern-1.5pt\hbox{\phantom{}\kern7pt{$\bullet^{\elbl{#2}}$}} } }$}}
\def\dlwng#1#2#3{\hbox{$\vcenter{\offinterlineskip{ \hbox{$\lelbl{#1}\circ$}
\kern-3.0pt\hbox{\phantom{}\kern6.0pt{$\Biggr\backslash$}}
\kern-0.5pt\hbox{\phantom{}\kern11pt{$\lelbl{#2}\circ$}}\kern-3.5pt
\hbox{\phantom{}\kern5.0pt {$\Biggr/$}}\kern-1.0pt\hbox{$\lelbl{#3}\circ$}}}$}
}
\def\rde#1#2#3{\hbox{\phantom{}\kern-4pt\hbox{$\vcenter{\offinterlineskip \hbox
{
\raise 4.5pt\hbox{\vrule height0.4pt width13pt depth0pt}
\kern-1pt\vbox{ \hbox{\drwng{#1}{#2}{#3}}} }}$ }} }
\def\rdet#1#2#3{\hbox{\phantom{}\kern-4pt\hbox{$\vcenter{\offinterlineskip \hbox
{
\raise 4.5pt\hbox{\vrule height0.4pt width13pt depth0pt}
\kern-1pt\vbox{ \hbox{\drwngt{#1}{#2}{#3}}} }}$ }} }
\def\lde#1#2#3{\hbox{$\vcenter{\offinterlineskip \hbox{
\dlwng{#1}{#2}{#3}\kern-4.2pt\lower0.4pt\hbox{$\vcenter{\hrule
width13pt}$}
\kern-8pt\phantom{} }} $}}
\def\rde{}{}{}{\rde{}{}{}}
\def\lde{}{}{}{\lde{}{}{}}
\def\rwngb{\hbox{$\vbox{\offinterlineskip{
\hbox{\phantom{}\kern6pt{$\bullet$}}\kern-2.5pt\hbox{$\Biggr/$}\kern-0.5pt
\hbox{\phantom{}\kern-5pt$\bullet$}\kern-3.0pt\hbox{$\Biggr\backslash$}
\kern-1.5pt\hbox{\phantom{}\kern6pt{$\bullet$}} }}$}}
\def\hbox{$\vbox{\offinterlineskip{ \hbox{$\bullet${\hbox{$\vbox{\offinterlineskip{ \hbox{$\bullet$}
\kern-3.0pt\hbox{\phantom{}\kern6.0pt{$\Biggr\backslash$}}
\kern-0.5pt\hbox{\phantom{}\kern11pt{$\bullet$}}\kern-3.5pt
\hbox{\phantom{}\kern5.0pt {$\Biggr/$}}\kern-1.0pt\hbox{$\bullet$} }}$}}
\def\dbrwng#1#2#3{\hbox{$\vcenter{ \offinterlineskip{
\hbox{\phantom{}\kern6pt{$\bullet^{\elbl{#3}}$}}
\kern-2.5pt\hbox{$\Biggr/$}\kern-0.5pt
\hbox{\phantom{}\kern-5pt$\bullet^{ \elbl{#1}}$}
\kern-3.0pt\hbox{$\Biggr\backslash$}
\kern-1.5pt\hbox{\phantom{}\kern6pt{$\bullet^{\elbl{#2}}$}} } }$}}
\def\dblwng#1#2#3{\hbox{$\vcenter{\offinterlineskip{ \hbox{$\lelbl{#1}\bullet$}
\kern-3.0pt\hbox{\phantom{}\kern6.0pt{$\Biggr\backslash$}}
\kern-0.5pt\hbox{\phantom{}\kern11pt{$\lelbl{#2}\bullet$}}\kern-3.5pt
\hbox{\phantom{}\kern5.0pt {$\Biggr/$}}\kern-1.0pt\hbox{$\lelbl{#3}\bullet$}}}
$} }
\def\rbde#1#2#3{\hbox{\phantom{}\kern-4pt\hbox{$\vcenter{\offinterlineskip \hbo
x{
\raise 4.5pt\hbox{\vrule height0.4pt width13pt depth0pt}
\kern-1pt\vbox{ \hbox{\dbrwng{#1}{#2}{#3}}} }}$ }} }
\def\lbde#1#2#3{\hbox{$\vcenter{\offinterlineskip \hbox{
\dblwng{#1}{#2}{#3}\kern-4.2pt\lower0.4pt\hbox{$\vcenter{\hrule w
idth13pt}$}
\kern-8pt\phantom{} }} $}}
\def\rbde{}{}{}{\rbde{}{}{}}
\def\lbde{}{}{}{\lbde{}{}{}}
\def\p#1{\phantom{-}#1}
\def\phantom{\ldots}{\phantom{\ldots}}
\def\left( \matrix{\p2&-1\cr-3&\p2\cr}\right) {\left( \matrix{\p2&-1\cr-3&\p2\cr}\right) }
\def\left(\matrix{2&1\cr3&2}\right){\left(\matrix{2&1\cr3&2}\right)}
\def\ddgu#1.#2.{\textfont2=\dynkfont \whtu0300{#1}\blku3000{#2}}
\def\ddgd#1.#2.{\textfont2=\dynkfont \whtd0300{#1}\blkd3000{#2}}
\def\eddgiu#1.#2.#3.{\textfont2=\dynkfont \whtu0100{#1}\whtu1300{#2}\blku3000{#3}}
\def\eddgid#1.#2.#3.{\textfont2=\dynkfont \whtd0100{#1}\whtd1300{#2}\blkd3000{#3}}
\def\eddgiiu#1.#2.#3.{\textfont2=\dynkfont \whtu0300{#1}\blku3100{#2}\blku1000{#3}}
\def\eddgiid#1.#2.#3.{\textfont2=\dynkfont \whtd0300{#1}\blkd3100{#2}\blkd1000{#3}}
\def\cmf{\left(\matrix{\p2&-1&\p0&\p0\cr-1&\p2&-1&\p0\cr\p0&-2&\p2&-1\cr
\p0&\p0&-1&\p2\cr}\right)}
\def\left(\matrix{2&3&2&1\cr3&6&4&2\cr4&8&6&3\cr2&4&3&2\cr}\right){\left(\matrix{2&3&2&1\cr3&6&4&2\cr4&8&6&3\cr2&4&3&2\cr}\right)}
\def\ddfu#1.#2.#3.#4.{\textfont2=\dynkfont \whtu0100{#1}\whtu1200{#2}\blku2100{#3}\blku1000{#4}
}
\def\ddfd#1.#2.#3.#4.{\textfont2=\dynkfont \whtd0100{#1}\whtd1200{#2}\blkd2100{#3}\blkd1000{#4}
}
\def\eddfiu#1.#2.#3.#4.#5.{\textfont2=\dynkfont \whtu0100{#1}\whtu1100{#2}\whtu1200{#3}\blku210
0{#4}\blku1000{#5}}
\def\eddfid#1.#2.#3.#4.#5.{\textfont2=\dynkfont \whtd0100{#1}\whtd1100{#2}\whtd1200{#3}\blkd210
0{#4}\blkd1000{#5}}
\def\eddfiiu#1.#2.#3.#4.#5.{\textfont2=\dynkfont \whtu0100{#1}\whtu1200{#2}\blku2100{#3}\blku11
00{#4}\blku1000{#5}}
\def\eddfiid#1.#2.#3.#4.#5.{\textfont2=\dynkfont \whtd0100{#1}\whtd1200{#2}\blkd2100{#3}\blkd11
00{#4}\blkd1000{#5}}
\def\cman{\left(\matrix{\p2&-1&\p0&\p0&\phantom{\ldots}&&&\cr-1&\p2&-1&\p0&\phantom{\ldots}&&&\cr
\p0&-1&\p2&-1&\phantom{\ldots}&&&\cr&&&&\phantom{\ldots}&&&\cr
&&&&\phantom{\ldots}&-1&\p2&-1\cr &&&&\phantom{\ldots}&\p0&-1&\p2\cr}\right)}
\def\ican{{1\over n+1}\left(\matrix{1n&1(n-1)&1(n-2)&&&1.1\cr
1(n-1)&2(n-1)&2(n-2)&&&2.1\cr
1(n-2)&2(n-2)&3(n-2)&&&3.1\cr
&&&&&\cr&&&&&\cr
1.2&2.2&3.2&&&(n-1)1\cr
1.1&2.1&3.1&&&n1\cr}\right)}
\def\ddanu#1.#2.#3.#4.#5.{\textfont2=\dynkfont \whtu0100{#1}\whtu1100{#2}\whtu1100{#3}\cdots
\whtu1100{#4}\whtu1000{#5}}
\def\ddand#1.#2.#3.#4.#5.{\textfont2=\dynkfont \whtd0100{#1}\whtd1100{#2}\whtd1100{#3}\cdots
\whtd1100{#4}\whtd1000{#5}}
\def\ddandte#1.#2.#3.#4.#5.{\textfont2=\dynkfont \blkd0100{#1}\whtd1100{#2}\blkd1100{#3}\cdots
\blkd1100{#4}\whtd1000{#5}}
\def\ddandto#1.#2.#3.#4.#5.{\textfont2=\dynkfont \blkd0100{#1}\whtd1100{#2}\blkd1100{#3}\cdots
\whtd1100{#4}\blkd1000{#5}}
\def\eddanu#1.#2.#3.#4.#5.{\textfont2=\dynkfont \whtu0100{#1}\whtu1100{#2}%
\up1{\whtr0000{#3}}\cdots\whtu1100{#4}\whtu1000{#5}}
\def\eddand#1.#2.#3.#4.#5.{\textfont2=\dynkfont \whtd0100{#1}\whtd1100{#2}%
\up1{\whtr0000{#3}}\cdots\whtd1100{#4}\whtd1000{#5}}
\def\eddaid#1.#2.{\textfont2=\dynkfont\whtd0400{#1}\hskip30pt\whtd4000{#2}}
\def\eddanid#1.#2.#3.#4.#5.{\textfont2=\dynkfont \whtd0200{#1}\whtd2100{#2}%
\whtd1100{#3}\cdots\whtd1200{#4}\blkd2000{#5}}
\def\eddaniu#1.#2.#3.#4.#5.{\textfont2=\dynkfont \whtu0200{#1}\whtu2100{#2}%
\whtu1100{#3}\cdots\whtu1200{#4}\blku2000{#5}}
\def\eddaniid#1.#2.#3.#4.#5.#6.{\hbox{$\vcenter{\hbox
{\textfont2=\dynkfont\hbox{$ \lbde{#1}{#2}{#3}\blkd1100{#4}\cdots%
\blkd1200{#5}\whtd2000{#6} $}} }$}}
\def\eddaniiu#1.#2.#3.#4.#5.#6.{\hbox{$\vcenter{\hbox
{\textfont2=\dynkfont\hbox{$ \lbde{#1}{#2}{#3}\blku1100{#4}\cdots%
\blku1200{#5}\whtu2000{#6} $}} }$}}
\def\dynk\blk0400\hskip30pt\wht4000{\textfont2=\dynkfont\blk0400\hskip30pt\wht4000}
\def\eddaiiid#1.#2.{\textfont2=\dynkfont\blkd0400{#1}\hskip30pt\whtd4000{#2}}
\def\cmbn{\left(\matrix{\p2&-1&\p0&\phantom{\ldots}\ins&&&&\cr-1&\p2&-1&&&&&\cr&&&&&&&\cr
&&&&-1&\p2&-1&\p0\cr&&&&\p0&-1&\p2&-1\cr&&&&\p0&\p0&-2&\p2\cr}\right)}
\def\icbn{\left(\matrix{1&\p1&\p1&\phantom{\ldots}\ins&&1&1/2\cr1&\p2&\p2&&&2&2/2\cr
1&\p2&\p3&&&3&3/2\cr&&&&&&\cr&&&&&&\cr
1&\p2&\p3&&&n-1&(n-1)/2\cr1&\p2&\p3&&&n-1&{n/2}\cr}\right)}
\def\ddbnu#1.#2.#3.#4.#5.{\textfont2=\dynkfont \whtu0100{#1}\whtu1100{#2}\whtu1100{#3}\cdots
\whtu1200{#4}\blku2000{#5}}
\def\ddbnd#1.#2.#3.#4.#5.{\textfont2=\dynkfont \whtd0100{#1}\whtd1100{#2}\whtd1100{#3}\cdots
\whtd1200{#4}\blkd2000{#5}}
\def\eddbnu#1.#2.#3.#4.#5.#6.{\textfont2=\dynkfont \lde{#1}{#2}{#3}\whtu1100{#4}\cdots
\whtu1200{#5}\blku2000{#6}}
\def\eddbnd#1.#2.#3.#4.#5.#6.{\textfont2=\dynkfont \lde{#1}{#2}{#3}\whtd1100{#4}\cdots
\whtd1200{#5}\blkd2000{#6}}
\def\cmcn{\left(\matrix{\p2&-1&\p0&\phantom{\ldots}&&&&\cr-1&\p2&-1&&&&&\cr&&&&&&&\cr
&&&&-1&\p2&-1&\p0\cr&&&&\p0&-1&\p2&-2\cr&&&&\p0&\p0&-1&\p2\cr}\right)}
\def\iccn{\left(\matrix{1&1&1&\phantom{\ldots}&&1&1\cr1&2&2&&&2&2\cr
1&2&3&&&3&3\cr&&&&&&\cr&&&&&&\cr1&2&3&&&n-1&n-1\cr{1/2}&
{2/2}&{3/2}&&&(n-1)/2&{n/2}\cr}\right)}
\def\ddcnu#1.#2.#3.#4.#5.{\textfont2=\dynkfont \blku0100{#1}\blku1100{#2}\blku1100{#3}\cdots
\blku1200{#4}\whtu2000{#5}}
\def\ddcnd#1.#2.#3.#4.#5.{\textfont2=\dynkfont \blkd0100{#1}\blkd1100{#2}\blkd1100{#3}\cdots
\blkd1200{#4}\whtd2000{#5}}
\def\eddcnu#1.#2.#3.#4.#5.#6.{\textfont2=\dynkfont \whtu0200{#1}\blku2100{#2}\blku1100{#3}
\blku1100{#4}\cdots
\blku1200{#5}\whtu2000{#6}}
\def\eddcnd#1.#2.#3.#4.#5.{\textfont2=\dynkfont \whtd0200{#1}\blkd2100{#2}\blkd1100{#3}
\cdots \blkd1200{#4}\whtd2000{#5}}
\def\cmdn{\left(\matrix{\p2&-1&\p0&&&&&\cr-1&\p2&-1&&&&&\cr&&&&&&&\cr
&&&&-1&\p2&-1&-1\cr&&&&\p0&-1&\p2&\p0\cr&&&&\p0&-1&\p0&\p2\cr}\right)}
\def\icdn{\left(\matrix{1&1&1&\phantom{\ldots}&&1&1/2&1/2\cr1&2&2&&&2&2/2&2/2\cr
1&2&3&&&3&3/2&3/2\cr&&&&&&&\cr&&&&&&&\cr
1&2&3&&&n-2&(n-2)/2&(n-2)/2\cr
1/2&2/2&3/2&&&(n-2)/2&n/4&(n-2)/4\cr
1/2&2/2&3/2&&&(n-2)/2&(n-2)/4&n/4\cr}\right)}
\def\dddnu#1.#2.#3.#4.#5.#6.{\hbox{$\vcenter{\hbox
{\textfont2=\dynkfont\hbox{$ \whtu0100{#1}\whtu1100{#2}\cdots%
\whtu1100{#3}\rde{#4}{#5}{#6} $}} }$}}
\def\dddnd#1.#2.#3.#4.#5.#6.{\hbox{$\vcenter{\hbox
{\textfont2=\dynkfont\hbox{$ \whtd0100{#1}\whtd1100{#2}\cdots%
\whtd1100{#3}\rde{#4}{#5}{#6} $}} }$}}
\def\dddndte#1.#2.#3.#4.#5.#6.{\hbox{$\vcenter{\hbox
{\textfont2=\dynkfont\hbox{$ \blkd0100{#1}\whtd1100{#2}\cdots%
\blkd1100{#3}\rdet{#4}{#5}{#6} $}} }$}}
\def\dddndto#1.#2.#3.#4.#5.#6.{\hbox{$\vcenter{\hbox
{\textfont2=\dynkfont\hbox{$ \whtd0100{#1}\blkd1100{#2}\cdots%
\blkd1100{#3}\rdet{#4}{#5}{#6} $}} }$}}
\def\dddiv#1.#2.#3.#4.{\hbox{$\vcenter{\hbox
{\textfont2=\dynkfont\hbox{$ \whtu0100{#1}\rde{#2}{#3}{#4}
$}} }$}}
\def\edddiv{\hbox{$\vcenter{\hbox{\textfont2=\dynkfont\hbox{$\wht0100
\up1{\wht0001}\dn1{\wht0010}\wht1111\wht1000 $}}}$}}
\def\edddnu#1.#2.#3.#4.#5.#6.#7.#8.{\hbox{$\vcenter{\hbox
{\textfont2=\dynkfont\hbox{$ \lde{#1}{#2}{#3}\whtu1100{#4}\cdots%
\whtu1100{#5}\rde{#6}{#7}{#8} $}} }$}}
\def\edddnd#1.#2.#3.#4.#5.#6.#7.#8.{\hbox{$\vcenter{\hbox
{\textfont2=\dynkfont\hbox{$ \lde{#1}{#2}{#3}\whtd1100{#4}\cdots%
\whtd1100{#5}\rde{#6}{#7}{#8} $}} }$}}
\def\edddniid#1.#2.#3.#4.#5.{\hbox{$\vcenter{\hbox
{\textfont2=\dynkfont\hbox{$ \blkd0200{#1}\whtd2100{#2}\whtd1100{#3}\cdots%
\whtd1200{#4}\blkd2000{#5} $}} }$}}
\def\edddniiu#1.#2.#3.#4.#5.{\hbox{$\vcenter{\hbox
{\textfont2=\dynkfont\hbox{$ \blku0200{#1}\whtu2100{#2}\whtu1100{#3}\cdots%
\whtu1200{#4}\blku2000{#5} $}} }$}}
\def\cmei{\left(\matrix{\p2&\p0&-1&\p0&\p0&\p0\cr\p0&\p2&\p0&-1&\p0&\p0\cr
-1&\p0&\p2&-1&\p0&\p0\cr\p0&-1&-1&\p2&-1&\p0\cr
\p0&\p0&\p0&-1&\p2&-1 \cr\p0&\p0&\p0&\p0&-1&\p2\cr}\right)}
\def\icei{{1\over3}\left(\matrix{4&3&5&6&4&2\cr3&6&6&9&6&3\cr5&6&10&12&8&4\cr
6&9&12&18&12&6\cr4&6&8&12&10&5\cr2&3&4&6&5&4\cr}\right)}
\def\ddei#1.#2.#3.#4.#5.#6.{\hbox{$\vcenter{\hbox
{\textfont2=\dynkfont \whtd0100{#1}\whtd1100{#3}%
\up1{\whtr0001{#2}}\whtd1110{#4}\whtd1100{#5}\whtd1000{#6}} }$}}
\def\ddeit#1.#2.#3.#4.#5.#6.{\hbox{$\vcenter{\hbox
{\textfont2=\dynkfont \whtd0100{#1}\blkd1100{#3}%
\up1{\blkr0001{#2}}\whtd1110{#4}\blkd1100{#5}\whtd1000{#6}} }$}}
\def\eddei#1.#2.#3.#4.#5.#6.#7.{\hbox{$\vcenter{\hbox
{\textfont2=\dynkfont \whtd0100{#1}\whtd1100{#3}%
\up1{\whtr0011{#2}}\up2{\whtr0001{#7}}\whtd1110{#4}\whtd1100{#5}%
\whtd1000{#6}} }$}}
\def\cmeii{\left(\matrix{\p2&\p0&-1&\p0&\p0&\p0&\p0\cr\p0&\p2&\p0&-1&\p0&\p0&\p0
\cr-1&\p0&\p2&-1&\p0&\p0&\p0\cr\p0&-1&-1&\p2&-1&\p0&\p0\cr
\p0&\p0&\p0&-1&\p2&-1&\p0\cr\p0&\p0&\p0&\p0&-1&\p2&-1\cr
\p0&\p0&\p0&\p0&\p0&-1&\p2\cr}\right)}
\def\iceii{{1\over2}\left(\matrix{4&4&6&8&6&4&2\cr4&7&8&12&9&6&3\cr
6&8&12&16&12&8&4\cr8&12&16&24&18&12&6\cr
6&9&12&18&15&10&5\cr4&6&8&12&10&8&4\cr
2&3&4&6&5&4&3\cr}\right)}
\def\ddeii#1.#2.#3.#4.#5.#6.#7.{\hbox{$\vcenter{\hbox
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\def{\cal E}}\def\ff{{\cal F}}\def\gg{{\cal G}}\def\hh{{\cal H}{{\cal E}}\def\ff{{\cal F}}\def\gg{{\cal G}}\def\hh{{\cal H}}
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\def{\rm End}\ {{\rm End}\ }
\def{\rm Aut}\ {{\rm Aut}\ }
\def{\rm Pic}{{\rm Pic}}
\def{\rm Nm}{{\rm Nm}}
\def{\rm Spin}{{\rm Spin}}
\def{\rm Grass}{{\rm Grass}}
\def{\rm gr}{{\rm gr}}
\def{\rm Fix}{{\rm Fix}}
\def{\cal G}{{\cal G}}
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\def{\rm Mat}{{\rm Mat}}
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\def{\rm Proj}\ {{\rm Proj}\ }
\def{\rm Sym}{{\rm Sym}}
\def{\rm rad}\ {{\rm rad}\ }
\def{\rm mult}{{\rm mult}}
\def{\rm ker \ }{{\rm ker \ }}
\def{\rm rank \ }{{\rm rank \ }}
\def{\rm trace}\ {{\rm trace}\ }
\def{\rm Pf}{{\rm Pf}}
\def{\rm char}\ {{\rm char}\ }
\def{\rm Sing}\ {{\rm Sing}\ }
\def{\rm Sec}\ {{\rm Sec}\ }
\def{\rm Seg}{{\rm Seg}}
\def{\rm Ver}{{\rm Ver}}
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\def\ses#1#2#3{0\rightarrow {#1}\rightarrow {#2}\rightarrow {#3}\rightarrow 0}
\def{\it Proof.}\ {{\it Proof.}\ }
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\begin{document}
\title{Trigonal curves and Galois ${\rm Spin}(8)$-bundles}
\author{W.M. Oxbury and S. Ramanan}
\date{}
\maketitle
\bigskip
The group ${\rm Spin}(8)$ occupies a special position among the complex simple Lie
groups, in having outer automorphism group $S_3$ and an appealing
Dynkin diagram:
$$
\textfont2=\dynkfont\dddiv{V}..{S^-}.{S^+}.
$$
The outer nodes are dual to three fundamental
8-dimensional representations, the standard representation $V$ on
which the group acts via the double cover ${\rm Spin}(8) \rightarrow SO(8)$,
and the half-spinor representations $S^{\pm}$. These spaces are
permuted by the action of $S_3$, and between any two
there is a Clifford multiplication to the third. It
is possible to identify all three spaces with the complex Cayley
algebra ${\mathbb O}}\def\bp{{\mathbb P}}\def\ratq{{\mathbb Q}$, and Clifford multiplication with multiplication of
octonions. In this sense ${\rm Spin}(8)$ is a member of the `exceptional'
club, and is closely related to the other exceptional groups
$G_2 = {\rm Aut}\ {\mathbb O}}\def\bp{{\mathbb P}}\def\ratq{{\mathbb Q},F_4,E_6$ etc.
\medskip
In this paper we study the moduli of principal holomorphic ${\rm Spin}(8)$-bundles
over an algebraic curve. However, in order to exploit triality, and to obtain
a moduli space with particularly nice properties,
we impose some additional constraints on our bundles.
We suppose that $X$ is an algebraic curve with an $S_3$-action. The
group $S_3$
then acts in two ways on ${\rm Spin}(8)$-bundles over a curve: by
pull-back
under the action on $X$, and by the triality action on the structure
group. We call a ${\rm Spin}(8)$-bundle {\it Galois} if these two actions coincide,
that is, if it is a fixed point of the group action
$$
F\mapsto u^*F^u,
\qquad
u\in S_3,
$$
on the moduli variety $\mm_X({\rm Spin}(8))$ of (semistable) ${\rm Spin}(8)$-bundles.
We denote the fixed-point set by $\ff_X \subset \mm_X({\rm Spin}(8))$, and
distinguish the subvariety of $\ff_X$ consisting of bundles which admit a
lift of the $S_3$-action (see Lemma \ref{galoisbundle} for the precise notion).
This subvariety has a partial desingularisation $\nn_X$
parametrising pairs $(F,\Lambda)$ where $F\in \ff_X$ and $\Lambda$ is an
$S_3$-lift, or more precisely a splitting $\Lambda: S_3 \rightarrow \gg_F$
of the Mumford group associated to $F$ (see Definition \ref{definition}).
$\nn_X$ is the main object of interest of this paper.
\medskip
In \S\ref{moduli} we compute the tangent spaces at stable points and the
semistable boundary of $\nn_X$ (see \ref{tgt-space} and \ref{ssboundary}).
The best situation, to which we mainly restrict ourselves, is where the quotient
$X/S_3$ is isomorphic to $\bp^1$, and in this case we show that $\nn_X$ is
smooth at stable Galois bundles and that the locus of
nonstable bundles can be identified with the moduli
space $\ss{\cal U}}\def\vv{{\cal V}}\def\xx{{\cal X}}\def\ww{{\cal W}_{X/\sigma}(2)$ of rank 2, trivial determinant vector bundles
on the quotient of $X$ by the involution $\sigma \in S_3$ which exchanges
the spinor representations (see \ref{notation} for notation).
Both properties are in marked contrast to
$\mm_X({\rm Spin}(8))$ itself, whose semistable boundary is more
complicated to describe
and whose stable points are not necessarily smooth but may have
finite quotient singularities.
\medskip
The condition $X/S_3 \cong \bp^1$ just means that $X = {\cal G} (C)$
is the Galois closure over $\bp^1$
of the trigonal curve $C = X/\sigma$, and conversely
one can construct $\nn_X$ starting from any
(non-cyclic) trigonal curve $C\map{3:1} \bp^1$. Indeed, this was
precisely our motivation
for the construction, and we shift attention to the trigonal point of view in
\S\ref{galois3}. The curve
$X = {\cal G} (C)$ is a branched double cover of $C$; we write
$\nn_C = \nn_X$; and the main result of the paper is the following:
\statement Theorem \ref{dimtheorem}. Given a (non-cyclic) trigonal curve
$C\map{3:1}
\bp^1$ there exists a projective moduli space $\nn_C$
(parametrising Galois ${\rm Spin}(8)$-bundles on ${\cal G}(C)$)
which admits an
inclusion ${\cal SU}_C(2) \hookrightarrow \nn_C$ such that $\nn_C$ is smooth and of
dimension $7g-14$ away from the image of ${\cal SU}_C(2)$.
(Cyclic trigonal curves need to be treated separately as the Galois group
is $\z/3$ rather than $S_3$, and although many of the computations work well
here---and we expect that the above theorem remains true---we
confine ourselves to brief remarks on this case at various points---see
the end of \S\ref{hypcase}.)
\medskip
The moduli space ${\cal SU}_C(2)$ rank 2 semistable vector bundles with trivial
determinant is an object of considerable interest.
As well as containing the Jacobian Kummer as its singular locus,
a fundamental geometric feature
is the so-called {\it Schottky configuration} of Prym Kummers.
(See for example \cite{vGP}, \cite{NR3}.)
That is, the group $J_C[2]$ of 2-torsion points in the Jacobian acts on
${\cal SU}_C(2)$ by tensor product; the fixed-point set of an element $\eta \in J_C[2]$
is a pair of (isomorphic)
Kummer varieties $(P_{\eta}/\pm) \cup (P^-_{\eta}/\pm) $ of dimension
$g-1$. $P_{\eta}$ is the Prym variety of the unramified double cover
$C_{\eta} \rightarrow C$ corresponding to $\eta$, and the map
$P_{\eta} \cup P^-_{\eta}\rightarrow {\cal SU}_C(2)$
is (up to a choice of $\eta^{1\over 2}$) the
direct image of line bundles from $C_{\eta}$. The incidence
relations among these Kummer varieties as $\eta$ varies, interpreted via
the embedding ${\cal SU}_C(2) \hookrightarrow |2\Theta|$ (where $\Theta$ is the Riemann
theta divisor in the Jacobian) correspond precisely to the Schottky-Jung-Donagi
identities among the thetanulls of $J_C$ and the Pryms.
\medskip
It turns out that when $C$ is trigonal
the Schottky configuration too has a `fattening' in
the moduli space $\nn_C$. By a beautiful and well-known construction of
Recillas \cite{Rec}, each trigonal Prym $P_{\eta}$ is isomorphic as a
principally polarised abelian variety to the Jacobian of a tetragonal
curve $R_{\eta} \map{4:1} \bp^1$. Thus a trigonal Schottky configuration
consists of Jacobian Kummers $J_{R_{\eta}}/\pm$. Each of these
is also the singular locus of a moduli space of bundles $\ss{\cal U}}\def\vv{{\cal V}}\def\xx{{\cal X}}\def\ww{{\cal W}_{R_{\eta}}(2)$.
We shall show:
\statement Theorem \ref{nonab-schottky}. Given a trigonal curve $C$ and
a nonzero 2-torsion point $\eta \in J_C[2]$ there exists
(up to a choice of $\eta^{1\over 2}$) a natural map $\ss{\cal U}}\def\vv{{\cal V}}\def\xx{{\cal X}}\def\ww{{\cal W}_{R_{\eta}}(2)
\hookrightarrow\nn_C$
for which the following diagram commutes:
$$
\begin{array}{ccc}
P_{\eta}& \mathbin{\hbox{$\,\widetilde\rightarrow\,$}} J_{R_{\eta}} \rightarrow &\ss{\cal U}}\def\vv{{\cal V}}\def\xx{{\cal X}}\def\ww{{\cal W}_{R_{\eta}}(2)\\
&&\\
\downarrow && \downarrow\\
&&\\
\ss{\cal U}}\def\vv{{\cal V}}\def\xx{{\cal X}}\def\ww{{\cal W}_C(2)&\hookrightarrow& \nn_C.\\
\end{array}
$$
One can view the right-hand side of this diagram, as $\eta$ varies,
as a nonabelian
Schottky configuration singular along the classical Schottky configuration
on the left-hand side.
\medskip
Finally, we wish to add a word about the motivation for these constructions.
Originally, this was our interest
the projective embedding (see \cite{BV}, \cite{vGI};
we assume $C$ is
nonhyperelliptic unless $g=2$)
$$
\phi: {\cal SU}_C(2) \rightarrow |2\Theta| = \bp^{2^g -1}.
$$
It is well-known that for $g=2$ the map $\phi$ is an isomorphism
${\cal SU}_C(2) \mathbin{\hbox{$\,\widetilde\rightarrow\,$}} \bp^3$ (see \cite{NR1});
and that for $g=3$ (see \cite{NR2}) the image $\phi({\cal SU}_C(2)) \subset \bp^7$
is the unique
Heisenberg-invariant quartic---the {\it Coble quartic}---singular
along the Kummer variety $J_C/\pm \subset
|2\Theta|$. When $g=4$ it was shown in \cite{OP} that in $|2\Theta| = \bp^{15}$
there exists a unique Heisenberg-invariant quartic singular along
the image $\phi({\cal SU}_C(2))$. It was shown, moreover, that the fixed-point set of
$\eta \in J_C[2]$ in this quartic is a pair of Coble quartics
$\ss{\cal U}}\def\vv{{\cal V}}\def\xx{{\cal X}}\def\ww{{\cal W}_{R_{\eta}}(2)$ of the corresponding Recillas curve. However, the
questions remained open: what is this 14-dimensional quartic as a moduli space?,
and is its singular locus equal to ${\cal SU}_C(2)$?
\medskip
The crucial point
here is that a general curve of genus 4 is trigonal (in two ways.
However, by Torelli's theorem $R_{\eta}$ is independent of the choice of
trigonal structure). In view of the results of the present paper
it is natural to expect that the genus 4 quartic in $\bp^{15}$ is $\nn_C$.
We hope to pursue this question in a sequel.
\medskip
\noindent
{\it Acknowledgments:}
The first author wishes to thank TIFR, Mumbai for its hospitality
during a visit in July 1998 when much of this work was carried out.
We are also grateful to Steve Wilson for pointing out Lemma
\ref{wilson}.
In writing this paper the authors were partially supported by EPSRC
grants GR/M03924 and GR/M36663.
\section{${\rm Spin}(8)$ and triality}
\label{spin}
We begin by recalling the triality story for ${\rm Spin}(8)$ from various points of
view. Roughly, \S\ref{group} will be used in the discussion of
Galois bundles of section \ref{trigonal};
\S\ref{2by2} and \S\ref{6quadrics} are needed for the
discussion of stability in section \ref{moduli}; and \S\ref{liealg} will be
used in the
computation of the dimension of the moduli space in section \ref{galois3}.
\subsection{The group ${\rm Spin}(8)$}
\label{group}
We shall always denote by
$V \cong \c^{2n}$ the standard orthogonal representation of
${\rm Spin}({2n})$, and by $S^{\pm} \cong \c^{2^{n-1}}$ its half-spinor
representations. We shall denote by $q$ the quadratic form on $V$;
if we fix a decomposition $V = N \oplus N^{\vee}$, where $N$ and
$N^{\vee}$ are maximal isotropic subspaces, dual via $q$, then by definition
\begin{equation}
\label{spinors}
\textstyle
S^{+} = \bigwedge^{\rm even}N,
\qquad
S^{-} = \bigwedge^{\rm odd}N.
\end{equation}
We need to recall some facts about
$S^{\pm}$
(see \cite{C}).
Start with the bilinear pairing
$$
\begin{array}{rcl}
r:\bigwedge N \otimes \bigwedge N & \rightarrow & \textstyle
\bigwedge^{n}N
\cong \c \\
s\otimes t & \mapsto & (\beta(s) \wedge t)_{n} \\
\end{array}
$$
where $\beta$ is the principal anti-involution of $\bigwedge N$, i.e.
it is the identity on $N$ and reverses multiplication; and where
$()_{n}$ denotes the component of top degree. When $n$ is odd this
form vanishes on each of $S^{\pm}$ and induces a nondegenerate
pairing $r:S^{+} \otimes S^{-} \rightarrow \c$. When $n$ is even it
restricts to a nondegenerate pairing on each of $S^{\pm}$, which is
symmetric precisely when $n\equiv 0(4)$.
We shall be
concerned with the case $n=4$;
here the pairing $r$ determines quadratic forms $q^{\pm}$ on
the spinor spaces $S^{\pm}$. Moreover, these determine an embedding
(in fact the projection into the product of any two factors is still
an embedding)
\begin{equation}
\label{spin8subgp}
{\rm Spin}(8) \hookrightarrow SO(V) \times SO(S^{+}) \times SO(S^{-}).
\end{equation}
There is a well-known triality relationship among
the three orthogonal factors here.
${\rm Spin}(8)$ has outer automorphism group $S_{3}$, and
we shall recall explicitly how $S_3$ acts on ${\rm Spin}(8)$, or more precisely
how to split the sequence
$$
1\rightarrow {\rm Inn}\ {\rm Spin}(8) \rightarrow {\rm Aut}\ {\rm Spin}(8)
\rightarrow S_3
\rightarrow 1.
$$
\begin{notat}\rm
\label{notation}
We shall use, here and in later sections,
generators $\sigma = (23)$ and $\tau = (123) \in S_{3}$;
thus $\sigma$ should interchange $S^{\pm}$, as an involution of
${\rm Spin}(8)$; while $\sigma \tau: V \leftrightarrow S^{+}$ and
$\sigma \tau^{2}: V \leftrightarrow S^{-}$.
\end{notat}
Choose unit vectors
$v_{0} \in V$, $s_{0}\in S^{+}$ and $t_{0}= v_{0} \cdot s_{0}\in
S^{-}$.
Given these choices we let $S_3$ act on the vector space $V\oplus
S^{+}\oplus S^{-}$ as follows.
$\sigma$ acts by reflection of $V$ in $v_{0}^{\perp}$ and by Clifford
multiplication $v_{0}: S^{+}\leftrightarrow S^{-}$. Similarly the
group elements $\sigma \tau$ and $\sigma \tau^{2}$ are represented by
$s_{0}$ and $t_{0}$ respectively.
Given these choices we shall denote the resulting representation by
$$
\rho : S_3 \rightarrow O(V\oplus S^+ \oplus S^-).
$$
This intertwines an $S_3$-action on
${\rm Spin}(8)$ by $g\mapsto g^u := \rho (u)^{-1}g
\rho(u)$ for $u\in S_3$ and $g \in
{\rm Spin}(8)\subset O(V\oplus S^+ \oplus S^-)$. Note that the subgroup
of $O(V\oplus S^+ \oplus S^-)$ generated by the images of $S_3$ and
${\rm Spin}(8)$ is isomorphic to the semi-direct product ${\rm Spin}(8) \mathbin{\hbox{\twelvesymb o}}
S_3$.
We note for later use that the group law on ${\rm Spin}(8) \mathbin{\hbox{\twelvesymb o}} S_3$ is
\begin{equation}
\label{sdp}
(g,u)(h,v)= (gh^u,uv)
\qquad
u,v \in S_3
\quad
g,h \in {\rm Spin}(8).
\end{equation}
In view of triality, there is a
multiplication map $S^{+}\otimes S^{-} \rightarrow V$
permuted by $S_3$ with the Clifford multiplications
$V\otimes S^{\pm} \rightarrow S^{\mp}$.
This can be
described in terms of a trilinear form:
\begin{equation}
\label{trilinear}
\begin{array}{rcl}
c: V\otimes S^{+}\otimes S^{-} & \rightarrow & \c \\
v\otimes s\otimes t & \mapsto & q^{-}(v\cdot s,t) = q^{+}(v\cdot
t,s)
\mathrel{\lower3pt\hbox{$\buildrel{\rm def}\over=$}} q(s\cdot t,v).
\end{array}
\end{equation}
The first equation is easily checked; the second defines the
Clifford multiplication by:
\begin{displaymath}
\begin{array}{ccc}
S^{+}\otimes S^{-} & \map{c} & V^{\vee} \\
&&\\
&.\searrow & \downarrow q \\
&&\\
& & V
\end{array}
\end{displaymath}
\begin{rem}\rm
One can use the three Clifford multiplications to define a
commutative algebra
structure on $V\oplus S^+ \oplus S^-$ by taking the product of two
vectors
in the same summand to be zero. The resulting algebra, called the
{\it Chevalley algebra}, is an example of a vertex operator algebra.
(See
\cite{FFH}.)
\end{rem}
\begin{lemm}
\label{outerautos}
The orthogonal action $\rho: S_{3} \rightarrow O(V\oplus
S^{+}\oplus S^{-})$
preserves the cubic form induced by the
trilinear form $c$, and hence also the Chevalley algebra structure,
and determines an $S_3$-action on
${\rm Spin}(8)$ as above. Moreover, every splitting $S_3 \rightarrow {\rm
Aut}\
{\rm Spin}(8)$ arises in this way.
\end{lemm}
\begin{rem}\rm
\label{oct1}
Given the above choices $V$ acquires an algebra structure
$$
\begin{array}{rcl}
V\times V & \rightarrow & S^{+}\times S^{-} \map{\cdot} V \\
(u,v) & \mapsto & (t_{0}\cdot u,s_{0}\cdot v)
\end{array}
$$
It is well-known
(and will follow from Remark \ref{oct2} below) that
this is precisely the (complex) Cayley algebra $V\cong {\mathbb O}}\def\bp{{\mathbb P}}\def\ratq{{\mathbb Q}$ with centre
$\<v_{0}\>$.
One deduces from this fact a characterisation of the image of
(\ref{spin8subgp}):
\begin{equation}
\label{spin8subgp2}
{\rm Spin}(8) = \{ (a,b,c)\in SO({\mathbb O}}\def\bp{{\mathbb P}}\def\ratq{{\mathbb Q}) \times SO({\mathbb O}}\def\bp{{\mathbb P}}\def\ratq{{\mathbb Q}) \times SO({\mathbb O}}\def\bp{{\mathbb P}}\def\ratq{{\mathbb Q}) \ |\
a(u)b(v) = c(uv)
\quad
\forall \ u,v\in {\mathbb O}}\def\bp{{\mathbb P}}\def\ratq{{\mathbb Q} \}.
\end{equation}
In this language the triality action is given by $\sigma:
(a,b,c)\mapsto (a',c',b')$ and $\tau: (a,b,c)\mapsto (b',c,a')$ where
$a'(u) = \overline{a(\overline u)}$.
\end{rem}
\begin{lemm}
\label{zero-norm}
For an isotropic subspace $U\subset V$ and an involution $g\in S_3$
not preserving $V$, consider the Clifford
multiplication maps:
$$
\mu: U\otimes U \map{{\rm id}\otimes g} V \otimes S^{\pm}
\map{\cdot} S^{\mp}
$$
If $\dim U =1$ then $\mu =0$, and if $\dim U =2$ then ${\rm rank \ } \mu \leq 1$.
\end{lemm}
{\it Proof.}\
Suppose first that $\dim U =1$.
Identifying $V,S^+,S^-$ with ${\mathbb O}}\def\bp{{\mathbb P}}\def\ratq{{\mathbb Q}$ as in Remark \ref{oct1} the map $\mu$
becomes $x\otimes x\mapsto x\overline x = \| x\| =0$. (See also Remark
\ref{oct2}). If $\dim U =2$ then the symmetric tensors $S^2 U \subset
U\otimes U$ are spanned by squares, and so are contained in kernel of
$\mu$. This shows that ${\rm rank \ } \mu \leq 1$.
{\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par}
Finally, note that the centre of ${\rm Spin}(8)$ is $Z({\rm Spin}(8)) = \z/2
\times \z /2$, consisting of the matrices of
$8\times 8$ blocks:
$$
\pmatrix{1&&\cr &1& \cr &&1\cr},
\pmatrix{1&&\cr &-1& \cr &&-1\cr},
\pmatrix{-1&&\cr &1& \cr &&-1\cr},
\pmatrix{-1&&\cr &-1& \cr &&1\cr}.
$$
\subsection{Triality in terms of $2\times 2$ matrices}
\label{2by2}
We can always split $V= \c^8$, as an orthogonal space,
into a direct sum of orthogonal
$\c^4$s. In turn $\c^4$ can be identified with ${\rm Hom}(A,B)$
where $A,B \cong \c^2$, with orthogonal structure given by the
determinant (geometrically, every smooth complex quadric surface is
the Segre $\bp^1 \times \bp^1$). The spaces $A^{\vee}$ and $B$ are then
the spinor representations, where ${\rm Spin}(4) = SL(2) \times SL(2)$.
We can view this as saying that $A,B$ carry fixed (complex)
orientations
$\lambda_A: \bigwedge^2 A \mathbin{\hbox{$\,\widetilde\rightarrow\,$}} \c$ and
$\lambda_B: \bigwedge^2 B \mathbin{\hbox{$\,\widetilde\rightarrow\,$}} \c$.
Let us generalise this situation for a moment. Suppose that $A,B
\cong \c^n$ and are each equipped with a fixed orientation. Then any
$u\in {\rm Hom}(A,B)$ has an {\it adjugate} homomorphism $\overline u \in
{\rm Hom}(B,A)$. This is just the transpose of the composition
$$
\textstyle
A^{\vee} \map{\lambda_A} \bigwedge^{n-1} A \map{\bigwedge^{n-1} u}
\bigwedge^{n-1} B \map{\lambda_B} B^{\vee},
$$
and with respect to chosen bases is given by the transpose matrix of
signed cofactors of the matrix representing $u$. Adjugacy is
therefore a natural birational
involution ${\rm Hom}(A,B) \leftrightarrow {\rm Hom}(B,A)$
for oriented vector spaces; when $n=2$ it is a linear involution.
(The reader may care to consider the next case $n=3$: projectively
the map blows up the Segre $\bp^2 \times \bp^2$ and contracts secant
lines down to the dual Segre fourfold.)
In the case $n=2$ the determinant of a map between oriented spaces
can be interpreted as
a quadratic form $\det: {\rm Hom}(A,B) \rightarrow \c$ whose polarisation is the
symmetric bilinear form
$$
\<u,v\> = {1\over 2} {\rm trace}\ u\overline v,
\qquad
u,v \in {\rm Hom}(A,B).
$$
We now return to our 8-dimensional orthogonal space, which we shall
decompose as
$$
V= {\rm Hom}(A,B) \oplus {\rm Hom}(C,D)
$$
where $A,B,C,D$ are oriented $\c^2$s.
It follows from Lemma \ref{spin-sum}, and the identifications $A\mathbin{\hbox{$\,\widetilde\rightarrow\,$}}
A^{\vee}$ etc given by the orientations, that the spinor spaces are
$$
\begin{array}{rcl}
S^+ & = & {\rm Hom}(B,C) \oplus {\rm Hom}(A,D), \\
S^- & = & {\rm Hom}(C,A)\oplus {\rm Hom}(B,D).
\end{array}
$$
It is straightforward to calculate the Clifford multiplication maps:
we shall take on $V$ the quadratic form $(a,b) \mapsto \det a - \det b$.
One then finds:
$$
\begin{array}{rcl}
V\otimes S^+ & \rightarrow & S^- \\
(a,b)\otimes (x,y) & \mapsto &
(\overline{a}\ \overline{x} +\overline{y}
b, bx + y\overline{a})
\end{array}
$$
with similar expressions for the other two maps. (The rule is: for
each of the two Hom summands, simply add the only possible
composites, allowing adjugates, so that the expression is well-defined.)
\begin{rem}\rm
\label{rankdrops}
The orientations on the spaces $A,B,C,D$ induce identifications
$\bigwedge^8 V \cong \bigwedge^8 S^+ \cong \bigwedge^8 S^-$, so the
determinant of each Clifford multiplication map is a well-defined scalar.
For each pair of $2\times 2$ matrices $(a,b) \in V = {\rm Hom}(A,B)\oplus
{\rm Hom}(C,D)$ one
finds that the linear map $m_{a,b} : S^+ \rightarrow
S^-$ satisfies:
$$
\det m_{a,b} = -(\det a - \det b)^4.
$$
This corresponds to the fact that the rank drops by 4 if
$\det a = \det b$, i.e. if $(a,b)\in V$ is isotropic. One can check this
directly (e.g. using Maple) by Gauss-Jordan elimination.
\end{rem}
\begin{rem}\rm
\label{oct2}
If we fix unimodular isomorphisms $A\mathbin{\hbox{$\,\widetilde\rightarrow\,$}} B \mathbin{\hbox{$\,\widetilde\rightarrow\,$}} C \mathbin{\hbox{$\,\widetilde\rightarrow\,$}} D$
we obtain a natural choice
of unit vectors $v_{0} = (1,0) \in V$, $s_{0} = (1,0) \in S^+$,
$t_{0} = (1,0) \in S^-$. (Note that $t_0 = v_0 s_0$ under the Clifford
multiplication above.) With this choice
the algebra structure of Remark \ref{oct1} becomes:
$$
\begin{array}{rcl}
V\times V & \rightarrow & V \\
(a,b)(c,d) & \mapsto & (ac+\overline{d} b,da+b\overline{c}).
\end{array}
$$
On the other hand, the chosen isomorphisms identify $V = {\rm End}\ A
\oplus {\rm End}\ A$ where ${\rm End}\ A \cong {\rm Mat}_{2}(\c) \cong
\h \otimes_{{\mathbb R}}\def\t{{\mathbb T}}\def\z{{\mathbb Z}}\c$, the complexification of the quaternions. Moreover,
quaternionic conjugation is precisely the map which sends a matrix to
its adjugate.
It follows that the above algebra structure on $V$ is exactly the
(split) Cayley-Dickson process applied to $\h \otimes_{{\mathbb R}}\def\t{{\mathbb T}}\def\z{{\mathbb Z}}\c$ (see
\cite{harvey} pp.105--106), i.e. identifies $V$ with ${\mathbb O}}\def\bp{{\mathbb P}}\def\ratq{{\mathbb Q}$.
\end{rem}
\subsection{6-dimensional quadrics}
\label{6quadrics}
Let $Q\subset
V$ and $Q^{\pm} \subset S^{\pm}$ be the quadrics
defined by the respective quadratic
forms on the basic representations of ${\rm Spin}(8)$.
Each quadric has two families of 4-dimensional isotropic
spaces, which we shall sometimes refer to as {\it $\alpha$-planes}
and
{\it $\beta$-planes}.
Such subspaces $A,B \subset Q$ belong to
opposite families if and only if
$$
\dim A\cap B \equiv 1
\quad
{\rm mod}\ 2.
$$
Moreover, these families are parametrised precisely by the quadrics $Q^{\pm}$
as follows.
Given vectors $s\in S^+$ and $t\in S^-$, consider the Clifford multiplication
maps $m_s: V \leftrightarrow S^-$ and $m_t: V \leftrightarrow S^+$. We have
observed in Remark \ref{rankdrops} that these maps have nonzero kernel
exactly when $s,t$ respectively are isotropic, and that then the rank is 4.
We define:
$$
\begin{array}{rcl}
A_s &=& {\rm ker \ } \{V\map{m_s} S^-\} = {\rm Im\ } \{S^- \map{m_s} V\} \subset Q \subset V,\\
B_t &=& {\rm ker \ } \{V\map{m_t} S^+\} = {\rm Im\ } \{S^+ \map{m_t} V\} \subset Q \subset V.\\
\end{array}
$$
One readily checks (see \cite{C}, or use the set-up of of \S\ref{2by2})
these equalities, that they define isotropic subspaces, and that:
$$
\dim A_s \cap B_t =
\cases{1& spanned by $s\cdot t \in V$ if $ s\cdot t \not=0$,\cr
3& if $ s\cdot t =0$.\cr}
$$
In particular $A_s, B_t$ are in opposite families.
By triality
(i.e. using (\ref{trilinear}) in \S\ref{spin}) we can make
the same constructions in each of $S^{\pm}$. We summarise our notation
in the following diagram:
\begin{equation}
\label{quad-pic}
\leavevmode
\epsfxsize=3in\epsfbox{quadrics.eps}
\end{equation}
Note that:
\begin{equation}
\label{prod=0}
s\cdot t =0
\quad \Longleftrightarrow\quad
s\in A_t
\quad \Longleftrightarrow\quad
t\in B_s
\quad \Longleftrightarrow\quad
\dim A_s \cap B_t = 3.
\end{equation}
Moreover, there are canonical dualities $A_s = B_s^{\vee}$ for $s\in Q^\pm$
or $Q$; this follows, for example, via the quadratic form on $V$, from
the exact sequences given by Clifford multiplication:
\begin{equation}
\label{ABsequences}
\begin{array}{c}
\ses{A_s}{V}{B_s},\\
\ses{B_t}{V}{A_t}.\\
\end{array}
\end{equation}
Geometrically, of course, $A_t$ is the set of $s\in Q^+$ for which
$\dim A_s \cap B_t = 3$, and so parametrises hyperplanes in $B_t$.
Similarly, for distinct $s,s' \in Q^+$ we have
\begin{equation}
\label{polarity}
\dim A_s \cap A_{s'} =
\cases{2 & if $\<s,s'\> =0$,\cr
0 & otherwise.\cr}
\end{equation}
Note that this fact identifies the Grassmannian ${\rm Grass}(2,A_s)$ with the
4-dimensional (tangent) quadric in $\bp (s^\perp /s)$.
We can generalise the above correspondence by defining,
for any isotropic subspace $U\subset V$, and using the
trilinear form (\ref{trilinear}):
$$
\begin{array}{rcl}
A_U = {\rm ker \ }\{S^- \map{c} (U\otimes S^+)^{\vee}\} & =&
{\rm Im\ } \{U\otimes S^+ \map{c} (S^-)^{\vee}\}^{\perp}, \\
B_U = {\rm ker \ }\{S^+ \map{c} (U\otimes S^-)^{\vee}\} & =&
{\rm Im\ } \{U\otimes S^- \map{c} (S^+)^{\vee}\}^{\perp}.\\
\end{array}
$$
Note that $A_U = \bigcap_{v\in U}A_v \subset Q^-$ unless $U$ is
an $\alpha$-plane $A_s$, in which case $A_U = B_s$---with similar
remarks holding for $B_U$. If $\dim U = 1,2,3$ then $\dim A_U = \dim B_U =
4,2,1$ respectively; if $U = B_t$ is a $\beta$-plane then $A_{U}$ is
1-dimensional and is spanned by $t \in S^-$.
By triality one can now
extend the notation of diagram (\ref{quad-pic}) to arbitrary {\it isotropic
subspaces} $v,s,t$. The following properties are essentially tautological:
\begin{lemm}
\label{AB1}
For any isotropic subspace $U\subset V$, $S^+$ or $S^-$, with $\dim U \not= 3$,
we have:
\par\indent \hangindent2\parindent \textindent{(i)}
$A_{B_U} = B_{A_U} =U$;
\par\indent \hangindent2\parindent \textindent{(ii)}
$A_{A_U} = B_U$ and $B_{B_U} = A_U$.
\end{lemm}
\begin{lemm}
\label{AB2}
Suppose that $U,R \subset V\oplus S^+ \oplus S^-$ are 2-dimensional
isotropic subspaces contained in different summands, and that
Clifford multiplication vanishes on $U\otimes R$. Then either $U=A_R$, $R=B_U$
or $U=B_R$, $R=A_U$.
\end{lemm}
Finally, fix a triality action $\rho :S_3 \rightarrow O(V\oplus S^+
\oplus S^-)$
as in \S\ref{group}.
\begin{lemm}
\label{AB3}
For any $g\in S_3$ and isotropic subspace $U\subset V$, $S^+$ or $S^-$
we have:
$$
g A_U = \cases{A_{gU}&\cr B_{gU}&\cr}
\qquad
g B_U = \cases{B_{gU}& if ${\rm sgn}\ g = +1$,\cr A_{gU}& if ${\rm sgn}\
g = -1$.\cr}
$$
\end{lemm}
{\it Proof.}\
Suppose that $U\subset V$ (the argument for $U\subset S^\pm$ is the same).
Then $A_U = \{ t\in S^- | c(u,s,t)=0\forall s\in S^+, u\in U \}$
using the trilinear form (\ref{trilinear}). Since this form is $S_3$-invariant
$gA_U$ consists of $x=gt \in V,S^+$ or $S^-$ (depending on $g$) such that
$c(gu,gs,x) =0$ for all $u\in U$, $s\in S^+$ (where we temporarily
disregard the order of
the arguments of $c$), i.e. $c(u',s',x)=0$ for all $u' \in gU$ and $s' \in
gS^+$. This shows that $gA_U = A_{gU}$ or $B_{gU}$; the assertion
that which one depends on
${\rm sgn}\ g$ follows easily from diagram (\ref{quad-pic}).
{\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par}
\subsection{Triality action on the Lie algebra}
\label{liealg}
We need next to understand how the Lie algebra
${\mathfrak g}}\def\gothh{{\mathfrak h} = {\mathfrak so}(8)$
decomposes under the action of $S_3$. We write ${\mathfrak g}}\def\gothh{{\mathfrak h} = \gothh \oplus
{\mathfrak g}}\def\gothh{{\mathfrak h}_+ \oplus {\mathfrak g}}\def\gothh{{\mathfrak h}_-$ where $\gothh$ is a Cartan subalgebra
with orthonormal basis ${\bf e}}\def\boldf{{\bf f}_1,{\bf e}}\def\boldf{{\bf f}_2,{\bf e}}\def\boldf{{\bf f}_3,{\bf e}}\def\boldf{{\bf f}_4$
and ${\mathfrak g}}\def\gothh{{\mathfrak h}_{\pm}$ are spanned by the positive and negative root
spaces
with respect to simple roots:
$$
\textfont2=\dynkfont\dddiv{\boldf_1={\bf e}}\def\boldf{{\bf f}_1 - {\bf e}}\def\boldf{{\bf f}_2}.{\boldf_0={\bf e}}\def\boldf{{\bf f}_2
- {\bf e}}\def\boldf{{\bf f}_3}.{\boldf_3={\bf e}}\def\boldf{{\bf f}_3 +
{\bf e}}\def\boldf{{\bf f}_4}.
{\boldf_2={\bf e}}\def\boldf{{\bf f}_3 - {\bf e}}\def\boldf{{\bf f}_4}.
$$
The 24 roots of ${\mathfrak g}}\def\gothh{{\mathfrak h}$ are $\pm
{\bf e}}\def\boldf{{\bf f}_{i} \pm {\bf e}}\def\boldf{{\bf f}_{j}$.
The group $S_3$ acts on the Dynkin diagram, and its action on ${\mathfrak g}}\def\gothh{{\mathfrak h}$ is
obtained by
examining the action on the root spaces. This can be made transparent by
rewriting the 12 positive roots as:
$$
\begin{array}{llll}
\boldf_1, & \boldf_0+\boldf_1, & \boldf_0+\boldf_2+\boldf_3,
& \boldf_0, \\
\boldf_2, & \boldf_0+\boldf_2, & \boldf_0+\boldf_1+\boldf_3,
&\boldf_0+\boldf_1+\boldf_2+\boldf_3, \\
\boldf_3, & \boldf_0+\boldf_3, &\boldf_0+\boldf_1+\boldf_2, &
2\boldf_0+\boldf_1+\boldf_2+\boldf_3.
\end{array}
$$
\medskip\noindent
\underline{\it $\sigma$-action}.
From the Dynkin diagram,
the involution $\sigma$ acts on $\gothh$ by ${\bf e}}\def\boldf{{\bf f}_{4}
\leftrightarrow -{\bf e}}\def\boldf{{\bf f}_{4}$, fixing the other ${\bf e}}\def\boldf{{\bf f}_{i}$. Thus
$\gothh = \three^{+} \oplus {\bf 1}}\def\two{{\bf 2}}\def\three{{\bf 3}}\def\four{{\bf 4}^{-}$ where the superscript denotes
the eigenvalue of each summand. Alternatively, $\sigma$ interchanges
$\boldf_2$, $\boldf_3$ and fixes $\boldf_0$, $\boldf_1$.
From this it follows that ${\mathfrak g}}\def\gothh{{\mathfrak h}_{\pm} =
\six^{+} \oplus (\three^{+} \oplus \three^-)$, and hence
$$
{\mathfrak so}(8) = \two{\bf 1}}\def\two{{\bf 2}}\def\three{{\bf 3}}\def\four{{\bf 4}^+ \oplus \seven^-
\quad
\hbox{under the action of $\sigma$.}
$$
\begin{rem}\rm
Alternatively, one can see this from the fact that ${\mathfrak so}(8) \cong\bigwedge^2
V$ where $\sigma$ acts on $V = \c^8$ by reflection in a hyperplane
$H\subset V$. Then $\two{\bf 1}}\def\two{{\bf 2}}\def\three{{\bf 3}}\def\four{{\bf 4}^+ = \bigwedge^2 H$ and $\seven^- = H$.
\end{rem}
\medskip\noindent
\underline{\it $\tau$-action}.
$\tau$ cyclically permutes
$\boldf_1,\boldf_2,\boldf_3$ and fixes
$\boldf_0$, so that
$\gothh = \two \oplus
{\bf 1}}\def\two{{\bf 2}}\def\three{{\bf 3}}\def\four{{\bf 4}^{\omega}\oplus {\bf 1}}\def\two{{\bf 2}}\def\three{{\bf 3}}\def\four{{\bf 4}^{\omega^2}$ where $\omega = e^{2\pi i/3}$.
For ${\mathfrak g}}\def\gothh{{\mathfrak h}_{\pm}$, we observe that
the entries of each of the first three columns of the array above
are cyclically permuted by $\tau$, while those of the last column are
invariant. It follows that each of
${\mathfrak g}}\def\gothh{{\mathfrak h}_{\pm} = \six \oplus \three^{\omega} \oplus \three^{\omega^2}$,
and hence:
$$
{\mathfrak so}(8) = {\bf 1}}\def\two{{\bf 2}}\def\three{{\bf 3}}\def\four{{\bf 4} \four \oplus \seven^{\omega} \oplus \seven^{\omega^2}
\quad
\hbox{under the action of $\tau$.}
$$
\begin{rem}\rm
The invariant 14-dimensional subspace is the Lie algebra of $G_2 =
{\rm Aut}\ {\mathbb O}}\def\bp{{\mathbb P}}\def\ratq{{\mathbb Q}$. This is well-known, and follows from (\ref{spin8subgp2}).
\end{rem}
\section{Galois ${\rm Spin}(8)$ bundles}
\label{trigonal}
After some some preliminary remarks on spin bundles in
\S\ref{spinbundles} we shall introduce the main objects of this paper,
Galois ${\rm Spin}(8)$-bundles, in \S\ref{bunS3} (see Definition
\ref{definition}).
\subsection{Spin bundles}
\label{spinbundles}
We next consider principal
$G$-bundles
$F\rightarrow X$ where $X$ is a curve and $G= {\rm Spin}(2n)$ an even
complex spin group. Given a representation $\rho : G \rightarrow
SL(W)$ we can form a vector bundle $W_F = F \times_{\rho} W$.
\begin{lemm}
\label{spin-sum}
If $F_1$ is a ${\rm Spin}(2m)$-bundle and $F_2$ is a ${\rm Spin}(2n)$-bundle,
then
there is a ${\rm Spin}(2m+2n)$-bundle $F_1 + F_2$ with
$$
\begin{array}{rcl}
V_{F_1 + F_2} &=& V_{F_1} \oplus V_{F_2},\\
S^+_{F_1 + F_2} &=& S^+_{F_1}\otimes S^+_{F_2} \oplus
S^-_{F_1}\otimes S^-_{F_2},\\
S^-_{F_1 + F_2} &=& S^+_{F_1}\otimes S^-_{F_2} \oplus
S^-_{F_1}\otimes S^+_{F_2}.\\
\end{array}
$$
\end{lemm}
\begin{rem}\rm
More generally,
${\rm Nm}(F_1 + F_2) = {\rm Nm}(F_1) \otimes {\rm Nm}(F_2)$ if $F_1$ and $F_2$ are Clifford
bundles and ${\rm Nm}$ is the spinor norm.
\end{rem}
{\it Proof.}\
Since the spin groups are simply connected the inclusion $SO(2m)\times
SO(2n) \hookrightarrow SO(2m+2n)$, $(g,h) \mapsto g\oplus h$ has a
unique lift $\lambda$ making the following diagram commute:
$$
\begin{array}{ccc}
{\rm Spin}(2m)\times {\rm Spin}(2n) & \map{\lambda} & {\rm Spin}(2m+2n) \\
&&\\
4:1\downarrow & & \downarrow 2:1 \\
&&\\
SO(2m)\times SO(2n) & \hookrightarrow & SO(2m+2n).\\
\end{array}
$$
The bundle $F_1 +F_2$ is obtained by applying $\lambda$ to transition
functions of $F_1$ and $F_2$ with respect to a sufficiently fine open
cover of the curve. The identification $V_{F_1 + F_2} =V_{F_1} \oplus
V_{F_2}$ is then immediate from the construction.
Denote by $V_{1}= N_{1}\oplus N_{1}^{\vee}$ and $V_{2}= N_{2}\oplus
N_{2}^{\vee}$
the orthogonal representations of ${\rm Spin}(2m)$ and ${\rm Spin}(2n)$
respectively, decomposed into
maximal isotropic subspaces. Then $N= N_{1}\oplus N_{2}$ is maximal
isotropic in $V = V_{1}\oplus V_{2}$ and
$$
\begin{array}{ccc}
\bigwedge^{\rm even}N & = &
\bigwedge^{\rm even}N_{1}\otimes \bigwedge^{\rm even}N_{2}
\oplus
\bigwedge^{\rm odd}N_{1}\otimes \bigwedge^{\rm odd}N_{2}, \\
\bigwedge^{\rm odd}N & = & \bigwedge^{\rm even}N_{1}\otimes
\bigwedge^{\rm odd}N_{2}
\oplus
\bigwedge^{\rm odd}N_{1}\otimes \bigwedge^{\rm even}N_{2}.
\end{array}
$$
From this it follows that the spinor bundles of $F_1 + F_2$ are as
asserted.
{\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par}
If $P\subset G$ is a parabolic subgroup we can form the bundle of
homogeneous
spaces $F/P$ with fibre $G/P$. The $G$-bundle $F$ is said to be {\it
stable} (resp.{\it semistable}) if for every maximal parabolic
$P\subset G$ and every section $s: X \rightarrow F/P$ one has
$$
\deg s^* T^{\rm vert}_{F/P} > 0
\qquad
\hbox{(resp. $\geq 0$)}
$$
where $T^{\rm vert}$ denotes the vertical tangent bundle. (See
\cite{Rth}.)
In the case of $G= {\rm Spin}(2n)$ there are (up to conjugacy) $n$
maximal parabolics and the spaces $G/P$ are the grassmannians of
isotropic subspaces
$U\subset V$ of dimension $d=1,\ldots, n-2$ and the two spinor
varieties of isotropic subspaces of dimension $d=n$. A section $s: X
\rightarrow F/P$
corresponds to an isotropic subbundle $\underline U \subset V_F$ and
it is easy to check that the stability condition reduces to the slope
inequality
$$
\mu(\underline U) < \mu(V_F)
\qquad
\hbox{
$\forall$ isotropic subbundles $\underline U \subset V_F$.}
$$
In other words, $F$ is a (semi)stable ${\rm Spin}(2n)$-bundle if and only
if $V_F$
is (semi)stable as an orthogonal vector bundle. (See also \cite{O},
Lemma 1.2.)
Recall that, just as for vector bundles, the moduli problem for spin
bundles requires an equivalence relation on semistable bundles
coarser than isomorphism, called S-equivalence. For the general
definition in the context of principal bundles we refer to \cite{KNR};
for our purposes the following remarks will be sufficient.
Suppose that $W$ is an orthogonal vector bundle with a
filtration
$$
U_1 \subset U_2 \subset \cdots \subset U_k \subset U_k^{\perp}\subset \cdots
U_2^{\perp}\subset U_1^{\perp}\subset W,
$$
where $U_{1},\ldots,U_k$ are destabilising subbundles (i.e. isotropic
of degree 0) and each `quotient'
$$
(U_{i+1}/U_i)\oplus (U_{i}^{\perp}/U_{i+1}^{\perp})
\cong
(U_{i+1}/U_i)\oplus (U_{i+1}/U_{i})^{\vee}
$$
is a stable orthogonal bundle, with its natural orthogonal structure
(we include $U_0 = 0$). Clearly such a filtration always exists, and
$k=0$ if and only if $W$ is stable. The graded orthogonal bundle
$$
{\rm gr}(W) = \bigoplus_{i=0}^{k-1}\left\{ (U_{i+1}/U_i)\oplus (U_{i+1}/U_{i})^{\vee}
\right\}\oplus U_k^{\perp}/U_k
$$
is independent of the filtration, and two orthogonal bundles $W,W'$ of the
same rank are said to be S-equivalent if and only if their
graded bundles are isomorphic as orthogonal bundles:
$$
W \sim W'
\qquad \Longleftrightarrow \qquad
{\rm gr}(W) \cong {\rm gr}(W').
$$
Recall that by \cite{R} Proposition 4.5 $V_F$ is stable as an
orthogonal bundle if and only if $V_F$ is polystable as a
vector bundle, i.e.
$$
V_F = V_1 \oplus \cdots \oplus V_k,
$$
where the summands $V_i$ are stable as vector bundles and nonisomorphic.
Let us now restrict to the case of ${\rm Spin} (8)$,
and suppose that the spinor bundles are polystable of the same shape,
that is, with stable summands of the same ranks as the $V_i$:
$$
\begin{array}{rcl}
S^+_F &=& S^+_1 \oplus \cdots \oplus S^+_k,\\
S^-_F &=& S^-_1 \oplus \cdots \oplus S^-_k.\\
\end{array}
$$
In this situation we shall need later on to understand the automorphism group
of the spin bundle $F$. This is determined by its action on the three
bundles above; by stability and orthogonality the automorphism group of
each of these is $\mu_2 ^k$ where $\mu_2 = \{\pm 1\}$. ${\rm Aut}\ F$ is then
described by the following diagram with exact rows:
\begin{equation}
\label{aut}
\begin{array}{ccccccc}
0\rightarrow &\mu_2& \map{\beta} & {\rm Aut}\ F & \map{\alpha} & (\mu_2)^k &
\rightarrow 0\\
&&&&&&\\
&\|&&\qquad \uparrow {\rm inclusion}&&\qquad \uparrow {\rm diagonal}&\\
&&&&&&\\
0\rightarrow &\mu_2& \rightarrow & Z({\rm Spin}(8)) & \rightarrow & \mu_2 &
\rightarrow 0.\\
\end{array}
\end{equation}
The map $\alpha$ is the representation on $V_{F}$, while $\beta(-1)$
acts as $-1$ on each of $S_F^{\pm}$ and as $+1$ on $V_F$.
Explicitly, the elements of ${\rm Aut}\ F$ are the $4\times 2^{k-1} = 2^{k+1}$
matrices (acting on
$V_F\oplus S_F^+ \oplus S_F^-$)
\begin{equation}
\label{split-autos}
\pmatrix{\varepsilon&&\cr &\varepsilon& \cr &&\varepsilon\cr},
\pmatrix{\varepsilon&&\cr &-\varepsilon& \cr &&-\varepsilon\cr},
\pmatrix{-\varepsilon&&\cr &\varepsilon& \cr &&-\varepsilon\cr},
\pmatrix{-\varepsilon&&\cr &-\varepsilon& \cr &&\varepsilon\cr},
\end{equation}
where $\varepsilon \in (\mu_2)^k$ satisfies $\prod \varepsilon_{i}
=1$, i.e. contains an even number of $-1$s (and in each matrix
denotes a diagonal block).
\subsection{Bundles with $S_3$-action}
\label{bunS3}
Let $X$ be a smooth curve acted on by the group $S_3$. We shall assume
that the action is faithful and
that each involution has nonempty fixed-point set.
The group $S_3$ acts in
two ways on isomorphism classes of principal
${\rm Spin}(8)$-bundles on $X$. First, it has a right
`triality' action $F\mapsto F^{u}$ (where $u \in S_3$) by
outer
automorphisms of the structure group. These are defined up to inner
automorphisms, as in Lemma \ref{outerautos},
but inner automorphisms preserve the isomorphism class
of the bundle. Second, the right action of $S_3$ on $X$ induces
by pull-back a left action on bundles, $F\mapsto u(F) := (u^{-1})^* F $.
We shall be interested in (semistable)
bundles for which these two actions agree, i.e.
the fixed-point set $\ff_X \subset \mm_X({\rm Spin}(8))$ of the
$S_3$-action $F\mapsto u^* F^u$, $u\in S_3$.
If $F\in \ff_X$ then there is a nontrivial {\it Mumford group} $\gg_F$
consisting of pairs $g\in S_3,\ \lambda:g(F)\mathbin{\hbox{$\,\widetilde\rightarrow\,$}} F^g$ with
multiplication law
$$
(g,\lambda)(h,\mu) = (gh, \lambda^h \circ g\mu).
$$
Here $\lambda^h : g(F^h)=g(F)^h \mathbin{\hbox{$\,\widetilde\rightarrow\,$}} F^{gh}$ and $g\mu : g h(F) \mathbin{\hbox{$\,\widetilde\rightarrow\,$}}
g(F^h)$ are the natural induced isomorphisms.
This group is an extension
\begin{equation}
\label{mumford}
1\longrightarrow {\rm Aut}\ F
\longrightarrow \gg_F
\longrightarrow S_3
\longrightarrow 1.
\end{equation}
Note in particular that there is an $S_3$-action on ${\rm Aut}\ F$ induced,
via conjugation in $\gg_F$, by this sequence. It is easy to verify
that the invariant subgroup $({\rm Aut}\ F)^{S_3}$ consists of $\alpha \in {\rm Aut}\
F$ commuting with $S_3$ in the sense that for all $g\in S_3 $ and
$\lambda: g(F)\mathbin{\hbox{$\,\widetilde\rightarrow\,$}} F^g$ the
following diagram commutes:
$$
\begin{array}{ccc}
g F & \map{\lambda} & F^g \\
&&\\
g\alpha \downarrow & & \downarrow \alpha^g \\
&&\\
g F & \map{\lambda} & F^g.\\
\end{array}
$$
\begin{rem}\rm
\label{S3-auto-remark}
If $F$ is stable as a
${\rm Spin}(8)$ bundle then the vector bundles $V_F, S_F^{\pm}$ are polystable
with $k$ stable summands, say. The automorphism group ${\rm Aut}\ F$ is then
the group of order $2^{k+1}$ described in the last section
((\ref{aut}) and (\ref{split-autos})).
The action of $S_3$ on ${\rm Aut}\ F$ determined by the sequence
(\ref{mumford}) permutes, for each $\varepsilon \in (\mu_2)^k$,
$\prod \varepsilon_{i}
=1$, the last three matrices of
(\ref{split-autos})) in the natural way. In particular we have
\begin{equation}
\label{S3-autos}
({\rm Aut}\ F )^{S_3} = {\rm ker \ } \{\Pi : (\mu_2)^k \rightarrow \mu_2 \}.
\end{equation}
\end{rem}
\begin{lemm}
\label{galoisbundle}
Suppose $F\in \ff_{X}$. Then the following data are equivalent:
\begin{enumerate}
\item
A splitting $\Lambda : S_3 \rightarrow \gg_F$ of the Mumford
sequence (\ref{mumford}).
\item
A lift of the $S_3$-action on $X$ to the principal bundle $F$, in the
sense that there is for each $u\in S_3$ a commutative diagram
$$
\begin{array}{ccc}
F & \map{\lambda(u)} & F^g \\
&&\\
\downarrow & & \downarrow \\
&&\\
X & \map{g\in S_3} & X
\end{array}
$$
and that $\lambda(u)^w \circ \lambda(w) = \lambda(uw)$
for all $u,w \in S_3$.
\item
An orthogonal lift of the $S_3$-action on $X$
to the {\it Chevalley bundle} $V_F \oplus S^+_F \oplus S^-_F$ such
that:
\par\indent \hangindent2\parindent \textindent{(i)}
the 3-cycle $\tau\in S_3$ permutes the summands cyclically $V_F \mathbin{\hbox{$\,\widetilde\rightarrow\,$}}
\tau^*S^+_F \mathbin{\hbox{$\,\widetilde\rightarrow\,$}} (\tau^2)^*S^-_F \mathbin{\hbox{$\,\widetilde\rightarrow\,$}} V_F$;
\par\indent \hangindent2\parindent \textindent{(ii)}
the involution $\sigma \in S_3$ lifts to an involution $V_F \mathbin{\hbox{$\,\widetilde\rightarrow\,$}}
\sigma^* V_F$,
acting in the fibre $(V_F)_x$ at a fixed point $x\in X$ of $\sigma$
as reflection in a hyperplane on which the quadratic form is nondegenerate,
and exchanges the spinor bundles $S^+_F \mathbin{\hbox{$\,\widetilde\rightarrow\,$}} \sigma^*S^-_F$.
\end{enumerate}
\end{lemm}
{\it Proof.}\
The equivalence of 1 and 2 is essentially obvious and we leave
its verification to the reader.
To show that 2 is equivalent to 3
we shall represent the bundle $F$ by a ${\rm Spin}(8)$-valued
Cech cocycle $\{g_{ij}\}$ with respect
to an open cover $\{U_i\}$ of the curve. We can assume that this open
cover
is invariant under the action of the finite group $S_3$, and we shall
denote the image of $U_i$ under $w\in S_3$ by $U_{i^w}$.
If $\{h_{ij}\}$ represents a second
bundle $F'$ then a bundle isomorphism $F\mathbin{\hbox{$\,\widetilde\rightarrow\,$}} F'$ is represented by a
cochain
$\{f_i\}$ satisfying
$$
f_i g_{ij} = h_{ij} f_j.
$$
For each $w\in S_3$ the bundle $w^* F$ has cocycle $\{w^*(g_{i^w
j^w})\}$;
while the bundle $F^{w^{-1}}$ has cocycle $\{w g_{ij} w^{-1}\}$, where the
conjugation
takes place in the semidirect product ${\rm Spin}(8) \mathbin{\hbox{\twelvesymb o}} S_3 \subset
O(V\oplus S^+ \oplus S^-)$. The condition that $F^{w^{-1}} \cong w^*
F$ is therefore:
\begin{equation}
\label{gal-condition1}
\hbox{\it $\exists$ ${\rm Spin}(8)$-valued cochain
$\{f_i\}$ satisfying
$f_i^{-1} w^*(g_{i^w j^w}) f_j = w g_{ij} w^{-1}$.
}
\end{equation}
Next now suppose that the vector bundle $V_F \oplus
S^+_F \oplus
S^-_F$ admits an orthogonal lift of the $S_3$-action. It is easy to
check
that necessary and sufficient conditions for this are
the existence, for each $w\in S_3$,
of an $O(V \oplus S^+ \oplus S^-)$-valued
cochain $\{{\widetilde f}_i \}$ satisfying
\begin{equation}
\label{gal-condition2}
{\widetilde f}_i g_{ij} = w^*(g_{i^w j^w}) {\widetilde f}_j,
\end{equation}
together with suitable compatibility
assumptions on the cochains $\{{\widetilde f}_i \}$ as $w\in S_3$ varies.
We shall show that (\ref{gal-condition2}) reduces to (\ref{gal-condition1})
when the lift
satisfies the conditions (i) and (ii) of the lemma.
Namely, these conditions
are equivalent to requiring that $\{{\widetilde f}_i \}$ takes values in
${\rm Spin}(8) \mathbin{\hbox{\twelvesymb o}} S_3 \subset O(V\oplus S^+ \oplus S^-)$ and projects to
the
element $w^{-1}\in S_3$. (Note that the 1-dimensional $-1$-eigenspaces
at the fixed points when $w$ is an involution
(3 part (ii) of the lemma) arise from the choices
of reflections of $V\oplus S^+ \oplus S^-$ needed to define $\rho: S_3
\hookrightarrow O(V\oplus S^+ \oplus S^-)$.)
Consequently, for each $w\in S_3$ the cochain defining the lift of
$w^{-1}$
to the Chevalley bundle can be expressed as ${\widetilde f}_i = (f_i,w^{-1})$
where $f_i$ is ${\rm Spin}(8)$-valued. Then (using (\ref{sdp}) in
\S\ref{spin})
$$
\begin{array}{rcl}
{\widetilde f}_i g_{ij}& =& (f_i,w^{-1})(g_{ij},1)\\
&=& (f_i w g_{ij}w^{-1},w^{-1})\\
\end{array}
$$
while
$$
\begin{array}{rcl}
w^*(g_{i^w j^w}) {\widetilde f}_j & =&(w^*(g_{i^w j^w}),1)(f_j,w^{-1}) \\
&=& (w^*(g_{i^w j^w})f_j,w^{-1}).\\
\end{array}
$$
So we see that (\ref{gal-condition2}) reduces to
(\ref{gal-condition1}) as required.
{\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par}
\begin{define}
\label{definition}
\par\indent \hangindent2\parindent \textindent{(i)}
By a {\it Galois ${\rm Spin}(8)$-bundle} we shall mean a pair $(F,\Lambda)$
where $F$ is a principal ${\rm Spin}(8)$-bundle on a curve $X$ on which
$S_3$ acts faithfully, and $\Lambda: S_3 \rightarrow \gg_F$ is a
lift of the group action to $F$ in the sense of Lemma \ref{galoisbundle}.
Semistability and S-equivalence of Galois
bundles will refer to the corresponding properties of the underlying
bundles $F$.
\par\indent \hangindent2\parindent \textindent{(ii)}
Let $\nn_X$ be the set of S-equivalence classes of semistable Galois
${\rm Spin}(8)$-bundles; and let $\ff_X \subset \mm_X({\rm Spin}(8))$ denote the
fixed point set of the
$S_3$-action $F\mapsto u^* F^u$, $u\in S_3$. For $F\in \ff_X$ we
denote by $\nn(F)$ the fibre of the forgetful map
$$
\nn_X \rightarrow \ff_X\subset \mm_X({\rm Spin}(8)).
$$
\end{define}
The set $\nn(F)$ of Galois structures, if nonempty, is a torsor over the
cohomology group $H^1(S_3, {\rm Aut}\ F)$.
\begin{lemm} {\capit (S.M.J. Wilson)}
\label{wilson}
Let $G$ be a finite group and $H \subset G$ be a normal subgroup of
coprime order and index. Suppose that $G$ acts on an abelian group
$A$ such that $H,A$ have coprime order. Then for all $q\geq 1$:
$$
\begin{array}{rcl}
H^{q}(H,A) & = & 0, \\
H^{q}(G,A) & \cong & H^{q}(G/H,A^{H}).\\
\end{array}
$$
\end{lemm}
{\it Proof.}\
As a $\z$-module $H^{q}(H,A)$ is annihilated by $|A|$ trivially, and
is also annihilated by $|H|$ since every element has order dividing
$|H|$ by \cite{Mac} p.117 Proposition 5.3. $H^{q}(H,A)$ is therefore
annihilated by ${\rm gcd}(|H|,|A|) = 1$, proving the first part. It
then follows from this and the Hochschild-Serre spectral sequence that
for all $q\geq 1$ the inflation maps $H^{q}(G/H,A^{H}) \rightarrow
H^{q}(G,A)$ are isomorphisms (see \cite{Mac} p.355 Exercise 3).
{\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par}
In our situation $G=S_3$, $H = \<\tau\>$, $A = {\rm Aut}\ F$, the
quotient $G/H$ acts trivially on $A^H$, and it follows that
$$
H^q(S_3, {\rm Aut}\ F) \cong {\rm ker \ } \{\Pi : (\mu_2)^k \rightarrow \mu_2 \}
\qquad
\hbox{for $q=1,2$.}
$$
From this we conclude:
\begin{prop}
\label{howmanylifts}
Suppose $F \in \ff_X$ is stable with $k$ summands (as a polystable
vector bundle).
\par\indent \hangindent2\parindent \textindent{(i)}
If $k=1$ then there exists a unique Galois structure on $F$.
\par\indent \hangindent2\parindent \textindent{(ii)}
If $k>1$ then $H^2(S_3,{\rm Aut}\ F) \cong (\mu_2)^{k-1}$. If the Mumford
sequence (\ref{mumford}) splits then the set $\nn(F)$ of Galois
structures on $F$ has cardinality $2^{k-1}$.
\end{prop}
\begin{rem}\rm
If $V_F$, and hence also $S_F^{\pm}$, are stable as
vector bundles (case (i) above) then one can see the uniqueness of the
Galois structure directly. Consider isomorphisms $\alpha, \beta:V_F \mathbin{\hbox{$\,\widetilde\rightarrow\,$}}
\sigma^* V_F$. By stability these differ by a scalar $\alpha =
\lambda \beta$, $\lambda \in
\c$, but at a fixed point both $\alpha, \beta$ act with 1-dimensional
$-1$-eigenspace in the fibre, and this forces $\lambda = 1$.
\end{rem}
\subsection{Further remarks}
\label{hypcase}
We should ask at this point how one can construct Galois bundles.
Note that since $F$ is determined by its
spinor bundles $S_F^{\pm}$ it is determined by the orthogonal bundle
$V_F$:
$$
S^+_F = \tau(V_F),
\qquad
S^-_F = \tau^2(V_F).
$$
(Because of the
relation $\sigma \tau = \tau^2 \sigma$ (recall \ref{notation} for notation)
these two bundles are necessarily
interchanged by $\sigma$.)
In the following lemma, which plays a key role in later sections, we
construct a `split' orthogonal bundle $V_F$ which gives rise to a Galois
spin bundle. We make use of the quotient
$$
\pi: X \map{2:1} Y = X/\sigma.
$$
\begin{lemm-def}
\label{su2trigonal}
Let $E\rightarrow Y = X/\sigma$ be a rank 2 vector bundle with $\det
E =
\oo_Y$.
Then there is a unique Galois ${\rm Spin}(8)$-bundle $F=F_E\rightarrow
X$ with
$$
V_{F} = \c^2 \otimes \pi^{*}E \oplus \tau(\pi^{*}E)\otimes
\tau^2(\pi^{*}E).
$$
\end{lemm-def}
{\it Proof.}\
Let $F_1 = \c^2 \otimes E'$ and $F_2 = \tau(E')\otimes
\tau^2(E')$ for any rank 2 bundle $E'\rightarrow X$ with trivial
determinant.
Each of $F_1,F_2$ is a ${\rm Spin}(4)$-bundle whose spinor bundles are its
two
factors.
So by Lemma \ref{spin-sum}, $F_1 \oplus F_2$ is the orthogonal
representation of a ${\rm Spin}(8)$ bundle $F$, and we have:
$$
\begin{array}{rcl}
V_{F} &=& F_1 \oplus F_2,\\
S^+_{F} &=& \c^2 \otimes \tau(E')\oplus \tau^2(E')\otimes
E' = \tau(F_1 \oplus F_2),\\
S^-_{F} &=& \c^2 \otimes \tau^2(E') \oplus E' \otimes
\tau(E') = \tau^2(F_1 \oplus F_2).\\
\end{array}
$$
It follows that $\sigma$ lifts to an isomorphism
$S_{F}^{+}\mathbin{\hbox{$\,\widetilde\rightarrow\,$}} \sigma^{*} S_{F}^{-}$.
Moreover, $\sigma$ lifts to an isomorphism $F_2 \mathbin{\hbox{$\,\widetilde\rightarrow\,$}} \sigma^* F_2$
having in the fibre at each fixed point $x\in X$ one-dimensional
$-1$-eigenspace $\bigwedge^2 (\tau E')_x$. This extends to an
isomorphism $V_F \mathbin{\hbox{$\,\widetilde\rightarrow\,$}} \sigma^* V_F$ with the same property provided
$E'$ is the pull-back of some bundle $E\rightarrow Y$; so we are done.
Note that in this construction we must have $\det E = \oo_Y$ since
$\det E' = \oo_X$:
by hypothesis $\sigma$ has nonempty fixed-point set,
so $\pi^*$ is
injective on line bundles.
{\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par}
\begin{rem}\rm
There are, of course, two more quotients of $X$ by choosing
$\sigma\tau$ or $\sigma\tau^2$ instead of $\sigma$. In fact these are
both isomorphic to $Y$: the $S_3$ action on $X$ induces an inclusion
$X \hookrightarrow Y\times Y\times Y$, and the three quotients are
then simply the projections on the three factors. Under the above
construction, choosing a different projection has the effect of
permuting
the bundles $V_F,S_F^{\pm}$.
\end{rem}
The existence of Galois bundles not of the form \ref{su2trigonal}
(and stable, in fact---note that the bundle
given by \ref{su2trigonal}) is semistable but not stable:
$\pi^* E \hookrightarrow V_F$ is a
destabilising subbundle) will be seen in section \ref{recillas}.
\medskip
Our next remark is a numerical observation that will reappear in later
(see Remark \ref{dimcount}(ii)).
We might relax the condition that $S_3$ acts faithfully on the curve $X$.
Suppose, in particular, that the normal subgroup $\< \tau \> \subset S_3$ acts
trivially, i.e. that $S_3$ acts by a single involution of $X$. Then
a Galois ${\rm Spin}(8)$-bundle $F$ requires isomorphisms $V_F \mathbin{\hbox{$\,\widetilde\rightarrow\,$}}
S^{\pm}_F$; let us interpret this as asking simply for a rank 8 orthogonal
bundle $V_F \rightarrow X$ together with a lift of the involution
$\sigma : X\leftrightarrow X$ which at fixed points acts in the fibre
with 1-dimensional $-1$-eigenspace.
Suppose in particular that $X\rightarrow \bp^1$ is a hyperelliptic
curve and $\sigma$ the sheet-involution. Then provided $X$ has genus
$g\geq 4$ there is a
projective moduli space $\mm$ of such $SO(8)$-bundles which is
described
explicitly in \cite{R} Theorem 1.
This says that
$$
\mm \cong {\rm Grass}_{g-4}(Q_1 \cap Q_2) /(\z/2)^{2g+2},
$$
where $Q_1,Q_2 \subset \bp^{2g+1}$ are quadrics, and ${\rm Grass}_{g-4}$
denotes the grassmannian of projective $g-4$-planes isotropic for
both quadrics.
In particular the dimension is easy to compute: $\mm$ is cut out in
the grassmannian by a pair of sections of the bundle $S^2
U^{\vee}$ where $U$ is the tautological bundle. Hence
$$
\begin{array}{rcl}
\dim \mm & = & (g-3)(g+5)-(g-3)(g-2) \\
& = & 7(g-3).
\end{array}
$$
\medskip
Finally, we make some remarks concerning
curves with $\z/3$-action---in particular, for example, cyclic trigonal
curves. In this case we can certainly imitate the above constructions replacing $S_3$ by $\z/3$, with triality action on ${\rm Spin}(8)$ determined by making a
choice (there are two) of embedding $\z/3 \hookrightarrow S_3$. The conjugation
action of $S_3$
on $\mm_X({\rm Spin}(8))$ then restricts to $\z/3$ and we can consider
the fixed-point-set $\ff_X^{\z/3} \subset \mm_X({\rm Spin}(8))$. A lifting
of the $\z/3$-action is determined by a splitting of the `restricted'
Mumford group
$$
\ses{{\rm Aut}\ F}{\gg_F^{\z/3}}{\z/3}.
$$
However, in this case the sequence {\it always splits uniquely} since by Lemma
\ref{wilson} $H^q(\z/3,{\rm Aut}\ F) =0$ for $q=1,2$. So for our moduli space of
$\z/3$-Galois bundles we can simply take $\nn_X = \ff_X$. Geometrically
a Galois bundle is now a ${\rm Spin}(8)$-bundle $F$ together with a lift of the
(chosen triality) $\z/3$-action in the sense of \ref{galoisbundle}(2),
or equivalently \ref{galoisbundle}(3) omitting the requirement (ii).
We expect that the results outlined in the introduction for the $S_3$ case
should hold also for $\z/3$. Indeed, the discussion of \S\ref{local}
in the next
section goes through unimpeded, as does the dimension calculation of the
moduli space in \S\ref{gen-dim} (see Remark \ref{cyclic-dim}). However,
the difficulty arises in computing the semistable boundary (\S\ref{boundary}),
where we make essential use of the elements of order 2 in $S_3$. Possibly
one could get round this and prove a corresponding result for $\z/3$,
but we have not pursued the question here.
\section{The moduli space}
\label{moduli}
We wish to construct a moduli space $\nn_X$ of Galois ${\rm Spin}(8)$
bundles on our $S_3$-curve $X$.
Using Proposition \ref{howmanylifts} this can be modelled on
the fixed-point set $\ff_{X} \subset \mm_{X}({\rm Spin}(8))$
of the `conjugation' action
$F\mapsto u^{-1}F^{u}= u^* F^{u}$, $u\in S_3$.
We shall show in this section that $ \nn_X$ inherits from $\ff_X$
the structure of an analytic space which is smooth over stable
spin bundles. $\nn_X$ and $\ff_X$ are locally isomorphic at stable
vector bundles (i.e. at $F$ such that $V_F$ is stable as a vector
bundle), while at polystable vector bundles $\nn_X$ resolves normal
crossing singularities in $\ff_X$:
\begin{center}
\leavevmode
\epsfxsize=3.2in\epsfbox{moduli.eps}
\end{center}
We analyse the semistable boundary (i.e. Galois bundles nonstable as
spin bundles) and show that, with the additional assumption that
$X/S_3 \cong \bp^1$, this
consists precisely of equivalence classes of bundles of the form
\ref{su2trigonal}.
\subsection{Local moduli}
\label{local}
To begin, we examine the $S_3$-action on the Kodaira-Spencer map at a
stable spin bundle in $\mm_X({\rm Spin}(8))$.
Suppose that $F$ is a principal $G$-bundle $F$, for some reductive group $G$,
represented by a Cech cocyle (transition functions) $\{g_{ij}\}$ with
respect an open cover $\{U_i \}$ of $X$. If ${\rm ad}\ F$ is the vector
bundle $F\times_{{\rm ad}\ } {\mathfrak g}}\def\gothh{{\mathfrak h}$ then a cohomology class $\xi \in
H^1(X,{\rm ad}\ F)$ is represented by a cocycle $\{ \xi_{ij} \}$ of
${\mathfrak g}}\def\gothh{{\mathfrak h}$-valued functions on $U_i \cap U_j$ satisfying
$$
0 = \xi_{ij}+\xi_{ji}^{g_{ij}}, \quad
0 = \xi_{ij}+\xi_{jk}^{g_{kj}}+ \xi_{ki}^{g_{ik}g_{kj}},
\qquad
\forall\ i,j,k;
$$
where $\xi^g := {\rm Ad}(g)\xi$ for $\xi\in {\mathfrak g}}\def\gothh{{\mathfrak h}$ and $g\in G$.
Exponentiating these conditions, i.e. applying $\exp: {\mathfrak g}}\def\gothh{{\mathfrak h}
\rightarrow G$, one gets precisely the cocycle conditions for
transition functions $\{g_{ij}\exp(\xi_{ij}) \}$. Denote the
corresponding $G$-bundle by $F_{\xi}$. If the bundle $F$ is stable
then so is $F_{\xi}$ in a neighbourhood of $0\in H^1(X,{\rm ad}\ F)$, and we
have a rational map
$$
{\rm ks} : H^1(X,{\rm ad}\ F) \rightarrow \mm_{X}(G),
$$
mapping $0\mapsto F$ and complete in the sense of \cite{Rth} Theorem
4.2. That is, ${\rm ks}$ induces an isomorphism from a neighbourhood of $0\in H^1(X,{\rm ad}\ F)
/\Gamma_F$, where $\Gamma_F = {\rm Aut}\ F/Z(G)$, to a neighbourhood of
$F\in \mm_X(G)$.
Now let $G = {\rm Spin}(8)$, and suppose that $(F,\Lambda)$ is a Galois
bundle (Definition \ref{definition}). Since the adjoint representation
of ${\rm Spin}(8)$ is preserved by triality it follows from Lemma
\ref{galoisbundle}(2) that $\Lambda$ induces an orthogonal lift of the
$S_3$-action on $X$ to the vector bundle ${\rm ad}\ F$. (This is completely
analogous to the lift of \ref{galoisbundle}(3) to the Chevalley
bundle; and indeed can be expressed in terms of \ref{galoisbundle}(3),
since as ${\rm Spin}(8)$-bundles ${\rm ad}\ F = \bigwedge^{2}V_F \cong
\bigwedge^{2}S^+_F \cong\bigwedge^{2}S^-_F$.)
In particular $\Lambda$ induces an action of $S_3$ on $H^1(X, {\rm ad}\ F)$.
\begin{prop}
Suppose that $S_3$ acts faithfully on the curve $X$ and that
$(F,\Lambda)$ (where $\Lambda \in \nn(F)$)
is a stable Galois ${\rm Spin}(8)$-bundle. Then the Kodaira-Spencer map
${\rm ks}$ is $S_3$-equivariant, where the $S_3$-action on $H^1(X,{\rm ad}\ F)$
is that induced by $\Lambda$ and the action on
$\mm_X({\rm Spin}(8))$ is $E\mapsto u^{*}E^u$ for $u\in S_3$.
\end{prop}
The proof is a straightforward calculation with transition functions,
using (\ref{gal-condition1}) and the cocycle conditions above for
elements $\xi \in H^{1}(X,{\rm ad}\ F)$, and we omit the details.
For each $(F,\Lambda) \in \nn_X$ we shall denote the $S_3$-invariant
subspace of $H^{1}(X,{\rm ad}\ F)$ by $H^{1}(X,{\rm ad}\ F)^{\Lambda}$.
It follows from the proposition that the Kodaira-Spencer map restricts
to a map
$$
{\rm ks}_{\Lambda}: H^{1}(X,{\rm ad}\ F)^{\Lambda} \rightarrow \ff_X.
$$
Suppose
first that $V_F$ is a stable vector bundle, so that (by Proposition
\ref{howmanylifts})
$\Lambda$ is unique.
The map ${\rm ks}_{\Lambda}$ factors to an injective map on the quotient
$H^{1}(X,{\rm ad}\ F)^{\Lambda}/\Gamma_F^{\Lambda}$ where $\Gamma_F^{\Lambda}
= ({\rm Aut}\ F)^{S_3}/Z({\rm Spin}(8))^{S_3}$. But from Remark \ref{S3-auto-remark}
we know
that $Z({\rm Spin}(8))^{S_3}= \{1\}$; and when $V_F$ is a stable vector bundle
$({\rm Aut}\ F)^{S_{3}}$ is also trivial.
We therefore have a commutative diagram (more precisely, the
horizontal maps are defined on neighbourhoods of~0):
$$
\begin{array}{ccc}
H^1(X,{\rm ad}\ F)/\Gamma_F & \hookrightarrow & \mm_X({\rm Spin}(8)) \\
&&\\
\uparrow && \uparrow \\
&&\\
H^1(X,{\rm ad}\ F)^{\Lambda} &\hookrightarrow & \ff_X. \\
\end{array}
$$
The spaces $\nn_{X}$ and $\ff_X$ are therefore locally isomorphic,
determining a smooth analytic structure on $\nn_X$ with tangent space
$H^1(X,{\rm ad}\ F)^{\Lambda}$.
\begin{rem}\rm
In \S\ref{recillas} we shall construct (for the case
$X/S_3 \cong \bp^1$) examples of
$(F,\Lambda) \in \nn_X$ for which the vector bundle $V_F$ is stable.
\end{rem}
More generally, for polystable $V_F$ with $k$ summands, consider the
set of Galois structures $\nn(F)
=\{\Lambda_{1},\ldots,\Lambda_{2^{k-1}}\}$. This set is acted on
transitively and faithfully by the group $({\rm Aut}\ F)^{S_{3}}$.
The subgroup of $\Gamma_F$
preserving each $H^1(X,{\rm ad}\ F)^{\Lambda_i}$ is therefore trivial and
so we have inclusions (from a neighbourhood of 0 to a
neighbourhood of $F$):
$$
H^{1}(X,{\rm ad}\ F)^{\Lambda_i} \hookrightarrow \ff_X;
\qquad
i=1,\ldots,2^{k-1}.
$$
Each space can be viewed as Zariski tangent space of $\nn_X$ at
$(F,\Lambda_i)$, and we conclude:
\begin{prop}
\label{tgt-space}
The moduli space $\nn_X$ is a complex analytic space mapping
holomorphically to $\ff_X\subset \mm_X({\rm Spin}(8))$.
At a stable Galois bundle $(F,\Lambda)$ it has Zariski tangent space
$$
T_{F,\Lambda} \nn_X = H^{1}(X,{\rm ad}\ F)^{\Lambda}.
$$
\end{prop}
We shall compute the dimension of the tangent space in section
\ref{gen-dim}, and deduce (in the case of interest $X/S_3 \cong \bp^1$)
that $\nn_X$ is
smooth at stable points (Theorem \ref{dimtheorem}).
\subsection{The semistable boundary}
\label{boundary}
Our next task is to describe Galois bundles which are semistable but
not stable. We shall assume from now on that the quotient $X/S_3$
is isomorphic to $\bp^1$, and under this assumption
it turns out that up to S-equivalence
these bundles are exactly those described in Lemma
\ref{su2trigonal}. To recall, let $E$ be a rank 2 vector bundle on $X$
pulled back from $X/\sigma$.
We consider the Galois ${\rm Spin}(8)$-bundle $F_E =
(V_{E},S^+_E,S_E^-)$ where
\begin{equation}
\label{split-form}
\begin{array}{rcl}
V_E & = & (\tau E)(\tau^2 E)\oplus E \c^2 ,\\
S^+_E & = & (\tau^2 E)E \oplus (\tau E) \c^2 , \\
S^-_E & = & E(\tau E) \oplus (\tau^2 E)\c^2 . \\
\end{array}
\end{equation}
We have written $E\c^2 = E\otimes \c^2$ etc for brevity. Note that if a
trivialisation of the determinant
line bundle $\det E$ is fixed then fibrewise we are in
the situation of \S\ref{2by2}: each rank 2 factor is naturally
self-dual, taking adjugates is a well-defined involution of each rank
4 summand, and we have an explicit description of the Clifford
multiplications among the three rank 8 bundles.
\begin{theo}
\label{ssboundary}
Suppose that $S_3$ acts faithfully on a curve $X$
with quotient $X/S_3 \cong \bp^1$.
A semistable Galois ${\rm Spin}(8)$ bundle $F\rightarrow X$ is unstable if
and only if it is of the form $F = F_E$ for some $E\in \ss{\cal U}}\def\vv{{\cal V}}\def\xx{{\cal X}}\def\ww{{\cal W}_{X/\sigma}(2)$
up to S-equivalence.
\end{theo}
We have seen in \S\ref{spinbundles} that stability of $F$ is
equivalent to stability of $V_F$ as an orthogonal bundle; so suppose
that $U\subset V_F$ is a destabilising subbundle. This means that $U$
is an isotropic subbundle of degree 0. In particular it is semistable
as a vector bundle and ${\rm rank \ } U \leq 4$.
Using the lift of the $S_3$-action on $X$ to the Chevalley bundle
$W_F = V_F \oplus S_F^+ \oplus S_F^-$ we can consider the image
bundles $gU \subset W_F$ for group elements $g\in S_3$. In addition,
we can consider the associated bundles (see \S\ref{6quadrics})
$$
\begin{array}{rcl}
A_U &=& {\rm Im\ } \{ U\otimes S_F^+ \rightarrow S_F^-\}^{\perp},\\
B_U &=& {\rm Im\ } \{ U\otimes S_F^- \rightarrow S_F^+\}^{\perp};\\
\end{array}
$$
as well as combinations of these constructions such as $A_{A_{U}}$,
$B_{gU}$ etc. We shall keep track of these subbundles of the
Chevalley bundle using the results of \S\ref{6quadrics}.
To begin, suppose that $U\subset V_F$ has
${\rm rank \ } U = 4$.
Then one of $A_U\subset S_F^-$ or $B_U \subset S_F^+$ is a
destabilising line
subbundle, and so by the action of $\tau \in S_3$ we get a
destabilising
line subbundle in $V_F$.
If, on the other hand, ${\rm rank \ } U =3$ then it is---at least locally---the
intersection
of a pair of rank 4 isotropic subbundles parametrising isotropic
subspaces of opposite families in the fibres. Globally we obtain a
rank 4 subbundle of the pull-back of $V_F$ to an \'etale double cover
of $X$; however, the class of this double cover is the Stiefel-Whitney
class $w_1(F)$, which is zero since $F$ is spin. We therefore have a
pair of rank 4 isotropic subbundles globally on $X$.
Since $U$ and $V_F$ are semistable these rank 4 bundles have degree 0
and so
again $V_F$ has (two) destabilising line subbundles.
It is therefore sufficient to assume that ${\rm rank \ } U \leq 2$. We shall
consider for any subbundle $U\subset V_F$ the Clifford
multiplication
$$
\begin{array}{rcc}
\tau\sigma U \otimes \tau^2 U & \hookrightarrow & S_F^+ \otimes S_F^- \\
&&\\
&\mu \searrow & \downarrow \\
&&\\
&&V_F\\
\end{array}
$$
\begin{lemm}
\label{mu=0}
If ${\rm rank \ } U =1$ then $\mu =0$; if ${\rm rank \ } U =2$ then ${\rm rank \ } \mu \leq 1$.
\end{lemm}
\begin{rem}\rm
In what follows, we shall freely interpret the results of
\S\ref{6quadrics} as (Galois) bundle theoretic statements. The
justification for this is that the Galois structure moves the fibres
in just the right way for the global Clifford multiplications to make
sense fibrewise. For example, if $U\subset V_F$ then consider the
fibres $U_x$ and $(\tau U)_x$ at a point $x\in X$. By definition
$$
(\tau U)_x = U_{\tau^{-1}(x)} \subset (V_F)_{\tau^{-1}(x)} =
(S_F^+)_x,
$$
so that there is a well-defined multiplication map $U_x \otimes (\tau U)_x
\rightarrow (S_F^-)_{x}$.
\end{rem}
{\it Proof of \ref{mu=0}.}
Apply Lemma \ref{zero-norm} to $\sigma U \subset V_F$ with $g = \tau
\sigma$. This says that the composition $\sigma U \otimes \tau U
\hookrightarrow V_F \otimes S_F^+ \rightarrow S_F^-$ has rank 0 if
$U$ is a line bundle and rank $\leq 1$ if $U$ has rank 2. Now apply
$\tau$ and use the $S_3$ equivariance of the Chevalley algebra
structure.
{\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par}
We consider first a destabilising subbundle $E\subset V_F$ of rank 2.
Observe that if $\mu$ has rank 1 then
the image of $\mu$ is a destabilising line subbundle; we postpone
this case until after Proposition \ref{rank2case} and assume that $\mu =0$.
(The fact that the image of $\mu$ is isotropic
can be seen from the description of the Clifford multiplication map
given in section \ref{2by2}.)
The second thing to observe is that, with respect to the quadratic
form,
$$
E^{\perp}/E \cong A_E \otimes B_E.
$$
This is because $E^{\perp}/E$ is an orthogonal bundle of rank 4 whose
isotropic 2-planes in the generic fibre are parametrised by $A_E,
B_E$ respectively.
(See \S\ref{6quadrics}).
\begin{prop}
\label{rank2case}
If $V_F$ has no destabilising line subbundle and
$E\subset V_F$ is a destabilising rank~2 subbundle then $F$ is
S-equivalent to $F_{E}$ (\ref{split-form}), where $E$ is the
pull-back of a stable rank~2 vector bundle on $X/\sigma$.
\end{prop}
{\it Proof.}\
We already have the S-equivalence
$V_F \sim E^{\vee} \oplus E \oplus A_E \otimes B_E$; we examine the
consequences of the vanishing of $\mu$.
Namely, $\tau^2 E = B_{\tau\sigma E}$ by Lemma \ref{AB2}, and so
$E=B_{\tau^2\sigma E}$ by Lemma \ref{AB3}. So by Lemma \ref{AB1} we have
$$
A_E = A_{B_{\tau^2\sigma E}} = \tau^2\sigma E.
$$
Similarly $\sigma E = \sigma B_{\tau^2\sigma E} = A_{\sigma\tau^2\sigma
E}= A_{\tau E}$ and so
$$
B_{\sigma E} = B_{A_{\tau E}} = \tau E.
$$
Next, $E^{\vee} \cong
\Delta^{-1}\otimes E$ where $\Delta = \det E$, and so
we have
$$
\begin{array}{rcl}
V_F &\sim & (\oo \oplus \Delta^{-1})E \oplus A_E B_E,\\
S^+_F &\sim & (\oo \oplus \Delta^{-1})A_E \oplus B_E E,\\
S^-_F &\sim & (\oo \oplus \Delta^{-1})B_E \oplus E\ A_E.\\
\end{array}
$$
We now impose
the Galois condition. $S^+_F = \tau V_F$ implies that $\Delta $ is
$\tau$-invariant ($\oo \oplus \Delta^{-1}$ is the only split rank 2
factor since $E,A_E,B_E$ are all stable). Similarly $\Delta$ is
$\sigma$-invariant since $V_F = \sigma V_F$; it therefore descends to a line
bundle on $\bp^1$, and since it has degree 0 this forces $\Delta = \oo_X$.
Similarly $\sigma E \cong E$, and it follows from the above
$A_E = \tau^2 E$, $B_E =
\tau E$. As before, we note that since the $-1$-eigenspaces at fixed points
of $\sigma$ lie in the fibres of $A_E B_E$, the involution must act trivially
in the fibres of $E$, which therefore descends.
{\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par}
The case of a destabilising line subbundle, to which we turn now,
involves a bit more work.
\begin{prop}
\label{prop.1}
Suppose that a line subbundle $L\subset V_F$ is
isotropic of degree 0, and that the Clifford multiplication $\tau L\otimes
\tau^2 L \rightarrow V_F$ vanishes. Then $F \sim F_E$ for
$E = L\oplus L^{-1}$.
\end{prop}
{\it Proof.}\
By (\ref{prod=0}) in \S\ref{6quadrics} and the hypothesis of the
proposition, we have an isotropic subbundle $U = A_{\tau L} \cap
B_{\tau^2 L}\subset V_F$ of rank 3. Using Lemma \ref{AB3}, moreover,
one sees that $L\subset U$. Accordingly we have an S-equivalence
$$
V_{F} \sim L\oplus L^{\vee} \oplus U/L \oplus (U/L)^{\vee} \oplus
U^{\perp}/U.
$$
We shall identify the last term first; note that $U^{\perp} = A_{\tau L} +
B_{\tau^2 L}$, so we have to compute the rank 1 quotients $A_{\tau
L}/U$ and $B_{\tau^2 L}/U$.
From (\ref{ABsequences}) it follows
that there is a short exact sequence of vector bundles
$$
\ses{\tau L \otimes A_{\tau L}}{\tau L \otimes V_F}{B_{\tau L}}.
$$
Twisting by $\tau L^{-1}$ and using the fact that $\tau L \otimes
\tau^2 L \rightarrow V_{F}$ is zero we get short exact sequences (in
which the vertical arrows are inclusions):
$$
\begin{array}{rcccccl}
0\rightarrow & A_{\tau L} & \rightarrow & V_F & \rightarrow & \tau
L^{-1}\otimes B_{\tau L} & \rightarrow 0 \\
& \uparrow & & \uparrow & & \uparrow & \\
0\rightarrow & U & \rightarrow & B_{\tau^2 L} & \rightarrow & \tau
L^{-1}\otimes \tau^2 L & \rightarrow 0, \\
\end{array}
$$
and similarly
$$
\begin{array}{rcccccl}
0\rightarrow & B_{\tau^2 L} & \rightarrow & V_F & \rightarrow &
\tau^2
L^{-1}\otimes A_{\tau^2 L} & \rightarrow 0 \\
& \uparrow & & \uparrow & & \uparrow & \\
0\rightarrow & U & \rightarrow & A_{\tau L} & \rightarrow & \tau^2
L^{-1}\otimes \tau L & \rightarrow 0. \\
\end{array}
$$
We deduce that
$$
V_F \sim (L \oplus \tau^2 L^{-1}\otimes \tau L \oplus U/L)
\oplus (\hbox{same bundle})^{\vee}.
$$
To impose the condition that $F$ is a Galois bundle we shall compute
directly the spinor bundles from this expression and (\ref{spinors})
in \S\ref{group}. First note that $\det A_{\tau L} = \det B_{\tau
L} = \tau L^2$, so from the exact sequences above $\det U = \tau
L \otimes \tau ^2 L$ and hence
$$
\det (U/L) = L^{-1}\tau L \tau ^2 L.
$$
Writing $N = L \oplus \tau L\tau^2 L^{-1} \oplus U/L$, the spinor
bundles are, up to a line bundle, $S_F^+ = \bigwedge^{\rm even} N$ and
$S_F^- = \bigwedge^{\rm
odd} N$. We find that to get trivial determinant one must twist by
$\tau L^{-1}$, and the result is then (writing $E = L\otimes (U/L)$
and using the fact that (since it has rank 2) $(U/L)^{\vee} = \tau L^{-1} \tau^2 L^{-1} \otimes
E$):
$$
\begin{array}{rcl}
V_F & \sim & L\oplus L^{-1} \oplus \tau L \tau^2 L^{-1}\oplus
\tau L^{-1} \tau^2 L \oplus U/L \oplus (U/L)^{\vee}\\
& = & L\oplus L^{-1} \oplus \tau L \tau^2 L^{-1}\oplus
\tau L^{-1} \tau^2 L \oplus \left( L^{-1}\otimes E\right) \oplus
\left( \tau L^{-1} \tau^2 L^{-1} \otimes E\right),\\
&&\\
S_F^{+} & \sim &\textstyle \bigwedge^{\rm even} N \\
& = & \tau L\oplus \tau L^{-1} \oplus \tau^2 (L) L^{-1}\oplus
\tau^2 (L^{-1}) L \oplus \left( \tau L^{-1}\otimes E\right)
\oplus \left( L^{-1}\tau^2
L^{-1} \otimes E \right), \\
&&\\
S_F^- & \sim & \textstyle \bigwedge^{\rm odd} N \\
& = & \tau^2 L\oplus \tau^2 L^{-1} \oplus L \tau L^{-1}\oplus
L^{-1} \tau L \oplus \left( \tau^2 L^{-1}\otimes E\right)
\oplus \left( L^{-1}\tau L^{-1}
\otimes E \right). \\
\end{array}
$$
Since $F$ is a Galois bundle these three vector bundles are permuted
cyclically by $\tau$, and it follows that the bundle $E = L \otimes
U/L$ is $\tau$-invariant. On the other hand, $E$ is also
$\sigma$-invariant (since $U$ is), and $\sigma$ acts as the identity
in the fibres of $E$ at fixed points since the $-1$-eigenspace in the
fibre of $V_F$ is 1-dimensional and nonisotropic. It follows that $E$
descends to a bundle on $\bp^1$ (in fact the trivial bundle, by
semistability of $V_F$, and can be written (notice that both summands
are trivial)
$$
E = \oo \oplus L\tau L \tau^2 L.
$$
It follows directly that
$$
V_F \sim (L \oplus L^{-1} \oplus \tau L \tau^2 L \oplus \tau L \tau^2
L^{-1})\oplus (\hbox{same bundle})^{\vee}
$$
and that $F$ has the desired form.
{\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par}
We next show that if $V_F$ has a destabilising line subbundle
then this can be chosen with the property of Proposition \ref{prop.1}.
\begin{lemm}
Suppose that a destabilising line subbundle $L\subset V_F$ is polar,
with respect to the quadratic form on $V_F$, to its image
$\sigma L \subset V_F$.
Then the Clifford multiplication $\tau L\otimes
\tau^2 L \rightarrow V_F$ vanishes.
\end{lemm}
{\it Proof.}\
If $\sigma L = L$ then the result follows from Lemma \ref{mu=0}, so
we may assume that $L, \sigma L$ are distinct subbundles. In this case
the bundle map $L\oplus \sigma L \rightarrow V_F$ is injective by
semistability (the image is isotropic since we assume the summands
are polar). In other words $L$ and $\sigma L$ are distinct in all
fibres.
If the multiplication $\tau L \otimes \tau^2 L \rightarrow V_F$ is
nonzero then it must be injective (again by semistability, since the
image is isotropic). This implies that $A_{\tau L} \cap B_{\tau^2 L}
\subset V_F$ has rank 1 (using (\ref{prod=0})). On the other hand it
contains $\sigma L$ (by Lemma \ref{mu=0}), and hence
$$
\sigma L = A_{\tau L} \cap B_{\tau^2 L}.
$$
We now consider the isotropic subbundles
$$
U = A_{\tau \sigma L} \cap B_{\tau^2 L},
\qquad
E = A_{\tau \sigma L} \cap A_{\tau L}.
$$
By Lemma \ref{mu=0} $U$ has rank 3. By \S\ref{6quadrics}
(\ref{polarity}), polarity of $L$ and $\sigma L$ we see also that
$E = \tau (A_{\sigma L} \cap A_{L})$ has rank 2. But $E,U$ are both
contained in the rank 4 bundle $A_{\tau \sigma L}$, so
$$
N = U\cap E = A_{\tau \sigma L}\cap A_{\tau L} \cap B_{\tau^2 L}
$$
is an isotropic line subbundle. ($E$ is not contained in $U$ since
then $A_{\tau L} \cap B_{\tau^2 L}$ would be $\geq 2$.) But $N\subset
A_{\tau L} \cap B_{\tau^2 L} = \sigma L$, so in particular $\sigma L
\subset A_{\tau \sigma L}$. Applying $\sigma$ (and using Lemma
\ref{AB3}) this shows that $L\subset B_{\tau^2 L}$. Applying $\tau^2$
then shows that $\tau^2 L\subset B_{\tau L}$, i.e. that $\tau L
\otimes \tau^2 L \rightarrow V_F$ is the zero map, a contradiction.
{\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par}
\begin{prop}
\label{prop.2}
If $V_F$ admits a destabilising line subbundle then it has such a line
subbundle $L \subset V_F$ for which $\tau L\otimes
\tau^2 L \rightarrow V_F$ vanishes.
\end{prop}
{\it Proof.}\
We assume that $L\subset V_F$ is a destabilising line subbundle
without this property, i.e. for which $\tau L\otimes
\tau^2 L \rightarrow V_F$ is injective; and by the previous lemma we
can assume that $L$ is nowhere polar to $\sigma L$, i.e. that the
homomorphism $L \otimes \sigma L \rightarrow \oo$ induced by the
quadratic form on $V_F$ is an isomorphism. By \S\ref{6quadrics}
(\ref{polarity}) it follows that in this case $V_F = A_{\tau \sigma
L} \oplus A_{\tau L}$. We let $U=A_{\tau \sigma
L} \cap B_{\tau^2 L}$, and we have a configuration of subbundles:
\begin{center}
\leavevmode
\epsfxsize=2in\epsfbox{subbundles.eps}
\end{center}
Note in particular that
$$
A_{\tau L} \cap B_{\tau^2 L} = \tau L\tau^2 L = \sigma L \cong L^{-1}.
$$
As in the proof of Proposition \ref{prop.1} we have exact sequences
(the vertical arrows are injective)
$$
\begin{array}{rcccccl}
0\rightarrow & A_{\tau \sigma L} & \rightarrow & V_F & \rightarrow & \tau
\sigma L^{-1}\otimes B_{\tau \sigma L} & \rightarrow 0 \\
& \uparrow & & \uparrow & & \uparrow & \\
0\rightarrow & U & \rightarrow & B_{\tau^2 L} & \rightarrow & \tau
\sigma L^{-1}\otimes \tau^2 L & \rightarrow 0. \\
\end{array}
$$
From this we see that $\det U = \tau^2 L \tau L^{-1}$ and
$$
\textstyle
\bigwedge^2 U = U^{\vee} \otimes \det U = U^{\vee} \otimes \tau^2 L \tau L^{-1}.
$$
Using these facts and \S\ref{group} (\ref{spinors}) we deduce that
$$
\begin{array}{rcl}
V_F & = & (L\oplus U) \oplus (L\oplus U)^{\vee} \\
S_F^+ & = & (\tau L \oplus L\tau L \otimes U) \oplus (\tau L \oplus L\tau
L \otimes U)^{\vee}\\
S_F^- & = & (\tau^2 L \oplus \tau L \otimes U ) \oplus
(\tau^2 L \oplus \tau L \otimes U )^{\vee}.
\end{array}
$$
We now impose the condition that $F$ is a Galois bundle to deduce
that
$$
\tau U = L \tau L \otimes U.
$$
Also $\sigma U = U$ since $\sigma$
interchanges $A_{\tau \sigma L}$ and $B_{\tau^2 L}$ (by Lemma
\ref{AB3}). At fixed points $\sigma$ acts trivially in the fibres of
$U$ since the latter is isotropic, and hence the projective bundle
$\bp(U)$ is $S_3$-invariant and descends to a $\bp^2$-bundle on
$\bp^1$. This is necessarily the projectivisation of a split vector
bundle, and hence
$U = P\oplus Q \oplus R$ for some line bundles $P,Q,R$ on $X$. Note
that these line bundles need not themselves descend to $\bp^1$ may be
permuted and twisted by the action of $S_3$. However, since $U$ is
$\sigma$-invariant at least one summand must also be
$\sigma$-invariant, and this summand will be a destabilising subbundle
of $V_F$ with the desired properties.
{\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par}
{\it Proof of Theorem \ref{ssboundary}.}
By Proposition \ref{rank2case} we are reduced to the case where
$V_F$ has a destabilising line subbundle
$L\subset V_F$. By Proposition \ref{prop.2} we can assume that Clifford
multiplication on $\tau L \otimes \tau^2 L$ is zero, and by \ref{prop.1}
this implies that $V_F \sim \c^2 E \oplus \tau E \tau^2 E$ with $E = L \oplus
L^{-1}$. We just have to check that $E$ descends to a bundle on $X/\sigma$.
On the other hand the Galois condition tells us that $V_F \sim \c^2 \sigma E
\oplus \tau^2 \sigma E \tau \sigma E$, and this implies that $\sigma E \cong
E$. At a fixed point $x\in X$ of $\sigma$ the 1-dimensional $-1$-eigenspace
of $\sigma$ in the fibre of $V_F$ is $\bigwedge^2 (\tau E_x)$ (since $\tau E$
and $\tau^2 E$ are interchanged by $\sigma$), and so in particular $\sigma$
acts trivially in the fibre $E_x$. This shows that $E$ is pulled back from a
bundle on $X/\sigma$, as required.
{\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par}
\section{Triality and trigonality}
\label{galois3}
In this section (in \S\ref{trig-gal-bun})
we shall introduce the main object of the paper,
which is the moduli space $\nn_C$ of Galois ${\rm Spin}(8)$-bundles on the Galois
closure of a trigonal curve $C\map{3:1} \bp^1$. Unless the curve is
cyclic this Galois curve is a connected double cover of $C$, and we are
interested in the Galois bundles on this double cover. The moduli
space $\nn_C$ is an extension of the variety ${\cal SU}_C(2)$, which contains its
singular locus.
\subsection{Galois trigonal curves}
\label{trig-pics}
Let $C$ be a nonhyperelliptic curve of genus $g\geq 3$ with a fixed
trigonal pencil $g^{1}_{3}$. We shall denote by $\bb
\subset \bp^{1}$ the branch divisor; by Riemann-Hurwitz this has
degree
$2g+4$. We shall suppose that $\bb$ contains $\delta$ points of
multiplicity two, i.e. at which the cover has cyclic triple
branching.
There are then $2g+4 -2\delta$ simple branch points:
\begin{center}
\leavevmode
\epsfxsize=2.5in\epsfbox{branch.eps}
\end{center}
We shall need to distinguish the case of a cyclic cover $C\map{3:1}
\bp^{1}$, in which case $\delta = g+2$. (We shall see in a moment
that
this is also a sufficient condition.)
We construct a Galois cover ${\cal G}(C) \rightarrow \bp^{1}$ as
follows. ${\cal G}(C)\subset C\times C$ is the closure of the set
$\{(q,r)\ |\ q\not= r,\ p+q+r \in g^{1}_{3}\ \hbox{for some }p\in
C\}$. This is a
double cover ${\cal G}(C) \rightarrow C$ by $(q,r) \mapsto p=
g^{1}_{3}-q-r$,
and is smooth with branching behaviour:
\begin{center}
\leavevmode
\epsfxsize=2.5in\epsfbox{galois.eps}
\end{center}
Notice that over each divisor $p+q+r \in g^{1}_{3}$ the points of
${\cal G}(C)$
correspond to the orderings of $p,q,r \in C$. Thus the permutation
group $S_{3}$ acts on ${\cal G}(C)$ on the right; in particular $C$ is
the
quotient of ${\cal G}(C)$ by the 2-cycle $\sigma = (23)$.
In general the general case
$S_3$ is the Galois group of ${\cal G} (C)$; however, in the cyclic
case ${\cal G} (C)$ breaks up into two connected components, each a copy
of $C$ acted on by $\z/3$.
Restricting for a moment to the Galois $S_3$ case, we find
by Riemann-Hurwitz that ${\cal G}(C)$ has genus $3g+1-\delta$.
Over simple branch points in $\bb \subset \bp^{1}$ the
three
ramification points
are respective fixed points of the three 2-cycles in $S_{3}$.
Over each double branch point there are two fixed points of the cyclic
subgroup $\<\tau\>\cong \z/3$, at which $\tau$ acts in the tangent
space
as $\omega$, $\omega^2$ respectively, where $\omega = e^{2\pi i/3}$
is a
primitive cube root of unity.
Consider now
the quotient $H' = {\cal G}(C)/\<\tau\>$. This is a hyperelliptic curve
branched over
$\bb \subset \bp^{1}$ and singular over the $\delta$ double points of
$\bb$.
By Riemann-Hurwitz (again, we are assuming ${\cal G} (C)$ is a connected
Galois
$S_3$ curve)
its normalisation $H$ has Euler characteristic $e(H) = 2\delta -2g$.
If
$\delta = g+2$ then $e(H) =4$ so $H$ is a union of two lines and
${\cal G} (C)$
is a pair of copies of $C$, a contradiction. We have therefore shown:
\begin{lemm}
$C$ is cyclic over $\bp^1$ if and only if $\delta = g+2$.
\end{lemm}
If $\delta \leq g+1$, on the other hand, then $g(H) = g+1 - \delta$.
We can summarise the situation in the following diagram.
\begin{equation}
\label{Gal(C)-diagram}
\begin{array}{ccccc}
& & ^{3g+1-\delta}{\cal G}(C)=X & & \\
& \swarrow & & \searrow\pi & \\
^{g+1-\delta}H' & & & & ^{g}C =Y \\
& \searrow & & \swarrow g^{1}_{3} & \\
& & \bp^{1} & & \\
\end{array}
\label{S3galois}
\end{equation}
In the rest of the paper we shall apply the discussion of the
previous
sections to $X = {\cal G}(C)$ and $Y=C$, in the general trigonal case,
and to $X=C$ when this is a cyclic trigonal curve. Whenever we
refer to ${\cal G}(C)$ we shall always assume that we are in the general
case.
\subsection{Trigonal Galois bundles}
\label{trig-gal-bun}
We now turn to the moduli space we are interested in. We assume that
$C$ is a non-cyclic trigonal curve and we let
$\nn_{C} := \nn_{{\cal G}(C)} \subset \mm_{{\cal G}(C)}({\rm Spin}(8))$.
(Though see Remark \ref{dimcount}(i) regarding the cyclic case.)
\begin{rem}\rm
If $F$ is such a Galois bundle then the sheet-involution of
${\cal G}(C)$ over $C$, corresponding to $\sigma \in S_3$, lifts to the
orthogonal bundle $V_F$; applying the 3-cycle $\tau \in S_3$ one
sees that
the fibres of $V_F$ over any $p+q+r \in g^1_3$ can be
identified
(though not canonically) with the three representations $V,S^+,S^-$.
If $F$ is semistable but not stable then $F = F_E$ for some
$E \in \ss{\cal U}}\def\vv{{\cal V}}\def\xx{{\cal X}}\def\ww{{\cal W}_C(2)$ (by Theorem \ref{ssboundary}).
The following table shows explicitly how, in this case, the
fibres of
$V_{F}, S_{F}^{\pm}$ are constructed from those of the rank 2 bundle
$E$ over
a divisor $p+q+r \in g^{1}_{3}$. We abbreviate
$\c^2 \otimes E_{p} \oplus E_{q}\otimes E_{r}$ to $2E_p+E_q E_r$:
$$
\begin{array}{c|c||c|c|c}
C & {\cal G}(C) & V_{F} & S^{+}_{F} & S^{-}_{F} \\
\hline
p & (q,r) & 2E_p+E_q E_r & 2E_q+E_r E_p & 2E_r+E_p E_q \\
& (r,q) & 2E_p+E_r E_q & 2E_r+E_q E_p & 2E_q+E_p E_r \\
\end{array}
$$
\end{rem}
\noindent
The following is the main result of this paper:
\begin{theo}
\label{dimtheorem}
For any non-cyclic trigonal curve $C$ the moduli space
$\nn_C$ is smooth of dimension $7g-14$ away from its semistable
boundary $\ss{\cal U}}\def\vv{{\cal V}}\def\xx{{\cal X}}\def\ww{{\cal W}_C(2)$.
\end{theo}
\begin{rems}\rm
\label{dimcount}
\par\indent \hangindent2\parindent \textindent{(i)}
We have already remarked in \S\ref{hypcase} that most of our constructions
work also for cyclic trigonal curves. to be precise,
if $C$ is cyclic we
define $\nn_{C}= \ff_C \subset \mm_{C}({\rm Spin}(8))$, the fixed-point-set
of the conjugation action of $\z/3$ after choosing an embedding $\z/3
\hookrightarrow S_3$. In this case both the smoothness (at stable points)
and the dimension statements
in Theorem \ref{dimtheorem} remain valid (see Remark \ref{cyclic-dim} below);
what is missing is the identification of the semistable boundary with
$\ss{\cal U}}\def\vv{{\cal V}}\def\xx{{\cal X}}\def\ww{{\cal W}_C(2)$ (though it is tempting to expect that this also remains true).
\par\indent \hangindent2\parindent \textindent{(ii)}
Notice that the moduli space of Galois bundles on the
hyperelliptic curve $H$ of diagram (\ref{Gal(C)-diagram})
has dimension $7g-14 - 7\delta$
(see \S\ref{hypcase}); when $\delta=0$ and
$H'= H$ is smooth this number is $7g-14$.
This raises the question of the relationship between these two moduli spaces.
Are they isomorphic, for example, or birational?
\end{rems}
In view of Proposition \ref{tgt-space}, Theorem \ref{ssboundary} and
the fact that in the trigonal situation the quotient $X/\sigma$ is
just the curve $C$ itself,
all that remains to prove of Theorem \ref{dimtheorem} is
the dimension statement; this will be
carried out next.
\subsection{Dimension calculation}
\label{gen-dim}
By Proposition \ref{tgt-space} we have to compute the dimension of the invariant subspace
$H^1({\cal G}(C),{\rm ad}\ F)^{S_3}$ (or in the cyclic case $H^1(C,{\rm ad}\
F)^{\z/3}$). We shall do a little more, in fact, and compute the
decomposition of $H^1({\cal G}(C),{\rm ad}\ F)$ into irreducible
representations under this group action. In the same way, of course,
we could as
easily compute $T_{F}\nn_X$ at a stable Galois bundle for any $X$, but
we shall stick to the trigonal situation.
We shall consider the $S_3$ case first and return to $\z/3$ in a
moment.
For use here and below
we recall the following result of \cite{AB} Theorem 4.12.
\begin{atiyah-bott}
\label{at-bott}
Suppose that $\gamma: X \rightarrow X$ is an automorphism of a
compact
complex manifold $X$ with a finite set ${\rm Fix}(\gamma)\subset X$ of
fixed
points,
and suppose that $\gamma$ lifts to a holomorphic vector bundle
$E\rightarrow X$.
Then
$$
\sum (-1)^p {\rm trace}\ \gamma|_{H^p(X,E)} = \sum_{x\in {\rm Fix}(\gamma)} {{\rm trace}\
\gamma|_{E_x}
\over \det(1-d\gamma_x)}.
$$
\end{atiyah-bott}
Suppose now that
$F\in \nn_C$ is a stable Galois ${\rm Spin}(8)$-bundle for a general
trigonal curve $C$.
We use Atiyah-Bott to compute
the trace of the group elements $\sigma,\tau \in S_3$ acting on
$H^1({\cal G}(C), {\rm ad}\ F)$. By stability $H^0({\cal G}(C), {\rm ad}\ F)=0$
and so for nontrivial
$\gamma \in S_3$:
$$
{\rm trace}\ \gamma|_{H^1({\cal G}(C), {\rm ad}\ F)} = - \sum_{x\in {\rm Fix}(\gamma)} {{\rm trace}\
\gamma|_{{\rm ad}\ F_x}
\over \det(1-d\gamma_x)}.
$$
\underline{$\gamma = \sigma$}:
$\sigma$ has $2g+4-2\delta$ fixed points, and at
each of these $d\sigma_x = -1$. On ${\rm ad}\ F_x$
we need
to compute ${\rm trace}\ \sigma$. But the bundle $F$ is Galois, so
this is just the trace of $\sigma$ in the triality action on ${\mathfrak so}_{8}$,
which by section \ref{liealg} is 14. Thus
$$
{\rm trace}\ \sigma |_{H^1({\cal G}(C), {\rm ad}\ F)} = -7(2g+4-2\delta).
$$
\underline{$\gamma = \tau$}:
we have seen that $\tau$ has $\delta$ {\it pairs} of fixed points, at
which in the tangent space it acts as $\omega$ and $\omega^2$
respectively. From section \ref{liealg}, on the other hand, ${\rm trace}\ \tau
=7$ in the adjoint representation. So by Atiyah-Bott
(and using $1+\omega +\omega^2 =0$) we get
$$
{\rm trace}\ \tau |_{H^1({\cal G}(C), {\rm ad}\ F)} = -7\delta \left\{
{1\over 1-\omega }+{1\over 1-\omega^2}
\right\} = -7\delta.
$$
Note that $\dim H^1({\cal G}(C), {\rm ad}\ F) = 28(3g-\delta)$ (since ${\cal G}(C)$
has genus $3g+1-\delta$); and so we have computed the character of
$H^1({\cal G}(C), {\rm ad}\ F)$ under
the $S_3$ action.
Recall that $S_3$ has three irreducible representations $\c$,
$\c_{\varepsilon}$, $\c^2_{\rho}$ and character table:
\begin{equation}
\label{S3-table}
\begin{array}{l|rrr}
&1&\sigma&\tau\\
\hline
{\bf 1}}\def\two{{\bf 2}}\def\three{{\bf 3}}\def\four{{\bf 4}&1&1&1\\
\varepsilon&1&-1&1\\
\rho&2&0&-1\\
\end{array}
\end{equation}
Here $\c = \c_{{\bf 1}}\def\two{{\bf 2}}\def\three{{\bf 3}}\def\four{{\bf 4}}$ is the trivial representation,
$\c_{\varepsilon}$ the
sign representation
and $\c^2_{\rho}$ the 2-dimensional reflection representation.
Denote the character of $H^1({\cal G}(C), {\rm ad}\ F)$ by $a{\bf 1}}\def\two{{\bf 2}}\def\three{{\bf 3}}\def\four{{\bf 4} +b\varepsilon +
c\rho$. Comparing with the character table gives:
$$
\begin{array}{lcl}
a+b+2c &=& 28(3g-\delta)\\
a-b &=& -7(2g+4-2\delta)\\
a+b-c &=& -7\delta .\\
\end{array}
$$
Solving these equations yields
$a= 7g-14$, $b=21g+14 -14\delta$, $c= 28g-7\delta$,
and hence an $S_3$-decomposition:
$$
H^1({\cal G}(C), {\rm ad}\ F) = \c^{7g-14} \oplus (21g+14-14\delta) \c_{\varepsilon}
\oplus
(28g-7\delta) \c^2_{\rho}.
$$
\begin{rem}\rm
\label{cyclic-dim}
One can make an analogous calculation in the case of a cyclic trigonal curve.
Here we apply the same method as above, with group $\< \tau \> \cong \z/3$
acting
on $C$ itself.
Pick a stable Galois ${\rm Spin}(8)$-bundle $F\in \nn_C$.
As above we compute
the trace of the group elements $\tau ,\tau^2$ acting on
$H^1(C, {\rm ad}\ F)$, where by stability $H^0(C, {\rm ad}\ F)=0$.
By Atiyah-Bott:
$$
\begin{array}{rcl}
{\rm trace}\ \tau |_{H^1(C, {\rm ad}\ F)} & = & -7(g+2)/(1-\omega), \\
{\rm trace}\ \tau^2 |_{H^1(C, {\rm ad}\ F)} & = & -7(g+2)/(1-\omega^2).\\
\end{array}
$$
Suppose that the eigenspace decomposition under $\tau$ is
$H^1(C, {\rm ad}\ F) = \c^a \oplus \c^b \oplus \c^c$ for eigenvalues
$1,\omega,\omega^2$ respectively. Then we have to solve the equations
$$
\begin{array}{lcl}
a+b+c &=& 28(g-1)\\
a+\omega b +\omega^2 c&=& -7(g+2)/(1-\omega)\\
a+\omega^2 b +\omega c &=& -7(g+2)/(1-\omega^2) .\\
\end{array}
$$
Adding the three equations gives again $a= 7g-14$.
\end{rem}
\section{A nonabelian Schottky configuration}
\label{recillas}
We shall show in this section that the moduli space $\nn_C$ contains a
nonabelian `fattening' of the classical Schottky configuration for
trigonal curves. The Schottky configuration is the union of Prym
Kummer varieties embedded in ${\cal SU}_C(2)$; when $C$ is trigonal two things
happen: (a) each Prym is, by the Recillas correspondence,
the Jacobian of a tetragonal curve
$R_{\eta}$, and (b) ${\cal SU}_C(2)$ is by Theorem \ref{dimtheorem}
the `singular locus' of a bigger moduli
space $\nn_C$. We show that $\nn_C$ contains a configuration of the
moduli varieties $\ss{\cal U}}\def\vv{{\cal V}}\def\xx{{\cal X}}\def\ww{{\cal W}_{R_{\eta}}(2)$---each singular, of course,
along the corresponding Prym Kummer.
\subsection{The Schottky configuration}
For each nonzero 2-torsion point $\eta \in J_C[2] \backslash
\{\oo\}$ we have an associated unramified double cover
$$f : C_{\eta} \rightarrow C.$$
We shall denote by $\varphi$ the involution of $C_{\eta}$ given by
sheet-interchange over~$C$; it will denote
also the induced involution of ${\rm Pic}(C_{\eta})$.
The kernel of the norm map on divisors has two isomorphic connected
components:
$$
{\rm Nm}^{-1}(\oo_C) = P_{\eta} \cup P_{\eta}^-,
$$
where $P_{\eta} = (1 - \varphi)J^0_{C_{\eta}}$ and $P_{\eta}^- = (1 -
\varphi)J^1_{C_{\eta}}$.
We shall also consider the 2-component subvariety
${\rm Nm}^{-1}(\eta) \subset J_{C_{\eta}}$; this is a torsor over ${\rm Nm}^{-1}(\oo_C)$
whose components are exchanged under translation by elements of $P_{\eta}^-$.
Note that since ${\rm ker \ } f^* = \{\oo,\eta\}$ and $f^* \circ {\rm Nm} = 1+\varphi$
it follows that the set of anti-invariant (degree 0) line bundles on $C_{\eta}$
is
$$
{\rm ker \ } (1+\varphi) = {\rm Nm}^{-1}(\oo_C)\cup {\rm Nm}^{-1}(\eta).
$$
$P_{\eta} \subset {\rm Nm}^{-1}(\oo_C)$ is distinguished as the component
containing the origin $\oo\in J_{C_{\eta}}$; the component
$P_{\eta}^- \subset {\rm Nm}^{-1}(\oo_C)$ contains the Prym-canonical curve
$C_{\eta} \map{1 - \varphi}P_{\eta}^-$. The components of $ {\rm Nm}^{-1}(\eta)$, on the other hand, are indistinguishable.
\begin{rem}\rm
The $[-1]$-action on $ J_{C_{\eta}}$ preserves each of these four
components (acting as sheet-involution $\varphi$): clearly it permutes
the components isomorphically, and so it suffices to note that each
component contains fixed points, i.e. points of $J_{C_{\eta}}[2]$.
\end{rem}
For each $\eta \in J_C[2] \backslash \{\oo\}$ there exist canonical maps
$$
{\rm Nm}^{-1}(\eta) \rightarrow {\cal SU}_C(2) \rightarrow |2\Theta|
$$
where the first map is just direct image $x \mapsto f_* x$; this gives a
semistable rank 2 vector bundle of determinant $\det f_* x = {\rm Nm}(x)\otimes
\det f_* \oo_{C_{\eta}} = \oo_C$. This map obviously factors through
the sheet-involution, and so the image is a pair of Prym-Kummer varieties.
As $\eta \in J_C[2]$ varies,
the configuration of these Kummers in ${\cal SU}_C(2) \subset |2\Theta |$, including
(for $\eta = \oo$) the Jacobian Kummer, is called the {\it Schottky
configuration}.
It is usually convenient to view the Kummer varieties of the Schottky
configuration
as the images of ${\rm Nm}^{-1}(\oo_C) = P_{\eta} \cup P_{\eta}^-$, though this
can only be done up to the translation action of $J_C[2]/\eta$. For this
we introduce the $J_C[2]$-torsor
$$
S(\eta) = \{\zeta\ |\ \zeta^2 = \eta\} \subset J_C.
$$
Notice that pull-back gives an
isomorphism of groups
$$
f^* : J_C[2]/\eta \mathbin{\hbox{$\,\widetilde\rightarrow\,$}} {\rm Nm}^{-1}(\oo_C)[2] := {\rm Nm}^{-1}(\oo_C)\cap
J_{C_{\eta}}[2],
$$
and correspondingly an isomorphism of torsors
$$
f^* : S(\eta)/\eta \mathbin{\hbox{$\,\widetilde\rightarrow\,$}} {\rm Nm}^{-1}(\eta)[2].
$$
Each choice of $\zeta \in S(\eta)/\eta$ determines an identification
${\rm Nm}^{-1}(\oo_C) \mathbin{\hbox{$\,\widetilde\rightarrow\,$}} {\rm Nm}^{-1}(\eta)$ by translation $x\mapsto f^*(\zeta)
\otimes x$. This sends 2-torsion points to 2-torsion points, and determines
a map
$$
{\rm Sch}_{\zeta}:
P_{\eta} \cup P_{\eta}^- \rightarrow {\cal SU}_C(2)
$$
by $x \mapsto f_*(f^*(\zeta)
\otimes x)=\zeta \otimes f_* x$. The image is
independent of the choice of $\zeta$ and is precisely the fixed-point
set of the involution $\otimes \eta$ of ${\cal SU}_C(2)$. Moreover, the
linear span of the image of $P_{\eta}$ in $|2\Theta|$
can be canonically identified with the linear
series $|2\Xi|$ where $\Xi$ is the canonical theta divisor on the
dual abelian variety, and represents the principal polarisation on
$P_{\eta}$ induced from that on $J(C_{\eta})$. Thus for each $\zeta \in
S(\eta)/\eta$
there is a commutative diagram
\begin{equation}
\label{prymkum}
\begin{array}{ccc}
P_{\eta} & \map{{\rm Sch}_{\zeta}} & {\cal SU}_C(2) \\
&&\\
\downarrow &&\downarrow\\
&&\\
|2\Xi| &\hookrightarrow & |2\Theta|.\\
\end{array}
\end{equation}
\subsection{$S_4$-curves over $\bp^1$}
When the curve $C$ is trigonal the Pryms of the Schottky configuration are
in fact Jacobians, by a well-known construction of Recillas.
We shall next review this construction, as formulated
by Donagi \cite{D}.
To any trigonal curve $C\map{3:1} \bp^{1}$
together
with a nontrivial 2-torsion point $\eta \in J_{C}[2]$ one can associate in a
natural way a tetragonal curve $R_{\eta}\map{4:1} \bp^{1}$ whose
polarised Jacobian is isomorphic to the Prym variety
of $C_{\eta} \rightarrow C$.
By definition $R_{\eta}$ is one of two isomorphic connected
components
of the fibre product
$$
\begin{array}{ccc}
R_{\eta}\cup R_{\eta}' &\subset & S^3 C_{\eta}\\
&&\\
\downarrow &&\downarrow \\
&&\\
\bp^1 &\hookrightarrow & S^3 C \\
\end{array}
$$
where $\bp^1 \hookrightarrow S^3 C$ is the trigonal pencil, and the
right-hand vertical map is the 8:1 cover induced from
$C_{\eta}\rightarrow C$.
Conversely $C_{\eta} = S^2_{\bp^1} R_{\eta} \subset S^2 R_{\eta}$
with
the obvious involution. In particular, $C_{\eta}$ and $R_{\eta}$ are
in
(3,2)-correspondence, and we shall denote the incidence curve by
${\cal I} \subset C_{\eta}\times R_{\eta}$.
We shall refer to $R_{\eta}$ as the {\it
Recillas curve}, and we shall consider its Galois closure
${\cal G}(R_{\eta})\rightarrow \bp^{1}$ with Galois group $S_{4}$. As in
section \ref{trig-pics} the points of ${\cal G}(R_{\eta})$ correspond to
orderings of the $g^{1}_{4}$-divisors in $R_{\eta}$.
\begin{notat}\rm
\label{S4notation}
We shall view $S_{4}$ as the permutation group of $\{0,1,2,3\}$,
containing $S_{3}$, the permutations of $\{1,2,3\}$, as a subgroup.
We shall fix elements $\sigma = (23)$, $\tau = (123)$ (as in earlier
sections) and also $\sigma' = (01)$, $\varphi = (0213)$.
(Note that $\varphi$ is uniquely determined up to inverse by the
condition
$\sigma \sigma'=\varphi^2$. The reason for using the same notation $\varphi$
as for the sheet-involution of $C_{\eta}$ will appear in a moment.)
We shall then refer to the subgroups `Klein', generated by the
conjugates of
$\sigma \sigma'=\varphi^2$, `dihedral' $= \<\sigma ,\varphi\>$ and
$A_{4}$ the alternating subgroup.
$S_4$ has five irreducible representations
and character table:
$$
\begin{array}{c|rrrrr}
& 1 & \sigma & \tau & \varphi & \sigma \sigma' \\
\hline
{\bf 1}}\def\two{{\bf 2}}\def\three{{\bf 3}}\def\four{{\bf 4} & 1 & 1 & 1 & 1 & 1 \\
\varepsilon & 1 & -1 & 1 & -1 & 1 \\
\rho & 3 & 1 & 0 & -1 & -1 \\
\rho \otimes \varepsilon & 3 & -1 & 0 & 1 & -1 \\
\two & 2 & 0 & -1 & 0 & 2
\end{array}
$$
Here $\rho$ is the 3-dimensional reflection representation
and $\two$ is the 2-dimensional reflection representation of
$S_{4}/{\rm Klein} \cong S_{3}$.
\end{notat}
The various curves in the Recillas construction are related by the
following
diagram which extends (\ref{S3galois}):
\begin{equation}
\label{S4galois}
\begin{array}{ccccc}
& & {\cal G}(R_{\eta}) & & \\
& \swarrow & & \searrow & \\
{\cal G}(C)\times_{C}C_{\eta} & & & & {\cal I} \\
&&&&\\
\downarrow & \searrow & &\textstyle 2:1\swarrow & \\
&&&&\\
{\cal G}(C) & & C_{\eta} & & \downarrow\textstyle 3:1 \\
&&&&\\
\downarrow & \searrow & \downarrow & & \\
&&&&\\
^{g+1-\delta}H & & ^{g}C & & ^{g-1}R_{\eta} \\
& \searrow & \downarrow & \swarrow & \\
&&&&\\
& & \bp^{1} & & \\
\end{array}
\end{equation}
By comparison of \S\ref{trig-pics} and \cite{D} p.74, one sees that
the Galois cover
${\cal G}(R_{\eta}) \rightarrow \bp^1$ has branching behaviour:
\begin{center}
\leavevmode
\epsfxsize=2.5in\epsfbox{recillas.eps}
\end{center}
By Riemann-Hurwitz it follows that ${\cal G}(R_{\eta})$
has
genus $12g+1-4\delta$, and in particular we see:
\begin{lemm}
\label{etale}
The 4:1 map
${\cal G}(R_{\eta})
\rightarrow {\cal G}(C)$ is unramified.
\end{lemm}
Diagram (\ref{S4galois}) is
Galois-dual to the tower of subgroups (and these diagrams
coincide with those of \cite{D}~p.73):
\begin{equation}
\label{S4galoisdual}
\begin{array}{ccccc}
& & 1 & & \\
& \swarrow & & \searrow & \\
\<\sigma \sigma'\> & & & & \<\sigma \> \\
\downarrow & \searrow & & \swarrow & \\
&&&&\\
{\rm Klein} & & \<\sigma, \sigma'\> & & \downarrow \\
&&&&\\
\downarrow & \searrow & \downarrow & & \\
&&&&\\
A_{4} & & {\rm dihedral} & & S_{3} \\
& \searrow & \downarrow & \swarrow & \\
&&&&\\
& & S_{4} & &
\end{array}
\end{equation}
Note that the quotient dihedral$/\<\sigma, \sigma'\> \cong \z/2$ is generated
by $\varphi$, which therefore acts as sheet-interchange of $C_{\eta}$ over $C$,
consistently with our earlier notation.
\begin{prop}
\label{Hodgedecomp}
As $S_{4}$-modules we have
$$
H^{1}({\cal G}(R_{\eta}),\oo) = (g+1-\delta)\c_{\varepsilon} \oplus
g \c^{2}_{\two} \oplus
(g-1) \c^{3}_{\rho} \oplus
(2g+1-\delta)\c^{3}_{\rho\otimes \varepsilon}.
$$
\end{prop}
{\it Proof.}\
One proceeds the same way as in the dimension computation of \S\ref{gen-dim}.
By Atiyah-Bott we have, for 2-cycles $\sigma \in S_4$,
$$
{\rm trace}\ \sigma|_{H^1({\cal G}(R_{\eta}),\oo)} =
1-{1\over 2}|{\rm Fix}(\sigma)|=
-2g-3+2\delta;
$$
for 3-cycles $\tau \in S_4$,
$$
{\rm trace}\ \tau|_{H^1({\cal G}(R_{\eta}),\oo)} =
1-\delta \left\{ {1\over 1-\omega} +{1\over 1-\omega^2} \right\}
=
1-\delta;
$$
while the trace of other nontrivial group elements is 1,
and the identity element of $S_4$ has trace $g({\cal G}(R_{\eta}))=
12g+1-4\delta$.
If the representation on $H^1({\cal G}(R_{\eta}),\oo)$ has character
$a{\bf 1}}\def\two{{\bf 2}}\def\three{{\bf 3}}\def\four{{\bf 4} + b\varepsilon + c\rho + d\rho \otimes \varepsilon + e\two$
for integers $a,b,c,d,e$, then
comparing with the character table \ref{S4notation}
we have to solve:
$$
\begin{array}{rrrrrrr}
12g+1-4\delta &=&a&+b&+3c&+3d&+2e \\
-2g-3+2\delta&=&a&-b&+c&-d& \\
1-\delta&=&a&+b&&&-e \\
1&=&a&-b&-c&+d& \\
1&=&a&+b&-c&-d&+2e \\
\end{array}
$$
These equations give $(a,b,c,d,e)=(0,g+1-\delta,g-1,2g+1-\delta,g)$.
{\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par}
We can now describe the polarised isomorphism $P_{\eta} \cong
J_{R_{\eta}}$ in this setting. Both abelian varieties embed by
pull-back in the Jacobian of the top curve ${\cal G}(R_{\eta})$; and
this Jacobian is acted on by the group-ring $\z[S_4]$.
\begin{prop}
\label{recillasisom}
The action of $1+\varphi^2 \in \z[S_4]$ on the Jacobian
$J_{{\cal G}(R_{\eta})}$ restricts to the polarised isomorphism of
Recillas
$\psi: J_{R_{\eta}}\mathbin{\hbox{$\,\widetilde\rightarrow\,$}} P_{\eta}$. Moreover, $2 \psi^{-1}:P_{\eta}
\rightarrow J_{R_{\eta}}$ is the restriction of $1+\tau + \tau^2 \in
\z[S_4]$.
\end{prop}
{\it Proof.}\
It is easy to identify the subspaces covering (the pull-backs of)
these two abelian varieties in the $S_{4}$-decomposition of
Proposition \ref{Hodgedecomp}: for $J_{R_{\eta}}$ we look for
$S_3$-invariant
vectors, while for $P_{\eta}$ we look for $\<\sigma, \sigma'\>$
invariant
vectors on which the sheet-involution $\varphi : C_{\eta}
\leftrightarrow
C_{\eta}$ over $C$ acts by $-1$.
One quickly finds that these are all in the component
$(g-1) \otimes \c^{3}_{\rho}$,
where
$$
\c^{3}_{\rho} = \{\boldx \in \c^4 \ |\ x_1 + x_2 + x_3 + x_4 =0\}
$$
with $S_4$ permuting coordinates. The two varieties are covered,
respectively,
by the subspaces
$$
\begin{array}{rclrcll}
H^1(R_{\eta}, \oo) &=& \c^{g-1}\otimes {\bf l}}\def\boldm{{\bf m}}\def\boldn{{\bf n}}\def\boldx{{\bf x}, &\textstyle
{\bf l}}\def\boldm{{\bf m}}\def\boldn{{\bf n}}\def\boldx{{\bf x} &=& {1\over 2}(3,-1,-1,-1)&\in
\c^{3}_{\rho},\\
H^1_-(C_{\eta},\oo) &=& \c^{g-1}\otimes \boldn, &
\boldn &=& (1,1,-1,-1)&\in \c^{3}_{\rho}.\\
\end{array}
$$
One notes at once (see \ref{S4notation}) that $(1+\varphi^2){\bf l}}\def\boldm{{\bf m}}\def\boldn{{\bf n}}\def\boldx{{\bf x} =
\boldn$
and
$(1+\tau + \tau^2)\boldn = 2 {\bf l}}\def\boldm{{\bf m}}\def\boldn{{\bf n}}\def\boldx{{\bf x}$.
This demonstrates the proposition as far as an {\it isogeny}
$\psi: J_{R_{\eta}}\rightarrow P_{\eta}$; we have to see that $\psi$
is the Recillas isomorphism as claimed.
But the latter, by construction, is induced from the
inclusion $C_{\eta} \subset S^2 R_{\eta}$, and this fits into a
commutative diagram:
$$
\begin{array}{ccc}
{\cal G}(R_{\eta}) & \map{1+\varphi^2} & S^2 {\cal G}(R_{\eta})\\
&&\\
\downarrow && \downarrow \\
&&\\
C_{\eta} & \hookrightarrow & S^2 R_{\eta}\\
\end{array}
$$
We conclude that via pull-back, $1+\varphi^2$ gives the polarised
isomorphism $\psi: J_{R_{\eta}}\mathbin{\hbox{$\,\widetilde\rightarrow\,$}} P_{\eta}$.
{\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par}
\subsection{Trigonal Schottky}
Given a choice of $\zeta \in S(\eta)/\eta$ we can identify $P_{\eta}$
with one component of ${\rm Nm}^{-1}(\eta)$. When $C$ is trigonal $P_{\eta}$
is canonically identified with the Jacobian $J_{R_{\eta}}$, and the latter
(more precisely, its Kummer) can therefore be viewed as a component
of the Schottky configuration:
$$
{\rm Sch}_{\zeta}:
J_{R_{\eta}} \rightarrow {\rm Nm}^{-1}(\eta) \rightarrow {\cal SU}_C(2).
$$
In Proposition \ref{rec_bundles}
we shall describe explicitly how rank 2 vector bundles on $C$
are constructed from line bundles on $R_{\eta}$ using diagrams
(\ref{S4galois}) and (\ref{S4galoisdual});
in the next section we will
then extend the construction to rank 2 vector bundles on $R_{\eta}$.
Let us denote by $\pi_R$ the map ${\cal G}(R_{\eta})\rightarrow R_{\eta}$; then
for any line bundle $L\in J_{R_{\eta}}$ we consider the collection of
bundles on ${\cal G}(R_{\eta})$ obtained by applying the
$S_4$-action
to $L_0 := \pi_R^* L$. Since $L_0$ is $S_3$-invariant there are just
four
isomorphism classes, and these are in 1:1 correspondence with the left
cosets
of $S_3 < S_4$. If we label the bundles as
$$
\begin{array}{rcl}
L_1 &=& \varphi^2 L_0 \\
L_2 &=& \varphi L_0 \\
L_3 &=& \varphi^3 L_0 \\
\end{array}
$$
then the Galois action of $S_3 < S_4$ will correspond to the usual
permutation action on the
subscripts $1,2,3$ (since $\varphi = (0213)$).
\begin{rem}\rm
Note that over a divisor $w+x+y+z \in g^{1}_{4} = \bp^{1}$ a point of
the Galois curve represents an ordering of $w,x,y,z$ and the fibres
of the bundles $L_{0},L_{1},L_{2},L_{3}$ over this point are
canonically
isomorphic, respectively, to the fibres $L_{w},L_{y},L_{x},L_{z}$ of $L$.
\end{rem}
We now consider the rank 2 vector bundle
$$
E'_{L,N} = NL_0 L_1 \oplus N^{-1} L_2 L_3
$$
where $N$ is a line bundle on ${\cal G}(R_{\eta})$ which will be chosen in a
moment.
First note that if $N$ is trivial then $E' = E'_{L,\oo}$ is invariant under
the dihedral group $\<\sigma , \varphi \>$, since $\sigma E' = L_0 L_1
\oplus L_3 L_2 = E'$ (acting trivially in the fibres at fixed points of
$\sigma$), and $\varphi E' = L_2 L_3 \oplus L_1 L_0 = E'$ (with no fixed points
of $\varphi$ to consider since $C$ is smooth). It follows that $E'_{L,\oo}$
descends to a bundle on~$C$.
More generally, the same is true of $E'_{L,N}$ provided $N$ is the pull-back
of a line bundle in
$$
{\rm ker \ }(1+\varphi) = {\rm Nm}^{-1}(\oo_C)\cup {\rm Nm}^{-1}(\eta)\subset J_{C_{\eta}},
$$
since then $N$ will be invariant
under $\sigma$ and anti-invariant under $\varphi$.
On the other hand, it follows from Proposition \ref{Hodgedecomp} that
$\pi_R^* J_{R_{\eta}}
\subset {\rm ker \ }(1+\varphi+\varphi^2+\varphi^3)$; hence $\det E'_{L,N} =
\oo_{{\cal G}(R_{\eta})}$. Consequently the bundle $E_{L,N} \rightarrow C$
which pulls back to $E'_{L,N}$ satisfies
$$
\det E_{L,N} \in \{\oo_C,\eta\}.
$$
It turns out that $\det E_{L,N} = \oo_C$ if and only if $N$ comes
from ${\rm Nm}^{-1}(\eta)$; and that we recover the Schottky map
${\rm Sch}_{\zeta}:
J_{R_{\eta}}
\rightarrow {\cal SU}_C(2)$ if we choose $N$ to be the pull-back of $\zeta \in
{\rm Nm}^{-1}(\eta)[2] \cong S(\eta)/\eta$:
\begin{prop}
\label{rec_bundles}
For each $\zeta \in S(\eta)/\eta$ the following diagram commutes:
$$
\begin{array}{rcl}
&&P_{\eta} \\
&&\\
&{\rm Recillas}\nearrow\qquad\quad & \downarrow {\rm Sch}_{\zeta} \\
&&\\
J_{R_{\eta}} & \rightarrow & {\cal SU}_C(2) \\
&&\\
L &\mapsto & E_{L,\zeta} \\
\end{array}
$$
where $E_{L,\zeta}$ is defined by $\pi_{C_{\eta}}^* f^*E_{L,\zeta}=
E'_{L,\pi_{C_{\eta}}^*\circ f^*(\zeta)}$, for the maps
$
{\cal G}(R_{\eta}) \map{\pi_{C_{\eta}}} C_{\eta} \map{f} C.
$.
\end{prop}
{\it Proof.}\
By Proposition \ref{recillasisom} the Recillas map $J_{R_{\eta}}
\rightarrow P_{\eta}$ is given, in the Jacobian of ${\cal G}(R_{\eta})$,
by $L \mapsto (1+\varphi^2)L = L_0L_1$. We shall view the latter as a line
bundle on $C_{\eta}$; we then have to show that its image $\zeta \otimes
f_*(L_o L_1) \in {\cal SU}_C(2)$ pulls back to $E'_{L,\pi_{C_{\eta}}^*\circ f^*(\zeta)}$.
But this pull-back is
$$
\begin{array}{rcl}
\pi_{C_{\eta}}^* f^*(f_*(f^*\zeta \otimes L_0 L_1))&=&
\pi_{C_{\eta}}^*((f^*\zeta)\otimes L_0 L_1 \oplus \varphi((f^*\zeta)\otimes
L_0 L_1) ) \\
&=& \pi_{C_{\eta}}^*((f^*\zeta)\otimes L_0 L_1 \oplus (f^*\zeta^{-1})\otimes
L_2 L_3) \\
&=& E'_{L,\pi_{C_{\eta}}^*\circ f^*(\zeta)}.
\end{array}
$$
{\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par}
\subsection{Nonabelian Schottky}
Extending the preceding construction, we can
use the diagrams (\ref{S4galois}) and
(\ref{S4galoisdual})
to construct Galois
${\rm Spin}(8)$-bundles from rank~2 vector bundles on the Recillas curve.
Given a rank~2 vector bundle $E\rightarrow R_{\eta}$ we consider the
bundles on ${\cal G}(R_{\eta})$ obtained by applying the
$S_4$-action
to $E_0 := \pi_{R}^* E$. Again, $S_3$-invariance implies that there are just
four
isomorphism classes in 1:1 correspondence with the left
cosets
of $S_3 < S_4$. We write
$$
\begin{array}{rcl}
E_1 &=& \varphi^2 E_0 \\
E_2 &=& \varphi E_0 \\
E_3 &=& \varphi^3 E_0. \\
\end{array}
$$
The Galois action of $S_4$ on the the bundles $E_0,E_1,E_2,E_3$ then
coincides with permutation of the indices.
\begin{lemm-def}
\label{rec_gal_bundles}
Fix an element $\zeta \in S(\eta)/\eta$, and denote its pull-back
to ${\cal G}(R_{\eta})$ by $\zeta'\in J_{{\cal G}(R_{\eta})}[2]$.
Given a semistable rank 2 vector bundle $E\rightarrow R_{\eta}$ with $\det E =
\oo$, the rank 8 orthogonal bundles (where $E_{0}E_{1} = E_{0}\otimes
E_{1}$ etc)
$$
\begin{array}{ccc}
V_{E,\zeta} & = & \zeta'\otimes (E_{0}E_{1}\oplus E_{2}E_{3}), \\
S^+_{E,\zeta} & = & \tau\zeta'\otimes (E_{0}E_{2}\oplus E_{3}E_{1}), \\
S^{-}_{E,\zeta} & = & \tau^2\zeta'\otimes (E_{0}E_{3}\oplus E_{1}E_{2}),
\end{array}
$$
are semistable and
descend to a Galois ${\rm Spin}(8)$-bundle $F_{E,\zeta} \in
\nn_C$.
\end{lemm-def}
{\it Proof.}\
By Proposition \ref{spin-sum} these three bundles are the standard
representations of a ${\rm Spin}(8)$-bundle, and by construction they are
permuted isomorphically
by the 3-cycle $\tau \in S_{4}$ under the Galois action.
The involution $\sigma$, on the other hand, lifts to an involution
of $V_F$ with 1-dimensional $-1$-eigenspace $\bigwedge^2 (E_2)_x$
at each fixed point $x\in {\cal G}(R_{\eta})$.
(So the dihedral invariance and descent to the curve $C$ of the
previous section {\it fail} in this rank 2 case.)
However, the triple is invariant under the Klein subgroup and
by Lemma \ref{etale} the map ${\cal G}(R_{\eta}) \rightarrow {\cal G}(C)$ is
unramified. It follows that the bundles descend to ${\cal G}(C)$ where they
satisfy the conditions of Proposition \ref{galoisbundle}.
{\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par}
\begin{rem}\rm
Observe that if we define $R_{0} = C \cup \bp^{1}$ and extend any
rank~2 bundle $E\rightarrow C$ to the trivial bundle on the component
$\bp^{1}$, then the above construction reduces to that of
Lemma~\ref{su2trigonal}.
\end{rem}
We shall write ${\rm Sch}_{\zeta}: \ss{\cal U}}\def\vv{{\cal V}}\def\xx{{\cal X}}\def\ww{{\cal W}_{R_{\eta}}(2) \rightarrow \nn_C$
for the map determined by \ref{rec_gal_bundles}, and verify that this
is consistent with the notation for Pryms:
\begin{theo}
\label{nonab-schottky}
For each $\eta \in J_C[2]$ and choice of $\zeta \in S(\eta)/\eta$
the following diagram commutes:
$$
\begin{array}{ccc}
P_{\eta}& \mathbin{\hbox{$\,\widetilde\rightarrow\,$}} J_{R_{\eta}} \rightarrow &\ss{\cal U}}\def\vv{{\cal V}}\def\xx{{\cal X}}\def\ww{{\cal W}_{R_{\eta}}(2)\\
&&\\
{\rm Sch}_{\zeta}\downarrow \qquad&& \qquad\downarrow{\rm Sch}_{\zeta}\\
&&\\
\ss{\cal U}}\def\vv{{\cal V}}\def\xx{{\cal X}}\def\ww{{\cal W}_C(2)&\longrightarrow& \nn_C\\
\end{array}
$$
\end{theo}
{\it Proof.}\
Let us fix the following notation for the
various maps which we will need:
$$
\begin{array}{ccccc}
&&{\cal G}(R_{\eta})&&\\
&\pi_{{\cal G}(C)}\swarrow&&\searrow \pi_{C_{\eta}}\\
{\cal G}(C)&&&&C_{\eta}\\
&\pi_C \searrow &&\swarrow f&\\
&&C&&\\
\end{array}
$$
Pick line bundles $L\in J_{R_{\eta}}$ and $N \in P_{\eta}$
corresponding
under the Recillas isomorphism. $L$ gives rise to a Galois bundle
$F_{L,\zeta} = (V_{L,\zeta},S_{L,\zeta}^+,S_{L,\zeta}^-)$ on ${\cal G}(C)$
(via the right-hand side of the diagram) which pulls back to the
triple
of orthogonal bundles on ${\cal G}(R_{\eta})$ (by \ref{rec_gal_bundles}):
\begin{equation}
\label{Lsplitting}
\begin{array}{rcl}
\pi_{{\cal G}(C)}^* V_{L,\zeta} & = & \zeta'\otimes\left\{
(L\oplus L^{-1})(\varphi^2 L\oplus \varphi^2 L^{-1})\oplus
(\varphi^3 L\oplus \varphi^3 L^{-1})(\varphi L\oplus \varphi L^{-1})\right\}
, \\
\pi_{{\cal G}(C)}^* S^+_{L,\zeta} & = & \tau\zeta'\otimes\left\{
(L\oplus L^{-1})(\varphi L\oplus \varphi L^{-1})\oplus
(\varphi^2 L\oplus \varphi^2 L^{-1})(\varphi^3 L\oplus \varphi^3
L^{-1})\right\}
, \\
\pi_{{\cal G}(C)}^* S^{-}_{L,\zeta} & = & \tau^2\zeta'\otimes\left\{
(L\oplus L^{-1})(\varphi^3 L\oplus \varphi^3 L^{-1})\oplus
(\varphi L\oplus \varphi L^{-1})(\varphi^2 L\oplus \varphi^2
L^{-1})\right\}.\\
\end{array}
\end{equation}
Via the left-hand side of the diagram, on the other hand,
$N$ gives
rise to a Galois bundle $F_{N,\zeta}\in \nn_C$ represented by a
triple $(V_{N,\zeta},S_{N,\zeta}^+,S_{N,\zeta}^-)$ and we shall show that these bundles split
under
pull-back to ${\cal G}(R_{\eta})$ into the same line bundle summands as
(\ref{Lsplitting}).
Starting with $N \in P_{\eta}$ we construct
$E_{N,\zeta} = \zeta\otimes f_* N \rightarrow C$; then
$V_{N,\zeta} =
\c^2 \otimes \pi_C^*E_{N,\zeta} \oplus \tau(\pi_C^*E_{N,\zeta})\otimes
\tau^2(\pi_C^*E_{N,\zeta})$ etc.
Now
$$
\begin{array}{rcl}
\pi_{{\cal G}(C)}^*(\pi_C^* E_{N,\zeta}) &=&
\pi_{C_{\eta}}^* f^* (\zeta\otimes f_* N) \\
&=& \zeta'\otimes\pi_{C_{\eta}}^* (N\oplus \varphi(N))\\
&=& \zeta'\otimes\pi_{C_{\eta}}^*(N\oplus N^{-1}),\\
\end{array}
$$
where we use the fact that $N$ is anti-invariant under the
sheet-involution
$\varphi: C_{\eta} \leftrightarrow C_{\eta}$. It follows (denoting
$\pi_{C_{\eta}}^* N$ by $N$ for simplicity, and using the fact that
$\zeta' \otimes \tau\zeta' \otimes \tau^2\zeta'=\oo$)
that
\begin{equation}
\label{Nsplitting}
\begin{array}{rcrcl}
\pi_{{\cal G}(C)}^* V_{N,\zeta} & = & \zeta'\otimes \left\{ \right.
2N \oplus 2N^{-1}&\oplus& (\tau N\oplus \tau N^{-1})
(\tau^2 N\oplus \tau^2 N^{-1})\left. \right\}, \\
\pi_{{\cal G}(C)}^* S^+_{N,\zeta} & = & \tau\zeta'\otimes \left\{ \right.
2\tau N \oplus 2\tau N^{-1}&\oplus& (\tau^2 N\oplus \tau^2 N^{-1})
(N\oplus N^{-1})\left. \right\}
, \\
\pi_{{\cal G}(C)}^* S^{-}_{N,\zeta} & = & \tau^2\zeta'\otimes \left\{ \right.
2\tau^2 N \oplus 2\tau^2 N^{-1}&\oplus& (N\oplus N^{-1})
(\tau N\oplus \tau N^{-1})\left. \right\}.
\\
\end{array}
\end{equation}
For each of the bundles $V_{L,\zeta},V_{N,\zeta}$ we expand the expression in
(\ref{Lsplitting})
and (\ref{Nsplitting}). Each line bundle summand is represented by a
vector
in $\c^{g-1}\otimes \c^3_{\rho} \subset H^1({\cal G}(R_{\eta}),\oo)$.
(See the proof of Proposition \ref{recillasisom}.)
By hypothesis $L,N$ are represented respectively by $\boldx \otimes
{\bf l}}\def\boldm{{\bf m}}\def\boldn{{\bf n}}\def\boldx{{\bf x},
\boldx \otimes \boldn$ for some $\boldx \in \c^{g-1}$, where
${\bf l}}\def\boldm{{\bf m}}\def\boldn{{\bf n}}\def\boldx{{\bf x} = {1\over 2}(3,-1,-1,-1)$ and $\boldn = (1,1,-1,-1)$.
We can therefore identify
each summand, as a vector in this representation,
using the $S_4$-action. For example, $\varphi L$
is given by the vector $\boldx \otimes \varphi {\bf l}}\def\boldm{{\bf m}}\def\boldn{{\bf n}}\def\boldx{{\bf x} = \boldx
\otimes
(-1,-1,3,-1)$, and $L \otimes \varphi L$ by
$\boldx \otimes ({\bf l}}\def\boldm{{\bf m}}\def\boldn{{\bf n}}\def\boldx{{\bf x} + \varphi {\bf l}}\def\boldm{{\bf m}}\def\boldn{{\bf n}}\def\boldx{{\bf x}) = \boldx \otimes
(2,-2,2,-2)$.
Computing in this way, we find that the summands of
$\pi_{{\cal G}(C)}^* V_{L,\zeta}$ are $\boldx$ tensored with the vectors:
$$
\begin{array}{l}
\pm (1,1,-1,-1),\\
\pm (1,1,-1,-1),\\
\pm (2,-2,0,0),\\
\pm (0,0,2,-2).\\
\end{array}
$$
We now play the same game with $\pi_{{\cal G}(C)}^* V_{N,\zeta}$, representing
$N$ by
$\boldx\otimes (1,1,-1,-1)$, $\tau N$ by $\boldx\otimes (1,-1,1,-1)$
etc.
After computing we find that $\pi_{{\cal G}(C)}^* V_{N,\zeta}$ is represented by
{\it exactly
the same eight vectors} as above. We conclude that $\pi_{{\cal G}(C)}^*
V_{L,\zeta} \cong
\pi_{{\cal G}(C)}^* V_{N,\zeta}$, and hence by construction $V_{L,\zeta}
\cong V_{N,\zeta}$.
Repeating this calculation for the spinor bundles shows likewise that
$S_{L,\zeta}^{\pm} \cong S_{N,\zeta}^{\pm}$, and hence $F_{L,\zeta} =
F_{N,\zeta}$.
{\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par}
\begin{rem}\rm
There is a natural `Heisenberg' action of $J_{C}[2]$ on the space
$\nn_{C}$: for $\eta \in J_{C}[2]$ we define
$$
\begin{array}{ccc}
V_{F} & \mapsto & \pi_C^{*}\eta \otimes V_{F}, \\
S_{F}^{+} & \mapsto & \tau(\pi_C^{*}\eta) \otimes S_{F}^{+} , \\
S_{F}^{-} & \mapsto & \tau^{2}(\pi_C^{*}\eta) \otimes S_{F}^{-}.
\end{array}
$$
The three image bundles are cyclically permuted by $\tau$, and the
last two exchanged by $\sigma$, and so they define a Galois
${\rm Spin}(8)$-bundle $\eta \cdot F \in \nn_C$. Moreover, one readily
checks, using the relation $\pi_C^{*}\eta \otimes \tau(\pi_C^{*}\eta)
\otimes
\tau^{2}(\pi_C^{*}\eta)= \oo$,
that the map $ \ss{\cal U}}\def\vv{{\cal V}}\def\xx{{\cal X}}\def\ww{{\cal W}_{C}(2) \hookrightarrow
\nn_{C}$ is equivariant where $J_{C}[2]$ acts by tensor product on
rank 2 vector bundles.
We expect that the fixed-point set of this involution is precisely
the `Schottky locus'
$\ss{\cal U}}\def\vv{{\cal V}}\def\xx{{\cal X}}\def\ww{{\cal W}_{R_{\eta}}(2) \cup \ss{\cal U}}\def\vv{{\cal V}}\def\xx{{\cal X}}\def\ww{{\cal W}_{R'_{\eta}}(2)$.
\end{rem}
|
1,116,691,499,308 | arxiv | \section{Introduction}
Gravitational-wave~(GW) measurements of the tidal deformability
of neutron stars~(NSs) have opened a new window to study
the properties of matter beyond the nuclear saturation point
within stellar cores~\cite{Hinderer:2018mrj,Chatziioannou:2020pqz}.
Equations of state with different stiffness provide tidal
deformabilities which may vary up to an order of magnitude. This
effect magnifies the details of the underlying microscopic model,
allowing to probe how fundamental interactions behave in extreme
regimes \cite{Lattimer:2006xb,Maselli:2018fay}.
In this paper, we argue that the very same situation can occur if boson stars~(BSs)~\cite{kaup1968,ruffini1969} exist in the Universe and form coalescing binaries within the horizon of current and future detectors. BSs are self-gravitating condensates of a bosonic field (see Refs.~\cite{liebling2017review,jetzer1992review} for some reviews). In their original and simplest proposal, they are solutions to Einstein gravity minimally coupled to a classical field theory for a complex scalar $\phi$:
\begin{equation}
{\cal L}_{\rm scalar}=\frac{1}{2}\partial_\mu\phi^\star\partial^\mu\phi+V\left(\left|\phi\right|\right)\,, \label{Lagrangian}
\end{equation}
where a star denotes complex conjugation and $V$ is the scalar self potential. The latter plays the same role as the equation of state for NSs: different microscopic interactions give rise to macroscopically different properties of the boson stars.
Depending on the mass of the boson field and on the self-interaction terms, BSs can exist in any mass range and can have a compactness comparable to or larger than that of a NS. It is intriguing that current GW measurements cannot exclude the existence of exotic compact objects other than black holes~(BHs) and NSs, especially for GW events in the low-mass~\cite{Abbott:2020khf} and high-mass gap, where neither BHs nor NSs are predicted in the standard scenario.
As a case study, in this paper we consider a simple class of quartic interactions [see Eq.~\eqref{potential:2}] and quantify the accuracy within which a GW detection of a BS coalescence can constrain the fundamental parameters (boson mass and coupling constants) of a given scalar field theory.
We focus on the inspiral phase, which can be accurately modeled with post-Newtonian~(PN) theory~\cite{blanchet2014review,PoissonWill}. Up to $1.5$ PN order (see below), the GW signal depends only on the masses and spins of the binary components and is, therefore, oblivious to the nature of the latter. However, the details of the coalescing bodies appear at higher PN order, notably through the effects of the spin-induced quadrupole moment (if the binary is spinning)~\cite{poisson1998quadrupole,PoissonWill}, a small tidal-heating term (if at least one of the binary components is a BH or can efficiently absorb radiation)~\cite{hartle1973,hughes2001,Maselli:2017cmm}, and most importantly through the tidal deformability contribution (the so-called tidal Love numbers~\cite{flanagan2008,Hinderer:2007mb,PoissonWill}) that becomes increasingly more relevant during the late stages of the inspiral and merger, as in the case of a binary NS coalescence (see, e.g., Refs.~\cite{Harry:2018hke,Chatziioannou:2020pqz} for some recent reviews).
Previous work considered the aforementioned effects independently and included in the waveform only a single effect at the time, focusing on the detectability of the tidal Love number~\cite{cardoso2017,sennett2017}, or of the spin-induced quadrupole moment~\cite{Kastha:2018bcr,krishnendu2017,krishnendu2019,krishnendu2019_2}, or of the tidal heating alone~\cite{Maselli:2017cmm,Datta:2019epe,Datta:2019euh,datta2020}. However, this approach neglects a crucial ingredient: for a given scalar field theory (i.e., fixing the potential in the Lagrangian~\eqref{Lagrangian}), both the tidal Love numbers and the spin quadrupole moment depend only on the masses and spins of the binary. Therefore, their concurrent inclusion does not increase the number of waveform parameters and alleviates their degeneracy.
As we shall show, the sensitivity of current detectors such as the Laser Interferometer Gravitational Wave Observatory (LIGO) and Virgo is not sufficient to place stringent constraints on the coupling constants of the theory. However, future facilities will provide much more stringent measurements. In particular, we consider the future Laser Interferometer Space Antenna~(LISA)~\cite{Audley:2017drz}, which can potentially detect supermassive binary BSs, and the Einstein Telescope~(ET)~\cite{Maggiore:2019uih}, a proposed third-generation~\cite{Hild:2010id} ground-based GW detector. Putative detections of binary BSs in different mass ranges can provide complementary constraints on the fundamental parameters of interacting scalar-field theories (see Fig.~\ref{fig:paramspace}), thus turning GW detectors into particle-physics laboratories~\cite{barack2019review,giudice2017}.
The plan of the work is as follows. Section~\ref{sec:boson stars} reviews the main properties of BSs in our model, in particular their mass, tidal deformability, and spin-induced quadrupole moment. Section~\ref{sec:waveform} reviews how those properties enter in the PN expansion of the waveform. Section~\ref{sec:fisher} reviews the Fisher matrix formalism to estimate statistical errors on the model parameters and discusses previous work on the topic. Section~\ref{sec:results} presents our results on the projected measurements of the fundamental coupling constant of the theory using GW detections of binary BSs. Finally, Appendix~\ref{appendix:1} summarizes our fits and useful relations between various BS parameters.
We use $G=c=1$ units, whereas we keep Planck's constant $\hbar$ explicit, defining the Planck mass as $M_P=\sqrt{\hbar}$.
\begin{figure}[t]
\includegraphics[width=0.48\textwidth]{parameter_space.pdf}
\caption{Parameter space of a scalar field theory with a quartic potential, Eq.~\eqref{potential:2}. The shaded area denotes the weak-coupling region, $\lambda<\mu^2$, whereas the three parallel inclined lines correspond to Eq.~\eqref{mmax:1} for different values of $M_{\rm max}$ and give an estimate of the region that can be probed by astrophysical BSs. The continuous black line corresponds to a model of self-interacting dark matter~\cite{giudice2017}, see Eq.~\eqref{selfDM}.}
\label{fig:paramspace}
\end{figure}
\section{BS binaries as GW sources}\label{sec:boson stars}
\subsection{Equilibrium configurations}
BSs are self-gravitating configurations of a complex scalar field described by the action
\begin{equation}
\label{action:1}
S=\int d^4x\sqrt{-g}\left[\frac{R}{16\pi}-{\cal L}_{\rm scalar}\right]\,.
\end{equation}
The shape of the potential in Eq.~\eqref{Lagrangian} determines the properties of the corresponding star. For a simple Klein-Gordon potential $V=\frac{\mu^2}{2}\left|\phi\right|^2$
initially considered in \cite{kaup1968,ruffini1969}, BSs have a maximum mass which scales with the boson mass $m_S\equiv\mu\hbar$ as $M_{\rm max}\sim M_P^2/m_S$. Unless the scalar field is ultralight ($m_S\ll 10^{-11}\,{\rm eV}$), the maximum mass is much smaller than the Chandrasekhar mass for compact astrophysical bodies and, therefore, such objects are often called \emph{mini} BSs. In this work we shall focus on the case of a quartic potential
\begin{equation}
\label{potential:2}
V\left(\left|\phi\right|\right)=\frac{\mu^2}{2}\left|\phi\right|^2+\frac{\lambda}{4}\left|\phi\right|^4\,.
\end{equation}
In the strong-coupling limit $\lambda\gg \mu^2$, the maximum mass of a \emph{static} BS scales as~\cite{colpi1986}
\begin{equation}
\label{mmax:1}
M_{\rm max}^{\rm static}\approx0.06\frac{\sqrt{\lambda \hbar}}{m_S^2}M_P^3\approx10^5 M_\odot\sqrt{\lambda \hbar}\left(\frac{{\rm MeV}}{m_S}\right)^2\,.
\end{equation}
Note that, if $\lambda\hbar=1$, varying the boson mass from few percents of a MeV up to a hundred MeV, one can cover the whole spectrum of physical BH masses.\footnote{Following Refs.~\cite{amaro2010,giudice2017}, it is also interesting to note that cosmological observations seem to suggest an interacting dark-matter component, with a cross section per unit mass $0.1 \text{cm}^2/\text{g}\lesssim\sigma/m_{dm}\lesssim 1 \text{cm}^2/\text{g}$, which, for the quartic potential~\eqref{potential:2}, translates to~\cite{giudice2017}
\begin{equation}
\label{selfDM}
\lambda\hbar \sim\left(\frac{m_S}{100\,{\rm MeV}}\right)^{3/2}\,.
\end{equation}
When $m_S\sim 10^3$~eV, this corresponds to supermassive BSs with $M\sim 10^5 M_\odot$ (see Fig.~\ref{fig:paramspace}).
} Hereafter we shall focus only on this regime, which allows for stellar-mass BSs [with mass $M=\mathcal{O}(10M_\odot)$] when $m_S=\mathcal{O}(10^2\,{\rm MeV})$, and for supermassive BSs [with mass $M={\cal O}(10^5 M_\odot$)] when $m_S={\cal O}({\rm MeV})$.
While the boson mass sets the scale of the system, in the strong-coupling limit all properties of the BS depend only on the following combination of the fundamental constants appearing in the action~\eqref{action:1},
\begin{equation}
\label{MB}
M_B \equiv \frac{\sqrt{\lambda}}{\mu^2} =\sqrt{\lambda \hbar} \frac{M_P^3}{m_S^2}\,,
\end{equation}
which has the dimension of a mass in our units.
\subsection{GW signatures}\label{sec:signatures}
The structure of a BS differs from that of a BH in several respects, which introduce distinctive features in the gravitational waveforms from coalescing BSs. The main differences can be summarized as follows~\cite{cardoso2019testing}:
\begin{enumerate}[i.]
\item BSs are less compact than BHs, as measured by their compactness $C=M/R$, where $R$ is the BS effective radius\footnote{At variance with NSs, BSs do not have a hard surface, as the scalar field is spread out
all over the radial direction. However, it decays exponentially and the configuration is highly localized in a radius $\sim1/\mu$. It is customary to
define the effective radius $R$ as the radius within which the $99\%$ of the total mass is contained.}. Nonspinning BHs have a compactness $C=M/R=1/2$, while massive BSs have a maximum compactness $C_{\rm max}\approx0.158$ \cite{amaro2010}, which is comparable to the typical compactness of a NS. This also implies that the ``contact" frequency of a binary BS is lower than in the binary BH case. For an equal-mass binary, the GW contact frequency in the point-particle limit reads
\begin{eqnarray}
\label{eq:f:contact}
f_{\rm contact} &=& \frac{1}{2^{3/2}\pi M_t}C^{3/2}\nonumber\\
&\approx& 1\,{\rm mHz} \left(\frac{10^6M_\odot}{M}\right) \left(\frac{C}{0.15}\right)^{3/2}\,,\label{fcontact}
\end{eqnarray}
which for $C\sim0.15$ is similar to the frequency of the innermost stable circular orbit~(ISCO) of a Schwarzschild BH with mass equal to the total mass $M_t=2M$ of the binary, $f_\text{ISCO}=(6^{3/2}\pi M_t)^{-1}$. The latter can be approximately assumed as the transition frequency between the inspiral and the merger in the binary BH case.
\item The spin $J$ of a BS is quantized\footnote{It has been recently shown that spinning mini BSs made of a scalar field are unstable and decay to their nonspinning state~\cite{sanchis2019}. The instability occurs on dynamical time scales, at least for large compactness. It is unknown whether self-interactions can cure this instability or make it phenomenologically irrelevant.} in units of its Noether charge, the latter existing due to the $U(1)$ symmetry of the Lagrangian~\eqref{Lagrangian}. Nonetheless, in the strong-coupling limit the quantization levels are extremely close to each other, and in practice the dimensionless spin $\chi=J/M^2$ can be treated as a continuous parameter~\cite{ryan1997}.
\item BSs have a nonvanishing tidal deformability, which expresses the tendency of the object to deform under the action of an external tidal field. In contrast, the tidal deformability --~as measured by the tidal Love numbers~-- of a nonspinning BH is zero\footnote{While this property holds true also for slowly spinning BHs in the axisymmetric case~\cite{Pani:2015hfa,Landry:2015zfa,Pani:2015nua}, it has been recently argued --~using analytical continuation methods~-- that the tidal Love numbers of a spinning BH are nonzero in the nonaxisymmetric case~\cite{LeTiec:2020spy}. Whether and how this affects the gravitational waveform is still an open question. At any rate, even if the tidal deformability of spinning BHs affects the waveform, its (small) fixed value can be used as a baseline for null-hypothesis tests, like the case of the spin-induced quadrupole.}~\cite{Damour:2009vw,binnington2009}.
\item The higher multipoles of a BH are uniquely determined by its mass $M$ and its dimensionless spin $\chi$. In particular, the quadrupole moment of a Kerr BH is $Q_{\rm Kerr}=-M^3\chi^2$. On the other hand, as we shall review in Sec.~\ref{sec:properties_spin}, the multipole moments of a BS are not necessarily quadratic in the spin and also vary with the compactness.
\item GWs interact weakly with the scalar field, so that effectively BSs do not absorb gravitational radiation. Therefore, at variance with BHs, tidal heating is absent~\cite{Maselli:2017cmm,Datta:2019epe}.
\end{enumerate}
In brief, the GW signatures of a BS are similar to those of a NS (see Refs.~\cite{Harry:2018hke,Chatziioannou:2020pqz} for some reviews), despite the fact that BSs can be supermassive (and, therefore, are potential exotic sources also for space-based detectors) and, in principle, highly spinning.
In the next subsections we shall review the above properties more quantitatively and provide useful fits for the quantities that constitute the basic ingredients for the PN waveform model that we shall later use.
\subsubsection{Tidal deformability}
\label{sec:properties_tidal}
The tidal deformability of (static) BSs was computed in \cite{cardoso2017,sennett2017}, which investigated the possibility of using GW measurements of tidal effects to distinguish BSs from BHs and from NSs (see also Ref.~\cite{wade2013}).
The (dimensionful) tidal deformability parameter $\lambda_T$ is defined by~\cite{flanagan2008}
\begin{equation}
\mathcal{Q}_{ij}=-\lambda_T\mathcal{E}_{ij} \label{Qij}
\end{equation}
where $\mathcal{E}_{ij}$ is the external tidal field and $\mathcal{Q}_{ij}$ is the induced asymptotic quadrupole moment. It is more convenient to work with the dimensionless tidal deformability (hereafter simply tidal deformability) $\Lambda=\lambda_T/M^5$, where $M$ is the mass of the object. A fitting formula for $\Lambda$ in terms of the mass $M$ of the BS and of the parameters $\mu$ and $\lambda$ was obtained in Ref.~\cite{sennett2017}. In the strong coupling limit, the fit reduces to\footnote{Notice that the scalar Lagrangian of \cite{sennett2017} differs from our Eq.~\eqref{action:1} by a factor of 2, which we took into account when rewriting Eq.~\eqref{fit:lambda}.}
\begin{equation}
\label{fit:lambda}
\frac{M}{M_B}=\frac{\sqrt{2}}{8\sqrt{\pi}}\left[-0.828+\frac{20.99}{\log\Lambda}-\frac{99.1}{\left(\log\Lambda\right)^2}+\frac{149.7}{\left(\log\Lambda\right)^3}\right]\,.
\end{equation}
The above expression can be inverted to find $\Lambda=\Lambda(M_B,M)$. Note that Eq.~\eqref{fit:lambda} has a stationary point for $\Lambda\approx289$, corresponding to $M/M_B\approx0.0611$, which is in agreement with Eq.~\eqref{mmax:1} for the maximum mass. Therefore, the tidal deformability of a massive BS is bounded from below by $\Lambda\gtrsim289$. Notice that $\Lambda$ spans many orders of magnitude as the mass deviates from its maximum value: for example, when $M/M_B=0.02$, $\Lambda\approx1.7\times10^6$. As we show in Appendix~\ref{appendix:1}, this is related to the fact that $\Lambda$ can be expressed in the form $\Lambda\sim k_2/C^5$, where $k_2={\cal O}(0.01-0.1)$ is a numerical factor~\cite{cardoso2017}, and the compactness $C$ can be as low as $0.03$ when $M/M_B=0.02$.
\subsubsection{Spin effects and the quadrupole moment}
\label{sec:properties_spin}
While $\mathcal{Q}_{ij}$ in Eq.~\eqref{Qij} is a \emph{tidal-induced} quadrupole moment, a spinning self-gravitating body has a \emph{spin-induced} quadrupole moment \cite{laarakkers1999}. An analysis of spinning BSs in the strong-coupling limit was carried out in Ref.~\cite{ryan1997}. The latter contains several important results, which we briefly summarize:
\begin{enumerate}[i.]
\item The maximum mass of a rotating BS is higher than in the nonspinning case. In Appendix~\ref{appendix:1} we show that $M_{\rm max}$ as a function of the spin is very well approximated by
\begin{equation}
\label{mmax:2}
M_{\rm max}\approx0.06\left(1+0.76\chi^2\right) M_B\,.
\end{equation}
\item In the axisymmetric case, the spin-induced quadrupole moment $\mathcal{Q}_{ij}$ can be written in terms of a single scalar quantity $Q$, which for a BS can be parametrized as
\begin{equation}
\label{quadrupole:moment}
Q = -\kappa(\chi,M/M_B)\chi^2 M^3\,.
\end{equation}
At variance with the BH case, the function $\kappa$ is not a constant ($\kappa_{\rm BH}=1$), but it depends on the spin and on the BS mass through the ratio $M/M_B$. The quantity $\kappa$ is shown in Fig.~\ref{fig:ryan:q} as a function of the dimensionless spin $\chi$ for some representative values of $M/M_B$.
It is worth noticing that $\kappa$ depends on the parameters of the potential only through the combination $M_B$ defined in~\eqref{MB}, as it is also the case for $\Lambda$. Hereafter we will refer to $\kappa$ as the ``reduced quadrupole moment".
%
A similar behavior is exhibited by the spin-induced octupole moment~\cite{ryan1997}. In the following, we will neglect this effect since it affects the GW waveform to higher PN order.
\end{enumerate}
\begin{figure}[hbt!]
\centering
\includegraphics[width=0.46\textwidth]{Ryan_k.pdf}
\caption{Spin-induced reduced quadrupole moment of a massive BS. The data are extracted and interpolated from Fig.~4 in Ref.~\cite{ryan1997}.}
\label{fig:ryan:q}
\end{figure}
For a BH, $\Lambda=0$ and $\kappa=1$, while for a compact BS $\Lambda\sim (10^2,10^6)$ and $\kappa(\chi\approx0)\sim(10,150)$. Therefore, there is a discontinuity gap between BHs and BSs, and the effects of these parameters on the waveform can be potentially large. Crucially, both $\Lambda$ and $\kappa$ depend only on the object mass (and spin) and on the coupling constant~\eqref{MB}. Therefore, for a given BS model $\Lambda$ and $\kappa$ are not independent quantities.
As shown in Fig.~\ref{fig:ryan:q}, the behavior of $\kappa$ as a function of $\chi$ for fixed values of $M/M_B$ is nonmonotonous at large spins, whereas the behavior of $\kappa_0\equiv \kappa(\chi\approx0)$ as a function of $M/M_B$ is simpler. As shown in Appendix~\ref{appendix:1}, after mapping the dependence of $M/M_B$ to $\Lambda$ one finds a linear fit for the logarithmic quantities,
\begin{equation}
\label{love:q:1}
\log\kappa_0 \approx 0.61+0.3\left(\log\Lambda\right)\,.
\end{equation}
In practice, this fit relates the reduced quadrupole moment of a slowly-spinning massive BS to its Love number. The full quadrupole moment is corrected by ${\cal O}(\chi^4)$ terms.
It is interesting that the $(\log\kappa_0)$-$(\log\Lambda)$ relation is approximately linear. This is reminiscent of the case of NSs, for which the ``Love-Q" relations have a similar form~\cite{yagi2013,yagi2013_2}. In that case, the coefficients of the fit are nearly universal for different equations of state of the star. In a future work, we will explore this issue for BSs with different scalar potentials~\cite{massimo}.
In the following, we will make use of Eqs.~\eqref{love:q:1} and \eqref{fit:lambda} to express all finite-size effects in the waveform in terms of the single parameter $M_B$.
In the next section, we review how these effects (as well as the tidal heating) enter in the PN approximation of the GW waveform.
\section{PN corrections to the waveform}
\label{sec:waveform}
We focus on the inspiral phase of the signal emitted by a compact binary coalescence, adopting the PN expanded TaylorF template in the frequency domain~\cite{Damour:2000zb,Damour:2002kr,Arun:2004hn}
\begin{equation}
\label{waveform:1}
h(f)=\mathcal{A}(f)e^{i\psi(f)}\,.
\end{equation}
The amplitude $\mathcal{A}$ and the phase $\psi$ are expanded as power series
in the orbital velocity $v=\left(\pi M_t f\right)^{1/3}$, where $M_t=M_1+M_2$ is the total mass of the binary and $M_i$ is the mass of the $i$th binary component (we assume $M_1\geq M_2$). A term proportional to $v^n$ corresponds to the $n/2$-PN order of the approximation.
In our analysis we retain only the dominant (Newtonian) term in amplitude, also
averaging over the orientation and the polarization angles, which specify the source's
position with respect to the detector, such that
\begin{equation}
\label{rms:amplitude}
\mathcal{A}=\frac{M_t^2}{D_L}\sqrt{\frac{\pi\eta}{30}}\left(\pi M_tf\right)^{-7/6}\ ,
\end{equation}
where $D_L$ is the luminosity distance and $\eta=M_1 M_2/M_t^2$ is the symmetric
mass ratio. For LISA, Eq.~\eqref{rms:amplitude} must also be multiplied by an
additional factor of $\sqrt{3}/2$ in order to account for the triangular geometry of the detector~\cite{Berti:2004bd}.
The BS signatures described in Sec.~\ref{sec:signatures}
affect the signal's phase at different PN orders, reflecting the frequency content of each effect. At small frequencies, where lower PN terms play a more significant role, sources behave as point particles and the details on their internal structure are effaced~\cite{damourEffac,blanchet2014review}. For larger frequencies, however, finite-size terms induced by spin-quadrupole and tidal effects become relevant.
Changes in the waveform due to the BS structure add linearly to the BH
phase, namely:
\begin{equation}
\psi(f)= \psi_{\textnormal{BH}}(f)+\psi_{\kappa}(f)+\psi_{\Lambda}(f)\ ,
\end{equation}
where $\psi_\kappa$ and $\psi_{\Lambda}$ identify the
modifications induced by the spin-quadrupole and by the tidal terms, respectively.
In our analysis, we consider a 3.5PN expanded phase $\psi_\textnormal{BH}$~\cite{mishra2016}, which includes spin-orbit, spin-spin (up to 3PN), and cubic spin
corrections~\cite{Khan:2015jqa,Isoyama:2017tbp}.
Furthermore, since BSs do not absorb GWs efficiently, for consistency in the ``BH part'' of the waveform ($\psi_{\textnormal{BH}}$), we ignore the tidal-heating term which enters the phase with a $v^5\log v$ (respectively, $v^8\log v$) correction for a spinning (nonspinning) BH.
The dominant tidal correction enters the waveform at 5PN order, hence it is
suppressed by a factor $v^3$ with respect to the 3.5PN phase $\psi_\textnormal{BH}$.
However, the potentially large values
of $\Lambda$ render tidal effects comparable with the rest of the point-particle expansion.
The leading tidal correction to the phase is given by~\cite{flanagan2008,favata2014}
\begin{equation}
\label{tidal:1}
\psi_{\Lambda}=-\frac{117}{256\eta}\tilde{\Lambda}\left(\pi M_t f\right)^{5/3}
\end{equation}
where
\begin{equation}
\label{tidal:2}
\tilde{\Lambda}=\frac{16}{13}\left[\left(1+\frac{12}{q}\right)\frac{M_1^5}{M_t^5}\Lambda_1+\left(1+12q\right)\frac{M_2^5}{M_t^5}\Lambda_2\right]
\end{equation}
is an effective total tidal deformability, and $q=M_1/M_2\geq1$ is the binary mass ratio; the normalization of $\tilde{\Lambda}$ is chosen so that $\tilde{\Lambda}=\Lambda_1=\Lambda_2$ for an equal mass binary. Indeed, note that $\Lambda_1$ and $\Lambda_2$ are not independent, since $\Lambda_i=\Lambda_i(M_i, M_B)$, so for a given theory they are fixed once the masses are known.\footnote{The same situation occurs for NSs once the equation of state is fixed. We remind that in the BS case the role of the equation of state is played by the potential $V(|\phi|)$.}
The next-to-leading tidal correction enters at 6PN order and depends on both $\tilde{\Lambda}$ and on a second combination
$\delta\tilde{\Lambda}$ of $\Lambda_1$ and $\Lambda_2$ \cite{Lackey:2014fwa}, and also on the magnetic tidal Love numbers~\cite{Yagi:2013sva,pani2018magnetic,abdelsalhin2018,forteza2018impact,Poisson:2020mdi}. The correction $\delta\tilde{\Lambda}$ has been, in general, neglected within previous studies on BS waveforms since it is subdominant
in the PN expansion, and if $\Lambda_1$ and $\Lambda_2$ are treated as independent quantities, it would introduce an extra waveform parameter . However, since in our model $\Lambda_i=\Lambda_i(M_B,M_i)$, including 6PN corrections does not introduce any extra parameter in the waveform, and at the same may actually lead to an overall improvement on the constraints that characterize the BS's structure. On the other hand, the effect of the magnetic tidal deformability is typically negligible since the magnetic tidal Love numbers are much smaller than the standard (electric) ones~\cite{forteza2018impact,cardoso2017}. For this reason, we shall neglect the magnetic-tidal contribution to the 6PN waveform.
The spin-induced quadrupole moments affect
the GW phase already at 2PN order~\cite{wade2013,krishnendu2017,Kastha:2018bcr,Raposo:2020yjy}.
For aligned spin binaries, the dominant quadrupole contribution reads
\begin{equation}
\label{phase}
\psi_{\rm\kappa} = -\frac{75}{64} \frac{\left(\kappa_1M_1^2\chi_1^2+\kappa_2M_2^2\chi_2^2\right)}{M_1 M_2} \left(\pi M_t f\right)^{-1/3}\,,
\end{equation}
where $\kappa_i$ is the spin-induced reduced quadrupole of the $i$th body (the binary BH case corresponds to $\kappa_1=\kappa_2=1).$
We incorporate also the first subdominant correction (appearing at 3PN) as given in Ref.~\cite{krishnendu2017}.
We neglect the contributions from the spin-induced octupolar corrections, which are subleading relative to the quadrupolar corrections.
Finally, we restrict our analysis to nonprecessing binaries, i.e., we assume that
the individual spins are aligned with the binary orbital angular momentum.
\section{Measurability of the binary parameters}
\label{sec:fisher}
In this section we briefly review the basic properties of
the Fisher information matrix formalism that we use to infer the uncertainties
on the waveform parameters \cite{cutler1994,vallisneri2008},
and we summarize the results of previous applications
of such formalism to the problem of distinguishing
BS binaries from their BH counterparts.
We assume that a detection criterion for a GW signal
$h(t,\vec{\theta})$ has been met, providing us with the
best estimates for the source parameters $\vec{\theta}_0$ (as masses, spins, distance, orientation angles, etc.).
Let $h(f,\vec{\theta}_0)$ be the waveform in the
frequency domain evaluated at $\vec{\theta}=\vec{\theta}_0$.
The signal-to-noise ratio (SNR) associated to the
detection is given by
\begin{equation}
\label{snr:1}
\left(\text{SNR}\right)^2=\left(h\left|\,h\right.\right)\ ,
\end{equation}
where
\begin{equation}
\label{inner:product}
\left(h_1\left|\,h_2\right.\right)=4\Re\int_{f_{\rm min}}^{f_{\rm max}} df\frac{h_1(f)^\star h_2(f)}{S_n(f)}
\end{equation}
is the waveform inner product defined over the detector's
noise spectral density $S_n(f)$.
For LISA, we choose the range of integration in the frequency domain as follows:
\begin{subequations}
\begin{align}
& f_{\rm min} = \text{\rm max}\left(10^{-4}\,\text{Hz},f_{\rm obs}\right)\\
& f_{\rm max} = \text{min}\left(1\,\text{Hz}, f_\text{ISCO}\right)\ ,
\end{align}
\end{subequations}
where we remind $f_\text{ISCO}=(6^{3/2}M_t\pi)^{-1}$, whereas
$f_{\rm obs}$ is determined by requiring that the observation lasted
$T_{\rm obs}=1$ year before the binary reached the ISCO frequency, namely (using a Newtonian approximation~\cite{berti2005})
\begin{equation}
\label{fobs}
f_{\rm obs}=4.149\times10^{-5}\left(\frac{\mathcal{M}}{10^6 M_\odot}\right)^{-5/8}\left(\frac{T_{\rm obs}}{1\,\text{yr}}\right)^{-3/8}\,\text{Hz}\,.
\end{equation}
On the other hand, for the analysis with the ET we use
\begin{equation}
f_{\rm min} = 3\,\text{Hz}\quad \ ,\quad
f_{\rm max} = f_\text{ISCO}\,.
\end{equation}
In the limit of large SNR, the best estimates $\vec{\theta}_0$
are unbiased, meaning that they approach the true values.
If we also assume that the instrumental noise is Gaussian,\footnote{Technically speaking, one also has to assume that the priors over $\theta^\mu$ are flat, which is approximately valid if the scale over which the priors change is smaller than the scale over which \eqref{p:theta:1} changes.} then the posterior distribution of the waveform parameters
$\vec{\theta}$ is described by a normal distribution peaked
around $\vec{\theta}_0$ \cite{vallisneri2008}:
\begin{equation}
\label{p:theta:1}
p\left(\theta\left|\mathbf{s}\right.\right)\propto e^{-\frac{1}{2}(\theta-\theta_0)^\mu\Gamma_{\mu\nu}(\theta-\theta_0)^\nu}
\end{equation}
where $\bf{\Gamma}$ is the Fisher information matrix defined
by
\begin{equation}
\label{fisher:1}
\Gamma_{\mu\nu}=\left(\frac{\partial h}{\partial\theta^\mu}\left|\,\frac{\partial h}{\partial\theta^\nu}\right.\right)\,.
\end{equation}
The 1-$\sigma$ uncertainty on $\Delta\vec{\theta}=\vec{\theta}-\vec{\theta}_0$ is given by
\begin{equation}
\label{sigma:1}
\sigma_j\equiv\sqrt{\left<(\Delta\theta^j)^2\right>}=\sqrt{\left(\bf{\Gamma}^{-1}\right)_{jj}}\,.
\end{equation}
The result~\eqref{sigma:1} is formally valid only in the limit of infinite SNR,
while for large SNR it is understood to provide only an order-of-magnitude estimate~\cite{vallisneri2008}. A rigorous approach would require a Bayesian analysis (see, e.g., \cite{rodriguez2013}). However, Bayesian simulations are computationally costly, while the Fisher formalism is extremely cheap; therefore, the latter is useful to understand if the problem is promising enough to be addressed with a more rigorous Bayesian approach.
Previous studies focused on waveforms' changes
induced by tidal interactions \cite{cardoso2017,sennett2017,Pani:2019cyc},
horizon absorption effects \cite{Maselli:2017cmm,Datta:2019epe}, and
spin-induced multipole moments \cite{krishnendu2017,Kastha:2018bcr,krishnendu2019,krishnendu2019_2,Kastha:2019brk}.
In particular, previous work considered only a single effect at the time, focusing for instance on the detectability of $\tilde\Lambda$ or $\kappa$. However, given a BS model, both $\Lambda$ and $\kappa$ (as well as all other inspiral waveform parameters) depend only on the masses and spins of the binary. Therefore, including all effects at once does not increase the number of waveform parameters, on the contrary, it might help breaking degeneracy and improving the accuracy of the template.
\begin{figure*}[hbt!]
\centering
\includegraphics[width=0.99\textwidth]{lisa.pdf}
\caption{Relative percentage errors $\sigma_{M_B}/M_B[\%]$ for binary BSs with a coupling parameter in the range $M_B\in1.7\times[10^5,10^7]M_\odot$, located at a luminosity distance $D_L=1$~Gpc and observed with LISA. From left to right, panels refer to different terms included in the PN waveform model, see text for details.}
\label{fig:results:1}
\end{figure*}
\begin{figure*}[hbt!]
\centering
\includegraphics[width=0.99\textwidth]{et.pdf}
\caption{Same as in Fig.~\ref{fig:results:1} but for binary BSs with coupling parameter in the range $M_B\in1.7\times[10,10^3]M_\odot$, located at a luminosity distance $D_L=500$~Mpc and observed with the ET.}
\label{fig:results:2}
\end{figure*}
\section{Results}
\label{sec:results}
Here we present the results of our Fisher-matrix analysis aimed
to assess the ability of measuring the fundamental coupling
constants of a scalar field theory using GW detections of binary BSs by LISA and the ET.
For concreteness, we restrict our attention to slightly unequal-mass
binaries, such that $M_1\approx0.06 M_B$ is approximately the maximum
mass for nonspinning BSs\footnote{From Eq.~\eqref{fit:lambda}, $\Lambda$ has an extremum at $M=M_{\rm max}^{\rm static}$, so its derivative with respect to $M/M_B$
diverges at that point, thus rendering the Fisher matrix ill-defined.
To overcome this problem, we choose $(M_{\rm max}^{\rm static}-M_1)/M_{\rm max}^{\rm static}=10^{-4}$.} and $M_2=0.05 M_B$,
with the mass ratio equal to $q=1.2$. We proceed by increasing levels of
complexity:
\begin{enumerate}[i.]
\item First, we consider the case in which the waveform has corrections only from the spin quadrupole moments, ignoring tidal deformabilities. We refer to this case as the ``3.5PN test" because the waveform is expanded only up to 3.5PN order.
\item Then, we consider the case in which the waveform has corrections only from the tidal deformabilities, ignoring spin-quadrupole moment terms. We refer to this case as the ``5PN$^\dagger$ test" because we include only the leading 5PN tidal correction, and the $\dagger$ refers to the absence of spin-quadrupole corrections.
\item In the third case, we incorporate both quadrupole and tidal corrections, the latter only at the leading 5PN order. We refer to this case as the ``5PN test".
\item Finally, we also include the 6PN tidal correction (though we neglect the 6PN contribution from the smaller magnetic tidal deformabilities). We refer to this case as the ``6PN test".
\end{enumerate}
For all cases, the independent parameters of the Fisher matrix are
\begin{equation}
\vec{\theta}=\left(\mathcal{A},t_c,\phi_c,\log\mathcal{M},\log\eta,\chi_s,\chi_a,M_B\right)\,,
\end{equation}
where $t_c$ and $\phi_c$ are the coalescence time and phase, $\mathcal{M}=(M_1M_2)^{3/5}/M_t^{1/5}$ is the chirp mass, and $\chi_{s,a}=(\chi_1\pm\chi_2)/2$ are the symmetric and antisymmetric spin components. By considering the amplitude $\mathcal{A}$ as an independent parameter,
it decouples from the other variables \cite{cutler1994},
such that the Fisher matrix becomes block-diagonal
$\Gamma_{i{\cal A}}=\delta_{i{\cal A}}$;
therefore we will not report the uncertainties for $\mathcal{A}$.
For each choice of $M_B$, we use Eq.~\eqref{fit:lambda} to compute the individual
tidal deformabilities, and Eq.~\eqref{love:q:1} to compute the corresponding
reduced quadrupole moments. This is not entirely consistent because
Eq.~\eqref{love:q:1} is valid only at small spins, while for generic spins
$\kappa$ has the complicate dependence shown in Fig.~\ref{fig:ryan:q}.
However, it is difficult to extract the dependence on $M_B$ from Fig.~\ref{fig:ryan:q},
especially due to the fact that data are not dense enough. Work in this direction is in progress and will be reported elsewhere~\cite{massimo}. Here, for simplicity, we assumed
$\kappa\approx\kappa_0$, which amounts to neglect ${\cal O}\left(\chi^4\right)$
corrections in the expression \eqref{quadrupole:moment} for the quadrupole moment. This approximation is valid for moderately small spins.
For LISA, we consider a coupling in the range $M_B\in1.7\times[10^5,10^7] M_\odot$, which corresponds to a maximum mass for nonspinning BSs $M_{\rm max}^{\rm static}\in[10^4,10^6]M_\odot$ at a fixed luminosity distance $D_L=1$Gpc. For the ET we consider $M_{\rm max}^{\rm static}\in[1,100]M_\odot$ at a fixed luminosity distance $D_L=500$Mpc. In both cases, we consider three different spin combinations $(\chi_1,\chi_2)=(0.1,0),(0.6,0.3),(0.9,0.8)$. Since we consider relatively small distances, the effect of the cosmological redshift is small, therefore, in the Fisher matrix we will use the values of the masses in the source frame.
Figures~\ref{fig:results:1} and \ref{fig:results:2} show the relative uncertainties on $M_B$ in the entire mass ranges considered, respectively, for LISA and the ET. We can understand the importance of the various contributions to the waveform as follows. First, tidal corrections have a larger impact than spin-quadrupole ones on the accuracy in $\sigma_{M_B}$: this can be seen by comparing the 3.5PN and 5PN$^\dagger$ panels, from which it is evident that for the 5PN$^\dagger$ model the uncertainties are smaller by an order of magnitude (and even more for small spins). The results of the 5PN$^\dagger$ model are in agreement with those of Ref.~\cite{cardoso2017}, as expected. In particular, we observe that they vary only mildly with the spin magnitude, as it is expected from the fact that the 5PN tidal correction to the phase does not depend explicitly on the spins.
Second, including the spin-induced quadrupole moment helps to break the degeneracy with the spin among different PN terms in the waveform, leading to an improvement in the overall measurability at moderate and high spins: indeed, by contrasting the full 5PN results with the reduced 5PN$^\dagger$, we can observe that the relative uncertainty in $M_B$ improves by an order of magnitude when $(\chi_1,\chi_2)=(0.6,0.3),(0.9,0.8)$. Finally, the inclusion of 6PN tidal corrections leads to a moderate variation around $40\%$ on $\sigma_{M_B}$, showing that the values inferred from the 5PN case are robust against the inclusion of the next-order tidal correction. This also confirms that the PN series is converging well also at $f\approx f_{\rm ISCO}$, suggesting that a PN waveform approximant is sufficiently accurate up to those frequencies.
Crucially, when all corrections are included, the uncertainties on $M_B$ are at subpercent and percent level, for LISA and the ET respectively, in the most optimistic configurations and exceed the value of $100\%$ only for the highest masses in the case of ET.
This means that, in the considered mass range, a putative binary BS detection can be used to measure the coupling $M_B=\sqrt{\lambda}/\mu^2$ with great precision. We remind that the errors coming from the Fisher analysis scales linearly with the luminosity distance, so assuming (say) a distance $D_L=10$~Gpc for LISA binaries would decrease the SNR and increase the errors by a factor of $10$.
Given the small errors in Figs.~\ref{fig:results:1} and \ref{fig:results:2}, even in the more conservative case in which the event is at ${\cal O}(10)\,{\rm Gpc}$ (for LISA) or at ${\cal O}(1)\,{\rm Gpc}$ (for the ET), the measurement of $M_B$ would still be accurate at ${\cal O}(10)\%$ level or better.
The dependence of the constraints on $M_B$ can be understood as follows. The detectability of a signal will be enhanced when the corresponding contact frequency falls within the most sensitive band of the corresponding detector. Using Eq.~\eqref{eq:f:contact} for $f_{\rm contact}$ and approximating $M\approx 2M_{\rm max}^{\rm static}\approx0.12M_B$, we obtain an order of magnitude estimate for the most sensitive regions of $M_B$. For example, the LISA detector will be most sensitive in the frequency band $[10^{-3},10^{-1}]$ Hz, corresponding to $M_B\approx[10^5,10^7] M_\odot$. On the other hand, the ET will be mostly sensitive in the band $[50,500]$ Hz, giving $M_B\approx[20,200] M_\odot$. This reasoning exemplifies why different detectors can probe different parameter regions, while also explaining the local minima in constraints of Figs.~\ref{fig:results:1} and \ref{fig:results:2}.
In Table~\ref{tab:fiducial} we also show the results of the Fisher analysis for LISA for a fiducial binary BS system with $M_B\approx8.3\times10^6 M_\odot$ ($M_{\rm max}^{\rm static}=5\times10^5 M_\odot$), corresponding to ${\rm SNR}\approx1.5\times 10^4$. Similarly, Table~\ref{tab:fiducial:2} shows the results for the ET, for a fiducial system with $M_B\approx8.3\times10^{2}M_\odot (M_{\rm max}^{\rm static}=50M_\odot$) and corresponding ${\rm SNR}\approx620$.
We have also checked whether, in the mass range considered for the ET, a successful measurement can be achieved with the advanced LIGO detector at design sensitivity. However, we find that the relative uncertainties in $M_B$, as measured with the aLIGO, always lie above $100\%$ with the exception of some high spins configurations which are only marginally detectable.
This is not only due to the globally lower SNR relative to ET, but more importantly to LIGO's low-frequency cutoff, which is higher than the ET's. Higher sensitivity at low frequency helps significantly to measure the low-PN parameters, and in turn to improve the measurement of high-PN parameters at higher frequency. We confirmed this expectation by artificially truncating the ET noise curve at higher low-frequency cutoff, finding that the constraints on $M_B$ quickly deteriorate as the low-frequency cutoff increases.
To summarize, our analysis suggests that future-generation GW observatories are necessary to make precise measurements of the fundamental couplings of a BS model.
\begin{table*}[p!]
\begin{ruledtabular}
\begin{tabular}{ccccccccc}
Test &$(\chi_1,\chi_2)$ &$\sigma_{t_c}$[sec] &$\sigma_{\phi_c}$[rad] &$\sigma_{\log\mathcal{M}}[\%]$ &$\sigma_{\log\eta}[\%]$ &$\sigma_{\chi_s}$ &$\sigma_{\chi_a}$ &$\sigma_{M_B}/M_B[\%]$\\
\hline
3.5PN &$(0.1,0)$ &$18.8$ &$0.302$ &$1.41\times10^{-2}$ &$8.90$ &$0.273$ &$2.42$ &$86.5$\\
5PN$^\dagger$ &$(0.1,0)$ &$28.8$ &$0.982$ &$6.57\times10^{-3}$ &$2.46$ &$9.66\times10^{-3}$ &$0.141$ &$8.27$\\
5PN &$(0.1,0)$ &$31.1$ &$1.05$ &$6.18\times10^{-3}$ &$2.16$ &$2.00\times10^{-2}$ &$0.254$ &$7.27$\\
6PN &$(0.1,0)$ &$28.2$ &$1.37$ &$1.23\times10^{-3}$ &$0.952$ &$7.68\times10^{-3}$ &$7.08\times10^{-2}$ &$3.23$\\
\hline
3.5PN &$(0.6,0.3)$ &$30.5$ &$4.78$ &$5.31\times10^{-3}$ &$0.378$ &$0.253$ &$2.23$ &$1.90$\\
5PN$^\dagger$ &$(0.6,0.3)$ &$33.4$ &$1.09$ &$6.14\times10^{-3}$ &$2.15$ &$1.02\times10^{-2}$ &$0.202$ &$7.23$\\
5PN &$(0.6,0.3)$ &$27.9$ &$4.54$ &$5.30\times10^{-3}$ &$0.246$ &$0.245$ &$2.16$ &$0.829$\\
6PN &$(0.6,0.3)$ &$35.4$ &$6.98$ &$6.86\times10^{-3}$ &$0.376$ &$0.368$ &$3.26$ &$1.28$\\
\hline
3.5PN &$(0.9,0.8)$ &$33.0$ &$5.49$ &$3.14\times10^{-3}$ &$2.28$ &$0.137$ &$1.01$ &$13.8$\\
5PN$^\dagger$ &$(0.9,0.8)$ &$35.6$ &$1.13$ &$5.89\times10^{-3}$ &$1.94$ &$1.84\times10^{-2}$ &$0.210$ &$6.52$\\
5PN &$(0.9,0.8)$ &$15.2$ &$0.985$ &$4.57\times10^{-3}$ &$0.297$ &$3.84\times10^{-2}$ &$0.287$ &$1.01$\\
6PN &$(0.9,0.8)$ &$25.2$ &$1.41$ &$5.43\times10^{-3}$ &$0.407$ &$5.14\times10^{-2}$ &$0.390$ &$1.39$\\
\end{tabular}
\caption{Fisher matrix uncertainties for a binary BS with fundamental couplings such that $0.06M_B=5\times10^5 M_\odot$, individual masses $M_1/M_B\approx0.06$ and $M_2/M_B=0.05$, located at a luminosity distance $D_L=1$~Gpc and observed with LISA. The corresponding SNR is $\approx1.5\times 10^4$.}
\label{tab:fiducial}
\end{ruledtabular}
\end{table*}
\begin{table*}[p!]
\begin{ruledtabular}
\begin{tabular}{ccccccccc}
Test &$(\chi_1,\chi_2)$ &$\sigma_{t_c}$[sec] &$\sigma_{\phi_c}$[rad] &$\sigma_{\log\mathcal{M}}[\%]$ &$\sigma_{\log\eta}[\%]$ &$\sigma_{\chi_s}$ &$\sigma_{\chi_a}$ &$\sigma_{M_B}/M_B[\%]$\\
\hline
3.5PN &$(0.1,0)$ &$8.71\times10^{-2}$ &$14.9$ &$0.914$ &$495$ &$14.7$ &$130$ &$4.74\times10^3$\\
5PN$^\dagger$ &$(0.1,0)$ &$0.122$ &$45.8$ &$0.353$ &$111$ &$0.237$ &$4.45$ &$373$\\
5PN &$(0.1,0)$ &$0.132$ &$48.4$ &$0.324$ &$92.1$ &$0.849$ &$11.0$ &$310$\\
6PN &$(0.1,0)$ &$9.22\times10^{-2}$ &$4.17$ &$4.36\times10^{-2}$ &$31.7$ &$0.258$ &$2.39$ &$107$\\
\hline
3.5PN &$(0.6,0.3)$ &$0.124$ &$197$ &$0.313$ &$16.6$ &$10.9$ &$96.0$ &$75.4$\\
5PN$^\dagger$ &$(0.6,0.3)$ &$0.140$ &$49.3$ &$0.324$ &$94.1$ &$0.528$ &$9.89$ &$317$\\
5PN &$(0.6,0.3)$ &$0.128$ &$201$ &$0.313$ &$11.1$ &$11.1$ &$97.7$ &$37.7$\\
6PN &$(0.6,0.3)$ &$0.197$ &$376$ &$0.487$ &$20.7$ &$20.2$ &$179$ &$70.4$\\
\hline
3.5PN &$(0.9,0.8)$ &$0.130$ &$218$ &$0.195$ &$90.1$ &$5.63$ &$41.3$ &$548$\\
5PN$^\dagger$ &$(0.9,0.8)$ &$0.148$ &$50.5$ &$0.309$ &$83.5$ &$0.899$ &$9.90$ &$281$\\
5PN &$(0.9,0.8)$ &$6.23\times10^{-2}$ &$37.7$ &$0.250$ &$12.4$ &$1.63$ &$11.9$ &$42.3$\\
6PN &$(0.9,0.8)$ &$0.113$ &$60.0$ &$0.322$ &$18.6$ &$2.38$ &$17.8$ &$63.5$\\
\end{tabular}
\caption{Same as Table~\ref{tab:fiducial} but for a binary BS with fundamental couplings such that $0.06M_B=50 M_\odot$, located at a luminosity distance $D_L=500$~Mpc and observed with the ET. In this case the corresponding SNR is $\approx620$.}
\label{tab:fiducial:2}
\end{ruledtabular}
\end{table*}
\section{Concluding remarks}
\label{sec:conclusion}
We have estimated the ability of measuring the fundamental coupling constants of a scalar field theory through GW measurements of binary BSs. If such binaries exist and merge, future detectors such as LISA and the ET have the potential to measure the properties of the component BSs with great precision. Such measurements can be mapped into constraints on the fundamental coupling constants of the scalar field theory. As depicted in Fig.~\ref{fig:paramspace}, stellar-mass and supermassive BSs probe different regions of the parameter space, so future LISA and ET measurements will be complementary.
Our estimates are obtained through a Fisher matrix analysis, which is only an approximate treatment (although, given that LISA will reach ${\rm SNR}\gg100$, the approximation should be less dramatic than in the case of, e.g., aLIGO). A more accurate parameter estimation would require using Bayesian techniques. The latter can be also used to compute Bayes factors and perform model selection between different hypotheses for the nature of the binary: BS-BS, BH-BH, BH-BS, or to perform model selection among different BS models. Another source of approximation is that we excluded the merger-ringdown part of the spectrum from our waveform model, and only included the PN corrections to the inspiral. Therefore, in the future, it would be important to incorporate phenomenological completions of the waveform to the merger-inspiral region, possibly informed by numerical simulations (see, e.g., Refs.~\cite{palenzuela2008,palenzuela2017,sanchis2019collisions}). The merger of two BSs shows some distinctive features compared to its BH counterpart, which can be used to confidently determine the nature of the coalescing binaries, and thus subsequently focus on a precision analysis of the inspiral signal to extract information about the model parameters.
Indeed, we have focused on massive BSs, but a natural extension of our work is to consider different classes of the scalar potential $V$, for example mini BSs~\cite{kaup1968,ruffini1969}, solitonic BSs~\cite{friedberg1987}, or the recently studied model of axion BSs~\cite{guerra2019axion,delgado2020rotating}. The latter two models allow for very compact BSs, which are expected to have properties closer to those of a BH.
When spinning, mini BSs are unstable and decay to their ground nonspinning case, at least for certain compactnesses~\cite{sanchis2019}. Strong self-interactions might cure this instability or make it phenomenologically irrelevant.
Likewise, BSs exist also for other integer-spin fundamental fields, such as Proca~\cite{brito2016proca,herdeiro2020tidal} and massive spin-$2$~\cite{Aoki:2017ixz} fields. Interestingly enough, at variance with their scalar counterpart, spinning Proca stars are not unstable, due to the spherical topology of the Proca field profile in the ground state as opposed to the toroidal topology of a scalar field~\cite{sanchis2019}. Our methods can be directly applied also to Proca stars or more generic BSs, provided a theoretical estimate of the tidal deformability and spin quadrupole is available for these models.
We also highlighted the existence of a simple phenomenological relation between the tidal deformability and the spin-induced quadrupole moment of static BSs described by the quartic potential~\eqref{potential:2}. The relation is a straight line in log-space, reminiscent of the approximately universal Love-Q relations for NSs~\cite{yagi2013,yagi2013_2}. We believe that such a remarkably simple behavior deserves further investigation. A computation of quadrupole moments for spinning mini BSs, solitonic BSs, or axion BSs would definitely reveal if such a relation is approximately universal also in the BS case, or it is just an accident of the potential~\eqref{potential:2}.
\begin{acknowledgments}
We thank Monica Colpi for the interesting discussion.
We acknowledge financial support provided under the European Union's H2020 ERC, Starting Grant Agreement No.~DarkGRA--757480, under the MIUR PRIN and FARE programmes (GW-NEXT, CUP:~B84I20000100001), and support from the Amaldi Research Center funded by the MIUR program `Dipartimento di Eccellenza" (CUP:~B81I18001170001).
\end{acknowledgments}
|
1,116,691,499,309 | arxiv | \section{Introduction}\label{sec:introduction}
Stochastic model-based evaluation is widely used to analyze the
performability, dependability and performance of complex systems.
A convenient technique to make model construction easier, and to promote an efficient solution process,
is based on the hierarchical submodels composition~\cite{SM91,DKS04}.
To overcome the heavy and error prone manual definition of many submodels, automated procedures are currently available to assist the modeller in generating the hierarchical composed model representing the
specific instance of the system. A general approach is based on the definition of primitives,
such as the \textit{Rep} operator proposed in the M{\"o}bius framework~\cite{DCCDDDSW02}, which can be used to represent different components, behaviors or
characteristics of a system, automatically replicating a single template
model. Each replica is indistinguishable by other replicas and then it is anonymous. Although satisfactory in general terms, such \textit{Rep} operator shows limitations from the efficiency point of view when employed in the modeling of systems composed of a (large) population of similar, but non anonymous, components.
In this paper, a new composition operator, called \textit{NARep}, is introduced to deal with efficient non-anonymous replication inside the M\"obius framework.
The new operator characteristics are identified and presented in the
context of performability and dependability
evaluation.
\section{The context}\label{sec:context}
The M\"obius modeling framework~\cite{DCCDDDSW02} is a powerful modular environment that supports multiple modeling formalisms and solution techniques.
Among the others, it supports the \ac{SAN} formalism,
composition operators \textit{Rep} and \textit{Join}~\cite{SM91}, performance variables
based on rate and impulse rewards (used also to define performability and dependability measures), analytical and simulative solvers.
\textit{Join} operator is a state-sharing composer that
brings together two or more submodels.
\textit{Rep} operator is a special
case of \textit{Join} that constructs a model consisting of identical copies (replicas) of a submodel.
\textit{Rep} and \textit{Join} operators are defined at level of \ac{AFI}~\cite{D00,DCCDDDSW02},
a common interface between model formalisms and solvers
that allows formalism-to-formalism and formalism-to-solver
interactions.
At \ac{AFI} level, places and activities
(transitions) of \ac{SAN} correspond to state variables and actions.
In the following, for sake of simplicity, we will use \ac{SAN}
models and \ac{SAN}
notation (places and activities) also to describe \textit{Rep} and \textit{Join} operators.
The state of a model is represented by the current value of all
the places defined in the model.
A place of a replicated \ac{SAN} model can be local (assume, at the same time, different values in different replicas)
or shared (same place, with only one value, that can be read or updated by each replica).
The \textit{Rep} operator can be used to represent multiple components, behaviors or
characteristics of a system having identical definition and
parameters.
\subsection{Non-anonymous replication in M{\"o}bius}
Non-anonymous replicas of a single \ac{SAN} model can be
automatically generated through \textit{Rep} operator via a
sophisticated use of local places and immediate activities as shown
in~\cite{CDGM14}.
Local places, representing replica indexes, are set up so that each replica becomes self-aware of its index.
The specific state, number of ``tokens'' or ``marks'' in Petri nets dialects, of the $i$-th non-anonymous
replica is represented by the $i$-th entry $P\text \textendash \textgreater
Index(i)\text \textendash \textgreater Mark()$ of a
shared array-type extended place $P$, i.e., an
extended place defined as an array of states, that is shared among \textit{all} the
replicas. Thus, each replica has read and write access to the state
of all the other replicas of the same \ac{SAN}.
In addition, a different submodel that shares the place $P$ with a \textit{Rep} has read and write access to
the state of each specific replica.
This approach allows to cope with dependability analysis of complex critical
systems, with a variety of interconnected components organized according to sophisticated topologies, such as electrical or transportation systems.
In~\cite{CDGM16}, which focuses on Smart Grids, a
stochastic modeling framework has been
proposed, which uses the \textit{Rep} operator as described above to quantitatively assess representative indicators of the resilience
and quality of service.
\subsection{Motivations for an enhanced replication operator}
Using an anonymous replicator to produce non-anonymous replicas
sounds unnatural, while on the other hand the \textit{Join} operator is impractical for large models. Moreover, dependencies introduced by the $n$-dimensional array-type extended
places shared among all the $n$ replicas of a \ac{SAN} model can
lead to a large time overhead when, during the initialisation phase
of the simulator, connectivity lists~\cite{W98} are
generated.
Implementing a non-anonymous replication with the \textit{Rep}
operator and with $n$-dimensional array-type extended places, shared
among $n$ replicas, slows down the connectivity lists
construction. In fact, $n^{2}$ checks are needed because each replica can
access to the state of \textit{all} the other replicas.
With simple toy models, varying $n$ from $10$ to $50$ the connectivity lists generation time overhead remains acceptable, but varying $n$ from
$100$ to $500$ the time overhead can increase of about $100$ times. Following these considerations, we propose the \textit{NARep} (\textit{non-anonymous replication}) new operator.
\section{\textit{NARep} definition}\label{sec:NARepDef}
\textit{NARep} automatically constructs a model
consisting of non-anonymous indexed replicas of a given submodel.
The index of a specific replica can be accessed from the generic \ac{SAN}
submodel using a new reserved keyword $repindex()$, corresponding to a method of the C++ \ac{SAN} class.
Each place $P$ included in a non-anonymous
model $aSAN$, replicated $n$ times,
corresponds to $n$
different place replicas via the \textit{NARep} operator, and place
replicas are accessed with $aSAN\text \textendash \textgreater Index(0)\text \textendash \textgreater P$,\dots,
$aSAN\text \textendash \textgreater Index(n-1)\text \textendash \textgreater P$. The \textit{NARep} operator can
be used to define whether the
replicas of each place $P$ are:
\begin{enumerate}[(i)]
\item local (default): read and write access to replica place
$aSAN\text \textendash \textgreater Index(i)\text \textendash \textgreater P$ limited only to the
corresponding $i$-th replica of the \ac{SAN} submodel. The generic
model $aSAN$, representing the generic replica $repindex()$, can
access to the corresponding replica $repindex()$ of the place $P$,
using: $aSAN\text \textendash \textgreater P\text \textendash \textgreater Mark()$, or equivalently
$aSAN\text \textendash \textgreater Index(repindex())\text \textendash \textgreater P\text \textendash \textgreater Mark()$.
\item place-shared: several place replicas correspond to a single
place. For example, if $aSAN\text \textendash \textgreater Index(0)\text \textendash \textgreater P\text \textendash \textgreater
Mark()$ is equal to $aSAN\text \textendash \textgreater Index(1)\text \textendash \textgreater P\text \textendash \textgreater
Mark()$, then the command $aSAN\text \textendash \textgreater Index(repindex())\text \textendash \textgreater P$
is referred to the same place replica each time that $repindex()$ is
equal to $0$ or $1$.
\item rep-shared: the same place replica is shared among several
replicas of the \ac{SAN} submodel. For example, the place replica
$aSAN\text \textendash \textgreater Index(i)\text \textendash \textgreater P$ can be rep-shared among the
$(i-1)$-th and $(i+1)$-th replicas of $aSAN$. In this case, the
generic model $aSAN$ can access to the rep-share places using
$aSAN\text \textendash \textgreater Index((repindex()-1)\%n)\text \textendash \textgreater P$, $aSAN\text \textendash \textgreater
Index(repindex())\text \textendash \textgreater P$ and $aSAN\text \textendash \textgreater
Index((repindex()+1)\%n)\text \textendash \textgreater P$. The
list of places that are rep-shared with a replica is
obtained using the command $P\text \textendash \textgreater repshared()$. Using this command, a generic \ac{SAN} model can
represent the access of the $i$-th replica only to those components
that are connected to the $i$-th replica.
\item up-shared: a local, place-shared or rep-shared place replica can
be shared among other submodels (not $aSAN$) at the level on top of
$NARep$. For example, $aSAN\text \textendash \textgreater Index(0)\text \textendash \textgreater P$ and
$aSAN\text \textendash \textgreater Index(1)\text \textendash \textgreater P$ can be local and up-shared with
the place $otherSAN\text \textendash \textgreater Q$, where the model $otherSAN$ is
composed with \textit{NARep}. In this case, the $Q$ place included
in $otherSAN$ is an extended place defined as a $2$ dimensional array,
each entry having the same type of the place $P$, such that $aSAN\text \textendash \textgreater
Index(0)\text \textendash \textgreater P$ is equal to $Q\text \textendash \textgreater Index(0)$ and
$aSAN\text \textendash \textgreater Index(1)\text \textendash \textgreater P$ is equal to $Q\text \textendash \textgreater Index(1)$.
\end{enumerate}
\section{Discussion}\label{sec:Discussion}
The proposed \textit{NARep} operator has the following advantages:
\begin{itemize}
\item Automatic generation of non-anonymous indexed
place and \ac{SAN}
replicas transparent to the modeler; this feature enlarges the framework expressive power.
\item A significant reduction of the connectivity lists generation
time overhead can be obtained exploiting only actually existing dependencies instead of the
assumed complete graph of dependencies. Therefore, the $n^{2}$ checks currently performed by the \textit{Rep} operator constitute the worst case for \textit{NARep}, when full connection is considered
\item Symmetries exploitation via the state-space lumping~\cite{DKS04} remains possible.
\end{itemize}
Future work will include:
\begin{itemize}
\item \textit{NARep} formalism definition,
\item integrate \textit{NARep} into M{\"o}bius framework,
\item evaluate the actual impact of \textit{NARep} on the
connectivity lists generation time overhead;
in particular, a significant improvement is expected for the Smart Grid model
presented in~\cite{CDGM16}, being the
electrical and communications networks sparse.
\end{itemize}
\bibliographystyle{IEEEtran}
|
1,116,691,499,310 | arxiv | \section{Introduction}
Recent ground and space-based observations have shown that multi-planet systems are abundant around main-sequence stars \citep[e.g.,][]{Howard+10,Howard+12,Borucki+11,Lissauer+11,Mayor+11,Youdin11,Batalha+13,Dressing+13,Petigura+13,Christiansen+15}. These studies reveal that the architecture of planetary systems can drastically vary from our solar system. For example, systems consisting of multiple low mass (sub-Jovian) planets with relatively compact orbits usually have periods that are shorter than Mercury's.
NASA's Kepler mission found an abundance of compact multi-planet super-Earths or sub-Neptune systems \citep[e.g.,][]{Mullally+15,Burke+15,Morton+16,Hansen17}. These systems seemed to have low eccentricities \citep[e.g.,][]{Lithwick+12,Van+15}. In addition, it was suggested that these many body systems are close to the dynamically stable limit \citep[e.g.,][]{Fang+13,Pu+15,Volk+15}. It was also proposed that single planet systems might be the product of systems initially consisting of multiple planets that underwent a period of disruption, i.e, collisions, resulting in reducing the number of planets (see \citet{Johansen+12} \citet{Becker+16}).
Giant planets may play a key role in forming inner planetary systems. Radial velocity surveys, along with the existence of Jupiter, have shown giants usually reside at larger distances from their host star than other planets in the system ($\geq 1$~AU)\citep[e.g.,][]{Knutson+14K,Bryan+16}. For example, there were some suggestions that the near-resonant gravitational interactions between giant outer planets and smaller inner planets can shape the configuration of an inner system's asteroid belt \citep[e.g.,][]{Williams+81,Minton+11}. Hence, it was suggested that our solar system's inner planets are of a second generation, which came after Jupiter migrated inward to its current orbit \citep[e.g.,][]{Batygin+15}. Furthermore, secular resonance may be the cause of water being delivered to earth \citep{1986A&A...170..138S,1994P&SS...42..301M}, the potential instability of Mercury's orbit \citep{1997A&A...318..975N,2009sf2a.conf..105F,2008ApJ...683.1207B,2011ApJ...739...31L}, and the obliquity of the exoplanets' in general \citep[e.g.,][]{Naoz11,Li+14Kepler}.
Recently, several studies showed that the presence of a giant planet can affect the ability to detect an inner system \citep[e.g.,][]{Hansen17,Becker+17,Mustill+17,Jontof+17,Huang+17}. Specifically, dynamical interactions from a giant planet, having a semi-major axis much greater than the planets in the inner system, can excite eccentricities and inclinations of the inner planets. A possible effect is that the inner system becomes incapable of having multiple transits or completely unstable. \citet{Volk+15} showed that observed multi-planet systems may actually be the remnants of a compact system that was tighter in the past but lost planets through dynamical instabilities and collisions \citep[see also][]{Petrovich+14}. Interestingly, verifying these problems could reconcile the Kepler dichotomy \citep[e.g.,][]{Johansen+12,Ballard+16,Hansen17}.
In this work, we investigate the stability of compact sub-Jovian inner planetary systems in the presence of a distant giant planet (see Figure \ref{fig:sailing} for an illustration of the system). Over long time scales, a distant giant planets' gravitational perturbations can excite the eccentricities of the inner planets' to high values, destabilizing the inner system. \citep[e.g.,][]{Naoz11}. However, if the frequency of angular momentum exchange between the inner planets is sufficiently high, then the inner system can stabilize. Below we derive an analytic stability criterion (Section \ref{sec:stab}). Then we analyze our nominal system in Section \ref{sec:Long} and provide specific predictions for Kepler systems in Section \ref{sec:Kepler}. Finally, we offer our conclusion in Section \ref{sec:dis}.
Near the completion of this work, we became aware of \citep{Pu+18} a complementary study of stability in similar systems. Here we provide a comprehensive stability criterion as a function of the companion's parameters. Furthermore, we provide a set of predictions for possible hidden companion orbits in several observed systems.
\begin{figure
\centering
\includegraphics[width=\linewidth]{Cartoon.pdf}
\caption{Here is a schematic of the systems being analyzed. The inclination of the third planet, $i_3$, is shown and measured relative to the z-axis, along the spin axis of the host star. $M, m_1, m_2$, and $m_3$ are the masses of the host star, and the first, second and third planets respectively. }
\label{fig:Catroon}
\end{figure}
\section{Analytical Stability criterion }\label{sec:stab}
Here we develop a generic treatment of the stability of two-body systems in the presence of an inclined outer planet.
Consider an inner system consisting of two planets ($m_1$ and $m_2$) orbiting around a host star $M$ with a relatively tight configuration (with semi-major axis $a_1$ and $a_2$, respectively). We introduce an inclined and eccentric companion that is much farther from the host star than the planets in the inner system ($m_3$ and $a_3$, see Figure \ref{fig:Catroon} for an illustration of the system). We initialize this system to have orbits far from mean-motion resonance, and have $a_1,a_2<<a_3$. The three planets' orbits have corresponding eccentricities $e_1$, $e_2$ and $e_3$. We set an arbitrary $z$-axis to be perpendicular to the initial orbital plane of the inner two planets, thus, the inclinations of $m_1$, $m_2$ and $m_3$ are defined as $i_1$, $i_2$ and $i_3$. Accordingly, for planets one, two, and three we denote the longitude of ascending nodes and the argument of periapse $\Omega_1$, $\Omega_2$, $\Omega_3$, $\omega_1, \omega_2$, and $\omega_3$.
The outer orbit can excite the eccentricities, via the Eccentric Kozai-Lidov mechanism (EKL) (\citet{Kozai}, \citet{Lidov}, see \citet{Naoz16} for review) on each planet in the inner system. The EKL resonance causes angular momentum exchange between the outer and inner planets, which in turn causes precession of the periapse of each of the inner orbits. However, angular momentum exchange between the inner two planets also induces precession of the periapse \citep["the so-called Laplace-Lagrange interactions, e.g.,][]{MD00}. If the inner orbits' angular momentum exchange takes place at a faster rate than that induced by the outer companion, then the system will not be disrupted by perturbations from the tertiary planet. The quadrupole approximation to the timescale of the EKL between the third planet $m_3$ and the second one, $m_2$ is given by: \begin{equation}\label{eq:Kozai}
\tau_{k(2,3)} \sim \frac{16}{15}\frac{a_3^3}{a_2^{3/2}} \frac{\sqrt{M+m_2+m_1}}{m_3\sqrt{G}}(1-e_3^2)^{3/2} \ ,
\end{equation}
\citep[e.g.,][]{Antognini15}
where $G$ is the gravitational constant. This is roughly the timescale at which the second planet's argument of periapse precesses. Note that here we considered the timescale of EKL excitations from the tertiary on $m_2$ since this timescale is shorter than the timescale of EKL excitations between $m_3$ and $m_1$. Planet $m_2$'s argument of periapse also precesses due to gravitational perturbations from the inner-most planet, $m_1$. The associated timescale is \citep[e.g.,][]{MD00}
\begin{eqnarray}
\tau_{LL}&\sim &\bigg[A_{22}+A_{21}\left(\frac{e_1}{e_2}\right)\cos(\varpi_2-\varpi_1)\nonumber \\ &-& B_{22}-B_{21}\left(\frac{i_1}{i_2}\right)\cos(\Omega_2-\Omega_1)\bigg]^{-1} \ ,
\end{eqnarray}
where $\varpi_j=\omega_j+\Omega_j$ for $j=1,2$. The $A_{i,j}$, and $B_{i,j}$ are laplace coefficients, which are determined by:
\begin{eqnarray}
A_{22} &=& n_2\frac{1}{4\pi}\frac{m_1}{M+m_2}\left(\frac{a_1}{a_2}\right)^2 f_{\psi} \ , \\
A_{21}&=&-n_2\frac{1}{4\pi}\frac{m_1}{M+m_2}\left(\frac{a_1}{a_2}\right) f_{2\psi} \ , \\
B_{22}&=&-n_2\frac{1}{4\pi}\frac{m_1}{M+m_2}\left(\frac{a_1}{a_2}\right)^2 f_{\psi} \ , \\
B_{21} &=& n_2\frac{1}{4\pi}\frac{m_1}{M+m_2}\left(\frac{a_1}{a_2}\right)f_{\psi}\ . \end{eqnarray}
and
\begin{eqnarray}
f_{\psi} &=& \int_{0}^{2\pi}\frac{\cos\psi}{ \left(1-2\left(\frac{a_1}{a_2}\right)\cos\psi+\left(\frac{a_1}{a_2}\right)^2\right)^\frac{3}{2} }d\psi \ , \\
f_{2\psi} &=&\int_{0}^{2\pi}\frac{\cos2\psi}{\left(1-2\left(\frac{a_1}{a_2}\right)\cos\psi+\left(\frac{a_1}{a_2}\right)^2\right)^\frac{3}{2}} d\psi \ .
\end{eqnarray}
The system will remain stable if angular momentum exchange between the two inner planets takes place faster than the precession induced by $m_3$. Accordingly, there exists two regions of parameter space: one contains systems where perturbations on $m_2$ are dominantly from $m_1$, and the other contains systems where $m_2$ is dominated by $m_3$. The former is called the Laplace-Lagrange region, and the ladder is called EKL region. The transition between the Laplace-Lagrange region and EKL region is determined by comparing the two timescales relating the frequencies of precession from each mechanism, i.e.,
\begin{equation}\label{eq:timescales}
\tau_k \sim \tau_{LL} \ .
\end{equation}
We have related the timescale for Kozai oscillations induced by $m_3$ on $m_2$ to the Laplace-Lagrange timescale between $m_1$ and $m_2$. Equating these timescales yields a simple expression for the critical eccentricity of the third planet, $e_{3,c}$, as a function of $a_2$, i.e.,
\begin{equation}\label{eq:e3c}
e_{3,c} \sim\left(1-\left[\frac{15}{16}\frac{m_3\sqrt{G}}{\sqrt{M_\odot+m_1+m_2}}\frac{a_2^{3/2}}{a_3^3}\frac{1}{f_{LL} }\right]^{\frac{2}{3}}\right)^{\frac{1}{2}} \ ,
\end{equation}
where
\begin{equation}\label{eq:fLL}
f_{LL}=A_{22}+A_{21} \frac{e_{1,ic}}{ e_{2,ic}}\cos(\varpi_1-\varpi_2)-B_{21}\frac{i_{1,ic}}{i_{2,ic}}\cos(\Omega_1-\Omega_2) -B_{22} \ .
\end{equation}
$e_{j,ic}$ and $i_{j,ic}$, are the initial eccentricity and inclination for the two inner planets, i.e., $j=1,2$.
As we show below, during the evolution of a stable system $\Omega_1\sim\Omega_2$, $\varpi_2\sim \varpi_1$, $e_1\sim e_2$, and $i_1\sim i_2$. Thus, we define a minimum stable configuration of $f_{LL,{\rm min}}$
\begin{equation}\label{eq:fLLmin}
f_{LL,{\rm min}}=A_{22}+\frac{e_{1,ic}}{ e_{2,ic}}A_{21} - \frac{i_{1,ic}}{i_{2,ic}}B_{21} -B_{22} \ .
\end{equation}
Numerically we find that during the evolution of an unstable system, $e_1/e2$, $i_1/i_2$, $\Omega_1/\Omega_2$, and $\varpi_2/ \varpi_1$ largely deviate from unity, where $\cos(\varpi_1-\varpi_2)$ can be negative. Thus, we also define the maximum stability as:
\begin{equation}\label{eq:fLLmax}
f_{LL,{\rm max}}=A_{22}-\frac{e_{1,ic}}{ e_{2,ic}}A_{21} -\frac{i_{1,ic}}{i_{2,ic}} B_{21} -B_{22} \ ,
\end{equation}
where the difference between Eq.~(\ref{eq:fLLmin}) and (\ref{eq:fLLmax}) is the sign of $A_{21}$.
The stability of systems transitions from the minimum to the maximum $f_{LL}$, as a function of $e_3$. In other words we find a band of parameter space, between $e_{3,c}(f_{LL,{\rm min}})$ and $e_{3,c}(f_{LL,{\rm max}})$, where systems are nearly unstable or completely unstable.
If the third planet's eccentricity is larger than the right hand side of Equation (\ref{eq:e3c}), then the inner system is more likely to become unstable. In the next section we test this stability criterion and show that it agrees with secular numerical integration.
We note that this stability criterion is based on the Laplace-Lagrange approximation, which assumes small eccentricities and inclinations for orbits in the inner system. Thus, it might break down for initial large eccentricities or mutual inclinations of the inner two planets. Furthermore, this stability criterion assumes that the inner system is compact and that the eccentricity excitations from the second planet on the first are suppressed. On the other hand, in the presence of these conditions the stability criterion depends only on the shortest EKL timescale induced by the far away companion and the corresponding Laplace-Lagrange timescale. Thus, it is straightforward to generalize it to more than two inner planets.
\section{Long term stability of multi-planet system}\label{sec:Long}
\subsection{The Gaussian averaging method }
To test our analytic stability criterion we utilize Gauss's Method. This prescription allows us to integrate the system over a long timescale (set to be $10$~Gyr, see below), in a time-efficient way. In this mathematical framework, the non-resonant planets' orbits are phase-averaged and are treated as massive, pliable wires interacting with each other. The line-density of each wire is inversely proportional to the orbital velocity of each planet. The secular, orbit-averaged method requires that the semi-major axes of the wires are constants of motion. We calculate the forces that different wires exert on each other over time.
This method was recently extended to include softened gravitational interactions by introducing a small parameter in the interaction potentials to avoid possible divergences while integrating \citep{Touma+09}. Furthermore, the method has been proven to be very powerful in addressing different problems in astrophysics, ranging from understanding the dynamics in galactic nuclei to describing how debris disks evolve. \citep[e.g.,][]{Touma+09,Sridhar+16,Nesvold+16,Saillenfest+17}.
\subsection{Stability test on a nominal system}
We tested our stability criterion by numerically integrating an ensemble of systems with identical initial conditions except for the semi-major axis and eccentricity of the second and third planets respectively. In this ensemble, we initialize all systems with a primary star $M=1$~M$_\odot$ orbited by three planets where the two inner planets, $m_1$, and $m_2$, each have masses of $1$~M$_\oplus$. The innermost planet, $m_1$, was placed in an orbit with a semi-major axis of $a_1=1$ au, and both inner planets were set initially on circular orbits in the same orbital plane. We set the third planet with a mass of $m_3=1$~M$_J$, at $a_3=100$ au. Throughout the ensemble, $a_2$ ranges from $1.4$ au to $5.9$ au, in steps of $0.5$ au, and $e_3$ ranges from zero to $0.95$ in steps of $0.05$. In all such systems, the third planet's inclination was set to be $45^\circ$ relative to the inner system. We integrated each system to $10$~Gyr, or until orbits crossed. From the results we made a grid labeling the stable/unstable systems of the ensemble. The grid revealed that stable orbits are bounded by the stability criterion. The same process was done on an ensemble which had the third planet's inclination set to $65^\circ$ instead; the result was the same.
\begin{figure}[!h!t]
\centering
\includegraphics[width=\linewidth]{phasestability1.png}
\caption{Here is the parameter space relating the third planets' eccentricity with the second planet' separation. The distant companion's inclination was set to $45^\circ$.
The color code shows $\log(1-\delta)$, Eq.~(\ref{eq:delta}). This parameter estimates the closest the two inner orbits came to each other during the evolution. A darker color characterizes far away separation while a lighter color characterizes orbit crossing.
The stability criterion, Eq.~(\ref{eq:e3c}), is plotted over the grid to show that stable orbits are bounded. The bottom green line represents $e_{3,c}(f_{LL,{\rm min}})$ and the top green line represents $e_{3,c}(f_{LL,{\rm max}})$, and thus the shaded band is the transition zone (see text). Systems above the zone undergo an instability episode while systems below the zone are stable. }
\label{fig:sailing}
\end{figure}
In Figure \ref{fig:sailing} we show the grid of systems we integrated in the parameter space relating $a_2$ and $e_3$. The color code is determined by a proxy that characterizes the stability of the system. The proxy is defined as $\log(1-\delta)$, where $\delta$ is a parameter which describes how close the two inner orbits came during the evolution:
\begin{equation}\label{eq:delta}
\delta={\rm min}\bigg [\frac{a_2(1+e_2) - a_1(1-e_1)}{a_2-a_1} \bigg ] \ .
\end{equation}
In Figure \ref{fig:sailing}, a lighter color means a smaller value of $\delta$, which when zero yields orbit crossing, while a darker color represents a stable configuration.
The criterion for stability is plotted over the grid to show that it is in agreement with the probability for orbits to cross. Above the stability curve, systems are more likely to destabilize during evolution.
\begin{figure*}
\centering
\includegraphics[scale=.9]{MultiPanle}\caption{ \upshape Here are the three prominent types of dynamics from each region. From left to right, stable, intermediate, and unstable, for our nominal system. Shown in the top panels are the apocenters and the pericenters of the innermost planets ($m_1$ in blue and $m_2$) in red. The middle panels show the inclinations of the inner planets ($i_1$, in blue and $i_2$, in red). {\it Note that this inclination is not with respect to the total angular momentum, but rather with respect to the initial angular momentum of the two inner planets.} The bottom panels show, colored in black, the difference between the longitude of ascending nodes of the two inner planets as $\cos (\Omega_1-\Omega_2)$, and, in yellow, the difference between the longitude of the periapsis as $\cos (\varpi_1-\varpi_2)$. These two parameters are present in Equation (\ref{eq:fLL}).
Recall that the nominal system has the following parameters: $M=1$~M$_\odot$, $m_1=m_2=1$~M$_\oplus$, $m_3=1$~M$_J$, $a_1=1$~au, $a_3=100$~au, and we set initially $\omega_1=\omega_2=\omega_3=\Omega_1=\Omega_2=\Omega_3=0$, $e_1=e_2=0.001$, $i_1=i_2=0.001$ and $i_3=45^\circ$. The only difference between each integrated system is the second planet's separation and the third planet's eccentricity. For the {\bf stable} system we chose $a_2=3$~au and $e_3=0.8$. For the {\bf intermediate} system we had $a_2=3$~au and $e_3=0.9$ (which placed it on the stability curve), and finally for the {\bf unstable} system we had $a_2=5.9$~au and $e_3=0.551$.
}\label{AAA}
\end{figure*}
The proxy also reveals a transitional zone of the parameter space. This zone agrees with our analytically determined zone (i.e., between $e_{3,c}(f_{LL,{\rm min}})$ and $e_{3,c}(f_{LL,{\rm max}})$), where, given the initial conditions in the numerical runs, we set $e_{1,ic}/e_{2,ic}\sim 1$ and $i_{1,ic}/i_{2,ic}\sim 1$. Closer to the stability curve, the orbits might get their eccentricities excited and periodically move closer to one another, but their orbits may never cross. Moreover, far into the top right of Figure 2, the instability will take place sooner in the evolution. In this region, the two inner orbits will undergo EKL evolution independently of each other.
The third planet can excite the inner planets' eccentricities on the EKL timescale. However, eccentricity excitations of each planet will not necessarily cause orbit crossing, as depicted in Figure \ref{fig:sailing}. In the parameter space for our ensemble, we have identified a region containing stable systems that seems to smoothly transition into the instability region. Hence, we identify three regimes: The stable regime, the intermediate regime, and the unstable regime. In the stable regime, the Laplace-Lagrange rapid angular momentum exchange between the two inner planets dominates over the gravitational perturbations of the outer orbit. The left column of Figure \ref{AAA} depicts this evolution; the system in this column resides in the bottom left of the parameter space in Figure \ref{fig:sailing}. The system remains stable for 10 Gyrs of evolution and the orbits never come close to one another. The two inner planets' inclinations oscillate together around the initial z-axis, due to the precession of the nodes (see below). In the intermediate region, systems might appear to be stable for a long time but the outer orbit perturbations become too dominant, causing instability of the inner system. We show the dynamics of this type of evolution in the middle panel of Figure \ref{AAA}. This system resides very close to the stability criterion plotted in Figure \ref{fig:sailing}, inside the transition zone between $e_{3,c}(f_{LL,{\rm min}})$ and $e_{3,c}(f_{LL,{\rm max}})$. In the unstable regime, gravitational perturbations from the third planet cause high amplitude eccentricity excitation in the inner planets, causing orbits to cross. We show this behavior in the right column \ref{AAA}. This system lies in the top right of the parameter space depicted in Figure \ref{fig:sailing}.
In the system's reference frame (see Figure \ref{fig:Catroon}), the inner planets' inclinations are the angles between their respective angular momenta and the normal to the {\it initial} orbits. Thus, the inclination modulation shown in the stable system (the left column in Figure \ref{AAA}, is due to precession of the nodes, which results in maximum inclination, which is $\sim 2\times i_3$, since we have started with $i_{3,0}=45^\circ$ \citep[e.g.,][]{Inn+97}.
In contrast, the unstable regime results in misalignment between the inner two planets.
\subsection{Applications: predictions for observed systems }\label{sec:Kepler}
\begin{table}
\begin{center}
\begin{tabular}{|l| c | c | c |}
\hline
name & SMA [au] & mass [M$_J$] & eccentricity \\
\hline
\hline
Kepler-36b & 0.12 & 0.015 & $<0.04$ \\
Kepler-36c & 0.13 & 0.027 &$<0.04$ \\
\hline\hline
Kepler-56b & 0.103 & 0.07 & - \\
Kepler-56c & 0.17 & 0.57 & -\\
\hline
\hline
Kepler-88b & 0.097 & 0.027 & 0.06 \\
Kepler-88c & 0.15 & 0.62 & 0.056 \\
\hline\hline
Kepler-109b & 0.07 & 0.023 & 0.21 \\
Kepler-109c & 0.15 & 0.07&0.03\\
\hline \hline
Kepler 419b & 0.37 & 2.5 & 0.83 \\
Kepler 419c & 1.68 & 7.3 & 0.18 \\
\hline\hline
Kepler-448b & 0.15 & 10 & 0.34 \\
Kepler-448c & 4.2 & 22 & 0.65\\
\hline
\end{tabular}
\end{center}
\caption{Observable parameters of the example systems. The observations are adopted from: {\it Kepler-36}: \citet{Carter+12}. {\it Kepler-56}: \citet{Huber+13} and \citet{Otor+16}. {\it Kepler-88} \citet{Nesvorny+13}, \citet{Barros+14} and \citet{Delisle+17}. {\it Kepler-109:} \citet{Marcy+14}, \citet{Van+15} and \citet{Silva+15}. {\it Kepler-419}:\citet{Dawson+12}, \citet{Dawson+14} and \citet{Huang+16}. {\it Kepler-448}: \citet{Bourrier+15}, \citet{Johnson+17} and \citet{Masuda17}.
}\label{table:obs} \vspace{-0.3cm}
\end{table}
The stability criterion derived here can be used to predict the parameter space in which a hidden companion can exist within a system without destabilizing it. As a proof-of-concept, we discuss the stability of a few non-resonant observed exoplanetary systems in the presence of inclined planets. Specifically, for some observed systems, we provide a set of predictions that characterize the ranges of parameter space available for hidden companions to exist in without disrupting inner orbits. We focus on the following systems: Kepler-419, Kepler-56, Kepler-448, and Kepler-36. These represent the few systems that characterize the extreme limits of two planet systems, from tightly packed super-earths such as Kepler-36 to hierarchical eccentric warm Jupiter systems such as Kepler-448. In Figure \ref{fig:Kepler} we show the relevant parameter space in which the systems can have a hidden inclined companion and remain stable. We chose the more conservative stability criterion for this exercise, i.e., $e_{3,c}(f_{LL,{\rm min}})$. Each line in the four panels of Figure \ref{fig:Kepler} shows the stability criterion for different companion masses. In particular, we consider companion masses of (from top to bottom), $0.1,0.5,1,5$ and $20$~M$_\odot$. For each companion mass, allowed system configurations lie below the curve, and unstable configurations are above the curve. We caution that very close to the stability curve (even below the curve) systems may still undergo eccentricity excitations that may affect the dynamics. The parameters of the inner, observed planets were taken from observations, see Table \ref{table:obs}. Below we discuss the specifics of the four example systems.
\begin{figure*}
\centering
\includegraphics[width=8cm]{kepler36final.png}
\includegraphics[width=8cm]{kepler56final.png}
\includegraphics[width=8cm]{kepler88.png}
\includegraphics[width=8cm]{kepler109final.png}
\includegraphics[width=8cm]{kepler448final.png}
\caption{{\bf The parameter space of hidden friends for a few observed systems.} We consider the companion's eccentricity $e_3$ and separation $a_3$ for five Kepler systems. For each of the systems we plot the stability criterion $e_{3,c}(f_{LL,{\rm min}})$, for the different companion masses. In particular, we consider companion mass of (from top to bottom), $0.1,0.5,1,5$ and $20$~M$_J$. The stable region exists below {\it each} curve and the instability region resides above each curve (For Kepler-448 we shaded both the stable and unstable regions below the curve corresponding to a $20$~M$_J$ companion and above the curve corresponding to a $0.1$~M$_J$ companion respectively). For each system, we used the observed parameters in Equation \ref{eq:fLLmin} to generate the contours. The observed parameters we used for the inner planets are specified in Table \ref{table:obs}. We note that a third companion was reported to Kepler-56, with a minimum mass of $5.6$~M$_J$, which yields a $3.1$~au separation \citep{Otor+16}. This constrains the eccentricity of the companion to lie on a vertical line of a constant semi-major axis in the parameter space. As such, we over-plotted Kepler-56d on the top right panel (dashed line). }
\label{fig:Kepler}\vspace{0.6cm}
\end{figure*}
\begin{itemize}
\item {\it Kepler-36} is an, approximately, one solar mass star orbited by two few-Earth mass planets on $13.8$ and $16.2$~days orbits \citep{Carter+12}. This compact configuration is expected to be stable in the presence of perturbations as shown in Figure \ref{fig:Kepler}, in the top left panel. For example, as can be seen in the Figure, an eccentric inclined $20$~M$_J$ brown dwarf can reside slightly beyond $2$~au.
\item {\it Kepler-56} is an evolved star ($M=1.37$~M$_\odot$) at the base of the red giant branch and is orbited by two sub-Jovian planets with low mutual inclinations \citep[e.g.,][]{Huber+13}. Most notably, asteroseismology analysis placed a lower limit on the two planets' obliquities of $37^\circ$. \citet{Li+14Kepler} showed that a large obliquity for this system is consistent with a dynamical origin. They suggested that a third outer companion is expected to reside far away and used the inner two planets' obliquities to constrain their inclinations. Follow-up radial velocity measurements estimated that a third companion indeed exists with a period of $1000$~days and a minimum mass of $5.6$~M$_J$ \citep[e.g.,][]{Otor+16}. Here we show that indeed a $\sim 5$~M$_J$ planet can exist at $\sim 3$~au, with a range of possible eccentricities up to $0.9$. A more massive planet can still exist at $\sim 3$~au with a possible range of eccentricities up to slightly below $e_3\sim 0.8$. This example of a tightly packed system yields the expected result, i.e, a large part of the parameter space is allowed to have an inclined companion, as depicted in Figure \ref{fig:Kepler} in the top right panel. In the presence of an inclined companion, the inner two planets will most likely have non-negligible obliquities, as was postulated by \citet{Li+14Kepler}.
\item {\it Kepler-88} is a star of, approximately, one solar mass. It has been observed to be orbited by two planets. The first planet is Earth-like ($m_1\sim 0.85M_{\oplus}$) and the second planet is comparable to Jupiter ($m_2\sim0.67$~M$_{J}$). Together, the two planets form a compact inner system with negligible eccentricities \citep[e.g.,][]{Nesvorny+13,Barros+14,Delisle+17}.
In Figure \ref{fig:Kepler}, left middle panel, we show that an inclined planet can exist in large parts of the parameter space. For example, a massive companion ($\sim 20$~M$_J$) can exist beyond $2$au with an eccentric orbit ($\sim 0.7$).
\item {\it Kepler-109} is a star that also has, approximately, one solar mass ($1.07$~M$_\odot$) and two planets orbiting it in a compact configuration, having a small mutual inclination \citep{Marcy+14,Van+15,Silva+15}. However, the eccentricity of the innermost planet is not as negligible as the secondary's ($e_1 \sim 0.21 > e_2 \sim 0.03$), but does not yield a violation of the approximation. We show the stability bounds within the $(e_3,a_3)$ parameter space for this system in the middle right panel of Figure \ref{fig:Kepler}. There it can be seen that this system can exist in the presence of an eccentric companion with $20$~M$_J$, so long as the companion is beyond $3$au.
\item {\it Kepler-419} is a $1.39$~M$_\odot$ mass star which is orbited by two Jupiter sized planets with non-negligible eccentricities and low mutual inclinations \citep[e.g.,][]{Dawson+12,Dawson+14,Huang+16}. The large, observed eccentricities of the planets violate the assumption that the eccentricities are sufficiently small. Recall that this assumption was used to derive the stability criterion in Eq.~(\ref{eq:e3c}). Furthermore, the quadrupole timescale for precession induced by Kepler-419c on Kepler-419b is comparable to that of Laplace-Lagrange's. Thus, a distant massive companion is expected to excite the two planets' eccentricities and inclinations. When the mutual inclination between the two inner planets is large, the second planet can further excite the innermost planets' eccentricity, thus rendering the system unstable. We have verified, numerically, the instability of the system is consistent with the breakdown of our criterion. We did this for several systems with far away companions with masses varying from $1,5,$ and $20$~M$_J$. Thus, we suggest that a massive inclined companion can probably be ruled out for this system. Since our stability criterion is violated here, it is not necessary to show it's parameter space in a plot. On the other hand, a smaller companion is not expected to cause secular excitations in the eccentricities of more massive planets, since the inner system will excite its eccentricity and inclination \citep[e.g., inverse EKL][]{Naoz+17,Zanardi+17,Vinson+18}. Therefore, the existence of a smaller inclined companion cannot be ruled out.
\item Kepler-448 is a $1.45$~M$_\odot$ star orbited by a $10$~M$_J$ warm Jupiter \citep{Bourrier+15,Johnson+17}. Recently, there was a reported discovery of a massive companion ($\sim 22$~M$_J$) with a rather hierarchical configuration \citep{Masuda17}. The hierarchical nature of the inner system yields a more limited range of separations to hide a companion, see Figure \ref{fig:Kepler}, in the bottom panel. On the other hand, the large masses of Kepler-448b and Kepler-448c imply that a small planetary inclined companion can still exist with negligible implications on the inner system. In fact, one would expect that the inner system will largely affect a less massive companion \citep[e.g.,][]{Naoz+17,Zanardi+17,Vinson+18}.
\end{itemize}
\section{discussion}\label{sec:dis}
We have analyzed the stability of four-body hierarchical systems, where the forth companion is set far away ($a_3\gg a_2,a_1$). Specifically, we focus on three planet systems, for which the two inner planets reside in a relatively close configuration and have an inclined, far away, companion. Observations have shown that multiple tightly packed planetary configurations are abundant in our Galaxy. These systems may host a far away companion, which might be inclined or even eccentric. An inclined perturber can excite the eccentricity of planets in the inner system via the EKL mechanism, which can ultimately destabilize the system.
We have analytically determined a stability criterion for two-planet systems in the presence of an inclined companion (Equation \ref{eq:e3c}). This criterion depends on the initial conditions of the inner planetary system and on the outer orbit's mass, eccentricity, and separation.
It is thus straightforward to generalize it to $n>2$ inner planetary systems.
We then numerically integrated a set of similar systems using the Gauss's averaging method, varying only the outer companion's eccentricity $e_3$ and the second planet's separation $a_2$. We have
characterized each numerical integrated systems' stability and showed that stable systems are consistent with our analytical criterion.
A system will remain stable if the timescale for angular momentum exchange between the two inner orbits takes place faster than eccentricity excitations that might be induced by a far away, inclined, companion. When the system is stable, the two inner orbits have minimal eccentricity modulations and their inclinations, with respect to their initial normal orbit, remained aligned to one another. An example of such as system is depicted in the left panels of Figure \ref{AAA}.
Assuming the normals of the inner orbits are initially parallel to the stellar axis allows precession of the nodes to be interpreted as obliquity variations. Thus, a non-negligible obliquity for two (or more) tightly packed inner orbits may be a signature of an inclined, distant, companion \citep[e.g.,][]{Li+14Kepler}. The obliquity, in this case, oscillates during the system's dynamical evolution, which can have large implications on the habitability of the planets
\citep[e.g.,][]{Shan+17}.
On the other hand, a system will destabilize if the precession induced by the outer companion is faster than the precessions caused by interactions between the inner bodies. In this case, the two inner planets will exhibit large eccentricity excitations accompanied with large inclination oscillations (see, for example, right panels in Figure \ref{AAA}). In this type of system, each planet undergoes nearly independent EKL oscillations, and thus extremely large eccentricity values can be expected, as well as chaos \citep[e.g.,][]{Naoz+13,Teyssandier+13,Li+14}.
We also showed (e.g., Figure \ref{fig:sailing}, green band) that the stability criterion includes a transition zone, where systems are likely to develop large eccentricity, leading to close orbits. Systems close to the transition zone, or in the transition zone, can be stable for long period of time, and develop instability very late in the evolution. We show an example for such as system in the middle column of Figure \ref{fig:sailing}. In this example, the system stayed stable for slightly more than a Gyr and developed an instability that leads to orbit crossing after $\sim 1.5$~Gyr.
We note that our analysis did not include General Relativistic or tidal effects between the planets and the star because, typically, including them will further stabilize systems. General relativistic precession tends to suppress eccentricity excitations if the precession takes place on a shorter timescale than the induced gravitational perturbations from a companion \citep[e.g.,][]{Naoz+12GR}. Also, tidal precession tends to suppress eccentricity excitations \citep{Dan,Liu+15}. These suppression effects become more prominent the closer the inner orbits are to the host star. Thus, if eccentricity excitations from the companion take place on a longer timescale than general relativity or tidal precession the system will remain stable.
Finally, as a proof-of-concept, we used our stability criterion to predict the parameter space in which a hidden inclined companion can reside for four Kepler systems (see Figure \ref{fig:Kepler}). The systems we consider were Kepler-419, Kepler-56, Kepler-448, and Kepler-36. These systems represent a range of configurations, from tightly packed systems with small or super-Earth mass planets to potentially hierarchical systems with Jupiter mass planets. A notable example is Kepler-56, where the recently detected third planet was reported to have a minimum mass of $5.6$~M$_J$, and a $\sim 1000$~days orbit. Such a system indeed resides in the predicted stable regime. Furthermore, given a mass for the Kepler-56d, we can limit its possible eccentricity.
\acknowledgements
PD acknowledges the partial support of the Weyl Undergraduate Scholarship. SN acknowledges the partial support of the Sloan fellowship.
\bibliographystyle{apj}
|
1,116,691,499,311 | arxiv | \section{Introduction}
Rotation-based ages of old Sun-like stars are rooted in a complex and intricate dependence on age, rotation, turbulent convection, structural variations and mass-loss due to magnetized winds \citep{skumanich72,reiners12,guerrero13, fionnagain18}. Classically, the age-dating method that relies on this phenomenon assumes that the rotational periods ($P_{\rm rot}$) can be expressed in well-defined functions of the age and mass (or a proxy of it), the so-called gyrochronology relations \citep{barnes07,mamajek08}. These relations had successfully confirmed the paradigm of rotation-activity-age coupling that powers the global dynamo evolution along the main-sequence \citep{barnes07,vidotto14,donascimento14,lorenzo16,lorenzo18} and reproduced the main features observed in open clusters spanning a wide range of ages \citep{meibom15}.
Apart from this inspiring agreement, some of the old \kepler\ field stars shows unexpected fast rotation, especially hotter ones with ages greater than 2$-$3 Gyr \citep{angus15,metcalfe16}. This tension led to idea that after a critical Rossby number \citep[$Ro \equiv P_{\rm rot}/\tau_{\rm CZ}$, where $\tau_{\rm CZ}$ is the convective turnover time;][]{noyes84} a drastic change of the stellar differential rotation (SDR) pattern might hamper the production and maintenance of magnetic field large-scale components over secular timescales. One of the most important (and accessible) effects of this drastic transition would be the presence of old and fast rotating stars with reduced angular momentum loss caused by magnetized winds. However, recent observational results gave us alternative hints about the possible smooth nature of the Sun-like rotational evolution in the light of \kepler\ asteroseismic data \citep{benomar18}.
Motivated by this up-to-date discussion about the smooth nature of the age-rotation relations, we can ask ourselves: \textit{what solar twins can tell us about the solar rotational evolution}? This letter uses solar twins to evaluate a recent claim \citep{vansaders16} about the solar rotational transition at a given critical Rossby number. Sec. \ref{sec:rotation_sample} describes our working sample, selection criteria of solar twin rotators and determination of \Prot through activity time series. In Sec. \ref{sec:age_rotation} we discuss the age-rotation evolution of solar twins and the suitability of standard rotational evolution models. The conclusions are drawn in Sec. \ref{sec:conclusions}.
\section{Solar twins rotation sample}\label{sec:rotation_sample}
We compiled 79 solar twins (plus the Sun) presented in \citet{spina17}. Our sample was extensively observed over the years (2011$-$2016) with HARPS spectrograph \citep{mayor03} fed by the 3.6 m telescope at La Silla Observatory, to search for planets around solar twins \citep[program 188.C-0265,][]{melendez17}. To prevent the inclusion of anomalously fast rotators due to binarity effects, 17 spectroscopic binary (SB) stars and other solar twins with a close companion within 4'' were discarded from the analysis \citep{santos17}. Other interesting solar twins spectroscopically analyzed in the past by our group were added to our sample \citep[HIP30503 and HIP78399 in][]{galarza16}. The resulting HARPS sample of this work is composed of 65 stars.
In order to estimate rotational velocities and other stellar parameters of interest, we made use of the updated atmospheric parameters ($T_{\rm eff}$, \logg, $[\mathrm{Fe/H}]$, and $[\mathrm{\alpha/Fe}]$) provided by \citet{spina17}. \gaia\ DR2 \textit{G} band photometry and parallaxes \citep{gaia18} were combined to the spectroscopic data to obtain stellar ages, masses, \logg\ and other evolutionary parameters following the procedures described in \citet{nolan18} (see their Sec. 6). For distant \kepler\ solar twins that will be described in the following sections, we used the reddening \textit{G} band corrections provided by \gaia\ DR2. The typical mass and metallicity of our solar twin sample agrees with solar parameters within $\pm$0.05 $M_{\odot}$ and $\pm$0.04 dex, respectively. Stellar atmospheric and evolutionary parameters, projected \Prot and other relevant information of the entire sample is shown in Table \ref{tab:data}. Projected rotational and macroturbulence velocities were determined through HARPS \textit{full width half maximum} of the \textit{cross correlation function} ($FWHM_{\rm CCF}$) vs. \vsini together with macroturbulence calibrations provided by \citet{santos16}. For details about the adopted procedures, see their Sec. 3. The CCF based \vsini errors are computed propagating the $FWHM_{\rm CCF}$ calibration and typical \vsini measurement errors estimated by \citet{santos16} yielding $\sigma$($v\sin i$)=0.23 km/s or $\sigma$($P_{\rm rot}/\sin i$) of 3 days (assuming 2\% of $\sigma (R/R_\odot)$).
\begin{table*}
\caption{Relevant parameters for our rotation sample of solar twins stars analyzed in this paper. This table is available in its entirety in machine readable
format at the CDS.}
\begin{tabular}{lcccccccl}
\hline
HIP & Age & Mass & \feh & \Psini & \Prot & $P_{\rm rot}^P$ & $P_{\rm rot}^P (Ro_{\rm crit}=2.0)$ & Remark \\
-- & [Gyr] & [$M/M_\odot$] & -- & [d] & [d] & [d] & [d] &-- \\
\hline
\hline
1954 & 4.3$_{-0.3}^{+0.3}$ & 0.97$\pm$0.03 & -0.090$\pm$0.003 & 26.6$\pm$3.1 & 24.1$\pm$0.2 & 25.2$\pm$2.7 & 24.8$\pm$2.7 & This work \\
7585 & 4.1$_{-0.2}^{+0.3}$ & 1.03$\pm$0.03 & 0.083$\pm$0.003 & 24.1$\pm$3.0 & 23.0$\pm$2.3 & 23.4$\pm$3.4 & 20.3$\pm$3.9 & \citet{see17} \\
22263 & 0.6$_{-0.6}^{+0.5}$ & 1.06$\pm$0.03 & 0.037$\pm$0.006 & 14.8$\pm$3.0 & 11.8$\pm$3.0 & 9.6$\pm$4.3 & 9.6$\pm$4.3 & \citet{suarez17} \\
30503 & 3.0$_{-1.4}^{+0.7}$ & 1.08$\pm$0.04 & 0.070$\pm$0.016 & 27.3$\pm$3.0 & 20.0$\pm$0.1 & 18.7$\pm$4 & 15.6$\pm$3.8 & This work \\
36515 & 0.3$_{-0.3}^{+0.3}$ & 1.03$\pm$0.03 & -0.029$\pm$0.009 & 11.9$\pm$3.0 & 4.6$\pm$1.2 & 7.2$\pm$2.3 & 7.2$\pm$2.3 & This work \\
\hline
\label{tab:data}
\end{tabular}
\end{table*}
Given that the rotation axes of the stars are randomly oriented in different inclination angles ($i$), our spectroscopic analysis is limited by the projection factor $\sin i$ \citep{gray05}. This feature skews the \Prot distribution towards higher values of rotation for a given age owing to the geometric factor 1/$\sin i$. On the other hand, if one analyses a progressively larger sample of stars, it is indeed expected that, for a given age/mass/${\rm [Fe/H]}$, the lower boundary of \Psini distribution asymptotically matches with the \textit{true} distribution of \Prot which is scattered by intrinsic effects (e.g. propagation of initial conditions and SDR effects). In fact, this approximation has an optimal applicability for samples of twin stars like ours, where mass and \feh effects, that might hamper the statistical corrections towards the true \Prot distributions, are mitigated.
Through 10$^6$ Monte-Carlo (MC) simulations we sought for an optimal selection criterion that closely unveils the distribution of stars with the highest chance of having ${\rm \sin i}\sim$1, for a given sub-sample size $N$. We ran $N$ MC simulations assuming random angle orientations ranging from 0 to $\pi$/2 for a range of angular rotational frequencies ($\Omega$). We assigned an intrinsic $\Omega$ error of 10\% to account for stellar differential rotation \citep[SDR,][]{epstein14}. For each simulation, the difference ($\Delta$) between the \textit{true} \Prot and the median \Psini distribution delimited by its upper 0.5(P50), 0.7(P70), 0.84(P84), and 0.975(P98) percentiles cut-offs. This procedure was repeated 10$^4$ times in order to estimate the best selection criteria (using the sample medians of each upper percentiles cut-offs) that minimizes $|\Delta|$ $-$ i.e. the stellar inclination selection bias. For a typical sub-sample size of 10 stars, we found that the average between P70 and P98 estimates closely converges into the centroid of the \textit{true} \Prot distribution, assuming rotation roughly constant along the age domain considered. The exclusion of the upper 2.5\% of each simulation prevents to systematically include unusually fast-rotating stars in each age-bin. Following this selection prescription, we bootstrapped the 1$\sigma$ confidence intervals within each age-bin of 2 Gyr through 10$^4$ repetitions. We found 2 Gyr age-bin as our optimal choice because it balances sampling and the ratio between the expected \Prot evolution and realistic \Prot errors due to intrinsic effects, at least for stars older than $\sim$1 Gyr. In fact, during the simulations, we also compute the age-\Prot correlation within each age bin and found negligible correlation between both variables (p-value$\gg$0.05). In Table \ref{tab:data}, we highlight the stars with probable $\sin~i\sim1$ along each age-bin. In Fig. \ref{fig:psini_prot_fig} (left panel) we show our subsample of 10 stars with both HARPS \Psini and \Prot measurements. Our \Psini estimates are consistent with the 1:1 relation represented by the red dashed lines. The intrinsic errors associated to \Prot measurements (10\%) are given by shaded region along the 1:1 identity line. As the inclination factor is encapsulated in the \Psini, the projected \Prot are slightly shifted towards higher rotation values albeit it marginally converges into the shaded region indicating that the majority of these stars have $\sin i\sim1$.
\begin{figure}
\centering
\begin{minipage}[t]{0.45 \linewidth}
\centering
\resizebox{\hsize}{!}{\includegraphics{PROT_PSINI.pdf}}
\end{minipage}
\centering
\begin{minipage}[t]{0.45 \linewidth}
\centering
\resizebox{\hsize}{!}{\includegraphics{FIG1.pdf}}
\end{minipage}
\caption{\textit{Left panel}: Comparison between \Prot and \Psini for a subsample of 10 solar twins observed by HARPS. The dashed red line stands for 1:1 relation and shaded region represents 10\% of \Prot intrinsic errors. \textit{Right panels}: \ion{Ca}{II} periodogram analysis of three solar twins with different ages. The most probable peaks for HIP30503 (upper panel), HIP1954 (middle panel), and HIP118115 (lower panel) are 20.0, 24.1, and 40.5 days. The dashed red lines stand for the false-alarm probability of 3 sigma.}
\label{fig:psini_prot_fig}
\end{figure}
Using our large \ion{Ca}{II} activity time-series of solar twins \citep{lorenzo18}, we determined \Prot of six stars using a Generalized Lomb-Scargle analysis \citep[GLS,][]{zechmeister09}: HIP 1954, 30503, 36515, 79672, 95962 and 118115. Our procedure is similar to the one described by \citet{suarez17}. In brief, for each star, we cleaned our time-series by removing the observations with low signal to noise ratio around the \ion{Ca}{II} lines (SNR$<$30). To avoid the inclusion of poor observations or stellar transient events such as flares, we removed from the activity time-series outliers placed above $\geq2.5\sigma$. Then, we look for the presence of strong signals with bootstrapped false-alarm probability (FAP) $\geq$3$\sigma$ at frequencies related to typical rotational timescales ($\leq$50 days). In other cases where no significant peaks were found, we detrended the time-series from eventual sinusoidal long-term signal (FAP$\geq3\sigma$) that are likely to be associated to stellar cycle modulations. In these cases, the \Prot measurements are determined in subsequent periodogram analysis of the detrended time-series. Signals matching with the expected window function in period space were not considered in our analysis. Figure \ref{fig:psini_prot_fig} (right panels) shows our periodogram analysis for three solar twins. Rotation periods for another four solar twins were gathered from the literature: HIP 7585, 22263, 42333 and 43297 \citep{petit08,wright11,suarez17,see17}.
Additionally, another four solar twins with \kepler\ \Prot were added to our analysis. Precise atmospheric parameters were obtained with Gemini/GRACES (ID:GN-2018B-FT-101) and KECK/HIRES high signal-to-noise ratio and high resolution observations conducted by our group \citep[][Yana Galarza et al. in prep.]{bedell17}: Kepler-11, KIC 10130039, 12404954 and 7202957 \citep{mcquillan13,mazeh15}. The relevant information about the \kepler\ solar twins are summarized in Table \ref{tab:data}. Finally, we collected literature data from three solar metallicity old open clusters observed by the \kepler\ mission spanning a critical age range of rotational evolution: NGC6811 ($\sim$1 Gyr, N=5 stars) NGC6819 ($\sim$2.5 Gyr, N=5 stars) and M67 ($\sim$4 Gyr, N=12 stars) \citep{meibom11, meibom15, barnes16, brandenburg18}. We restricted our sample selection only to those stars with near-solar \Teff based on ${\rm (B-V)}_0$ index \citep[5600 $\leq$ \Teff $\leq$ 5900 K,][]{casagrande10}. For NGC6811, the $\rm{(B-V)}$ colors were estimated inverting the $\rm{(g-r)}$ vs. $\rm{(B-V)}$ calibration equation by \citet{bilir05}. The average \Prot of $\sim$1 solar mass star in these clusters are 10.2$\pm$0.6, 18.1$\pm$0.5 and 24.0$\pm$2.4 days for NGC6811, NGC6819 and M67, respectively.
\section{Age-Rotation relation}\label{sec:age_rotation}
In Fig. \ref{fig:omega_fig} we highlight the rotational evolution of our sample of selected solar twins (red triangles), open clusters (black squares) and the Sun (in black, represented by the $\odot$ symbol). Stars with measured \Prot are represented by the black circles. We denoted as black crosses the centroid of each one of the 2 Gyr age-bin based on its respective age-\Prot average and dispersion (see Sec. \ref{sec:rotation_sample}). The solar twin sample is composed of those stars with \Prot errors within 1$\sigma$ from its respective age-\Prot cluster centroid. All selected stars are fully consistent with expected dispersion due to intrinsic measurement errors ($\sim$10\%). The only outlier is the 6 Gyr-old KIC 10130039 which deviates from the expected \Prot distribution by more than 50\%. Thanks to our extensive radial-velocity monitoring together with detailed mapping of chemical anomalies and activity levels \citep{santos17,spina17,lorenzo18}, we found that the stars placed considerably below the lower limit of rotation rate for a given age are, in fact, spectroscopic binaries. Two illustrative examples from \citet{santos16} are the 4.0 and 7.2 Gyr-old SB HIP 19911 and 67620 which rotate at 4.1 km/s and 2.7 km/s level (\Psini$\sim$12 and 20 days), respectively. We found that these stars are not likely to be representative of the rotational sample with a significance higher than 3$\sigma$.
\begin{figure}
\centering
\begin{minipage}[t]{0.85 \linewidth}
\centering
\resizebox{\hsize}{!}{\includegraphics{FIG3_PROT_RO.pdf}}
\end{minipage}
\caption{\textit{Left panel}: Modified Kawaler rotation track for 1.00 M$_\odot$ and solar metallicity in solid black line without considering the weakened braking efficiency scenario. We assign $Ro_{\rm critical}$ at the turn-off as 2.6 ($Ro_{\rm TO}$). Dash dotted lines stand for 10\% of \Prot error given by intrinsic effects. \kepler\ OCs are represented by black squares. Solar twins with measured \Prot are the black empty circles. Selected twin sample and its respective age-\Prot centroid stand for red triangles and black crosses, respectively. The Sun is represented by $\odot$. \textit{Right panel}: Modified Kawaler wind law model for $Ro_{\rm critical}$=2.0.}
\label{fig:omega_fig}
\end{figure}
To test the hypothesis of weakened magnetic braking at a given Rossby number threshold \citep{vansaders16}, we constructed rotational evolution tracks adopting the well-known modified Kawaler wind-law \citep[assuming N=1.5,][]{kawaler88,krishnamurthi97} and the updated \textrm{YaPSI} grid of stellar tracks provided by \citet{spada17} for different masses and metallicities. The angular momentum equation is solved assuming a negligible moment of inertia change ($dI/I \rightarrow 0$) along the main-sequence:
\begin{equation}\label{eq:kw}
\frac{dJ}{dt} = -K_w\Omega^3\Bigg(\frac{R}{R_\odot}\Bigg)^{0.5}\Bigg(\frac{M}{M_\odot}\Bigg)^{-0.5}, \qquad\qquad \rm{for}\quad \Omega>\Omega_{sat}
\end{equation}
The constant $K_w$ is fine-tuned to match the unsaturated rotational evolution model to the solar properties (\Prot=25.4 days at 4.57 Gyr), considering 1.0 M/M$_\odot$ and solar metallicity track. The saturated value of rotation rate ($\Omega_{sat}$) is scaled by the solar convective turnover time following \citet{krishnamurthi97}. The rotational evolution model is modified whenever the star approaches into the turn-off (${\rm TO}$) region or a given critical Rossby number. After this stage, there is a dominance of structural changes over the magnetic braking terms in the angular momentum evolution (i.e. $dJ/dt\approx0$). Thus, we fix the angular momentum ($J_{\rm TO/crit}$), leaving only the moment of inertia to vary towards the main-sequence turn-off \citep{vansaders13}. For the Sun, our turn-off threshold is at $Ro_{\rm TO}\gtrsim$2.6 (or $\sim$8 Gyr)
Predicted \Prot were derived for 22 Sun-like members of the OCs NGC6811, NGC6819 and M67 using Eq. \ref{eq:kw}. Effective temperatures were adopted as mass proxy using $(B-V)$ calibration by \citet{casagrande10} and literature spectroscopic \feh of each OC \citep{leebrown15,liu16b,netopil16}. In other words, the rotational tracks were built as function of age, $T_{\rm eff}$ and ${\rm [Fe/H]}$. In Fig. \ref{fig:age_mass_oc}, we test the consistency of our predicted \Prot as a function of $T_{\rm eff}$, fixing the age of each OC. The 1$\sigma$ confidence bands set by 10\% of \Prot uncertainty are represented by the shaded regions around each OC age-\Teff diagram prediction. Visually, our predictions are in good agreement with the existent \Prot data. For NGC6811, we show a rotational track for an age of 0.85 Gyr which is 15\% younger than the canonical age of 1 Gyr.
We show in Fig. \ref{fig:omega_fig} the $Ro_{\rm TO}\sim$2.6 (left panel) and $Ro_{\rm crit}\sim$2 (right panel) scenarios for the rotational evolution of the Sun. Visually, both approaches seem to be in reasonable agreement. The magnetic weakened braking scenario favors the faster rotators placed in the lower boundary of the \Prot distribution. On the other hand, the model with smooth rotational evolution follows more closely the average trend observed, especially towards the oldest stars.
\begin{figure}
\centering
\begin{minipage}[t]{0.85 \linewidth}
\centering
\resizebox{\hsize}{!}{\includegraphics{FIGURE_M67.pdf}}
\end{minipage}
\caption{Consistency check of our models using OC data. Blue, red and black lines are the rotational tracks for M67, NGC6819 and NGC6811 with ages of 3.9, 2.4 and 0.85 Gyr, respectively. Each OC is represented by its correspondent color. The Sun is represented by $\odot$.}
\label{fig:age_mass_oc}
\end{figure}
We quantified the suitability of both approaches by computing for each star the \Prot probability density function that depends on the age, mass, \feh for a given $Ro_{\rm crit}$. The errors are always assumed to follow Gaussian distributions. The adopted theoretical \Prot is represented by its median ($P_{\rm rot}^P$) and 16$-$84\% percentiles based on \Prot cumulative distribution function ($\sigma_{P_{\rm rot}}^P$). For instance, we derived $Ro_{\odot}$=2.13 ($P_{\rm rot, \odot}^P$=25.3 days). We calculated the Bayesian Information Criterion ($BIC$). $BIC$ accounts the trade-off between the fitting quality ($\hat{L}$), number of fitting parameters ($k$) of a given model (M) and the sample size (N): $BIC({\rm M}) = -2\ln\hat{L}(M)+k\ln{N}$, where $\hat{L}(M)$ is the product of likelihood of each data point with the composite errors computed through quadratic propagation of the individual measured and predicted \Prot errors. The $BIC$ difference ($\Delta BIC({\rm M2},{\rm M1}) \equiv BIC_{\rm M2} - BIC_{\rm M1}$) derived from 2 different models indicates which one is more likely. Defining M1 as the smooth rotational evolution model (Fig. \ref{fig:omega_fig}, left panel), we calculated $\Delta$BIC(M2,M1) where M2 stands for models with progressively larger $Ro_{\rm crit}$ ranging from 1.5 up to the subgiant branch, where both assumptions converge into the same \Prot solution ($Ro_{\rm crit}\rightarrow Ro_{\rm TO}$). The OC stars were used in this work as a consistency check of our models so our statistical tests are only based on field stars.
To find the optimal $Ro_{\rm crit}$ for our sample, all the possible uncertainties were considered (10\% error related to SDR, model errors due to stellar parameters and measured \Prot errors). The best fit in terms of $Ro_{\rm crit}$ is $2.6_{-0.1}^{+\infty}$ with the corresponding 1$\sigma$ lower age limit of $t_{\rm crit}\gtrsim8.8$, $\gtrsim6.5$, $\gtrsim4.2$, and $\gtrsim2.5$ Gyr for 0.95, 1.00, 1.05 and 1.10 solar mass and metallicity stars, respectively. On the other hand, at 95\% confidence level, our result marginally approaches to the solar properties with the $Ro_{\rm crit}$ ranging from 2.3 to $Ro_{\rm TO}$ ($t_{\rm crit}\gtrsim$5.3 Gyr, for a solar mass/metallicity star). In all cases, the $\Delta$BIC analysis indicates values greater than +2. For $Ro_{\rm crit,\odot}$, we found $\Delta$BIC=+9.2 indicating a strong evidence favoring the smooth rotation evolution model, at least until the solar rotational level ($\gtrsim$25 days). For a solar mass star, if the magnetic weakened braking scenario is taking place at these $Ro$ thresholds, we should only detect unusually fast rotators (i.e. stars that depart the gyrochronology relations, considering the measurement errors) at ages considerably older than the Sun (say $\gtrsim$6 Gyr). Unfortunately, there is a lack of \Prot detections in this age range.
Given the statistical difficulties to disentangle the two scenarios, we tend to favor the simplest assumption of the smooth rotational evolution. Other possibility is that maybe the magnetic transition might occur, if it happens at all, at later evolutionary stages than it was hypothesized before \citep{vansaders16}. All in all, we conclude with a marginal level of confidence that, considering the available data of solar twins, no indisputable indication emerged about the weakened magnetic braking scenario. On the other hand, we are aware that this phenomenon might be correlated to other manifestations such as drastic changes in stellar cycle morphology and also in stellar differential rotation profile, as some \kepler\ data suggests. For a comprehensive discussion of these possibilities, see \citet{metcalfe17}. Even though, we stress that more data of similar stars is still needed to clarify this issue and firmly establish at what level should we trust on rotation-based ages.
\section{Conclusions}\label{sec:conclusions}
The goal of this paper is to test different rotational evolution scenarios using a selected sample of solar twins characterized with the HARPS, HIRES and GRACES spectrographs. Stellar ages and other evolutionary parameters were estimated through HR diagram analysis with the help of new \gaia\ DR2 G band photometry and parallaxes and precise atmospheric parameters. Measured \Prot of 14 solar twins were collected from the literature and/or estimated in this work through \ion{Ca}{II} H \& K activity time-series. To trace the rotational evolution of solar twins, we build a grid of rotational evolutionary tracks based on modified Kawaler wind law and structural models. We compute these rotation tracks for a large range of critical Rossby number ($Ro_{\rm crit}$) to account for the magnetic weakened braking phenomena observed by \citet{vansaders16}. We found a marginal statistical evidence favoring the smooth rotation evolution. In the light of magnetic weakened braking scenario, the lower limit of critical Rossby number would be $Ro_{\rm crit}\gtrsim2.3$ (at 95\% confidence level) which intercepts an age range somewhat older than the Sun and the end of the main-sequence. This result highlights the difficulty to statistically discern both scenarios with the current sample of solar twins. Therefore, it is desirable that other works also approach this issue by determining new \Prot of old solar twin stars to clarify the past and the future of the solar rotational evolution.
\section*{Acknowledgements}
We would like to acknowledge the anonymous referee, whose comments have unquestionably led to an improved paper. DLO is grateful to the Brazilian workers and taxpayers, whose effort provided in the past years the stability for young scientists to independently develop their scientific ideas. DLO and JM thanks support from FAPESP (2016/20667-8; 2018/04055-8). LAdS acknowledges the financial support from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program ({\sc Four Aces}; grant 724427). MA acknowledges funding from the Australian Research Council (grant DP150100250)
\bibliographystyle{mnras}
|
1,116,691,499,312 | arxiv | \section{Introduction}
We work over the complex number field $\mathbb C$ throughout,
and we denote the Picard number of $X$ by $\rho(X) = \rank\Pic(X)$.
\begin{definition}
A normal projective 3-fold $X$ is called a {\em Fano 3-fold} if and
only if $X$ has $\mathbb Q$-factorial terminal singularities, $-K_X$ is ample,
and $\rho(X) = 1$.
The {\em Fano index $f=f(X)$} of a Fano 3-fold $X$ is
\[
f(X) = \max \{ m \in \mathbb Z_{> 0} \st -K_X = mA
\textrm{ for some Weil divisor }A \}.
\]
where equality of divisors denotes linear equivalence of some multiple.
A Weil divisor $A$ for which $-K_X = fA$ is called
a {\em primitive ample divisor}.
\end{definition}
Fano 3-folds are sometimes called $\mathbb Q$-Fano $3$-folds to distinguish
them from the classical nonsingular case.
By \cite{Su}, Theorem 0.3, we know that $f\le 19$.
The case $f=1$ is the main case with several hundred confirmed families
and much work towards classification ongoing.
In the case $f\ge 4$, \cite{Su1} contains a near classification of
about 80 families (only a handful of which remain in doubt).
We study the case $f=2$ here.
Typical classical examples
include the cubic 3-fold in $\mathbb P^4$ and the intersection of two
quadrics in $\mathbb P^5$; see \cite{encyc}, Table~12.2, for example,
in which $r$ denotes the Fano index.
Note that the anti-canonical divisor $-K_X$ is only expected to be
a Weil divisor, although necessarily it will be $\mathbb Q$-Cartier.
A Fano $3$-fold $X$ with primitive ample divisor $A$ has a graded ring
\[
R(X, A) = \bigoplus_{n \geq 0} H^{0}(X, \mathcal O_{X}(nA)).
\]
This graded ring is finitely generated. The {\em Hilbert series of $X,A$}
is defined to be that of the graded ring $R(X,A)$. A choice of minimal
(homogeneous) generating set $x_0,\dots,x_N\in R(X,A)$ determines
an embedding
\begin{equation}
\label{eq!emb}
X\hookrightarrow \mathbb P^N = \mathbb P(a_0,\dots,a_N)
\end{equation}
for some weighted projective space (wps) $\mathbb P^N$, where
$x_i\in H^{0}(X, \mathcal O_{X}(a_iA))$.
With this embedding in mind, we say that $X,A$ has {\em codimension} $N-3$.
A Fano 3-fold is {\em Bogomolov--Kawamata stable},
or simply {\em stable}, if
\begin{equation*}
(-K_X)^3 \leq 3(-K_X c_2(X)).
\end{equation*}
Fano 3-folds satisfying this condition fall into one half of Kawamata's
argument on the boundedness of Fanos \cite{Ka2}.
Our first result is a list of the possible Hilbert series of graded rings
$R(X,A)$ for Fano $3$-fold of Fano index $2$ and their analysis in low
codimension.
\begin{thm}
\label{thm!main}
There are at most 1492 power series that are the Hilbert series
of some $X,A$ with $X$ a Fano 3-fold of Fano index~2 and $A$ a primitive
ample divisor, of which 1413 could correspond to stable Fano 3-folds.
Of these power series, 36 can be realised by some stable $X,A$ of
codimension $\le 3$.
\end{thm}
The proof is explained in section~\ref{sec!pf}; arguments of Kawamata
\cite{Ka1}, \cite{Ka2} and Suzuki \cite{Su} impose conditions
on geometrical data of $X,A$ which are then analysed by computer.
The 36 cases in low codimension are listed in
Tables~\ref{tab!cod1}--\ref{tab!cod3} in section~\ref{sec!cod123}.
In the case $f=1$, there are many Fano 3-folds with with empty
anti-canonical linear system. This does not happen in index~2:
\begin{thm}
\label{thm!nonvan}
If $X,A$ is a Fano 3-fold of Fano index $2$
then $H^0(X,-K_X)\not= 0$.
\end{thm}
We showed in \cite{BS3} that $-K_X$ also has sections whenever
$f\ge 3$, and so this result shows that $-K_X$ has a section
whenever $f\not=1$.
When $f\ge 3$, the Riemann--Roch formula is a function of
the singularities of $X$. Fano 3-folds with $f\le 2$ have
an extra parameter, their {\em genus}, $g=\dim H^0(X,A)-2$.
We have sharp bounds on the degree for various genera:
\begin{thm}
\label{thm!bounds}
Let $X,A$ be a Fano 3-fold of Fano index $2$ with $-K_X = 2A$.
Let $g=\dim H^0(X,A)-2$.
Then $-2\le g \le 9$, and the degree $A^3$ of $X$ is bounded below
by $1/165$; according to small $g$ the sharp bounds are:
\[
\begin{array}{r|ccccc}
\text{genus $g$} & -2 & -1 & 0 & 1 & 2 \\
\hline
\text{lower bound for $A^3$} & 1/165 & 1/35 & 1/3 & 1 & 2
\end{array}
\]
Moreover, all these lower bounds are achieved by stable Fano 3-folds.
\end{thm}
The graded ring approach to building classifications is well known;
we describe it in this case in section~\ref{sec!pf}.
There are two main points to be aware of.
First, although the the Hilbert series of any $R(X,A)$ will indeed
appear in the list of Theorem~\ref{thm!main}, there may be power
series in the list that do not correspond to such a graded ring:
the list comprises {\em candidates} for the Hilbert series of
Fano 3-folds, and an appearance on the list does not imply that a
Fano 3-fold exists with that Hilbert series.
Second, we cannot say exactly which rings occur---for there will be
many degenerations, as in \cite{B2}, and classifying them will be
difficult. But in many
cases we can predict at least the smallest possible codimension
of such a ring together with the weights of the corresponding wps.
When the proposed ring has codimension $\le 3$, then we are able
to construct it. Gorenstein rings in codimension~4 are more subtle.
In section~\ref{sec!cod4} we describe a first classification into 35 cases.
In section~\ref{sec!examples} we discuss some typical examples.
Then section~\ref{sec!series} contains the proofs after first
assembling the ingredients: the appropriate Riemann--Roch formula,
bounds on the singularities and degree, and Kawamata's boundedness result.
In section~\ref{sec!lists} we study the converse question, that of
constructing Fano 3-folds with given Hilbert series, listing results in
Tables~\ref{tab!cod1}--\ref{tab!cod4}.
It is our pleasure to thank Miles Reid for his help and
encouragement throughout this work.
\section{Examples}
\label{sec!examples}
We work out some examples, pointing out where each Ingredient~1--5
of section~\ref{sec!series} fits in.
We use our standard notation of Notation~\ref{stdnotn} below.
To run the graded ring method, we need a basket of singularities
and an integer $g\ge -2$. In the first example, we choose $g=-2$.
Consider the basket of quotient singularities
\[
\mathcal B =
\left\{
\frac{1}{3}(1,2,2),
\frac{1}{5}(1,4,2),
\frac{1}{11}(3,8,2)
\right\},
\]
where $\frac{1}{r}(a,-a,2)$ denotes the germ of the quotient
singularity $\mathbb C^3/(\mathbb Z/r)$ by the action
\[
(x,y,z) \mapsto (\varepsilon^a x,\varepsilon^{-a} y, \varepsilon^2 z)
\quad
\textrm{where $\varepsilon=\exp(2\pi i/r)$}.
\]
(The third component 2 is forced since such a singularity
will be polarised locally by the canonical class.)
We want to make a Fano 3-fold $X,A$ of Fano index~2 such
that $X$ has exactly the singularities of $\mathcal B$
and $\dim H^0(X,A) = g + 2$.
(It would be enough that $X$ has terminal singularities
which contribute to Riemann--Roch as though they were the
singularities of $\mathcal B$, but in practice a general element of
any family we construct has quotient singularities.)
According to Ingredient~\ref{ing!basket} in section~\ref{sec!series}
below, the basket $\mathcal B$ is a possibility since
\[
\sum_{\mathcal B} r - 1/r =
(3-1/3) + (5-1/5) + (11-1/11) < 24.
\]
Applying the formula of Ingredient~\ref{ingred!hs} with data $\mathcal B,g$
describes Hilbert series
\[
P(t) =
\frac{1-t^{38}}{(1-t^2)(1-t^3)(1-t^5)(1-t^{11})(1-t^{19})}.
\]
Certainly any hypersurface of the form
\[
X = X_{38} \subset \mathbb P(2,3,5,11,19)
\]
will have Hilbert series $P_X(t)$ equal to $P(t)$.
One can check that a general member of this family is
indeed a quasismooth index~2 Fano 3-fold with singularities exactly
$\mathcal B$ and $H^0(X,A)=0$.
This is one of eight hypersurfaces that are Fano 3-folds of
Fano index~2: all eight are listed in Table~\ref{tab!cod1}
of section~\ref{sec!cod123}.
Notice that any quasismooth Fano 3-fold of this form has a K3 elephant;
that is, the section $S\in \mid{-}K_X\mid$ is a K3 surface,
$S_{38}\subset \mathbb P(3,5,11,19)$.
\paragraph{Examples from del Pezzo surfaces}
A classical example is the Fano 3-fold $X\subset\mathbb P^6$
whose hyperplane section is the del Pezzo surface of degree~5.
The equations of a general such 3-fold are well known: they are the
five maximal Pfaffians of a skew $5\times 5$ matrix of general
linear forms on $\mathbb P^6$.
Extensions of other del Pezzo surfaces also give rise
to index~2 Fano 3-folds. In \cite{cascade}, such series of
families is called a `cascade' and other examples using log
del Pezzo surfaces are described there.
The main case of \cite{cascade} is a series of families which,
in order of increasing codimension (although, as with blowups
of $\mathbb P^2$, the opposite order is also natural) begins
with the hypersurface $X_{10} \subset \mathbb P(1^2, 2, 3, 5)$,
followed by $X_{4,4} \subset \mathbb P(1^3, 2^2, 3)$ in codimension~2
and a family in codimension~3, $X\subset \mathbb P(1^4, 2^2, 3)$.
These families arise using $\mathcal B=\{\frac{1}{3}(1,2,2)\}$
and allowing $g$ to vary, starting at $g=0$ and increasing.
When $g=0$, the formula of Ingredient~\ref{ingred!hs} with
data $\mathcal B,g$ describes Hilbert series
\[
P(t) = (1-t^{10})/(1-t)^2(1-t^2)(1-t^3)(1-t^5),
\]
and, as before, this corresponds to a family of Fano hypersurfaces
above of degree $A^3=1/3$. When $g=1$, we get
\[
P(t) = (1-t^4)^2/(1-t)^3(1-t^2)^3(1-t^3),
\]
corresponding to the codimension~2 complete intersection above
of degree $4/3=1/3+1$.
Increasing $g$ again to $g=2$, and thus $A^3=7/3=2/3+2$ by
Ingredient~\ref{ing!mindeg} below, describes Hilbert series
\[
P(t) = (t^4 + t^3 + 3t^2 + t + 1)/(1-t)^3(1-t^3).
\]
This description is not as revealing as previous ones, but it is
easy to remedy. The numerator indicates that a further generator
in degree~1 is needed, and with an extra factor of $1-t$, we see that
\[
P(t) = (-t^5 - 2t^3 + 2t^2 + 1)/(1-t)^4(1-t^3).
\]
Now it is clear that two new generators in degree~2 are
required---these also serve to polarise the index~3 singularity in
the basket. The final result is
\[
P(t) = (-t^9 + 2t^6 + 3t^5 - 3t^4 - 2t^3 + 1)/(1-t)^4(1-t^2)^2(1-t^3).
\]
This Hilbert series is realised by a graded ring with generators
$x_{1}, x_{2}, x_{3}$, $y_{1}, y_{2}$, $z$ in degrees 1, 1, 1, 2, 2, 3
and relations generated by the five maximal Pfaffians
of the following $5 \times 5$ skew matrix (where as usual we
omit the leading diagonal of zeroes and leave the skew lower half
implicit)
\[
M = \begin{pmatrix}
x_{1} & x_{2} & b_{14} & b_{15}\\
& x_{3} & b_{24} & b_{25}\\
&& b_{34} & b_{35}\\
&&& z
\end{pmatrix}
\quad
\textrm{of degrees}
\quad
\begin{pmatrix}
1 & 1 & 2 & 2\\
& 1 & 2 & 2\\
&& 2 & 2\\
&&& 3
\end{pmatrix}.
\]
The $b_{ij}$ are forms of degree~2 in the $x$ and $y$ variables.
It is easy to check that for general $b_{ij}$, this defines the
graded ring of a Fano 3-fold in $\mathbb P^6(1^4,2^2,3)$ as required, and
we think of the $b_{ij}$ as being parameters defining a flat family
of Fano 3-folds of index~2 with given basket $\mathcal B$ and genus $g$.
\section{Hilbert series of Fano 3-folds}
\label{sec!series}
We describe in sections~\ref{sec!RR}--\ref{sec!bddness} the five
ingredients that go into the raw list of power series that is the
basis for our classification. Our raw list will include the Hilbert
series of every index~2 Fano 3-fold, although it may contain power
series not of this form. The derivation of the list is explained
in section~\ref{sec!pf}.
Although it is not necessary for everything below, we suppose throughout
that Fano 3-folds $X$ treated here will have Fano index $f(X)=2$.
\subsection{Riemann-Roch formula}
\label{sec!RR}
We explain the notion of a {\em basket of singularities}; see
\cite{YPG} or \cite{B} section~2.1.
Let $P\in X$ be a 3-dimensional terminal singularity of index $r > 1$
and $(U, P)$ a germ at $P$. (In particular, $rK_U$ is Cartier on $U$.)
If $(U, P)$ is a quotient singularity, then it isomorphic to the germ
at the origin of some $\mathbb C^3/(\mathbb Z/r)$, with action
\[
\varepsilon\cdot(x,y,z)\mapsto (\varepsilon^ax, \varepsilon^ay, \varepsilon^cy).
\]
We denote this by $\frac{1}{r}(a, b, c)$
and recall that $\hcf(r,abc) = 1$ and $a+b=r$ up to permutations of $a,b,c$.
When $X$ has Fano index~2, then we may suppose that
$(a,b,c) = (a,r-a,2)$. In particular, $r$ cannot be even.
If $(U, P)$ is not a quotient singularity, then
$(U, P)$ can be deformed to a unique finite collection of
terminal quotient singularities, say $\{Q_1,\dots, Q_{n(P)}\}$ (where,
as is usual, this is a set with possible repetitions) for some
number $n(P) \geq 1$. Each point $Q_i$ is some quotient singularity
$\frac{1}{r_{i}}(a_i, -a_i,2)$
where $r_i$ and $a_i$ are coprime,
$r_i \geq 2$, and $r=\lcm\{r_1,\dots r_{n(P)}\}$.
We call the set $\mathcal B(U, P) := \{Q_{1},\dots Q_{n(P)}\}$ the
{\em basket of singularities} of $(U, P)$.
In the global case, we assemble all local baskets into one.
\begin{definition}
Let $X$ be a 3-fold with terminal singularities and $\{P_1,\dots,P_m\}$
the set of singular points of $X$ of index $\geq 2$.
Denoting germs $P_{i} \in U_{i} (\subset X)$,
we define the {\em basket of singularities of $X$}
to be the disjoint union $\mathcal B(U_{1},P_{1}) \cup\dots\cup \mathcal B(U_m,P_m)$
(a set with possible repetitions.)
\end{definition}
\begin{notn}
\label{stdnotn}
We denote a Fano 3-fold of index~$f=2$ for which $-K_X=2A$ by $X,A$.
The basket of singularities of $X,A$ is denoted $\mathcal B$.
A typical singularity of $\mathcal B$ is denoted $\frac{1}{r}(a,-a,2)$,
and we use this notation whenever taking a sum over the elements of $\mathcal B$.
\end{notn}
\begin{thm}[\cite{Ka1}]
Let $X$ be a Fano $3$-fold with basket of singularities $\mathcal B$.
Then $-K_Xc_2(X) > 0$, the Euler characteristic $\chi(\mathcal{O}_X)=1$
and
\[
\chi(\mathcal{O}_{X}) =
\frac{-K_{X}c_{2}(X)}{24} + \sum\frac{r^{2} -1}{24r},
\]
the sum taken over $\mathcal B$ (see Notation~\ref{stdnotn}).
\end{thm}
This theorem gives the following bounds on the
singularities in the basket:
\begin{ingredient}[Basket bound]
\label{ing!basket}
If $\mathcal B$ is the basket of a Fano 3-fold $X$, then
\[
\sum\frac{r^{2} -1}{r} < 24
\quad
\textrm{and}
\quad
-K_{X}c_{2}(X) = 24 - \sum\frac{r^{2} -1}{r}
\]
where each sum is taken over $\mathcal B$ (see Notation~\ref{stdnotn}).
\end{ingredient}
For the next theorem, recall that the plurigenera of a polarised
variety $X,A$ are denoted $P_n(X,A) = \dim H^0(X,nA)$.
\begin{thm}[{\cite{Su}}]
\label{thm!RR}
Let $X$ be a Fano $3$-fold of Fano index $f$ and basket of singularities
$\mathcal B$. Let $A$ be a primitive Weil divisor with $-K_{X} = fA$. Then
\begin{eqnarray*}
\chi(\mathcal{O}_{X}(nA)) & = &
\chi(\mathcal{O}_{X}) + \frac{n(n+f)(2n+f)}{12}A^{3}
+ \frac{nAc_{2}(X)}{12} \\
&&
+ \sum_\mathcal B \left( -i_n\frac{r^{2} -1}{12r}
+ \sum_{j=1}^{i_n-1} \frac{\overline{bj}(r-\overline{bj}) }{2r}\right)
\end{eqnarray*}
where the sum is taken over points $P=\frac{1}{r}(a,-a,2)$
in $\mathcal B$ (see Notation~\ref{stdnotn}) using notation:
$i_n \in [0, r-1]$ is the local index of $nA$ at $P$ (see \cite{YPG}),
$b \in [0, r-1]$ satisfies $ab \equiv 2 \mod r$ and
$\overline{c} \in [0, r-1]$ is the residue of $c$ $\mod r$.
By Kawamata--Viehweg vanishing, $\chi(nA) = h^{0}(nA)$ for all $n > -f$.
So the Hilbert series $P_{X,A}(t)$ of $X$ is:
\begin{eqnarray*}
P_{X,A}(t) & = & \sum_{n = 0}^{\infty} P_{n}(X,A)t^{n}\hspace*{1cm}\\
& = &
\frac{1}{1-t} +
\frac{(f^{2}+3f+2)t+(-2f^{2}+ 8)t^{2}+(f^{2}-3f + 2)t^{3}}{12(1-t)^{4}}A^{3}\\
& & \qquad + \frac{t}{(1-t)^{2}}\frac{Ac_{2}(X)}{12} + \sum_{P\in\mathcal B} c_P(t)
\end{eqnarray*}
where, for a point $P=\frac{1}{r}(a,-a,2)$ in $\mathcal B$,
\begin{eqnarray*}
c_P(t) & = & \frac{1}{1-t^{r}}
\left(
\sum_{k=1}^{r-1}
\left(-i_k\frac{r^2-1}{12r} +
\sum_{j=1}^{i_k-1}\frac{\overline{bj}(r-\overline{bj})}{2r}\right)t^k
\right).
\end{eqnarray*}
\end{thm}
Setting $f=2$ in the Hilbert series above gives the closed formula:
\begin{ingredient}[Hilbert series]
Using Notation~\ref{stdnotn} and the expression for $c_P(t)$
from Theorem~\ref{thm!RR}, the Hilbert series is
\label{ingred!hs}
\[
P_{X,A}(t) = \frac{1}{1-t} + \frac{t}{(1-t)^{4}}A^{3}
+ \frac{t}{(1-t)^{2}}\frac{Ac_{2}(X)}{12} + \sum c_P(t)
\]
where the sum is taken over $\mathcal B$.
\end{ingredient}
Setting $n=1$ in the Riemann--Roch formula, we compute the minimum
possible value of the degree $A^3$.
\begin{ingredient}[Minimum degree]
\label{ing!mindeg}
Using Notation~\ref{stdnotn},
\[
0 < A^3 =
- 1 -
\frac{Ac_2(X)}{12} -
\sum \left( -i_1\frac{r^{2} -1}{12r}
+ \sum_{j=1}^{i_1-1} \frac{\overline{bj}(r-\overline{bj}) }{2r}\right)
+ N
\]
for some integer $N\ge 0$, and where the sum is taken over $\mathcal B$.
\end{ingredient}
Since the Riemann--Roch formula also holds when $n=-1$, we have an
additional constraint.
\begin{ingredient}[Polarisation condition]
\label{ing!pol}
Using Notation~\ref{stdnotn},
\[
1 +
\sum \left( -(i_{-1})\frac{r^{2} -1}{12r}
+ \sum_{j=1}^{(i_{-1})-1} \frac{\overline{bj}(r-\overline{bj}) }{2r}\right) =
\frac{Ac_{2}(X)}{12}.
\]
where the sum is taken over $\mathcal B$.
\end{ingredient}
\subsection{Kawamata Boundedness}
\label{sec!bddness}
Following \cite{Ka2}, let $X$ be a Fano 3-fold with
Fano index $2$ and $\mathcal E = (\Omega_{X}^{1})^{**}$ the double dual of
the sheaf of K\"ahler differentials of $X$.
We do not define $\mu$-semistability (with respect to $-K_{X}$) here,
since we do not use it further, but note the role it plays in
strengthening the following bound on the degree.
\begin{thm}[{\cite{Ka2}}]
\label{thm:kaw}
Let $X$ be a Fano $3$-fold with Fano index $2$.
Then $(-K_{X})^{3} \leq \frac{16}{5}(-K_{X}c_{2}(X))$.
If, furthermore, the sheaf $\mathcal E$ described above is $\mu$-semistable,
then $(-K_{X})^{3} \leq 3(-K_{X}c_{2}(X))$.
\end{thm}
\begin{ingredient}
\label{ing!kawa}
Using Notation~\ref{stdnotn}, $A^3 \le \frac{4}{5}Ac_2(X)$.
(In the stable case, the upper bound is $A^3\le \frac{3}{4}Ac_2(X)$.)
\end{ingredient}
Applying the upper bound for $Ac_2$ of Ingredient~\ref{ing!basket},
this bound implies $A^3\le 48/5$ (and $A^3\le 9$ in the stable case.)
In fact, once all possible baskets are calculated, the sharp bound
is $A^3\le 9$ achieved using the empty basket and $g=9$; the next
largest degree is $A^3=25/3$ achieved by the basket $\{\frac{1}{3}(1,2,2)\}$
and $g=8$ (which is not stable since $Ac_2=32/3$).
In particular, for a fixed basket there are at
most 9 different values for $g$ that give Fano Hilbert series.
Comparing with \cite{cascade}, we regard this as a bound on the
number of blowups (or projections) that we can make from maximal $g$.
A familiar instance of this bound is the maximum number of blowups of
$\mathbb P^2$ that is a del Pezzo surface; again, 8 is the limit.
In \cite{cascade}, such a cascade of 8 log del Pezzo surfaces is
constructed linking the hypersurface $S_{10}\subset\mathbb P(1,2,3,5)$ with
a log del Pezzo of degree $25/3$.
\subsection{Proofs of Theorems~\ref{thm!main}--\ref{thm!bounds}}
\label{sec!pf}
These proofs use some computer calculations which we describe rather
than reproducing; we use the computer algebra system Magma \cite{M}
for our calculations, and a short file with Magma code that
can be run to generate these results is available at \cite{code}.
\paragraph{Listing the Hilbert series}
The first step is to construct the list of all power series according
to the five ingredients assembled in section~\ref{sec!series}.
This requires little comment. We simply compute all possible baskets
satisfying Ingredients~\ref{ing!basket} and \ref{ing!pol}, together
with all possible values for $N$ using Ingredients~\ref{ing!mindeg}
and \ref{ing!kawa}, and then apply the Riemann--Roch formula
of Ingredient~\ref{ingred!hs}.
The result is all power series that could be the Hilbert series
of a Fano 3-fold of index~2. There are 1492 such series in all.
(If we impose the lower `stable' bound of Ingredient~5, this
number reduces to 1413.)
\paragraph{Bounds on the degree}
It is easy to compute both upper and lower bounds on the degree
(and also on $Ac_2(X)$) by computer check on the list.
In Theorem~\ref{thm!bounds} we list lower bounds for small $g$, since
our methods of construction reveal Fano 3-folds that realise these
bounds. The following table includes all the degree bounds for each
genus $g$, even though we do not know whether all of them are sharp
or not.
\begin{center}
$
\begin{array}{|r|ccccc}
\hline
\text{genus $g$} & -2 & -1 & 0 & 1 & 2 \\
\hline
\text{lower bound for $A^3$} & 1/165 & 1/35 & 1/3 & 1 & 2\\
\text{upper bound for $A^3$} & 11/15 & 32/21 & 89/39 & 64/21 & 19/5\\
\hline
\text{total number of series} & 337& 470& 303& 174& 97\\
\text{(of which are unstable)} & 6& 14& 14& 15& 11\\
\hline
\end{array}
$
\hfill
\vspace{3mm}
\\ \hfill
$
\begin{array}{ccccccc|}
\hline
3 & 4 & 5 & 6 & 7 & 8 & 9 \\
\hline
3 & 4 & 5 & 6 & 7 & 8 & 9 \\
68/15 & 16/3 & 6 & 48/7 & 38/5 & 25/3 & 9\\
\hline
54& 28& 14& 8& 4& 2& 1\\
7& 5& 2& 2& 2& 1& 0\\
\hline
\end{array}
$
\end{center}
\paragraph{Nonempty anticanonical system}
The proof of Theorem~\ref{thm!nonvan} is straightforward: since we
do not discard candidate Hilbert series unless they are proved not
to come from a Fano 3-fold, it is enough to calculate
the coefficient of $t^2$ in each one. Again, this is done by
a computer on all 1492 Hilbert series.
Since $H^0(X,-K_X)\not= 0$, we can ask whether the linear system
$\vert -K_X \vert$ contains a K3 surface. Such a K3 section is sometimes
impossible because its singular rank would be too big---see \cite{B},
Proposition~4. For example, $\mathcal B=\{\frac{1}{21}(10,11,2)\}$ with $g=0$
satisfy all our numerical conditions (including stability)---they predict
$A^3 = 19/21$ and $(1/12)Ac_2(X) = 8/63$. But a K3 section cannot exist
because the corresponding surface singularity $\frac{1}{21}(10,11)$ has
20 exceptional curves in its resolution, pairwise orthogonal in
the Picard group, which cannot happen in $H^2(S)$ for a K3 surface $S$.
There are 171 such cases (of which 9 are unstable). (These examples
all appear to be in high codimension, and we do not analyse them further.)
When the singular rank is $\le 19$, we get an estimate of the degrees
of generators of a model $R(X,A)$ for a Fano 3-fold by comparing with
K3 surfaces appearing in the K3 database \cite{B}. This gives an idea of
how to understand the result of Theorem~\ref{thm!main}, just as the estimates
of \cite{cascade} 3.2.5 do in the case $f\ge 3$. But this list should not
be taken as more than a guide. Using this K3 comparison, the number
of Hilbert series per codimension is as follows.
\begin{center}
$
\begin{array}{|r|cccccccc}
\hline
\textrm{estimated codimension} & 1&2&3&4&5&6&7&8\\
\hline
\text{total number of series} &
8 & 26 & 2 & 35 & 13 & 59 & 25 & 99\\
\text{(of which are unstable)} &
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
\hline
\end{array}
$
\hfill
\vspace{3mm}
\\ \hfill
$
\begin{array}{cccccccccc|}
\hline
9&10&11&12&13&14&15&16&17&18\\
\hline
51 & 163 & 93 & 227 & 126 & 255 & 48 & 78 & 8 & 3 \\
1 & 2 & 4 & 6 & 8 & 30 & 6 & 11 & 0 & 0\\
\hline
\end{array}
$
\end{center}
We describe the 36 candidates in codimension~$\le 3$ in
section~\ref{sec!cod123} and the 35 candidates in codimension~4
in section~\ref{sec!cod4} below.
\section{Constructing lists of Fano $3$-folds}
\label{sec!lists}
The remaining claim is that 36 of the 1492 families can be
realised in codimension $\le 3$. This is achieved by the
examples in Tables~\ref{tab!cod1}--\ref{tab!cod3} below.
We also explain the list in Table~\ref{tab!cod4} of codimension~4
candidates and describe the role of unprojection methods.
\subsection{Models in low codimension}
\label{sec!results}
\paragraph{Finding the first generators}
The following standard analysis of a Hilbert series bounds the
minimum number of generators below. Since we are concentrating
on codimension $\le 3$, once we have confirmed that the ring must
have at least eight generators if it exists, then we do not pursue
it further.
Consider the data $\mathcal B = \{ \frac{1}{9}(1,8,2) \}$, $g=1$.
Suppose there is a Fano $X,A$ with this data; we begin
to describe $R(X,A)$.
(Such $X$ would have $A^3=13/9$ and $Ac_2=17/27$.)
The Hilbert series expressed as a power series is
\[
1 + 3t + 8t^2 + 17t^3 + 32t^4 + 54t^5 + 85t^6 + \cdots.
\]
Any graded ring having this Hilbert series must have exactly
three generators in degree~1. These span at most a 6-dimensional
subspace in degree~2, so there are at least two new generators
in degree~2. If this number is exactly two, then a similar argument
shows that there must be at least one new generator in degree~3.
The only alternative is to have three new generators in degree~2.
Either way, we already have at least six generators in the ring.
Now we turn to the singularities. Certainly $X$ has a terminal
singularity of index~9. Whether it is a quotient singularity
or not, locally it must be a quotient by $\mathbb Z/9$ acting with
at least one eigencoordinate of weight 8---every case of index~9 in
the classification of terminal singularities \cite{mori} and
their baskets \cite{YPG} has local eigencoordinates $a,-a$,
where $a=1$ in this case. So $R(X,A)$ must
have at least one generator of weight divisible by 9 and another
of weight 8 modulo 9. So $R(X,A)$ is in codimension $\ge~4$.
(One could also argue on the denominator of the Hilbert series:
as a rational function, it has cyclotomic polynomial of degree~9
in its denominator, so the ring needs a generator in degree some
multiple of 9 to cancel this.)
\paragraph{Constructing varieties and additional generators}
We carry out the analysis above systematically using a computer.
When it suggests that a ring may be in codimension $\le 3$, then we
attempt to construct it as in the examples of section~\ref{sec!examples}.
The construction is straightforward, although there is one small twist:
it happens frequently that we can find a graded ring with the right
Hilbert series but that does not correspond to a Fano 3-fold.
For example, with $\mathcal B = \{ \frac1{11}(2,9,2) \}$, $g=-1$,
the Hilbert series is
\[
P(t)=\frac{(1-t^6)(1-t^8)(1-t^{10})}{\prod(1-t^a)}
\]
where the product is taken over $a\in\{1,2,2,2,3,5,11\}$.
This is the Hilbert series of any $X_{6,9,10}\subset\mathbb P(1,2,2,3,3,5,11)$.
But such a variety is not a Fano 3-fold: the equations cannot involve
the variable of degree 11, so it will have a cone singularity at
the index~11 point.
Moreover, there is no variable of degree 9 to polarise the singularity
as we wanted. In this case, the solution is clear:
adding a generator of degree~9 to the ring gives a codimension~4 model
$X\subset\mathbb P(1,2,2,2,3,5,9,11)$.
The family of complete intersections comprise a component of the
Hilbert scheme that does not contain the desired varieties.
\subsection{Classification in codimensions 1 to 3}
\label{sec!cod123}
The lists in Tables~\ref{tab!cod1}--\ref{tab!cod3} contain families
whose general element lies in
codimension at most 3. (We denote the singularity $\frac{1}{3}(1,2,2)$
simply by $\frac{1}{3}$.) In addition to these, there will be
degenerations of low codimension families that occur in higher
codimension. For example, a Fano $X_{6,18}\subset\mathbb P(1,2,3,5,6,9)$
for which the variable of degree~6 does not appear in the equation
of degree~6 is not listed in the table of codimension~2 Fano
3-folds; its Hilbert series is that of the degree~$1/15$ hypersurface,
and the degeneration occurs as the two singularities of index~3
in that hypersurface come together.
It is easy to confirm that the general element in each case
of Tables~\ref{tab!cod1}--\ref{tab!cod3} is a Fano 3-fold with
the indicated properties.
\begin{table}[h]
\[
\begin{array}{lccc}
\quad\textrm{Fano hypersurface} & \textrm{Basket $\mathcal B$} & A^3 & Ac_2/12 \\
\hline
\hline
X_3\subset\mathbb P(1,1,1,1,1) & \textrm{nonsingular} & 3 & 1 \\
X_4\subset\mathbb P(1,1,1,1,2) & \textrm{nonsingular} & 2 & 1 \\
X_6\subset\mathbb P(1,1,1,2,3) & \textrm{nonsingular} & 1 & 1 \\
X_{10}\subset\mathbb P(1,1,2,3,5) & \frac{1}{3} \textrm{ ($= \frac{1}{3}(1,2,2)$)} & 1/3 & 8/9 \\
\hline
X_{18}\subset\mathbb P(1,2,3,5,9) &
2\times\frac{1}{3},\ \frac{1}{5}(1,4,2) & 1/15 & 26/45 \\
X_{22}\subset\mathbb P(1,2,3,7,11) &
\frac{1}{3},\ \frac{1}{7}(3,4,2) & 1/21 & 38/63 \\
X_{26}\subset\mathbb P(1,2,5,7,13) &
\frac{1}{5}(2,3,2),\ \frac{1}{7}(1,6,2) & 1/35 & 18/35 \\
X_{38}\subset\mathbb P(2,3,5,11,19) &
\quad\frac{1}{3},\ \frac{1}{5}(1,4,2),\ \frac{1}{11}(3,8,2)\quad & 1/165 & 116/495\\
\hline
\end{array}
\]
\caption{Fano 3-folds in codimension 1\label{tab!cod1}}
\end{table}
\begin{table}[t]
\[
\begin{array}{lccc}
\qquad\textrm{Fano 3-fold} & \textrm{Basket $\mathcal B$} & A^3 & Ac_2/12 \\
\hline
\hline
X_{2,2}\subset\mathbb P(1,1,1,1,1,1) &
\textrm{nonsingular}
& 3 & 1 \\
X_{4,4}\subset\mathbb P(1,1,1,2,2,3) &
\frac{1}{3}
& 4/3 & 8/9 \\
X_{4,6}\subset\mathbb P(1,1,2,2,3,3) &
2\times \frac{1}{3}
& 2/3 & 7/9 \\
X_{6,6}\subset\mathbb P(1,1,2,2,3,5) &
\frac{1}{5}(2,3,2)
& 3/5 & 4/5 \\
X_{6,8}\subset\mathbb P(1,1,2,3,4,5) &
\frac{1}{5}(1,4,2)
& 2/5 & 4/5 \\
\hline
X_{6,6}\subset\mathbb P(1,2,2,3,3,3) &
4\times\frac{1}{3}
& 1/3 & 5/9 \\
X_{6,8}\subset\mathbb P(1,2,2,3,3,5) &
2\times\frac{1}{3},\ \frac{1}{5}(2,3,2)
& 4/15 & 26/45 \\
X_{6,10}\subset\mathbb P(1,2,2,3,5,5) &
2\times \frac{1}{5}(2,3,2)
& 1/5 & 3/5 \\
X_{8,10}\subset\mathbb P(1,2,2,3,5,7) &
\frac{1}{3},\ \frac{1}{7}(2,5,2)
& 4/21 & 38/63 \\
X_{10,14}\subset\mathbb P(1,2,2,5,7,9) &
\frac{1}{9}(2,7,2)
& 1/9 & 17/27 \\
\hline
X_{8,10}\subset\mathbb P(1,2,3,4,5,5) &
\frac{1}{3},\ 2\times\frac{1}{5}(1,4,2)
& 2/15 & 22/45 \\
X_{8,12}\subset\mathbb P(1,2,3,4,5,7) &
\frac{1}{5}(1,4,2),\ \frac{1}{7}(3,4,2)
& 4/35 & 18/35 \\
X_{10,12}\subset\mathbb P(1,2,3,4,5,9) &
\frac{1}{3},\ \frac{1}{9}(4,5,2)
& 1/9 & 14/27 \\
X_{12,14}\subset\mathbb P(1,2,3,4,7,11) &
\frac{1}{11}(4,7,2)
& 1/11 & 6/11 \\
X_{10,12}\subset\mathbb P(1,2,3,5,6,7) &
2\times\frac{1}{3},\ \frac{1}{7}(1,6,2)
& 2/21 & 31/63 \\
X_{14,16}\subset\mathbb P(1,2,5,7,8,9) &
\frac{1}{5}(2,3,2),\ \frac{1}{9}(1,8,2)
& 2/45 & 58/135 \\
\hline
X_{10,12}\subset\mathbb P(2,2,3,5,5,7) &
2\times\frac{1}{5}(2,3,2),\ \frac{1}{7}(2,5,2)
& 2/35 & 11/35 \\
X_{10,14}\subset\mathbb P(2,2,3,5,7,7) &
\frac{1}{3},\ 2\times\frac{1}{7}(2,5,2)
& 1/21 & 20/63 \\
X_{12,14}\subset\mathbb P(2,2,3,5,7,9) &
\frac{1}{3},\ \frac{1}{5}(2,3,2),\ \frac{1}{9}(2,7,2)
& 2/45 & 43/135 \\
X_{14,18}\subset\mathbb P(2,2,3,7,9,11) &
2\times\frac{1}{3},\ \frac{1}{11}(2,9,2)
& 1/33 & 32/99 \\
X_{18,22}\subset\mathbb P(2,2,5,9,11,13) &
\frac{1}{5}(1,4,2),\ \frac{1}{13}(2,11,2)
& 1/65 & 17/65 \\
\hline
X_{10,12}\subset\mathbb P(2,3,3,4,5,7) &
4\times\frac{1}{3},\ \frac{1}{7}(3,4,2)
& 1/21 & 17/63 \\
X_{12,14}\subset\mathbb P(2,3,4,5,7,7) &
\frac{1}{5}(2,3,2),\ 2\times\frac{1}{7}(3,4,2)
& 1/35 & 8/35 \\
X_{14,16}\subset\mathbb P(2,3,4,5,7,11) &
\frac{1}{3},\ \frac{1}{5}(2,3,2),\ \frac{1}{11}(4,7,2)
& 4/165 & 116/495 \\
X_{18,20}\subset\mathbb P(2,4,5,7,9,13) &
\frac{1}{7}(2,5,2),\ \frac{1}{13}(4,9,2)
& 1/91 & 16/91 \\
X_{18,20}\subset\mathbb P(2,5,6,7,9,11) &
\frac{1}{3},\ \frac{1}{7}(2,5,2),\ \frac{1}{11}(5,6,2)
& 2/231 & 103/693 \\
\hline
\end{array}
\]
\caption{Fano 3-folds in codimension 2}
\end{table}
\begin{table}
\[
\begin{array}{lccc}
\qquad\textrm{Fano 3-fold} & \textrm{Basket $\mathcal B$} & A^3 & Ac_2/12 \\
\hline
\hline
X_{2,2,2,2,2}\subset\mathbb P(1,1,1,1,1,1,1) & \textrm{\quad nonsingular\quad} & 5 & 1 \\
X_{3,3,4,4,4}\subset\mathbb P(1,1,1,1,2,2,3) & \frac{1}{3} & 7/3 & 8/9 \\
\hline
\end{array}
\]
\caption{Fano 3-folds in codimension 3\label{tab!cod3}\label{tab!cod2}}
\end{table}
\subsection{Classification in codimension 4}
\label{sec!cod4}
We continue our analysis of rings into higher codimension,
although the method becomes more complicated and we do not
check all possibilities rigorously. (Complete results are on the
webpage \cite{code}.) We list in Table~\ref{tab!cod4} the 35
Hilbert series with proposals for models in codimension~4---it is
conceivable that there are other examples (other than degenerations)
but we do not expect them. We list them according to the weights
of the ambient wps $\mathbb P^7$, that is, the degrees of minimal
generators of $R(X,A)$; for full details, see the webpage at \cite{code}.
The task is to construct these Fano 3-folds.
\paragraph{Projection guided by a K3 section}
If $|{-}K_X|$ contains a K3 surface $S$ we may compare our results
with those of \cite{B} as a guide to the constructions we might make.
Consider, the following codimension~4 candidate:
\[
X\subset\mathbb P(2,2,3,5,5,7,12,17)
\]
with $\mathcal B=\{\frac{1}{17}(5,12,2)\}$.
A section in degree~2 is a known K3 surface known with a
Type~1 projection from a $\frac{1}{17}(5,12)$ point:
\[
\left(
S\subset\mathbb P(2,3,5,5,7,12,17)
\right)
\dashrightarrow
\left(
T_{10,12,14,15,17}\subset\mathbb P(2,3,5,5,7,12)
\right).
\]
The K3 surface $T$ can be constructed easily, and moreover it can
be forced to contain a linear $\mathbb P(5,12)$. This curve can be
unprojected to construct $S$.
This is how we proceed with cases in codimension~4 here, but there is
a twist.
When we project from a quotient singularity---say the point
$\frac{1}{17}(5,12,2)$ in the example above---the result will
always contain a line of index~2 singularities. That is, projection
automatically incurs canonical singularities that contribute to
Riemann--Roch. (Recall from \cite{ABR} that projection
of 3-folds results in slightly worse singularities that we
usually work with, but that typically these are Gorenstein and
do not contribute in the Riemann--Roch formula.)
And so the result of projection will not have Hilbert series
already on our lists. This obstructs the inductive approach
exemplified by \cite{B}.
There are two ways out. One would be to mimic classical
methods and make a double projection. In the example above, after
projecting from $\frac{1}{17}(5,12,2)$, we can project from the resulting
$\frac{1}{12}(5,7,2)$ singularity. This second projection contracts
the index~2 line, and we land in the family
\[
Z_{10,12}\subset\mathbb P(2,2,3,5,5,7).
\]
To proceed, we would need an analysis of the exceptional locus of
such double projections and a Type~III-style unprojection
result, following \cite{kinosaki} e.g.~9.16.
The approach we take is to consider also weak Fano 3-folds with
canonical singularities. This avoids the bottleneck in codimension~$3$
for general unprojection methods: by admitting some canonical
singularities, in particular lines of
index~2 with up to two non-isolated points of higher index on them,
we see many more weak Fano 3-folds in codimension~3.
We impose appropriate surfaces in these
3-folds and the apply general results of unprojection;
this method is explained in detail in \cite{BKR}. In short, we
construct the example above by imposing the plane $\mathbb P(2,5,12)$
as a linear subspace in the codimension~3 weak Fano 3-fold
\[
\mathbb P(2,5,12)\subset Y_{10,12,14,15,17}\subset\mathbb P(2,2,3,5,5,7,12).
\]
Such $Y$ has a non-isolated singularity $\frac{1}{12}(2,5,7)$ on
a line of index~2 singularities, which are all strictly canonical
singularities.
\paragraph{Codimension~4 candidates having no Type~I projection}
The projection method outlined above constructs examples for
33 of the 35 candidates in codimension~4. The remaining two
cases, which we do not construct, are:
\[
X\subset\mathbb P^7(2,3,3,4,5,5,6,7)
\]
with $\mathcal B=\{ 5\times\frac{1}{3}(1,2,2), \frac{1}{5}(1,4,2)\}$,
$A^3 = 1/15$, $\frac{1}{12}Ac_2(X) = 11/45$ and Hilbert numerator
$1 - t^8 - t^9 - 2t^{10} - t^{11} - 2t^{12} + t^{14} +\cdots+ t^{33}$;
and
\[
X\subset\mathbb P^7(2,3,5,6,7,7,8,9)
\]
with
$\mathcal B=\{ 3\times\frac{1}{3}(1,2,2), \frac{1}{5}(2,3,2), \frac{1}{7}(1,6,2)\}$,
$A^3 = 1/35$, $\frac{1}{12}Ac_2(X) = 19/105$ and Hilbert numerator
$1 - t^{12} - t^{13} - 2t^{14} - t^{15} - 2t^{16} - t^{17} - t^{18}
+ t^{19} +\cdots + t^{45}$.
\begin{table}
$
\begin{array}{lccc}
\quad\textrm{Ambient $\mathbb P^7$} & \textrm{Basket $\mathcal B$} & A^3 & Ac_2/12 \\
\hline
\hline
\mathbb P( 1, 1, 1, 1, 1, 1, 1, 1 ) & \textrm{nonsingular}& 6& 1 \\
\mathbb P( 1, 1, 1, 1, 1, 2, 2, 3 ) & \frac{1}{3}& 10/3& 8/9 \\
\mathbb P( 1, 1, 1, 2, 2, 2, 3, 3 ) & 2\times\frac{1}{3}& 5/3& 7/9 \\
\mathbb P( 1, 1, 1, 2, 2, 2, 3, 5 ) & \frac{1}{5}(2,3,2)& 8/5& 4/5 \\
\mathbb P( 1, 1, 1, 2, 2, 3, 4, 5 ) & \frac{1}{5}(1,4,2)& 7/5& 4/5 \\
\hline
\mathbb P( 1, 1, 2, 2, 2, 3, 3, 3 ) & 3\times\frac{1}{3}& 1& 2/3 \\
\mathbb P( 1, 1, 2, 2, 2, 3, 3, 5 ) & \frac{1}{3}, \frac{1}{5}(2,3,2)& 14/15& 31/45 \\
\mathbb P( 1, 1, 2, 2, 2, 3, 5, 7 ) & \frac{1}{7}(2,5,2)& 6/7& 5/7 \\
\mathbb P( 1, 1, 2, 2, 3, 3, 4, 5 ) & \frac{1}{3},\frac{1}{5}(1,4,2)& 11/15& 31/45 \\
\mathbb P( 1, 1, 2, 2, 3, 3, 4, 7 ) & \frac{1}{7}(3,4,2)& 5/7& 5/7 \\
\hline
\mathbb P( 1, 1, 2, 3, 4, 5, 6, 7 ) & \frac{1}{7}(1,6,2)& 3/7& 5/7 \\
\mathbb P( 1, 2, 2, 3, 3, 3, 4, 5 ) & 3\times\frac{1}{3},\frac{1}{5}(1,4,2)& 2/5& 7/15 \\
\mathbb P( 1, 2, 2, 3, 3, 3, 4, 7 ) & 2\times\frac{1}{3}, \frac{1}{7}(3,4,2)& 8/21& 31/63 \\
\mathbb P( 1, 2, 2, 3, 3, 4, 5, 5 ) & \frac{1}{3}, \frac{1}{5}(1,4,2), \frac{1}{5}(2,3,2)& 1/3& 22/45 \\
\mathbb P( 1, 2, 2, 3, 3, 4, 5, 7 ) & \frac{1}{5}(2,3,2), \frac{1}{7}(3,4,2)& 11/35 & 18/35 \\
\hline
\mathbb P( 1, 2, 2, 3, 3, 5, 8, 11 ) & \frac{1}{11}(3,8,2)& 3/11& 6/11 \\
\mathbb P( 1, 2, 2, 3, 4, 5, 5, 7 ) & \frac{1}{5}(1,4,2), \frac{1}{7}(2,5,2)& 9/35& 18/35 \\
\mathbb P( 1, 2, 3, 4, 4, 5, 5, 5 ) & 3\times\frac{1}{5}(1,4,2)& 1/5& 2/5 \\
\mathbb P( 1, 2, 3, 4, 4, 5, 5, 9 ) & \frac{1}{5}(1,4,2), \frac{1}{9}(4,5,2)& 8/45& 58/135 \\
\mathbb P( 1, 2, 3, 4, 4, 5, 9, 13 ) & \frac{1}{13}(4,9,2)& 2/13& 6/13 \\
\hline
\mathbb P( 1, 2, 3, 4, 5, 5, 6, 7 ) & \frac{1}{3}, \frac{1}{5}(1,4,2), \frac{1}{7}(1,6,2)& 17/105& 127/315 \\
\mathbb P( 1, 2, 3, 4, 5, 5, 6, 11 ) & \frac{1}{3}, \frac{1}{11}(5,6,2)& 5/33& 43/99 \\
\mathbb P( 1, 2, 3, 4, 5, 6, 7, 7 ) & \frac{1}{7}(1,6,2), \frac{1}{7}(3,4,2)& 1/7& 3/7 \\
\mathbb P( 1, 2, 3, 5, 6, 7, 8, 9 ) & 2\times\frac{1}{3}, \frac{1}{9}(1,10,2)& 1/9& 11/27 \\
\mathbb P( 1, 2, 5, 7, 8, 9, 10, 11 ) & \frac{1}{5}(2,3,2), \frac{1}{11}(1,10,2)& 3/55& 19/55 \\
\hline
\mathbb P( 2, 2, 3, 3, 4, 5, 5, 5 ) & \frac{1}{3},3\times\frac{1}{5}(2,3,2)& 2/15& 13/45 \\
\mathbb P( 2, 2, 3, 3, 4, 5, 5, 7 ) & 2\times\frac{1}{3}, \frac{1}{5}(2,3,2), \frac{1}{7}(2,5,2) &13/105 & 92/315 \\
\mathbb P( 2, 2, 3, 3, 4, 5, 7, 9 ) & 3\times\frac{1}{3}, \frac{1}{9}(2,7,2)& 1/9& 8/27 \\
\mathbb P( 2, 2, 3, 5, 5, 7, 12, 17 ) & \frac{1}{17}(5,12,2)& 1/17& 5/17 \\
\mathbb P( 2, 3, 3, 4, 5, 5, 6, 7 ) & 5\times\frac{1}{3},\frac{1}{5}(1,4,2)& 1/15& 11/45 \\
\hline
\mathbb P( 2, 3, 3, 4, 5, 7, 10, 13 ) & 2\times\frac{1}{3}, \frac{1}{13}(3,10,2)& 2/39& 28/117 \\
\mathbb P( 2, 3, 4, 5, 5, 6, 7, 7 ) & \frac{1}{3},\frac{1}{5}(1,4,2),\frac{1}{5}(2,3,2),\frac{1}{7}(3,4,2)& 1/21& 64/315 \\
\mathbb P( 2, 3, 4, 5, 5, 6, 7, 9 ) & 2\times\frac{1}{3}, \frac{1}{5}(2,3,2), \frac{1}{9}(4,5,2)& 2/45& 28/135 \\
\mathbb P( 2, 3, 5, 6, 7, 7, 8, 9 ) & 3\times\frac{1}{3}, \frac{1}{5}(2,3,2), \frac{1}{7}(1,6,2)& 1/35& 19/105 \\
\mathbb P( 2, 5, 5, 6, 7, 8, 9, 11 ) & 2\times\frac{1}{5}(2,3,2), \frac{1}{11}(5,6,2)& 1/55& 8/55 \\
\hline
\end{array}
$
\caption{Fano 3-folds in codimension 4\label{tab!cod4}}
\end{table}
|
1,116,691,499,313 | arxiv | \section{Introduction}
Over the past twenty years, photometric and spectroscopic data have radically changed our picture of Galactic globular clusters (GCs). One of the most baffling discoveries is the presence of multiple stellar populations (mPOPs) in GCs \citep[see Anderson 1997;][and references therein]{1999Natur.402...55L, 2004ApJ...605L.125B, 2009A&A...505..139C, 2009A&A...505..117C, 2015AJ....149...91P, 2015MNRAS.454.4197R, 2018ARA&A..56...83B, 2012A&ARv..20...50G, 2019A&ARv..27....8G,2020A&ARv..28....5C}. To shed light on the formation and evolution of mPOPs, the ``HST Legacy Survey of Galactic Globular Clusters: Shedding UV Light on Their Populations and Formation" \citep[GO-13297, PI: Piotto;][]{2015AJ....149...91P} was devised. UltraViolet (UV) and optical data of this and other \textit{Hubble Space Telescope} (\textit{HST}) programs have allowed the creation of color-magnitude (CMDs) and color-(pseudo-)color diagrams, which in turn have provided essential elements to better understand the mPOP phenomenon in Galactic GCs \citep[e.g.,][]{2013MNRAS.431.2126M,2017MNRAS.464.3636M}.\looseness=-4
Other recent important observational findings concern the internal kinematics of GCs. Studies on the internal motions of these systems have revealed that GCs are characterized by complex internal kinematic properties including velocity anisotropy, rotation, and partial energy equipartition \citep[see the review of][and references therein]{2018ComAC...5....2V}. These results have provided the motivation for new theoretical studies aimed at building the theoretical framework necessary to interpret these observational findings \citep[e.g.,][]{2017MNRAS.471.1181B,2018MNRAS.475L..96B,2016MNRAS.461..402T,2017MNRAS.469..683T,2018MNRAS.475L..86T,2019MNRAS.487.5535T,2017MNRAS.471.2778B,2021MNRAS.502.4762B,2019ApJ...887..123S,2021MNRAS.504L..12P,2022MNRAS.509.3815P}. Most of what we have learned on this topic comes from internal motions in the plane of the sky obtained with \textit{HST} and \textit{Gaia} data. These two space observatories have different characteristics that make them best suited for specific, yet complementary, investigations of GCs.\looseness=-4
The \textit{Gaia} mission has revitalized astrometry. The availability of high-precision proper motions (PMs) over the entire sky has enabled a large variety of investigations, for example, large-scale structures in the Galaxy, Galaxy kinematics, stellar streams, tidal tails \citep[e.g.,][]{2018A&A...616A..11G,2019ApJ...872..152I,2021ApJ...914..123I}. The internal kinematics of GCs has benefited from the \textit{Gaia} PMs as well \citep[e.g.,][]{2018MNRAS.481.2125B,2019MNRAS.487.3693J,2021MNRAS.505.5978V,2021arXiv210910998E}, but these \textit{Gaia}-based analyses are focused on (and limited to) bright stars outside the centermost regions. Crowding \citep{2017MNRAS.467..412P} and faintness (i.e., access to low-mass stars) are two hurdles complex (or impossible) to overcome even in the next \textit{Gaia} data releases. However, there is important information in the core of GCs and in their faint members that is necessary to properly characterize GCs, and one of the few ways to obtain these necessary data is with \textit{HST}.\looseness=-4
Here we combine the wealth of information available in the \textit{HST} archive to compute high-precision PMs with the goal of analyzing the internal motions within GCs \citep[following][]{2014ApJ...797..115B}. The internal kinematics have a lot to tell, not only about GCs as a whole, but also about their mPOPs. Indeed, present-day trends in the velocity-dispersion and anisotropy radial profiles are the results of the different initial conditions of first- and second-generation stars. Thus, measuring the internal motions of the mPOPs can help us to shed light on key aspects of the mPOP formation and evolution \citep[e.g.,][]{2013ApJ...779...85M,2016ApJ...823...61M,2015MNRAS.450.1164H,2016MNRAS.455.3693T,2019MNRAS.487.5535T,2013MNRAS.429.1913V,2021MNRAS.502.4290V,2021MNRAS.502.1974S}. This is one of the remaining goals of the GO-13297 program. We present the PM catalogs for the 56 globular and one open clusters targeted by the GO-13297 program, and an overview of the kinematics of their brightest (and more massive) members. To showcase the quality of our catalogs, we also analyzed in great detail the internal kinematics of NGC~5904.\looseness=-4
Together with the PM catalogs, we release several photometric catalogs that are useful to select high-quality objects for studying the internal motions. Recently, \citet{2018MNRAS.481.3382N} published the final version of the photometric catalogs of this project. The photometry presented in this work is not meant to replace the one published by \citet{2018MNRAS.481.3382N}. Indeed, although the photometric precision is comparable, the completeness of the catalogs presented here is lower because they contain only sources with positions measured in at least two epochs to enable the determination of the PM. Also, we release only the photometric catalogs of the images in the filters/cameras actually used for the PM computation.\looseness=-4
\section{Data sets and reduction}\label{datared}
One of the goals of this project is to provide high-precision, \textit{homogeneous} PMs from \textit{heterogeneous} \textit{HST} data (i.e. observations with different cameras and programs) for stars in the central regions of GCs. We aimed at computing reliable PMs for as many stars as possible, with particular care for sources in the most crowded regions of the field of view (FoV) and for stars as faint as possible that we can detect with the available \textit{HST} archival images. No two data sets are alike, thus a careful data reduction has been specifically tailored to each GC. Although the data-reduction process is detailed in various papers \citep{2018ApJ...853...86B, 2018ApJ...861...99L, 2019ApJ...873..109L, 2018MNRAS.481.3382N}, we provide here a brief overview and the main differences in our reduction with respect to these works. Two clusters were processed independently as part of other publications: NGC~362 \citep{2018ApJ...861...99L} and NGC~6352 \citep{2019ApJ...873..109L}. For them, we refer to the corresponding papers for the detailed description of the data reduction.\looseness=-4
We made use of \texttt{\_flt} images (which are dark and bias corrected, and have been flat-fielded, but not resampled) taken with the Wide-Field Channel (WFC) and High Resolution Camera (HRC) of the Advanced Camera for Surveys (ACS), and with the Ultraviolet-VISible (UVIS) and InfraRed (IR) channels of the Wide-Field Camera 3 (WFC3) taken before 2019\footnote{For NGC~6121, we excluded data from the GO-12911 program, which are part of a more detailed analysis in progress on this cluster.}. In the case of ACS/WFC and WFC3/UVIS data, images were pipeline-corrected for the charge-transfer-efficiency (CTE) defects, as described in \citet{2010PASP..122.1035A}. As discussed in \citet{2014ApJ...797..115B}, not all filters/cameras are suitable for PMs. However, some parts of the reduction process (e.g., the second-pass photometry described below) take advantage from a large number of images to better detect faint sources. For this reason, we chose to include all images at disposal in the first part of the data reduction\footnote{The data sets used in this work are collected at \protect\dataset[10.17909/gajx-kf45]{http://dx.doi.org/10.17909/gajx-kf45}. All our data products are available at MAST as a High Level Science Product via \protect\dataset[10.17909/jpfd-2m08]{\doi{10.17909/jpfd-2m08}}. See also: \href{https://archive.stsci.edu/hlsp/hacks}{https://archive.stsci.edu/hlsp/hacks}.}.\looseness=-4
Our data reduction is a combination of first- and second-pass photometric stages. First-pass photometry was used to create an initial set of positions and fluxes for the brightest and most isolated sources in each exposure via effective-point-spread-function (ePSF) fitting through a single wave of finding. The ePSFs were specifically tailored to each image, starting from the publicly-available, spatially-variable library of \textit{HST} ePSFs\footnote{\href{http://www.stsci.edu/~jayander/}{http://www.stsci.edu/$\sim$jayander/}}. Source positions were also corrected for geometric distortion by means of the distortion solutions provided by \citet{2004acs..rept...15A, 2006acs..rept....1A}, \citet{2016wfc..rept...12A}, \citet{2009PASP..121.1419B} and \citet{2011PASP..123..622B}.\looseness=-4
Bright, unsaturated stars in the single-image catalogs were cross-matched with those in the \textit{Gaia} Data Release 2 (DR2) catalog \citep{2016A&A...595A...1G, 2018A&A...616A...1G}. This step was necessary to setup a common, pixel-based reference frame with specific axis orientation (X and Y axes toward west and north, respectively) and pixel scale (40 mas pixel$^{-1}$). The center of the clusters (from \citealt{2010AJ....140.1830G} with the exception of NGC~5897 and NGC~6791 for which we used the coordinates from the \citealt{1996AJ....112.1487H} catalog, 2010 edition, and Simbad database\footnote{\href{http://simbad.u-strasbg.fr/simbad/}{http://simbad.u-strasbg.fr/simbad/}}, respectively) were placed at a specific position, for example (5000, 5000), so to always have positive master-frame coordinates (the exact coordinates of the center of the clusters is provided in the header of the published catalogs). As in \citet{2018ApJ...853...86B} and \citet{2019ApJ...873..109L}, a master frame was created for every filter, camera and epoch. Then, we iteratively cross-identified the same stars in all images, and applied six-parameter linear transformations to transform stellar positions in each single-image catalog on to the master frame. Once on the same reference system, positions and instrumental magnitudes (rescaled to the magnitude of the longest exposure in each epoch/camera/filter) were averaged to create a preliminary astro-photometric catalog.\looseness=-4
The second-pass photometry was obtained with the software \texttt{KS2} \citep[Anderson et al., in preparation; see also][]{2017ApJ...842....6B}. \texttt{KS2} makes use of all images at once to increase the signal of faint objects that would otherwise be undetected in a single image. Starting from the brightest sources, \texttt{KS2} progressively finds fainter stars, and measures their position and flux via ePSF fitting after all detected close-by neighbors have been subtracted from the image. This step is particularly important in crowded environments like the core of GCs. We run \texttt{KS2} separately for different epochs, by grouping data taken 1--2 yrs apart, to retain stars that have moved by more than one pixel from one epoch to another \citep[see][]{2018ApJ...853...86B}. \texttt{KS2} allows us to define a specific set of images that can be ``stacked" together and used to find sources in the FoV. There is not one-size-fits-all solution for selecting these reference exposures. For each cluster, we selected a combination of cameras/detectors/filters that provided a good compromise between using a large number of exposures to more easily identify faint stars, and avoiding spurious detections (caused, for example, by the mix of filters with very different wavelength coverage).\looseness=-4
Instrumental magnitudes in the \texttt{KS2}-based catalogs were converted on to the VEGA-mag system. Photometry obtained with ACS/WFC F606W, ACS/WFC F814W, WFC3/UVIS F336W or WFC3/UVIS F438W filters was registered on to the VEGA-mag system by computing the zero-point difference with the corresponding photometry in the catalogs of \citet{2018MNRAS.481.3382N}. The zero-point
takes into account for the normalization to 1-s exposure time, the aperture correction and the VEGA-mag zero-point. Photometry in all other cameras/filters was calibrated as described in \citet{2017ApJ...842....6B} by means of \texttt{\_drz} (for ACS/HRC) or \texttt{\_drc} (for ACS/WFC or WFC3/UVIS) images and the official aperture corrections and VEGA-mag zero-points\footnote{See the resources provided here: \href{https://www.stsci.edu/hst/instrumentation/acs/data-analysis}{https://www.stsci.edu/hst/instrumentation/acs/data-analysis}, \href{https://www.stsci.edu/hst/instrumentation/wfc3/data-analysis/photometric-calibration}{https://www.stsci.edu/hst/instrumentation/wfc3/data-analysis/photometric-calibration}.}.
Finally, \texttt{KS2} provides position and flux of all detected sources in the raw reference-frame system of each image \citep{2018ApJ...853...86B}. We made use of these \texttt{KS2}-based single catalogs to compute our PMs.\looseness=-4
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{pmcorrm3.png}
\caption{Local PM maps of NGC~5272 before (panels a, b, c, d) and after (panels e, f, g, h) the a-posteriori corrections. The four leftmost/rightmost panels are obtained using only stars brighter/fainter than instrumental F606W magnitude $-10$ (signal-to-noise ratio $\sim$100; $m_{\rm F606W} \sim 21.6$). Points are color-coded according to the color bar on the left (in units of mas yr$^{-1}$). See the text for details.}
\label{fig:pmcorrm3}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{pmcorrngc1261.png}
\caption{Similar to Fig.~\ref{fig:pmcorrm3}, but for the GC NGC~1261. The effects of residual, uncorrected CTE are clearly visible in panels (c) and (d).}
\label{fig:pmcorrngc1261}
\end{figure*}
\section{Proper motions}\label{pms}
PMs were computed following the procedures and caveats described in \citet{2014ApJ...797..115B}. Briefly, positions in the \texttt{KS2}-based raw catalogs were corrected for geometric distortion and then transformed (with six-parameter linear transformations) on to the same reference-frame system defined in Sect.~\ref{datared}. Only cameras/detectors/filters best suited for astrometry\footnote{Besides the filters for which we do not have an ad-hoc geometric-distortion correction, we did not use filters bluer than F336W for the WFC3/UVIS detector and F330W for the ACS/HRC camera, respectively. We also excluded all WFC3/IR data given the worse resolution of the WFC3/IR detector.} \citep[see][]{2014ApJ...797..115B} were used in the PM computation. Then, positions as a function of time were fit with a least-squares straight line. The slope of this straight line provides a direct estimate of the PM of the source. The PM errors are the uncertainties of the PM fit obtained using the actual residuals of the points about the fitted line \citep{2014ApJ...797..115B}\footnote{Objects with a peculiar motion that cannot be modeled by a simple straight-line fit, like wide binaries, would result in large PM errors. However, these objects are hard to be discerned from single stars with poorly-measured proper motions from the information in our catalogs alone. A systematic search and accurate PM estimate for objects with very-peculiar motions would require an ad-hoc analysis, which is outside the scope of this project.}.\looseness=-4
For each source, the six-parameter linear transformations used in the PM computation were obtained by using a set of close-by, bright, well-measured cluster stars. Thus, our PMs are computed relative to the bulk motion of each GC at that given specific location in the FoV, and the cluster PM distribution is centered on the origin of the vector-point diagram (VPD). In addition, we also provide the PM zero-point needed to transform these relative PMs on to an absolute reference system (Appendix~\ref{abspm}).\looseness=-4
Another important feature of our PM derivation is that any signature of internal cluster rotation in the plane of the sky is removed from cluster stars, but it is present, with opposite sign, in all other sources \citep{2017ApJ...844..167B}. Thus, we cannot directly infer cluster rotation from the kinematics of cluster members. The same argument is also valid for the parallax effect \citep[e.g.][]{2018ApJ...861...99L,2018ApJ...854...45L}.\looseness=-4
Small spatially-variable and magnitude-dependent systematic errors are present in our uncorrected PMs. As in \citet{2018ApJ...853...86B}, we notice two main types of systematic errors:\looseness=-4
\begin{itemize}
\item a low-frequency effect correlated with the temporal baseline, as well as number and type of images, used to compute the PMs. To remove this time-dependent systematic, we divided our sample in $N$ sub-groups based on the PM temporal baseline. $N$ varies from cluster to cluster due to the heterogeneous data sets used. Then, we computed the median PM of each sub-group, which should be zero by construction. If it is not zero, we subtracted this median PM value to the PM of each star in the sub-group;\looseness=-4
\item a high-frequency, spatially- and magnitude-dependent systematic error. By construction, the average PM of cluster stars should be zero regardless their magnitude and location in the field. This is not always true locally, mainly because of a combination of CTE and geometric-distortion residuals. These residual systematic errors were corrected using the median PM of the closest, both spatially and in magnitude, $N$ well-measured cluster members (target star excluded). The closeness criterion and the number of reference stars $N$ were tailored to each cluster to reach the best compromise between mapping the variations as locally as possible and the need of large statistics. For very-bright(faint) objects, we set up a magnitude threshold above(below) which reference stars are used for the correction instead of a fixed $\Delta$mag. This was done to increase the statistics at the extreme ends of the magnitude range.\looseness=-4
\end{itemize}
The errors we report in our catalogues for the PMs thus corrected include the propagated contributions from the uncertainties in the corrections themselves. When no enough reference stars were available for the high-frequency correction, this correction was not applied.\looseness=-4
Figure~\ref{fig:pmcorrm3} shows maps of raw and a-posteriori, locally-corrected PMs for the GC NGC~5272. The FoV has been divided in square cells of 100 WFC3/UVIS pixels per side. In each cell, we selected the 50 well-measured, cluster stars closer to center of the cell, and computed the average PM in each direction. Panels (a) to (d) present the local PM maps obtained by means of the raw, uncorrected PMs, while panels (e) to (h) show the corrected PM maps. In each row, the two leftmost panels refer to stars brighter than F606W instrumental magnitude equal to $-10$ (signal-to-noise ratio of $\sim$100; $m_{\rm F606W} \sim 21.6$), while the maps for fainter stars are shown in the two rightmost panels. The comparison between top and bottom panels clearly highlights the effectiveness of the a-posteriori corrections. However, some residual high-frequency systematic errors are still present among the faintest objects. Thus, caution is advised when using these objects.\looseness=-4
The raw PMs of some clusters, mainly those including the ACS/WFC data of the GO-14235 program (PI: Sohn), present larger systematics related to uncorrected CTE. The CTE that affects the \textit{HST} detectors has worsen over time, and the official pipeline is not always able to completely correct it. An example is shown for NGC~1261 in the top panels of Figure~\ref{fig:pmcorrngc1261}, where it is clear that (i) bright and faint stars have different systematic errors in the PMs, and (ii) the local PM map of faint stars presents a discontinuity in the FoV along the chip gap of the GO-14235 data set. For these specific cases, we applied an additional a-posteriori correction prior to that for high-frequency systematics. Briefly, we divided our sample of well-measured stars in four magnitude bins. In each sub-sample, we computed the average PM (in each direction) of cluster members in 125-pixel-wide bins along the direction perpendicular to that of the CTE systematic in the local PM map. The correction to the PM of each star was computed by interpolating among these binned values. As for the other a-posteriori corrections, the errors of this CTE-related correction are included in the corrected-PM error budget.\looseness=-4
\begin{figure*}[t!]
\centering
\includegraphics[width=\textwidth]{overview.png}
\caption{Overview of the PM catalog of NGC~6652. (a): VPD of the relative, corrected PMs. Stars within the red circle (centered on the origin of the VPD, and with radius of 1 mas yr$^{-1}$) are likely cluster members, the blue circle marks the location of Sagittarius-Dwarf objects, while the green ellipse highlights a group of Bulge stars. Gray dots are all sources outside any of these three selections (likely Bulge objects or cluster stars with large PM uncertainties). (b): $m_{\rm F606W}$ versus $(m_{\rm F606W}-m_{\rm F814W})$ CMD. Black dots are likely cluster members; blue triangles are members of the Sagittarius Dwarf; green crosses are Bulge objects (gray points in the VPD are not shown for clarity). (c) and (d): $\mu_\delta$ and $\mu_\alpha \cos\delta$ PMs as a function of $m_{\rm F606W}$. Only stars member of NGC~6652 are shown (black points). The red points (with error bars) are the median values of the PMs in 0.5-mag bins. The azure line is set to zero. (e) and (f): $\mu_\delta$ and $\mu_\alpha \cos\delta$ PMs as a function of $(m_{\rm F606W}-m_{\rm F814W})$. Panels (c) to (f) show no obvious trends of our corrected PMs with stellar magnitude and color. (g) 1D corrected-PM error (the sum in quadrature of the PM errors in each direction divided by $\sqrt{2}$) as a function of $m_{\rm F606W}$. The red line is set at the median value of the 1D PM errors of bright, well-measured unsaturated stars.}
\label{fig:overview}
\end{figure*}
The steps described above generally remove the majority of systematic errors included in the raw PMs. However, these corrections are not perfect, especially for very faint stars, and we advise users to carefully check PMs for magnitude/color/spatial systematics on a cluster-by-cluster basis.\looseness=-4
\begin{figure*}
\centering
\includegraphics[width=\columnwidth]{pietime.png}
\includegraphics[width=\columnwidth]{pienimg.png}
\caption{Pie charts of the total temporal baselines (left) and number of images used to compute PMs (right) in our catalogs.}
\label{fig:pies}
\end{figure*}
Figure~\ref{fig:overview} provides an overview of the PM catalog of NGC~6652. This GC is located in the outer Bulge, projected on the Baade's Window \citep{2015MNRAS.450.3270R}, and in foreground of the Sagittarius Dwarf spheroidal. The VPD obtained with our corrected PMs is presented in panel (a). On the basis of their location in the VPD, we arbitrarily defined three groups of objects: cluster stars (points within the red circle), Sagittarius-Dwarf members (points within the blue circle), and Bulge stars (objects within the green ellipse). In the CMD in panel (b), we highlighted members of NGC~6652 in black (stars within the red circle in the VPD), Bulge objects in green and stars associated with the Sagittarius Dwarf in azure. This is a simple example of one of the possible applications of our PM catalogs. Corrected PMs in each coordinate as a function of $m_{\rm F606W}$ are plotted in panels (c) and (d), and as a function of $(m_{\rm F606W}-m_{\rm F814W})$ color in panels (e) and (f). Only cluster members are shown. The red points (with error bars) are the median values of the PMs in 0.5-mag bins. The azure line is set to zero. These plots do not show any significant magnitude- and color-dependent systematics in our corrected PMs, but very blue and red objects hint at the presence of some residual color-dependent systematics. However, stars with these extreme colors are very faint (see panel b), and we expect both PMs and their corrections to be still affected by small systematic residuals. We stress, though, that no quality selections other than the membership were applied here. Finally, panel (g) shows the 1D corrected-PM error as a function of $m_{\rm F606W}$. The red, horizontal line is set at the median 1D PM error of bright, well-measured stars, i.e., 25.6 $\mu$as yr$^{-1}$.\looseness=-4
A description of the final PM and photometric catalogs (together with some caveats about their usage) is provided in Appendix~\ref{cats}, Tables~\ref{tab:pmcat} and \ref{tab:photcat}, respectively. The precision of our PMs varies from cluster to cluster and depends on temporal baseline, as well as on the number and depth of images. Figure~\ref{fig:pies} presents pie charts of the temporal baselines and the number of images used to compute PMs in all our catalogs, respectively. About $\sim$33\% of the PMs are computed with only 10--20 images and a temporal baseline between 7 and 9 yr, i.e., by combining only GO-10775 and GO-13297 data. The remaining of the PMs results from the mix of heterogeneous data sets. For bright, well-measured stars, the median raw-PM precision is between about 7 and 60 $\mu$as yr$^{-1}$, while that of corrected PMs ranges between about 13 and 120 $\mu$as yr$^{-1}$.\looseness=-4
\section{Internal kinematics of stellar clusters}\label{kin}
As a benchmark of our PM catalogs, here we provide velocity-dispersion and anisotropy radial profiles of the clusters in our project. The kinematic profiles of the cores complement those in the outskirts derived with the Gaia Early Data Release 3 (EDR3) catalog \citep[e.g.,][]{2021MNRAS.505.5978V}. We also include the kinematic profile of the open cluster NGC~6791.\looseness=-4
We started by selecting well-measured objects in each epoch/filter/camera combination as follows (see Table~\ref{tab:photcat} for the explanation of the parameters):
\begin{enumerate}[(i)]
\item magnitude rms lower than the 90-th percentile of the distribution at any given magnitude. An object with a magnitude rms better than 0.01 mag is always included in the well-measured sample, while all sources with magnitude rms larger than 0.15 mag are excluded.
\item \texttt{QFIT} value larger than the 90-th percentile of the distribution at any given magnitude (note that the closer to 1 is the \texttt{QFIT} parameter, the better the PSF fit). Again, we retained all objects with a \texttt{QFIT} larger than 0.99 and discarded those with a value lower than 0.75.
\item $|$\texttt{RADXS}$|$ value lower than the 90-th percentile of the distribution at any given magnitude, but keeping all objects with a $|$\texttt{RADXS}$|$ lower than 0.01 and rejecting those with a value larger 0.15;
\item the photometric $N_{\rm u}^{\rm phot}/N_{\rm f}^{\rm phot}$ ratio (see Table~\ref{tab:photcat}) is greater than 0.75;
\item $o < 1$;
\item flux at least 3.5$\sigma$ above the local sky.
\end{enumerate}
These (empirically-derived) thresholds were chosen as a compromise between rejecting bad measurements and keeping a large sample of stars for the subsequent analyses. To avoid crowding bias, we measured the 90-th-percentile trends for magnitude rms, \texttt{QFIT} and \texttt{RADXS} in a region outside the core of each cluster where sources are more isolated and applied these cuts to all stars across the FoV. When not enough stars are available outside the core of the cluster (for example, for the photometry obtained with ACS/HRC full-frame or WFC3/UVIS subarray images), all stars were used, regardless their location in the FoV.\looseness=-4
For each epoch, all criteria from (i) to (vi) have to be fulfilled in at least two filters if a star was measured in at least two filters, otherwise in the only filter through which it was been detected. Finally, an object that passed all previous selections is defined as ``well-measured" if it passes all criteria in at least two epochs.\looseness=-4
The PMs also provide useful parameters to select trustworthy sources for the kinematic analysis. In addition to all previous criteria, we also removed all objects that have the astrometric $N_{\rm u}^{\rm PM}/N_{\rm f}^{\rm PM}$ ratio (see Table~\ref{tab:pmcat}) smaller than 0.8--0.9 (the exact value changes from cluster to cluster), $\chi^2_{\mu_\alpha \cos\delta}$ and $\chi^2_{\mu_\delta}$ larger than 1.25--1.5, PM error larger than 0.5 mas yr$^{-1}$, or for which the a-posteriori PM correction was not computed. Finally, we also excluded stars with a PM error larger than $f$ times the local velocity dispersion $\sigma_\mu$ of a sub-sample of well-measured, close-by cluster stars. For each cluster, we compared the velocity-dispersion radial profiles of RGB stars obtained by varying $f$ from 0.5 to 0.9, with steps of 0.1, finding a general good agreement (within 1$\sigma$) between the inferred kinematics. If enough stars are left after all our quality selections, we used $f = 0.5$. For 9 GCs (NGC~5053, NGC~5466, NGC~5897, NGC~6144, NGC~6366, NGC~6496, NGC~6535, NGC~6584 and NGC~6717), we used a value of $f = 0.8$, while for NGC~6981 we set $f = 0.9$. We included these 10 clusters in our analysis, but because their PM errors are of the order of their $\sigma_\mu$, we advise caution in the interpretation of their velocity-dispersion profiles.\looseness=-4
All these criteria are designed to obtain a good compromise between statistics, quality and completeness. The only two exceptions to the strategy described above are NGC~362 and NGC~6352, for which we adopted the quality selections described in \citet{2018ApJ...861...99L} and \citet{2019ApJ...873..109L}, respectively.\looseness=-4
GCs are old, collisional systems that, after many two-body relaxation times, present a (partial) degree of energy equipartition. Because of the heterogeneous mass range covered by our PM catalogs, we restricted the analysis of the kinematic profiles to stars brighter than the MS turn-off along the sub-giant and red-giant branches (SGB and RGB, respectively). Since in the post-MS evolutionary stages the evolutionary lifetimes are quite shorter than the core H-burning one, and in any case shorter than the typical two-body relaxation time, SGB and RGB stars can be safely considered as having the same kinematic mass (see also Sect.~\ref{weight}).\looseness=-4
The velocity-dispersion profile of each cluster was obtained by dividing the sample of massive cluster members\footnote{The cluster membership was inferred by means of CMDs and PMs. The PM threshold for each cluster was defined as the best compromise between including genuine members with large observed dispersion (which includes the contribution of both the PM errors and intrinsic velocity dispersion), and removing field objects with PMs similar to that of the cluster.} into equally-populated radial bins, with at least 100 stars per bin. In case of low statistics, we lowered the number of stars per bin to ensure at least 3 radial bins, regardless of the number of stars within each bin. When statistics allowed, we also imposed a radial bin comprising only the centermost 5--10 arcsec. The velocity dispersion in each radial bin was computed similarly to what described in \citet{2020ApJ...895...15R}, i.e., by maximizing the following likelihood:
\begin{equation}\label{eq:vdisp}
\begin{aligned}
\ln \mathcal{L} = -\frac{1}{2} \sum_{n} \Bigl[&\frac{(v_{\rm rad,\,n}-v_{\rm rad})^2}{\sigma_{\rm rad}^2+\epsilon_{\rm rad,\,n}^2} + \ln(\sigma_{\rm rad}^2 + \epsilon_{\rm rad,\,n}^2) + \\
& \frac{(v_{\rm tan,\,n}-v_{\rm tan})^2}{\sigma_{\rm tan}^2+\epsilon_{\rm tan,\,n}^2} + \ln(\sigma_{\rm tan}^2 + \epsilon_{\rm tan,\,n}^2) \Bigr] \mathrm{ ,}
\end{aligned}
\end{equation}
where $(v_{\rm rad,\,n}, v_{\rm tan,\,n})$ are the radial and tangential components of the PM of the $n$-th star, $(\epsilon_{\rm rad,\,n}, \epsilon_{\rm tan,\,n})$ are the radial and tangential components of the PM uncertainty of the $n$-th star, $(v_{\rm rad}, v_{\rm tan})$ are the radial and tangential mean motions of the cluster, and $(\sigma_{\rm rad}, \sigma_{\rm tan})$ are the radial and tangential velocity dispersions of the cluster. We also computed the combined velocity dispersion $\sigma_\mu$ in each radial bin using the same likelihood in Eq.~\ref{eq:vdisp} but with $\sigma_{\rm rad}=\sigma_{\rm tan}=\sigma_\mu$. We used the affine-invariant Markov Chain Monte Carlo (MCMC) method \texttt{emcee} \citep{2013PASP..125..306F} to sample the parameter space and obtain the posterior probability distribution functions (PDFs) for $\sigma_\mu$, $\sigma_{\rm rad}$ and $\sigma_{\rm tan}$. We run the MCMC chain with 20 walkers for 5000 steps, and rejected the first 200 steps. The best-fit values correspond to the median of the PDFs, while the corresponding errors are defined as the average between the 16-th and 84-th percentile about the median. Finally, the velocity dispersions were corrected as described in \citet[but see also Appendix~A of \citealt{2006A&A...445..513V}]{2015ApJ...803...29W} to take into account that maximum-likelihood estimators are biased and underestimate the true dispersion of the velocity distribution. The difference between corrected and uncorrected $\sigma_\mu$ is, on average, $\sim$0.6\%, it never reaches 3\% and it is more important for bins with less than 100 stars.\looseness=-4
Figures~\ref{fig:kin1}--\ref{fig:kin4} in Appendix~\ref{app:kin} show the result of our analysis for the 57 stellar clusters in the GO-13297 project. These profiles are also available at our website\footnote{\href{https://archive.stsci.edu/hlsp/hacks}{https://archive.stsci.edu/hlsp/hacks}}. For each cluster, the $m_{\rm F606W}$ versus $(m_{\rm F606W}-m_{\rm F814W})$ CMD is shown on the rightmost panel. Well-measured members of the GC brighter than the MS turn-off (highlighted by the azure, dashed horizontal line) are plotted as red points, all other objects that passed the quality criteria are shown as black points.\looseness=-4
\begin{table}[t!]
\caption{\textit{Gaia}-based velocity dispersions for NGC~6791.}
\centering
\label{tab:ngc6791}
\begin{tabular}{cccc}
\hline
\hline
Radius & $\sigma_\mu$ \\
$[$arcsec$]$ & $[$mas yr$^{-1}$$]$ \\
\hline
73.55 & $0.096 \pm 0.005$ \\
144.58 & $0.074 \pm 0.005$ \\
222.38 & $0.075 \pm 0.005$ \\
311.43 & $0.075 \pm 0.005$ \\
487.58 & $0.066 \pm 0.005$ \\
\hline
\end{tabular}
\end{table}
The velocity dispersion $\sigma_\mu$ as a function of distance from the center of the cluster is presented in the bottom-left panel. Black, filled points are obtained from this work, while black, open points refer to the measurements in the GC database of Holger Baumgardt\footnote{\href{https://people.smp.uq.edu.au/HolgerBaumgardt/globular/}{https://people.smp.uq.edu.au/HolgerBaumgardt/globular/}} obtained with the Gaia EDR3 PMs, which were presented by \citet{2021MNRAS.505.5978V}. The only exception is NGC~6791, which is not included in the work of \citet{2021MNRAS.505.5978V}. For this cluster, we independently derived \textit{Gaia}-based $\sigma_\mu$ using our tools\footnote{We considered only cluster's stars (selected by means of PMs and CMD) within 750 arcsec from the center of NGC~6791. We rejected all sources that have a re-normalized unit weight error (\texttt{RUWE}) greater than 1.4, astrometric excess noise larger than 0.4, number of bad along-scan observations exceeding 1.5\% of the total number of along-scan observations, or 2D PM error worse than 0.3 mas yr$^{-1}$. We also excluded all objects fainter than the MS turn-off or brighter than $G = 13$.} (Table~\ref{tab:ngc6791}). The black dashed lines are set at the core ($r_{\rm c}$; obtained using the definition in \citealt{1987degc.book.....S}, see Eq.~1-34, which is similar to the definition of the King scale radius $r_0$) and the projected half-light ($r_{\rm h}$) radii provided in the Baumgardt's database. For NGC~6791, we used $r_{\rm c}$ and $r_{\rm h}$ from \citet{2015MNRAS.449.1811D} and \citet{2019MNRAS.483.2197K}, respectively.\looseness=-4
\begin{figure*}[t!]
\centering
\includegraphics[width=\textwidth]{comp_watkins.png}
\caption{Comparison of the velocity-dispersion radial profiles obtained with \textit{HST}-based PMs in this paper (black points) and in \citet[red points]{2015ApJ...803...29W}.}
\label{fig:comp_watkins}
\end{figure*}
\citet{2021MNRAS.505.5978V} used all stars at disposal regardless of their magnitude. Thus, the \textit{Gaia}-based velocity dispersions for close-by clusters were derived from stars with various masses and, because of the effects of energy equipartition, could be systematically higher than our \textit{HST}-based profiles. However, \citet{2021MNRAS.505.5978V} argued that the \textit{Gaia} uncertainties for faint stars are likely underestimated, and included a scaling factor for the PM errors to obtain consistent $\sigma_\mu$ values between different magnitude intervals. Furthermore, the authors applied various quality cuts to their samples prior to the determination of the PM velocity dispersions (see their Sect.~3) that likely excluded from the fits faint (and so low-mass) objects with large PM uncertainties. For this reason, we choose to directly compare our \textit{HST} profiles in the cores with these \textit{Gaia} velocity dispersions outside the cores. Overall, we find a good agreement between \textit{HST} and \textit{Gaia} profiles, with a few exceptions. NGC~5466 and NGC~6981 have \textit{HST} PM uncertainties of the same order of the $\sigma_\mu$ (we used a large value of $f$ for the analysis), thus the inferred values of $\sigma_\mu$ should be interpreted cautiously. For NGC~6934, most of the \textit{Gaia} stars used by \citet{2021MNRAS.505.5978V} have PM errors larger than the intrinsic $\sigma_\mu$ of the cluster, and the \textit{Gaia} measurements might not be completely reliable. The \textit{HST} profile of NGC~6304 seems higher than the one from the \textit{Gaia} PMs. However, the \textit{HST}-based kinematics is in agreement with the line-of-sight (LOS) velocities in the Baumgardt's database.\looseness=-4
We fit these points with a 4-th order polynomial forced to be flat at the center. The coefficients of the 4-th-order polynomial are obtained with a maximum-likelihood approach. For each cluster in Figures~\ref{fig:kin1}--\ref{fig:kin4}, the blue line in the bottom-left panel is obtained with the best-fit (median) values of the polynomial fit, while the cyan lines are 100 random solutions of the polynomial fit. We used these polynomial functions to derive the $\sigma_\mu$ at the center of each cluster, at $r_{\rm c}$ and at $r_{\rm h}$. We provide all these values in Table~\ref{tab:vdisp}.\looseness=-4
The polynomial fits show again the agreement between \textit{HST} and \textit{Gaia} data for most of the clusters. There are a few points that are outliers with respect to the polynomial-fit predictions. Most of these outliers refer to the velocity dispersion in the innermost bins. A rise of the velocity dispersion in the innermost region can be a proxy of the presence of an intermediate-mass black hole \citep[and references therein]{2020ARA&A..58..257G}, but also of issues related to crowding/blending. Indeed, if two sources are blended and confused as one, or if the light contamination from the neighbors is high (which can still happen even with our data reduction), the position measured can be shifted from the real position. The offset is different for every image/filter/camera. Hence, the net result is that its PM is likely larger than what it should, thus increasing the $\sigma_\mu$ of the sources in the very-crowded region \citep[see the discussion in][]{2014ApJ...797..115B}. Finally, the shape and the abrupt drop of the polynomial functions in the outermost parts of the FoV do not correspond to physical effects and are just plotted for completeness.\looseness=-4
The anisotropy ($\sigma_{\rm tan}/\sigma_{\rm rad}$) as a function of distance from the center of the cluster is presented in the top-left panels. Anisotropy is discussed in \citet{2021MNRAS.505.5978V}, but their values are not publicly available. Thus, only \textit{HST}-based data are shown in the top-left panels of Figs.~\ref{fig:kin1}--\ref{fig:kin4} (black, filled dots). The red, dashed, horizontal line is set to 1 and marks the isotropic case. Most clusters are isotropic in the core, but a few objects shows a radial anisotropy outside about one $r_{\rm h}$. We will discuss the kinematic anisotropy in detail in the next Section.\looseness=-4
\citet{2019MNRAS.487.3693J} analyzed 10 GCs using \textit{Gaia} DR2 data and made publicly available their velocity dispersion and anisotropy radial profiles. Although their profiles do not cover the centermost region observed by our \textit{HST} data, we find an agreement at the 1$\sigma$ level between their $\sigma_\mu$ and anisotropy values for the nine clusters in common with our data set.\looseness=-4
We also compared our kinematic profiles with those of \citet{2015ApJ...803...29W}, see Fig.~\ref{fig:comp_watkins}, where a previous version of the \textit{HST} PM catalogs was used \citep{2014ApJ...797..115B}\footnote{The main differences between the PMs computed in this work and in \citet{2014ApJ...797..115B} are the following: (i) more data and longer temporal baseline at disposal; (ii) we now include a second-pass photometry stage to improve measurements for faint stars and in crowded environments; (iii) PM errors now include the contribution of the a-posteriori systematic corrections.}. There is an overall agreement between the profiles at the 3$\sigma$ level. The profiles of \citet{2015ApJ...803...29W} for NGC~1851, NGC~6441 and NGC~7078 are generally higher than those in our work. For NGC~7078, we also do not see the drop of the velocity dispersion in the centermost region seen by \citet{2015ApJ...803...29W}. A similar discrepancy is found when comparing the profiles of NGC~2808, NGC~6681 and NGC~6715, although at a less extent. The reason of these discrepancies might be the better treatment for crowding in our data reduction and/or the additional quality selections applied in our work. The profile of NGC~7099 is instead completely different, but the PMs used by \citet{2015ApJ...803...29W} were computed with a temporal baseline of only two years \citep[see][]{2014ApJ...797..115B}, and their quality is worse than that of our PMs.\looseness=-4
The velocity-dispersion profile of NGC~6441 was also studied by \citet{2021MNRAS.503.1490H}. The PMs computed by \citet{2021MNRAS.503.1490H} are obtained from the combination of data taken with ACS/HRC@\textit{HST} and NACO@VLT detectors, which are better suited for probing the centermost arcsec of this very-crowded cluster. Our PM profile seem to suggest a more moderate increase of the velocity dispersion towards the center, although the values in the region in common within 10 arcsec from the center are in agreement at the 1$\sigma$ level. The central velocity dispersion inferred in our work with the polynomial fit is $\sigma_\mu^{r=0} = (0.285 \pm 0.012)$ mas yr$^{-1}$, while their $\sigma_\mu$ measured at $r \sim 0.76$ arcsec is $(0.316 \pm 0.034)$ mas yr$^{-1}$. It is hard to clarify the nature of the discrepancy between these profiles given the different resolutions of the instruments used to infer the PMs in the centermost arcsec of the cluster.\looseness=-4
Finally, we compared our PM velocity dispersions with those based on LOS velocities in the Baumgardt database, which are taken from various sources in the literature. There is an overall agreement between the PM- and LOS-velocity-based profiles, although there are discrepancies in some cases (NGC~5053, NGC~5272, NGC~5286, NGC~5466, NGC~6093, NGC~6101, NGC~6144, NGC~6205, NGC~6584, NGC~6681, NGC~6809, NGC~6981 and NGC~7099) where our PM-based $\sigma_\mu$ are higher than the $\sigma_{\rm LOS}$, either in general or only in the centermost bins. The origin of the differences between PM- and LOS-velocity-based profiles might be related to systematics in either data sets, errors in the cluster distance (see Sect.~\ref{kdist}), or be instead a proxy of a peculiar kinematic state of the cluster. However, a detailed comparison between these $\sigma_\mu$ is outside the scope of the paper.\looseness=-4
\section{General kinematic properties}
The collection of kinematic profiles in Figures~\ref{fig:kin1}--\ref{fig:kin4} allows us to analyze some general properties of stellar clusters, similarly to what already done in the literature \citep[e.g.,][]{2015ApJ...803...29W}, but with a larger sample. We cross-correlated the kinematic pieces of information derived by means of \textit{HST} and \textit{Gaia} data with the structural properties of the GCs.\looseness=-4
\begin{figure}[t!]
\centering
\includegraphics[width=\columnwidth]{trel.png}
\caption{Anisotropy at the half-light radius as a function of the ratio of the cluster age to the half-mass relaxation time. An isotropic system is characterized by $\sigma_{\rm tan}/\sigma_{\rm rad} = 1$. Red points refer to core-collapsed GCs. All other systems are shown as black dots. The three solid lines show the time evolution of $\sigma_{\rm tan}/\sigma_{\rm rad}$ from three Monte Carlo simulations with initial $W_0$ and filling factor (defined as the ratio of the half-mass to tidal radius) equal to (5, 0.09; green line), (5, 0.18; blue line), (7, 0.06; orange line); the three systems reach core collapse, respectively, at Age$/t_{\rm rh} \simeq$ 10.9, 24.7, 6.8.}
\label{fig:trel}
\end{figure}
A quantity of particular interest in the kinematic characterization of star clusters is the anisotropy of the velocity distribution. Simulations following the evolution of star clusters during the initial violent-relaxation phase and including the effects of the tidal field of the host galaxy have shown that these systems emerge from this early evolutionary phase with an isotropic velocity distribution in the core, a radially anisotropic distribution in the intermediate regions, and an isotropic or slightly tangentially anisotropic distribution in the outermost regions \citep[see][]{2014MNRAS.443L..79V}. This specific configuration depends on the initial conditions at the formation of the cluster, on the surrounding environment, and it changes during the subsequent long-term evolution; in particular during a cluster's long-term evolution, the effects of two-body relaxation and mass loss lead to a gradual decrease of the radial anisotropy imprinted during the early dynamical phases \citep[][see that paper also for the possible development of radial anisotropy in tidally underfilling clusters with an initial isotropic velocity distribution]{2016MNRAS.455.3693T}.\looseness=-4
\begin{figure*}[th!]
\centering
\includegraphics[width=\textwidth]{anisotropy.png}
\caption{Anisotropy as function of distance from the center of the cluster in units of $r_{\rm h}$. The red, dashed, horizontal line is set at 1. Gray dots refer to the individual measurements shown in Figs.~\ref{fig:kin1}--\ref{fig:kin4}. Black points, with error bars, are the 3.5$\sigma$-clipped average values of the the anisotropy in 1-$r_{\rm h}$-wide bins (steps of 0.25 $r_{\rm h}$; only bins with at least 5 points are considered). From top to bottom, the plots refer to clusters with Age/$t_{\rm rh} \ge 10$, $7 \le$ Age/$t_{\rm rh}<10$ and Age/$t_{\rm rh} < 7$. Left panels refer to non-core-collapsed clusters, while those on the right shows the results for core-collapsed clusters.}
\label{fig:anisotropy}
\end{figure*}
Figure~\ref{fig:trel} shows the ratio between the tangential and radial components of the velocity dispersions measured at the half-light radius as a function of the clusters' dynamical age as measured by the ratio of their physical age\footnote{Ages are mainly from \citet{2010ApJ...708..698D}. For NGC~1851, NGC~2808, NGC~6388, NGC~6441, NGC~6656 and NGC~6715, we considered the values from \citet{2014ApJ...785...21M}, while for NGC~5897 we refer to \citet{2014A&A...565A..23K}. Finally, the age of NGC~6791 is from \citet{2021A&A...649A.178B}.} to their half-mass relaxation time ($t_{r{\rm h}}$; from the Baumgardt catalog\footnote{For NGC~6791, we considered a half-light radius of 4.1 arcmin from \citet{2019MNRAS.483.2197K}.}). The values of $\sigma_{\rm tan}$ and $\sigma_{\rm rad}$ at the half-light radius were derived by fitting a 4-th order polynomial function to the corresponding profile as described in Sect.~\ref{kin}. We excluded from the plot all GCs for which the \textit{HST} data do not reach the half-light radius (so the anisotropy ratio would need to be extrapolated). Red points represent core-collapsed GCs, all other systems are presented as black points. Core-collapsed GCs (regardless if their are considered as `possible' or `post' core collapse) were labelled as such according to \citet{1995AJ....109..218T} or, if not included in their list, to the Harris catalog\footnote{There are some discrepancies in the list of core-collapsed GCs in \citet{1995AJ....109..218T} and the Harris catalog, specifically NGC~6717 and NGC~6723. In the following, we consider the classification of \citet{1995AJ....109..218T}, for which NGC~6717 is a possible core-collapsed GC and NGC~6723 is not.}. Our analysis shows that dynamically older clusters (Age/$t_{\rm rh}$ $\gtrsim$ 10) tend to be isotropic even at the half-light radius, while dynamically-young systems are characterize by a radially-anisotropic velocity distribution at the half-light radius. A transition between the two regimes happens at Age/$t_{\rm rh}$ between 7 and 10. These findings are consistent with the theoretical expectations discussed above \citep[see, e.g., ][]{2014MNRAS.443L..79V,2016MNRAS.455.3693T,2017MNRAS.471.1181B} and previous observational works \citep{2015ApJ...803...29W}.\looseness=-4
To further illustrate the theoretical expectations concerning the evolution of the radial anisotropy, in Fig.~\ref{fig:trel} we show the time evolution of $\sigma_{\rm tan}/\sigma_{\rm rad}$ (calculated at the projected half-light radius) from a few Monte Carlo simulations run with the MOCCA code
\citep{2013MNRAS.431.2184G}. The simulations follow the dynamical evolution of a few simple stellar systems composed of 500k stars with masses following a \citet{2001MNRAS.322..231K} initial mass function between 0.1 and 0.8 $M_{\odot}$, and spatially distributed according to the density profiles of \citet{1966AJ.....71...64K} models with values of the central dimensionless potential equal to $W_0=5$ and $W_0=7$ (corresponding, respectively, to $c \simeq 1.03$ and $c \simeq 1.53$). The systems are characterized by an initial anisotropic velocity distribution following the Osipkov-Merrit profile \citep[see, e.g.,][]{2008gady.book.....B}, $\beta = 1-\sigma_{\rm tan}^2/(2\sigma_{\rm rad}^2)=1/(1+r_{\rm a}^2/r^2)$ with $r_{\rm a}$ equal to the half-mass radius. As shown in this plot, the initial anisotropy of the clusters at the half-light radius gradually decreases during the cluster's long-term evolution. For the tidally filling system, the enhanced rate of star loss lead to a more rapid isotropization of the velocity dispersion. At the time the system reaches core collapse, the cluster's radial anisotropy slightly increases and then continues its gradual decrease towards isotropy \citep[see also][for a study of the evolution of anisotropy for systems with a variety of different initial conditions]{2016MNRAS.455.3693T}. The differences between the anisotropy of systems which have similar dynamical ages but are in the pre- or post-core-collapsed phase is small and within the uncertainty of the observed values.\looseness=-4
In Fig.~\ref{fig:anisotropy}, to further investigate the kinematic anisotropy in stellar clusters, we divided the sample into three groups, i.e., clusters with a Age/$t_{\rm rh} \ge 10$, between 7 and 10 (the Age/$t_{\rm rh}$ transition region found in Fig.~\ref{fig:trel}), and lower than 7. In each group, we also separated core-collapsed clusters (right panels) from all other systems (left panels). We collected all anisotropy measurements shown in Figs.~\ref{fig:kin1}--\ref{fig:kin4}, and plotted them as a function of distance from the center of the cluster in units of $r_{\rm h}$. Gray points are the individual measurements, while black dots are the moving average of those points. Clusters with Age/$t_{\rm rh} \ge 10$ are isotropic at all distances within our FoV, regardless of their core-collapsed status, which is what we expect. Clusters with Age/$t_{\rm rh}$ between 7 and 10 show again to be isotropic at all distances, although the non-core-collapsed sample hints a marginal radial anisotropy at $r \gtrsim r_{\rm h}$. Finally, dynamically-young clusters clearly present the expected radial anisotropy at large radii.\looseness=-4
Thus, clusters that underwent a core collapse seem to have a similar velocity field as that of the other GCs with similar dynamical ages; this appears to be consistent with what suggested by the results of the simulations presented in Fig.~\ref{fig:trel}, which show that core collapse has only a relatively small effect on the radial anisotropy measured at the half-light radius. Additional simulations and a larger observational sample of core-collapsed clusters are necessary to further explore this issue. In particular, it is worth noticing that the core-collapsed sample with Age/$t_{\rm rh} < 7$ is composed by only four clusters: NGC~6541, NGC~6752, NGC~7099 and NGC~7078. While the first three objects show an isotropic field even slightly further than the half-light radius, the latter present a strong radial anisotropy. This feature for NGC~7078 was also noted by \citet{2014ApJ...797..115B} and \citet{2021MNRAS.505.5978V}. Among these four GCs, NGC~7078 (i) is more massive, (ii) is further from the center of the Galaxy, and (iii) has had less interactions with the Galactic potential of Bulge/Bar and Disk \citep[see][]{2021MNRAS.505.5978V}. The combination of these properties could have preserved some of the original radial anisotropy in the innermost regions \citep[e.g.,][]{2014MNRAS.443L..79V}.\looseness=-4
\begin{figure}[t!]
\centering
\includegraphics[width=\columnwidth]{ratios.png}
\caption{(Bottom panel): ratio between the velocity dispersion at the core and half-light radii (from the Baumgardt catalog) as a function of concentration index $c$. The values of $c$ are obtained from \citet{2005ApJS..161..304M}. For GCs not analyzed in that work, we used the values from the \citet[2010 edition]{1996AJ....112.1487H}. Red points are GCs marked as core-collapsed in the Harris catalog; green dots are measurements from \citet{2021AJ....161...41C}; black dots are all other objects. (Top panel): as in the bottom panel, but using the velocity dispersion at $r_0$ and $r_{\rm h}$ \citep[radii from][]{2005ApJS..161..304M}. Some core-collapsed GCs are missing in the plot because they were not analyzed by \citet{2005ApJS..161..304M}. The results obtained with the \texttt{\textsc{LIMEPY}} models are shown as an azure, dashed line.}
\label{fig:ratios}
\end{figure}
Core-collapsed GCs are not only more spatially concentrated than other GCs, but their velocity-dispersion radial profile is also steeper. This has been shown by \citet{2015ApJ...803...29W} and \citet{2021AJ....161...41C}.
In Fig.~\ref{fig:ratios}, we provide an updated version of this finding, and a comparison with theoretical models. The bottom panel of Fig.~\ref{fig:ratios} shows the ratio between the velocity dispersion at the core and half-light radii ($\sigma_{r_{\rm c}}/\sigma_{r_{\rm h}}$), computed as described before at $r_{\rm c}$ and $r_{\rm h}$ from the Baumgardt catalog, as function of the concentration index $c$. The values of $c$ are taken from \citet{2005ApJS..161..304M} and are defined as the $\log(r_{\rm t}/r_0)$, where $r_{\rm t}$ and $r_0$ are the tidal and King scale radii, respectively. We selected $c$ obtained from the fit of a \citet{1966AJ.....71...64K} profile. If $c$ is not provided for a GC in the work of \citet{2005ApJS..161..304M}, we used the value in the \citet[2010 edition]{1996AJ....112.1487H}. Red, filled circles mark core-collapsed GCs, while all other systems are plotted as black dots. Green points are clusters taken from the sample of \citet{2021AJ....161...41C}. This panel highlights a different location in the plot for core-collapsed GCs and the other stellar systems. This Figure confirms the trend between the spatial concentration and the steepness of the radial profile of the velocity dispersion.\looseness=-4
In order to further explore this trend and carry out a consistent comparison with theoretical models, we show in the top panel of Fig.~\ref{fig:ratios} the ratio of the values of the the velocity dispersions calculated at the King scale radius and at the half-light radius \citep[this time obtained at $r_0$ and $r_{\rm h}$ of][for consistency]{2005ApJS..161..304M} versus the concentration $c$ together with a line showing the expected variation of these quantities along the sequence of King models
\citep[calculated using the \texttt{\textsc{LIMEPY}} software\footnote{\href{https://github.com/mgieles/limepy}{https://github.com/mgieles/limepy}};][]{2015MNRAS.454..576G}. The overall agreement between the observed and the theoretical sequence shown in the top panel of Fig.~\ref{fig:ratios} further supports the idea that more concentrated GCs have a steeper velocity-dispersion radial profile according to what is expected for the family of King models with different values of $c$. It is interesting to notice the increasing deviation of observational data from the theoretical King sequence at larger concentrations ($c \gtrsim 1.6$), including some of those from the sample of \citet[green points in Fig.~\ref{fig:ratios}]{2021AJ....161...41C} that, according to those authors, are in an advanced dynamical state, being close to (or recently undergone) core collapse. Although this issue requires further investigation, it may represent a kinematic manifestation of the deviation of clusters from the dynamical properties of King models as they approach core-collapse.\looseness=-4
There are a few exceptions in the overall picture described in the top panel of Fig.~\ref{fig:ratios}. The core-collapsed GC with $c<2.0$ and $\sigma_{r_{\rm c}}/\sigma_{r_{\rm h}}<1.2$ is NGC~362. NGC~362 is a post-core-collapsed GC \citep[e.g.,][]{2013ApJ...778..135D,2018ApJ...861...99L}, and its peculiar position could be explained by the structural and dynamical evolution of post-core-collapsed GCs driven by gravothermal oscillations \citep[e.g.,][]{1996ApJ...471..796M}.\looseness=-4
Three clusters shown with black dots seem to have characteristics similar to core-collapsed GCs. These GCs are NGC~5272, NGC~6652 and NGC~6715. \citet{2005ApJS..161..304M} pointed out that NGC~5272 shows deviations from the classical King isotropic model, and can be better fit with other models \citep[see also][]{1976ApJ...206..128D,1979AJ.....84..752G}. NGC~6652 is not considered a core-collapsed GC, but it is very concentrated and has a very steep surface-brightness profile, typical of core-collapsed systems \citep{2006AJ....132..447N}. NGC~6715 is a cluster at the center of the Sagittarius Dwarf, an environment that could explain its peculiar core-collapsed-like concentration.\looseness=-4
\subsection{Kinematic distances}\label{kdist}
The Baumgardt's GC database also contains LOS velocity measurements. We made use of them to estimate the distance of GCs by using the simple relation between the velocity dispersion along the LOS and in the plane of the sky:
\begin{equation}\label{eq:distance}
\begin{aligned}
\sigma_{\rm LOS} = 4.7404\,d\,\sigma_\mu\,;
\end{aligned}
\end{equation}
where $d$ is the distance in kpc, $\sigma_{\rm LOS}$ and $\sigma_\mu$ are the velocity dispersions along the LOS (in km s$^{-1}$) and in the plane of the sky (from PMs in mas yr$^{-1}$), respectively, and 4.7404 (km\,yr\,kpc$^{-1}$\,mas$^{-1}$\,s$^{-1}$) is the conversion factor. The only assumption here is that GCs are isotropic. We computed the distance $d$ by comparing the values of the velocity dispersion at the same distance, i.e., at the center of the cluster, obtained for both LOS and PM measurements by fitting a polynomial function to the corresponding velocity-dispersion radial profiles as described in Sect.~\ref{kin}. Our results are summarized in Table~\ref{tab:distance}. We considered only clusters with enough LOS velocities to solve the polynomial fit.\looseness=-4
Figure~\ref{fig:distance} shows the comparison between our distances and those from \citet[panel a; using \textit{Gaia} parallaxes, the comparison between PM and LOS velocities, and/or star counts]{2021MNRAS.505.5957B}, the Harris catalog (panel b; collected from various sources in the literature), the work of \citet[panel c; obtained with an approach similar to that used in our paper]{2015ApJ...812..149W} for a sample of 14 GCs, and the estimates from \citet[panel d; using the luminosity level of the zero-age horizontal branch\footnote{Distance moduli in \textit{HST} Wide-Field Planetary Camera 2 (WFPC2) F555W filter of \citet{2005A&A...432..851R} were converted in distances in kpc after correcting for extinction using the extinction coefficients $A_{\rm F555W}$ provided in \citet{1995PASP..107..156H} and the $E(B-V)$ reddening in the Harris catalog.}]{2005A&A...432..851R}. The median difference between $d$ in our work and those in the literature shown in Fig.~\ref{fig:distance} are $(-0.01 \pm 0.16)$ \citep{2021MNRAS.505.5957B}, $(-0.05 \pm 0.18)$ kpc (Harris catalog), $(-0.19 \pm 0.17)$ kpc \citep{2015ApJ...812..149W}, and
$(-0.23 \pm 0.20)$ kpc \citep{2005A&A...432..851R}, respectively. All distance estimates are in agreement with these literature values at the $\sim$1$\sigma$ level. This is another proof of the goodness of our PM measurements. At large distances ($>$10 kpc), the difference between our distances and those from the literature increases, and so does the scatter of the points in Fig.~\ref{fig:distance}. The lower velocity dispersion of some of these clusters, as well as the larger uncertainties both in the PMs, LOS velocities and parallaxes, are likely the reasons of these discrepancies.\looseness=-4
\begin{table}[t!]
\centering
\caption{GC kinematic distances.}
\label{tab:distance}
\begin{tabular}{cc|cc}
\hline
\hline
Cluster & $d$ & Cluster & $d$ \\
& $[$kpc$]$ & & $[$kpc$]$ \\
\hline
NGC~104 & $ 4.34 \pm 0.06$ & NGC~6218 & $ 5.23 \pm 0.30$ \\
NGC~288 & $ 9.08 \pm 0.79$ & NGC~6254 & $ 5.37 \pm 0.21$ \\
NGC~362 & $ 9.33 \pm 0.31$ & NGC~6304 & $ 7.50 \pm 1.04$ \\
NGC~1261 & $13.14 \pm 1.23$ & NGC~6341 & $ 7.68 \pm 0.66$ \\
NGC~1851 & $11.66 \pm 0.25$ & NGC~6362 & $ 9.02 \pm 0.93$ \\
NGC~2808 & $10.07 \pm 0.24$ & NGC~6388 & $11.69 \pm 0.30$ \\
NGC~3201 & $ 4.73 \pm 0.15$ & NGC~6397 & $ 2.25 \pm 0.11$ \\
NGC~4590 & $10.44 \pm 1.28$ & NGC~6441 & $13.15 \pm 0.63$ \\
NGC~4833 & $ 7.21 \pm 0.93$ & NGC~6541 & $ 7.36 \pm 0.38$ \\
NGC~5024 & $14.86 \pm 1.69$ & NGC~6624 & $ 7.91 \pm 0.59$ \\
NGC~5272 & $ 8.16 \pm 0.52$ & NGC~6656 & $ 3.03 \pm 0.09$ \\
NGC~5286 & $ 9.65 \pm 0.61$ & NGC~6681 & $11.10 \pm 0.68$ \\
NGC~5897 & $15.35 \pm 3.04$ & NGC~6715 & $25.32 \pm 2.31$ \\
NGC~5904 & $ 7.42 \pm 0.21$ & NGC~6723 & $ 7.65 \pm 0.81$ \\
NGC~5927 & $ 9.33 \pm 0.75$ & NGC~6752 & $ 3.85 \pm 0.13$ \\
NGC~5986 & $10.61 \pm 1.28$ & NGC~6809 & $ 4.62 \pm 0.37$ \\
NGC~6093 & $ 9.34 \pm 0.41$ & NGC~7078 & $10.81 \pm 0.21$ \\
NGC~6121 & $ 1.85 \pm 0.10$ & NGC~7089 & $10.36 \pm 0.33$ \\
NGC~6171 & $ 6.36 \pm 0.54$ & NGC~7099 & $ 9.50 \pm 0.32$ \\
NGC~6205 & $ 6.16 \pm 0.44$ \\
\hline
\end{tabular}
\end{table}
\section{Possible applications}
We choose the GC NGC~5904 to showcase some scientific applications enabled by our PMs\footnote{NGC 5904 shows one of the cleanest and better-defined rotation curves obtained so far in a GC \citep{2018ApJ...861...16L}, thus suggesting that no significant residual rotation affects its kinematics in the plane of the sky.}. Note that not all PM catalogs provide the same PM precision and overall quality, and some of the examples described below cannot be applied.\looseness=-4
\begin{figure}[t!]
\centering
\includegraphics[width=\columnwidth]{distance.png}
\caption{Comparison between our distances and those in the literature: \citet[panel a]{2021MNRAS.505.5957B}, the Harris catalog (panel b), \citet[panel c]{2015ApJ...812..149W} and \citet[panel d]{2005A&A...432..851R}. Only error bars are shown for clarity (no error bars are available for distances from the Harris catalog). The red lines are at 0.}
\label{fig:distance}
\end{figure}
\subsection{Internal kinematics of mPOPs}\label{mpops}
\begin{figure*}[t!]
\centering
\includegraphics[width=0.997\textwidth]{ngc5904_mpoprgb.png}
\caption{Analysis of the internal kinematics of the mPOPs along the RGB of NGC~5904. We used the chromosome map of \citet{2017MNRAS.464.3636M} (top-right panel) and an Hess diagram (bottom-right panel) to select three groups of stars: 1G POPa (red points), 2G POPb and POPc (azure and green points, respectively). The left panels show the velocity-dispersion (bottom) and anisotropy (top) radial profiles for each population. 1G and 2G stars are kinematically isotropic and have the same velocity dispersions within our FoV. The gray solid and dashed vertical lines are set the core and half-light radii, respectively. The black, dashed horizontal line in the top-left panel is set to 1 (isotropic case). The small panels next to the left plots present the average velocity dispersion and anisotropy of each population across the entire FoV.\looseness=-4}
\label{fig:mpops}
\vspace{0.5 cm}
\includegraphics[width=0.997\textwidth]{ngc5904_mpopspatdistr.png}
\caption{Analysis of the spatial distribution of mPOPs along the RGB of NGC~5904. The left panels show the $m_{\rm F814W}$ versus $\Delta^*_{\rm C\,F275W,F336W,F438W}$ CMDs for stars in each radial bin (distance from the center of the cluster increasing from bottom to top). The central panels present the corresponding kernel-density distributions (gray line) and the triple-Gaussian functions (1G POPa: red; 2G POPb: azure; 2G POPc: green). The sum of the three Gaussians is plot as a black line. The right panel shows the spatial distribution of the three mPOPs. The horizontal lines (and their shaded regions) corresponds to the ratio of each mPOP using all stars in the field. The vertical, gray line is set at $r_{\rm c}$ ($r_{\rm h}$ is outside the plot boundaries).\looseness=-4}
\label{fig:mpopspatdistr}
\end{figure*}
Kinematic differences between mPOPs can be a proxy of a different initial formation and evolution of first- (1G) and second-generation (2G) stars \citep{2010ApJ...724L..99B,2013ApJ...779...85M,2016ApJ...823...61M,2015MNRAS.450.1164H,2016MNRAS.455.3693T,2019MNRAS.487.5535T,2013MNRAS.429.1913V,2021MNRAS.502.4290V,2019MNRAS.489.3269C,2021MNRAS.502.1974S}. Recently, numerous observational efforts have investigated the kinematic properties of 1G and 2G stars to help us understanding the mPOP phenomenon. We now know that 1G and 2G stars share similar kinematic features in dynamically-old GCs, or at least in the centermost regions of GCs where the two-body relaxation time is short, because two-body processes have already erased any kinematic differences between mPOPs \citep{2010ApJ...710.1032A,2018ApJ...861...99L,2019ApJ...873..109L}. The outermost regions of GCs are instead less relaxed, and some fingerprints of the initial kinematic differences between mPOPs can still be detected \citep{2013ApJ...771L..15R,2015ApJ...810L..13B,2018ApJ...853...86B,2018MNRAS.479.5005M,2018ApJ...859...15D,2018ApJ...864...33D,2019ApJ...884L..24D,2020ApJ...889...18C,2020ApJ...898..147C}.\looseness=-4
Evidence of the presence of mPOPs in NGC~5904 has been shown both spectroscopically \citep{2001AJ....122.1438I,2009A&A...505..117C,2013A&A...549A..41G} and photometrically \citep{2017MNRAS.464.3636M,2017ApJ...844...77L,2021ApJ...918L..24L}. To identify the mPOPs in our field, we made use of the pseudo color-color diagram (``chromosome map") computed by \citet{2017MNRAS.464.3636M} for this stellar cluster. We cross-correlated our PM catalog of NGC~5904 with that of \citet{2017MNRAS.464.3636M} and used their chromosome map to select 1G and 2G stars along the RGB of this cluster (right panels of Fig.~\ref{fig:mpops}). We identified three groups of stars: a 1G population (hereafter POPa in red) and two 2G groups (POPb and POPc in azure and green, respectively). The 1G-2G tagging was obtained similarly to what shown in Fig.~4 of \citet{2017MNRAS.464.3636M}. The two 2G sub-populations were arbitrarily identified using the Hess diagram in Fig.~\ref{fig:mpops}.\looseness=-4
We then measured\footnote{In addition to the criteria described in Sect.~\ref{kin}, stars that were analyzed in this section passed the photometric criteria described in \citet{2017MNRAS.464.3636M}.} the velocity dispersions of each population in various radial bins of at least 50 stars each. The velocity-dispersion (bottom-left panel) and anisotropy (top-left panel) radial profiles show that 1G and 2G stars have similar kinematic temperature and are isotropic within our FoV (there is only a marginal hint of a radially and tangentially anisotropic POPa and POPc, respectively). This is expected given that our field covers out to about the half-light radius of the cluster, a region where two-body encounters have likely removed any initial kinematic difference between mPOPs. Our findings are in agreement with those obtained by \citet{2020ApJ...889...18C} with the \textit{Gaia} DR2 PMs.\looseness=-4
This result is in contrast with the finding of \citet{2021ApJ...918L..24L} that measured the analog of POPc stars more spatially concentrated than the other two populations even within the half-light radius. However, analogously to the internal motions, differences in the spatial segregation of mPOPs can be preserved in regions where the relaxation time is long enough to preserve them. Thus, a complete mixing and similar kinematic features are likely expected for the mPOPs in the core of NGC~5904.\looseness=-4
To shed light on this disagreement, we computed the spatial distribution of the three populations in our FoV as follows. First, we divided our sample\footnote{We removed the constraint on the PM error to increase the statistics at disposal.} of RGB stars in three equally-populated bins and computed a kernel-density distribution of the $\Delta^*_{\rm C\,F275W,F336W,F438W}$ color for the stars in each bin. The $\Delta_{\rm C\,F275W,F336W,F438W}$ color was obtained \citet{2017MNRAS.464.3636M} after rectifying the RGB sequences in the $m_{\rm F814W}$ versus ${\rm C_{F275W,F336W,F438W}}$ CMD. However, the sequences have been rectified using all stars in the FoV and, in different radial bins, they can show some deviations from being exactly vertical. Thus, we fine-tuned the $\Delta_{\rm C\,F275W,F336W,F438W}$ color in each radial bin using two fiducial lines (one at the red and blue sides of the RGB sequence, respectively) drawn by hand and then computing a new color $\Delta^*_{\rm C\,F275W,F336W,F438W}$ as in \citet[see left panels of Fig.~\ref{fig:mpopspatdistr}]{2019ApJ...873..109L}:\looseness=-4
\begin{equation}
\begin{split}
\Delta^*_{\rm C\,F275W,F336W,F438W} = \\
= \frac{(\Delta_{\rm C\,F275W,F336W,F438W} - \rm fiducial_{red})}{(\rm fiducial_{blue} - \rm fiducial_{red})} \, .
\end{split}
\end{equation}
The kernel-density distribution\footnote{The kernel-density estimation was obtained with the \texttt{python} dedicated tools in \texttt{scikit-learn} \citep{scikit-learn}, and assuming a `Epanechnikov' kernel with bandwidth of 0.1. These parameters were chosen as a good compromise between smoothness and preserve the features in the mPOP distribution.} of the $\Delta^*_{\rm C\,F275W,F336W,F438W}$ color in each bin was fitted with a triple-Gaussian function (central panels of Fig.~\ref{fig:mpopspatdistr}), and the fraction of stars in each population was estimated following the same approach of \citet{2013ApJ...765...32B}. To ensure the density distribution was robust against the small statistics along the RGB, we obtained the fraction of each mPOP by bootstrapping with replacement the sample of stars for 1\,000 times. The final value for the fractions of stars and their errors were determined as the median and the 68.27-th percentile of the distribution about the median, respectively. The ratios of the three mPOPs are shown in the right panel of Fig.~\ref{fig:mpopspatdistr}. At odds with the finding of \citet{2021ApJ...918L..24L}, we can see that the fractions of each mPOP do not vary as function of distance from the center of the cluster within our FoV, as expected.\looseness=-4
\subsection{Energy equipartition}\label{eeq}
The energy-equipartition state of a GC is one of the most challenging measurements to obtain because it requires precise PMs along a (relatively) wide range of masses, i.e., for faint stars. Nowadays, this is one of the few applications that only \textit{HST} can allow us to investigate in detail.\looseness=-4
In collisional systems in a certain state of energy equipartition, there is a relation between the stellar mass $m$ and the velocity dispersion $\sigma_\mu$: $\sigma_\mu \propto m^{-\eta}$, where $\eta$ is the level of energy equipartition of the system. Theoretical works \citep{2013MNRAS.435.3272T,2016MNRAS.458.3644B,2017MNRAS.464.1977W} have shown that a complete state of energy equipartition ($\eta = 0.5$) is never reached because of the so-called Spitzer instability. Recent observational works on this topic verified the goodness of this prediction \citep{2010ApJ...710.1032A,2018ApJ...853...86B,2018ApJ...861...99L,2019ApJ...873..109L}.\looseness=-4
We made use of the exquisite PMs of NGC~5904 to measure its state of energy equipartition using both the parameter $\eta$ defined above and the equipartition mass parameter $m_{\rm eq}$ introduced by \citet{2016MNRAS.458.3644B}\footnote{According to \citet{2016MNRAS.458.3644B}, the relation between the velocity dispersion $\sigma_\mu$ and the stellar mass is the following:
\begin{equation*}
\sigma (m) = \left\{
\begin{array}{lc}
\sigma_0 \exp(-\frac{1}{2}\frac{m}{m_{\rm eq}}) & {\rm if}~{m \le m_{\rm eq}} \\
\sigma_0 \exp(-\frac{1}{2}) (\frac{m}{m_{\rm eq}})^{-\frac{1}{2}} & {\rm if}~{m > m_{\rm eq}} \\
\end{array}
\right. \, ,
\end{equation*}
where $\sigma_0$ is the velocity dispersion at $m = 0$. To avoid unphysical values of the level of energy equipartition, there is cut-off at $m = m_{\rm eq}$.}. As a reference, a high degree of energy equipartition is characterized by a large value of $\eta$ and a small value of $m_{\rm eq}$.\looseness=-4
We started by inferring the mass of stars along the MS below the MS turn-off by means of the updated isochrones of the ``A Bag of Stellar Tracks and Isochrones" \citep[BaSTI;][]{2018ApJ...856..125H}. The parameters for NGC~5904 were chosen from the recent work of \citet{2019MNRAS.483.4949G}: a solar-scaled, 12.15 Gyr old isochrone for $[$Fe$/$H$] = -1.33$, $Y = 0.2478$, accounting for atomic diffusion, with a distance of 7.4 kpc. The fit is shown in Fig.~\ref{fig:eeq}. Although not perfect for the faintest portion of the MS due to still-existing shortcomings in this mass regime on the color-effective temperature relationship, the fit is good enough to assign a mass to each star.\looseness=-4
\begin{figure*}[t!]
\centering
\includegraphics[width=0.99\textwidth]{ngc5904_eeqfit.png}
\caption{Measurement of the level of energy equipartition in NGC~5904. In the CMD on the right, we show in red the isochrone used to estimate the mass of the stars along the MS. In the left panels, we show the velocity dispersion $\sigma_\mu$ as function of mass for MS stars. The gold line is the best fit obtained with the scale parameter $m_{\rm eq}$ (bottom panel) or the classical formalism of $\eta$ (middle panel). The yellow regions correspond to the 1$\sigma$ errors of the fit. The top-left panel presents the median 1-D PM error as a function of stellar mass for three samples: all objects in the catalog (red line), sources that passed our photometric quality selections; yellow line), and stars that survived both the photometric and astrometric cuts (green line).\looseness=-4}
\label{fig:eeq}
\includegraphics[width=0.99\textwidth]{ngc5904_eeqfit_rad.png}
\caption{Local levels of energy equipartition of NGC~5904. Left panels show the value of $m_{\rm eq}$ as a function of distance from the center of the cluster, while right panels present the same trend for $\eta$. Bottom panels are obtained by considering the combined velocity dispersion $\sigma_\mu$. The gold line and the yellow region represent the global value of the level of energy equipartition and its 1$\sigma$ region measured in Fig.~\ref{fig:eeq}. In the top panels, we highlight the trends obtained by estimating the level of energy equipartition using either the radial component $\sigma_{\rm rad}$ (red points) or the tangential component $\sigma_{\rm tan}$ (blue points).\looseness=-4}
\label{fig:eeq_rad}
\end{figure*}
We then computed the $\sigma_\mu$ in 35 bins of 1077 stars each along the MS. The bottom-left panel of Fig.~\ref{fig:eeq} shows the result obtained by fitting the mass-dependent exponential relation of \citet{2016MNRAS.458.3644B} with a maximum-likelihood approach. We find a global $m_{\rm eq} = 1.18 \pm 0.07$ $M_\odot$. \citet{2016MNRAS.458.3644B} provided a relation between $m_{\rm eq}$ and the ratio between the cluster age and its core relaxation time. Using the age of \citet{2019MNRAS.483.4949G} and a core relaxation time of 0.19 Gyr \citep[2010 edition]{1996AJ....112.1487H}, we expect a value of $m_{\rm eq} = 1.67^{+0.32}_{-0.28}$ $M_\odot$, which is in agreement with our estimate at the 2$\sigma$ level.\looseness=-4
The middle-left panel presents the linear fit in the same log-log plane. We found a value of $\eta = 0.25 \pm 0.01$. As for the value of $m_{\rm eq}$, this finding is consistent with the theoretical predictions of a partial state of energy equipartition even in an advanced stage of the cluster evolution.\looseness=-4
As a reference, the median 1-D PM error as a function of stellar mass is plot in the top-left panel of Fig.~\ref{fig:eeq}. The three lines correspond to the median PM trends of: all objects in the catalog (red line), sources that passed our photometric quality selections (see Sect.~\ref{kin}; yellow line), and stars that survived both the photometric and astrometric cuts (always described in Sect.~\ref{kin}; green line).
Our data set allowed us to push this investigation even further. We divided our FoV in four equally-populated radial bins, and measured the level of energy equipartition in each bin using the velocity dispersions measured in ten equally-populated magnitude bins. The result is shown in the bottom panels of Fig.~\ref{fig:eeq_rad} (left for $m_{\rm eq}$ and right for $\eta$). We can notice a marginal evidence of the innermost regions of the cluster that have a higher degree of energy equipartition (low $m_{\rm eq}$ and high $\eta$) than the outskirts. The innermost point within $r_{\rm c}$ is the most uncertain because the fit is obtained with a smaller mass baseline.\looseness=-4
In the top panels, we reproduced a similar analysis, but this time measuring the level of energy equipartition using the radial and the tangential components of the velocity dispersion separately. We find that the levels of energy equipartition from $\sigma_{\rm rad}$ and $\sigma_{\rm tan}$ are consistent with each other at all radii, with only marginal differences at the 1--2$\sigma$ level. The larger difference between the level of energy equipartition in the two components is shown in the innermost bin. The discrepancy is mainly the result of a poor fit, especially in the case of the exponential fit. This is in agreement with the simulations of \citet{2021MNRAS.504L..12P,2022MNRAS.509.3815P} who found the degree of energy equipartition in the two velocity components to be similar in the region within the half-light radius. Future studies will extend the investigation of the energy equipartition in the tangential and radial dispersions to the outer regions of GCs where, according to \citet{2021MNRAS.504L..12P,2022MNRAS.509.3815P}, the degree of energy equipartition in these two velocity components may differ.\looseness=-4
\begin{figure*}[t!]
\centering
\includegraphics[width=\textwidth]{ngc5904_bss_wd.png}
\caption{The optical CMD of NGC~5904 is shown in the right panel. Black dots are the reference population used in the analysis (Sect.~\ref{weight}), blue dots are BSs, and green points correspond to WDs. All other objects are shown as gray dots. In the bottom-left panel, we show the velocity-dispersion radial profiles of the the reference population (in black), BSs (in blue) and WDs (in green). The red line is the polynomial fit to the $\sigma_\mu$ of the reference population. The top-left panel presents the anisotropy radial profile for the same stars. The red, dashed horizontal line is set to 1. In all the left panels of this Figure, the vertical lines mark the $r_{\rm c}$ (solid) and $r_{\rm h}$ (dashed).\looseness=-4}
\label{fig:bss_wd}
\end{figure*}
\paragraph{A note on the PM errors for faint stars} Despite our careful data reduction and PM computation, the PM errors might be under/over estimated. For example, a higher velocity dispersion for faint stars could be caused by underestimated PM uncertainties. We repeated the analysis shown in Fig.~\ref{fig:eeq} by using only stars brighter than (1) $m_{\rm F606W} = 22.1$ ($\sim 0.5 M_\odot$) and (2) $m_{\rm F606W} = 23.2$ ($\sim 0.42 M_\odot$). We find:\looseness=-4
\begin{itemize}
\item $m_{\rm F606W} \le 22.1$ ($M \gtrsim 0.5 M_\odot$): $m_{\rm eq} = 1.60 \pm 0.18$ $M_\odot$ , $\eta = 0.21 \pm 0.02$ ;
\item $m_{\rm F606W} \le 23.2$ ($M \gtrsim 0.42 M_\odot$): $m_{\rm eq} = 1.31 \pm 0.10$ $M_\odot$ , $\eta = 0.24 \pm 0.02$ .
\end{itemize}
Although the results are in agreement at the 1--2$\sigma$ level, it seems that the value of $m_{\rm eq}$ is larger ($\eta$ is smaller) the brighter is the magnitude cut, i.e., when we exclude the faintest objects with large PM errors. However, the exponential fit is less robust when the mass interval is small (the associated errors increase). The variation of $\eta$ is instead mild.\looseness=-4
Furthermore, the selections applied can bias the analyzed sample and the inferred kinematics, as extensively discussed in \citet{2014ApJ...797..115B}. In the case of NGC~5904, we find that the selection on the PM reduced $\chi^2$ applies a more significant cut on the PM errors, as shown in the top panel of Fig.~\ref{fig:eeq} by the comparison of the yellow and green lines. As a test, we repeated the same analysis as before upon rescaling the PM errors of the stars used in the measurement of the level of energy equipartition by the ratio between the median PM errors obtained with sources that passed our photometric quality selections (yellow line in the top panel of Fig.~\ref{fig:eeq}), and with stars that survived both the photometric and astrometric cuts (green line). We find a global level of energy equipartition of $m_{\rm eq} = 1.50 \pm 0.11$ $M_\odot$ and $\eta = 0.20 \pm 0.01$. Since we increased the PM errors, the intrinsic velocity dispersion of faint stars decreased, but again all estimates are in agreement within 2--3$\sigma$.\looseness=-4
It is hard to say if these tests are a proxy of underestimated PM errors for faint stars, or they are biased by the less-constrained fit, or what we see is just due to the intrinsic kinematics of our tracers. Nevertheless, these examples highlight how important it is to understand the data set used. Once again, we advise users to carefully check the PMs, and test their quality selections on cluster-by-cluster basis, especially when the PM errors are of the order of the intrinsic velocity dispersion of the stars.\looseness=-4
\subsection{Kinematic mass determination}\label{weight}
The knowledge of the relation between mass and velocity dispersion allows us to measure the kinematic mass of stars, similarly to what done by \citet{2016ApJ...827...12B} and \citet{2018ApJ...861...99L,2019ApJ...873..109L}. The unknown mass of a group of stars X ($M_{\rm X}$) with velocity dispersion $\sigma_{\rm X}$ can be obtained as follows:
\begin{equation}
\alpha_{\rm X} = \frac{\sigma_{\rm X}}{\sigma_{\rm ref}} = \left(\frac{M_{\rm X}}{M_{\rm ref}}\right)^{-\eta} \, ,
\end{equation}
where $M_{\rm ref}$ and $\sigma_{\rm ref}$ are the known mass and velocity dispersion of a reference population. For this analysis, we choose to measure the mass of two groups of stars: blue stragglers (BSs) and white dwarfs (WDs). Our reference population are stars brighter than MS turn-off along SGB and RGB (same stars analyzed in Fig.~\ref{fig:kin2}), for which we assumed $M_{\rm ref}$ to be equal to the mass of stars at the MS turn-off. Using the isochrone in Sect.~\ref{eeq}, we defined $M_{\rm ref} = 0.78$ $M_\odot$.\looseness=-4
We selected BSs and WDs in various optical and UV CMDs (right panel of Fig.~\ref{fig:bss_wd}). We split BSs into two (of 22 and 21 stars, respectively) and WDs into one (of 14 stars) groups, respectively, and measured their velocity dispersions $\sigma_\mu$. The result is shown in the bottom-left panel of Fig.~\ref{fig:bss_wd}. Blue points depict the velocity dispersion of the BSs, while the green dot refers to the WD kinematics. Black points are the reference population. The effect of the energy equipartition is shown in the plot: BSs, which are more massive than RGB, SGB and MS stars, seem to be slightly kinematically colder (lower $\sigma_\mu$) than the reference population. WDs have a mass of about 0.5 $M_\odot$ \citep[e.g.,][and references therein]{2019MNRAS.488.3857B}. Being less massive than the reference population, their $\sigma_\mu$ is instead marginally higher. Finally, the top-left panel shows that BSs and WDs are kinematically isotropic as the other stars at the $\sim$1$\sigma$ level.\looseness=-4
We previously fit the $\sigma_\mu$ of the reference population with a polynomial function (see Sect.~\ref{kin}). To obtain the mass of the BSs, we repeated the same computation but we also solved for $\alpha_{\rm BS}$ by rescaling the polynomial of the reference population to fit the BS $\sigma_\mu$ at the same time. By assuming the global value of $\eta = 0.25 \pm 0.01$, we obtain $M_{\rm BS} = 0.84 \pm 0.28$ $M_\odot$. However, we can notice that BSs are preferentially located in regions where the local level of energy equipartition is higher than the global level. If we use the average value of $\eta$ in these regions ($0.34 \pm 0.06$), the mass becomes $M_{\rm BS} = 0.82 \pm 0.20$ $M_\odot$. The difference between the two values is small since the value of $\alpha_{\rm BS}$ is close to the unity. Both estimates are in agreement with the BS mass of NGC~5904 obtained by \citet[$0.82^{+0.29}_{-0.18}$ $M_\odot$]{2016ApJ...827...12B}. The mass of the BSs in this cluster is slightly lower than the average BS mass \citep[1.0--1.6 $M_\odot$; e.g.,][]{2018ApJ...860...36F}. However, it is worth noticing that our sample is mainly composed of BSs close to the MS turn-off, which are less massive than those on the bright-end of the BS sequence \citep[e.g.,][]{2019ApJ...879...56R}.\looseness=-4
For the WDs, a robust analysis cannot be obtained because we have only one point at disposal to obtain $\alpha_{\rm WD}$. Nevertheless, we repeated the same analysis to qualitatively assess the mass of the WDs. If we use the global value of $\eta$, we obtain $M_{\rm WD} = 0.39 \pm 0.25$ $M_\odot$, which is lower than the average mass of WDs in GCs. However, if we assume the local level of energy equipartition at the average WD distance from the center of the cluster, we find $M_{\rm WD} = 0.48 \pm 0.21$ $M_\odot$.\looseness=-4
\section{Conclusions}
We computed PMs for 57 stellar clusters studied in the GO-13297 program. The astro-photometric catalogs we made publicly available represent the most complete, homogeneous collection of PMs of stars in the core of stellar clusters, and more than double the number of clusters for which high-precision, \textit{HST}-based PMs are available \citep{2014ApJ...797..115B}. Furthermore, the astrometric information we release are complementary to those provided by the current (and future) \textit{Gaia} data releases. At the dawn of a new era in astronomy with the first light of the \textit{James Webb Space Telescope} (\textit{JWST}), the legacy that these PM catalogs offer is further enhanced, since they can serve as an important astrometric benchmark for \textit{JWST}-based data reduction and tools.\looseness=-4
We described the data reduction and, in great detail, the quality selections needed to select reliable objects for any kinematic analysis. We stress again that the data used for each cluster are different, thus any correction or selection should be tailored on a cluster-by-cluster basis. This is particularly important for stars with PM errors of the same order of the amplitude of the kinematic features one wants to measure, for example for stars along the MS of GCs.\looseness=-4
We made use of our catalogs to study the general kinematic properties of bright, massive stars in our clusters. We provided additional evidence supporting early findings that dynamically-young systems have a radially anisotropic velocity distribution at the half-light radius, while in dynamically older clusters the velocity distribution is isotropic at the same distance from the center of the cluster. This trend is consistent with the theoretical results of simulations showing that initially radially anisotropic clusters evolve toward an isotropic velocity distribution during their long-term evolution. Interestingly, core-collapsed clusters show similar properties as the non-core-collapsed systems, although a larger sample of core-collapsed GCs is necessary to confirm the similarities with non-core-collapsed clusters (in particular for the group with longer relaxations times).\looseness=-4
Finally, we showcased our PM catalogs using the GC NGC~5904. We separated the mPOPs along the RGB of the cluster and showed that, within our FoV, 1G and 2G stars have the same kinematics, are kinematically isotropic and have the same flat radial distributions. A detailed analysis of the kinematics of mPOPs will be the subject of another paper of this series.\looseness=-4
We investigated in detail the level of energy equipartition of NGC~5904. This cluster is in an advance stage of its dynamical evolution, yet it has reached only a partial state of energy equipartition, as predicted by theoretical simulations. The knowledge of the level of energy equipartition also allowed us to measure the kinematic mass the BSs and WDs, finding a good agreement with the typical mass of these objects obtained with different methods in the literature.\looseness=-4
\clearpage
\section*{Acknowledgments}
The authors thank the anonymous referee for the detailed suggestions that improved the quality of our work. ML and AB acknowledges support from GO-13297, GO-15857 and GO-16298. EV acknowledges support from NSF grant AST-2009193. AA acknowledges support from the Spanish Agencia Estatal del Ministerio Ciencia e Innovaci\'on (AEI-MICINN) under grant PID2020-115981GB-I00. LB acknowledges the funding support from Italian Space Agency (ASI) regulated by ``Accordo ASI-INAF n. 2013-016-R.0 del 9 luglio 2013 e integrazione del 9 luglio 2015 CHEOPS Fasi A/B/C". LRB acknowledges support by MIUR under PRIN program \#2017Z2HSMF and PRIN-INAF-2019-PI:BEDIN. SC acknowledges support from INFN (Iniziativa specifica TAsP), and from PLATO ASI-INAF agreement n.2015-019-R.1-2018. FRF, ED e BL acknowledge funding from Italian MIUR throughout the PRIN-2017 grant awarded to the project Light-on-Dark (PI:Ferraro) through contract PRIN-2017K7REXT. ML thanks Drs. Peter Zeidler, Laura Watkins and Silvia Raso for the useful discussions in various aspects of this project.
This research was pursued in collaboration with the HSTPROMO (High-resolution Space Telescope PROper MOtion) collaboration, a set of projects aimed at improving our dynamical understanding of stars, clusters and galaxies in the nearby Universe through measurement and interpretation of proper motions from \textit{HST}, \textit{Gaia}, and other space observatories. We thank the collaboration members for the sharing of their ideas and software.
Based on observations with the NASA/ESA \textit{HST}, obtained at the Space Telescope Science Institute, which is operated by AURA, Inc., under NASA contract NAS 5-26555. This work has made use of data from the European Space Agency (ESA) mission {\it Gaia} (\url{https://www.cosmos.esa.int/gaia}), processed by the {\it Gaia} Data Processing and Analysis Consortium (DPAC, \url{https://www.cosmos.esa.int/web/gaia/dpac/consortium}). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the {\it Gaia} Multilateral Agreement. This research made use of \texttt{astropy}, a community-developed core \texttt{python} package for Astronomy \citep{astropy:2013, astropy:2018}, \texttt{\textsc{LIMEPY}} \citep{2015MNRAS.454..576G}, \texttt{emcee} \citep{2013PASP..125..306F}, \texttt{scikit-learn} \citep{scikit-learn}, and of the SIMBAD database \citep{2000A&AS..143....9W}, operated at CDS, Strasbourg, France.
\subsubsection*{#1}}
\pagestyle{headings}
\markright{Reference sheet: \texttt{natbib}}
\usepackage{shortvrb}
\MakeShortVerb{\|}
\begin{document}
\thispagestyle{plain}
\newcommand{\textsc{Bib}\TeX}{\textsc{Bib}\TeX}
\newcommand{\texttt{#1}\def\filedate{#2}\def\fileversion{#3}}}{\texttt{#1}\def\filedate{#2}\def\fileversion{#3}}}
\begin{center}{\bfseries\Large
Reference sheet for \texttt{#1}\def\filedate{#2}\def\fileversion{#3}}\ usage}\\
\large(Describing version \fileversion\ from \filedate)
\end{center}
\begin{quote}\slshape
For a more detailed description of the \texttt{#1}\def\filedate{#2}\def\fileversion{#3}}\ package, \LaTeX\ the
source file \texttt{#1}\def\filedate{#2}\def\fileversion{#3}}\texttt{.dtx}.
\end{quote}
\head{Overview}
The \texttt{#1}\def\filedate{#2}\def\fileversion{#3}}\ package is a reimplementation of the \LaTeX\ |\cite| command,
to work with both author--year and numerical citations. It is compatible with
the standard bibliographic style files, such as \texttt{plain.bst}, as well as
with those for \texttt{harvard}, \texttt{apalike}, \texttt{chicago},
\texttt{astron}, \texttt{authordate}, and of course \texttt{#1}\def\filedate{#2}\def\fileversion{#3}}.
\head{Loading}
Load with |\usepackage[|\emph{options}|]{|\texttt{#1}\def\filedate{#2}\def\fileversion{#3}}|}|. See list of
\emph{options} at the end.
\head{Replacement bibliography styles}
I provide three new \texttt{.bst} files to replace the standard \LaTeX\
numerical ones:
\begin{quote}\ttfamily
plainnat.bst \qquad abbrvnat.bst \qquad unsrtnat.bst
\end{quote}
\head{Basic commands}
The \texttt{#1}\def\filedate{#2}\def\fileversion{#3}}\ package 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.
\begin{quote}
\begin{tabular}{l@{\quad$\Rightarrow$\quad}l}
|\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)
\end{tabular}
\end{quote}
\head{Multiple citations}
Multiple citations may be made by including more than one
citation key in the |\cite| command argument.
\begin{quote}
\begin{tabular}{l@{\quad$\Rightarrow$\quad}l}
|\citet{jon90,jam91}| & Jones et al. (1990); James et al. (1991)\\
|\citep{jon90,jam91}| & (Jones et al., 1990; James et al. 1991)\\
|\citep{jon90,jon91}| & (Jones et al., 1990, 1991)\\
|\citep{jon90a,jon90b}| & (Jones et al., 1990a,b)
\end{tabular}
\end{quote}
\head{Numerical mode}
These examples are for author--year citation mode. In numerical mode, the
results are different.
\begin{quote}
\begin{tabular}{l@{\quad$\Rightarrow$\quad}l}
|\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]
\end{tabular}
\end{quote}
\head{Suppressed parentheses}
As an alternative form of citation, |\citealt| is the same as |\citet| but
\emph{without parentheses}. Similarly, |\citealp| is |\citep| without
parentheses. Multiple references, notes, and the starred variants
also exist.
\begin{quote}
\begin{tabular}{l@{\quad$\Rightarrow$\quad}l}
|\citealt{jon90}| & Jones et al.\ 1990\\
|\citealt*{jon90}| & Jones, Baker, and Williams 1990\\
|\citealp{jon90}| & Jones et al., 1990\\
|\citealp*{jon90}| & Jones, Baker, and Williams, 1990\\
|\citealp{jon90,jam91}| & Jones et al., 1990; James et al., 1991\\
|\citealp[pg.~32]{jon90}| & Jones et al., 1990, pg.~32\\
|\citetext{priv.\ comm.}| & (priv.\ comm.)
\end{tabular}
\end{quote}
The |\citetext| command
allows arbitrary text to be placed in the current citation parentheses.
This may be used in combination with |\citealp|.
\head{Partial citations}
In author--year schemes, it is sometimes desirable to be able to refer to
the authors without the year, or vice versa. This is provided with the
extra commands
\begin{quote}
\begin{tabular}{l@{\quad$\Rightarrow$\quad}l}
|\citeauthor{jon90}| & Jones et al.\\
|\citeauthor*{jon90}| & Jones, Baker, and Williams\\
|\citeyear{jon90}| & 1990\\
|\citeyearpar{jon90}| & (1990)
\end{tabular}
\end{quote}
\head{Forcing upper cased names}
If the first author's name contains a \textsl{von} part, such as ``della
Robbia'', then |\citet{dRob98}| produces ``della Robbia (1998)'', even at the
beginning of a sentence. One can force the first letter to be in upper case
with the command |\Citet| instead. Other upper case commands also exist.
\begin{quote}
\begin{tabular}{rl@{\quad$\Rightarrow$\quad}l}
when & |\citet{dRob98}| & della Robbia (1998) \\
then & |\Citet{dRob98}| & Della Robbia (1998) \\
& |\Citep{dRob98}| & (Della Robbia, 1998) \\
& |\Citealt{dRob98}| & Della Robbia 1998 \\
& |\Citealp{dRob98}| & Della Robbia, 1998 \\
& |\Citeauthor{dRob98}| & Della Robbia
\end{tabular}
\end{quote}
These commands also exist in starred versions for full author names.
\head{Citation aliasing}
Sometimes one wants to refer to a reference with a special designation,
rather than by the authors, i.e. as Paper~I, Paper~II. Such aliases can be
defined and used, textual and/or parenthetical with:
\begin{quote}
\begin{tabular}{lcl}
|\defcitealias{jon90}{Paper~I}|\\
|\citetalias{jon90}| & $\Rightarrow$ & Paper~I\\
|\citepalias{jon90}| & $\Rightarrow$ & (Paper~I)
\end{tabular}
\end{quote}
These citation commands function much like |\citet| and |\citep|: they may
take multiple keys in the argument, may contain notes, and are marked as
hyperlinks.
\head{Selecting citation style and punctuation}
Use the command |\bibpunct| with one optional and 6 mandatory arguments:
\begin{enumerate}
\item the opening bracket symbol, default = (
\item the closing bracket symbol, default = )
\item the punctuation between multiple citations, default = ;
\item the letter `n' for numerical style, or `s' for numerical superscript
style, any other letter for
author--year, default = author--year;
\item the punctuation that comes between the author names and the year
\item the punctuation that comes between years or numbers when common author
lists are suppressed (default = ,);
\end{enumerate}
The optional argument is the character preceding a post-note, default is a
comma plus space. In redefining this character, one must include a space if
one is wanted.
Example~1, |\bibpunct{[}{]}{,}{a}{}{;}| changes the output of
\begin{quote}
|\citep{jon90,jon91,jam92}|
\end{quote}
into [Jones et al. 1990; 1991, James et al. 1992].
Example~2, |\bibpunct[; ]{(}{)}{,}{a}{}{;}| changes the output of
\begin{quote}
|\citep[and references therein]{jon90}|
\end{quote}
into (Jones et al. 1990; and references therein).
\head{Other formatting options}
Redefine |\bibsection| to the desired sectioning command for introducing
the list of references. This is normally |\section*| or |\chapter*|.
Define |\bibpreamble| to be any text that is to be printed after the heading but
before the actual list of references.
Define |\bibfont| to be a font declaration, e.g.\ |\small| to apply to
the list of references.
Define |\citenumfont| to be a font declaration or command like |\itshape|
or |\textit|.
Redefine |\bibnumfmt| as a command with an argument to format the numbers in
the list of references. The default definition is |[#1]|.
The indentation after the first line of each reference is given by
|\bibhang|; change this with the |\setlength| command.
The vertical spacing between references is set by |\bibsep|; change this with
the |\setlength| command.
\head{Automatic indexing of citations}
If one wishes to have the citations entered in the \texttt{.idx} indexing
file, it is only necessary to issue |\citeindextrue| at any point in the
document. All following |\cite| commands, of all variations, then insert
the corresponding entry to that file. With |\citeindexfalse|, these
entries will no longer be made.
\head{Use with \texttt{chapterbib} package}
The \texttt{#1}\def\filedate{#2}\def\fileversion{#3}}\ package is compatible with the \texttt{chapterbib} package
which makes it possible to have several bibliographies in one document.
The package makes use of the |\include| command, and each |\include|d file
has its own bibliography.
The order in which the \texttt{chapterbib} and \texttt{#1}\def\filedate{#2}\def\fileversion{#3}}\ packages are loaded
is unimportant.
The \texttt{chapterbib} package provides an option \texttt{sectionbib}
that puts the bibliography in a |\section*| instead of |\chapter*|,
something that makes sense if there is a bibliography in each chapter.
This option will not work when \texttt{#1}\def\filedate{#2}\def\fileversion{#3}}\ is also loaded; instead, add
the option to \texttt{#1}\def\filedate{#2}\def\fileversion{#3}}.
Every |\include|d file must contain its own
|\bibliography| command where the bibliography is to appear. The database
files listed as arguments to this command can be different in each file,
of course. However, what is not so obvious, is that each file must also
contain a |\bibliographystyle| command, \emph{preferably with the same
style argument}.
\head{Sorting and compressing citations}
Do not use the \texttt{cite} package with \texttt{#1}\def\filedate{#2}\def\fileversion{#3}}; rather use one of the
options \texttt{sort} or \texttt{sort\&compress}.
These also work with author--year citations, making multiple citations appear
in their order in the reference list.
\head{Long author list on first citation}
Use option \texttt{longnamesfirst} to have first citation automatically give
the full list of authors.
Suppress this for certain citations with |\shortcites{|\emph{key-list}|}|,
given before the first citation.
\head{Local configuration}
Any local recoding or definitions can be put in \texttt{#1}\def\filedate{#2}\def\fileversion{#3}}\texttt{.cfg} which
is read in after the main package file.
\head{Options that can be added to \texttt{\char`\\ usepackage}}
\begin{description}
\item[\ttfamily round] (default) for round parentheses;
\item[\ttfamily square] for square brackets;
\item[\ttfamily curly] for curly braces;
\item[\ttfamily angle] for angle brackets;
\item[\ttfamily colon] (default) to separate multiple citations with
colons;
\item[\ttfamily comma] to use commas as separaters;
\item[\ttfamily authoryear] (default) for author--year citations;
\item[\ttfamily numbers] for numerical citations;
\item[\ttfamily super] for superscripted numerical citations, as in
\textsl{Nature};
\item[\ttfamily sort] orders multiple citations into the sequence in
which they appear in the list of references;
\item[\ttfamily sort\&compress] as \texttt{sort} but in addition multiple
numerical citations are compressed if possible (as 3--6, 15);
\item[\ttfamily longnamesfirst] makes the first citation of any reference
the equivalent of the starred variant (full author list) and subsequent
citations normal (abbreviated list);
\item[\ttfamily sectionbib] redefines |\thebibliography| to issue
|\section*| instead of |\chapter*|; valid only for classes with a
|\chapter| command; to be used with the \texttt{chapterbib} package;
\item[\ttfamily nonamebreak] keeps all the authors' names in a citation on
one line; causes overfull hboxes but helps with some \texttt{hyperref}
problems.
\end{description}
\end{document}
|
1,116,691,499,314 | arxiv | \section{Algorithm}
\subsection{Computing $q$-gram Frequencies on Uncompressed Strings}
\label{subsection:uncompressed}
We describe two algorithms (Algorithm~\ref{algo:naive}
and Algorithm~\ref{algo:naive_sa}) for computing
the $q$-gram frequencies of a given uncompressed string $T$.
A na\"ive algorithm for computing the
$q$-gram frequencies is given in Algorithm~\ref{algo:naive}.
The algorithm constructs an associative array, where keys consist
of $q$-grams, and the values correspond to the occurrence frequencies
of the $q$-grams.
The time complexity depends on the implementation of the associative
array,
but requires at least $O(q|T|)$ time
since each $q$-gram is considered explicitly, and
the associative array is accessed $O(|T|)$ times:
e.g. $O(q|T|\log|\Sigma|)$ time and $O(q|T|)$ space using a
simple trie.
\begin{algorithm2e}[t]
\SetKwInput{KwOut}{Report}
\SetKw{KwReport}{Report}
\KwIn{string $T$, integer $q \geq 1$}
\KwOut{$(P, |\mathit{Occ}(T,P)|)$ for all $P\in\Sigma^q$ where
$\mathit{Occ}(T,P) \neq \emptyset$.}
$\mathbf{S} \leftarrow \emptyset$\tcp*[l]{empty associative array}
\For{$i\leftarrow 1$ \KwTo $|T|-q+1$}{
$\mathit{qgram} \leftarrow T[i:i+q-1]$\;
\lIf{$\mathit{qgram} \in\mathrm{keys}(\mathbf{S})$}{\label{algo:naive:increment}
$\mathbf{S}[\mathit{qgram}] \leftarrow
\mathbf{S}[\mathit{qgram}] + 1$\;
}
\lElse{
$\mathbf{S}[\mathit{qgram}] \leftarrow 1$\tcp*[l]{new $q$-gram}\label{algo:naive:addweight}
}
}
\lFor{$\mathit{qgram}\in\mathrm{keys}(\mathbf{S})$}{\KwReport{$(\mathit{qgram},\mathbf{S}[\mathit{qgram}])$}}
\caption{A na\"ive algorithm for computing $q$-gram frequencies.}
\label{algo:naive}
\end{algorithm2e}
\begin{algorithm2e}[t]
\SetKwInput{KwOut}{Report}
\SetKw{KwReport}{Report}
\SetKw{KwOr}{or}
\SetKw{KwAnd}{and}
\KwIn{string $T$, integer $q \geq 1$}
\KwOut{
$(i, |\mathit{Occ}(T,P)|)$
for all $P\in\Sigma^q$ and some position $i\in \mathit{Occ}(T,P)$.
}
$\mathit{SA} \leftarrow \mathit{SUFFIXARRAY}(T)$;
$\mathit{LCP} \leftarrow \mathit{LCPARRAY}(T,SA)$;
$\mathit{count} \leftarrow 1$\;
\For{$i\leftarrow 2$ \KwTo $|T|+1$}{
\If{$i = |T|+1$ \KwOr $\mathit{LCP}[i] < q$}{
\lIf{$\mathit{count} > 0$}{
\KwReport{$(\mathit{SA}[i-1], \mathit{count})$}; $\mathit{count} \leftarrow 0$\;
}
}
\lIf{$i \leq |T|$ \KwAnd $\mathit{SA}[i] \leq |T| - q + 1$}{
$\mathit{count} \leftarrow \mathit{count}+1$\;\label{algo:naive_sa:increment}
}
}
\caption{A linear time algorithm for computing $q$-gram frequencies.}
\label{algo:naive_sa}
\end{algorithm2e}
The $q$-gram frequencies of string $T$ can be calculated in
$O(|T|)$ time using suffix array $\mathit{SA}$ and lcp array $\mathit{LCP}$,
as shown in Algorithm~\ref{algo:naive_sa}.
For each $1\leq i\leq |T|$, the suffix $\mathit{SA}[i]$ represents
an occurrence of $q$-gram $T[\mathit{SA}[i]:\mathit{SA}[i]+q-1]$,
if the suffix is long enough, i.e. $\mathit{SA}[i] \leq |T| - q +1$.
The key is that since the suffixes are lexicographically
sorted, intervals on the suffix array where the values in the lcp array
are at least $q$ represent occurrences of the same $q$-gram.
The algorithm runs in $O(|T|)$ time, since $\mathit{SA}$ and $\mathit{LCP}$ can be
constructed in $O(|T|)$. The rest is a simple $O(|T|)$ loop.
A technicality is that we encode the output for a $q$-gram as
one of the positions in the text where the $q$-gram occurs,
rather than the $q$-gram itself.
This is because there can be a total of $O(|T|)$ different $q$-grams,
and if we output them as length-$q$ strings,
it would require at least $O(q|T|)$ time.
\subsection{Computing $q$-gram Frequencies on SLP}
We now describe the core idea of our algorithms,
and explain two variations which utilize variants of the two algorithms for uncompressed strings
presented in Section~\ref{subsection:uncompressed}.
For $q = 1$, the $1$-gram frequencies are simply the frequencies of
the alphabet and the output is
$(a,\sum\{\mathit{vOcc}(X_i)\mid X_i = a\})$ for each $a\in\Sigma$,
which takes only $O(n)$ time.
For $q \geq 2$, we make use of Lemma~\ref{lemma:mkSLP} below.
The idea is similar to the {\em $mk$ Lemma}~\cite{charikar05:_small_gramm_probl},
but the statement is more specific.
\begin{lemma}
\label{lemma:mkSLP}
Let $\mathcal{T} = \{X_i\}_{i=1}^n$ be an SLP that represents string $T$.
For an interval $[u:v]$ $(1\leq u < v \leq |T|)$,
there exists exactly one variable $X_i=X_\ell X_r$
such that for some $[u':v'] \in \mathit{itv}(X_i)$,
the following holds:
$[u:v] \subseteq [u':v']$,
$u \in [u':u'+|X_\ell|-1] \in \mathit{itv}(X_\ell)$
and
$v \in [u'+|X_\ell|:v'] \in \mathit{itv}(X_r)$.
\end{lemma}
\begin{proof}
Consider length $1$ intervals $[u:u]$ and $[v:v]$
corresponding to leaves in the derivation tree.
$X_i$ corresponds to the lowest common ancestor
of these intervals in the derivation tree.
\hfill $\Box$
\end{proof}
\begin{wrapfigure}[9]{r}{0.36\textwidth}
\vspace{-1.43cm}
\centerline{\includegraphics[width=0.35\textwidth]{SLP_ngram.eps}}
\caption{
Length-$q$ intervals corresponding to $X_i = X_\ell X_r$.
}
\label{fig:SLP-kgram}
\end{wrapfigure}
From Lemma~\ref{lemma:mkSLP}, each occurrence of
a $q$-gram ($q \geq 2$) represented by some length-$q$ interval of $T$,
corresponds to a single variable $X_i = X_{\ell}X_r$, and is split in
two by intervals corresponding to $X_{\ell}$ and $X_r$.
On the other hand, consider all length-$q$ intervals that
correspond to a given variable. Counting the frequencies of the $q$-grams
they represent, and summing them up for all variables give the
frequencies of all $q$-grams of $T$.
For variable $X_i = X_\ell X_r$, let
$t_i = \mathit{suf}(X_\ell, q-1) \mathit{pre}(X_r,q-1)$.
Then, all $q$-grams represented by length $q$ intervals that
correspond to $X_i$ are those in $t_i$. (Fig.~\ref{fig:SLP-kgram}).
If we obtain the frequencies of all $q$-grams in $t_i$, and then
multiply each frequency by $\mathit{vOcc}(X_i)$,
we obtain frequencies for the $q$-grams occurring in all intervals
derived by $X_i$.
It remains to sum up the $q$-gram frequencies of $t_i$
for all $1\leq i\leq n$.
We can regard it as obtaining the {\em weighted} $q$-gram
frequencies in the set of strings $\{t_1,\ldots,t_n\}$,
where each $q$-gram in $t_i$ is weighted by $\mathit{vOcc}(X_i)$.
We further reduce this problem to a weighted $q$-gram frequency
problem for a single string $z$ as in Algorithm~\ref{algo:slpmain}.
String $z$ is constructed by concatenating $t_i$
such that $q \leq |t_i| \leq 2(q-1)$,
and the weights of $q$-grams starting at each position in $z$
is held in array $w$.
On line~\ref{algo:slpmain:append0}, $0$'s instead of $\mathit{vOcc}(X_i)$ are appended to
$w$ for the last $q-1$ values corresponding to $t_i$.
This is to avoid counting unwanted $q$-grams that are generated by the
concatenation of $t_i$ to $z$ on line~\ref{algo:slpmain:zappend},
which are not substrings of each $t_i$.
The weighted $q$-gram frequency problem for a single string
(Line~\ref{algo:slpmainweightedfreqs}) can be solved
with a slight modification of Algorithm~\ref{algo:naive} or \ref{algo:naive_sa}.
The modified algorithms are shown respectively in
Algorithms~\ref{algo:weighted_naive} and \ref{algo:weighted_naive_sa}.
\begin{theorem}
Given an SLP ${\mathcal T} = \{X_i\}_{i=1}^n$ of size $n$
representing a string $T$,
the $q$-gram frequencies of $T$ can be computed in $O(qn)$ time
for any $q > 0$.
\end{theorem}
\begin{proof}
Consider Algorithm~\ref{algo:slpmain}.
The correctness is straightforward from the above arguments, so we
consider the time complexity.
Line~\ref{algo:slpmain:varocc} can be computed in $O(n)$ time.
Line~\ref{algo:slpmain:prefsuf} can be computed in $O(qn)$ time by a simple dynamic
programming.
For $\mathit{pre}()$:
If $X_i=a$ for some $a\in\Sigma$, then $\mathit{pre}(X_i,q-1) = a$.
If $X_i=X_{\ell}X_r$ and $|X_{\ell}| \geq q-1$, then $\mathit{pre}(X_i, q-1) = \mathit{pre}(X_{\ell}, q-1)$.
If $X_i=X_{\ell}X_r$ and $|X_{\ell}| < q-1$, then $\mathit{pre}(X_i, q-1)
= \mathit{pre}(X_{\ell}, q-1)\mathit{pre}(X_r, q - 1 - |X_{\ell}|)$.
The strings $\mathit{suf}()$ can be computed similarly.
The computation amounts to copying $O(q)$ characters for each
variable, and thus can be done in $O(qn)$ time.
For the loop at line~\ref{algo:slpmain:mainloop}, since the length of
string $t_i$ appended to $z$, as well as the number of elements
appended to $w$ is at most $2(q-1)$ in each loop,
the total time complexity is $O(qn)$.
Finally, since the length of $z$ and $w$ is $O(qn)$,
line~\ref{algo:slpmainweightedfreqs} can be calculated in $O(qn)$
time
using the weighted version of Algorithm~\ref{algo:naive_sa} (Algorithm~\ref{algo:weighted_naive_sa}).
\hfill $\Box$
\end{proof}
Note that the time complexity for using the weighted version of
Algorithm~\ref{algo:naive} for line~\ref{algo:slpmainweightedfreqs}
of Algorithm~\ref{algo:slpmain} would be at least $O(q^2n)$:
e.g. $O(q^2n\log|\Sigma|)$ time and $O(q^2n)$ space using a trie.
\begin{algorithm2e}[t]
\caption{Calculating $q$-gram frequencies of an SLP for $q\geq 2$}
\label{algo:slpmain}
\SetKw{KwAnd}{and}
\SetKwInput{KwOut}{Report}
\KwIn{SLP ${\mathcal T} = \{X_i\}_{i=1}^n$ representing string $T$, integer $q\geq 2$.}
\KwOut{all $q$-grams and their frequencies which occur in $T$.}
\SetKw{KwReport}{Report}
Calculate $\mathit{vOcc}(X_i)$ for all $1\leq i\leq n$\; \label{algo:slpmain:varocc}
Calculate $\mathit{pre}(X_i,q-1)$ and $\mathit{suf}(X_i,q-1)$ for all $1\leq
i\leq n-1$ \; \label{algo:slpmain:prefsuf}
$z \leftarrow \varepsilon$; $w \leftarrow []$\;
\For{$i \leftarrow 1$ \KwTo $n$}{\label{algo:slpmain:mainloop}
\If{$X_i = X_{\ell}X_r$ \KwAnd $|X_i| \geq q$}{
$t_i = \mathit{suf}(X_\ell,q-1)\mathit{pre}(X_r,q-1)$;
$z$.append($t_i$)\; \label{algo:slpmain:zappend}
\lFor{$j \leftarrow 1$ \KwTo $|t_i|-q+1$}{
$w$.append($\mathit{vOcc}(X_i)$)\;
}
\lFor{$j \leftarrow 1$ \KwTo $q-1$}{
$w$.append(0)\; \label{algo:slpmain:append0}
}
}
}
\KwReport
$q$-gram frequencies in $z$, where each $q$-gram
$z[i:i+q-1]$
is {\em weighted} by $w[i]$.\label{algo:slpmainweightedfreqs}
\end{algorithm2e}
\begin{algorithm2e}[t]
\caption{A variant of Algorithm~\ref{algo:naive} for weighted $q$-gram frequencies.}
\label{algo:weighted_naive}
\SetKwInput{KwOut}{Report}
\SetKw{KwReport}{Report}
\KwIn{string $T$, array of integers $w$ of length $|T|$, integer $q \geq 1$}
\KwOut{$(P, \sum_{i\in\mathit{Occ}(T,P)}w[i])$ for all $P\in\Sigma^q$ where
$\sum_{i\in\mathit{Occ}(T,P)}w[i] > 0$.}
$\mathbf{S} \leftarrow \emptyset$\tcp*[l]{empty associative array}
\For{$i\leftarrow 1$ \KwTo $|T|-q+1$}{
$\mathit{qgram} \leftarrow T[i:i+q-1]$\;
\lIf{$\mathit{qgram} \in\mathrm{keys}(\mathbf{S})$}{
$\mathbf{S}[\mathit{qgram}] \leftarrow
\mathbf{S}[\mathit{qgram}] + w[i]$\;
}
\lElse{\lIf{$w[i]>0$}{$\mathbf{S}[\mathit{qgram}] \leftarrow w[i]$\tcp*[l]{new $q$-gram}}}
}
\lFor{$\mathit{qgram}\in\mathrm{keys}(\mathbf{S})$}{\KwReport{$(\mathit{qgram},\mathbf{S}[\mathit{qgram}])$}}
\end{algorithm2e}
\begin{algorithm2e}[t]
\caption{A variant of Algorithm~\ref{algo:naive_sa} for weighted
$q$-gram frequencies.}
\label{algo:weighted_naive_sa}
\SetKwInput{KwOut}{Output}
\SetKw{KwReport}{Report}
\SetKw{KwOr}{or}
\SetKw{KwAnd}{and}
\KwIn{string $T$, array of integers $w$ of length $|T|$, integer $q \geq 1$}
\KwOut{$(i, \sum_{i\in\mathit{Occ}(T,P)}w[i])$ for all $P\in\Sigma^q$ where
$\sum_{i\in\mathit{Occ}(T,P)}w[i] > 0$
and some position $i\in\mathit{Occ}(T,P)$.}
$\mathit{SA} \leftarrow \mathit{SUFFIXARRAY}(T)$;
$\mathit{LCP} \leftarrow \mathit{LCPARRAY}(T,SA)$;
$\mathit{count} \leftarrow 1$\;
\For{$i\leftarrow 2$ \KwTo $|T|+1$}{
\If{$i = |T|+1$ \KwOr $\mathit{LCP}[i] < q$}{
\lIf{$\mathit{count} > 0$}{
\KwReport{$(\mathit{SA}[i-1], \mathit{count})$}; $\mathit{count} \leftarrow 0$\;
}
}
\lIf{$i \leq |T|$ \KwAnd $\mathit{SA}[i] \leq |T| - q + 1$}{
$\mathit{count} \leftarrow \mathit{count}+w[\mathit{SA}[i]]$\;
}
}
\end{algorithm2e}
\section{Applications and Extensions}
We showed that for an SLP ${\mathcal{T}}$ of size $n$ representing string $T$,
$q$-gram frequency problems on $T$ can be reduced to
{\em weighted} $q$-gram frequency problems on a string $z$ of length $O(qn)$,
which can be much shorter than $T$.
This idea can further be applied to obtain efficient compressed
string processing algorithms for
interesting problems which we briefly introduce below.
\subsection{$q$-gram Spectrum Kernel}
A string kernel is a function that computes the inner product between two strings
which are mapped to some feature space.
It is used when classifying string or text
data using methods such as Support Vector Machines (SVMs), and is usually the
dominating factor in the time complexity of SVM learning and classification.
A $q$-gram spectrum kernel~\cite{leslie02:_spect_kernel} considers the
feature space of $q$-grams.
For string $T$, let $\phi_q(T) = (|\mathit{Occ}(T,p)|)_{p\in\Sigma^q}$.
The kernel function is defined as
$K_q(T_1,T_2) =
\langle\phi_q(T_1),\phi_q(T_2)\rangle =
\sum_{p\in\Sigma^q} |\mathit{Occ}(T_1,p)| |\mathit{Occ}(T_2,p)|$.
The calculation of the kernel function amounts to
summing up the product of occurrence frequencies
in strings $T_1$ and $T_2$ for all
$q$-grams which occur in both $T_1$ and $T_2$.
This can be done in $O(|T_1|+|T_2|)$ time using suffix arrays.
For two SLPs ${\mathcal T_1}$ and ${\mathcal T_2}$ of
size $n_1$ and $n_2$ representing strings $T_1$ and $T_2$,
respectively, the $q$-gram spectrum kernel $K_q(T_1,T_2)$ can be computed
in $O(q(n_1+n_2))$ time by a slight modification of our algorithm.
\subsection{Optimal Substring Patterns of Length $q$}
Given two sets of strings, finding string patterns that are
frequent in one set and not in the other, is an important
problem in string data mining, with many problem formulations and the
types of patterns to be considered,
e.g.: in Bioinformatics~\cite{brazma98:_approac},
Machine Learning (optimal patterns~\cite{arimura98:_fast_algor_discov_optim_strin}),
and more recently KDD
(emerging patterns~\cite{chan03:_minin_emerg_subst}).
A simple optimal $q$-gram pattern discovery problem can be defined as follows:
Let $\mathbf{T_1}$
and $\mathbf{T_2}$
be two multisets of strings.
The problem is to find the $q$-gram $p$ which gives
the highest (or lowest) score according to some scoring function
that depends only on $|\mathbf{T_1}|$, $|\mathbf{T_2}|$,
and the number of strings respectively in $\mathbf{T_1}$ and $\mathbf{T_2}$ for
which $p$ is a substring.
For uncompressed strings, the problem can be solved in $O(N)$ time,
where $N$ is the total length of the strings in both $\mathbf{T_1}$ and
$\mathbf{T_2}$, by applying the algorithm of~\cite{HuiCPM92} to two sets of strings.
For the SLP compressed version of this problem,
the input is two multisets of SLPs,
each representing strings in $\mathbf{T_1}$ and $\mathbf{T_2}$.
If $n$ is the total number of variables used in all
of the SLPs, the problem can be solved in $O(qn)$ time.
\subsection{Different Lengths}
The ideas in this paper can be used to consider all substrings of length {\em not only} $q$,
but {\em all lengths up-to} $q$, with some modifications.
For the applications discussed above,
although the number of such substrings increases to $O(q^2n)$,
the $O(qn)$ time complexity can be maintained
by using standard techniques of suffix
arrays~\cite{gusfield97:_algor_strin_trees_sequen,Kasai01}.
This is because there exist only $O(qn)$ substring with distinct frequencies
(corresponding to nodes of the suffix tree),
and the computations of the extra substrings can be summarized with
respect to them.
\ignore{
\subsection{Collage System}
Collage system~\cite{KidaCollageTCS} is a more general framework for modeling
various compression methods.
In addition to the simple concatenation operation used in SLPs,
it includes operations for repetition and prefix/suffix truncation
of variables.
For example, while a LZ77 encoded representation of size $m$ may require
$O(m^2\log m)$ size when represented as an SLP,
it can be represented as a collage system of size $O(m\log m)$~\cite{GasieniecSWAT96}.
Our algorithm can be extended to run in $O((q+h)n)$ time on
collage system of size $n$, where $h$ is the height of the derivation tree.
The increase in complexity comes from the handling of truncated
variables. Details will be presented in a forthcoming paper.
}
\section{Conclusion}
We presented an $O(qn)$ time and space algorithm for calculating all
$q$-gram frequencies in a string, given an SLP of size $n$
representing the string.
This solves, much more efficiently, a more general problem than
considered in previous work.
Computational experiments on various real texts showed that the algorithms
run faster than algorithms that work on the
uncompressed string, when $q$ is small.
Although larger values of $q$ allow us to capture longer character dependencies,
the dimensionality of the features increases, making the space of
occurring $q$-grams sparse.
Therefore, meaningful values of $q$ for
typical applications can be fairly small in practice (e.g. $3\sim 6$),
so our algorithms have practical value.
A future work is extending our algorithms that work on SLPs,
to algorithms that work on collage systems~\cite{KidaCollageTCS}.
A Collage System is a more general framework for modeling various
compression methods.
In addition to the simple concatenation operation used in SLPs,
it includes operations for repetition and prefix/suffix truncation
of variables.
\ignore{
For example, while a LZ77 encoded representation of size $m$ may require
$O(m^2\log m)$ size when represented as an SLP,
it can be represented as a collage system of size $O(m\log m)$~\cite{GasieniecSWAT96}.
}
This is the first paper to show the potential of the compressed
string processing approach in developing efficient and {\em practical} algorithms
for problems in the field of string mining and classification.
More and more efficient algorithms for various processing of
text in compressed representations are becoming available.
We believe texts will eventually be stored in compressed form by default,
since not only will it save space, but it will also have the added benefit of being able to conduct
various computations on it more efficiently later on, when needed.
\section{Computational Experiments}
We implemented 4 algorithms (NMP, NSA, SMP, SSA) that
count the frequencies of all $q$-grams in a given text.
NMP (Algorithm~\ref{algo:naive}) and NSA (Algorithm~\ref{algo:naive_sa})
work on the uncompressed text.
SMP (Algorithm~\ref{algo:slpmain} +
Algorithm~\ref{algo:weighted_naive})
and
SSA (Algorithm~\ref{algo:slpmain} +
Algorithm~\ref{algo:weighted_naive_sa})
work on SLPs.
The algorithms were implemented using the C++ language.
We used {\tt std::map} from the Standard Template Library (STL)
for the associative array implementation.
\footnote{We also used {\tt std::hash\_map} but omit the results due
to lack of space. Choosing the hashing function to use is difficult,
and we note that its performance was unstable and sometimes very bad
when varying $q$.}
For constructing suffix arrays, we used the divsufsort
library\footnote{\url{http://code.google.com/p/libdivsufsort/}}
developed by Yuta Mori. This implementation is not linear time
in the worst case, but has been empirically shown to be one of
the fastest implementations on various data.
All computations were conducted on a Mac Xserve (Early 2009)
with 2 x 2.93GHz Quad Core Xeon processors and 24GB Memory,
only utilizing a single process/thread at once.
The program was compiled using the GNU C++ compiler ({\tt g++}) 4.2.1
with the {\tt -fast} option for optimization.
The running times are measured in seconds, starting from after reading
the uncompressed text into memory for NMP and NSA, and
after reading the SLP that represents the text into memory for SMP and
SSA.
Each computation is repeated at least 3 times, and the average is taken.
\subsection{Fibonacci Strings}
\begin{wrapfigure}[12]{r}{0.48\textwidth}
\vspace{-1.2cm}
\begin{center}
\includegraphics[width=0.47\textwidth]{graph_data_exp_fib.32.0115.csvk=50.eps}
\caption{Running times of NMP, NSA, SMP, SSA on Fibonacci strings
for $q=50$.}
\label{fig:graph_fib}
\end{center}
\end{wrapfigure}
The $i$ th Fibonacci string $F_i$ can be represented by the following
SLP:
$X_1 = \mathtt{b}$, $X_2 = \mathtt{a}$,
$X_i = X_{i-1} X_{i-2}$ for $i > 2$,
and $F_i = \mathit{val}(X_i)$.
Fig.~\ref{fig:graph_fib} shows the running times on
Fibonacci strings $F_{20}, F_{25}, \ldots, F_{95}$, for $q=50$.
Although this is an extreme case since Fibonacci strings can
be exponentially compressed, we can see that SMP and
SSA that work on the SLP are clearly faster than NMP and NSA which
work on the uncompressed string.
\subsection{Pizza \& Chili Corpus}
\begin{wrapfigure}[14]{r}{0.5\textwidth}
\vspace{-1.0cm}
\begin{center}
\includegraphics[width=0.47\textwidth]{speedUp_pizza.eps}
\caption{Time ratios NMP/SMP and NSA/SSA
plotted against ratio $|z|/|T|$.}
\label{fig:speed_up}
\end{center}
\end{wrapfigure}
We also applied the algorithms on texts
XML, DNA, ENGLISH, and PROTEINS, with sizes 50MB, 100MB, and 200MB,
obtained from the Pizza \& Chili
Corpus\footnote{\url{http://pizzachili.dcc.uchile.cl/texts.html}}.
We used RE-PAIR~\cite{LarssonDCC99} to obtain SLPs for this data.
Table~\ref{table:pizza} shows the running times for all algorithms and
data, where $q$ is varied from $2$ to $10$.
We see that for all corpora,
SMP and SSA running on SLPs are actually faster than
NMP and NSA running on uncompressed text, when $q$ is small.
Furthermore, SMP is faster than SSA when $q$ is smaller.
Interestingly for XML, the SLP versions are faster even for $q$
up to $9$.
Fig.~\ref{fig:speed_up} shows the same results
as time ratio: NMP/SMP and NSA/ SSA, plotted against ratio:
(length of $z$ in Algorithm~\ref{algo:slpmain})/(length of uncompressed text).
As expected, the SLP versions are basically faster than their uncompressed
counterparts, when $|z|/\mbox{(text length)}$ is less than $1$,
since the SLP versions run the weighted versions of the uncompressed algorithms
on a text of length $|z|$.
SLPs generated by other grammar based compression algorithms showed
similar tendencies (data not shown).
\begin{table}[t]
\caption{Running times in seconds for data from the Pizza \& Chili Corpus.
Bold numbers represent the fastest time for each data and $q$.
Times for SMP and SSA are prefixed with $\triangleright$,
if they become fastest when all algorithms start from the SLP representation,
i.e., NMP and NSA require time for decompressing the SLP (denoted
by decompression time).
The bold horizontal lines show the boundary where $|z|$ in
Algorithm~\ref{algo:slpmain}
exceeds the uncompressed text length.
}
\label{table:pizza}
\begin{center}
\scriptsize
\setlength{\tabcolsep}{1pt}
\renewcommand{\rmdefault}{ptm}
\renewcommand{\sfdefault}{phv}
\renewcommand{\ttdefault}{pcr}
\normalfont
\input{table1.tex}
\end{center}
\end{table}
\section{Introduction}
A major problem in managing large scale string data is its sheer size.
Therefore, such data is normally stored in compressed form.
In order to utilize or analyze the data afterwards,
the string is usually decompressed, where we must again confront the
size of the data.
To cope with this problem, algorithms that work directly on compressed
representations of strings without explicit decompression
have gained attention, especially for the string pattern matching problem~\cite{amir92:_effic_two_dimen_compr_match}
where
algorithms on compressed text can actually run faster than
algorithms on the uncompressed text~\cite{ShibataCIAC2000}.
There has been growing interest in what problems can be
efficiently solved in this kind of
setting~\cite{lifshits07:_proces_compr_texts,hermelin09:_unified_algor_accel_edit_distan}.
Since there exist many different text compression schemes,
it is not realistic to develop different algorithms for each scheme.
Thus, it is common to consider algorithms on texts represented as
{\em straight line programs} (SLPs)~\cite{NJC97,lifshits07:_proces_compr_texts,hermelin09:_unified_algor_accel_edit_distan}.
An SLP is a context free grammar in the Chomsky normal form that
derives a single string.
Texts compressed by any grammar-based compression algorithms
(e.g.~\cite{SEQUITUR,LarssonDCC99}) can be represented as SLPs,
and those compressed by the LZ-family (e.g.~\cite{LZ77,LZ78}) can be quickly
transformed to SLPs~\cite{rytter03:_applic_lempel_ziv}.
Recently, even {\em compressed self-indices} based on SLPs have
appeared~\cite{claudear:_self_index_gramm_based_compr},
and SLPs are a promising representation of compressed strings
for conducting various operations.
In this paper, we explore a more advanced field of
application for compressed string processing:
mining and classification on
string data given in compressed form.
Discovering useful patterns hidden in strings as well as
automatic and accurate classification of
strings into various groups, are important problems in
the field of data mining and machine learning with many
applications.
As a first step toward {\em compressed} string mining and
classification, we consider the problem of
finding the occurrence frequencies for all
$q$-grams contained in a given string.
$q$-grams are important features of string data,
widely used for this purpose in many fields such as text and natural
language processing, and bioinformatics.
In~\cite{inenaga09:_findin_charac_subst_compr_texts},
an $O(|\Sigma|^2n^2)$-time $O(n^2)$-space algorithm
for finding the {\em most frequent} $2$-gram from an
SLP of size $n$ representing text $T$ over alphabet $\Sigma$ was presented.
In~\cite{claudear:_self_index_gramm_based_compr},
it is mentioned
that the most frequent $2$-gram can be found in $O(|\Sigma|^2n\log n)$-time
and $O(n\log|T|)$-space, if the SLP is pre-processed and a self-index is built.
It is possible to extend these two algorithms
to handle $q$-grams for $q > 2$, but would respectively require
$O(|\Sigma|^qqn^2)$ and $O(|\Sigma|^qqn\log n)$ time,
since they must essentially enumerate and count the occurrences of all
substrings of length $q$,
regardless of whether the $q$-gram occurs in the string.
Note also that any algorithm that works on the uncompressed text
$T$ requires exponential time in the worst case, since $|T|$ can be as
large as $O(2^n)$.
The main contribution of this paper is an $O(qn)$ time and space
algorithm that computes the occurrence frequencies for {\em all}
$q$-grams in the text, given
an SLP of size $n$ representing the text.
Our new algorithm solves the more general problem
and greatly improves the computational complexity compared to previous work.
We also conduct computational experiments on various real texts,
showing that when $q$ is small, our algorithm and its variation actually run faster
than algorithms that work on the uncompressed text.
Our algorithms have profound applications in the field of string mining
and classification, and several applications and extensions are discussed.
For example, our algorithm leads to an $O(q(n_1+n_2))$ time algorithm for
computing the $q$-gram spectrum kernel~\cite{leslie02:_spect_kernel}
between SLP compressed texts of size $n_1$ and $n_2$.
It also leads to an $O(qn)$ time algorithm for finding
the optimal $q$-gram (or emerging $q$-gram) that discriminates between
two sets of SLP compressed strings, when $n$ is the total size of the
SLPs.
\subsubsection{Related Work}
There exist many works on {\em compressed text
indices}~\cite{navarro07:_compr},
but the main focus there is on fast search for a {\em given} pattern.
The compressed indices basically replace or simulate operations on
uncompressed indices using a smaller data structure.
Indices are important for efficient string processing,
but note that simply replacing the underlying index used in a mining
algorithm will generally increase time complexities of the algorithm
due to the extra overhead required to access the compressed index.
On the other hand, our approach is a new mining algorithm which exploits
characteristics of the compressed representation to achieve faster
running times.
Several algorithms for finding characteristic sequences from compressed texts have been proposed, e.g.,
finding the longest common substring of two strings~\cite{matsubara_tcs2009},
finding all palindromes~\cite{matsubara_tcs2009},
finding most frequent substrings~\cite{inenaga09:_findin_charac_subst_compr_texts}, and
finding the longest repeating substring~\cite{inenaga09:_findin_charac_subst_compr_texts}.
However, none of them have reported results of computational
experiments, implying that this paper is the first to show the
practical usefulness of a compressed text mining algorithm.
\input{preliminaries.tex}
\input{algorithm.tex}
\input{applications.tex}
\input{experiments.tex}
\input{conclusion.tex}
\bibliographystyle{splncs03}
\section{Preliminaries}
Let $\Sigma$ be a finite {\em alphabet}.
An element of $\Sigma^*$ is called a {\em string}.
For any integer $q > 0$, an element of $\Sigma^q$ is called an \emph{$q$-gram}.
The length of a string $T$ is denoted by $|T|$.
The empty string $\varepsilon$ is a string of length 0,
namely, $|\varepsilon| = 0$.
For a string $T = XYZ$, $X$, $Y$ and $Z$ are called
a \emph{prefix}, \emph{substring}, and \emph{suffix} of $T$, respectively.
The $i$-th character of a string $T$ is denoted by $T[i]$ for $1 \leq i \leq |T|$,
and the substring of a string $T$ that begins at position $i$ and
ends at position $j$ is denoted by $T[i:j]$ for $1 \leq i \leq j \leq |T|$.
For convenience, let $T[i:j] = \varepsilon$ if $j < i$.
For a string $T$ and integer $q \geq 0$, let $\mathit{pre}(T,q)$ and $\mathit{suf}(T,q)$
represent respectively, the length-$q$ prefix and suffix of $T$.
That is, $\mathit{pre}(T,q) = T[1:\min(q,|T|)]$ and
$\mathit{suf}(T,q) = T[\max(1,|T|-q+1):|T|]$.
For any strings $T$ and $P$,
let $\mathit{Occ}(T,P)$ be the set of occurrences of $P$ in $T$, i.e.,
$\mathit{Occ}(T,P) = \{k > 0 \mid T[k:k+|P|-1] = P\}$.
The number of elements $|\mathit{Occ}(T,P)|$ is called
the \emph{occurrence frequency} of $P$ in $T$.
\subsection{Straight Line Programs}
\begin{algorithm2e}[t]
\caption{Calculating $\mathit{vOcc}(X_i)$ for all $1\leq i \leq n$.}
\label{algo:varocc}
\KwIn{SLP ${\mathcal T} = \{X_i\}_{i=1}^n$ representing string $T$.}
\KwOut{$\mathit{vOcc}(X_i)$ for all $1\leq i\leq n$}
$\mathit{vOcc}[X_n] \leftarrow 1$\;
\lFor{$i\leftarrow 1$ \KwTo $n-1$}{
$\mathit{vOcc}[X_i] \leftarrow 0$\;
}
\For{$i\leftarrow n$ \KwTo $2$}{
\If{ $X_i = X_{\ell}X_r$ }{
$\mathit{vOcc}[X_{\ell}] \leftarrow \mathit{vOcc}[X_{\ell}] + \mathit{vOcc}[X_i]$;
$\mathit{vOcc}[X_r] \leftarrow \mathit{vOcc}[X_r] + \mathit{vOcc}[X_i]$\;
}
}
\end{algorithm2e}
\begin{wrapfigure}[13]{r}{0.5\textwidth}
\vspace{-1.56cm}
\centerline{\includegraphics[width=0.48\textwidth]{slp.eps}}
\caption{
The derivation tree of
SLP $\mathcal T = \{X_i\}_{i=1}^{7}$ with $X_1 = \mathtt{a}$, $X_2 = \mathtt{b}$, $X_3 = X_1X_2$,
$X_4 = X_1X_3$, $X_5 = X_3X_4$, $X_6 = X_4X_5$, and $X_7 = X_6X_5$,
representing string $T = \mathit{val}(X_7) = \mathtt{aababaababaab}$.
}
\label{fig:SLP}
\end{wrapfigure}
A {\em straight line program} ({\em SLP}) $\mathcal T$ is a sequence of assignments
$X_1 = expr_1, X_2 = expr_2, \ldots, X_n = expr_n$,
where each $X_i$ is a variable and each $expr_i$ is an expression, where
$expr_i = a$ ($a\in\Sigma$), or $expr_i = X_{\ell} X_r$~($\ell,r < i $).
Let $\mathit{val}(X_i)$ represent the string derived from $X_i$.
When it is not confusing, we identify a variable $X_i$
with $\mathit{val}(X_i)$.
Then, $|X_i|$ denotes the length of the string $X_i$ derives.
An SLP $\mathcal{T}$ {\em represents}
the string $T = \mathit{val}(X_n)$.
The \emph{size} of the program $\mathcal T$ is the number $n$ of
assignments in $\mathcal T$. (See Fig.~\ref{fig:SLP})
The substring intervals of $T$ that each variable derives can
be defined recursively as follows: $\mathit{itv}(X_n) = \{ [1:|T|]\}$, and
$\mathit{itv}(X_i) =
\{ [u + |X_\ell|:v] \mid X_k = X_\ell X_i, [u:v] \in \mathit{itv}(X_k) \}
\cup
\{ [u:u+|X_i|-1] \mid X_k = X_i X_r, [u:v] \in \mathit{itv}(X_k) \}
$ for $i < n$.
For example, $\mathit{itv}(X_5) = \{ [4:8], [9:13]\}$ in Fig.~\ref{fig:SLP}.
Considering the transitive reduction of set inclusion,
the intervals $\cup_{i=1}^n \mathit{itv}(X_i)$ naturally form a
binary tree (the derivation tree).
Let $\mathit{vOcc}(X_i) = |\mathit{itv}(X_i)|$ denote
the number of times a variable $X_i$ occurs in the derivation of $T$.
$\mathit{vOcc}(X_i)$ for all $1\leq i\leq n$ can be computed in $O(n)$
time by a simple iteration on the variables,
since $\mathit{vOcc}(X_n) = 1$ and for $i < n$,
$\mathit{vOcc}(X_i) =
\sum \{ \mathit{vOcc}(X_k) \mid X_k = X_\ell X_i \}
+
\sum \{\mathit{vOcc}(X_k) \mid X_k = X_iX_r \}$.
(See Algorithm~\ref{algo:varocc})
\subsection{Suffix Arrays and LCP Arrays}
The suffix array $\mathit{SA}$~\cite{manber93:_suffix} of any string $T$
is an array of length $|T|$ such that
$\mathit{SA}[i] = j$, where $T[j:|T|]$ is the $i$-th lexicographically smallest suffix of $T$.
The \emph{lcp} array of any string $T$ is an array of length $|T|$ such that
$\mathit{LCP}[i]$ is the length of the longest common prefix of
$T[\mathit{SA}[i-1]:|T|]$ and $T[\mathit{SA}[i]:|T|]$ for $2 \leq i \leq |T|$,
and $\mathit{LCP}[1] = 0$.
The suffix array for any string of length $|T|$
can be constructed in $O(|T|)$
time~(e.g.~\cite{Karkkainen_Sanders_icalp03})
assuming an integer alphabet.
Given the text and suffix array, the lcp array can also be calculated
in $O(|T|)$ time~\cite{Kasai01}.
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